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Heat loss from a model deer in a wind tunnel and in forest stands Sagar, Robert M. 1994

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HEAT LOSS FROM A MODEL DEER IN A W I N D T U N N E L A N D IN FOREST STANDS By Robert M. Sagar B. Sc. (Meteorology) Pensylvania State University M. Sc. (Agronomy) University of Nebraska  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF D O C T O R OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF SOIL SCIENCE  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA  November 1993 © Robert M. Sagar, 1993  In  presenting  degree  at  this  the  thesis  in  partial  fulfilment  of  University  of  British  Columbia,  1 agree  freely available for copying  of  department publication  this or of  reference  thesis by  this  for  his thesis  and  study.  scholarly  or for  her  I further  purposes  gain shall  permission.  (Signature)  Department  of  foil  The University of British Vancouver, Canada  DE-6 (2/88)  S f j P . D f C Columbia  requirements that  agree  may  representatives.  financial  the  It not  be is  that  the  Library  permission  granted  by  understood be  for  allowed  an  advanced  shall make for  the that  without  it  extensive  head  of  my  copying  or  my  written  Abstract  A realistically dimensioned polystyrene model of a black-tailed deer was constructed and tested in two forest stands and a wind tunnel to determine heat transfer relationships for the boundary layer and coat. Heat transfer from the elliptically cross-sectioned model deer trunk without a coat, in cross flow, was nearly the same as that for a circular cylinder. Heat transfer from the model in longitudinal flow was somewhat larger than in cross flow. Boundary layer conductance was not significantly different when the model was covered by a real deer coat. Turbulence in the forest stands enhanced conductance by about 30% for the cross flow orientation. Insulation provided by the deer's coat was much larger than that provided by the boundary layer, except in nearly calm conditions. The depth of a coat was found to be an important determinant of its insulation value and thus piloerection may be an important mechanism of thermoregulation. Free convection accounted for a significant proportion of heat transfer within the coat, while radiative transfer through the coat and conduction along individual hairs was relatively unimportant.  Forced convection had  only a limited effect on heat transfer within the coat at wind speeds less than 8 m s - 1 . There was no evidence of any turbulent enhancement of coat conductance in the forest stands at the low wind speeds observed. In order to estimate the radiative conductivity through the deer coat in the case of piloerection, it was necessary to used a numerical integration procedure. An approximate method for determining radiative conductivity, recommended in the literature, was found to be unsatisfactory for the case of piloerection. A model which predicts deer standard operative temperature and metabolic rates for  11  various forest habitats was tested. The model illustrated the importance of the deer's coat insulation in limiting heat loss and demonstrated the need for more research on the coat conductance of live deer. For the winter data set used in model testing, daily average metabolic requirements for a deer were similar in an old-growth stand and an adjacent open area. It is desirable however, to calculate hourly outputs for different habitats to determine the optimal microclimates for deer at different times of the day.  iii  Table of C o n t e n t s  Abstract  ii  List of Tables  viii  List of F i g u r e s  x  List of S y m b o l s  xix  Acknowledgements 1  2  xxiv  Introduction  1  1.1  5  Literature Cited  B o u n d a r y Layer C o n d u c t a n c e  8  2.1  Introduction  8  2.2  Experimental Methods  9  2.2.1  Model Deer Design and Construction  9  2.2.2  Measurement Theory  13  2.2.3  Experimental Sites and Instrumentation  14  2.2.3.1  Wind Tunnel  14  2.2.3.2  Field Sites  17  2.3  Results and Discussion  19  2.3.1  Laminar Flow  19  2.3.2  Turbulent Flow  32  iv  3  2.4  Conclusions  36  2.5  Literature Cited  37  Coat Conductance  40  3.1  Introduction  40  3.2  Theory  42  3.3  3.4  3.2.1  Mechanisms of Heat Transport Through Animal Coats  42  3.2.2  Models of Heat Transfer Through the Coat  46  Experimental Methods  48  3.3.1  Coat Covered Model Deer  48  3.3.2  Experiments and Instrumentation  54  3.3.2.1  Wind Tunnel  54  3.3.2.2  Field Site  59  Results and Discussion  61  3.4.1  Coat Conductance of a Deer Standing in Still Air  61  3.4.2  Effect of Wind Speed and Deer Orientation on Coat Conductance  73  3.4.2.1  Deer in Cross Flow  73  3.4.2.2  Deer in Longitudinal Flow  76  3.4.2.3  Comparison of Coat Conductance Measured in the Forest with That Measured in the Wind Tunnel  3.4.3  4  Comparison of Boundary Layer and Coat Conductance  80 86  3.5  Conclusions  89  3.6  Literature Cited  90  Thermal Radiation  93  4.1  Introduction  93  4.2  Theory  94 v  4.3  5  4.2.1  The Interception Function p  94  4.2.2  An Approximate Method for Determining Radiative Conductance  99  4.2.3  Numerical Integration to Determine kr  102  Results and Discussion  105  4.3.1  105  The Effect of Piloerection on Heat Transfer through a Deer Coat .  4.4  Conclusions  110  4.5  Literature Cited  Ill  D e e r H e a t Loss M o d e l  112  5.1  Introduction  112  5.2  The Model  112  5.2.1  A Brief Description of the Model  113  5.2.2  Some Examples of Model Outputs  119  5.2.2.1  Site Description  119  5.2.2.2  Instrumentation  120  5.2.2.3  Comparison of Model Outputs Using Relationships Found in this Thesis with Those from Parker and Gillingham (1990)  5.2.2.4  120  Comparison of Deer Heat Loss in Forested and Open Habitats  6  122  5.3  Conclusions  126  5.4  Literature Cited  128  Conclusions  130  Appendices  132  A Net Radiation  132 vi  B Time C o n s t a n t s  135  C I R T Calibration  136  D Turbulent S p e c t r a  139  E Equivalence of Equations. 4.9 and 4.10  141  F Numerical Integration  143  G Model Code  145  H Statistics  154  VII  List of Tables  3.1  Typical values of coat parameters for various animals taken from Cena and Clark (1973). The radiative conductivity was calculated using Eqs. 3.2 and 3.3 at a temperature of 20 °C  3.2  44  Summary of coat parameters measured at various positions on the mule deer hide as well as calculations of hair angle (<f> = arccos(/// s )) and the thermal conductance of a layer of still air at 20 °C which is the same depth as the coat (ga)  3.3  52  Typical values of the time constant for Tsk or ATC for step changes in either power density or wind speed imposed on the model deer  3.4  Comparison of mean coat conductances (<jFc) for positions 2 and 3 for the longitudinal and cross flow cases  A.l  58  81  Summary of view factors for positions on the top and sides of the model deer which was oriented in either the cross flow (cross) or longitudinal flow (long) orientations  B.l  134  Some time constants ( r ) for the response of surface temperature of the bare model deer to step changes in wind speed in the wind tunnel  C.l  135  Comparisons of surface temperatures measured with an Everest Interscience, Model 4000 IRT (TJRT)  with the temperature of a blackbody  calibration block during the second wind tunnel experiment with the coat covered model deer  138  viii  F.l  Values of the integrand, F(r), over the specified range of r values, above and below the plane AB as shown in Figure 4.4  H.l  144  Statistics for linear regressions of logy against log a; (logy = a + The original functions were of the form y = Axh, a = log A.  blogx).  Hypothesis  testing was carried out at the 0.05 significance level H.2 Statistics for non-linear regressions of the form y = constant  IX  155 + axb.  ...  156  List of F i g u r e s  1  Schematic of the model deer  10  2  Specification of measurement positions on the model deer  12  3  Orientations of the model deer during the wind tunnel experiments. . . .  16  4  Temperature difference (Ts — Ta) as a function of angular position (at longitudinal position 2) around the bare model deer exposed to cross flow in the wind tunnel at two wind speeds  5  20  Local Nusselt number (Nu#) vs. Reynolds number (Re) for the stagnation point (0°) at position 2, comparing the published results of Giedt (1949) ( • ) , Bosch (1936) (o), Schmidt and Wenner (1941) (O) and Schmidt and Wenner (1943), as referenced by Sandborn (1972) ( A ) with wind tunnel results from this study for a bare (•) and a coat covered (•) model deer. The solid line is a regression through the aforementioned published results. The linear regression line through the bare model deer points (not shown) is given by the equation Nue=0.90Re°' 5 °. Regression statistics are shown in Appendix H  6  21  Same as Figure 2.5 except data presented is for the top point (90°). The linear regression line through the bare model deer points (not shown) is given by the equation Nug=0.18Re°- 61 . Regression statistics are shown in Appendix H  22  x  2.7  Same as Figure 2.5, except data presented is for the lee point (180°). The linear regression line through the bare model deer points (not shown) is given by the equation Nu^=0.12Re°- 69 . Regression statistics are shown in Appendix H  2.8  23  Local boundary layer conductance (gb) vs. angular position (at longitudinal position 2) for the bare model deer trunk in cross flow at different wind speeds in the wind tunnel  2.9  25  Local boundary layer conductance for positions on the side of the model deer in longitudinal flow as a function of straight line distance downwind of position 5  27  2.10 Local Nusselt number Nu x plotted against local Reynolds number Re^ for points on the side of the bare model deer in longitudinal flow in the wind tunnel (•). Also shown are pos. 5 points (•), where d = 0.01m was assumed. The equation of the regression line through the pos. 5 d a t a is Nuj;=0.371Re 0 4 6 0 . Regression statistics are shown in Appendix H  28  2.11 Comparison of Nu vs. Re relationships for bare model deer in longitudinal and cross flow orientations in the wind tunnel. Regression statistics are shown in Appendix H  30  2.12 Comparison of overall <#, vs. wind speed relationships for bare model deer in longitudinal and cross flow orientations in the wind tunnel. Regression statistics are shown in Appendix H  31  2.13 Nu# vs. Re for the bare model deer in an old growth stand (longitudinal position 1) at the stagnation point with wind tunnel regression line (longitudinal position 2) for comparison. Regression statistics are shown in Appendix H  33  XI  2.14 Same as Figure 2.13, except data presented are for the top position and field data are for longitudinal position 3, while wind tunnel data are for position 2. Regression statistics are shown in Appendix H  34  2.15 Same as Figure 2.13, except data presented are for the lee position at longitudinal position 2 in both cases. Regression statistics are shown in Appendix H 3.1  35  Alternative resistance models for heat transfer through animal coats proposed by McArthur and Monteith (1980b). Model one assumes that all mechanisms act in parallel, while model two assumes that forced and free convection are in series, with forced convection acting to some wind penetration depth, t  3.2  47  Diagram showing how thermocouples were installed to measure skin surface temperature  3.3  50  Response of the temperature difference ATC for the model in cross flow at position 3-0 (stagnation) to a step change in wind speed from 2.6 to 5.3 m s - 1 . The solid line is a non-linear least squares fit through the data points  3.4  57  Example of skin temperature (Tsk) change at position 1-180 on the coat covered model deer in cross flow after a step change in wind speed from 5.3 to 0 m s - 1 . Solid line is a non-linear least squares fit through all the data points, while dashed line is the resulting fit when only the first 15 data points were used  3.5  60  Orientation of the model deer to receive cross flow winds at the Browns River Site and typical diurnal wind directions  xn  62  6  The temperature difference across the model deer coat (ATC) as a function of power flux density for position 2-90 on top of the model deer with u = 0 m s - 1 . The data points have been fitted using a non-linear regression of the form ATC = aPb where a = 0.324 and 6 = 0.91  7  63  The relationship between coat conductance and temperature difference for position 2 on the top of the model deer in still air ( • ) . The • symbol is for the case where the model deer was turned upside down and the same coat area (pos.2) monitored as before being inverted. The solid line is a non-linear regression line. Regression statistics are shown in Appendix H.  8  64  Coat conductance and its components as a function of ATC for position 2-90 (top) at a wind speed of O r n s - 1 .  The still air value, based on a  coat depth of 9.8 mm, is shown for comparison. The solid lines through the total and convective points are non-linear regressions of the form g = a + bAT°. Radiative and hair conductances were calculated using Eq. 3.13 and convective conductance was calculated using Eq. 3.12.  Regression  statistics are shown in Appendix H 9  66  Typical example of the magnitudes ( W m - 2 ) of the energy fluxes through the coat and boundary layer due to the mechanisms of conduction, convection and radiation. The example is for position 2-90 (top) under no wind conditions. Numbers in parentheses represent the percentage of the total flux due to a particular mechanism. Temperatures measured at the skin surface, coat surface and free stream air are shown on the right. . . .  xiii  67  3.10 Mean convective coat conductance for longitudinal positions 2 and 3 as a function of angular position with M = 0 m s " ' and P = 5 2 W m - 2 .  The  • symbol indicates that the measurement was made on the same portion of coat as the 90° position with the model turned upside down. The o symbols connected by the dashed line indicate the still air conductance at the various positions  69  3.11 Mean convective conductivity for longitudinal positions 2 and 3 as a function of angular position with u = 0 m s _ 1 and P = 52 W m - 2 . The • symbol indicates t h a t the measurement was made on the same portion of coat as the 90 ° position with the model turned upside down  70  3.12 Comparison of the coat conductance as a function of wind speed for the stagnation (0°), top (90°) and lee (180°) positions on the model deer in cross flow.  Longitudinal positions 2 and 3 were averaged to obtain  the points for each angular position. Regression statistics are shown in Appendix H  74  3.13 Coat conductance and its components as a function of wind speed for position 3-0 (stagnation) on the model deer in cross flow. The still air value is shown for comparison and is based on a coat depth of 18.1 mm. The solid lines through the total and convective points are non-linear regressions of the form g = a + buc. Radiative and hair conductances were calculated using Eq. 3.13 and convective conductance was calculated using Eq. 3.12. Regression statistics are shown in Appendix H  75  3.14 Mean coat conductance of the model deer trunk in cross flow as a function of wind speed. Solid line is a non-linear regression line. Regression statistics are shown in Appendix H  xiv  77  3.15 (a) Coat conductance on the top of the coat covered model deer in longitudinal flow as a function of wind speed. Solid lines are for the rear end facing the wind (orientation r), while dashed lines are for the head facing the wind (orientation h). (b) Same as above except for the 180° side positions  78  3.16 (a). Typical diurnal pattern of wind speed measured at 0.8 m above the ground by a hot wire anemometer at the Browns River site on 7 and 8 August 1990. (b). Wind direction measured at 1.5 m above the ground at the same site, for the same time period. Also shown are the mean wind speed and direction (u and WD)  along with their respective standard  deviations (au and <J\YD) f ° r the day and night periods  82  3.17 (a) Coat conductance as a function of wind speed at position 3-90 on model deer in cross flow at Browns River site. This is a typical nighttime data set taken from 2040, 7 July 1990 to 0540, 8 July 1990. (b) Same as above except data are for position 2-0 for the period 1250-1730, 8 August 1990.  84  3.18 Comparison of coat conductances at wind speeds ranging from 0 to 1.2 m s - 1 for various positions on the model deer in cross flow, measured outdoors at Browns River (black bars) with those measured in the wind tunnel (grey stippled bars). The numbers above the bars are the power densities ( W m ~ 2 ) used  85  3.19 Comparison of coat conductances at wind speeds ranging from 0 to 1.2 m s " 1 for various positions on the model deer in longitudinal flow, measured outdoors at Browns River (black bars) with those measured in the wind tunnel (grey stippled bars). Measurements taken with the head facing into the wind are labeled h, while those taken with the rear end facing the wind are labeled r  87 xv  3.20 Comparison of the mean coat conductance (ljc) with the mean boundary layer conductance (7jb) as a function of wind speed for the model deer in cross flow. Boundary layer conductances were measured on the bare model deer. Solid lines are regression lines. Regression statistics are shown in Appendix H 4.1  88  Geometrical relationship between a single hair oriented in the direction given by the unit vector ra in the xz plane and radiation emitted by the skin surface in the direction given by the unit vector h  4.2  95  Geometrical relationship between the hair direction vector, m, and the radiation transmission vector, n, showing that the length of the projection of a single hair onto a plane perpendicular to h is ls sin /? where ls is the hair length  4.3  97  Diagram showing the geometrical relationships for determining p (adapted from Kreith, 1973). The radius of the hemisphere corresponds to a unit thickness of coat  4.4  100  Illustration of the method for determining kr of a dry coat with a uniform temperature gradient across the coat  4.5  Comparison of p (numerical integration) with p± (Cena and Monteith's method) as a function of hair angle for position 2-90 on the model deer. .  4.6  103  106  Results of a simulation showing the effects of piloerection on radiative conductance calculated using numerical integration and using Cena and Monteith's approximate equations with p±_ and p for position 2-90  xvi  108  7  Simulation of the effect of piloerection on the components of coat conductance at position 2-90 (top) on the model deer with u = 0 m s - 1 . (a) Cena and Monteith's approximation with p± was used to calculate radiative conductance, (b) The numerical integration method was used to calculate the radiative conductance  1  109  Mean boundary layer conductance (gb) as a function of wind speed determined in this study for the model deer trunk with a turbulent enhancement factor of 1.3 (line a ) , g~h determined using the equation recommended by Campbell (1977) with an enhancement factor of 1.43 (line b) and gb using Campbell's equation without enhancement (line c)  2  115  Mean coat conductance (gc) as a function of wind speed for mule deer hide determined in this study (line a) and gc determined using equations recommended by Parker and Gillingham (1990) for winter with T e = 0 ° C (lineb)  3  118  Hourly solar radiation (Rs), 1.5 m air temperature (Ta) and wind speed (u) recorded at the open site near Woss, B.C. for the period 1 January 1989 to 11 February 1989. Also shown are periods when snow cover exceeded 15 cm (heavy horizontal lines) for the open and old growth sites  4  121  Comparison of mean daily standard operative temperature (Tes) in the old groth stand calculated using equations developed in this study and equations recommended by Parker and Gillingham (P & G) (1990) . Also shown is mean daily air temperature (T a )  5  123  Comparison of the metabolic rate (M) calculated using the equations developed in this study (line a) and that calculated using Parker and Gillingham's equations (line b)  124  xvii  Comparison of the standard operative temperature (Tes) for a deer within the old growth stand, with that for a deer at the open site and at the open site with the wind speed increased by a factor of 10  125  Comparison of the metabolic rate (M) of a deer within the old growth, with that of a deer at the open site and at the open site with the wind speed increased by a factor of 10  127  Plot showing comparison of Everest Interscience, Model 4000 IRT {TJUT) with a blackbody calibration block (Tbb)- The solid line is a regression through the data points  137  Typical power spectra of the streamwise (u) velocity component observed at a height of 2 m above the the forest floor at (a) the second growth stand on 19 July 1990 between 1330 and 1400 P S T and (b) the old growth stand on 9 August 1989 between 1315 and 1415 PST  xviii  140  List of S y m b o l s  A  surface area of model deer ( m - 2 )  Ai  area of given region on model deer (m 2 )  Ap  projected area of hair onto plane perpendicular to n (m 2 )  a  temperature gradient in layer of coat ( " C m " 1 )  B  blackbody radiation ( W m - 2 )  b  slope of blackbody radiation curve (4crT 3 , W m - 2 K - 3 )  cp  specific heat of air (J k g - 1 K - 1 )  d  mean hair diameter (m)  d  characteristic dimension of model deer (m)  di  characteristic dimension of given region on model deer (m)  F  net radiative flux through a layer of coat (W m - 2 )  Fa  flux  density of radiation emitted by a unit area of hemisphere (W m - 2 )  /  view factor  ga  conductance of still air in coat (mm s - 1 )  gb  boundary layer conductance ( m s - 1 or m m s - 1 )  gc  coat conductance ( m m s - 1 )  gc  mean coat conductance over whole model deer coat ( m m s - 1 )  gcon  convective conductance (free and forced) ( m m s - 1 )  gcs  mean coat conductance over whole model deer coat at standard low wind speed ( m m s - 1 )  xix  gfc  coat conductance due to forced convection ( m m s - 1 )  gfT  coat conductance due to free convection ( m m s - 1 )  gh  hair conductance ( m m s - 1 )  gm  thermal conductance due to molecular conduction, including hair and still air ( m m s - 1 )  gr  radiative conductance ( m m s - 1 )  H  sensible heat flux density at model deer surface (W m - 2 )  ka  area weighted thermal conductivity of still air ( m W m - 1 K - 1 )  kair  thermal conductivity of still air (25 m W m - 1 K - 1 @ 20 °C)  kc  thermal conductivity of coat ( m W m - 1 K - 1 )  kcon  convective conductivity including free and forced convection (mW m - 1 K - 1 )  kh  thermal conductivity of hair ( m W m - 1 K - 1 )  k0  thermal conductivity of organic matter ( 2 5 0 m W m - 1 K - 1 @ 20°C)  kr  radiative conductivity ( m W m - 1 K - 1 )  /  mean coat depth normal to skin surface (cm)  ls  mean hair length (cm)  M  basal metabolic rate ( W m - 2 )  m  unit vector in direction of hair  N  radiance of a unit area of hemisphere ( W m - 2 s r - 1 )  n  number of hairs per unit area of skin surface ( m - 2 )  h  unit vector in direction of radiation transmission  Nu  Nusselt Number  Nu  mean Nusselt Number for model deer trunk  Nu,-  spatially averaged Nusselt Number for a given region of model deer used in determining Nu  Nuj;  local Nusselt Number for model deer in longitudinal flow  Nu0  local Nusselt Number for model deer in cross flow  n<t>aa  power spectra variable (m 2 s - 2 )  p  radiation interception function ( c m - 1 )  P  power density supplied to model deer surface (W m - 2 )  p  mean probability that radiation will be intercepted in a unit depth of coat ( c m - 1 )  p±  probability of interception for radiation which is traveling perpendicular to skin surface ( c m - 1 )  R  circuit resistance of nichrome heating wire on model deer trunk (7.81 Vt)  r  radius of hemispherical shell (m)  Re  Reynold Number  Re^  local Reynolds Number for model deer in longitudinal flow  Rabs  net absorbed flux density of radiative energy (W m - 2 )  fh  mean boundary layer resistance for whole model deer trunk ( s m - 1 )  rc  mean coat resistance for whole model deer trunk ( s m - 1 )  rcs  value of rc at standard low wind speed ( s m - 1 )  re  parallel combination of boundary layer and radiative resistances ( s m - 1 )  res  value of r e at standard low wind speed ( s m - 1 )  THb  whole body resistance ( s m - 1 )  ^Hbs  value of rub at standard low wind speed (0.1 m s - 1 ) ( s m - 1 )  Rs  solar radiation flux density ( W m ~ 2 )  rt  resistance of body tissue to heat flow ( s m _ 1 )  S  rate of heat storage per unit trunk area (W m - 2 )  t  elapsed time (min)  t  penetration depth of wind into coat (m)  T;,  deep body temperature of deer (°C)  Tbt  temperature of thermocouple in blackbody calibration block (°C)  Tc  ceiling temperature in wind tunnel (°C)  Te  operative temperature (°C)  Teff  effective environmental temperature (°C)  Tes  standard operative temperature (°C)  Tf  floor temperature in wind tunnel (°C)  TIRT  temperature registered by infrared thermometer when pointed at the blackbody calibration block (°C)  Ts  surface temperature of model deer (painted surface, Chapter two or fur surface, Chapter three) (°C)  Tsk  skin surface temperature (°C)  Tskf  equilibrium value of Tsk after step change in wind speed or power (°C)  Tw  wall temperature in wind tunnel (°C)  u  mean horizontal wind speed ( m s - 1 )  V  voltage supplied to model deer (V)  WD  wind direction (deg)  Wk  fraction of coat cross-section occupied by hair  w0  fraction of hair cross-section occupied by solid  a  angle between projections of m and h vectors (deg)  xxn  ah  thermal diffusivity of air (m 2 s  x  /?  the angle between the rh and n vectors (deg)  ATC  Tsk - T3 (°C)  ATcf  equilibrium value of ATC after step change in wind speed or power (  ATC{  initial value of ATC before step change in wind speed or power (°C)  e  emissivity of a surface  ee  emissivity of environment  es  emissivity of model deer  A  wavelength of turbulent eddies (m)  v  kinematic viscosity ( m 2 s _ 1 )  <f>  angle between normal to skin surface and hair or hair angle (deg)  p  density of moist air (kg m~ 3 )  a  Stephan-Boltzman constant (5.67 x 1 0 - 8 W m _ 2 K - 4 )  crgc  standard deviation of coat conductance (mm s _ 1 )  au  standard deviation of mean horizontal wind speed ( m s - 1 )  (TWD  standard deviation of wind direction (deg)  r  fraction of radiation transmitted through a coat layer of thickness  )  z (m) r  time constant (min)  <f  angle between the z axis and the radiation direction vector, n (deg)  xxiii  Acknowledgements  My most sincere thanks to my thesis supervisor, Dr. T.A. Black, for his expert guidance and financial support of my research. I also appreciate the guidance and editorial comments provided by my thesis committee members, Drs. T.M. Ballard, M.D. Novak and F.L. Bunnell. I would like to extend a special thanks to Dr. Jing Ming Chen for his generous help in explaining difficult concepts. I am grateful to my friend and colleague Dr. Xuhui Lee for his collaboration during this research. Many thanks to my good friend, Rick Ketler for his technical help and for his companionship in the mountains when I needed a break. I appreciate the friendship and assistance of my fellow graduate students Ralph Adams, Rob Fleming, Chuck Bulmer, Andrea Ryan and others. Thanks to Reka Vasarhelyi, John Janmaat, and David Loh for their assistance during field work. I am grateful to Dr. I.S. Gartshore, Department of Mechanical Engineering, for his advice and allowing me to use the department's wind tunnel. I wish to acknowledge the financial support provided by the Department of Soil Science, the University Graduate Fellowships of B.C., the Natural Science and Engineering Research Council of Canada, Canadian Forest Products Inc. and MacMillan Blodel Limited. Finally, I would like to thank Leet Mueller for her love and support during the final stages of my thesis.  xxiv  Chapter 1  Introduction  Loss of habitat due to encroaching development and resource exploitation is the most serious problem facing wildlife species today. A major concern for wildlife managers in British Columbia is the loss of wildlife habitat due to the forestry and mining industries. On Vancouver Island, there is concern that continued cutting of old-growth stands is destroying prime black-tailed deer (Odocoileus hemionus  columbianus)  wintering range  (Nyberg et al., 1986). This thesis is part of a multidisciplinary study to examine the effects of the loss of old-growth habitat and to assess the suitability of managed secondgrowth stands as deer winter range. In order to consider the suitability of habitats for animal survival we need to look at an animal's overall energy budget, including the cost of activities such as locomotion, gain of energy through food intake and losses of energy to the environment by sensible heat and radiation. Energy losses to the environment may become especially important under severe winter conditions of deep snow and poor food availability (Parker and Gillingham, 1990). These energy losses may be critical in determining future survival and reproduction. For example, trade-offs exist between a deer staying in good thermal cover (favorable microclimate) or leaving it and spending energy walking through deep snow in search of food. In this respect, the quality of the habitat (old vs. second-growth or clearcuts) is very important (Nyberg et al., 1986). To quantify deer energy losses to the environment, we need a sound understanding of  1  Chapter 1.  Introduction  2  the microclimate beneath old and second growth stands. An old-growth and a secondgrowth Douglas-fir stand were studied as part of the deer winter range project. Wind and temperature regimes within the old and second-growth Douglas-fir stands were studied by Lee and Black (1993a, b and c). The long and shortwave radiation regimes beneath the second growth stand were discussed by Black et al. (1991). Chen and Black (1991 and 1992) developed a procedure for quantifying the leaf area index and canopy architecture necessary for radiation penetration models in the second growth stand. Lee and Black (1993b and c) report on t r a m measurements to obtain spatial averages of net radiant fluxes beneath the old growth and second growth stands, respectively. These papers give us a good basis to predict the general microclimatic conditions that a deer would experience within these stands. In addition, we need to understand how these conditions affect deer energy losses. Early work which contributed to our understanding of heat loss from mammals involved measurements of the insulation values for various wild and domestic mammals using isolated fur samples (Scholander et al. 1950, Hammel 1955). Moote (1955) and Tregar (1965) studied the effects of wind on the thermal insulation of animal fur. Much of the work which attempts to look at energy losses from whole animals has centered on domestic farm animals (Joyce et al., 1966; Wiersma and Nelson, 1967, McArthur and Monteith, 1980 a, b and c). These studies were carried out on live and model animals which avoid the shortcoming of assuming that coat conductances measured on isolated fur samples are valid for whole animals. Many of the studies on animal energy balances are carried out in chambers under controlled conditions which may not be representative of the conditions which animals experience in their natural habitats. This concern was voiced by Cena and Monteith (1975). It is also commonly assumed that for purposes of determining boundary layer thermal conductances, animal body parts can be thought of as consisting of regular  Chapter 1.  Introduction  3  geometric shapes such as circular cylinders or spheres. This assumption allows for the use of engineering heat transfer relationships for these shapes which have been determined in wind tunnels, such as those determined by Giedt (1949) for circular cylinders. Mitchell (1974) asserted that boundary layer heat transfer for a wide variety of animal shapes can be determined by assuming that they are spherical and using the cube root of their volume as a characteristic dimension. Biologists desire easily obtainable environmental variables which relate to animal heat loss in various habitats. Parker and Gillingham (1990) state that in many studies, the only variable used is air temperature (for example, Repasky, 1991). Unfortunately, air temperature alone does not fully describe an animal's thermal environment. Porter and Gates (1969) combined air temperature, windspeed, net absorbed radiation and animal factors such as metabolic rate and thermal resistances on a two dimensional graph which they called an animal's climate space. The climate space defines the limits of the above environmental variables within which an animal must stay in order to survive. Morhardt and Gates (1974) combined the environmental variables used in a climate space diagram into a single variable called the equivalent blackbody temperature, T e , or operative temperature. Operative temperature can be defined as the temperature at which an isothermal blackbody chamber would need to be at in order to provide the same net radiative and convective exchange as present in the animal's natural habitat (Campbell, 1977). A variation of Te also described by Bakken (1981) is the standard operative temperature, Tes, which is similar to Te except t h a t the convective conditions within the blackbody chamber are at some standard low value. Both Te and Tes have been used extensively in ecological studies (Parker and Robbins 1984, Mahoney and King 1977, Bakken 1980), which is a significant improvement over using air temperature alone. These values, however, are only as good as the estimates of environmental variables that went into them. The work of Parker and Gillingham (1990)  Chapter 1.  4  Introduction  is a particularly relevant example, since it deals with thermally critical environments for mule deer (Odocoileus hemionus hemionus  hemionus),  of which black-tailed deer  (Odocoileus  columbianus) is a subspecies.  Environmental variables such as air temperature, windspeed and solar radiation were either measured or allowed to vary between the extremes expected for a particular environment inhabited by the deer. A particular concern in this thesis is the way in which boundary layer and coat resistances are determined. Parker and Gillingham calculated boundary layer resistance using a relationship derived for flat plates, recommended by Campbell (1977) as being a good average for cylinders, plates and spheres. They calculated coat resistance using a general relationship recommended in Campbell et al. (1980) as being good for many different animal coats. T h e overall objective of this dissertation is to develop and test heat transfer relationships for the coat and boundary layers of black-tailed deer in their natural environment. The work was carried out using a realistically dimensioned model deer in natural (old and second-growth Douglas-fir stands) and controlled (wind tunnel) environments. Chapter two describes the effect of windspeed on boundary layer conductance of the model deer and tests the hypothesis that heat transfer from its trunk can be modeled as that from a circular cylinder. In Chapter three, the effect of windspeed on coat conductance is examined along with the various mechanisms of heat transfer through the coat. Chapter four focuses on the determination of the transfer of radiative energy through the coat. Chapter five uses the findings of this thesis in a model which predicts deer heat loss in various forest habitats. Finally, Chapter six summarizes the important findings of this thesis.  Chapter 1.  1.1  Introduction  5  Literature C i t e d  Bakken, G.S. (1980) The use of standard operative temperature in the study of the thermal energetics of birds. Physiol. Zool. 5 3 , 108-119. Bakken, G.S. (1981) How many equivalent black-body temperatures are there? J.  Therm.  Biol. 6, 59-60. Black, T.A., J.M. Chen, X. Lee, and R.M.Sagar (1991) Characteristics of shortwave and longwave irradiances under a Douglas-fir forest stand. Can. J. For. Res. 2 1 , 10201028. Campbell, G.S. (1977) An Introduction  to Environmental  Biophysics.  Springer-Verlag,  New York, 159 pp. Campbell, G.S., A.J. McArthur and J.L., Monteith , 1980: Windspeed dependence of heat and mass transfer through coats and clothing. Boundary-Layer  Meteorol. 18,  485-493. Cena, K. and J.L. Monteith (1975) Transfer processes in animal coats: II. Conduction and convection. Proc. R. Soc. Lond. B. 188, 395-411. Chen, J.M. and T.A. Black (1991) Measuring leaf area index of plant canopies with branch architecture. Agric. For. Meteorol. 5 7 , 1-12. Chen, J.M. and T.A. Black (1992) Foliage area and architecture of plant canopies from sunfleck size distributions. Agric. For. Meteorol. 60, 249-266. Giedt, W.H. (1949) Investigation of variation of point unit-heat-transfer coefficient around a cylinder normal to the air stream. Trans. ASME. 7 1 , 375-381. Hammel, H.T. (1955) Thermal properties of fur. Am. J. Physiol.  1 8 2 , 369-376.  Chapter 1.  Introduction  6  Joyce, J.P., K.L. Blaxter and C. Park (1966) the effect of natural outdoor environments on the energy requirements of sheep. Res. Vet. Set. 7, 342-359. Lee, X. and T.A. Black (1993a) Atmospheric turbulence within and above a Douglas-fir stand. P a r t I: Statistical properties of the velocity field. Boundary-Layer  Meteorol.  64, 149-174. Lee, X. and T.A. Black (1993b) Atmospheric turbulence within and above a Douglas-fir stand. P a r t II: Eddy fluxes of sensible heat and water vapour.  Boundary-Layer  Meteorol., (in press). Lee, X. and T.A. Black (1993c) Turbulence near the forest floor of an old growth Douglasfir stand on a south-facing slope. For. Sci. 3 9 , 211-230. Mahoney, S.A. and J.R. King (1977) The use of the equivalent blackbody temperature in the thermal energetics of small birds. J. Therm. Biol. 2, 115-120. McArthur, A.J. and J.L. Monteith (1980a) Air movement and heat loss from sheep. I. Boundary layer insulation of a model sheep, with and without fleece. Proc. R. Soc. Lond. B. 2 0 9 , 187-208. McArthur, A.J. and J.L. Monteith (1980b) Air movement and heat loss from sheep. II. Thermal insulation of fleece in wind. Proc. R. Soc. Lond. B. 2 0 9 , 209-217. McArthur, A.J. and J.L. Monteith (1980c) Air movement and heat loss from sheep. III. Components of insulation in a controlled environment. Proc. R. Soc. Lond,  B.  2 0 9 , 219-237. Mitchell, D. (1974) Convective heat transfer from man and other animals. In Heat Loss from Animals  and Man (Edited by Monteith, J.L. and Mount, L.E.). pp. 59-76.  Butterworths, London.  Chapter 1.  7  Introduction  Moote, I. (1955) The thermal insulation of caribou pelts. Text. Res. J. 2 5 , 796-801. Morhardt, S.S. and D.M. Gates (1974) Energy-exchange analysis of the Belding ground squirrel and its habitat. Ecol. Mongr. 44, 17-44. Nyberg, J.B., F.L. Bunnell, D.W. Janz and R.M. Ellis (1986) Managing young forests as black-tailed deer winter ranges. Ministry of Forests. Land Man. Report 37, Victoria, B.C. 48 pp. Parker, K.L. and C.T. Robbins (1984) Thermoregulation in mule deer and elk. Can. J. Zool. 62, 1409-1422. Parker, K.L. and M.P. Gillingham (1990) Estimate of critical thermal environments for mule deer. J. Range Manage. 4 3 , 73-81. Porter, W.P. and D.M. Gates (1969) Thermodynamic equilibria of animals with environment. Ecol. Mongr. 3 9 , 227-244. Repasky, R.R. (1991) Temperature and the northern distributions of wintering birds. Ecology 72, 2274-2285. Scholander, P.F., Hock, R., Walters, V., Johnson F., and Irving, L. (1950) Heat regulation in some arctic and tropical mammals and birds. Biol. Bull. 9 9 , 237-258. Tregar, R.T. (1965) Hair density wind speed and heat loss in mammals. J. Appl.  Physiol.  20, 796-801. Wiersma, F. and G.L. Nelson (1967) Nonevaporative convective heat transfer from the surface of a bovine. Trans. ASAE 10, 733-737.  Chapter 2  B o u n d a r y Layer C o n d u c t a n c e of a M o d e l D e e r in a W i n d T u n n e l and Douglas-fir S t a n d s  2.1  Introduction  The overall objective of this chapter was to determine the sensible heat transfer relationships for the laminar boundary layer of black-tailed deer {Odocoileus  hemionus  columbianus)  Sensible  in old-growth and managed second-growth Douglas-fir stands.  heat loss is one of the two main mechanisms for energy loss from deer. It is controlled by three resistances, which are the boundary layer, coat and subcutaneous tissue (Campbell, 1977). Of these, the boundary layer resistance is most affected by the wind regime. Wind regime in the forest stand in turn is affected by stand density and structure (Lee, 1992). Boundary layer resistance also controls the rate of evaporation from a wet pelage (Parker, 1988). Local boundary layer conductance relationships for smooth circular cylinders in laminar cross flow have been well established in wind tunnel studies (e.g.  Giedt 1949).  However, real animals are not smooth circular cylinders living in wind tunnels. Mitchell (1974) cautions against assuming that a real animal can be treated as a smooth cylinder without experimental confirmation. Several studies have measured boundary layer conductances on model animals including Wiersma and Nelson (1967) for cattle and McArthur and Monteith (1980) for sheep. The latter study showed that the turbulence outdoors can cause significant enhancement of heat loss over that measured in laminar  8  Chapter 2. Boundary  Layer  Conductance  9  flow. Few studies have been attempted with live animals, such as Mitchell (1985) for birds tethered in a wind tunnel and McArthur (1977) for sheep outdoors. Due to the obvious logistical difficulties in working with a live deer it was decided to build a realistically dimensioned model deer. The specific objectives were (i) to compare wind tunnel results for local boundary layer conductance of the model deer to those previously published for circular cylinders, (ii) to determine whether the presence of a coat affects boundary layer conductance, (iii) to determine the effects of deer orientation on boundary layer conductance and (iv) to compare results obtained in the laminar flow conditions of the wind tunnel with those obtained in the turbulent conditions of a forest stand.  2.2  Experimental Methods  2.2.1  Model Deer Design and Construction  The model deer (Figure 2.1) consisted of a head and trunk carved out of expanded polystyrene resting on a wooden and acrylic stand. The density of the expanded polystyrene was 24 k g m - 3 and the specific heat was 1130 J k g - 1 ° C _ 1 . The shape and dimensions used were those of an adult black-tailed deer. Dimensions were obtained from Parker (1989, personal communication). An elliptical trunk cross-section was decided on after observing captive black-tailed deer and examining photographs of them. Deer metabolic heat was simulated by passing current through 18 gauge nichrome heating wire (resistance/unit length = 1.33 0, m _ 1 ) , which was wrapped around the trunk of the model deer at a density of 2 winds c m - 1 over the surface area (A) of 0.74 m 2 . The nichrome wire was configured as five sections of 39.1 Q each, wired in parallel to give a circuit resistance (R) of 7.81 0 . The wire was covered by tightly wrapped masking tape  Chapter 2. Boundary Layer Conductance  10  heating wire nichrome (18 AWG) density 2 winds/cm  {  0.80 m  •  I\ll\I l i m '  Power Supply +  12/24 Volts  m  0.32 m  —>i0.25mi<«—  wooden legs - 5 cm  Figure 2.1: Schematic of the model deer.  trunk cross section  Chapter 2. Boundary  Layer  Conductance  11  and painted white with an oil based exterior primer to provide a surface with a high longwave emissivity (0.95) and high shortwave reflectivity. High emissivity is necessary in order to make accurate measurements of surface temperature, while a high shortwave reflectivity helps to minimize errors in calculated boundary layer conductance due to errors in measuring the flux density of net radiation (see Eqs. 2.1 and 2.2). Power was supplied by one or two 12 V, recreational vehicle, lead-acid batteries connected in series, giving a power density (P) of 25 W m - 2 or 100 W m - 2 , respectively. It was calculated from V2/(RA),  where the voltage supplied to the model (V) was measured  continuously at the nichrome terminals on the model. The bare model was run at both 12 and 24 V, while the hide covered model was run at only 12 V which resulted in a skin temperature of 30-35 °C. Changes in stored heat were measured every 5 minutes by 32 chromel-constantan thermocouples embedded at representative locations throughout the polystyrene trunk. Power leads and thermocouple wires ran radially through the polystyrene to a hollow central cavity and then out of the model between the neck and trunk. The central cavity was filled with polyurethane foam chips to minimize free convection. Positions at which measurements were made on the model deer trunk have been specified as angular and longitudinal positions as shown in Figure 2.2 where 0° always denotes the stagnation point. An example of the notation used to specify a unique position on the model deer is 3-0, which means longitudinal position 3 at angular position 0° (stagnation). Surface thermocouple positions coincided with these positions, while embedded thermocouples were along radial lines from these positions to the central cavity. A tanned deer hide was tailored to fit the model deer trunk. The hide was secured snugly to the model by lacing string through holes on either side of the line on the ventral surface where the hide had been cut. For more detail on the deer hide used in this experiment see Chapter 3.  Chapter 2. Boundary Layer Conductance  12  Stagnation Point  Wind Direction  270  Longitudinal Positions  Angular Positions  Figure 2.2: Specification of measurement positions on the model deer.  Chapter 2. Boundary  Layer  Conductance  13  Measurements made on the model were similar to those made by McArthur and Monteith (1980) on a model sheep. The main difference between the two models is that the model deer was constructed of polystyrene, which stored very little heat in comparison to the 22 gauge aluminum sheet metal construction of the model sheep. The model sheep represents a constant temperature model while the deer was a constant flux model. This design allows steady state to be achieved more quickly than for the model sheep.  2.2.2  Measurement Theory  The sensible heat flux density, H, from the trunk surface to the air is given by H = P-Rn-S  (2.1)  where P is the power density, Rn is the flux density of net radiation from the trunk and S is the rate of heat storage per unit trunk area. The Nusselt number, Nu, which is a non-dimensional boundary layer heat conductance, is given by  pcpoch{Ts - Ta) where d is the characteristic dimension for the model deer, p is the density of air, cp is the specific heat of air, a^ is the thermal diffusivity of air, Ts is the surface temperature at a given location and Ta is the free stream air temperature. The Reynolds number, Re, is a non-dimensional wind speed given by Re = —  (2.3)  where u is the mean horizontal wind speed and v is the kinematic viscosity of air. Forced convection relationships of the form Nu = aRe 6 have been published for cylinders oriented at various angles with respect to the mean wind flow.  Chapter 2. Boundary  Layer  Conductance  14  It is convenient to express the ability of a medium to transport heat in terms of conductance, which is the reciprocal of resistance. The boundary layer conductance, <#,, is expressed as ii  pcP(Ts - To) and can be related to Nu by combining eqs. 2.2 and 2.4 to give gb = ahNu/d  2.2.3 2.2.3.1  (2.5)  E x p e r i m e n t a l Sites and I n s t r u m e n t a t i o n W i n d Tunnel  Wind tunnel experiments were conducted in a large open-ended wind tunnel of the blowdown type, (see Iqbal et al. 1977) belonging to the Department of Mechanical Engineering, University of British Columbia. The wind tunnel cross-section was 2.4 m wide by 1.6 m high with a working section 24.4 m long. The purpose of using the wind tunnel was to provide laminar flow conditions so that these measurements of heat transfer could be compared with other wind tunnel studies for circular cylinders and with measurements made under turbulent outdoor conditions. Experiments were conducted approximately 5 m from the start of the working section. Turbulence intensity measured at a height of 67 cm with a three dimensional hot film anemometer probe (Dantec Electronics Inc., Mahwah, NJ, Model 55R91) was less than 2% and wind velocity was essentially constant over the cross-section of the wind tunnel to within a few centimeters of the walls. The free stream wind velocity and air temperature were measured 1.1 m upstream of the model deer by pairs of Thornthwaite sensitive cup anemometers (C.W. Thornthwaite Associates, Centerton, NJ, Model 901-LED) and welded fine wire 13 [im diameter, chromel-constantan thermocouples, respectively situated at 0.8 and 1.2 m above the wind tunnel floor. Surface temperatures  Chapter 2. Boundary  Layer  Conductance  15  were monitored at various positions on the model deer trunk using an (Everest Interscience Inc., Fullerton, California, Model 112) infrared thermometer (IRT) mounted on a tripod. A miniature net radiometer (Swissteco Pty. Ltd., Melbourne, Australia, Model ME-1) was used to measure net radiative flux density at various positions; however, it was difficult to make accurate measurements due to shadow and view factor problems. Instead, net radiative flux density was calculated using measurements of deer surface temperature and wind tunnel wall surface temperature (see Appendix A). Wind tunnel wall surface temperature was measured using another infrared thermometer (Everest Interscience Inc., Model 4000) . Data was recorded using a d a t a logger (Campbell Scientific Inc., Logan, Utah, Model 21X). Experiments were carried out with both a bare and coat covered model in cross flow and longitudinal orientations (Figure 2.3). For the bare model, wind speed was varied from 0 . 6 m s " 1 to 5.4 m s - 1 . For the cross flow configuration, measurements were made at various angular positions for longitudinal position 2 in the middle of the trunk and position 5 on the rear end. For the longitudinal flow orientation, measurements were made at longitudinal positions 1 , 2 , and 3 on one side and the top of the trunk and longitudinal position 5. Several different measurement procedures were used. The first procedure was to keep the IRT stationary at a position and take measurements for five minute periods at each of 4 windspeeds. The second procedure was to hold wind speed constant for a longer time to ensure equilibrium has been reached before making surface temperature measurements at all the positions. Surface temperature equilibrated rapidly to a change in wind speed (see Appendix B) so that the first procedure was adequate for determining boundary layer conductances. However, when making measurements of coat conductance, the second procedure was desirable because the skin surface temperature required a much longer time to equilibrate to a change in wind speed or power supply voltage. More details on  Chapter 2. Boundary Layer Conductance  16  Wind Tunnel Top Views  A CZZ5=> KJ A  A  >..  A  4  A  wind direction  wind direction  Cross Flow  Longitudinal Flow  Figure 2.3: Orientations of the model deer during the wind tunnel experiments.  Chapter 2. Boundary  Layer  Conductance  17  the wind tunnel experiment using the coat covered model deer are presented in Chapter 3. It proved more difficult to obtain accurate measurements of g\, for the coat covered model due to smaller differences between surface and air temperature.  Temperature  differences for the bare model ( P = 24 V) were about 3°C for u = 5 . 4 m s " 1 and 8.5 °C for u = 0 . 6 m s " 1 compared with 1 °C and 5°C, respectively, for the coat covered model ( P = 12V). Estimates of gi, made with temperature differences of less than 2 ° C were difficult, because an error of only 0.1 °C in the surface or air temperature measurements could result in a 20 % error in g^. To minimize errors, surface temperature measurements were corrected by making periodic comparisons with a black body calibration block in the wind tunnel (see Appendix C).  2.2.3.2  Field Sites  Field testing of the bare model was carried out in an old-growth, Douglas-fir stand (30 m tall, 500 stem h a - 1 ) located near Woss, Vancouver Island, during the summer of 1989. A second experiment using the coat covered model was carried out in a thinned and pruned second- growth, Douglas-fir stand (16.7m tall, 575 stems h a - 1 ) near Courtenay, Vancouver Island during the summer of 1990. The old-growth stand contained patchy understory vegetation less than 0.7m tall (mainly salal), while the second growth stand had a sparse understory less than 0.5 m tall (salal, Oregon grape, and huckleberry). Both sites were situated on hillsides, where the diurnal wind pattern in fair weather was upslope during the day and downslope at night. Consequently, the model deer was oriented along the slope so that prevailing winds would provide cross flow conditions. Wind speeds were measured with hot wire anemometers located at 0.2, 0.4, 0.8 and 2.0 m above the ground, as described by Lee and Black (1993). Air temperature was measured with fine wire thermocouples at all four heights in the old-growth stand and  Chapter 2. Boundary  Layer  Conductance  18  only at 0.8 m in the second-growth stand. Air temperature and wind speed at the 0.8 m height were used in calculations of conductances because this was the mid trunk height of the model deer. Turbulence intensity (cru/u, where <ru is the standard deviation of the mean horizontal wind speed, (u) was measured in the forest using a 3-dimensional sonic anemometer (Applied Technologies Inc., Boulder, Colorado, Model SWS-211/3V). Radiative surface temperature and net radiative flux density were measured at selected locations on the model deer trunk using a hand-held Everest Interscience infrared thermometer (model 112) and a Swissteco miniature net radiometer (model ME-1), respectively. The hand held IRT and net radiometer were mounted on tripods and positioned to monitor a single location on the model.  Generally, a single location was  monitored all day or all night. The hand-held IRT and net radiometer were periodically removed from their tripods to take readings at all the positions within a short time period (approximately 3 minutes). The model deer was powered at either 12 or 24 V in the old-growth stand and at only 12 V in the second-growth stand due to unrealistically high temperatures in the polystyrene when higher voltages were applied to the coat covered model. It was desirable to keep the model out of direct sunlight to minimize the error in calculating conductances.  Natural shade was sufficient most of the time in the old-  growth stand but due to the openness of the second growth stand, artificial shade was needed. This was accomplished by suspending a 1 x 2 m rectangle of black plastic film from a cable strung east-west between the trees at 6 m above the ground on the south side of the model deer. The rectangle was moved manually along the cable during the day to keep the model in shade. This arrangement served to minimize any modification of the wind regime near the model.  Chapter 2. Boundary  2.3  Layer  Conductance  19  R e s u l t s and D i s c u s s i o n  2.3.1  Laminar F l o w  Cross flow: Figure 2.4 shows the distribution of the difference between surface and air temperature as a function of angular position around the the bare model deer at longitudinal position 2, exposed to cross flow in the wind tunnel. The temperature difference is largest on the top (90°) and bottom (270°) of the model indicating a thicker boundary layer and less effective heat transfer than at the stagnation (0 °) or lee positions (180 °). These temperature differences were used to calculate local Nusselt numbers, Nu# (d = 0.284 m), for comparison with published Nu# for long circular cylinders in cross flow. The results of these comparisons are shown in Figures 2.5, 2.6, and 2.7 for the stagnation, top and lee positions, respectively.  The regression statistics are reported in  Appendix H. The data points from this study fall on or slightly below the regression line through the published data for the stagnation and lee positions. However, the data for the top position lie above the regression line through the published data. This may be a result of the point of boundary layer separation for an elliptical cylinder being different from that of a circular cylinder. The point of boundary layer separation corresponds to a relative minimum in Nu (see discussion of Figure 2.8 below). Also shown in Figures 2.5, 2.6 and 2.7 are data for a coat covered model. As was mentioned previously, it was difficult to get many reliable measurements of Nu# for the coat covered model due to small temperature differences between the coat surface and air. Only one data point has been plotted in Figure 2.5 because the others had small temperature differences which resulted in unacceptably large errors in Nu^. Some of the data points fall above and some below the bare model values. Overall, it is difficult to make a case for the Nu# of the coat covered model being significantly different from the  Chapter 2. Boundary  8  Layer  20  Conductance  1  1  1  1  90 Wb  l  Direction  n  (  >  f  + 180  \  —  2TO  u = 1.5 m/s / \J  ~~\J  4 u = 5.4 m / s ^ ^ U—  u  2 -  0  1  0  1  1  90 180 Angular Position (degrees)  1  270  Figure 2.4: Temperature difference (Ts — Ta) as a function of angular position (at longitudinal position 2) around the bare model deer exposed to cross flow in the wind tunnel at two wind speeds.  Chapter 2. Boundary Layer Conductance  21  1000 NUQ = 0.729Re  CD  3  100-  Stagnation 10 10"  10  10"  10  Re Figure 2.5: Local Nusselt number (Nug) vs. Reynolds number (Re) for the stagnation point (0°) at position 2, comparing the published results of Giedt (1949) ( • ) , Bosch (1936) (o), Schmidt and Wenner (1941) (O) and Schmidt and Wenner (1943), as referenced by Sandborn (1972) (A) with wind tunnel results from this study for a bare (•) and a coat covered (M) model deer. The solid line is a regression through the aforementioned published results. The linear regression line through the bare model deer points (not shown) is given by the equation Nu0=O.9ORe0-50. Regression statistics are shown in Appendix H.  Chapter 2. Boundary Layer Conductance  22  1000  <p  I ioo  10  Figure 2.6: Same as Figure 2.5 except data presented is for the top point (90°). The linear regression line through the bare model deer points (not shown) is given by the equation Nu^=0.18Re°-61. Regression statistics are shown in Appendix H.  Chapter 2. Boundary  Layer  23  Conductance  1000  <r>  I 100  10 Re  Figure 2.7: Same as Figure 2.5, except data presented is for the lee point (180°). The linear regression line through the bare model deer points (not shown) is given by the equation Nug=0.12Re°- 69 . Regression statistics are shown in Appendix H.  Chapter 2. Boundary  Layer  Conductance  24  bare model. McArthur and Monteith (1980) presented data showing that the regression line of overall Nusselt number, Nu vs. Re (log-log plot) for a fleece covered model had a greater slope than for the corresponding bare model. For Re < 6 xlO 4 , Nu for the fleece covered model was lower, while for Re > 6 x l 0 4 , it was higher than for the bare model. This was attributed to increased boundary layer turbulence at the higher Re. A pattern similar to this can be envisaged in Figures 2.6 and 2.7, but measurement uncertainties preclude firm conclusions. Figure 2.8 shows local gb as functions of position and wind speed for the bare model in the wind tunnel for cross flow. Local gt, decreases with increasing angle from the stagnation point as the boundary layer thickens, reaching a minimum at the point of boundary layer separation around 90° and then increases to a second maximum at 180° in the lee. This lee maximum becomes more pronounced as wind speed increases. High gb in this region is due to the action of turbulent eddies in the wake zone. This pattern corresponds to that described in Kreith (1973) for a circular cylinder in cross flow and d a t a he presents from Giedt (1949). A wind speed u = 5.4 m s " 1 corresponds to Re ~ 110,000 which is nearing the transition to a turbulent boundary layer. However, it appears that this point has not been reached because according to Kreith (1973) we would expect to see a maximum of local boundary layer conductance between 100 and 120° and a minimum at around 145° if the boundary layer had become turbulent. Longitudinal flow: There is much less literature pertaining to heat transfer from a cylinder in longitudinal flow than in cross flow. Much of what is available is from aerospace research pertaining to heat flow from rockets, missiles and airplanes at very high Re. In one such reference, Chauvin and de Moraes (1951), report that the local equation for turbulent heat transfer from a flat plate at subsonic speeds agreed well with that observed for a parabolic body at Mach numbers up to 2.48. In other words, the local  25  Chapter 2. Boundary Layer Conductance  30 90  S  20 u = 5.4 m/s  10 u = 2.7 m/s u = 1.5 m/s u = 0.6 m/s  0  0  90 180 270 Angular Position (degrees)  Figure 2.8: Local boundary layer conductance (<?;,) vs. angular position (at longitudinal position 2) for the bare model deer trunk in cross flow at different wind speeds in the wind tunnel.  Chapter 2. Boundary  Layer  Conductance  26  heat transfer coefficient decreased with increasing distance downwind from the leading edge. Figure 2.9 shows that g\> for the bare model decreased monotonically with increasing distance from position 5. Figure 2.10 shows the local Nusselt number for longitudinal flow ( N ^ ) plotted against the local Reynolds number (Re^) for positions 1, 2 and 3 on one side of the deer and position 5 on the rear end. Wind speeds ranged from 0.6 to 5.5 m s - 1 for the d a t a shown. The straight line distance from position 5, parallel to the model deer's long axis, was used as the characteristic dimension, except at position 5 where a small number, 0.01m, was used instead of 0 to determine whether position 5 points fell onto the same line as the other positions. The equation of the regression line through the side points for longitudinal flow is N u r = 0.143Re°' 679  (2.6)  which can be compared with the equations given by Wigley and Clark (1973) for iso-flux flat plates in laminar and turbulent flow, respectively Nu x = 0.453Pr a 3 3 Re°- 5  (2.7)  Nu* = 0.045Pr°- 33 Re°- 84  (2.8)  where Pr is the Prandtl number (Pr 0 - 33 = 0.89 for air). The exponent on Re x in Eq. 2.6 suggests that flow over the side of the model deer tends toward turbulent flow. This is also supported by the increase in Nu^ when proceeding from the rear end points to the side points which may be the so called turbulent j u m p which is sometimes observed when a flow changes from laminar to turbulent. A similar result was obtained for points on top of the model. Since the position 5 points do not fall on the same line as the side points it appears that it is not appropriate to think of position 5 as the leading edge of a  Chapter 2. Boundary  C/3  Layer  27  Conductance  35 30  O  25  u = 5.4 m/s  O  a o U 5-1  20 15  ;? 10 -a  § o PQ  Wind Direction 4  5 0  u = 1.5 m/s  0 5  3 2 Distance Downwind (m) or Longitudinal Position  0.6  Figure 2.9: Local boundary layer conductance for positions on the side of the model deer in longitudinal flow as a function of straight line distance downwind of position 5.  28  Chapter 2. Boundary Layer Conductance  1000  Nu x = 0.143Re  100 side data X  £  10 rear end data  10'  10 Re.  10'  10  Figure 2.10: Local Nusselt number Nu^ plotted against local Reynolds number Re^ for points on the side of the bare model deer in longitudinal flow in the wind tunnel (•). Also shown are pos. 5 points (•), where d = 0.01m was assumed. The equation of the regression line through the pos. 5 data is Nu :E =0.371Re 0460 . Regression statistics are shown in Appendix H.  Chapter 2. Boundary  Layer  Conductance  29  flat plate. It may be more appropriate to consider this position as being the stagnation point of a sphere. Average Nu for deer's trunk: Figure 2.11 compares the Nu (the average Nusselt number for the entire trunk) vs. Re relationships for the model deer trunk in longitudinal and cross-flow cases. Nu was calculated using area weighted spatial averages of Ts for the trunk of the model in the following equation.  m  =jt^  (2-9)  where Nu; is the spatially averaged Nu for region i of the model, A is the total surface area of the model, Ai is the surface area of region i, d is the characteristic dimension for the model as a whole, di is the characteristic dimension of region i and n is the number of regions (two, including the main part of the trunk and the rear end section, consisting of the last 12 cm). It can be seen that Nu is higher for the longitudinal flow case. The regression equations for the cross flow and longitudinal cases, respectively, are Nu" = 0.226Re° 6 1 3  (2.10)  Nu" = 0.155Re°- 692  (2.11)  These regression equations may be compared with that of McArthur and Monteith (1980) for a model sheep in cross flow Nu" = 0.095Re° 6 8 4  (2.12)  Caution should be used in interpreting the higher Nu for longitudinal flow as enhanced convective heat transfer because a larger characteristic dimension (d = 0.8 m vs. d = 0.284m) was used. A better way to compare the effectiveness of convective heat transfer is by looking at mean boundary layer conductance, gb, as shown in Figure 2.12. This graph shows that  Chapter 2. Boundary  Layer  30  Conductance  1000 Longitudinal Flow Nu = 0.155Re  Cross Flow 0.613 Nu = 0.226Re  10 10  10 Re  10  Figure 2.11: Comparison of Nu vs. Re relationships for bare model deer in longitudinal and cross flow orientations in the wind tunnel. Regression statistics are shown in Appendix H.  Chapter 2. Boundary  Layer  Conductance  31  24 Longitudinal Flow  20 h  0.691  Cross Flow 0.610 gb = 7.183u  2 4 Wind Speed (m/s) Figure 2.12: Comparison of overall gt, vs. wind speed relationships for bare model deer in longitudinal and cross flow orientations in the wind tunnel. Regression statistics are shown in Appendix H.  Chapter 2. Boundary  Layer  32  Conductance  heat loss from the model deer is enhanced in longitudinal flow by as much as 25 % when the windspeed reaches 5.4 m s " 1 . This may be caused by the development of a turbulent boundary layer in the longitudinal flow case as suggested by Figure 2.10. No data were taken for longitudinal flow with the model deer's head facing the wind. The results would be affected by the presence of the head and neck causing eddies in their lee.  2.3.2  Turbulent F l o w  Figures 2.13, 2.14 and 2.15 show Nu# vs. Re for stagnation, top and lee positions, respectively, of the model deer situated in an old-growth stand.  As indicated earlier, the  prevailing wind directions gave cross-flow conditions. The wind tunnel results for the same position on the model deer in cross flow are also shown for comparison. The very low windspeed in the old growth stand made comparison with the wind tunnel difficult. In the old growth, Nu# for the top and stagnation positions was significantly larger than t h a t observed in the wind tunnel (at the 5 % significance level, see Appendix H). Enhancement where the two data sets overlapped was 7 5 % for the top position, 14% for the stagnation position and negligible for the lee position. Turbulent enhancement of heat transfer depends on turbulence intensity and the turbulence scale (Van der Hegge Zijnen, 1958). According to the literature reviewed by Kestin (1966), turbulent enhancement of 50 to 100% is not uncommon when increasing the turbulence intensity from 1 or 2 % up to 30 %. In this study, turbulence intensity for the horizontal component (u) of wind velocity was approximately 5 0 % for wind speeds above 0.2 m s _ 1 . This was much higher than the turbulence intensities of less than 2 % observed for laminar flow conditions in the wind tunnel. The importance of eddy size to heat transfer is discussed in Hinze (1959) and confirmed experimentally for a flat plate at various angles to the flow by Chen et al. (1988).  Chapter 2. Boundary  Layer  33  Conductance  1000  Nu e = 1.47Re  CD  I ioo wind tunnel NUQ = 0.90Re" n Field Data • Wind Tunnel Data  10 10"  10 Re  10'  Figure 2.13: Nu# vs. Re for the bare model deer in an old growth stand (longitudinal position 1) at the stagnation point with wind tunnel regression line (longitudinal position 2) for comparison. Regression statistics are shown in Appendix H.  Chapter 2. Boundary  Layer  34  Conductance  1000  field 0.437 Nu e = 1.54Reu  #100  Wind Tunnel  Figure 2.14: Same as Figure 2.13, except data presented are for the top position and field data are for longitudinal position 3, while wind tunnel d a t a are for position 2. Regression statistics are shown in Appendix H.  35  Chapter 2. Boundary Layer Conductance  1000  05  I  100  Nu e =1.852Re  a Field Data • Wind Tunnel  10 Re Figure 2.15: Same as Figure 2.13, except data presented are for the lee position at longitudinal position 2 in both cases. Regression statistics are shown in Appendix H.  Chapter 2. Boundary  Layer  Conductance  36  They found that when turbulent eddies correspond in size to those in the wake of an object, which are in turn of similar size to the object, large increases in heat transfer can occur due to a resonance effect. In the forest, a deer's trunk may be subject to this resonance effect due to its size being similar to tree trunks. In the old-growth stand, mean tree trunk diameter was about 1 m, while in the second-growth stand it was only about 0.25m. The distribution of eddy sizes observed in the old-growth and second-growth stands is shown in Appendix D.  2.4  Conclusions  Local Nusselt number (Nu#) vs. Re relationships for an elliptically cross-sectioned trunk of a bare model deer in cross flow in the laminar flow conditions of a wind tunnel were found to agree well with those for circular cylinders at the stagnation and lee positions. Agreement was not as good at the top position, with Nu# measurements from this study being somewhat larger than those for circular cylinders obtained in previous studies. Boundary layer conductances for the coat covered model deer were not significantly different than those for the bare model.  Overall heat transfer from the trunk of the  model deer was larger when it was placed in longitudinal flow (rear end facing the wind) than in cross-flow. This may have been due to the existence of a turbulent boundary layer in the longitudinal case. In an old-growth Douglas-fir stand, boundary layer conductance for the trunk of the bare model was enhanced by an average of about 30 % for all positions, over that measured in the wind tunnel, due to much higher turbulence intensities in the forest and possibly turbulence resonance caused by the similar scale of the trees and the deer trunk.  Chapter 2. Boundary  2.5  Layer  37  Conductance  Literature C i t e d  Bosch, M.T. (1936) Die Wdrmeubertragung. Campbell, G.S. (1977) An Introduction  3rd ed. Springer Verlag, Berlin.  to Environmental  Biophysics.  Springer-Verlag,  New York, 159 pp. Chauvin, L.T. and C.A. de Moraes (1957) Correlation of supersonic convective heat transfer coefficients from measurements of skin temperature of a parabolic body of revolution. NACA RM L51A18. Chen, J.M., A. Ibbetson and J.R. Milford (1988) Boundary-layer resistances of artificial leaves in turbulent air II. Leaves inclined to the mean flow. Boundary-Layer  Mete-  orol 4 5 , 371-390. Giedt, W.H. (1949) Investigation of variation of point unit-heat-transfer coefficient around a cylinder normal to the air stream. Trans. ASME. 7 1 , 375-381. Hinze, J.O. (1959) Turbulence: An Introduction  to Its Mechanism and Theory. McGraw-  Hill, Toronto, 586 pp. Iqbal, M., A.K. Khatry and B. Senguin (1977) A study of the effects of multiple windbreaks. Boundary-Layer  Meteorol. 1 1 , 187-203.  Kestin, J. (1966) The effect of free-stream turbulence on heat transfer rates. In Advances in Heat Transfer (Edited by Irvine, T.F. and Hartnett, J.P.). pp. 1-31. Academic Press, New York. Kreith, F. (1973) Principles pp.  of Heat Transfer, 3rd ed. Harper and Row, New York, 656  Chapter 2. Boundary  Layer  38  Conductance  Lee, X. and T.A. Black (1993) Turbulence near the forest floor of an old growth Douglasfir stand on a south-facing slope. For. Sci. 39, 211-230. McArthur, A.J. (1977) Heat loss from sheep. Univ. of Nottingham, Ph.D. thesis, 186 pp. McArthur, A.J. and J.L. Monteith (1980) Air movement and heat loss from sheep. I. Boundary layer insulation of a model sheep, with and without fleece. Proc. R. Soc. Lond. B. 2 0 9 , 187-208. Mitchell, D. (1974) Convective heat transfer from man and other animals. In Heat Loss from Animals  and Man (Edited by Monteith, J.L. and Mount, L.E.). pp. 59-76.  Butterworths, London Mitchell, M.A. (1985) Measurement of forced convective heat transfer in birds: a wind tunnel calorimeter. J. Therm. Biol. 10, 87-95. Parker, K.L. (1988) Effect of heat, cold and rain on coastal black-tailed deer. Can.  J.  Zool. 6 6 , 2475-2483. Sanborn, V.A. (1972) Resistance  Temperature  Transducers.  Meteorology Press, Fort  Collins, Colorado, 545 pp. Schmidt, E. and K. Wenner (1941) Warmeabgabe iiber den Umfang eins angeblasenen geheizten Zylinders. Forschg Ing.-Wes.  12, 65-73.  Van der Hegge Zijnen, B.G. (1958) Heat transfer from horizontal cylinders to a turbulent air flow. Appl. Sci. Res. A 7 , 205-223. Wiersma, F . and G.L. Nelson (1967) Nonevaporative convective heat transfer from the surface of a bovine. Trans. ASAE 10, 733-737.  Chapter 2. Boundary  Layer Conductance  39  Wigley, G. and J.A. Clark (1974) Heat transport coefficients for constant energy flux models of broad leaves. Boundary-Layer  Meteorol. 7, 139-150.  Chapter 3 C o a t C o n d u c t a n c e of a M o d e l D e e r in a W i n d Tunnel a n d a Douglas-fir Stand  3.1  Introduction  The ability of deer and other mammals to to maintain their core body temperature at a nearly constant level is due in no small part to the insulation or resistance to heat loss provided by their coats. Humans have relatively little natural coat, so they must rely on artificial insulation in the form of clothing. As such, a lot of research has been conducted in an attempt to quantify and understand the mechanisms of insulation. It is often convenient to talk about coat conductance which is the ability of fur to transfer energy, i.e. the reciprocal of coat resistance. We sometimes use the term 'coat' to mean collectively the skin or hide and the attached hair, but when talking about the coat conductance or resistance we refer only to heat transfer from the outer skin surface to the tips of the hair. Scholander et al. (1950) measured the thermal resistance of isolated coat samples from various animals. Moote (1955) and Tregar (1965) attempted to quantify the effect of wind speed and penetration on the coat conductance of isolated coat samples. Cena and Clarke (1973) developed theory to describe radiative transfer through animal coats. Cena and Monteith (1975a and b) made measurements on coat samples and attempted to quantify the relative importance of radiation, convection and conduction in transferring energy through the coat. They suggested, however, that measurements should be made  40  Chapter 3. Coat  Conductance  41  on real or model animals in the field to determine how well wild or domestic animals are coupled to their habitats. One of the few examples of an outdoor study is that of McArthur (1977) using a live sheep. It was found that overall coat conductance was greater outdoors than in the wind tunnel. Numerous studies have been conducted on model animals in wind tunnels such as those by Wiersma and Nelson (1967) for cattle and Mitchell (1985) for birds. Several important issues arise from examining the literature. These include the importance of free and forced convective heat transfer through the coat and the effect of coat structure and part of the body being considered on these transfer modes. Another issue is the form of the functional relationship between coat conductance and wind speed. Cena and Monteith (1975b) concluded that free convection is an important mechanism of heat transfer through animal coats; however, Davis and Birkebak (1975) attempted to show through theoretical calculations that free convection is not important. Both studies concluded that forced convection is an important transfer mechanism depending upon how much the wind disturbs the coat. Campbell et al. (1980) argued that coat thermal conductance (or coat conductance, g c ) is a linear function of wind speed (u). They concluded this after statistical analysis of published data from many sources, where the exponent on u ranged from 0.2 to 4.0. However, Bakken (1991) argued that n (the exponent on u) should range from 0.5 to 2.0 because convective heat transfer is proportional to u 0 ' 5 , while mechanical force is approximately proportional to u2. In this chapter, the objectives were (i) to determine the relative roles of free and forced convection, radiation and conduction in transporting heat through the coat, (ii) to determine the relationships between gc and mean wind speed for various positions on the trunk, (iii) to determine whether the turbulence in the outdoor environment (specifically that in a forest stand) causes an enhancement in coat conductance, and (iv)  Chapter 3.  Coat  42  Conductance  to determine the importance of coat conductance relative to boundary layer conductance in transporting heat away from the deer.  3.2  Theory  3.2.1  M e c h a n i s m s of H e a t Transport T h r o u g h A n i m a l C o a t s  Monteith and Unsworth (1990) list the three mechanisms which contribute to heat transport through animal coats as conduction, radiation and convection. The relative importance of these mechanisms depends on the physical properties of the coat and ambient meteorological conditions. Conduction:  It takes place in either the hairs or the still air between them.  The  thermal conductivity (heat flux density per unit temperature gradient) of still air at 20 °C is 2 5 m W m _ 1 K _ 1 , while that of the solid portion of an individual hair is an order of magnitude greater at about 250 m W m - 1 K _ 1 . Despite this large thermal conductivity, Cena and Monteith (1975b) concluded that heat conduction through hair is small when compared with the total heat transfer through the coat. This is because the proportion of the cross-sectional area of the coat occupied by hair is small (on the order of 2 to 10 % for most mammals) and much of the cross-sectional area of an individual hair consists of air. Typical values of hair conductivity (kk) range from 1 to 10 m W m _ 1 K _ 1 . Scholander et al. (1950) and Hammel (1955) measured the coat thermal conductivity (kc) of samples from numerous species of wild animals and found values ranging from 36 to 9 4 m W m - 1 K _ 1 , but typically in the range 40 to 6 0 m W m - 1 K _ 1 . Since these values significantly exceed the still air value, we must look to radiation and convection to explain the excess. Coat conductance can be calculated from kc using  *=h  <31)  Chapter 3.  Coat  43  Conductance  where p and cp are conventionally taken to be the density and specific heat of air, and / is the mean coat depth normal to the skin surface. Radiation:  Theory for predicting the transfer of thermal radiation through animal  coats was developed by Cena and Clark (1973) and Cena and Monteith (1975a and b). The radiative conductivity (kr) of an animal coat which is assumed to be of uniform composition and to have a linear temperature gradient across it can be expressed as  kr = \i  (3.2)  Sp where b is the slope of the blackbody radiation curve, 4crT 3 , (T is taken to be the mean absolute temperature of the coat), <r is the Stefan-Boltzmann constant and the parameter p is the mean probability that radiation will be intercepted in a unit depth of coat. Instead of using p in Eq. 3.2, the authors used an approximate value which we shall call p±. This value of p considers only radiation which is emitted in a direction normal to the skin surface and is calculated using p± = nJtan(arccos(/// s ))  (3-3)  where n is the number of hairs per unit area of skin, d is the mean hair diameter and ls is the mean hair length. Some typical values of coat parameters for a few animals are given in Table 3.1 below. Unfortunately, the above equations for calculating kr prove to be invalid for the physically important case of piloerection, when an animal attempts to conserve heat by erecting its hair perpendicular (or nearly so) to the skin surface, making / = ls.  As  / approaches ls, p±_ goes to zero in Eq. 3.3 and Eq. 3.2 predicts an infinite radiative conductance. The value of p calculated in Eq. 3.3 only considers the interception of radiation leaving the skin surface in a direction perpendicular to it. The authors assumed that this value is a good estimate of the mean interception function p for radiation  44  Chapter 3. Coat Conductance  Table 3.1: Typical values of coat parameters for various animals taken from Cena and Clark (1973). The radiative conductivity was calculated using Eqs. 3.2 and 3.3 at a temperature of 20 °C . Animal Sheep Rabbit Badger Cow Goat Fox Deer  n (cm" 2 ) 1460 4200 240 1260 110 3600 520  d {urn) 42 31 71 44 83 20 150  /  /,  (cm) 5.0 2.0 1.8 0.6 2.5 1.4 1.5  (cm) 6.0 2.5 4.5 2.0 3.0 2.0 3.7  P± (cm" 1 ) 4.6 9.7 3.9 17.8 0.7 7.2 17.9  A-y  (mWm^K-1) 16.8 8.0 19.8 4.3 110.5 10.7 4.3  Chapter 3. Coat  45  Conductance  emitted in all directions. With this assumption, the numerical integration to determine the interception of long wave radiation by the coat is avoided. This integration is described in part by Cena and Monteith (1975b) and will be fully described in the next chapter. For now, it will suffice to say that a mathematical expression for p, along with an expression for long wave radiance, was numerically integrated to determine a net radiative flux density (F) through a coat layer with a given temperature gradient (a). The radiative conductivity is then given by F kr = —  (3.4)  a Convection:  The third mechanism of heat transfer in an animal coat is convection.  The two types of convection which can occur are free and forced convection. Free convection is due to air movement induced by temperature gradients, while forced convection is due to wind. As mentioned in the preceeding section, the functional relationship between gc and u has been the subject of debate in the literature. Several earlier works such as Joyce et al. (1966) and Campbell (1977) proposed that  g? = ^(O)"1 - an05  (3.5)  where <7C(0) is the coat conductance under still air conditions and a is an empirical constant.  Campbell et al. (1980) critically examined the work of many authors and  concluded that coat conductance varied linearly with wind speed as follows gc = gc(0) + cu  (3.6)  where c is independent of wind speed, but possibly a function of coat depth and the parameter gc(0) is comprised of the hair, still air and radiative conductances. Campbell et al. (1980) noted that the values predicted by Eq. 3.6 may be too low if free convection is present in the coat because the equation only applies to forced convection.  Chapter 3. Coat  46  Conductance  Free convection has been found to be an important mechanism of heat transfer in animal coats by Cena and Monteith (1975b) and McArthur and Monteith (1980a and b). Free convective heat loss from many objects, irrespective of their shape, has been found to obey the five-fourths power law, i.e. convective heat loss is proportional to AT" 5 / 4 , where A T is the temperature difference between the object and the air. A plausible expression for the coat conductance due to free convection, gjr, is gfr = A(Tsk - T s ) 0 - 25  (3.7)  where A is an empirically determined constant which may be related to the physical properties of the coat, Tsk is the skin surface temperature and Ts is the coat surface temperature. For convenience, the temperature difference, Tsk — Ts will sometimes be referred to as AT C .  3.2.2  M o d e l s of H e a t Transfer T h r o u g h t h e Coat  McArthur and Monteith (1980b) proposed the two alternative electrical analogue models of heat transfer through an animal coat shown in Figure 3.1. Model 1 assumes that the heat transfer mechanisms, molecular conduction (which includes hair and still air), radiation, forced convection and free convection act in parallel. Model 2 assumes that radiation and molecular conduction act in parallel with each other and the series combination of free and forced convection. The latter model assumes that the wind penetrates to a depth t in the upper coat layer, where forced convection is the dominant mode of convection and mechanical mixing eliminates any temperature gradient, while free convection dominates in the lower layer near the skin surface. McArthur and Monteith show that t is given by  t = —I 9c  (3.8)  Chapter 3. Coat  Skin Surface  Conductance  Coat Surface  molecular conduction v^W^/V^  radiation WNA/vWv  Model One forced convection WvVWVVV" free convection  molecular conduction /WVWA  Model! Two forced  WVWvW convection  • A W W W  convection  radiation  vwwwv  Figure 3.1: Alternative resistance models for heat transfer through animal coats proposed by McArthur and Monteith (1980b). Model one assumes that all mechanisms act in parallel, while model two assumes that forced and free convection are in series, with forced convection acting to some wind penetration depth, t.  Chapter 3.  Coat  48  Conductance  where c is the empirical constant from eq. 3.6. Models 1 and 2 respectively can be expressed mathematically as follows gc = gm+9r gc = 9m+gr  + 9fc + 9}r  (3.9)  + 77—7-7-7—  (3.10)  where gm is the molecular conductance, gr is the radiative conductance, and gfc the forced convective conductance.  3.3  Experimental Methods  3.3.1  Coat Covered Model Deer  Deer coat description:  The design and construction of the model deer trunk was described  in Chapter 2. Some details are repeated here along with a complete description of the deer coat installation and instrumentation. Although the trunk had been constructed according to the dimensions of an adult black-tailed deer (Odocoileus hemionus columbianus), the trunk proved to be too large to be completely covered by either of the available tanned black-tailed deer coats. Complete coverage of the trunk was desirable to avoid lateral heat flow within the polystyrene. A larger commercially tanned mule deer [Odocoileus hemionus hemionus)  coat was used in  lieu of the black-tailed deer coat to cover the model deer trunk. The mule deer coat was determined to be a winter coat based on the time of year it was taken (November) and descriptions found in Rue (1989) and Walmo (1981). The coat was characterized by a grayish tan color as opposed to the more ruddy hue of a summer coat. There were woolly under hairs present and the intermediate guard hairs (see Raddi, 1967) which comprised the bulk of the coat were hollow and relatively brittle, both of which are characteristic of a winter coat.  Chapter 3.  Coat  Conductance  49  Preparation of the coat consisted of cutting and sewing (mainly along ventral surface) so t h a t it would fit around the trunk. The coat was secured snugly to the model by lacing string through holes on either side of a line on the ventral surface where the coat had been cut. This allowed for good thermal contact between the coat and trunk, while allowing for easy removal and installation.  A total of twelve thermocouples were installed to  monitor skin surface temperature for longitudinal positions 1 to 4 (see Figure 2.2) at the stagnation (0°), top (90°) and lee positions (180°). The naming convention for the various positions is the same as that used in Chapter 2 for the bare model deer. For example, position 2-0 refers to a point defined by longitudinal position 2 and an angular position of 0° (stagnation). The thermocouples were constructed of Teflon coated 30 gauge chromel-constantan wire and attached to on the outer skin surface as shown in Figure 3.2. The technique used was to thread the thermocouple through the hide from below with a needle, pull the thermocouple into place and fasten it there with two small dabs of five minute epoxy glue as shown. A single loop of thread was used to ensure that thermocouple was in good contact with the skin surface. This technique caused very little disturbance of the hair. No measurements were taken along the bottom (270°) or the rear end (pos. 5) of the coat covered model because of the join between the two sides of the coat. A series of measurements to characterize the mule deer coat were made after the completion of the field and wind tunnel experiments. The measurements were needed for calculating the magnitudes of radiative and conductive heat transfer in the coat. Hair samples were taken near six different measurement positions on the coat which were angular positions 0 ° , 90° and 180° for longitudinal positions 2 and 3. One hair was plucked from the coat at each of four points which were evenly spaced on a 3 cm radius circle centered at the measurement position. The measurement positions were marked by a small dot of orange dye on the fur tips directly above the skin surface thermocouples.  Chapter 3. Coat  50  Conductance  rzzzzzzzzzzzzzzzzzzzzzzzz to data logger T glue  J  \  glue  thread  Teflon insulated 30 gauge chromel-constantan wire  Figure 3.2: Diagram showing how thermocouples were installed to measure skin surface temperature.  Chapter 3. Coat  Conductance  51  The length of each of the hairs was measured and averaged for each position. Coat depth was measured with the probe end of a vernier caliper for each position where a hair was plucked from the hide. This depth is different from hair length because hairs do not stand straight up; rather they lie at an angle to the skin surface. A sample of twelve other hairs was selected for making an estimate of mean hair diameter. The hairs were examined under a compound microscope with a 6 p  resolution  length scale on the eyepiece. The mean diameter of each hair was determined as the average of the diameters at the base, middle and end of each hair. Several thin hair crosssections were examined under the microscope to make an estimate of the proportions of air space and solid (cell wall) material. Hair density was determined for two locations on the mule deer coat representing the top and sides. This was done by cutting 2 cm by 2 cm pieces of fur from the coat with a scalpel from the side of the coat without hair to avoid cutting off hair. The hair was then pulled off the coat and counted. A summary of these hair measurements is provided in Table 3.2. Calculation of coat conductance:  The outer surfaces for radiative and convective energy  exchange are assumed to be identical and the distance below the hair tips negligible compared to the coat depth. At steady state, the total or effective coat conductance is given by <7C = — — pcP{Tsk - Ts)  (3.11)  where P is the power density of the nichrome heating wire wound on the polystyrene just below the hide and is assumed to be equal to the flux density of energy reaching the coat surface. Heat storage in the coat was assumed to be negligible after a sufficiently long time (more on this in the next section). The small correction to P due to the coat surface area being slightly larger than that of the bare model surface area where the  Chapter 3.  Coat  Conductance  52  Table 3.2: Summary of coat parameters measured at various positions on the mule deer hide as well as calculations of hair angle (<f> = arccos(//Z s )) and the thermal conductance of a layer of still air at 20 °C which is the same depth as the coat (ga)-  Position 1-0 2-0 3-0 1-90 2-90 3-90 1-180 2-180 3-180  /(mm) 13.8 17.1 18.1 7.5 9.8 13.4 14.7 17.1 19.5  ls (mm) -  49.4 50.3 -  43.2 50.0 -  50.2 53.2  (f> (deg) ^ ( m m s " 1 ) 1.51 1.22 69.7 68.9 1.15 2.78 76.9 2.13 74.5 1.55 1.42 70.1 1.22 68.5 1.07  Chapter 3. Coat  Conductance  53  power density is measured was neglected. Partitioning  Coat Conductance:  In the data analysis, gc was considered to be made up  of three component conductances acting in parallel as follows.  9c = 9r + Qh + 9con  (3.12)  where gr and g^ are the radiation and hair conductances, respectively and gcon is the conductance of the air either still or moving (free or forced convection). The relationships of these conductances to their respective conductivities are gr =  "V  "'h  i  j , gh = T- and pCpl pCpls  gcon =  "'con  pCpl  in 1 r>\  (3.13)  where p is taken to be the density of air in all cases. The corresponding relationship for still air is  *=h  (314)  The partitioning of component conductances in Eq. 3.12 differs from that in models 1 and 2 (Eqs. 3.9 and 3.10) in several ways. First, molecular conduction through hair is considered to be a separate component from conduction through still air between the hairs. Secondly, forced and free convection have been lumped together in one component, with molecular conduction through still air being included. The reason for not separating out a conductance of still air as was done in Eq. 3.9, is that it is difficult to imagine a layer of still air the thickness of the coat, coexisting with convective activity within the coat. The view was adopted that if gcon = ga, then the air in the coat was still and that if gcon > ga, then some degree of forced or free convection was present in addition to molecular conduction through the air. The radiative conductivity (kr) was calculated using a numerical integration procedure described in Chapter 4 and Eq. 3.4. The area weighted thermal conductivities for  Chapter 3.  Coat  54  Conductance  hair and still air, respectively, can be calculated as follows fa = wh(w0k0  + (1 - w0)kair)  (3.15)  and ka = (1 - wh)kair  (3.16)  where &at> is the unweighted thermal conductivity of still air ( 2 5 m W m _ 1 K _ 1 at 20 °C), Wh is the fraction of coat cross-section occupied by hair, w0 is the fraction of the crosssectional area of hair that is solid and k0 is the thermal conductivity of organic matter ( 2 5 0 m W m - 1 K " " 1 ) . For the mule deer coat used in this experiment, the hair density, n, ranged from 205 to 404 c m - 2 and the mean hair diameter, d, was 0.019 cm. From the data, w^ was calculated to be 0.116. Microscopic examination of hair cross-sections yielded an estimate of 0.2 for w0. Using the above weighting factors and conductivities it was calculated that kh = 8 . 0 m W m - 1 K _ 1 and ka = 22.1 m W m - 1 K - 1 .  3.3.2 3.3.2.1  Experiments and Instrumentation W i n d Tunnel  Two wind tunnel experiments were conducted on the coat covered model deer, one for the longitudinal flow orientation and one for the cross flow orientation. The wind tunnel used was a large (2.4 m wide by 1.6 m high) open-ended tunnel of the blow-down type (see Iqbal et al., 1977) which provided laminar flow conditions (see Chapter 2). The first wind tunnel experiment used the same instrumentation as described for the boundary layer conductance experiment in Chapter 2. The power source for the model deer was a single 12 V car battery, yielding a power flux density of about 2 4 W m - 2 . For this experiment the model deer was oriented with its long axis parallel to the air flow direction (longitudinal flow). Measurements were made with both the head facing the wind and the rear end facing the wind.  Chapter 3.  Coat  Conductance  55  The instrumentation used during the second wind tunnel experiment was slightly different from that used in the first experiment. Wind speed and air temperature were measured upstream of the model at heights of 0.8 m and 1.2 m above the floor with Thornthwaite sensitive cup anemometers (C.W. Thornthwaite Associates, Centerton, NJ, Model 901-LED) and fine wire thermocouples. The same infrared thermometers (IRT's) were used as in the first experiment except their roles were reversed. The Model 4000 (Everest Interscience Inc., Fullerton, California) was used to monitor surface temperature on the model deer, while the hand held Everest Model 112 IRT was used to monitor wind tunnel wall temperatures. The model 4000 was periodically pointed at a black body calibration block with an embedded thermocouple to check its accuracy. Instead of recreation vehicle batteries, power was supplied to the model deer by a homemade power supply with a 110 V AC input and a variable DC voltage output. Measurements were made at 17.4, 11.8 and 8.7 V. A voltage of 17.4V yielded skin temperatures of 40-50°C, while the lower voltages produced skin temperatures of 25-35°C. Varying the voltage allowed the effect of different temperature gradients through the coat to be studied (i.e. to determine whether free convection occured). Another procedure which was used to diagnose the presence of free convection was to turn the model upside down and measure coat conductance at a position that was previously on the top of the model deer. This measurement was compared with that for the same position when the model was right side up. This eliminated any differences due to coat variability or non uniform power flux density. Any difference in coat conductance between the two configurations was attributed to free convection. During the second experiment, the model deer was placed with its long axis perpendicular to the air flow (cross flow). Tests were conducted at wind speeds of 1.04, 2.6, 5.3 and 8 . 1 m s - 1 and equilibration times of more than one hour were sometimes necessary for steady state to be achieved. This long equilibration time was mainly due to the time  Chapter 3. Coat  56  Conductance  it took for the skin to reach steady state. Once steady state was achieved, the Model 4000 IRT was moved around to measure surface temperature at other locations on the coat surface. Factors affecting equilibration time were found to include wind speed, position on the model deer, coat thickness and power flux density. Slower equilibration times resulted from small step changes at low wind speeds (e.g. 0 to 1.04m s _ 1 ) and to low power levels (e. g. 17 to 12 V). Equilibration times for the model deer in cross flow were longer at the lee positions than the stagnation positions because the coat was thicker on the lee side and there was little penetration of the wind into the coat. Equilibration time was defined in terms of the change in t h e temperature difference, ATC, or skin temperature, Tsk, in response to a step change in wind speed or power. The data were fitted with a non-linear regression of the form ATC = ATcf - (ATcf - ATci)e-^T  (3.17)  where t is t h e elapsed time since the step change and ATci is t h e initial value of ATC. The equilibrium or final temperature difference, ATcf, and t h e time constant, r , were determined by non-linear regression. The above curve fitting procedure was used to verify that ATcf had been attained. In cases where ATcf was not reached, its calculated value was used. Figure 3.3 shows an example of equilibration of ATC in response to a step change in wind speed. Table 3.3 gives the time constants for various situations for the model deer in cross flow. Some of the time constants were defined in terms of the change in Tsk instead of AT C , due to the lack of coat surface temperature data or unsteady coat surface temperature. In the longitudinal flow case, it was discovered that insufficient time (only 10-17 minutes) was allowed for equilibration of ATC. was used to determine Tsk or ATcf  This became apparent when Eq. 3.17  and gave unexpectedly low r values (generally less  Chapter 3. Coat  57  Conductance  19 non-linear least squares fit of form: ATc = ATcf ATci = 18.6C ATcf=15.6°C T=17.9min  15  0  20  40 60 Elapsed Time (min)  80  Figure 3.3: Response of the temperature difference ATC for the model in cross flow at position 3-0 (stagnation) to a step change in wind speed from 2.6 to 5.3 m s " 1 . The solid line is a non-linear least squares fit through the data points.  Chapter 3. Coat  Conductance  58  Table 3.3: Typical values of the time constant for Tsk or ATC for step changes in either power density or wind speed imposed on the model deer. Change Wind Speed 0 to 1.6ms - 1 2.6 to 5.3 m s - 1 Power Density** 52 t o 2 0 W m ~ 2 *ATC was used instead of Tsk ** At 3 m s " 1  Time Constant (minutes) Stagnation Top Lee 33.5* 17.5 28.4  17.7  30.8  Chapter 3. Coat  Conductance  t h a n 10 minutes) resulting in an overestimate or underestimate of ATcf.  59  Values of r  were expected to be similar to those for the cross flow case because similar values were observed in the case of the bare model deer in both orientations. T h e problem is illustrated in Figure 3.4, which shows the response of Tsk at position 1-180 to a change in windspeed from 5.3 m s " 1 to 0 m s - 1 for the the coat covered model deer in cross flow. In this case, enough time was allowed for the model deer to reach equilibrium. The non-linear regression line for the entire d a t a set (solid line) gives a time constant of r = 18.5 minutes. The imperfect fit of the line to the data indicates that ATC did not show a perfect first order response. This pattern is typical of all the plots made using data from experiments with the coat covered model deer and was likely caused by the combination of the relatively short time constant for the coat and the longer time constant for the skin layer below it. Non steady air temperatures in the wind tunnel may also have played a part in this. The dashed line shows the result of a non -linear regression when only the first 15 data points were used. This clearly shows that although the fit to the d a t a is good over the first 15 points, it diverges from the d a t a at this point giving a final ATcf value that represents only 65% of the actual change and a r of only 7.2 minutes. To obtain a reasonable estimate of the final skin temperature  (Tskf),  it was calculated using Eq. 3.17, with the first and last measured values of Tsk or ATC and assuming that r = 22 minutes. This assumed value of r was based on the fact that time constants for the coat covered model deer in cross flow generally varied from 15 to 30 minutes.  3.3.2.2  Field Site  Field testing of the coat covered model deer took place during the summer of 1990 in a thinned and pruned second-growth, Douglas-fir stand (16.7m tall, 575 stems h a - 1 ) near Courtenay, Vancouver Island. For a more detailed site description see Lee (1992) and Lee  Chapter 3.  48  Coat  60  Conductance  fit to all data points T=18.5min T„ tf = 47.4°C  15th point  60 90 Elapsed Time (min)  150  Figure 3.4: Example of skin temperature (Tsk) change at position 1-180 on the coat covered model deer in cross flow after a step change in wind speed from 5.3 to O r n s - 1 . Solid line is a non-linear least squares fit through all the data points, while dashed line is the resulting fit when only the first 15 d a t a points were used.  Chapter 3.  Coat  Conductance  61  and Black (1993). The site was gently sloping downhill to the east-northeast. During the experimental period (20 July to 15 August 1990), the weather was fair, with an upslope/sea breeze during the day and downslope/land breeze at night. The model deer was oriented along the slope as shown in Figure 3.5, so that it received cross flow winds during both day and night. Typical wind directions were very close to perpendicular to the length of the deer. For a period of time, the deer was also oriented 90 ° from the orientation shown in Figure 3.5 with its rear end facing downhill to receive longitudinal flow. The instrumentation used in this experiment was described in detail in Chapter 1. The model deer was powered by one 12 V recreational vehicle battery, generally yielding skin temperatures between 25 and 35 °C. Due to the open nature of the forest stand, the model needed to be shaded from direct sunlight to minimize errors in calculating conductances (see Chapter 2).  3.4 3.4.1  R e s u l t s and D i s c u s s i o n Coat C o n d u c t a n c e of a D e e r S t a n d i n g in Still Air  The first case examined was the situation,in which the mean wind speed w = 0 m s _ 1 . This eliminated forced convection as a heat transfer mechanism and made it easier to determine whether free convection was an important heat transfer mechanism. Figure 3.6 shows a slightly non-linear relationship between power density (P) and the temperature difference, AT C , for position 2-90 on the top of the model deer. Figure 3.7 shows the relationship between the coat conductance, gc and ATC.  The solid line is a non-linear  regression line with the equation gc = 2.96A71CCU0. There was a 14% increase in gc over the measured range of ATC (3.4 to 12.1 °C). This indicates that free convection may play a role in coat heat transport. The closed square is the result of a measurement made  Chapter 3. Coat Conductance  62  N  c  360  typical night wind direction 300'  model deer cross flow orientation  o  W 270  >90  E  typical daytime vwind direction 115° 202 rear 180 S Figure 3.5: Orientation of the model deer to receive cross flow winds at the Browns River Site and typical diurnal wind directions.  63  Chapter 3. Coat Conductance  12  u = Om/s AT =0.342 P  U <  60 Power Flux Density (W/m ) Figure 3.6: The temperature difference across the model deer coat (ATC) as a function of power flux density for position 2-90 on top of the model deer with u = Oms - 1 . The data points have been fitted using a non-linear regression of the form ATC = aPb where a = 0.324 and 6 = 0.91.  Chapter 3.  Coat  64  Conductance  u = Om/s  S 3 O  -S  gc =2.96AT, o.io  0  2  G O  U  I  1  U 0  0  10  15  ATC ( C) Figure 3.7: The relationship between coat conductance and temperature difference for position 2 on the top of the model deer in still air ( • ) . The • symbol is for the case where the model deer was turned upside down and the same coat area (pos.2) monitored as before being inverted. The solid line is a non-hnear regression line. Regression statistics are shown in Appendix H.  Chapter 3.  Coat  Conductance  65  after turning the model upside down and measuring AT C for the same portion of coat used to obtain the other data points. The fact that gc was lower when the model was inverted suggests that free convection was present. Experimental data for heated flat plates shows that heat transfer from a plate facing downward is only half as efficient in transferring heat with a given temperature gradient as that for a plate facing upward (Kreith, 1965). In the case of the model deer, the curvature of the trunk would tend the aid free convection on the bottom, making coat conductance more than half of that on the top. In Figure 3.8, the coat conductance for position 2-90 as a function of ATC has been partitioned into its component conductances.  Hair and radiative conductivities were  calculated using Eqs. 3.15 and 3.4 (following numerical integration to obtain F), respectively. They were converted to conductances using Eq. 3.13. The convective conductance was calculated by subtracting the sum of the hair and radiative conductances from the coat conductance (Eq. 3.12), which was measured. The radiative and hair conductances made up only a small portion of the coat conductance. In addition, the sum of the hair, radiative and that conductance which could be attributed to still air (shown as a dashed line) was still considerably less than the coat conductance. Since there was no wind, this implies that free convection was responsible for transferring a significant quantity of heat through the coat. Non-linear regression through the d a t a points assuming an intercept of 1.93 mm s _ 1 (the still air value) yielded the following equation gcon = 1.93 + 0.69 ATC0-28  (3.18)  where gcon is the convective conductance. The exponent 0.28 is close to the 0.25 value predicted by the 5/4's power law (see eq. 3.7). Figure 3.9 gives a typical example of the flux densities of energy transported through  Chapter 3. Coat Conductance  66  4 - u = Om/s  measured gc —  -00-  0  0  8 ATC (°C)  radiative —o-o -°f hair  12  Figure 3.8: Coat conductance and its components as a function of ATC for position 2-90 (top) at a wind speed of O m s - 1 . The still air value, based on a coat depth of 9.8 mm, is shown for comparison. The solid lines through the total and convective points are non-linear regressions of the form g = a -\- bAT£. Radiative and hair conductances were calculated using Eq. 3.13 and convective conductance was calculated using Eq. 3.12. Regression statistics are shown in Appendix H.  67  Chapter 3. Coat Conductance  example for top of deer u = Om/s T = 25C  I  boundary layer T = 31 C  hair  skin wires  = 43C P = 53 (100)  Figure 3.9: Typical example of the magnitudes ( W m - 2 ) of the energy fluxes through the coat and boundary layer due to the mechanisms of conduction, convection and radiation. The example is for position 2-90 (top) under no wind conditions. Numbers in parentheses represent the percentage of the total flux due to a particular mechanism. Temperatures measured at the skin surface, coat surface and free stream air are shown on the right.  Chapter 3.  Coat  Conductance  68  the coat and boundary layer via the different mechanisms for position 2-90 with no wind. Numbers in parentheses are the percentages of the total flux density attributed to a particular mechanism. In the coat 4 6 W m " 2 of the 53 W m - 2 (87%) of power supplied was transported through by convection. The radiative and hair pathways accounted for 9 and 4 % of the total flux density, respectively. The situation changed dramatically in the boundary layer where radiation became the dominant mode of heat transfer, accounting for 74% of the loss, while convection decreased to a smaller but still significant 26% of the total. The temperatures measured at the skin surface, hair surface and in the free stream air are shown on the right hand side of the figure. Figure 3.10 shows coat convective conductance (gcon) as a function of angular position around the deer. The conductances at each angular position are the result of averaging longitudinal positions 2 and 3. There were significant differences between the various angular positions. These differences cannot be explained by experimental error, which was estimated to be no greater than 10% in the wind tunnel. Conductance was high at 90° because the coat depth was smaller than on the sides (0 and 180°). The average coat depths for the different angular positions were 17.6mm for 0 ° , 11.6mm for 90° and 18.3mm for 180°. In Figure 3.11, the conductances of the previous figure have been converted to conductivities using Eq. 3.13. Conductivity indicates a medium's intrinsic ability to transfer heat without regard for the distance over which heat is being transferred. The figure shows only small differences among the 0, 90 and 180° positions which are within the range of experimental error. The convective conductivity at the 270 ° position is expected to be slightly lower than the 90° position because the measurement was made on the same portion of coat with the model turned upside down (see Figure 3.7). In comparing Figures 3.11 and 3.10, it is apparent that coat depth is a more important determinant  Chapter 3.  1  Coat  69  Conductance  u = Om/s  still air conductance  °  1 180  0 o  270  >  (3 o  0  90 180 Angular Position (degrees)  270  Figure 3.10: Mean convective coat conductance for longitudinal positions 2 and 3 as a function of angular position with u = 0 m s _ 1 and P = 52 W m~ 2 . The • symbol indicates that the measurement was made on the same portion of coat as the 90° position with the model turned upside down. The o symbols connected by the dashed line indicate the still air conductance at the various positions.  70  Chapter 3. Coat Conductance  40  1  i  i  1  -  30 -  -  still air •1—>  o 20  —_  _—m  90  T-t  o >  10  u = 0m/s  -  0  + 180-  °{  —  I  0  i  l  270  90 180 Angular Position (degrees)  i  270  Figure 3.11: Mean convective conductivity for longitudinal positions 2 and 3 as a function of angular position with u = flms_1 and P = 52 W m - 2 . The • symbol indicates that the measurement was made on the same portion of coat as the 90 ° position with the model turned upside down.  Chapter 3.  Coat  71  Conductance  of coat conductance than any differences in the strength of free convection at the various positions. Differences in the effectiveness of free convection at the various positions should appear as variations in the conductivities in Figure 3.11. As differences in conductivities are not significant, the large variations in conductances at the various positions in Figure 3.10 can only be explained by variations in coat depth, / (see Eq. 3.13). This is supported by the good correlation of convective conductance with still air conductance (a function of coat depth) in Figure 3.10. To examine the possible structures and flow patterns of the free convection within the coat, the appropriate non-dimensional number is the Rayleigh number (Ra) which is the product of the Grashof (Gr) and Prandtl (Pr) numbers. The Grashof number for the coat is given by G r = ^ ^  (3.19)  where a is the coefficient of thermal expansion of air (1/273K) and g is the gravitational constant. The Prandtl number for air is 0.71. Theoretical and experimental evidence presented in Plate (1971) shows that up to a critical Ra value, Ra c , sensible heat transfer will be by molecular conduction only. Above this value, buoyancy forces overcome viscous forces and air movement due to free convection enhances heat transfer. The value of Ra c for a system with a rigid boundary, such as the skin surface in the deer coat, has been determined to be 1700. Values of Ra for the deer coat used in this experiment, taking coat depth (/) to be the characteristic dimension, range from 400 to 14000, with most values exceeding 3000. Temperature differences across the coat (AT C ) ranged from 3 to 20 °C. The few low values of Ra were mainly a result of the small coat depth at positions on the top of the deer and small temperature differences across the coat at low power densities. These results suggest that free convective eddies on the size scale of the coat thickness are probable.  Chapter 3.  Coat  Conductance  72  During the winter, with air temperatures of 0°C or less, it is likely that ATC values on a deer's trunk region would be greater than 20°C, causing enhanced free convection and ensuring that Ra would exceed the critical value for the thinner sections of the coat. It should be noted that the presence of hair is not accounted for in the Ra calculation and would tend to inhibit free convection once started, but does not affect the stability calculation, i.e. the value of Ra c . Free convective eddies of the same size scale as the trunk diameter, manifested by a vertical flow up the sides of the deer trunk and culminating in a plume above the trunk are likely to cause air movement in the boundary layer. An example of this flow can be seen in Figure 7-7 of Kreith (1973) which can be inferred from the temperature field around the heated circular cylinder. Lewis et al. (1969) show illustrations of free convective flow patterns over a human head and leg (figures also shown in Monteith and Unsworth (1990)). This flow pattern in the boundary layer may extend into the upper part of the coat layer where it is coupled with the smaller eddies in the coat. This flow over the deer trunk could explain the enhancement of convective heat transfer over the still air value at position 2-90 (Figure 3.8) where Ra values are below or only marginally above the critical Ra value. Based on Gr values on the order of 106 or 10 7 , the free convective flow within the coat should be laminar (flow becomes turbulent when Gr > 10 9 (Monteith and Unsworth, 1990)). The dependence of coat conductance on coat depth seen in this study is related to the fact that molecular conduction through the coat is the dominant mode of heat transfer (roughly 2/3 of the total). The view of Campbell et al. (1980) that convection and conduction can occur simultaneously and without interaction in an animal coat suggests that convection can be taken as being in parallel with conduction as in models 1 and 2 of Figure 3.1.  Chapter 3.  3.4.2 3.4.2.1  Coat  73  Conductance  Effect of W i n d S p e e d a n d D e e r Orientation on C o a t C o n d u c t a n c e D e e r in Cross F l o w  Figure 3.12 compares the coat conductances at different angular positions (average of longitudinal positions 2 and 3) for the model deer in cross flow. Changes in wind speeds up to 8 m s " 1 had little if any effect on coat conductance at the top and lee positions of the model deer, while they had a marked effect at the stagnation point. This would suggest penetration of the wind into the coat at the stagnation point, but little if any penetration at the other positions. It is difficult to explain the lack of wind penetration at the top and lee postions when, as shown in Figure 3.8, free convection is very likely to be occurring within the fur. This free convection necessitates that there be air exchange across the coat surface. This apparent contradiction may be resolved if, only air moving in certain directions with respect to the fur surface can penetrate it. The equation of the line fitted through the points for the stagnation position is gc = 2.48 + 0.036u 1 8 3  (3.20)  The exponent on u of n = 1.83, which has a small standard error of 0.003 (see Appendix H), lends credence to the assertion by Bakken (1991) that n may lie between 0.5 and 2.0 with values close to two being due to mechanical disruption of the coat. The differing values of gc at u = 0 m s _ 1 in Figure 3.12 are due to differences in coat depth at the angular positions, which mainly affect still air conductance. Figure 3.13 shows the components of coat conductance as a function of wind speed at position 3-0 (stagnation point) for the model deer in cross flow. Non-linear regression lines were fitted to the total and convective conductance data points. The substantial difference between the convective conductance and the still air conductance (dashed line) again shows that free convection is an important mechanism of heat transfer through the  74  Chapter 3. Coat Conductance  1  1 P = 52 W/m 2 C/3  43  1-  •  ••  1  1  V  1  L 8 3  "<=3.24  n  gc = 2.32 >  2  '  ^0° a9QO  n  •*->  o 3  '  gc = 2.48 + O.Cl 3 6 u n  o  •  o  •  v  ^  Q  lbO  o  U  •4-J  uO  1 0  .  0  1  2  .  1  >  l  4 6 Wind Speed (m/s)  i  l  .  8  Figure 3.12: Comparison of the coat conductance as a function of wind speed for the stagnation (0°), top (90°) and lee (180°) positions on the model deer in cross flow. Longitudinal positions 2 and 3 were averaged to obtain the points for each angular position. Regression statistics are shown in Appendix H.  Chapter 3. Coat Conductance  75  measured gc  _ P = 52 W/m CO  convective CD O  g COB =1.60 + 0.01u z  o  -g a  still air  o U 1  0?  0  2  -<D-  -00-  -n>  -CD-  4 6 Wind Speed (m/s)  radiative  -o  rn hair 8  Figure 3.13: Coat conductance and its components as a function of wind speed for position 3-0 (stagnation) on the model deer in cross flow. The still air value is shown for comparison and is based on a coat depth of 18.1mm. The solid lines through the total and convective points are non-linear regressions of the form g = a + buc. Radiative and hair conductances were calculated using Eq. 3.13 and convective conductance was calculated using Eq. 3.12. Regression statistics are shown in Appendix H.  Chapter 3.  Coat  76  Conductance  coat when there is little or no wind. The mean coat conductance (gc) of the trunk as a function of wind speed for the model deer in cross flow is shown in Figure 3.14. These values of~gcwere calculated using the mean values of ATC for positions 2 and 3 at angular positions 0, 90 and 180°. Values of gc for positions 2-270 and 3-270 (bottom of deer) were measured for only the wind speeds 0 m s " 1 and 2 . 6 m s " 1 so they were not included in the data for all wind speeds presented in Figure 3.14. Calculations of g~c including these two positions gave a value that was only 3-4% larger than those not including the positions. The exponent in the non-linear regression equation reflects the influence of the stagnation point on the mean; however, the 0.006 coefficient results from the other positions not responding to changes in wind speed. This causes gc to increase only weakly with wind speed.  3.4.2.2  D e e r in L o n g i t u d i n a l F l o w  Figure 3.15 shows total coat conductance as a function of wind speed for positions on the top (90°) of the model deer (a) and on the 180° side (lee side in cross flow experiment) (b) for the deer oriented with the rear end facing the wind (orientation r) and for the head facing the wind (orientation h).  The differences in conductance at the various  positions for a given wind speed are due mainly to differences in the coat depths (/) which were shown in Table 3.2. Coat depth increased from the front (position 1) to the rear (position 3). Furthermore, coat depth tended to be larger on the sides than on the top of the trunk. Coat conductance was largest at position 1 and smallest at position 3 on the side and the top for both orientations. With the exception of position 1-180 at 0.7ms"" 1 and 5 . 3 m s " 1 , the coat conductances for the top positions were always significantly larger than the corresponding positions on the side. As an example of the effects of coat depth, consider the positions 1-90 and 2-90 (top positions) at 5.3 m s - 1 . The measured coat conductance at 1-90 is gc = 5 . 5 0 m m s _ 1 , compared with gc = 4 . 8 2 m m s _ 1  Chapter 3.  Coat  2.11  00  S  77  Conductance  gc = 2.45 + 0.006u 3  O O  •3 G O  U O  U a  P = 52 W/m 0  0  2  4 6 Wind Speed (m/s)  8  Figure 3.14: Mean coat conductance of the model deer trunk in cross flow as a function of wind speed. Solid line is a non-linear regression line. Regression statistics are shown in Appendix H.  Chapter 3.  Coat  78  Conductance  8  tW  O  o o O C3  O  U  0  2 4 Wind Speed (m/s)  6  Figure 3.15: (a) Coat conductance on the top of the coat covered model deer in longitudinal flow as a function of wind speed. Solid lines are for the rear end facing the wind (orientation r), while dashed lines are for the head facing the wind (orientation h). (b) Same as above except for the 180° side positions.  Chapter 3.  Coat  Conductance  79  at position 2-90, a difference of 0.69mm s _ 1 . The still air conductances of a layer with the same thickness as the coat at positions 1-90 and 2-90 (Table 3.2) are 2.48mm s - 1 and 1.9 m m s _ 1 , respectively, giving a difference of 0.58 mm s _ 1 . This accounts for most of the difference between the coat conductances at the two positions, while the remainder could easily be accounted for by the uncertainty in the coat depth measurements of ± 1 mm. Coat conductance increased by about 20% as the wind speed increased from 0.6 to 5.3 m s - 1 at positions 1-90 and 2-90 for orientation r (Figure 3.15a).  Increases were  also observed for these positions when the model was in orientation h. No wind speed dependence was observed at 3-90. This may be due to more dense or stiffer fur at this position not allowing increasing penetration of the wind with increasing wind speed. For the two wind speeds studied in orientation h, orientation r coat conductances were as much as 10% larger than those for orientation h. This may be explained by more effective wind penetration into the coat for orientation r because of the hair lying with its tips pointing toward the rear end of the deer. Figure 3.15b shows that there was very little wind speed dependence or difference between the two orientations at positions 2-180 and 3-180. Position 1-180, however, showed a very high value of coat conductance at a wind speed of 5.3 m s _ 1 for orientation r. This high conductance may be explained by a raised area of hair which was observed directly above the skin thermocouple at this position. This raised area of hair would tend to be lifted in orientation r, especially at the higher wind speeds, allowing more penetration of the wind and increasing coat conductance. This illustrates how effective wind penetration can be in reducing the insulation value of the coat. We might expect a dramatic increase in coat conductances for other positions in orientation r at some higher wind speed if the hair became lifted. This would not be the case for orientation h, because the hair would tend to be pushed down. This pushing down may cause modest increases in coat conductance due to the reduction of the depth of the still air layer.  Chapter 3.  Coat  Conductance  80  Coat conductances for the 2-90 and 3-90 positions (top) and the 2-180 and 3-180 positions (lee side in cross flow experiment) were somewhat larger in the two longitudinal flow orientations than in the cross flow orientation (Table 3.4). During the longitudinal flow experiment, coat conductance was not measured on the side of the model deer that was facing the wind in the cross flow (stagnation side) experiment but presumably would be the same as that on the lee side given similar coat characteristics. Consequently, the values of coat conductance for the lee side were taken to be representative of both the lee and stagnation sides and compared with the means for these positions in cross flow (boldfaced values in Table 3.4). There was little difference between the heat loss on the sides of the deer in cross flow and longitudinal flow because the higher coat conductances on the stagnation side of the deer in cross flow were balanced by lower conductances on the lee side. Coat conductance was however 12 to 28% higher for the top of the model deer in longitudinal flow (orientation r) than for the top of the deer in cross flow. Neglecting heat loss from the head and legs, the deer would feel warmer standing in cross flow than in longitudinal flow.  3.4.2.3  C o m p a r i s o n of C o a t C o n d u c t a n c e M e a s u r e d in t h e Forest w i t h T h a t M e a s u r e d in t h e W i n d Tunnel  A typical example of the diurnal variation of wind speed and direction at the Browns River site is shown in Figure 3.16a and b for 7 and 8 July 1990. These values are 5-minute averages. Figure 3.16a shows that nighttime wind speeds were more constant than the daytime wind speeds which were high in the middle of the day when surface heating was strongest and low during the morning and evening. Figure 3.16b shows that wind direction, WD,  at night was generally very steady (standard deviation, <TWD = 22.4°)  near a mean of 300° (NW), while the daytime wind direction had a larger variation (CWD = 2 9 . 4 ° )  about the mean of 112° (SE). The nighttime standard deviation would  Chapter 3.  Coat  Conductance  81  Table 3.4: Comparison of mean coat conductances (gc) for positions 2 and 3 for the longitudinal and cross flow cases. Angular Position (degrees) 0 180 mean 1 90 0 180 mean 1 90 0 180 mean 1 90 0 180 90 x  u(ms  5.3 5.3 5.3 1.5 1.5 1.5 0.62 0.62 0.62 0 0 0  l  )  Coat Conductance (mm s 1) Cross Flow Longitudinal Flow rear into wind head into wind 3.24 2.29 2.71 2.77 3.17 4.05 2 2.54 2.41 2.31 2.37 2.42 3.30 3.67 3.53 2 2.50 2 2.32 2.41 2.36 2.41 2 3.24 3.41 3.63 3 3 2.48 2.48 2.48 3 3 2.32 2.32 2.32 3 3 3.21 3.21 3.21  Mean of 0° and 180° positions, shown in boldface type. Value estimated from functions shown on Figure 3.12. 3 Value measured during cross flow experiment. 2  Chapter 3.  Coat  82  Conductance  i—•—r  T—'—r ->(-« -night  day 1 •Fu = 0.65m/s a.. = 0.168m/s  u = 0.78m/s a u = 0.126m/s  1.2c«  13  ^  0.8  CO 13  I °-4 d J  0 360  T  i  L  «—r  night-  mfffijff^^  bo 270  •day-  •* H-  WD=112Cc aWD = 29.0  w  a a D  %  180  T3  g  WD = 293c ^WD = 22.4  90  0  1400  1800  2200  200  600  7 July  1000  1400  1800  8 July Time (PST)  Figure 3.16: (a). Typical diurnal pattern of wind speed measured at 0.8m above the ground by a hot wire anemometer at the Browns River site on 7 and 8 August 1990. (b). Wind direction measured at 1.5 m above the ground at the same site, for the same time period. Also shown are the mean wind speed and direction (u and WD) along with their respective standard deviations {ou and awv) for the day and night periods.  Chapter 3.  Coat  Conductance  83  have been lower if it were not for the one-half hour episode during which wind speed dropped and wind direction shifted to along the slope at around 0130 P S T . Figures 3.17a and b show the calculated 5 minute average coat conductances for two positions on the model deer as a function of wind speed.  These figures correspond,  respectively, to the daytime and nighttime portions of Figures 3.16a and b. The data shown are fairly typical of the coat conductance data collected during these experiments. Little if any evidence of a wind speed dependence of coat conductance was seen even at the stagnation positions. This can probably be attributed to the fact that the wind speed rarely exceeded 1.2 m s - 1 at 0.8 m above the forest floor. At these wind speeds, little penetration into a deer coat is expected. It can be see t h a t the daytime data are much more scattered (agc = 0.141 mm s _ 1 ) than the nighttime data {agc = 0.031 mm s - 1 ) . Since there is no wind speed dependence, the daytime scatter of coat conductance is probably a result of the larger wind direction variability during the day. Despite the scatter observed during the daytime, reasonable values of coat conductance for low wind speed (0- 1.2ms" 1 ) were obtained by taking the mean values of coat conductances after waiting for some adjustment period following the sudden shift in wind direction during the night/day transition. Figure 3.18 compares coat conductances made at various positions on the model deer coat in the wind tunnel and the forest in cross flow conditions. The forest data were collected over a range of wind speeds from 0 to 1.2 m s - 1 , with data collection periods of 8 to 12 hours (Figures 3.17a and b are typical examples). Wind tunnel data were collected at discrete wind speeds within the range of wind speeds observed in the forest for periods of up to three hours. In some cases, d a t a taken at the same power density were not available for comparison. Despite the slightly different methods of data collection, the comparison between the forest and wind tunnel data is valid, since coat conductance was invariant with wind speed for both data sets and was only a weak  Chapter 3. Coat  84  Conductance  Position 3-90 night  O  HB  L/EftcF  g. = 2.26mm/s a „ = 0.03 mm/s inSor-n  j&  1  o  i  -  0 4  1 r Position 2-0 day  O  U o U  D  a  O  DD„a  ea * J• ^ap h ^  0  0  0.2  g. = 2.66 mm/s a„, = 0.141 mm/s  0.4 0.6 0.8 1.0 0.8m Wind Speed (m/s)  1.2  Figure 3.17: (a) Coat conductance as a function of wind speed at position 3-90 on model deer in cross flow at Browns River site. This is a typical nighttime data set taken from 2040, 7 July 1990 to 0540, 8 July 1990. (b) Same as above except data are for position 2-0 for the period 1250-1730, 8 August 1990.  Chapter 3. Coat Conductance  85  Wind Tunnel Forest (Browns River)  Pos. 2 52 24 m  Stagnation  Pos. 3  Lee  52  Pos. 3 52  24 R  Top  Figure 3.18: Comparison of coat conductances at wind speeds ranging from 0 to 1.2 m s " 1 for various positions on the model deer in cross flow, measured outdoors at Browns River (black bars) with those measured in the wind tunnel (grey stippled bars). The numbers above the bars are the power densities (W m - 2 ) used.  Chapter 3. Coat  Conductance  86  function of power density (Figures 3.6 and 3.7). Differences in coat conductance between the forest and wind tunnel were not significant, considering the maximum estimated measurement error of 10% for the wind tunnel data and 1 5 % for the forest data. The somewhat larger differences between wind tunnel measurements taken at P — 52 W m - 2 and forest measurements at F = 2 4 W m - 2 are probably due to some enhancement of free convection at the higher P values in the wind tunnel. Figure 3.19 compares coat conductances in the forest and wind tunnel for the model deer in longitudinal flow for the same wind speed range as above. The power density was 2 4 W m - 2 for both cases. As was the case for cross flow, there was no significant difference between coat conductance in the forest and the wind tunnel for wind speeds up to 1.2 m s - 1 , given the estimated measurement errors. There was very little if any penetration of wind into the coat at the low wind speeds observed in the forest, despite the high turbulence intensities. It is possible, however, that at higher wind speeds when penetration does occur, turbulent enhancement of coat conductance may be observed. Also, the wind speed at which penetration occurs, may be lower in the turbulent outdoor environment.  3.4.3  C o m p a r i s o n of B o u n d a r y Layer a n d C o a t C o n d u c t a n c e  Figure 3.20 compares the mean coat conductance of the trunk as a function of wind speed for the model deer in cross flow with the mean boundary layer conductance. All data were collected in the wind tunnel. The figure illustrates the much greater importance of the coat as an insulator than the laminar boundary layer. For moderate wind speeds of about 5 m s " 1 , the insulation provided by the coat is more than 6 times that provided by the boundary layer. A small part of the difference may be due to the higher power flux density supplied in the case of the bare model ( l O O W m - 2 vs. 5 0 W m - 2 ) which could enhance free convection somewhat. Based on the results presented previously in  Chapter 3. Coat  87  Conductance  Wind Tunnel Forest (Browns River) P = 24 W/m2 for all cases  Pos. 1 h  Pos. 1  Pos. 2 r  Side  Top  Figure 3.19: Comparison of coat conductances at wind speeds ranging from 0 to 1.2 m s " 1 for various positions on the model deer in longitudinal flow, measured outdoors at Browns River (black bars) with those measured in the wind tunnel (grey stippled bars). Measurements taken with the head facing into the wind are labeled h, while those taken with the rear end facing the wind are labeled r.  Chapter 3. Coat Conductance  24  —i  1  88  1  1  1  1  r  D coat covered model deer (18V)  20  •  bare model deer (24V)  gb = 7.01u  16  0.635  12 8 2.11  gc = 2.45 + O.Olu  4 II  0  0  B-  -B-  •a  4 6 Wind Speed (m/s)  8  Figure 3.20: Comparison of the mean coat conductance (gc) with the mean boundary layer conductance (gb) as a function of wind speed for the model deer in cross flow. Boundary layer conductances were measured on the bare model deer. Solid lines are regression lines. Regression statistics are shown in Appendix H.  Chapter 3.  Coat  Conductance  89  this chapter, the enhancement would only be on the order of 0.5 m m s 1 .  3.5  Conclusions  Radiative and hair conductances alone were not large enough to account for the amount by which coat conductance (gc) exceeded still air conductance; therefore, free convection must have occurred within the coat when there was no wind. Differences in the magnitude of free convection at the various angular positions were not large enough to account for the observed differences in convective coat conductances. These differences were a result of the variation in coat depth among the positions causing still air conductance to change. The lack of wind speed dependence exhibited by coat conductance at all positions except the stagnation point in cross flow indicates that wind speeds up to 8 m s _ 1 are not effective in penetrating the deer coat on the top and lee sides. There was some evidence of wind speed dependence at some positions on the model deer in longitudinal flow. Coat conductances were found to be slightly higher for the deer oriented with the rear end into the wind, indicating more effective penetration of the coat by the wind. This was due to the hair tips being oriented towards the rear of the animal. Penetration of the deer coat by the wind was also found to be dependent on coat depth and the occurrence of raised areas of hair. Reduction of the coat thickness caused by the force of the wind on the hair may be a cause of wind speed dependence. There was no evidence of turbulent enhancement of coat conductance occurring outdoors at wind speeds up to 1.2 m s " 1 ; however, it may occur at higher wind speeds. The insulating value of a mule deer winter pelage is significantly larger than that provided by the laminar boundary layer, even at wind speeds as low as 0.6 m s - 1 . At wind speeds of about 5 m s _ 1 the insulation value of the deer coat is more than 6 times t h a t of the laminar boundary layer.  Chapter 3. Coat  3.6  Conductance  90  Literature C i t e d  Bakken, G.S. (1991) Wind speed dependence of overall thermal conductance of fur and feather insulation. J. Therm. Biol. 16, 121-126. Campbell, G.S. (1977) An Introduction  to Environmental  Biophysics.  Springer-Verlag,  New York, 159 pp. Campbell, G.S., A.J. McArthur and J.L., Monteith (1980) Windspeed dependence of heat and mass transfer through coats and clothing. Boundary-Layer  Meteorol. 18, 485-  493. Cena, K. and J.A. Clark  (1973) Thermal radiation from animal coats: Coat structure  and measurements of radiative temperature. Phys. Med. Biol. 18, 432-443. Cena, K. and J.L. Monteith (1975a) Transfer processes in animal coats.  I. Radiative  transfer. Proc. R. Soc. Lond. B. 188, 377-393. Cena, K. and J.L. Monteith (1975b) Transfer processes in animal coats: II. Conduction and convection. Proc. R. Soc. Lond. B., 188, 395-411. Davis, L.B. and R.C. Birkebak (1975) Convective energy transfer in fur. In of Biophysical  Ecology(Edited  Perspectives  by Gates, D.M. and Schmerl, R.B.), pp.525-548.  Springer-Verlag, New York. Hammel, H.T. (1955) Thermal properties of fur. Am. J. Physiol.  1 8 2 , 369-376.  Iqbal, M., A.K. Khatry and B. Senguin (1977) A study of the effects of multiple windbreaks. Boundary-Layer  Meteorol. 11, 187-203.  Joyce, J.P., K.L. Blaxter and C. Park (1966) The effect of natural outdoor environments on the energy requirements of sheep. Res. Vet. Sci. 7, 342-359.  Chapter 3. Coat Conductance  91  Kreith, F. (1973) Principles of Heat Transfer, 3rd ed. New York, 656 pp. Lee, X. (1992) Atmospheric turbulence within and above a coniferous forest. Univ. of British Columbia, Ph.D. Thesis. 176 pp. Lee, X. and T.A. Black (1993) Atmospheric turbulence within and above a Douglas-fir stand. Part I: Statistical properties of the velocity field. Boundary-Layer Meteorol. 64, 149-174. Lewis, H.E., A.R. Forster, B.J. Mullan, R.N. Cox and R.P. Clark (1969) Aerodynamics of the human microenvironment. Lancet, 1, 1273-1277. McArthur, A.J. (1977) Heat loss from sheep. Univ. of Nottingham. Ph.D. 186 pp. McArthur, A.J. and J.L. Monteith (1980a) Air movement and heat loss from sheep. II. Thermal insulation of fleece in wind. Proc. R. Soc. Lond. B. 209, 209-217. McArthur, A.J. and J.L. Monteith (1980a) Air movement and heat loss from sheep. III. Components of insulation in a controlled environment. Proc. R. Soc. Lond. B. 209, 219-237. Mitchell, M.A. (1985) Measurement of forced convective heat transfer in birds: a wind tunnel calorimeter. J. of Therm. Biol. 10, 87-95. Monteith, J.L. and M.H. Unsworth (1990) Principles of Environmental Physics. 2nd ed. Edward Arnold, London, 291 pp. Moote, I. (1955) The thermal insulation of caribou pelts. Text. Res. J. 25, 796-801. Plate, E.J. (1971) Aerodynamic Characteristics of Atmospheric Boundary Layers, U.S. Atomic Energy Commission, Oak Ridge, Tennessee. 190pp.  Chapter 3.  Coat  Conductance  Raddi, A.G. (1967) The pelage of the black-tailed deer.  92  Univ. of British Columbia.  Ph.D. 215 pp. Rue, L.R. (1989) The Deer of North America, Outdoor Life Books, Grolier Book Clubs Inc., Danbury, Conn., 544 pp. Scholander, P.F., Hock, R., Walters, V., Johnson F., and Irving, L. (1950) Heat regulation in some arctic and tropical mammals and birds. Biol. Bull. 9 9 , 237-258. Tregar, R.T. (1965) Hair density wind speed and heat loss in mammals. J. Appl.  Physiol.  20, 796-801. Walmo, O.C. (1981) Mule and Black-tailed Deer of North America,  Univ. of Nebraska  Press, Lincoln, Nebraska, 605 pp. Wiersma, F. and G.L. Nelson (1967) Non-evaporative convective heat transfer from the surface of a bovine. Trans. ASAE 10, 733-737.  Chapter 4 Transfer of T h e r m a l R a d i a t i o n T h r o u g h a D e e r C o a t  4.1  Introduction  For many mammals such as deer, the dense coat covering the body represents the most important component of insulation in maintaining the animal's thermal equilibrium. Not only is the coat important in keeping the animal warm in cold environments, but also it can serve to protect the animal from high solar radiative loads as has been suggested by Macfarlane (1968) and Cena and Monteith (1975a) for domestic sheep and cattle. Heat is transferred away from an animal's skin through the coat by a combination of conduction, convection and radiation.  For animals with thick coats, such as deer  and sheep, radiative transfer away from the skin is almost exclusively in the form of thermal radiation. A theory of the transfer of thermal radiation through animal coats was developed by Cena and Clark (1973) and Cena and Monteith (1975a and b). These authors recommended a simple algebraic equation to calculate radiative conductivity (kr).  The only parameters needed were hair length (/ 5 ), coat depth (/), hair density (n)  and the temperature gradient through the coat (a). The authors recognized that the theory was not applicable to the case where straight hair was perpendicular to the skin surface (the theory predicts infinite radiative conductivity in this case), however this situation was dismissed as physically unrealistic in a biological system. In the context of the present study on heat transfer through deer coats however, this situation is does occur when cold stressed deer attempt to conserve  93  Chapter 4. Thermal  Radiation  94  heat through piloerection (Parker and Gillingham, 1990). Piloerection is the process by which the hair, normally inclined at angle of 10-30° to the skin surface, is raised torward perpendicular to the skin surface. In this chapter, the groundwork laid by Cena and Clark (1973) along with Cena and Monteith (1975a and b) is utilized in order to (i) develop a method for determining kr that is generally applicable to hair at all inclination angles and (ii) apply this method to calculating predicting the radiative conductivity of a deer coat t h a t is piloerected.  4.2  Theory  4.2.1  T h e I n t e r c e p t i o n Function p  Cena and Clark (1973) proposed the following model for transmission of thermal radiation in uniform animal coats where the hairs are assumed to have an emissivity of unity T(Z) = exp(-pz)  (4.1)  where T(Z) is the fraction of radiation transmitted through a coat layer of thickness z (cm) and p ( c m - 1 ) is the mean probability that the radiation will be intercepted within a unit depth of coat. In order to obtain p, which considers the interception of radiation which was emitted in all directions above the skin surface from a unit area on the skin surface, we must first obtain a general expression for the value of p, when radiation is emitted in a given direction. The derivation of an expression for p is as follows. First consider a coat of uniform density and with all hairs oriented in the same direction. Figure 4.1 shows the geometrical relationship between a single hair and a ray of radiation originating from the skin surface, traveling in a given direction. The coordinate system is constructed such that  Chapter 4.  Thermal  Radiation  95  z ^ - y  Figure 4.1: Geometrical relationship between a single hair oriented in the direction given by the unit vector rh in the xz plane and radiation emitted by the skin surface in the direction given by the unit vector h.  Chapter 4. Thermal  Radiation  96  the unit vector ra, describing the hair direction, lies in the xz plane, while the unit vector n, describing the radiation transmission direction, may originate from any positive z direction. The angles describing the directions of the m and h vectors are as follows: a - The angle between the projections of vector ra and h onto the xy plane (skin surface) or azimuthal angle. <j> - The angle between the z axis and the hair direction vector, ra or hair angle. £ - The angle between the z axis and the radiation direction vector, h. /3 - The angle between the vectors ra and h on the plane formed by those two vectors. The probability of interception (p) of radiation emitted in a direction specified by the radiation vector, h, is related to the projected area of a hair per unit coat depth onto a plane normal to n. The geometry for this is shown in Figure 4.2. The length of a single hair is ls and the length of its projection is ls sin /?. If the mean diameter of a hair is denoted by d, the projected area, Ap may be written as Ap = dls sin (3  (4.2)  The value of p for a given angle /3 is then . . ndlssm/3 P\P) = i /  =  ndsin/? J"  (4-3)  COS (p  where n is the number of hairs per unit skin area and / is the mean coat thickness. Another way of describing p(/3) is that it is the fraction of rays leaving a unit area of skin surface, traveling in a given direction with respect to the hairs (specified by /?), that are intercepted per unit depth of coat.  Chapter 4. Thermal Radiation  97  Figure 4.2: Geometrical relationship between the hair direction vector, m, and the radiation transmission vector, h, showing that the length of the projection of a single hair onto a plane perpendicular to n is ls sin 0 where ls is the hair length.  Chapter 4.  Thermal  Radiation  In order to integratep(fl)  98  for radiation emitted in all positive z directions, sin/? needs  to be expressed in terms of the angles <f>, £ and a. The vector components of m and n are as follows: m = sin <f>i + Oj + cos 4>k A  A  (4-4) A  h = sin £ cos m + sin £ sin aj + cos £&  (4-5)  Using the trigonometric relationship, cos 2 /3 + sin 2 /? — 1, we can write sin ft = J\ - cos 2 /?  (4.6)  Since the dot product of two unit vectors is equal to the cosine of the angle between  them, cos ft = m • h = sin <f> sin £ cos a + cos <f> cos £  (4-7)  Substituting Eq. 4.7 into Eq. 4.6 , we have sin /? = [1 — (sin <^ sin £ cos a + cos ^ cos i)2]1^2  (4-8)  Now we can make use of Eq. 4.8 to rewrite Eq. 4.3 for a given <f> as p(£, a) =  - [ 1 — (sin <j) sin £ cos a + cos <j> cos £ ) 2 ] ^ 2  (4.9)  This expression can be compared with Eq. 10 from Cena and Clark (1973), which is p'((, a) = nd[(l + tan 2 <j>){\ + tan 2 § - (1 + tan <^ tan ( cos a ) 2 ] 1 / 2  (4.10)  Eqs. 4.9 and 4.10 are not as different as they first appear. After algebraic and trigonometric manipulations (see appendix E), Eq. 4.9 can be expressed as p{(,a)  = ndcos£[(l  + t a n 2 ^ ) ( l + tan2£) - ( 1 + t a n ^ t a n £ c o s a ) 2 ] 1 / 2  (4.11)  This is identical to Eq. 4.10, except that Eq. 4.11 has been multiplied by l / c o s £ to get Eq. 4.10. Cena and Clark multiplied by pathlength per unit coat depth ( 1 / cos £) in order  Chapter 4. Thermal  Radiation  99  to account for the pathlength of radiation through the coat. In other words, Eq. 4.10 is not on a per unit pathlength basis but Eq. 4.11 is. We now wish to find a mean value of p(£, a) for radiation being emitted in all directions from the skin surface. To do this we must integrate Eq. 4.11 over all a from 0 to 2w and £ from 0 to TT/2. The radiation geometry which defines the integral is in spherical coordinates and is illustrated in Figure 4.3. T h e integral is as follows: 1  r2ir  riv/2 v(t  p=-  /  a)  ^Vcos£sin£dad£  7T Ja=0 J£=0  (4.12)  COS £  The base of the hemisphere corresponds to the skin surface and its radius corresponds to a unit thickness of the coat. Sin xi, cos £ and 1/ cos £ can be thought of as weighting factors, where sin £ weights for the decreasing size of the elemental area dA as £ decreases when the top of the hemisphere is approached, cos£ weights for Lambert's Law, i.e., angular emittance decreases as the angle between the emitted ray and the normal to the skin surface increases and 1/ cos £ accounts for the radiation pathlength between the skin and the coat surface increasing as £ increases (increasing t h e pathlength will increase the probability of interception). Equation 4.12 reduces to I  (lit  P= 7T Ja=0  4.2.2  /-TT/2  /  p(£a)sin£dad£  (4.13)  J£=0  A n A p p r o x i m a t e M e t h o d for D e t e r m i n i n g R a d i a t i v e C o n d u c t a n c e  Cena and Monteith (1975b) argued that the radiative conductivity ( i r , W m " l 0 K _ 1 ) through an animal coat can be approximated by kr = ^b/p  (4.14)  where b — 4aT3 ( W m - 2 ° K - 1 ) (T is the mean absolute air temperature in the coat and a is the Stefan-Boltzman constant) and p ( m _ 1 ) is an appropriate value of the interception  Chapter 4. Thermal Radiation  100  dA = sin^docd^  Figure 4.3: Diagram showing the geometrical relationships for determining p (adapted from Kreith, 1973). The radius of the hemisphere corresponds to a unit thickness of coat.  Chapter 4. Thermal  function.  Radiation  101  Some of the assumptions implicit in Eq. 4.14 will be discussed in the next  section. Cena and Monteith (1975a) argued that the appropriate value of p to be used in Eq. 4.14 is p, the exact value of which can be obtained by performing the integration given in Eq. 4.13. However, they proposed that using the value of p for radiation emitted perpendicular to the skin surface (£ = 0, a = 0) will give an adequate estimate of kr, thus eliminating the necessity of performing a numerical integration. This value of p which we will call p±_ can be obtained by simplifying Eq. 4.10 to the following p± = nc?tan[arccos(/// s )] = nd tan <j>  (4-15)  Cena and Monteith (1975a) supported this with experimental evidence. In their experiment, the transmission of direct longwave radiation, through the coats of various animal species, which were clipped to successively shorter depths, was measured with a collimating radiometer which was oriented perpendicular to the skin surface. It is not surprising that p± gave a good estimate of the measured longwave radiation flux for sheep coats where l/ls values were near 0.8 [cj> = 36.8 °; hair tending toward perpendicular to the skin surface) because most of the radiation reaching the bottom of the collimator represented radiation from a narrow range of angles centered around £ = 0 ° , which was not greatly impeded by the hair. The results obtained for animals such as the calf of a domestic cow (l/ls  = 0.3; <f> = 72.5°) were not as good. Unfortunately, no experimental verification  was performed for the transmission of diffuse longwave radiation (radiation from all £) , which represents the bulk of the radiation transmitted from the skin to the coat surface. Furthermore, despite the fact that p± seems to work well for coats with high l/ls, it predicts a p value of 0 when / / / , = 1 (<j) = 0 ° ) . Using Eq. 4.14, this would give a radiative conductivity of kr = oo, which is clearly not physically realistic.  Chapter 4.  4.2.3  Thermal  Radiation  102  N u m e r i c a l Integration t o D e t e r m i n e kT  A more physically realistic value of kr for the piloerection case and one that is generally applicable for all hair angles can be obtained through numerical integration of an equation similar to Eq. 4.13 considering radiation emitted in all directions. The principle is given by Cena and Monteith (1975b); however, their simplifying assumptions lead to Eqs. 4.14 and 4.15 which are not generally applicable for all hair angles. Figure 4.4 illustrates the method of calculating kr in a layer of dry hair with a uniform temperature gradient. We wish to consider the net flux of longwave radiation reaching a point X on a plane AB (parallel to skin surface) which originated from a hemispherical shell of radius r surrounding X. We will assume that the temperature gradient (a = dT/dz)  is constant and negative (z = 0 at skin surface) throughout the coat. This means  that the skin is always warmer than the coat surface. The hair and air temperatures are assumed to be equal at a given z. The temperature on AB will be denoted by T and the blackbody radiation emitted from it as B(T).  The temperature at any point on  the hemispherical shell can be specified as Tz = T — ar cos £ and likewise the blackbody radiation as Bz = B{T) — abr cos £. The fraction of radiation traveling in a given direction through a hemispherical shell of infinitesimal thickness, dr, that is intercepted by the shell is pdr. Assuming that the hairs act as blackbodies, i.e., the fraction of radiation intercepted by the hairs equals the fraction of radiation emitted by the hair, we can invoke Kirchoff's Law to deduce that the effective emissivity of the layer is also given by pdr.  The flux density of radiation  (Fa) emitted by a unit area of the hemisphere can be written as Fa = [B(T) - abr cos t]p(£, a)dr  (4.16)  and the corresponding flux density of radiation per unit solid angle (radiance) arriving  Chapter 4. Thermal  103  Radiation  Coat Surface  Skin Surface  z=0  Figure 4.4: Illustration of the method for determining kr of a dry coat with a uniform temperature gradient across the coat .  Chapter 4. Thermal  104  Radiation  at X from direction £ is N = ~[B(T)  - abr cos(\p{(,a)e-v(^rAr  (4.17)  IT  The term e~pr is the transmissivity which attenuates radiation as it travels through the coat. Now we can make use of an integral which is similar to that used to calculate p (Eq. 4.13) in order to calculate the flux density of radiation, F(r) crossing AB that originates from the hemispherical shell. 1  f2ir  F(r) = IT Ja=0  rir/2  da  p(£, a)e-pti>a>dr[B(T)  - abr cos {] sin { cos £d£  (4.18)  J£=0  The preceding equation is not easily integrable, so numerical integration was performed to evaluate it. The flux density of radiation arriving at AB from a coat layer of infinite depth can be obtained by summing F(r) for all r values from 0 to oo or /•oo  F+ =  / i?(r)dr Jo  (4.19)  In practice, we do not need to sum for all r values out to infinity because most of the radiation arriving at X comes from hair within 0.5 cm of the point, as dictated by e~pr. For example, the lowest p values (least attenuation) obtained for the coat on the model deer were around 20 c m - 1 , which means that approximately 99% of the radiation reaching X originates from within a distance of 0.23 cm. The flux arriving at AB from a layer of coat above it (F~) is determined in exactly the same way as F+ except that Fa = [B(T) + abr cos £]. This means that F~ will be slightly less that F + , resulting in a net radiative flux (F = F+ — F~) across AB toward the coat surface. The radiative conductivity of the coat can then be determined using F kr = (4.20) a The preceding equation replaces Eq. 4.14 which Cena and Monteith (1975b) obtained by integrating Eq. 4.18, assuming that p was invariant with £ and a and equal to p±.  Chapter 4. Thermal  Radiation  105  The numerical integration to determine kr values was carried out using a spreadsheet. Values of the flux density of radiation contributed by a hemispherical shell, F(r), were determined by incrementing a and ( by 10° (0.175rad) between their respective limits. The function was evaluated for r ranging from 0 to 1 cm with a A r of 0.05cm and a starting value of 0.025 cm. Starting at 0.025 cm means evaluating F(r) at the midpoint of each A r interval, which because F(r) is a decreasing function, minimizes the overestimation of F + and F~ resulting from using finite steps. A reasonable temperature gradient of a — —400°C/m was chosen although it was found that changing a did not affect the result. The temperature at AB was chosen as T = 3 0 0 K ; however, it was found that changing T did affect the results because b is calculated at this temperature and the result varies linearly with b. A simple correction was applied by calculating b at the mean coat temperature, i.e. kr(b) = kr(b300)-^-  (4.21)  0300  where kr{b) is the corrected value of kr at the mean coat temperature, kT(b3oo) is the value of kr calculated when b was determined at 300 K and 6300 is the value of b at 300 K. A sample numerical integration (values of F(r) at different r) is given in appendix F .  4.3 4.3.1  Results and Discussion T h e Effect of P i l o e r e c t i o n on H e a t Transfer t h r o u g h a D e e r C o a t  Figure 4.5 compares p±, calculated using Eq. 4.15, with p determined through numerical integration of Eq. 4.13 for position 2-90 on the model deer as a function of hair angle. Both curves exhibit the same general shape with p generally exceeding p± by 10-15 c m - 1 . The value of pL goes to zero when the hair is standing straight up which would suggest that no thermal radiation is intercepted by the coat. The finite value of p when the hair angle is zero is more realistic, indicating that some radiation is still being intercepted by  106  Chapter 4. Thermal Radiation  I 40 *? X)  \cu  0  20  40 60 Hair Angle (degrees)  80  Figure 4.5: Comparison of p (numerical integration) with p± (Cena and Monteith's method) as a function of hair angle for position 2-90 on the model deer.  Chapter 4.  Thermal  Radiation  107  the coat. Figure 4.6 illustrates the effect of piloerection on radiative conductance for position 2-90 as calculated using (i) Eqs. 4.14 and 4.15, (ii) Eqs. 4.13 and 4.14 and (iii) numerical integration (Eqs. 4.18, 4.19 and 4.20). The integrated value of radiative conductance remains almost constant over the entire range of hair angles, despite the fact that there is a significant change in p over the same range. This can be explained by the fact that the decrease in p is compensated by the decrease in coat thickness as hair angle increases. The value of gr calculated using Cena and Monteith's approximate equation with p is also fairly constant with increasing hair angle; however, this estimate is about 2/3 of that using numerical integration. The fact that this value of gr (calculated using p) is constant, indicates that p does describe the main feature of the physics of diffuse radiative transmittance through the coat. Radiative conductance calculated using Cena and Monteith's approximation (p±) is quite similar to the integrated value until the hair angle reaches 30°. At this point, the radiative conductance begins to increase sharply toward infinity at a hair angle of 0 ° , illustrating the shortcoming of this approach. Figure 4.7a and b simulates the effect of piloerection on the components of coat conductance at position 2-90 (top) on the model deer, with u = 0 m s - 1 .  Cena and  Monteith's approximation (Eqs. 4.14 and 4.15) was used to calculate kr in Figure 4.7a, while numerical integration (Eqs. 4.18, 4.19 and 4.20) was used in Figure 4.7b.  The  components of coat conductance were assumed to act in parallel (they can be added). The value of total coat conductance (gc) at the largest hair angle (<f> = 11°) was measured. Radiative conductivity was calculated for the varying hair angles as described above and converted to radiative conductance (gT = kr/(lpcp)).  Hair conductance (gh) was the same  for all hair angles and was calculated as described in Chapter 3. Convective conductance (<7con) was calculated at <f> = 77° by subtracting the sum of gr and gh from gc. At other hair angles, gcon was calculated by assuming that it decreased from its value at </> = 77°,  Chapter 4.  Thermal  108  Radiation  C/i  O  o  4  o U >  Hair Angle (degrees) Figure 4.6: Results of a simulation showing the effects of piloerection on radiative conductance calculated using numerical integration and using Cena and Monteith's approximate equations with p± and p for position 2-90.  Chapter 4. Thermal  Radiation  109  a  c/i  O  O  -§ o U 3 gc  -  -o  convective o—  1 hair  radiative  0? 0  -D*f  20  40  60  80  Hair Angle (degrees) Figure 4.7: Simulation of the effect of piloerection on the components of coat conductance at position 2-90 (top) on the model deer with u = O m s - 1 . (a) Cena and Monteith's approximation with p± was used to calculate radiative conductance, (b) The numerical integration method was used to calculate the radiative conductance.  Chapter 4.  Thermal  Radiation  110  by an amount equal to the reduction in conductance through a layer of still air which had increased its thickness by the same amount as the coat (i.e. gcon is assumed to be proportional to coat thickness). The total coat conductance (gc) was calculated for <f> < 77° by adding up the component conductances (gc = gcon + gh + gr)In Figure 4.7a, as <f> decreases from 77° to 50°, gc falls because of decreasing gcon; however it levels off and then begins to rise sharply as <f> approaches zero due to increasing gr. In Figure 4.7b, gc behaves more reasonably and only decreases due to the change in gcon while gr and gh remain constant. This analysis assumes that piloerection does not increase coat conductance due to the change in coat structure causing an increase in free convection. It is possible that under windy conditions, piloerection would allow more effective penetration of the wind into the coat, which would tend to offset decreases in coat conductance due to the larger depth of the insulating layer.  4.4  Conclusions  The approximate method for determining radiative conductivity proposed by Cena and Clark (1973) and Cena and Monteith (1975a and b) using (kr = (4/3)(6//>j_) is satisfactory for hair angles (the angle to which the hair is inclined with respect to the normal to the skin surface) as small as 30 °; however, for smaller hair angles (which may be observed in the case of piloerection) it gives unrealistically high values of radiative conductivity. Using p in this equation and converting to conductance, indicates that gr is virtually constant with hair angle, in agreement with the numerical integration method. The case of piloerection and all hair angles can be treated using the numerical integration method. This method shows that raising the hair (decreasing the hair angle) results in decreasing interception of thermal radiation per unit depth of coat. However, since the total coat depth is increasing with piloerection, there are only negligible changes in  Chapter 4. Thermal  Radiation  111  radiative conductance. It is likely that the deer would receive the greatest benefit from piloerection under calm wind conditions. The benefits of piloerection may become negligible at some higher wind speed due to increased penetration of wind into the coat (more forced convection).  4.5  Literature Cited  Cena, K. and J.A. Clark (1973) Thermal radiation from animal coats: Coat structure and measurements of radiative temperature. Phys. Med. Biol. 18, 432-443. Cena, K. and J.L. Monteith (1975a) Transfer processes in animal coats.  I. Radiative  transfer. Proc. R. Soc. Lond. B. 188, 377-393. Cena, K. and J.L. Monteith (1975b) Transfer processes in animal coats. II. Conduction and convection. Proc. R. Soc. Lond. B. 188, 395-411. Macfarlane, W.V. (1968) Weather, climate and domestic animals. In Agricultural teorology (Proc. of the W.M.O. Seminar), 1, pp.  Me-  119-161. Melbourne: World  Meteorological Organization. Parker, K.L. and M.P. Gillingham (1990) Estimates of critical thermal environments for mule deer. J. Range Manage. 4 3 , 73-81.  Chapter 5 T e s t i n g a M o d e l of H e a t Loss from D e e r in Forest H a b i t a t s  5.1  Introduction  In this chapter, the findings of this thesis are used in a computer model to make predictions of deer heat loss in forest habitats. The model represents a framework within which the focused objectives of this research can be placed in a broader context.  5.2  The Model  The model, called DEERCLIM, was developed as part of the Managed Stands for Deer Winter Range Program (see Bunnell et al., 1991) and incorporated into a larger model which simulates the effects of forestry practices on deer winter range carrying capacity. A more complete description of DEERCLIM than will be provided here is given as an appendix to Bunnell et al. (1991) entitled "Microclimate Model". T h e code was written by J.M. Chen and is given in Appendix G. The first version of DEERCLIM used equations for boundary layer and coat conductance suggested by Parker and Gillingham (1990) which are quite general and somewhat empirical. In this thesis, equations for boundary layer and coat conductances have been developed which are based on direct measurements on a model deer. One objective of this chapter is to compare results of DEERCLIM using Parker and Gillingham's equations with those using the equations developed in this thesis. A further objective is to compare model predictions using winter data collected in forest with those based on data from an  112  Chapter 5. Deer Heat Loss Model  113  adjacent open area. It should be noted that no attempt has been made to completely test the model for all conditions; rather, we wish as the opening paragraph states, to put the work of this thesis in a broader context.  5.2.1  A Brief D e s c r i p t i o n of t h e M o d e l  The inputs of weather data to the model are daily values of global solar radiation, mean wind speed, mean wind direction, maximum and minimum air temperature, and snow depth. In addition, the site parameters latitude, longitude, slope angle, slope orientation, overstory leaf area index, average tree height, average understory height, and stand density are needed. The model accepts inputs of solar radiation and wind speed above old-growth and second-growth Douglas-fir stands and then estimates their values mean at the heights of a deer standing or lying down. The wind speed esimations are made possible by the work of Lee and Black (1993a, b and c), while the radiation predictions are based on the results of Chen and Black (1992), Chen and Black (1991), Black et al. (1991) and Lee and Black (1993b and c). The outputs from the model are the daily means of operative temperature, T e , standard operative temperature, Tes and basal metabolic rate, M, which were calculated by averaging the 24 hourly values. These hourly values were calculated by simulating the diurnal course of each input variable from its daily value. This procedure helped to minimize the error which is introduced if daily outputs are calculated using mean daily inputs. Qualitatively, Te is the air temperature adjusted for the effects of radiative and convective exchange imposed at the coat surface. Tes is similar to Te except that it has been calculated at a standard low wind speed (0.1 m s - 1 ) and it also considers the effect of the tissue and coat resistances as well as the imposed conditions.  114  Chapter 5. Deer Heat Loss Model  The operative temperature is given by Te = Ta+  Ve{Rabs  ~ ™Tt)  (5.1)  pcp where Ta is the air temperature, Rabs is the incident radiative flux absorbed per unit deer surface area and re is the parallel combination of the mean boundary layer (r&) and longwave radiative resistances ( r r ) . Mean boundary layer resistance was calculated by Parker and Gillingham (1990) for mule deer using  rb = 307^/dJH  (5.2)  where d is the characteristic dimension (m) of the deer, u is the wind speed ( m s - 1 ) and fb has units of s m _ 1 . Parker and Gillingham multiplied Eq. 5.2 by 0.7 to account for the turbulent enhancement of heat transfer in the outdoor environment. Equation 5.2 was originally developed for heat transfer from a flat plate and was subsequently recommended by Campbell (1977) as representing a good average value for spheres and cylinders. Wind tunnel measurements for the bare model deer in cross flow (see Chapter 2) yielded the following equation for mean boundary layer resistance of the trunk (same units as in Eq. 5.2) J0.39  n = 228—  (5.3)  Based on data collected in a second growth Douglas-fir stand (see Chapter 2), Eq. 5.3 was multiplied by 0.77 to account for turbulent enhancement. Figure 5.1 shows mean boundary layer conductances (gb = l/r&) using the reciprocals of Eqs. 5.2 and 5.3 multiplied by their respective enhancement factors and Eq. 5.2 alone as functions of wind speed. There is little difference between the two enhanced lines at low wind speeds ( < 1 m s - 1 ) ; however, they begin to diverge at higher wind speeds with the enhanced Eq. 5.3 giving a higher conductance. Both enhanced lines show significantly higher conductances than the non-enhanced line from Eq. 5.2.  Chapter 5. Deer Heat Loss Model  115  Wind Speed (m/s) Figure 5.1: Mean boundary layer conductance (gb) as a function of wind speed determined in this study for the model deer trunk with a turbulent enhancement factor of 1.3 (line a), gb determined using the equation recommended by Campbell (1977) with an enhancement factor of 1.43 (line b) and gb using Campbell's equation without enhancement (line c).  Chapter 5. Deer Heat Loss Model  116  The standard operative temperature is given by T  (rHbs + res)(Tb - Te)  T  les = lb  -,  ;  v  {rm + re)  (5.4)  where rub is the whole body resistance, rubs and res are the values of rjjb and r e calculated at the standard low wind speed (0.1 m s - 1 ) and 7& is core body temperature. The whole body resistance is the sum of the peripheral tissue resistance, rt and the mean coat resistance, r c . Parker and Gillingham determined rt in s m " 1 using the empirical relationships rt = 63.95-5.58T e (°C) (winter) or rt = 2 0 4 . 4 3 - 6 . 4 9 7 ; (°C) (summer) from Webster (1974). They determined mean coat resistance with the equation  fc  = TT^MU  (5 5)  -  where rcs is the mean coat resistance at the standard low wind speed (0.1 m s _ 1 ) and u is in m s " 1 . The value of rcs was determined from the equation rcs = rnbs — ft where ^(sm  - 1  ) was calculated using 857.9-29.9T e -0.19T e 2 +0.006T e 3 (°C) (winter) or 603.29-  4.62T e — 0.30ZIe2(°C) (summer). The preceeding equations for r^bs came from polynomial fits to data for mule deer collected by Parker and Robbins (1984). For consistency with Chapter 3, it is convenient to rewrite Eq. 5.5 in terms of conductance gc = gcs{\ + 0.08w) where gcs (l/rcs)  (5.6)  is the mean coat conductance at the standard low wind speed and u is  in m s - 1 . This equation is of the form recommended by Campbell et al. (1980), where the exponent of u is 1. Mean coat conductance as a function wind speed for the winter coat of a mule deer was experimentally determined in the wind tunnel (see Chapter 3) to be gc = 2.45 + 0.006u 2 1 1  (5.7)  Chapter 5. Deer Heat Loss Model  117  where gc is in m m s - 1 and u is in m s _ 1 . Figure 5.2 compares mean coat conductances calculated using Parker and Gillingham's equations for winter (@ Te = 0 °C) with that calculated from Eq. 5.7. The figure shows that the results of this study give a significantly larger g~c than Parker and Gillingham's equations.  This is mainly due to the lower  value of gcs which is calculated using their empirical equations for rt and rnbs-  At  an operative temperature of 15 °C the value of gca predicted by Parker and Gillingham's winter equations would be approximately 2.45 mm s _ 1 , so that the two lines would predict nearly the same coat conductance; however, at temperatures lower than 0 °C, which are likely in winter, the difference between the two lines would be larger. The dependence of gcs on Te can only be explained by changing coat depth but Parker and Robbins (1984) indicate that piloerection does not begin for an adult mule deer in winter until Te drops to almost -20°C. Parker (1988) found that for one position on the flank of a live blacktailed deer, coat depth was a weak function of air temperature (coat depth increased as air temperature decreased); however, this change in coat depth is not enough to account for the change in gcs as a function of Te indicated by Parker and Gillingham's winter equations. The final output from the model, metabolic rate, M, is given by M =  l 2 c  - P v(Tb-Tes)  fHbs + res  where the 1.2 mulitiplier accounts approximately for heat lost by the deer due to latent heat loss from the skin and through respiration (see Campbell, 1977). Metabolic rate was not calculated by Parker and Gillingham; however, changes in the calculations of boundary layer and coat resistance will change M. A possible shortcoming with the model described here is t h a t it was necessary to combine the empirical equations (i.e. those for tissue resistance) from another study with the equations for coat and boundary layer conductance derived in this study to  Chapter 5. Deer Heat Loss Model  1  118  1  1  1  nn, •  1 a  This study  3 "  b  _—-^P  Parker and Gillingham (1990) Winter Equations — T e = 0°C  0  1  0  1  1  1  4 6 8 Wind Speed (m/s)  I  10  Figure 5.2: Mean coat conductance (gc) as a function of wind speed for mule deer hide determined in this study (line a) and gc determined using equations recommended by Parker and Gillingham (1990) for winter with r e = 0 ° C (line b ) .  Chapter 5. Deer Heat Loss Model  119  calculate Tes and M. It is important to remember that Eqs. 5.3 and 5.7 were derived from measurements on a model deer consisting of a heated trunk only. The same methodology could easily be used to determine the relationships for the boundary layer and coat conductances of the legs, neck and head. However, the complex physiology of these regions, which was beyond the scope of this study, makes it difficult to determine actual heat loss from these regions.  5.2.2 5.2.2.1  S o m e E x a m p l e s of M o d e l O u t p u t s Site D e s c r i p t i o n  Data for use in model testing were collected within an old-growth Douglas-fir suga menziesii  (Pseudot-  (Mirb.) Franco) stand and an adjacent open site during the winters of  1988-89 and 1989-90. The experimental site was located at an elevation of 510 m on a 30-40% south facing slope near Woss, British Columbia, Northern Vancouver Island ( 5 0 ° 6 5 N , 1 2 6 ° 3 8 W ) . The open site consisted of a 15 m wide strip between the logging road running along the slope and the old-growth stand above. A 20-year-old Douglasfir stand extended down the slope from the logging road. Vegetation in the open area consisted of salal (Gaultheria  shallon Pursh) and huckleberry (Vaccinium  parvifolium)  which was generally less than 1 m tall along with scattered taller Douglas-fir trees. The old growth measurement site was about 100 m uphill of the open site and was characterized by a 200+ years old Douglas-fir stand of 500-700 stems/ha and average tree height of greater than 25 m (Lee and Black, 1993c). The predominant understory species was salal which was less than 0.7 m tall.  Chapter 5. Deer Heat Loss Model  5.2.2.2  120  Instrumentation  At both the open and old growth sites, air temperature was measured at a height of 1.5 m above the ground by a thermistor housed inside a Gill 12 plate shield (R.M. Young Co., Traverse City, Michigan). Unshielded thermistors measured air temperature at heights of 15, 40 and 75 cm at each site and enabled the determination of time periods when snow cover exceeded these depths. Wind speed and direction were measured at a height of 2 m by an R.M. Young wind monitor (R.M. Young Co., Traverse City, Michigan, Model 05103) at the open site. The wind speed was logarithmically corrected to a height of 0.8 m to represent that at deer height. Wind speed at the old growth site was estimated using regression relationships between measurements made there and at the open site during the summer of 1989 (Lee and Black, 1993c). Solar radiation was measured at the open site with a pyranometer (Li-Cor Inc., Lincoln, Nebraska, Model LI-200S). Within stand solar radiation was estimated by the model using that measured at the open site as an input. Data from both sites were recorded by a data logger (Campbell Scientific Inc., Logan, Utah, Model 21X).  5.2.2.3  C o m p a r i s o n of M o d e l O u t p u t s U s i n g R e l a t i o n s h i p s F o u n d in this T h e s i s w i t h T h o s e from Parker a n d G i l l i n g h a m ( 1 9 9 0 )  A winter d a t a set from 1 January 1989 to 11 February 1989 was chosen for testing of the DEERCLIM model. Figure 5.3 summarizes the prevailing weather conditions during the period showing the hourly air temperatures, solar radiation, and wind speeds for the open site, as well as periods when snow cover exceeded 15 cm at the open and old growth sites. The first 30 days were characterized by mostly cloudy skies and temperatures generally between + 5 and -5 °C and several snowfalls. This was followed by 3-4 days with minimum temperatures as low as -17°C, accompanied by clear skies. The final week featured clear  Chapter 5. Deer Heat Loss Model  121  500 250  uUlfllllL LA . IIUJI hl.JyUJJn  cs  B ^  &  0  10 U C3  0  -10  snow depth >15cm  old growth — — — t/i  0 01  08  15 Jan  22  29  05 Feb  12  Date Figure 5.3: Hourly solar radiation (Rs), 1.5 m air temperature (Ta) and wind speed (u) recorded at the open site near Woss, B.C. for the period 1 January 1989 to 11 February 1989. Also shown are periods when snow cover exceeded 15 cm (heavy horizontal lines) for the open and old growth sites.  Chapter 5. Deer Heat Loss Model  122  skies and temperatures gradually warming up to above freezing. Wind speeds were light throughout the period, never exceeding 1.5 m s - 1 . The snow interception capacity of the old growth stand is illustrated by the fact that there was only 15 days with a snow depth greater than 15 cm within the stand and 26 days at the open site. Figure 5.4 compares mean daily air temperatures with calculations of mean daily Tes output from DEERCLIM using Parker and Gillingham's equations and the equations developed in this study. For the given conditions in this case, the Tes values calculated using Parker and Gillingham's equations were about 1 °C lower than mean daily air temperature, while Tes calculated using equations from this study were about 2°C lower than mean daily air temperature; however, this wasn't the case for all conditions. The differences in Tes were larger during the period of cold weather. Te was virtually the same as Ta in the old growth stand due to low wind speeds and solar radiation. Figure 5.5 compares the metabolic rates from DEERCLIM calculated using Parker and Gillingham's equations with those calculated using equations from this study. The results from this study gave an estimated metabolic rate which was nearly double that calculated using Parker and Gillingham's equations. The difference is mainly due to the lower value of rnbs calculated in this study.  The values of M calculated in this  study increased as Tes decreased; however, this was not the case when using Parker and Gillingham's equations because coat conductance decreased as Tes decreased.  5.2.2.4  C o m p a r i s o n of D e e r H e a t Loss in Forested a n d O p e n H a b i t a t s  Figure 5.6 compares values of Tes (using equations from this study) for the three following cases: (i) the old-growth stand, (ii) the open site and (iii) the open site but with the measured wind speeds multiplied by a factor of 10. Despite the higher values of solar radiation at the open site, Tes was quite similar to that inside the old growth. This is because increased longwave radiation losses to the sky offset any gains due to higher  Chapter 5. Deer Heat Loss Model  I  123  I  I  I  I  I  01  08  15 Jan  22  29  I  05 Feb  l_J  12  Date Figure 5.4: Comparison of mean daily standard operative temperature (Tes) in the old groth stand calculated using equations developed in this study and equations recommended by Parker and Gillingham (P & G) (1990) . Also shown is mean daily air temperature (T a ).  Chapter 5. Deer Heat Loss Model  124  120  Date Figure 5.5: Comparison of the metabolic rate (M) calculated using the equations developed in this study (line a) and that calculated using Parker and Gillingham's equations (line b ) .  Chapter 5. Deer Heat Loss Model  125  0  •10 U 05  -20  -30 01  08  15 Jan  22  Date  29  05 Feb  12  Figure 5.6: Comparison of the standard operative temperature (Tes) for a deer within the old growth stand, with that for a deer at the open site and at the open site with the wind speed increased by a factor of 10.  Chapter 5. Deer Heat Loss Model  126  solar radiation. Caution should be used when interpreting these results, because daily Tes values were being used. A deer might minimize heat losses by spending daytime hours in the open area when shortwave radiation gains outweigh longwave losses and nighttime hours in the forest to reduce longwave losses. For example, if Tes were computed for the conditions outside the stand during the day and for inside the stand during the night, the resulting mean daily Tes would be larger than in either environment alone. The predicted Te3 values for the high wind case were significantly lower (5-15 °C) than for the other two cases. On 2 February, the highest wind speeds during the period, combined with the lowest air temperature to produce a Tes of -31 °C for the high wind case (5.8 m s - 1 ) , despite the sunny weather. Figure 5.7 compares the metabolic rates predicted by DEERCLIM for the same cases shown in Figure 5.6. As with Tes, there was little difference between the old growth and open sites until a higher wind speed was assumed. In this case, increasing the wind speed by a factor 10 resulted in only a 10-15% increase in M.  5.3  Conclusions  A model which estimates standard operative temperatures and deer metabolic rates in forest habitats was tested for a limited range of environmental conditions. Comparison of the model output using the equations of Parker and Gillingham (1990) and those developed in this study showed that the latter estimated metabolic rates which were as much as twice those estimated by the former. The differences between these estimates is due to the larger coat conductances calculated in this study. There is sketchy evidence available that live deer may gradually piloerect their hair as air temperature decreases (Parker, 1988). This might explain in part the higher coat conductances measured in this study on a non-living tanned deer hide.  Chapter 5. Deer Heat Loss Model  01  08  15 Jan  127  22  29  05 Feb  12  Date  Figure 5.7: Comparison of the metabolic rate (M) of a deer within the old growth, with that of a deer at the open site and at the open site with the wind speed increased by a factor of 10.  Chapter 5. Deer Heat Loss Model  128  Sample runs of the model DEERCLIM using real winter data show that under some conditions, metabolic requirements of deer in the old growth stand and the open are similar, at least on a daily basis. However, it seems likely that if the model was used to calculate outputs on an hourly basis, it would show that a deer could minimize its heat loss by spending part of the day in one habitat and part in another. Increasing wind speeds by a factor of 10 only increased the metabolic rate by 10-15%. This is due to the weak wind speed dependence of mean coat conductance at wind speeds as high as 8ms"1.  5.4  Literature Cited  Bunnell, F.L., T.A. Black, J.M. Chen, L.L. Kremsater, X. Lee and R.M. Sagar (1991a) Managed stands for deer winter range program: Final progress report.  NSERC  File # 661-02/87. Black, T.A., J.M. Chen, X. Lee, and R.M. Sagar (1991) Characteristics of shortwave and longwave irradiances under a Douglas-fir forest stand. Can. J. For. Res. 2 1 , 10201028. Campbell, G.S. (1977) An Introduction  to Environmental  Biophysics.  Springer-Verlag,  New York, 159 pp. Campbell, G.S., A.J. McArthur and J.L. Monteith (1980) Windspeed dependence of heat and mass transfer through coats and clothing. Boundary-Layer  Meteorol. 1 8 , 485-  493. Chen, J.M. and T.A. Black (1991) Measuring leaf area index of plant canopies with branch architecture. Agric. For. Meteorol. 5 7 , 1-12.  Chapter 5. Deer Heat Loss Model  129  Chen, J.M. and T.A. Black (1992) Foliage area and architecture of plant canopies from sunfleck size distributions. Agric. For. Meteorol. 60, 249-266. Lee, X. and T.A. Black (1993a) Atmospheric turbulence within and above a Douglas-fir stand. P a r t I: Statistical properties of the velocity field. Boundary-Layer  Meteorol.  64, 149-174. Lee, X. and T.A. Black (1993b) Atmospheric turbulence within and above a Douglas-fir stand. Part II: Eddy fluxes of sensible heat and water vapour.  Boundary-Layer  Meteorol., (in press) Lee, X. and T.A. Black (1993c) Turbulence near the forest floor of an old growth Douglasfir stand on a south-facing slope. For. Sci. 3 9 , 211-230. Parker, K.L. (1988) Effect of heat, cold and rain on coastal black-tailed deer. Can.  J.  Zool. 66, 2475-2483. Parker, K.L. and C.T. Robbins (1984) Thermoregulation in mule deer and elk. Can. J. Zool. 62, 1409-1422. Parker, K.L. and M.P. Gillingham (1990) Estimate of critical thermal environments for mule deer. J. Range Manage. 4 3 , 73-81. Webster, A.J.F. (1974) Heat loss from cattle with particular emphasis on the effects of cold. In Heat Loss From Animals and Man (Edited by Monteith, J.L. and Mount, L.E.) pp. 205-231. Butterworths, London.  Chapter 6  General Conclusions  Heat transfer through the boundary layer of the elliptically cross-sectioned model deer trunk in cross flow did not differ significantly from that expected for a circular cylinder. Mean boundary layer conductance was increased slightly when the model deer was exposed to longitudinal flow. There was no evidence to suggest that the presence of fur on a deer will significantly increase boundary layer heat transfer above that expected from a smooth circular cylinder. Turbulent air flow in the trunk space of an old-growth, Douglas-fir stand caused about a 30% enhancement in boundary layer conductance. The insulation provided by the deer's coat was significantly larger than that provided by the boundary layer at all but the lowest ( < 0 . 3 m s _ 1 ) wind speeds. There was little wind penetration into the coat of a deer in cross flow, except at the stagnation point. This was evidenced by a lack of dependence of coat conductance on wind speed at positions other than the stagnation point. There was some indication of wind penetration into the coat of a deer in longitudinal flow especially when the deer's rear end was facing into the wind and at higher wind speeds when ruffling of the fur occurred. Free convection was found to be an important mechanism of heat transfer within the coat, while radiative transfer and heat conduction along individual hairs were relatively unimportant. Coat depth was found to be an important determinant of coat conductance. It was concluded that forced convection within the coat is minimal at the low wind speeds typically observed in coastal forested deer habitats. A numerical integration procedure was developed to allow accurate determination of  130  Chapter 6.  Conclusions  131  radiative transfer through the coat when piloerection is occurring. Simulations showed t h a t radiative transfer remained unimportant when hairs are standing up due to the longer p a t h length radiation must travel to escape the coat. T h e simulation showed that piloerection can significantly increase the insulation provided by the coat through an increase in the depth of still air; however the effects of piloerection on free and forced convection were not investigated. Output from the deer heat loss model, DEERCLIM, with the equations for deer coat conductance developed in this study predict basal metabolic rates which were nearly twice those determined using the existing equations from Parker and Gillingham (1990). There was some evidence to suggest that coat conductance measured in this study was higher than that which would be measured on a live deer, because a live deer may raise its hair to increase insulation even when it is not cold stressed. More research should be carried out on live deer to measure the response of coat depth to changing environmental conditions. With an adequate knowledge of changes in deer coat depth, changes in coat conductance can be estimated as was shown in Chapter 4. Model simulations using real winter data showed that metabolic rates were similar for the open and old growth sites, despite higher solar radiation at the open site, due to increased longwave losses at the open site. Metabolic rates increased by only 10-15% when wind speeds at the open site increased by a factor of 10 because even the higher windspeeds were relatively ineffective in penetrating the coat. Using hourly inputs to DEERCLIM would illustrate how a deer could minimize heat loss by moving to favorable habitats depending on the time of day.  Literature C i t e d Parker, K.L. and M.P. Gillingham (1990) Estimate of critical thermal environments for mule deer. J. Range Manage. 4 3 , 73-81.  Appendix A  D e t e r m i n a t i o n of t h e N e t R a d i a t i v e F l u x D e n s i t y for Various P o s i t i o n s on t h e M o d e l D e e r in t h e W i n d T u n n e l  Net radiative flux density (Rn)  was calculated for the various positions on the model  deer trunk (loss from the deer is positive Rn) during the wind tunnel experiments. It was necessary to use view factor theory to accurately calculate incoming longwave radiation to a given position on the model deer because the wind tunnel wasn't isothermal. The surface temperatures of the walls, ceiling and floor were monitored periodically with a hand held infrared thermometer. The ceiling was sometimes more than one degree C warmer than the floor. During the experiments, interior lighting of the wind tunnel was turned off so shortwave radiation was negligible when compared with longwave radiation. View factors ( / ) were calculated for positions on the top, sides and bottom of the model deer when it was in the longitudinal and cross flow orientations (see Figure 2.3). The positions on the side of the model deer for which view factors were calculated, were assumed to be halfway between the floor and ceiling (they were actually slightly lower than halfway) to simplify calculations. Similarly, points on the top of the model deer for which view factors were calculated were assumed to be at the midpoint of the wind tunnel cross-section. The head of the model deer was assumed to have no effect on these positions. The following view factors were calculated by analytically integrating view factor equations (see Howell, 1982 and Chapman, 1969). The model deer was situated approximately 5 m from the blowing end of the wind tunnel which has a cross-section of 1.6 m  132  Appendix  A. Net  Radiation  133  high by 2.4 m wide at this point. Table A.l summarizes the view factors. The view factor / is defined as the fraction of the radiation leaving an elemental area dA on the deer body that is intercepted by a particular part of the wind tunnel, e.g. the floor. With a knowledge of these view factors, along with the appropriate surface temperatures, we can proceed to calculate the net radiative flux density for a given position on the model deer. For example, to calculate Rn for a point on the side of the model deer in cross flow, we start with the following equation: Teff = 0.3182) + 0.36471™ + 0.318TC  (A.l)  where T e / / is the effective environmental temperature ( ° C ) , 7 ) the surface temperature of the floor, Tw the surface temperature of the walls and Tc the surface temperature of the ceiling. The net radiative flux density from the deer is then computed using Rn = esaTf - eeaT?ff  (A.2)  where Ts is the surface temperature of the model deer, es is the emissivity of the model deer (es = 0.95 and 0.97 for the bare and coat covered model, respectively) and ee is the emissivity of the environment (assumed to be 1).  Literature C i t e d Chapman, A.J. (1969) Heat Transfer. Macmillan Co., New York, 406 pp. Howell, J. R (1982) A Catalog of Radiation Configuration Factors. McGraw-Hill, Toronto, 243 pp.  Appendix A. Net Radiation  134  Table A.l: Summary of view factors for positions on the top and sides of the model deer which was oriented in either the cross flow (cross) or longitudinal flow (long) orientations. Position and Orientation side, cross side, long top, cross and long bottom, cross and long  View Factors (/) floor walls 0.318 0.364 0.203 0.594 0.121 0.098 0.902  ceiling 0.318 0.203 0.879  Appendix B  S o m e E x a m p l e s of T i m e C o n s t a n t s for Equilibration of Surface T e m p e r a t u r e of t h e B a r e M o d e l D e e r t o S t e p C h a n g e s in W i n d S p e e d  The following table shows some representative time constants for t h e response of surface temperature on t h e bare model deer to step changes in wind speed. Time constants are shown for both the cross flow (cross) and longitudinal flow (long) orientations of the model deer. Table B . l : Some time constants ( r ) for the response of surface temperature of the bare model deer to step changes in wind speed in the wind tunnel. Position 2-180 2-180 2-180 2-180 2-180 2-180 2-0 2-90 2-90 2-90 3-90  Deer Orientation cross cross cross long long long cross cross cross long long  wind speed change (ms _ 1 ) 2.6-1.5 1.5-5.3 5.3-0.6 2.6-1.5 1.5-5.3 5.3-0.6 5.3-1.5 1.5-5.3 5.3-0.6 5.3-1.5 5.3-0.6  135  r (min) 1.8 1.9 2.7 2.0 1.7 2.3 2.6 1.4 2.8 1.8 2.4  Appendix C  C o m p a r i s o n of t h e E v e r e s t M o d e l 4 0 0 0 I R T w i t h a B l a c k b o d y Calibration B l o c k D u r i n g t h e S e c o n d W i n d Tunnel E x p e r i m e n t I n v o l v i n g t h e C o a t Covered Model Deer  During the second wind tunnel experiment using the coat covered model deer, surface temperature on the model deer was monitored using an Everest Interscience Inc. (Fullerton, CA), Model 4000 infrared thermometer (IRT). Periodically, during the experiment, the IRT was compared with a blackbody calibration block. The calibration block was made from a 4.3 cm long piece of cylindrical (7.6 cm diameter) aluminum bar stock. A 1.7 cm deep by 4.9 cm diameter cavity was machined into one end of the block to accept the IRT. A chromel-constantan thermocouple was embedded just below the bottom surface of the cavity, which was coated with Parson's optical black paint. The temperature measured by the thermocouple (Tbb) was compared with that measured by the IRT (TIRT)  when it was inserted into the cavity. T h e data collected between  28 June and 1 July 1991 is shown in Table C.l. In Figure C.l the difference between Tbb and TIRT (Tbb — TIRT)  is plotted against blackbody temperature (Tbb)- The equation of  the linear regression line through the data points is Tbb - TIRT = 0.035T66 - 0.85 The surface temperature measurements were corrected by subtracting (Tbb ~ TIRT) measured surface temperature where (Tbb — TIRT) using Eq. C.l. 136  (C.l) from  was calculated at that temperature  Appendix  C. IRT  1  Calibration  1  137  1  0.24 -  '  i  1  i  1  1 D  -  •  0.18  •  -  /  regression line D  -  -  0.12 -  a  -  —  a  "  0.06 0  -  a  1  23  a  /  , / l  ,  25  1  27 Tbb (°C)  i  ...  1  29  '  1  31  Figure C.l: Plot showing comparison of Everest Interscience, Model 4000 IRT {TIRT) with a blackbody calibration block (Tbb)- The solid line is a regression through the data points.  Appendix  C. IRT  Calibration  138  Table C.l: Comparisons of surface temperatures measured with an Everest Interscience, Model 4000 IRT (TIRT) with the temperature of a blackbody calibration block during the second wind tunnel experiment with the coat covered model deer.  Date 28 June 29 June 29 June 29 June 30 June 30 June 30 June 30 June 1 July 1 July 1 July  Time 1917 0903 1341 1756 0945 1543 1720 1854 1107 1445 1639  TIRT  (°C)  27.95 24.55 26.83 27.48 24.68 28.09 28.86 28.91 27.36 29.37 30.08  Tbb(°C) 28.08 24.60 26.93 27.63 24.69 28.18 29.02 29.10 27.44 29.57 30.32  Tbb-TmT(°C) 0.13 0.05 0.10 0.15 0.01 0.09 0.16 0.19 0.08 0.20 0.24  Appendix D Turbulent P o w e r S p e c t r a in t h e Old G r o w t h and S e c o n d G r o w t h S t a n d s  Typical examples of the turbulent spectra of the u (streamwise) component of wind velocity for the old growth and second growth stands are shown in Figures D.la and b , respectively. The data were collected at a height of 2 m above the forest floor by a 3-dimensional sonic anemometer (see Chapter 2) at a sampling frequency of 10 Hz. The second growth data were collected for a period of one-half hour while the old growth data were collected for one hour. The power spectra variable, n<f>aa, which indicates the amount of turbulent kinetic energy contained in eddies of a given sized is plotted on the ordinate, while wavelength (A), which indicates eddy size, is plotted on the abscissa. Both spectra exhibit a double peaked pattern with one peak near A = 100 m and one near A = 1 m. T h e peak near A = 1 m has been attributed to vortex shedding by tree trunks (Lee and Black, 1993). These vortices are of similar size to those which are shed from a deer (characteristic dimension = 0.30m). As was discussed in Chapter 2, this may lead to turbulent enhancement of heat transfer through the deer's boundary layer. However, it should be noted that the larger eddies (100m size) contain 10-100 times more energy than the l m eddies and therefore make a larger contribution to the observed turbulence intensity.  Literature Cited Lee, X. and T.A. Black (1993) Turbulence near the forest floor of an old growth Douglasfir stand on a south-facing slope For. Sci. 3 9 , 211-230.  139  Appendix D. Turbulent Spectra  140  *•  •• ••  •  10"  second-growth  — • •  •• •  • •  10" r<i  •  • • • • • • • • • • • • • • • ••• m  —  a Cfl  10'  <N  10"  8 8  • •  G 10-  2  10-  3  i  i  10'  10'  •• •  ••  i  0  10'  *.  old-growth •  •  • • •  b -4  10  ., ., L,  103  \ i  102  i  101  i  10°  Wave length (m) Figure D.l: Typical power spectra of the streamwise (u) velocity component observed at a height of 2 m above the the forest floor at (a) the second growth stand on 19 July 1990 between 1330 and 1400PST and (b) the old growth stand on 9 August 1989 between 1315 and 1415PST.  Appendix E Algebraic a n d T r i g o n o m e t r i c M a n i p u l a t i o n s S h o w i n g t h a t E q u a t i o n s 4.9 a n d 4.10 are Equivalent  The expression for the radiation interception function p(£, a), as derived in Chapter 4 (Eq. 4.9) is as follows: p(£, a) = ——^[1 — (sin <f> sin £ cos a + cos <j> cos if) 2 ] 1 ' 2  (E-l)  The geometric relationships of the angles are shown in Figures 4.1 and 4.2. We wish to show that Eq. E.l is equivalent ot Eq. 4.10, which was derived by Cena and Clark (1973) (Eq. 10). Equation 4.10 is as follows: p(£,a)  = nd[(l +tan 2 <^)(l + t a n 2 £ ) - ( 1 + t a n < £ t a n £ c o s a ) 2 ] 1 / 2  (E.2)  We begin by manipulating the expression for sin/? (Eq. 4.8), which is sin /? = [1 - (sin <f> sin £, cos a + cos <f> cos £ ) 2 ] ^ 2  (E.3)  Factoring cos 2 ^ c o s 2 £ out of the squared term in Eq. E.3 gives . ^ r, o , o J. / sin d> sin £ cos a sin/?= [l-cos2<£cos2£ , + ± cos <p cos £  cos ^>cos£. 0l1/0 7 7f 1/2 cos <p cos £  ,,-, N (E.4)  Simplifying this equation we get sin/3 = [1 - c o s 2 < ^ c o s 2 £ ( t a n < ^ t a n £ c o s a + l ) 2 ] 1 / 2  (E.5)  Now we factor cos <j> cos £ out of the square brackets to obtain sin ,5 = cos^cos^f—— — - (tan 6 tan £ cos a + I)2]1/2 cos 2 <p cos 2 £ 141  (E.6)  Appendix  E. Equivalence  of Equations.  4.9 and 4.10  142  We can can make use of the following trigonometric identity to further simplify Eq. E.6. cos 2 0 =  1  —— 1 + tan 2 6  (E.7) y '  Now, substituting Eq. E.7 into Eq. E.6 for the cos 2 <j> and cos 2 £ terms we have sin/3 = cos</>cos£[(l +tan 2 <^)(l + t a n 2 £ ) - (tan </> t a n £ cos a + 1) 2 ] 1 / 2  (E.8)  Finally, this expression for sin j3 can be substituted into Eq. 4.3 (p(fl) = ndls sin /?//) and using the fact that cos (j> = l/la we get p(£,ct) = ndcos{[(l  + tan 2 <^)(l + t a n 2 £ ) - ( 1 + t a n ^ t a n £ c o s < * ) 2 ] 1 / 2  which is the same as Eq. 4.11.  (E.9)  Appendix F  A Sample Numerical Integration to Determine kr  Shown below are the results of a numerical integration to determine kr for position 3-180 (lee side). The values of F(r) are a result of evaluation of Eq. 4.18, which is: F(r) = - I* &a T 7T Ja=0  p(^a)e-p^'a^rdr[B(T)  - abr cos £] sin ( cos £d{  (F.l)  J£=0  This equation is valid for calculating the radiative flux which arrives at plane AB in Figure 4.4 from below. To obtain the radiative flux reaching plane AB from above, abr cos £ was added to B{t) in Eq. F.l, instead of subtracted. The parameters for this numerical integration were as follows: nd = 391.5m- 1 / = 19.5mm ls = 53.2 mm a = -400 K m " 1 6=6.2Wm-2K"1 r = 0.00025 -> 0.00975m dr = 0.0005m  da = 0.17453 rad or 10° d£ = 0.17453rad or 10° The results of the integration are given in the table below The net radiative flux (F) crossing the plane AB in this case is 4.05Wm - 1 , therefore the radiative conductivity (K) calculated using Eq. 4.20 is 10.01 mWm" 1 K _1 . 143  Appendix  F. Numerical  Integration  144  Table F . l : Values of the integrand, F(r), over the specified range of r values, above and below the plane AB as shown in Figure 4.4. below  above  r(m) .00025 .00075 .00125 .00175 .00225 .00275 .00325 .00375 .00425 .00475 .00525 .00575 .00625 .00675 .00725 .00775 .00825 .00875 .00925 .00975  F(r)  F(r)  160.90 102.19 65.50 42.42 27.80 18.47 12.46 8.55 5.98 4.27 3.11 2.31 1.75 1.36 1.07 0.86 0.71 0.59 0.49 0.42  160.60 101.64 64.91 41.89 27.36 18.12 12.19 8.34 5.81 4.14 3.00 2.23 1.69 1.31 1.03 0.83 0.67 0.56 0.47 0.40  Sums  461.23  457.18  Appendix G  T h e Quick Basic C o m p u t e r Code for t h e Deer Heat Loss Model D E E R C L I M  145  Appendix G. Model Code I * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *  MICROCLIMATE MODEL FOR DEER WINTER RANGE IN DOUGLAS-FIR FOREST STANDS ' Programmed by Jing Ming Chen, Andy Black and Bob Sagar of the Department ' of Soil Science at UBC. Xuhui Lee was consulted for wind ' regime relationships. ' This model computes hourly and daily values of operative temperature, ' standard operative temperature and basal metabolic rates for ' standing and lying deer in various habitats.  '***********DECLARE FUNCTIONS AND SUBROUTIONS ***************** DECLARE FUNCTION Wind.on.Slope! (Ufl, Zs!, Asl!, Au!) DECLARE FUNCTION Uw.to.Uh! (Uwhl!, H!) DECLARE FUNCTION Rhb.deer.standing! (U!, Tel!) DECLARE FUNCTION Metabolic.Iying! (Tell!, Tesll!, Rhbs!, Res!, Tal!, SD) DECLARE FUNCTION deer .body .temperature! (Tel!) DECLARE FUNCTION whole.body.resistance! (Tel!) DECLARE FUNCTION whole.body.resistance.wind!(Tel!, U!) DECLARE FUNCTION rhb.deer.statnding! (U!, Tal!) DECLARE FUNCTION Qabs.deer.lying! (Sdrul!, Sdfiil!, Rdul!, Tal!, Zal!, SD) DECLARE FUNCTION Te.deer.lying! (Tal!, Rel!, Qabsll!) DECLARE FUNCTION Ra.deer.standing! (U!) DECLARE FUNCTION Re! (Rr!, Ral!) DECLARE FUNCTION Ra.deer! (U!) DECLARE FUNCTION Qabs.deer.standing! (Sdrul!, Sdfiil!, Rdul!, Tal!, Z!, SD) DECLARE FUNCTION Te.deer.standing! (Tal, Rel!, Qabsl) DECLARE FUNCTION Tes.deer.standing! (Tel, Tal!, U!, Rhb!, Rel!) DECLARE FUNCTION R.radiative! (Tal!) DECLARE FUNCTION Res.heat.loss! (Tal!, RH1!) DECLARE FUNCTION Metabolic.rest! (Tel, Tesl!, Rhbs!, Res!) DECLARE SUB windspeed (Uhl!, Au!, H!, Hus!, lai!, Asl!, Udsl, Udll, SD) DECLARE FUNCTION sunset.hour! (Phi!, D!, ST) DECLARE FUNCTION power! (x!, Y!) DECLARE FUNCTION clear.sky.direct! (Z!) DECLARE FUNCTION clear.sky .diffuse! (Z!) DECLARE FUNCTION extra.global! (Z!) DECLARE FUNCTION cloudy.diffuse! (Z!, Sgl!) DECLARE FUNCTION sky .longwave! (Z!, Tal!, Sgl!) DECLARE FUNCTION G.fiinction! (IA!) DECLARE FUNCTION below.overstory.direct! (Sdrl!, IA1, lai!) DECLARE FUNCTION below.overstory.diffiise! (Sdrl!, Sdfl!, lai!) DECLARE FUNCTION sky.view.factor! (lai!) DECLARE FUNCTION below.overstory .longwave! (SLW!, lai!, Tal) DECLARE FUNCTION day .number! 0 DECLARE SUB incidence.on.slope(A!, Z!, Asl!, Zs!, incident.angle!) DECLARE FUNCTION daynumber% (day%, month %) DECLARE FUNCTION eqn.of.time! (theta!) DECLARE FUNCTION theta! (day«) DECLARE FUNCTION decimal.hour! (PST%) DECLARE FUNCTION solar.time! (DH, longitude, longref, E) DECLARE FUNCTION declination! (thetap!) DECLARE FUNCTION Zenith! (H!, D!, Phi!) DECLARE FUNCTION Azimuth! (H!, D!, Z!) OPTION BASE 1 'declare some constants CONST NUM.OF.VAR% = 30, here* = 3, CGA% = 2 CONST pi = 3.14159 CONST sigma = 5.67E-08 CONST etad = .95 'thermal emissvity of deer surface '********* LIST OF VARIABLES AND DIMENSIONED ARRAYS ************ 'a=solar azimuth angle (radians) 'Asl =orrientation of slope (input in degrees) 'Au=mean daily wind direction (degrees) 'B=maximum global irradiance (W/m2) 'CSdf= clear sky diffuse irradiance (W/m2) 'CSdr=clear sky direct irradiance (W/m2) 'D=solar declination (radians) 'day % = day of month 'DH=decimal hour 'dn% =Julian day number 'E=equation of time  Appendix G. Model Code 'G=G-fiinction of Douglas-fir canopies 'H=average tree height (m) 'LAI=leaf area index 'latitude = latitude of site (degrees) 'longitude=longitude of site (degrees) 'month % = month of year (e.g. for January, month % = 1) 'Msm=mean of Ms (Metabolic rate for standing deer) (W/m2) 'Mlm=mean of Ml (Metabolic rate for lying deer) (W/m2) 'Phi=radians of latitude 'Ras=standard aerodynamic resistance 'Rel = environmental resistance 'Res=standard environmental resistance (all resistances) 'Rhbs=standard whole body resistance of deer (s/m) 'Rr=radiative resistance 'SD=snow depth (m) 'SgO=extraterrestrial irradiance on a horizontal surface (W/m2) 'SRHr=sun rise hour (real) 'SRH% = sun rise hour (integer) 'SSH%= sunset hour (integer) 'ST=Solar time 'Stotal=total daily solar radiation (MJ/m2) 'SVF=sky view factor 't=radians of Julian day Tam=daily mean air temperature (C) 'Tem—daily mean operative temperature for standing deer (C) Telm=daily mean operative temperature for lying deer (C) Tesm=daily mean standard operative temperature for standing deer (C) 'Teslm=daily mean standard operative temperature for lying deer (C) Tmin=daily minimum air temperature (C) 'Tmax=daily maximum air temperature (C) 'Udlm=mean of Udl (windspeed at lying deer height) (m/s) 'Udsm=mean of Uds (windspeed at standing deer height) (m/s) 'Uhm=mean of Uh (windspeed at mean tree height) (m/s) 'Uw=windspeed at a weather station (m/s) 'Zs=slope DIM Sg(l TO 24) 'hourly global solar radiation W/m2 DIM Sdr(l TO 24) 'hourly direct solar irradiance DIM Sdf(l TO 24) 'hourly diffuse solar irradiance DIM SLW(1 TO 24) 'hourly downward sky longwave irradiance " DIM Q(l TO 24) 'hourly Quantum flux density E/m2 DIM Ta(l TO 24) 'hourly air temperature C DIM RH(1 TO 24) 'hourly relative air humidity % DIM Sdfu(l TO 24) 'hourly below-overstory diffuse irradiance W/m2 DIM Sdru(l TO 24) 'hourly below-overstory direct irradiance " DIM Rdu(l TO 24) 'hourly below-overstory downward longwave radiation " DIM Uwh(l TO 24) 'hourly mean windspeed at a weather station m/s DIM Uh(l TO 24) 'hourly mean windspeed at the average tree height " DIM Uds(l TO 24) 'hourly mean windspeed at the standing deer height 0.8m " DIM Udl(l TO 24) 'hourly mean windspeed at the lying deer height 0.15m " DIM Ra(l TO 24) 'hourly averages of deer boundary layer resistance at standing pos. DIM Ral(l TO 24) 'hourly averages of deer boundary layer resistance at lying pos. DIM Rhb(l TO 24) 'hourly averages of deer coat resistance DIM Te(l TO 24) 'hourly averages of deer environmental temperature DIM Tel(l TO 24) 'hourly averages of deer envitonmental temperature at lying position DIM Tes(l TO 24) 'hourly averages of deer standard environmental temperature at standing position DIM Tesl(l TO 24) 'hourly averages of deer standard environmental temperature at lying position DIM Qabs(l TO 24) 'hourly total radiative energy absorbed by the deer at standing position DIM Qabsl(l TO 24) 'hourly total radiative energy absorbed by the deer at lying position DIM Ms(l TO 24) 'hourly averages of deer metabolic heat production 'in standing position DIM Ml(l TO 24) 'hourly averages of deer metabolic heat production 'in lying position DIM IAS(1 TO 24) 'hourly solar incident angle on slope DIM Za(l TO 24) 'hourly solar zenith angle  '* '*  * BEGINNING OF SIMULATION MODEL  '* 'open input and output files OPEN "I", 01, "WOSSDAT" OPEN "A", #2, "DEERVAR" ' Initialize some variables for Woss old growth site (Norman Rd.) longitude = 126.5 latitude = 50.17 lai = 0 Zs = 15: Asl = 180 H = 0 SD = 0  *  *  Appendix  G. Model  Code  BEGIN.OF.MODEL: CLS Top.of.Input.Block: 'read in data from input files INPUT #1, day%, month%, Tmax, Tmin, Uw, Stotal, Au, SD CLS Zs = Z s * p i / 180 Asl = (Asl + 0 ) * p i / 180 SVF = sky.view.factor(lai) Phi = latitude * pi / 180 Au = Au * pi / 180 'initialize some variables for calculation of means Tern = 0!: Telm = 0!: Tesm = 0!: Teslm = 0!: Msm = 0!: Mlm = 0! Tam = 0!: Uhm = 0!: Udsm = 0!: Udlm = 0! 'determine sunrise and sunset times dn% = daynumber%(day%, month %) t = theta(dn%) E = eqn.of.time!(t) DH = decimal.hour!(1200) ST = solar.time!(DH, longitude, 120!, E) D = declination(t) SSHr = sunset.hour(Phi, D, ST): SSH9& = INT(SSHr + .5) SRH% = INT(2 * ST - SSHr + .5)'sun rise hour 'computing global radiation between sunrise and sunset FOR i% = SRH% TO SSH% B = Stotal * 1000000! / 2 / 3600 / (SSH% - SRH%) * pi Sg(i%) = B * COS(pi / (SSH% - SRH%) * (12 - i%)) IF Sg(i%) < 0 THEN Sg(i%) = 0 NEXT LOCATE 6, 36: PRINT "HOUR ="; FOR Time% = 1 TO 24 LOCATE 6, 42: PRINT Time% KEYCHECKS = INKEY$ IF Time% < SRH% OR Time% > SSH% THEN Sg(Time%) = 0! 'Compute hourly average air temperatures in C \% = Time* Ta(i») = (Tmax + Tmin) / 2 + .5 * (Tmax - Tmin) * SIN(pi / 12 * (20 - i%)) 'Compute hourly mean windspeed m/s Uwh(i«) = Uw + .6 * Uw * SIN(pi / 12 * (20 - i%)) Uds(i%) = Uwh(i%) IF Uds(i%) < .1 THEN Uds(i%) = .1 Udl(i%) = Uds(i%) * .2 IF Udl(i%) < .1 THEN Udl(i%) = .1 'Ul = Uw.to.Uh(Uwh(I%), 20) 'Uh(I%) = Wind.on.Slope(Ul,Zs, Asl, Au) 'CALL windspeed(Uh(I%), Au!, H!, .3, lai!, Asl!, Uds(I%), Udl(I%), SD) 'Compute hourly direct irradiance on slope E = eqn.of.time!(t) DH = decimal.hour!(i% * 100) ST = solar.time!(DH, longitude, 120!, E) D = declination!(t) Za(i%) = Zenith!(ST, D, Phi) A = Azimuth!(ST, D, Za(i%)) CALL incidence.on.slope(A,Za(i%), Asl, Zs, IAS(i%)) IF IASCt%) < 0! THEN IAS(i«) = pi / 2 - .01 'Compute hourly radiative components below the overstory of the stand IFZa(i%) > 0THEN CSdr = clear.sky.direct(ZaG%)) CSdf = clear.slcy.diffuse(Za(i%)) SgO = extra .global(Za(i%)) SdfCi%) = cloudy .diffuse(Za(i%),Sg(i%)) SdrO%) = SgCi%) - Sdf(i%) ELSE Sdfi3%) = 0! Sdr(i%) = 0! END IF SLW(i%) = sky.longwave(Phi- D, TaO%), Sg(12)) G = G.function(IAS(i%)) SdruCi%) = below.overstory.direct(Sdr(i%),IAS(i%), lai) Sdfufi%) = below.overstory.diffuse(SdrCi%),SdfCi%), lai)  Appendix  G. Model  Code  Rdu(j%) = below.overstory.longwave(SLW(i%),lai, Ta(i%)) 'Compute hourly values of energy balance variables for standing deer Ra(i%) = Ra.deer.standing(Uds(i%)) Qabs(i56) = Qabs.deer.standing(Sdru(i%),Sdfu(i%), Rdu(i96), Ta(i%), Zafi%), SD) Rr = R.radiative(Ta(i%)) Rel = Re(Rr, Ra(i%)) Te(i%) = Te.deer.standing(Ta(i%),Rel, Qabs(i%)) Rhb(i%) = whole.body.resistance.wind(Te(i%),Uds(i%)) Rhbs = whole.body.resistance(Te(i%)) Ras = Ra.deer.standing(.l) Res = Re(Rr, Ras) Tes(i%) = Tes.deer.standing(Te(i%),Res, Rhbs, Rel, Rhb(i%)) Ms(i%) = Metabolic.rest(Te(i%), Tes(i%), Rhbs, Res) 'Compute hourly values of energy balance variables for lying deer RalCi*) = Ra.deer.standing(Udl(i%)) Qabsl(i%) = Qabs.deer.lying(Sdru(i%),Sdfu(i«), Rdu(i«), Ta(i%), ZaCi%), SD) Rr = R.radiative(Ta(i%)) Rel = Re(Rr, Ral(i%)) Tel(i5&) = Te.deer.lying(Ta(i%),Rel, Qabsl(i%)) Rhbl = whole.body.resistance.wind(Tel(i%),Udl(i%)) Rhbs = whole.body .resistance(Tel(i %)) Ras = Ra.deer.standing(.l) Res = Re(Rr, Ras) Tesl(i%) = Tes.deer.standing(Tel(i%),Res, Rhbs, Rel, Rhbl) Ml(i%) = Metabolic.lying(Tel(i%),Tesl(i%), Rhbs, Res, Ta(i%), SD) 'Compute daily averages of wind speed, deer metabolic rate and operative 'temperature Tem = Tern + Te(i%): Telm = Telm + Tel(i%) Tesm = Tesm + Tes(i%): Teslm = Teslm + Tesl(i%) Msm = Msm + Ms(i%): Mlm = Mlm + Ml(i%) Tam = Tam + Ta(i%) Uhm = Uhm + Uh(i%): Udsm = Udsm + Uds(i%) Udlm = Udlm + Udl(i%) END.OF.MODEL: NEXT Time*: Time% = Time% - 1 Tem = Tem / 24: Telm = Telm / 24 Tesm = Tesm / 24: Teslm = Teslm / 24 Msm = Msm / 24: Mlm = Mlm / 24 Tam = Tam / 24 Uhm = Uhm / 24: Udsm = Udsm / 24 Udlm = Udlm / 24 'Write output to a file called DEERVAR PRINT »2, USING "###.##"; day%; month%; Tam; Uhm; Udsm; Udlm; Tem; Tesm; Msm; Telm; Teslm; Mlm CLS LOCATE 8 1* PRINT "SUMMARY OF THE DAILY MEAN VALUES" LOCATE 10, 1: PRINT USING "Daily mean air temperature (C): ###.##"; Tam LOCATE 11,1: PRINT USING "Daily mean windspeed at tree height (m/s): ##.##"; Uhm LOCATE 12, 1: PRINT USING "Daily mean windspeed at standing deer height (m/s): ##.##"; Udsm LOCATE 13, 1: PRINT USING "Daily mean windspeed at lying deer height (m/s): HUM"; Udlm LOCATE 14, 1: PRINT USING "Daily mean Te for standing deer (C): ###.##"; Tem LOCATE 15, 1: PRINT USING "Daily mean Tes for standing deer (C): ###.##"; Tesm LOCATE 16,1: PRINT USING "Daily mean metabolic rate of standing deer (W/m2): ###.##"; Msm LOCATE 17, 1: PRINT USING "Daily mean Te for lying-down deer (C): ###.##"; Telm LOCATE 18,1: PRINT USING "Daily mean Tes for lying-down deer (C): ###.##"; Teslm LOCATE 19, 1: PRINT USING "Daily mean metabolic rate of lying-down deer (W/m2): ###.##"; Mlm PRINT : PRINT GOTO Top.of.Input.Block END , « » * » * * * « » « * * * * » * « * * * « * * J ^ N C T J O N S y^jrj SUBROUTINES ************************* FUNCTION Azimuth (ST, D, Z) STATIC'solar azimuth angle 'ST:solar.time, D:declination, Z:zenith angle A = COS(D) * SIN((ST - 12) • pi / 12) / SIN(Z) IF ABS(ABS(A) - 1) < .001 THEN A = ABS(A) - .001 Azimuth = ATN(A / SQR(1 - A * A)) + pi END FUNCTION FUNCTION below.overstory.diffuse (Sdrl, Sdfl, lai) STATIC IF Sdfl < .5 THEN below.overstory .diffuse = 0! GOTO label 1  Appendix  G. Model  Code  END IF Gd = .78 - .084 * Sdrl / Sdfl below.overstory .diffuse = Sdfl * EXP(-Gd * lai * .7) label!: END FUNCTION FUNCTION below.overstory.direct (Sdrl, IA, lai) STATIC G = G.function(IA) below.overstory.direct = Sdrl * EXP(-G * lai * .7 / COS(IA)) IF IA > = pi / 2 - .02 THEN below.overstory.direct = 0! END FUNCTION FUNCTION below.overstory.longwave (SLW, lai, Tal) STATIC SVF = sky.view.factor(lai) below.overstory .longwave = SLW * SVF + sigma * powerfTal + 273, 4) * (1 - SVF) END FUNCTION FUNCTION clear.sky.diffuse (Z) STATIC cosZ = COS(Z) IF COS(Z) < .01 THEN cosZ = .01 clear.sky.diffuse = 1360 * COS(Z) * (.271 - .294 * power(.7, 1! / cosZ)) END FUNCTION FUNCTION clear.sky.direct (Z) STATIC cosZ = COS(Z) IF cosZ < .01 THEN cosZ = .01 clear.sky .direct = 1360 * COS(Z) * power(.7, 1! / COS(Z)) END FUNCTION FUNCTION cloudy .diffuse (Zl, Sgl) STATIC Tl = Sgl / extra.global(Zl) IFT1 < = .22 THEN T2 = 1!- .09*T1 ELSEIF Tl K. 8 THEN T2 = .95 - .1604 * Tl + 4.388 * Tl * Tl - 16.638 * Tl * Tl * Tl + 12.336 * Tl * Tl * Tl * Tl ELSE T2 = .165 END IF 'PRINT T l , T2, Sgl, extra.global(Zl) cloudy .diffuse = T2 * Sgl END FUNCTION FUNCTION daynumber% (day%, month%) STATIC DIM montha%(0TO 12) 'number of days in a month dn% = 0: montha%(0) = 0 montha%(l) = 31: montha%(2) = 28: montha%(3) = 31: montha%(4) = 30: montha%(5) = 31 montha%(6) = 30: montha%(7) = 31: montha%(8) = 31: montha%(9) = 30 montha%(10) = 31: montha%(ll) = 30: montha%(12) = 31 F O R i » = 0TOmonth%- 1 dn% = dn% + montha%(i%) NEXT daynumber% = dn% + day% END FUNCTION FUNCTION decimal.hour (PST%) STATIC'converts HrMin to decimal hour decimal.hour = INT(PST% / 100) + (PST% / 100 - INT(PST% / 100)) * 10 / 6 END FUNCTION FUNCTION declination (thetap) STATIC'solar declination at noon t = thetap dl = .006918 - .399912 * COS(t) + .070257 • SIN© d2 = -.006758 • COS(2 * t) + .00907 * SIN(2 * t) - 0 d3 = -.002697 * COS(3 * t) + .00148 * SIN(3 * t) declination = dl + d2 + d3 END FUNCTION FUNCTION deer .body .temperature (Tel) STATIC deer.body.temperature = 37.95 + 6! / (1 + EXP(-.103 * (Tel - 53.65))) END FUNCTION FUNCTION eqn.of.time (thetap) STATIC t = thetap eq = .000075 + .001868 * COS(t) - .032077 * SIN(t) - .014615 * COS(2 * t) - .040849 * SIN(2 * t) eqn.of.time = eq * 24 / 2 / 3.14159 END FUNCTION FUNCTION extra .global (Zl) STATIC extra.global = 1380 * COS(Zl) END FUNCTION  Appendix  G. Model  Code  FUNCTION G.function (IA) STATIC G.function = .5 IF IA < .8 THEN G.fiinction = (2.5 + 1.5 * IA) / 4.6 ELSE G.fiinction = (3.7 - 4.81 * (IA - .8)) / 4.6 END IF ' END FUNCTION SUB incidence.on.slope (A, Z, Asl, Zs, incident.angle) STATIC'A:solar azimuth angle 'Z: solar zenith 'Asl: slope azimuth 'Zs: slope zenith B = SIN(Z) * SIN(Zs) * COS(A) * COS(Asl) + SIN(Z) * SIN(Zs) * SIN(A) * SIN(Asl) + COS(Zs) * COS(Z) i.a = ATN(SQR(1 - B * B) / B) IF B < = 0 THEN incident.angle = pi / 2 - .01 ELSE incident.angle = i.a END IF END SUB FUNCTION Metabohc.lying (Tell, Tesll, Rhbs, Res, Tal, SD) Tb = deer.body.temperature(Tell) Mil = 1200 * (Tb - Tesll) / (Rhbs + Res) 'assuming floor temperature Tg = 0 on snow 'assuming coat resistance= 800 s/m Tg = Tal IF SD > .3 THEN Tg = 0 conduction = 1200 * (Tb - (Tb + Tg) / 2) / 800 Metabolic.lying = 1.2 * (.7 * Mil + .3 * conduction) END FUNCTION FUNCTION Metabolic.rest (Tel, Tesl, Rhbs, Res) STATIC Tb = deer.body.temperature(Tel) Metabolic.rest = 1200 * 1.2 * (Tb - Tesl) / (Rhbs + Res) END FUNCTION FUNCTION power (x, Y) power = EXP(Y * LOG(x)) END FUNCTION FUNCTION Qabs.deer.lying (Sdrul, Sdful, Rdul, Tal, Zal, SD) STATIC IF COS(Zal) < .01 THEN Z = 1.52'avoid overflow of l/cos(Z) bellow 'assuming Ap/A=0.2 for cylinder-sphere ends, G.S. Campbell shortwave = .2 / .7 * Sdrul / COS(Zal) + (.5 * 3 / 4 + .2 * 1.5 / 4) / .7 * Sdful reflectivity = .2 IF SD > .3 THEN reflectivity = .5 shortwave.reflected = reflectivity * (.5 / 4 + .2 * 2.5 / 4) / .7 * (Sdrul + Sdful) 'estimate forest floor temperature Tg Tg = Tal + (Sdrul + Sdrfl) * .01 longwave = (Rdul * (.5 * 3 / 4 + .2 * 1.5 / 4) / .7 + (.5 / 4 + .2 * 2.5 / 4) / .7 * sigma * power(273 + Tg, 4)) 'assuming 30% of the deer surface area is in contact with the floor when lying Qabs.deer.lying = .8 * (shortwave + shortwave.reflected) + etad * longwave END FUNCTION FUNCTION Qabs.deer.standing (Sdrul, Sdful, Rdul, Tal, Z, SD) STATIC IF COS(Z) < .01 THEN Z = 1.52'avoid overflow of l/cos(Z) bellow 'for cylinder-sphere ends Ap/A=0.3 approx., see Pp80 G.S. Campbell shortwave = .2 * Sdrul / COS(Z) + .5 * Sdful reflectivity = .2 IF SD > .3 THEN reflectivity = .5 shortwave.reflected = reflectivity * .5 * (Sdrul + Sdful) 'estimate forest floor temperature Tg Tg = Tal + (Sdrul + Sdrfl) * .01 longwave = .5 * (Rdul + sigma * power(273 + Tg, 4)) '80% absorptivity of deer surface to shortwave, etad=0.95 longwave emissivity Qabs.deer.standing = .8 * (shortwave + shortwave.reflected) + etad * longwave END FUNCTION FUNCTION R.radiative (Tal) STATIC R.radiative = 1200 / 4 / sigma / power(273 + Tal, 3) END FUNCTION FUNCTION Ra.deer.standing (U) STATIC A = 1.3 'enhancement due to turulence gb = (A * 7.183 * U * .61) / 1000 'boundary layer conductance for deer 'in cross flow (m/s) Ra.deer.standing = 1 / gb END FUNCTION FUNCTION Re (Rr, Ral)  'environmental resistance  Appendix G. Model Code Re = Rr * Ral / (Rr + Ral) END FUNCTION FUNCTION Res.heat.loss (Tal, RH1) STATIC Res.heat.loss = 1 END FUNCTION FUNCTION sky .longwave (Z, Tal, Sgl) STATIC SgO = clear.sky.direct(Z) + clear.sky.diffuse(Z) Ratio = Sgl / SgO si = (1.22 * sigma * powerfTal + 273, 4) - 171) * Ratio + (1 - Ratio) * sigma * power IF si < 200 THEN sky .longwave = 200 ELSE sky .longwave = si END IF END FUNCTION FUNCTION sky.view.factor (lai) STATIC 'below a forest stand of LAI N% = 20 LT = 0! FORJ% = 1 TON% - 1 Alpha = J% * pi / 2 / N « G = G.ftinction(Alpha) LT = LT + SIN(Alpha) * COS(Alpha) * EXP(-G * lai * .7 / COS(Alpha)) NEXT sky.view.factor = 2 * LT * pi / 2 / N% END FUNCTION FUNCTION solar.time (DH, longitude, longref, E) STATIC solar.time = DH + (longitude - longref) / 15 + E END FUNCTION FUNCTION sunset.hour (Phi, D, ST) STATIC SST = -TAN(Phi) * TAN(D) SS = ATN(SQR(1 - SST * SST) / SST) IF SST < 0 THEN SS = pi - ABS(SS) sunset.hour = ST + 12 * SS / pi END FUNCTION FUNCTION Te.deer.lying (Tal, Rel, Qabsll) STATIC 'deer operative environmental temperature at lying-down position '70% of the deer surface have longwave radiative exchange with the environment IF Tal > 200 THEN Tal = Tal - 273 Te.deer.lying = Tal + Rel * (Qabsll - sigma * etad * power(273 + Tal, 4)) / 1200 END FUNCTION FUNCTION Te.deer.standing (Tal, Rel, Qabs) STATIC 'deer operative environmental temperature at standing position Te.deer.standing = Tal + Rel * (Qabs - sigma * etad * power(273 + Tal, 4)) / 1200 END FUNCTION FUNCTIONTes.deer.standing(Tel, Resl, Rhbsl, Rel, Rhbl) STATIC 'standard deer operative environment temperature in standing position Tb = deer.body.temperaturefTel) 'deer body temperature Tes.deer.standing = Tb - (Tb - Tel) * (Rhbsl + Resl) / (Rhbl + Rel) END FUNCTION FUNCTION theta (dn%) STATIC theta = .0172 * dn% END FUNCTION FUNCTION Uw.to.Uh (Uwhl, H) STATIC 'transform of windspeed at a weather station to that at the average 'forest stand height 'setting up roughness of the forest stand ZOf and 'roughness of the weather station ZOw ZOf = .1 * H ZOw = .1  ZOratio = ZOf/ZOw D = .6 * H Uw.to.Uh = Uwhl * power(Z0ratio, .07) • LOG((H - D) / Z0Q / LOG(10 / ZOw) END FUNCTION FUNCTION whole.body.resistance (Tel) STATIC Rhtl = 63.95 - 5.58 * Tel Rhcsl = 408 Rhbsl = Rhcsl + Rhtl whole.body.resistance = Rhbsl END FUNCTION  Appendix  G. Model  Code  FUNCTION whole.body.resistance.wind(Tel, U) STATIC 'Rhbsl = 857.9 - 29.9 * Tel - .19 * Tel * Tel + .006 * Tel * Tel * Tel 'IF Tel > 0! THEN Rhbsl = 857.9 'Rhbsl = 1000 Rhtl = 63.95 - 5.58 * Tel gc = (2.45'+ .006 * U A 2.11) / 1000 'coat conductance (m/s) Rhcl = 1 / gc Rhcsl = 408 Rhbsl = Rhcsl + Rhtl whole.body.resistance.wind = Rhtl + Rhcl END FUNCTION FUNCTION Wind.on.Slope (Uf, Zs, Asl, Au) STATIC 'Uf windspeed on a flat terrian 'Zs = slope 'Asl = slope arrientation 'Au=wind direction A = -COS(Asl - Au) Aw = A * SIN(Zs) / COS(Zs) SELECT CASE Aw CASE IS < -.09 Omega = 1.6 CASE -.09 TO 0 Omega = 1 - 6.7 * Aw CASE 0 TO .09 Omega = 1 - 5.5 * Aw CASE IS > .091 Omega = .5 CASE ELSE Omega = I END SELECT Wind.on.Slope = Uf * Omega END FUNCTION SUB windspeed (Uhl, Au, H, Hus, lai, Asl, Udsl, Udll, SD) STATIC 'Urf windspeed at reference height 0.2H 'setting up parameters Alpha = .4 * lai 'attenuation coefficient ZOu = .1 * Hus du = .7 * Hus IF SD > .3 THEN 'SD:Snow depth ZOu = .02 du = 0! END IF Urf = Uhl * EXP(-Alpha * (1! - .2)) Zrf = .2 * H Zds = .8 'standing deer height Zdl = .15 'lying deer height Udsl = Urf * LOG((Zds - du) / ZOu) / LOG((Zrf - du) / ZOu) IF Zdl < = du THEN Udll = .2 * Udsl GOTO label2 END IF Udll = Urf * LOG((Zdl - du) / ZOu) / LOG((Zrf - du) / ZOu) label2: END SUB FUNCTION Zenith (ST, D, Phi) STATIC 'solar zenith angle on horizontal plane 'H:solar.time, D.declination, Phi:latitude Zl = SIN(Phi) * SIN(D) + COS(Phi) * COS(D) • COS((ST - 12) * pi / 12) Zenith = ATN(SQR(1 - Zl * Zl) / Zl) END FUNCTION  Appendix H Statistical A n a l y s i s of Linear a n d N o n - l i n e a r R e g r e s s i o n s  This appendix presents statistical analysis of various linear and non-linear regressions which were associated with figures shown in this thesis. In Table H . l , standard errors of the regression, Syx and those of the regression coefficients, Sa and Sb, are reported for linear regressions. In addition, hypothesis testing was carried out at the 0.05 significance level on the pairs of lines listed for each figure to determine if differences between the lines were statistically significant. Calculations were made in a spreadsheet and the statistical theory is given in Zar (1984). In Table H.2, standard errors are reported for the regression coefficients of non-linear regressions. The Systat software package (Systat Inc., Evanston, Illinois, V. 5.01) was used to calculate the regression coefficients and standard errors.  Literature Cited Zar, J.H. (1984) Biostatistical  Analysis,  3rd ed., Prentice-Hall, Englewood Cliffs, N.J.,  718 pp.  154  Appendix  H.  Statistics  155  Table H.l: Statistics for linear regressions of log y against log a; (logy = a + b\ogx). The original functions were of the form y = Axb, a — log A. Hypothesis testing was carried out at the 0.05 significance level. Fig. # 2.5 2.5 2.6 2.6 2.7 2.7 2.10 2.10 2.11 2.11 2.12 2.12 2.13 2.13 2.14 2.14 2.15 2.15 1 2 3 4  y Var./ Comment Nu#/ l p.w. Nu<?/ t.s. Nu#/ p.w. Nu,?/ t.s. Nu#/ p.w. Nu#/ t.w. Nufl/side Nutf/rear N u / 2 c.f. N u / l.f. N u / c.f. N u / l.f. Nuffield Nu<?/ 3 w.t. Nuffield Nu#/ w.t. Nuffield Nu#/ w.t.  Reg. Coeff. a b -0.137 0.531 -0.045 0.500 -1.471 0.730 -0.733 0.606 -0.735 0.651 -0.907 0.689 -0.845 0.679 -0.431 0.466 -0.647 0.613 -0.809 0.692 0.856 0.610 0.891 0.691 0.167 0.462 -0.045 0.500 0.188 0.437 -0.733 0.606 0.268 0.411 -0.907 0.689  Standard Errors Sb Oyx Sa 0.021 0.045 0.0093 0.0089 0.059 0.013 0.12 0.27 0.055 0.011 0.072 0.016 0.041 0.089 0.20 0.0076 0.050 0.011 0.064 0.016 0.013 0.016 0.017 0.022 0.014 0.090 0.020 0.027 0.20 0.039 0.014 0.0090 0.020 0.017 0.027 0.039 0.094 0.029 0.024 0.0089 0.059 0.013 0.237 0.061 0.073 0.072 0.011 0.016 0.023 0.13 0.032 0.0076 0.050 0.011  p.w. - previous workers; t.s. - this study c.f. - cross flow; l.f. - longitudinal flow w.t. - wind tunnel diff. - different  R2 0.995 0.999 0.916 0.999 0.948 0.999 0.997 0.995 0.998 0.994 0.998 0.994 0.888 0.999 0.526 0.999 0.837 0.999  Hypoth. Test Slope Elev. same Miff. same  diff.  same  same  diff.  diff.  same  diff.  same  diff.  same  diff.  same  diff.  diff.  same  Appendix  H.  156  Statistics  Table H.2: Statistics for non-linear regressions of the form y — constant  Fig.  y Var.  1  constant  #  3.7 9c 3.8 9con 3.8 9c 3.12 9c(0°) 3.13 9c 3.14 Tc 3.20 9b 1  0 1.93 2.40 2.48 2.27 2.45 0  Reg. a 2.955 0.688 0.655 0.036 0.015 0.006 7.009  + axb.  Coeff. Standard Errors R2 b sa sb 0.099 0.167 0.028 0.810 0.278 0.11 0.078 0.817 0.087 0.806 0.300 0.12 1.824 0.037 0.003 0.999 2.189 0.001 0.042 0.999 2.102 0.002 0.15 0.991 0.635 0.25 0.028 0.997  The constant was not an unknown in the non-linear regression. It represents a measured or assumed value of the {/-intercept.  

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