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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 WIND TUNNEL AND 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 DOCTOR 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 this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, 1 agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my wri t ten permission. (Signature) Department of foil S f j P . D f C The University of British Columbia Vancouver, Canada DE-6 (2/88) 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 Contents Abstract ii List of Tables viii List of Figures x List of Symbols x ix Acknowledgements xx iv 1 Introduct ion 1 1.1 Literature Cited 5 2 Boundary Layer Conductance 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 2.4 Conclusions 36 2.5 Literature Cited 37 3 Coat Conductance 40 3.1 Introduction 40 3.2 Theory 42 3.2.1 Mechanisms of Heat Transport Through Animal Coats 42 3.2.2 Models of Heat Transfer Through the Coat 46 3.3 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 3.4 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 80 3.4.3 Comparison of Boundary Layer and Coat Conductance 86 3.5 Conclusions 89 3.6 Literature Cited 90 4 Thermal Radiat ion 93 4.1 Introduction 93 4.2 Theory 94 v 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 4.3 Results and Discussion 105 4.3.1 The Effect of Piloerection on Heat Transfer through a Deer Coat . 105 4.4 Conclusions 110 4.5 Literature Cited I l l 5 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) 120 5.2.2.4 Comparison of Deer Heat Loss in Forested and Open Habitats 122 5.3 Conclusions 126 5.4 Literature Cited 128 6 Conc lu s ions 130 A p p e n d i c e s 132 A N e t R a d i a t i o n 132 vi B Time Constants 135 C IRT Calibration 136 D Turbulent Spectra 139 E Equivalence of Equations. 4.9 and 4.10 141 F Numerical Integrat ion 143 G Model Code 145 H Statistics 154 V I I 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 44 3.2 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) 52 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 58 3.4 Comparison of mean coat conductances (<jFc) for positions 2 and 3 for the longitudinal and cross flow cases 81 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 134 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 135 C.l Comparisons of surface temperatures measured with an Everest Inter-science, 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 4.4 144 H.l Statistics for linear regressions of logy against log a; (logy = a + blogx). The original functions were of the form y = Axh, a = log A. Hypothesis testing was carried out at the 0.05 significance level 155 H.2 Statistics for non-linear regressions of the form y = constant + axb. . . . 156 IX List of Figures 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 20 5 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 21 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 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 23 2.8 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 25 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 27 2.10 Local Nusselt number Nux 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 Nuj;=0.371Re0460. 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 (lon-gitudinal 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 35 3.1 Alternative resistance models for heat transfer through animal coats pro-posed 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 pen-etration depth, t 47 3.2 Diagram showing how thermocouples were installed to measure skin sur-face temperature 50 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 57 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 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 60 3.5 Orientation of the model deer to receive cross flow winds at the Browns River Site and typical diurnal wind directions 62 xn 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 63 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-linear regression line. Regression statistics are shown in Appendix H. 64 8 Coat conductance and its components as a function of ATC for position 2-90 (top) at a wind speed of Orns - 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 66 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, con-vection 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. . . . 67 xiii 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 func-tion 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 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 func-tion of wind speed. Solid line is a non-linear regression line. Regression statistics are shown in Appendix H 77 xiv 3.15 (a) Coat conductance on the top of the coat covered model deer in lon-gitudinal 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 (Wm~ 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 out-doors 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 88 4.1 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 95 4.2 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 97 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 100 4.4 Illustration of the method for determining kr of a dry coat with a uniform temperature gradient across the coat 103 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. . 106 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 2-90 108 xvi 7 Simulation of the effect of piloerection on the components of coat conduc-tance 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 con-ductance, (b) The numerical integration method was used to calculate the radiative conductance 109 1 Mean boundary layer conductance (gb) as a function of wind speed deter-mined 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) 115 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 T e =0°C ( l ineb) 118 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 121 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 (Ta) 123 5 Comparison of the metabolic rate (M) calculated using the equations de-veloped in this study (line a) and that calculated using Parker and Gilling-ham'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 PST and (b) the old growth stand on 9 August 1989 between 1315 and 1415 PST 140 xviii List of Symbols A surface area of model deer ( m - 2 ) Ai area of given region on model deer (m2) Ap projected area of hair onto plane perpendicular to n (m2) a temperature gradient in layer of coat ( "Cm" 1 ) B blackbody radiation ( W m - 2 ) b slope of blackbody radiation curve (4crT3, 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 mW 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 (m2 s - 2 ) p radiation interception function (cm - 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 (cm - 1 ) p± probability of interception for radiation which is traveling perpendicular to skin surface (cm - 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 re 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 (Wm~ 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 (m2 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 10 - 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 A c k n o w l e d g e m e n t s 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 com-ments 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 Introduct ion 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 second-growth 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 second-growth 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 ram 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 in-volved 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, Te, 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 that 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. Introduction 4 is a particularly relevant example, since it deals with thermally critical environments for mule deer (Odocoileus hemionus hemionus), of which black-tailed deer (Odocoileus hemionus 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 envi-ronment 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 calcu-lated coat resistance using a general relationship recommended in Campbell et al. (1980) as being good for many different animal coats. The overall objective of this dissertation is to develop and test heat transfer relation-ships 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. Introduction 5 1.1 Literature Cited 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. 21 , 1020-1028. 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. 57 , 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. 182, 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. Par 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. Par 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 Douglas-fir stand on a south-facing slope. For. Sci. 39, 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. 209, 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. 209, 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. 209, 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. Introduction 7 Moote, I. (1955) The thermal insulation of caribou pelts. Text. Res. J. 25 , 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, Victo-ria, 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. 43 , 73-81. Porter, W.P. and D.M. Gates (1969) Thermodynamic equilibria of animals with envi-ronment. Ecol. Mongr. 39, 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 regula-tion in some arctic and tropical mammals and birds. Biol. Bull. 99 , 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 Conductance of a M ode l Deer in a W i n d Tunnel and Douglas-fir Stands 2.1 Introduct ion The overall objective of this chapter was to determine the sensible heat transfer re-lationships for the laminar boundary layer of black-tailed deer {Odocoileus hemionus columbianus) in old-growth and managed second-growth Douglas-fir stands. Sensible 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 (Camp-bell, 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 lam-inar 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 cylin-der 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 Exper imenta l M e t h o d s 2.2.1 M o d e l Deer Des ign and Construct ion 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 polysty-rene 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 m2 . 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 Power Supply + 12/24 Volts {heating wire nichrome (18 AWG) density 2 winds/cm 0.80 m • I \ l l \ I l i m ' m0.32 m wooden legs -5 cm —>i0.25mi<«— trunk cross section Figure 2.1: Schematic of the model deer. 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 con-nected 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 Longitudinal Positions Wind Direction 270 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 = aRe6 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 (2.5) 2.2.3 Exper imenta l Sites and Instrumentat ion 2.2.3.1 W i n d Tunnel Wind tunnel experiments were conducted in a large open-ended wind tunnel of the blow-down type, (see Iqbal et al. 1977) belonging to the Department of Mechanical Engineer-ing, 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 sec-tion. 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 Thornth-waite sensitive cup anemometers (C.W. Thornthwaite Associates, Centerton, NJ, Model 901-LED) and welded fine wire 13 [im diameter, chromel-constantan thermocouples, re-spectively 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 Inter-science 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 In-terscience Inc., Model 4000) . Data was recorded using a data 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 .6ms" 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 CZZ5=> A A >.. wind direction Cross Flow A KJ A 4 A wind direction 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 .4ms" 1 and 8.5 °C for u = 0 . 6ms" 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 se-lected 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), re-spectively. The hand held IRT and net radiometer were mounted on tripods and po-sitioned 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 pe-riod (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 Layer Conductance 19 2.3 Resul ts and Discuss ion 2.3.1 Laminar Flow 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 longi-tudinal 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 bound-ary 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 stag-nation, 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 Layer Conductance 20 8 4 -2 -0 1 1 u = 1.5 m/s / \J ~~\J u = 5.4 m / s ^ ^ U— u 1 1 W b l n ( Direction \ 1 90 f > 2TO 1 + 180 1 — 1 0 90 180 Angular Position (degrees) 270 Figure 2.4: Temperature difference (Ts — Ta) as a function of angular position (at longi-tudinal 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 CD 3 100-10 NUQ = 0.729Re Stagnation 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 refer-enced 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 aforemen-tioned 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 Conductance 23 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 xlO4 , 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 data 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 Chapter 2. Boundary Layer Conductance 25 30 S 20 10 0 90 0 u = 5.4 m/s u = 2.7 m/s u = 1.5 m/s u = 0.6 m/s 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 data 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 Nu 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 Nux = 0.453Pra33Re°-5 (2.7) Nu* = 0.045Pr°-33Re°-84 (2.8) where Pr is the Prandtl number (Pr0-33 = 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 jump 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 Layer Conductance 27 C/3 O O a o U 5-1 ;? -a § o PQ 35 30 25 20 15 10 5 0 Wind Direction 4 0 5 u = 5.4 m/s u = 1.5 m/s 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. Chapter 2. Boundary Layer Conductance 28 1000 100 X £ 10 Nu x = 0.143Re side data rear end data 10' 10 10' 10 Re. 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=0.371Re0460. 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°613 (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°684 (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 Conductance 30 1000 Longitudinal Flow Nu = 0.155Re 10 Cross Flow Nu = 0.226Re 0.613 10 10 Re 10 Figure 2.11: Comparison of Nu vs. Re relationships for bare model deer in longitudi-nal and cross flow orientations in the wind tunnel. Regression statistics are shown in Appendix H. Chapter 2. Boundary Layer Conductance 31 24 20 h Longitudinal Flow 0.691 Cross Flow gb = 7.183u 0.610 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 Conductance 32 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 low Figures 2.13, 2.14 and 2.15 show Nu# vs. Re for stagnation, top and lee positions, re-spectively, 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 that observed in the wind tunnel (at the 5 % significance level, see Appendix H). En-hancement where the two data sets overlapped was 75% 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 50% 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 con-firmed experimentally for a flat plate at various angles to the flow by Chen et al. (1988). Chapter 2. Boundary Layer Conductance 33 1000 CD I ioo 10 Nue = 1.47Re wind tunnel NUQ = 0.90Re" n Field Data • Wind Tunnel Data 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 Conductance 34 1000 # 1 0 0 field Nue = 1.54Reu 0.437 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 data are for position 2. Regression statistics are shown in Appendix H. Chapter 2. Boundary Layer Conductance 35 1000 05 I 100 Nue=1.852Re 10 Re a Field Data • Wind Tunnel 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 res-onance 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 Layer Conductance 37 2.5 Literature Cited Bosch, M.T. (1936) Die Wdrmeubertragung. 3rd ed. Springer Verlag, Berlin. Campbell, G.S. (1977) An Introduction to Environmental Biophysics. Springer-Verlag, New York, 159 pp. Chauvin, L.T. and C.A. de Moraes (1957) Correlation of supersonic convective heat trans-fer coefficients from measurements of skin temperature of a parabolic body of rev-olution. 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 45 , 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 wind-breaks. 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 of Heat Transfer, 3rd ed. Harper and Row, New York, 656 pp. Chapter 2. Boundary Layer Conductance 38 Lee, X. and T.A. Black (1993) Turbulence near the forest floor of an old growth Douglas-fir 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. 209, 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. 66 , 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 Coat Conductance of a M ode l D e e r in a W i n d Tunnel and a Douglas-fir Stand 3.1 Introduct ion 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 at tempt 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) at tempted 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 im-portance 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, gc) 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 u0 '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 Conductance 42 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 Mechanisms of Heat Transport Through Animal Coats Monteith and Unsworth (1990) list the three mechanisms which contribute to heat trans-port through animal coats as conduction, radiation and convection. The relative impor-tance 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 Conductance 43 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, 4crT3, (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 Chapter 3. Coat Conductance 44 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 Conductance 45 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 expres-sion 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 convec-tion 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 Conductance 46 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 - Ts)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 ATC. 3.2.2 Models of Heat Transfer Through the 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 in-cludes 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 gradi-ent, while free convection dominates in the lower layer near the skin surface. McArthur and Monteith show that t is given by t = —I (3.8) 9c Chapter 3. Coat Conductance Skin Surface molecular conduction v ^ W ^ / V ^ radiation W N A / v W v Model One forced convection W v V W V V V " free convection Coat Surface molecular conduction / W V W A Model! Two W V W v W convection forced • 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 Conductance 48 where c is the empirical constant from eq. 3.6. Models 1 and 2 respectively can be expressed mathematically as follows gc = gm+9r + 9fc + 9}r (3.9) gc = 9m+gr + 7 7 — 7 - 7 - 7 — (3.10) where gm is the molecular conductance, gr is the radiative conductance, and gfc the forced convective conductance. 3.3 Exper imenta l M e t h o d s 3.3.1 Coat Covered Mode l 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 that 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 Conductance 50 rzzzzzzzzzzzzzzzzzzzzzzzz to data logger T glue \ glue J 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 cross-sections 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 = — — (3.11) pcP{Tsk - Ts) 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(//Zs)) 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) -69.7 68.9 -76.9 74.5 -70.1 68.5 ^ ( m m s " 1 ) 1.51 1.22 1.15 2.78 2.13 1.55 1.42 1.22 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 "V "'h i "'con in 1 r>\ gr = j , gh = T- and gcon = (3.13) pCpl pCpls pCpl 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 proce-dure described in Chapter 4 and Eq. 3.4. The area weighted thermal conductivities for Chapter 3. Coat Conductance 54 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 cross-sectional area of hair that is solid and k0 is the thermal conductivity of organic matter (250mWm - 1K"" 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 Exper iments and Instrumentat ion 3.3.2.1 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 tem-peratures 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 perpen-dicular 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 Conductance 56 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 the 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 the elapsed time since the step change and ATci is the initial value of ATC. The equilibrium or final temperature difference, ATcf, and the 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 ATC, 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. This became apparent when Eq. 3.17 was used to determine Tsk or ATcf and gave unexpectedly low r values (generally less Chapter 3. Coat Conductance 57 19 15 non-linear least squares fit of form: ATc = ATcf ATci = 18.6C ATcf=15.6°C T=17.9min 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 to20Wm~ 2 Time Constant (minutes) Stagnation 33.5* 17.5 28.4 Top 17.7 Lee 30.8 *ATC was used instead of Tsk ** At 3 m s " 1 Chapter 3. Coat Conductance 59 than 10 minutes) resulting in an overestimate or underestimate of ATcf. 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. The 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 da ta 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 data is good over the first 15 points, it diverges from the data 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 F ie ld S i t e 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. Coat Conductance 60 48 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 Orns - 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. 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 Resul ts and Discuss ion 3.4.1 Coat Conductance of a Deer Standing 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, ATC, 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' W 270 model deer cross flow orientation > 9 0 E typical daytime vwind direction 115° o 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. Chapter 3. Coat Conductance 63 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 Conductance 64 S 3 O 0 -S 2 G O U I 1 U 0 u = Om/s 0 gc =2.96AT, o.io 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 ATC 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), respec-tively. 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 con-ductance. 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 data 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 0 0 measured gc — -00-radiative —o-o -°f hair 8 ATC (°C) 12 Figure 3.8: Coat conductance and its components as a function of ATC for position 2-90 (top) at a wind speed of Oms - 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. Chapter 3. Coat Conductance 67 example for top of deer u = Om/s I boundary layer hair skin wires T = 2 5 C T = 31 C = 4 3 C P = 53 (100) Figure 3.9: Typical example of the magnitudes (Wm - 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 con-ductivities 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. Coat Conductance 69 1 ° 1 o > (3 o u = Om/s still air conductance 0 180 270 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. Chapter 3. Coat Conductance 70 40 30 • 1 — > o T-t o > 20 10 0 --—_ — -1 u = 0m/s I i still air i i °{ l 90 270 1 -_—m + 180-i 0 90 180 Angular Position (degrees) 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 Conductance 71 of coat conductance than any differences in the strength of free convection at the vari-ous 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 conduc-tivities 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, Rac , 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 Rac 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 (ATC) 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 Rac . 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 107, the free convective flow within the coat should be laminar (flow becomes turbulent when Gr > 109 (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. Coat Conductance 73 3.4.2 Effect of W i n d Speed and Deer Orientation on Coat Conductance 3.4.2.1 Deer in Cross Flow 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 + 0.036u183 (3.20) The exponent on u of n = 1.83, which has a small standard error of 0.003 (see Ap-pendix 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 Chapter 3. Coat Conductance 74 C/3 u 4 -o •*-> o 3 o U •4-J O 3 2 1 0 1 1 • P = 52 W/m2 n n V n > o v . 1 . 1 gc 1 - • •• 1 = 2.48 + O.C •Q > 1 ' 1 ' l 3 6 u L 8 3 ^ 0 ° " < = 3 . 2 4 a 9 Q O gc = 2.32 ^ lbO l i l . 0 2 4 6 Wind Speed (m/s) 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 CO CD O o -g a o _ P = 52 W/m measured gc U 1 0? 0 convective gCOB=1.60 + 0.01uz still air -<D--00-radiative -n> -CD--o rn hair 2 4 6 Wind Speed (m/s) 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 Conductance 76 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 ~gc were 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 Deer in Longitudinal F low 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 .3ms" 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 .50mms _ 1 , compared with gc = 4 .82mms _ 1 Chapter 3. Coat Conductance 77 00 S 3 O O •3 G O U O U a gc = 2.45 + 0.006u 2.11 0 P = 52 W/m 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 Conductance 8 78 tW O o o O C3 O U 0 2 4 6 Wind Speed (m/s) Figure 3.15: (a) Coat conductance on the top of the coat covered model deer in longitu-dinal 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 mm 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 (bold-faced 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 Comparison of Coat Conductance Measured in t h e Forest wi th T h a t Measured in the 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 mean1 90 0 180 mean1 90 0 180 mean1 90 0 180 90 u ( m s l) 5.3 5.3 5.3 1.5 1.5 1.5 0.62 0.62 0.62 0 0 0 Coat Conductance (mm s 1) Cross Flow 3.24 2.29 2.77 3.17 22.54 2.31 2.42 3.30 22.50 22.32 2.41 23.24 2.48 2.32 3.21 Longitudinal Flow rear into wind 2.71 4.05 2.37 3.67 2.41 3.63 32.48 32.32 33.21 head into wind -2.41 3.53 2.36 3.41 32.48 32.32 33.21 xMean of 0° and 180° positions, shown in boldface type. 2Value estimated from functions shown on Figure 3.12. 3Value measured during cross flow experiment. Chapter 3. Coat Conductance 82 1.2-c« 13 ^ 0.8 CO 13 I °-4 d 0 360 bo 270 w a % 180 T3 g 90 0 i—•—r T — ' — r -night u = 0.78m/s au = 0.126m/s ->(-« day 1 •Fu = 0.65m/s a.. = 0.168m/s J i L T «—r night-•* H- •day-mfffijff^^ WD = 293c ^WD = 22.4 WD=112c aWD = 29.0C a D 1400 1800 1800 2200 200 600 1000 1400 7 July 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 PST. 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 attr ibuted 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 that 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, data 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 Conductance 84 j& o i O U o U 1 -0 4 0 Position 3-90 night O H B L/EftcF g. = 2.26mm/s a „ = 0.03 mm/s inSor-n 1 r Position 2-0 day D aO DD„a ea *J^ph^ • a g. = 2.66 mm/s a„, = 0.141 mm/s 0 0.2 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) Stagnation Pos. 2 52 24 m Pos. 3 52 Pos. 3 52 24 R Lee Top Figure 3.18: Comparison of coat conductances at wind speeds ranging from 0 to 1.2 ms" 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 15% 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 Comparison of Boundary Layer and Coat Conductance 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 ( lOOWm - 2 vs. 5 0 W m - 2 ) which could enhance free convection somewhat. Based on the results presented previously in Chapter 3. Coat Conductance 87 Wind Tunnel Forest (Browns River) P = 24 W/m2 for all cases Pos. 1 h Pos. 2 r Side Pos. 1 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). Mea-surements 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 88 24 20 16 12 8 4 0 —i 1 1 1 1 1 r D coat covered model deer (18V) • bare model deer (24V) II B- - B -gb = 7.01u 0.635 gc = 2.45 + O.Olu 2.11 •a 0 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 mm 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 out-doors 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 that of the laminar boundary layer. Chapter 3. Coat Conductance 90 3.6 Literature Cited 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 Perspectives of Biophysical Ecology(Edited 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. 182, 369-376. Iqbal, M., A.K. Khatry and B. Senguin (1977) A study of the effects of multiple wind-breaks. 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 92 Raddi, A.G. (1967) The pelage of the black-tailed deer. 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 regula-tion in some arctic and tropical mammals and birds. Biol. Bull. 99 , 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 Thermal Radiat ion Through a D e e r Coat 4.1 Introduct ion 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 con-ductivity 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 at tempt 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 that is piloerected. 4.2 Theory 4.2.1 The Intercept ion 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 ( cm - 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 geomet-rical 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 vec-tor 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 ndsin/? P\P) = i = 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 radi-ation 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 98 In order to integratep(fl) 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 (4-4) A A A h = sin £ cos m + sin £ sin aj + cos £& (4-5) Using the trigonometric relationship, cos2 /3 + sin2 /? — 1, we can write sin ft = J\ - cos2 /? (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 + tan2 <j>){\ + tan2 § - (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 trigono-metric manipulations (see appendix E), Eq. 4.9 can be expressed as p{(,a) = ndcos£[(l + t a n 2 ^ ) ( l + tan 2£) - ( 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 /cos£ 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. The integral is as follows: 1 r2ir riv/2 v(t a) p=- / ^ V c o s £ s i n £ d a d £ (4.12) 7T Ja=0 J£=0 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 the pathlength will increase the probability of interception). Equation 4.12 reduces to I (lit /-TT/2 P = - / p ( £ a ) s i n £ d a d £ (4.13) 7T Ja=0 J£=0 4.2.2 A n Approx imate M e t h o d for Determin ing Radiat ive Conductance 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 Radiation 101 function. 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 experi-ment, the transmission of direct longwave radiation, through the coats of various animal species, which were clipped to successively shorter depths, was measured with a collimat-ing 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 pre-dicts 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. Thermal Radiation 102 4.2.3 Numerical Integration to 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 Radiation 103 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 Radiation 104 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 rir/2 F(r) = - da p(£, a)e-pti>a>dr[B(T) - abr cos {] sin { cos £d£ (4.18) IT Ja=0 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 (4.19) Jo 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 over-estimation 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=300K; 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 Resul t s and Discuss ion 4.3.1 T h e Effect of P i loerect ion on Heat Transfer through a Deer Coat 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 Chapter 4. Thermal Radiation 106 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 Radiation 108 C/i O o 4 o U > Hair Angle (degrees) Figure 4.6: Results of a simulation showing the effects of piloerection on radiative conduc-tance 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 c/i O O -§ o U a 3 -1 -gc - convective -o o— radiative hair 0? -D*f 0 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 Conc lu s ions 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 satisfac-tory 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 integra-tion 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 negli-gible 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 Me-teorology (Proc. of the W.M.O. Seminar), 1, pp. 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 Test ing a M o d e l of Heat Loss from Deer in Forest Habi ta t s 5.1 Introduct ion In this chapter, the findings of this thesis are used in a computer model to make pre-dictions 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 Mode l 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". The 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 conduc-tance 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 Descr ipt ion of the 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, Te, stan-dard 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. Chapter 5. Deer Heat Loss Model 114 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 = 2 2 8 — (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 bound-ary 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 T (rHbs + res)(Tb - Te) les = lb -, ; v (5.4) {rm + re) where rub is the whole body resistance, rubs and res are the values of rjjb and re calcu-lated 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, rc . Parker and Gillingham determined rt in s m " 1 using the empirical relationships rt = 63.95-5.58T e(°C) (winter) or rt = 204 .43-6 .497; (°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 ^ ( s m - 1 ) was calculated using 857.9-29.9Te-0.19Te2+0.006Te3(°C) (winter) or 603.29-4.62Te — 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) (5.6) where gcs (l/rcs) 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 + 0.006u211 (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 black-tailed 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-2Pcv(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 that 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 118 3 " 0 1 1 1 1 1 1 nn, • a This study b _—-^P Parker and Gillingham (1990) Winter Equations — Te = 0°C 1 1 1 I 0 4 6 8 Wind Speed (m/s) 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 S o m e Examples of M ode l Outputs 5.2.2.1 Site Descript ion Data for use in model testing were collected within an old-growth Douglas-fir (Pseudot-suga menziesii (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 (50°65N, 126°38W). 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 Douglas-fir 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 character-ized 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 120 5.2.2.2 Ins trumentat ion 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 Comparison of Mode l Outputs Us ing Relat ionships Found in this Thes is wi th Those from Parker and Gil l ingham (1990) A winter data 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 U C3 10 0 -10 uUlfllllL LA . IIUJI hl.JyUJJn snow depth old growth >15cm — — — 01 cs 500 B 250 ^ 0 & t/i 08 12 0 15 22 29 05 Jan Feb 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 Comparison of Deer Heat Loss in Forested and Open Habitats 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 123 I I I I I I I l_J 01 08 15 22 29 05 12 Jan Feb 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 recom-mended by Parker and Gillingham (P & G) (1990) . Also shown is mean daily air temperature (Ta). Chapter 5. Deer Heat Loss Model 124 120 Date Figure 5.5: Comparison of the metabolic rate (M) calculated using the equations devel-oped in this study (line a) and that calculated using Parker and Gillingham's equations (line b) . Chapter 5. Deer Heat Loss Model 125 U 05 0 •10 -20 -30 01 08 15 Jan 22 Date 29 05 12 Feb 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 Conc lus ions 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 127 01 08 15 Jan 22 Date 29 05 Feb 12 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 8 m s " 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 , 1020-1028. 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. Chen, J.M. and T.A. Black (1991) Measuring leaf area index of plant canopies with branch architecture. Agric. For. Meteorol. 57 , 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. Par 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 Douglas-fir stand on a south-facing slope. For. Sci. 39, 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. 43 , 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 ex-posed 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 po-sitions 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 con-vection was found to be an important mechanism of heat transfer within the coat, while radiative transfer and heat conduction along individual hairs were relatively unimpor-tant . 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 that radiative transfer remained unimportant when hairs are standing up due to the longer path length radiation must travel to escape the coat. The 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 Cited Parker, K.L. and M.P. Gillingham (1990) Estimate of critical thermal environments for mule deer. J. Range Manage. 43 , 73-81. Appendix A Determinat ion of the N e t Radiat ive F lux D e n s i t y for Various Posi t ions on the M ode l Deer in t h e W i n d Tunnel 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 approx-imately 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 tempera-tures, 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 Cited 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 0.318 0.203 0.902 walls 0.364 0.594 0.121 0.098 ceiling 0.318 0.203 0.879 Appendix B S o m e Examples of T i m e Constants for Equil ibration of Surface Temperature of the Bare M ode l Deer to Step Changes in W i n d Speed The following table shows some representative time constants for the response of surface temperature on the 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 r (min) 1.8 1.9 2.7 2.0 1.7 2.3 2.6 1.4 2.8 1.8 2.4 135 A p p e n d i x C Comparison of the Everest M ode l 4000 IRT wi th a B l a c k b o d y Calibration Block During t h e Second W i n d Tunnel Exper iment Involving t h e Coat Covered M ode l D e e r 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. (Fuller-ton, 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 di-ameter) 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 mea-sured by the IRT (TIRT) when it was inserted into the cavity. The 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 (C.l) The surface temperature measurements were corrected by subtracting (Tbb ~ TIRT) from measured surface temperature where (Tbb — TIRT) was calculated at that temperature using Eq. C.l . 136 Appendix C. IRT Calibration 137 0.24 -0.18 0.12 -0.06 -0 1 ---" 1 1 1 ' 1 regression line -a a / , / l , 1 D a i a i ... 1 • • / 1 i D ' 1 ---— 1 23 25 27 Tbb (°C) 29 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 Power Spectra in the Old Growth and Second Growth Stands 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. The 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 Douglas-fir stand on a south-facing slope For. Sci. 39, 211-230. 139 Appendix D. Turbulent Spectra 140 r<i Cfl <N 10" 10" 10' • — • — a * • • • • • second-growth • • • • • • • • • • • • • • • • • • • • • m • • • • i i i 10" 10' 10' 10' 0 8 8 G 10 10 10 -2 -3 -4 • • • • • • • b ., ., L, *. old-growth • • • • • \ i i i 103 102 101 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 and Trigonometric Manipulat ions Showing that Equat ions 4.9 and 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 +tan2<^)(l + tan 2 £) - ( 1 + tan<£tan£cosa) 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 cos2 ^cos 2 £ out of the squared term in Eq. E.3 gives . ^ r, o , o J. / sin d> sin £ cos a cos ^>cos£.0 l1 /0 ,,-, N s i n / ? = [ l - cos 2 <£cos 2 £ ± , + 7 7 f 1 / 2 (E.4) cos <p cos £ cos <p cos £ 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 (E.6) cos2 <p cos2 £ 141 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. cos2 0 = 1—— (E.7) 1 + tan2 6 y ' Now, substituting Eq. E.7 into Eq. E.6 for the cos2 <j> and cos2£ terms we have sin/3 = cos</>cos£[(l +tan2<^)(l + tan 2 £) - (tan </> tan £ 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 + tan2<^)(l + tan 2 £) - ( 1 + tan^ tan£cos<*) 2 ] 1 / 2 (E.9) which is the same as Eq. 4.11. 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 p(^a)e-p^'a^rdr[B(T) - abr cos £] sin ( cos £d{ (F.l) 7T Ja=0 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 Km" 1 6=6 .2Wm- 2 K" 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. r(m) .00025 .00075 .00125 .00175 .00225 .00275 .00325 .00375 .00425 .00475 .00525 .00575 .00625 .00675 .00725 .00775 .00825 .00875 .00925 .00975 below 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 above F(r) 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 The Quick Basic Compute r Code for the Deer Heat Loss Model DEERCLIM 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 Tl , 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 FORi» = 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 Statist ical Analys is of Linear and Non- l inear Regress ions 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 co-efficients, 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 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 Nu / 2 c.f. Nu/ l.f. Nu / c.f. Nu / l.f. Nuffield Nu<?/ 3 w.t. Nuffield Nu#/ w.t. Nuffield Nu#/ w.t. Reg. Coeff. a -0.137 -0.045 -1.471 -0.733 -0.735 -0.907 -0.845 -0.431 -0.647 -0.809 0.856 0.891 0.167 -0.045 0.188 -0.733 0.268 -0.907 b 0.531 0.500 0.730 0.606 0.651 0.689 0.679 0.466 0.613 0.692 0.610 0.691 0.462 0.500 0.437 0.606 0.411 0.689 Standard Errors Oyx 0.021 0.0089 0.12 0.011 0.089 0.0076 0.016 0.016 0.014 0.027 0.014 0.027 0.029 0.0089 0.073 0.011 0.023 0.0076 Sa 0.045 0.059 0.27 0.072 0.20 0.050 0.064 0.017 0.090 0.20 0.0090 0.017 0.094 0.059 0.237 0.072 0.13 0.050 Sb 0.0093 0.013 0.055 0.016 0.041 0.011 0.013 0.022 0.020 0.039 0.020 0.039 0.024 0.013 0.061 0.016 0.032 0.011 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 same same same diff. same same same same diff. Elev. Miff. diff. same diff. diff. diff. diff. diff. same 1 p.w. - previous workers; t.s. - this study 2 c.f. - cross flow; l.f. - longitudinal flow 3 w.t. - wind tunnel 4 diff. - different Appendix H. Statistics 156 Table H.2: Statistics for non-linear regressions of the form y — constant + axb. Fig. # 3.7 3.8 3.8 3.12 3.13 3.14 3.20 y Var. 9c 9con 9c 9c(0°) 9c Tc 9b 1constant 0 1.93 2.40 2.48 2.27 2.45 0 Reg. Coeff. a 2.955 0.688 0.655 0.036 0.015 0.006 7.009 b 0.099 0.278 0.300 1.824 2.189 2.102 0.635 Standard Errors sa 0.167 0.11 0.12 0.037 0.001 0.002 0.25 sb 0.028 0.078 0.087 0.003 0.042 0.15 0.028 R2 0.810 0.817 0.806 0.999 0.999 0.991 0.997 1 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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