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Spouted bed hydrodynamics at temperatures up to 580 C Li, Yang 1992

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SPOUTED B E D HYDRODYNAMICS AT TEMPERATURES U P TO 580° C by YANG LI B. Eng. East China University of Chemical Technology, 1985 M. Eng. East China University of Chemical Technology, 1988  A THESIS SUBMITTED IN PARTIAL FULFILLMENT O F THE REQUIREMENTS FOR THE DEGREE O F MASTER OF A P P L I E D SCIENCE  in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CHEMICAL  ENGINEERING  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA  August 1992 © YANG LI, 1992  In  presenting  degree freely  at  this  the  thesis  in  partial  fulfilment  of  University  of  British  Columbia,  I agree  available for  copying  of  department publication  this or of  reference  thesis by  this  for  his thesis  and  study.  scholarly  or for  her  financial  of &  The University of British Vancouver, Canada  Date  DE-6 (2/88)  faiytj  Columbia  purposes  gain  shall  requirements that  agree  may  representatives.  permission.  Department  I further  the  It not  be is  that  the  Library  permission  granted  by  understood be  for  allowed  an  advanced  shall for  the that  without  make  it  extensive  head  of  my  copying  or  my  written  Abstract  A study of the hydrodynamics of spouted beds at temperatures ranging from room temperature to 580°C was carried out using a 0.156 m I.D. stainless steel conical-cylindrical half-column. Five narrowly sized fractions of Target sand with reciprocal mean diameters of 0.915 m m , 1.010 mm, 1.200 m m , 1.630 m m and 2.025 m m , and three orifices with internal diameters of 12.70 mm, 19.05 m m and 26.64 m m were used. The main purpose of the present work was to obtain a wide range of experimental data at high temperatures and compare the results with existing equations, to establish new correlations under different circumstances. Aspects studied included minimum spouting velocity, Ums, maximum spoutable bed height, Hm, and average spout diameter, Ds. It was found that the stability of spouting decreased with increasing temperature. The value of Ums increased with increasing temperature, especially for the large particles. The best of several empirical equations developed for Ums is one which uses the free-settling terminal velocity of the particles as a correlating parameter. The McNab - Bridgwater equation for Hm overpredicted Hm substantially at room temperatures and underpredicted Hm slightly at high temperatures. tion with a slightly smaller value of Um/Umf  A similar equa-  than that recommended by McNab and  Bridgwater gives better overall results. The Wu et al. non-dimensional equation for Ds, which explicitly includes the effect of gas density and gas viscosity, hence of gas temperature, gave better absolute prediction of the average spout diameter, Ds, than did the dimensional McNab equation, especially at elevated bed temperatures.  ii  Table of C o n t e n t s  Abstract  ii  List of Tables  vii  List of Figures  viii  Acknowledgement 1  2  xi  Introduction  1  1.1  Rationale for the Present Work  1  1.2  Objectives of the Present Work  3  Literature R e v i e w  4  2.1  General Information  4  2.2  Spoutability  5  2.3  Minimum Spouting Velocity  5  2.3.1  Mathur and Gishler equation  6  2.3.2  Correlation of Grbavcic et al.  9  2.3.3  Wu et al. Modification of Equation (2.1)  10  2.3.4  The Maximum Value of Ums  11  2.4  Maximum Spoutable Bed Height  11  2.5  Spout Diameter  15  2.6  Pressure Drop and Pressure Distribution  17  iii  3  4  Experimental Apparatus  19  3.1  Equipment  19  3.1.1  Choice and Description of Equipment  19  3.1.2  Heaters  20  3.2  Instrumentation  23  3.3  Bed Material  26  E x p e r i m e n t a l P r o c e d u r e s and C o n d i t i o n s  29  4.1  Operating procedure  29  4.1.1  Operation  29  4.1.2  Measurement  30  4.1.3  Calculation of Ums  31  Experimental Conditions  34  4.2.1  Range  34  4.2.2  Experimental Error Calculation  34  4.2  5  Results: Minimum Spouting Velocity  36  5.1  Measurement difficulties  36  5.2  Effect of Particle Diameter  37  5.3  Effect of Orifice Diameter  37  5.4  Effect of Bed Height  42  5.5  Effect of Temperature  42  5.6  Data Correlation  47  5.6.1  First Option  47  5.6.2  Second Option  48  5.6.3  Third Option  49  5.6.4  Fourth Option  51 iv  6  Results: Maximum Spoutable Bed Height  58  6.1  Spoutability  58  6.2  Maximum Spoutable Bed Height  60  6.2.1  Effect of Particle diameter on Hm  60  6.2.2  Effect of Orifice Diameter on Hm  64  6.2.3  Effect of Temperature on Hm  64  7 Results: Average Spout Diameter  8  74  7.1  Effect of Bed Temperature o n D s  74  7.2  Effect of Bed Height o n D s  74  7.3  Comparison with Existing Correlations  77  Conclusions  81  Bibliography  87  Appendices  92  A Calibration of Rotameters  92  B Derivation of the Expression for —^-  96  C Experimental Conditions  97  D Experimental Data  100  E Fortran Programs E.l Program on Ums correlation  109 109  E.2 Program to calculate average spout diameter  v  119  F E r r o r % for t h e Ums Values P r e d i c t e d by Four Equations  121  G E r r o r % for t h e Spout D i a m e t e r  128  List of Tables  3.1  Typical measurement of sand particles  27  3.2  Mean diameters of sand particles  28  5.3  Constants in Equation (5.47) and root mean square errors for three correlations  5.4  48  Comparison of average and root mean square errors of Ums by equations of Mathur and Gishler, Wu et al., Grbavcic et al. and best fit by present work  54  6.5  Spoutability of sand particles  59  6.6  Change of critical value of dp with temperature  63  vn  List of Figures  1.1  Schematic diagram of a spouted bed  2.2  Typical pressure drop versus velocity curve for a spouted bed of coarse particles  2  7  3.3  Details of the spouted bed column  21  3.4  Dimensions of the orifice plates  22  3.5  Schematic of the experimental equipment  24  3.6  Calibration curve for — APS versus — APa  4.7  Simplified flow diagram of the apparatus  5.8  Effect of particle diameter on Ums. (DC=156 m m , Z),=19.05 m m , H=0.2 m) 38  5.9  Effect of particle diameter on Ums. (Z) c =156 m m , Z?,=12.70 m m , H=0.2 m) 39  26 32  5.10 Effect of orifice diameter on Ums. (Dc=156 mm, d p =2.025 mm, H=0.3 m)  40  5.11 Effect of orifice diameter on Ums. (£) c =156 mm, d p =1.010 m m , H=0.2 m)  41  5.12 Effect of bed height on Ums. (£>c=156 mm, A - 2 6 . 6 4 m m , dp=2.025 m m )  43  5.13 Effect of temperature on Ums. (Dc=156 mm, 2}, = 19.05 mm)  44  5.14 Effect of temperature on Ums. (DC=156 mm, Z?,=26.64 mm)  45  5.15 Effect of temperature on Ums. (Z) c =156 mm, £>,=12.70 mm)  46  5.16 Experimental values of Ums vs. values predicted by Equation (5.75). . . .  53  5.17 Comparison of correlations for Ums with experimental data. (£> c =156 m m , A = 19.05 m m , <fp=2.025 mm)  56  viii  5.18 Comparison of correlations for Ums with experimental data. (D c =156 mm, A - 1 2 . 7 0 mm, d p =1.630 mm)  57  6.19 Comparison between experimental data (points), prediction by Equation 2.10a (solid line) and prediction by modified equation (broken line). . . .  61  6.20 Comparison between experimental data (points), prediction by Equation 2.10a (solid line) and prediction by modified equation (broken line). . . . 6.21 Effect of particle diameter on Hm.  62  Points represent experimental data,  lines represent McNab - Bridgwater equation. ( J D C = 1 5 6 m m , A = 2 6 . 6 4 m m ) 65 6.22 Effect of particle diameter on Hm.  Points represent experimental data,  lines represent McNab - Bridgwater equation. (7J C =156 m m , Z?j=19.05 m m ) 66 6.23 Effect of particle diameter on Hm.  Points represent experimental data,  lines represent McNab - Bridgwater equation. (Dc=156 m m , D s =12.70 mm) 67 6.24 Effect of orifice diameter on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. (Z?c=156 m m , d p =2.025 mm)  .  68  6.25 Effect of orifice diameter on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. (DC=156 m m , <ip=1.630 m m )  .  68  6.26 Effect of orifice diameter on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. (DC=156 m m , d p =1.200 mm)  .  69  6.27 Effect of orifice diameter on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. (Z) c =156 m m , dp=1.0W 6.28 Effect of temperature on Hm.  mm)  69  Points represent experimental data, lines  represent McNab - Bridgwater equation. (Z) c =156 m m , 7J,=26.64 m m ) 6.29 Effect of temperature on Hm.  .  .  71  Points represent experimental data, lines  represent McNab - Bridgwater equation. (Z?c=156 m m , Z) t =19.05 mm)  ix  .  72  6.30 Effect of temperature on Hm.  Points represent experimental data, lines  represent McNab - Bridgwater equation. (Dc=156 m m , Z),=12.70 m m ) .  73  7.31 Effect of temperature on Ds. (Dc=156 m m , D,=19.05 m m , d p =1.630 m m )  75  7.32 Effect of temperature on Ds. (Dc=156 m m , £> t =26.64 m m , d p =2.025 m m )  76  7.33 Effect of bed height on Ds. (Z>c=156 m m , A - 2 6 . 6 4 m m , d p =2.025 mm)  78  7.34 Comparison of Ds measured experimentally with Ds predicted by McNab equation. (Z) c =156 mm)  79  7.35 Comparison of Ds measured experimentally with Ds predicted by Wu et al. equation. (Z) c =156 mm)  80  A.36 Schematic set-up for rotameter calibration  93  A.37 Calibration curve (small rotameter)  94  A.38 Calibration curve (large rotameter)  95  x  Acknowledgement  I would like to express my appreciation to Dr. N. Epstein and Dr. C. J. Lim, under whose supervision and guidance this work was carried out. Thanks are also due to the people of the Department of Chemical Engineering Workshop and Stores for their invaluable assistance. Finally, to my parents for their encouragement and support.  XI  Chapter 1  Introduction  1.1  R a t i o n a l e for t h e P r e s e n t Work  The spouted bed technique was developed by Mathur and Gishler [24] for drying wheat in the 1950's. Since then, spouted beds have been used as an alternative to fluidized beds for gas contacting of coarse particles (dp > 1 mm). Figure 1.1 illustrates schematically a typical cylindrical spouted bed column with a conical base. Under the condition of stable spouting, the spouted bed consists essentially of two regions: a dilute phase central core of upward moving gas and particles called the spout and a surrounding dense phase region of downward moving particles and upward percolating gas known as the annulus.  In a bed filled with coarse particles, fluid, usually  gas, is injected vertically from the bottom of the bed through a centrally located small opening called the orifice. Particles are entrained in the spout by the gas at high velocity, and then penetrate somewhat above the bed level in a region called the fountain, where they fall back onto the annulus surface. In the annulus, particles slowly move downwards by gravity and, to some extent, radially inwards as a loosely packed bed. These particles are re-entrained into the spout through the spout wall over the entire bed height. The fluid from the spout seeps through the annular solids as it travels upwards. This systematic movement of the fluid and the solids leads to effective contact between them. A complete review of spouted bed technology was presented in the monograph by Mathur and Epstein [1]. More recent reviews are given by Epstein and Grace [2] and by  1  Chapter 1.  2  Introduction  FOUNTAIN  P« « C e  00  BED SURFACE SPOUT  ill'.  ANNULUS SPOUT-ANNULUS INTERFACE  CONICAL b  0r  FLUID INLET  Figure 1.1: Schematic diagram of a spouted bed.  BASE  Chapter 1.  3  Introduction  Bridgwater [3]. As pointed out by Lim et al. [11], spouted beds exhibit some advantages over conventional fluidized beds. They have been used for various physical and chemical processes and have achieved increasing recognition. Recently, high temperature spouting has attracted some attention because of its industrial applications, particularly in the energy field. These applications include not only carbonization of caking coal [4, 5, 6], drying of granular solids, slurries and solutions, and tablet coating [1], but also gasification , pyrolysis and combustion of caking coals [7, 8, 11, 14, 20, 21], and combustion of low heating value fuels and wastes [11, 15, 16, 17, 18]. While the hydrodynamics of spouted beds at ambient temperatures have been well studied in the past, knowledge of spouted bed hydrodynamics at high temperature is far from sufficient yet. The fragmentry information available on high temperature spouting differs from one worker to another and is sometimes even contradictory. The present work involves a detailed study of certain hydrodynamic features of spouted beds at high temperature.  1.2  O b j e c t i v e s of t h e P r e s e n t W o r k  Some important hydrodynamic parameters of spouted beds are: spoutability, minimum spouting velocity, maximum spoutable height, spout shape and diameter, overall bed pressure drop, pressure profiles, fluid and particle velocities in the spout and annulus. The primary objective of the present research is to collect experimental data on some of these hydrodynamic parameters at varying operating conditions, including a temperature range from room temperature to 580°C.  Using the data obtained, the  validity of existing equations can be examined and, where indicated, new correlations can be developed and explanations offered for unanticipated results of the present work.  Chapter 2 Literature R e v i e w  2.1  General Information  With the increasing development of the spouted bed as a high temperature reactor, the need for better understanding of spouted bed hydrodynamics at high temperatures has become evident. Gas spouting at ambient conditions has been well studied in most aspects and many equations are available for predicting hydrodynamic parameters. Mathur and Epstein [1] and Epstein and Grace [2] have given complete reviews of spouted bed technology. However, information on high temperature spouting is scarce. A few published articles on this subject were mainly about reactor performance characteristics [8, 10, 11], reaction kinetics [12, 13,14], and combustion models [15,16, 17, 18,19, 20, 21]. The hydrodynamics at high temperature are not well understood. Stanley Wu [22] studied the hydrodynamics of spouted beds at temperatures up to 420°C. The temperature range of Wu's study was thus limited and only three particle sizes were investigated. Bogang Ye [21] made some investigations on spouted bed hydrodynamics in a 0.15 m internal diameter half-column spout-fluid bed at high temperature, by burning Minto coal. However, the combustion inside the spouted bed made it diffucult to study the hydrodynamics precisely. Minto coal caused serious sintering problems because of the poor micro-circulation of solids and the limited bed-to-wall heat transfer coefficient with air as external coolant. The limestone used for sulphur capture underwent a large change in its mean particle diameter after several hours of experimental operation, thus affecting  4  Chapter 2. Literature  Review  5  the mean diameter of the bed solids. The equations originally developed at room temperature conditions have been applied at high temperatures, with the assumption that these equations do not change significantly at elevated temperatures. Often, however, modification of the existing equations are required when they are used at elevated temperatures. It is thus important that the real features of gas spouting at high temperature, including the hydrodynamics, be investigated systematically.  2.2  Spoutability  Spoutability refers to those conditions for which stable spouting occurs in a spouted bed. Increasing bed temperature could shift the flow regime from stable spouting to pulsatory spouting [20, 22]. Chandnani and Epstein [23] proposed that stable spouting can occur only if Di/dp < 25.4. This criterion does not predict any effect of the bed temperature. Wu [22] showed that at some temperatures below 420° C, this criterion sometimes failed. Zhao et al. [20] found that the hydrodynamic pattern and even the flow regime changed substantially with temperature, particularly with smaller particles. Hydrodynamic patterns of spouted beds are influenced by such conditions as fluid flow rates, solids properties, bed height and fluid properties, the last of which are affected by increasing the temperature. Particle density apparently has a negligible effect on spoutability [23].  2.3  M i n i m u m Spouting Velocity  The minimum superficial fluid velocity at which a spouted bed will remain in the stable spouting state is called the minimum spouting velocity, Ums. It is determined experimentally by reducing the fluid flow rate to a point at which a further decrease of flowrate will  Chapter 2. Literature  6  Review  cause the spout to collapse and the bed pressure drop to increase suddenly. The spouting velocity at this point is taken as the minimum spouting velocity. It is sometimes only a relatively narrow region above incipient spouting where stable behavior prevails. Figure 2.2 shows a typical curve of pressure drop versus superficial velocity for spouting of coarse particles (dp > 1 mm).  In a typical run, the fluid flowrate is first increased until  point C is reached, which indicates stable spouting. However, this point is bed-history dependent and is not exactly reproducible. By decreasing the flowrate to point B, at which a further decrease of flowrate will cause the spout to collapse and the bed pressure to increase suddenly, it has been found that the velocity at this point is reproducible. Hence the minimum (superficial) spouting velocity Ums is represented by point B. It is generally known that Ums depends on solid and fluid properties, column geometry and bed depth. For a given bed material and given fluid properties, Ums increases with increasing bed depth and fluid inlet diameter, and with decreasing column diameter. For a given column geometry and bed height, Ums increases with increasing particle diameter and decreasing fluid density.  2.3.1  M a t h u r and Gishler e q u a t i o n  The Mathur and Gishler [24] equation is the most widely used empirical equation for predicting the minimum spouting velocity [1]. This empirical equation was derived from data for both gas and liquid spouted beds with diameters up to 0.6 m starting with dimensional analysis. The equation is:  Urns —  dp  \Dt]  [DC\ [DJ  1/3  2gH(pp-pfJ  N  Pi  Ghosh [29] derived a similar theoretical equation based on a momentum exchange  Chapter 2. Literature  Review  7  a  O cc a  uu cc ID CO CO UJ  cc a.  SUPERFICIAL  VELOCITY  Figure 2.2: Typical pressure drop versus velocity curve for a spouted bed of coarse particles.  Chapter 2. Literature  8  Review  between the entering fluid and the entrained particles fan Ums  dp  \Di  Dc  \.Del\  2gH(Pp -  PI)  (2 2)  PJ  The main difference between Equation (2.1) and Equation (2.2) is the exponent on the Dif Dc term, its value being | in the empirical equation as against unity in the theoretical. The term ^/ff is likely to be a function of Di/Dc  [29].  Both Equations (2.1) and (2.2) predict Ums to be directly proportional to Hb, with b equal to 0.5, which was confirmed experimentally by other authors such as Thorley et al [45] and Cowan et al. [37]. This value was justified theoretically by Madonna et al. [38]. Smith and Reddy [35] obtained Ums = aH°-50~x'7&(DilDc\  showing from their experiments  that b was smaller than 0.5. Lim and Grace [27] found b in the range 1.0 — 1.4 for a large diameter bed. Green and Bridgwater [30] also indicated that the exponent on H is greater in larger diameter vessels. These facts show that the value of b is not well established and probably depends on the geometry of the system. The proportionality between Ums and dp has been verified by other authors working with beds of closely sized materials [45, 34] and with beds containing a wide spread of particle sizes [35]. Manurung [36], working with materials consisting of both closefractions and mixed sizes, obtained Ums a e^ 6 2 for otherwise fixed conditions, using the reciprocal mean diameter for dv. As noted by Mathur and Epstein [1], Equation (2.1) underestimated the minimum spouting velocity by a factor of nearly 2 for a single measurment (on wheat) in a 0.91 m diameter vessel. Wu et al. [39], using a column of 156 mm I.D found that for air spouting at room temperature, Equation (2.1) underestimated Ums with a deviation up to 30%, while at higher temperatures the equation actually worked better. Ottawa sand with a particle diameter range from 0.945 mm to 1.665 mm and orifices with diameters from 12.70 mm to 26.64 mm were used in Wu's work. The change of temperature was  Chapter 2. Literature  9  Review  reflected in a change of both gas density and gas viscosity: when temperature increases, the gas density decreases and the gas viscosity increases. The effect of fluid density in Equation (2.1) is such that Ums increases with increasing temperature. The absence of fluid viscosity in this equation has, however, been questioned by Charlton et al. [26]. Fane and Mitchell [25] proposed an empirical dimensional correction to Equation (2.1) based on experimental data in a 1.1 m diameter column and claimed that Ums first falls and then begins to rise as bed diameter is increased, the latter being in a direction opposite to that suggested by Equation (2.1). This claim was supported by both Lim et al. [27] and He et al. [28]. Thus Equation(2.1) has not been very successful for large columns.  2.3.2  C o r r e l a t i o n of G r b a v c i c et al.  Using the model of Mamuro and Hattori [31] at maximum spoutable bed height, Grbavcic et al. [32] proposed the following correlation for predicting Ums for spherical particles: Ums UmS  a.  1 a  3  l  =l~  (2 3)  -JT  -  where as is defined as the ratio of the area of the spout to that of the column. Since as is much smaller than 1 in most cases, Equation (2.4) can be further simplified to  ams  -.  Umf  l-  H  3  Hm.  (2.4)  where Umf is given by the Ergun (1952) equation: - (%)  = {PP ~ Pf)(l - emf)g = fiUm, + Wis  (2.5)  with f\ and f2 given by //(l - e m / ) 2 h = 150 2  (H) £/  h = 1-75  Pfi1 ~ ernf)  Since the Grbavcic equation was verified only for water spouted beds at room temperature, its application to high temperature air spouted beds has yet to be examined.  Chapter 2. Literature  2.3.3  10  Review  W u et al. Modification of Equation (2.1)  A modified form of the Mathur and Gishler [24] equation, with best fit values of the coefficient and of the exponents on the dimensionless groups for conditions including elevated temperature, was given by Wu et al. [39]: U„.  V2gB  = 10.6  dp  [Dc\  1.05  0.256  [A] 0.266 r/n - 0 . 0 9 5 Wc\  LAJ  Pp~  .  PS  Pf  (2.6) .  The most significant difference from the original Mathur and Gishler equation is that the exponent on (pp — Pf)/pj (pp — Pf)/pf  is 0.256 instead of 0.5. Unlike Mathur and Gishler [24],  and 2gH were not grouped as one parameter.  As pointed by Ye et al. [21], both the Mathur and Gishler equation and the Wu equation underpredicted Ums at very high temperature, though the latter equation worked better than the former. Ye et al. showed in his experimental data that Ums decreases with increasing temperature for smaller particles and increases with increasing temperature for larger particles. The effects of dp and temperature appeared to be much more complex than predicted. T h e problem encountered in spouted beds is inherently more complex than in fluidization, for which Umf always increases with an increase in temperature [43]. Most of the existing equations mentioned above have not paid much attention to the change of viscosity due to the change of temperature. Gas viscosity increases with temperature [40]. Deficiencies in predictions may be due to inadequate knowledge of how to include dp in the above equations, and the absence of fluid viscosity. A detailed study on the effects of different independent variables on minimum spouting velocity at high temperature is thus of some importance.  Chapter 2. Literature  2.3.4  Review  11  T h e M a x i m u m Value of Ums  The value of Ums at the maximum spoutable bed height is termed Um, the maximum value of the minimum spouting velocity [41]. For many materials, Um is expected to coincide with the minimum fluidization velocity since beyond Hm a spouted bed transforms into a fluidized bed. Umf.  Experimental data by previous workers show that Um often exceeds  In the case of sand (dp = 0.42 — 0.83 mm) in a spouted bed of 152 mm I.D. at  room temperature, Um is approximately equal to Umj, while it is 33% higher than Umf for wheat (dp = 3.2 — 6.4 mm) and 45% for semicoke (dp = 1—5 mm) [1]. Values of Um exceeding Umf by 10-33% have been reported by Becker [41] for a variety of uniform size materials. Differences in the properties of the solid materials and in spouting vessel geometry might affect the ratio Um/Umf.  For a fixed Di/Dc  ratio, Um increases with  increasing column diameter, while for a fixed value of Dc, it increases with increasing orifice diameter.  2.4  M a x i m u m Spoutable Bed Height  The maximum spoutable bed height, Hm, is the maximum height at which steady stable spouting can be maintained.  For bed heights above Hm,  the bed will sometimes be  partitioned into an internal spouting zone and an upper level fluidization region. Mathur and Epstein [1] suggested three distinct mechanisms for spout termination beyond i.e., 1. Fluidization of Annular Solids 2. Choking of the Spout 3. Growth of Instability at the Spout-Annulus Interface At the maximum spoutable bed height, Equation (2.1) becomes  Hm.  Chapter 2. Literature  12  Review  U„  [A] 1/3  dp  [DC\  i  2gHm(pp-pf)  (2.1a)  Pf  As mentioned above, Um has a close relationship with Umf. In general,  U„  u,mf On the other hand, Umf  ca  = h = 1.0 -  1.5  (2.7)  n be estimated from the Ergun [42] equation on sub-  stitution of the empirical approximations of Wen and Yu [43], i.e. \/<f>tmj = 14 and (1 - e ™ / ) / ^ 2 ^ = 11, which yields  Rtmf  = dpUyPf P  =  33.7(Vl + 35.9 x 1 0 - M r - 1)  (2.8)  where d l(pp-pf)gpf y Ar _= v V-1  (2.9)  Equations (2.1a), (2.7) and (2.8) are combined to eliminate Um and Umf, the result being Hm =  dp  \DC1 2/3 "5686?" {Vl + 35.9 x lO" 6 Ar - 1) : Ar [Dil  (2.10)  McNab and Bridgwater [44] found that Equation (2.10) gave the best fit to existing experimental data for Hm in gas spouted beds with b\ — 1.11. Thorley et ol. [45] were able to predict values of Hm, though only approximately, under a variety of conditions by simultaneously solving an equation for Ums with an equation for Umj. This approach was subsequently adopted by other workers with variations in the particular equations used for calculating spouting and fluidization velocities. The majority of the empirical and semi-empirical models for predicting Hm were listed by Mamuro and Hattori [31]. Mathur and Epstein [1] listed the empirical equations for predicting Hm and made a comparison to decide which of the various calculation methods  Chapter 2. Literature  Review  13  proposed are suitable for predictive purposes. The Malek and Lu [46] equation is the most simple correlation based on a sufficiently wide range of variables to be of practical interest. It is given by H„  = 0.105  Dc  'De'  0.75  [•Del  0.4  A2"  2  kJ  dp  (2.11)  where A is a shape factor with values ranging from 1.0 for millet, sand and timothy seed to 1.65 for gravel, while pp is the particle density in  g/cm3.  Lefroy and Davidson [47] derived the following expression for Hm by extending their force balance equations on spouted beds:  8DsHm  (2.12)  -tawy = 0.36  Grbavcic et al. [32] proposed an empirical equation to calculate Hm in their correlation for Ums based on data for water-spouted beds of spherical glass particles: Hrr,  Dr  = 0.347  (tf(f)  0.31  (2.13)  Littman et al. [48, 49] developed two models using monodispersed spherical particles. The first of these models states that  »~DDi  0.215 + 5 ^ A  / o r / t > 0  .02  (2.14)  where A is defined by pUmfUt  (2.15)  (PP ~ Pf)gDi Umf is calculated from Equation (2.8), and Ut is estimated from the following: Ar = l&Ret + 2.7Re}-b*'; Ret < 1000 (2.16) U5  Ret = 1.745Ar ; Ret > 1000  Chapter 2. Literature  Review  14  The second model states that HmDs  0.345  Dl - D]  DA DJ  -0.384  (2.17)  The first model was derived from momentum considerations. It was established that the A-parameter linked the maximum jet penetration to the momentum exiting the inlet orifice. The latter model (Hm — Ds relationship) follows from a solution of the vectorial form of Ergun's equation for the annular flow field. In that analysis McNab's [50] relationship was used to predict the spout diameter, while the Lefroy and Davidson [47] pressure profile was assumed to hold at the spout-annulus interface. Wu [22] compared some of the existing Hm correlations, such as the equation of McNab and Bridgwater, Littman's first model for Hm, and that of Malek and Lu, with his experimental data at temperatures up to 420° C. The results showed that the McNab and Bridgwater relation, Equation (2.10), gave the best prediction. Equation (2.10) does not take the temperature effect explicitly into consideration. The effect of temperature can be determined by differentiating Equation (2.10) with respect to Ar while other variables are kept constant (see Appendix B): dHn dAr  x  Ci 1 Ar3  '-?- + 35.9 x 10- 6 - W-JAr  V Ar  Ar  3  1 + 35.9 x 10~ 6 Ar 4 > 0  for  Ar > 0  The above equation shows that for all value of Ar, dHm/dAr  (2.18)  > 0, which indicates that  Hm always increases with increasing Ar. For gas spouting, when temperature increases, fluid density decreases while viscosity increases, which results in decreasing the value of Ar if dp is fixed. Therefore, Hm also decreases with increasing temperature. phenomenon was verified in both Wu's [22] and Zhao's [20] experiments.  This  Chapter 2. Literature  Review  15  The effect of temperature was further investigated experimentally by Wu et al. [39] by looking at the effect of changing gas density at constant viscosity and vice versa. Wu et al. found that Hm increased with increasing p/ and with decreasing p; in other words, Hm is higher if the spouting gas is more dense and less viscous. Thus correlations which contain the effect of both p and p seem to work better than those which ignore p,. The particle diameter effect on Hm can be examined by substituting Equation (2.9) into Equation (2.10) and then differentiating the latter with respect to dp, setting dHm/d(dp)  equal to zero; then {dp)crit = 60.6  1/3  f  (2.19)  .9(Pp~Pf)Pf. where (<fp)cr,( is the critical value of dp, below which Hm increases with dp and above which Hm decreases as dp increases. This critical value changes with temperature. The qualitative effect of increasing dp was observed by Wu at temperatures up to 420°C.  2.5  Spout Diameter  The spout is the central core of the bed and is a region of high fluid velocity and low solids concentration. Knowledge of the spout diameter is necessary for an understanding of the dynamics of the bed and for design purposes. There are many equations available for estimating the average spout diameter [1, 53]. However, attempts to apply principles of solids flow mechanics to the determination of Ds have achieved only qualitative success. Bridgwater and Mathur [51] developed a simplified theoretical model which was derived from a force balance analysis. Their theoretical equation is WO'®  =  1  (220)  This dimensionless equation was reduced to a more manageable dimensional form based on a number of approximations; in SI units of kg, m and s,  Chapter 2. Literature  Review  16  £0.500.75  Ds = 0.384  Pi25  (2.21)  This result is primarily restricted to air spouting, and it was later pointed out by McNab and Bridgwater [52] that the model of Bridgwater and Mathur was oversimplified. The longitudinal average value of spout diameter, Ds, has been correlated empirically by a dimensional correlation over a wide range of experimental data by McNab [50], applying statistical analysis to the data. The following expression is the result: £ 0'0.49 . 4 9 n0.68 0  Da = 2.0  PV1  (2.22)  in the same units as for Equation (2.21). The McNab equation and that of Bridgwater and Mathur have the same variables and the exponent on each of the variables has the same order of magnitude. The main difference is in the modifying coefficient. McNab's equation was later found by Wu et al. [39] to be unsuitable for estimating Ds at elevated bed temperatures, because it overpredicted the effect of temperature on average spout diameter. A more restrictive equation, which applies only to beds at their maximum spoutable height, but which has the virtue of being dimensionally consistent, is given by Littman and Morgan [49]. The most recent approach to determine the average spout diameter was carried out by Wu et al. [39], who developed the following expression for Ds by applying a least squares fit to their data using the theoretical model of Bridgwater and Mathur [51]: " £0.4333£)0.5832 0.1334'  Ds = 5.606  (,W)0'2834  (2.23)  This equation showed relatively good agreement with Wu's experimental data. Besides, it was dimensionally consistent, which was another advantage over the McNab expression. Wu found that the effect of bed temperature on Ds was not very significant. At a constant  Chapter 2. Literature  Review  17  bed height and a constant value of Us/Ums, Ds was observed to decrease slightly with increasing bed temperature. Ye et al. [21] compared his expermental data with Equations (2.21), (2.22) and (2.23) and found that all three equations underpredicted Ds, but that Equation (2.23) of Wu et al. was the best of the three. Krzywanski et al. [54] developed a relationship giving spout diameter as a function of bed level for both two dimensional and cylindrical spouted beds. This approach requires no prior knowledge of the pressure and particle/gas velocity fields in either the spout or the annulus. However, it does require input information about the average spout diameter, which can be obtained from standard correlations.  2.6  P r e s s u r e D r o p and P r e s s u r e D i s t r i b u t i o n  Equations for the longitudinal pressure profile in the annulus and the overall bed pressure drop were put forward by Epstein and Levine [55] using the Ergun equation [42] and force balance analysis of Mamuro and Hattori [31]. This is the only model that has a theoretical basis and also fits the experimental data reasonably well. Other equations were developed by Manurung [36] and Lefroy and Davidson [47], as well as by Morgan and Littman [56]. Manurung's equation [36] for pressure drop was developed by considering APS, the absolute spouting pressure drop to approach the nuidised bed pressure drop as the bed depth increases to infinity. Lefroy and Davidson [47] presented an empirical correlation based on their pressure measurements at the spout-annulus interface. Morgan and Littman [56] developed general pressure drop correlations based on a number of experimental pressure measurements reported in the literature. Wu et al. [39] showed that the bed temperature had no observable effect on the pressure drop and the shape of the longitudinal and radial pressure profiles. In general,  Chapter 2. Literature  Review  18  the radial profiles in the cylindrical section were flat [1] and the longitudinal profiles could be described by the quarter cosine curve of Lefroy and Davidson [47]. It has also been shown [57] that the particle shape and voidage coefficients developed by Wen and Yu [43] for use in the Ergun equation [42], which is applied in some of the above spouted bed pressure drop relationships, remain unchanged even at high temperatures.  Chapter 3  Experimental Apparatus  3.1  Equipment  3.1.1  C h o i c e a n d D e s c r i p t i o n of E q u i p m e n t  Experiments were carried out in a half column spouted bed. The use of a half column allows visual observation and direct measurement of such hydrodynamic parameters as maximum spoutable bed height and spout diameter. The validity of using a half column for the present measurements has been justified by Whiting and Geldart [58], Geldart et al [59] and Lim [60]. The spouted bed column was constructed of 316 stainless steel and consists of two parts: (1) a half cylindrical section of 0.156 m I.D. and 1.06 m height with a wall of 6.4 mm thickness. This section was also furnished with solids input and discharge lines; (2) a truncated 60° included angle half conical section 0.13 m high with a semi-circular orifice as the spouting gas inlet. A flat stainless steel panel on which three 1/4 inch thick transparent fused quartz glass plates were mounted for direct observation served as the front. The quartz glass was able to withstand high temperature. On top of the column, a sand feed system was built which had a conical container and a ball valve. The feed line was then connected to the air exhaust pipe. When feeding the sand into the column, a low flow rate of spouting air was maintained so that sand could get into the spouted bed column by gravity. If no spouting air was maintained, the bed of solids was packed too tightly, which made the initial spouting very difficult. The feed control valve was  19  Chapter 3. Experimental  Apparatus  20  then closed during the whole experiment period. There was one sand discharge line 0.2 m above the cylindrical base and eight measuring ports, one 0.38 mm above the orifice in the conical section and seven in the cylindrical section with vertical separations of 100 mm. These ports were all used for measuring pressure during the experiment. The fluid inlet section was a 26.64 mm I.D. half pipe with a straight vertical length of 0.300 m. This is shown in Figure 3.3. Three different orifice diameters were used in the experiments, namely 12.7, 19.05 and 26.64 mm, respectively. In order to get a more stable spouting than otherwise, all orifices had a converging nozzle-type bottom and an extended collar 3.2 mm high at the top, as shown in Figure 3.4. A very fine stainless steel wire screen was placed underneath the orifice so as to prevent sand particles from falling down into the inlet pipe. A high temperature insulating material (970-J paper supplied by Plibrico Limited of Canada) was used as the gasket material between the glass and the steel panel. The thickness of gasket material used was such that the internal surface between the quartz glasses and the steel panel was sufficiently smooth to avoid disturbing the flow pattern in the bed. The spouted bed was externally insulated by ceramic fibre insulation of thickness 1 inch to prevent heat loss to the surroundings. The ceramic fibre was also used to cover the quartz glass windows, and these covers were only removed momentarily for visual observation.  3.1.2  Heaters  Three cylindrical electric heaters (Watlow Ceramic Fiber Heaters), each with a maximum power rating of 3.6 kw, were mounted on the outside of 2-inch 316 stainless steel pipes. These heaters may be operated up to 1100°C with suitable control. Ceramic rings were packed inside the pipes to enhance heat transfer. High temperature gaskets (supplied by  Chapter 3. Experimented  Apparatus  21  304  \  I? 6  I 9  *  Side View  II  Front View  1. Spouting flow line. 2. Pressure port. 3. Conical base. 4. Solids discharge lines. 5. Measuring port. 6. Halfcolumn. 7. Gas exhaust line. 8. Port for thermocouple. 9. Front panel. 10. Quartz glass window. 11. Orifice. (All dimensions are in mm.)  Figure 3.3: Details of the spouted bed column.  Chapter 3. Experimental Apparatus  Dimension (mm) Size A B C D E S 50.8 41.0 12.70 3.2 3.2 M 50.8 41.0 19.05 3.2 3.2 L 50.8 41.0 26.64 3.2 3.2  22  F 9.5 9.5 9.5  G 1.6 1.6 1.6  Figure 3.4: Dimensions of the orifice plates.  Chapter 3. Experimental  Apparatus  23  A.R. Thompson Ltd.) were used in the joint sections of the pipes. All three heaters were controlled by monitoring the temperature using thermocouples in the gap between the outside wall of the pipe and the inside wall of the heater. The heaters were housed in a metal box and blanketed by the ceramic glass fibre insulation. Another small heater (supplied from Thermacraft Ltd.) with a power rating of 1.2 kw was mounted on the fluid inlet section to further heat the inlet air to the desired temperature.  3.2  Instrumentation  The schematic flow diameter of the experimental setup is shown in Figure 3.5. Air flow from the building compressor passed through one of the two rotameters, which were used to control and adjust the flow rate. Calibration curves for the rotameters at standard conditions are given in Appendix A. The measured flow rates are then converted to the actual conditions in the spouted bed. The detailed calculation of the volumetric flow rate through the spouted bed, Vs, and the minimum spouting velocity are presented in the next chapter. From the rotameter, air flowed into the heating units and was raised to the desired temperature before it was admitted into the spouted bed. The high temperature air from the bed was discharged into the surrounding atmosphere outside the building through an exhaust hose. Temperatures were measured and monitored by seven Chromel-Alumel type thermocouples, four of which showed their readings on the temperature controllers for the four heaters. The rest were connected to a digital display through a selecting switch. One was positioned in the outlet of the large heating unit and the other in the inlet section of the spouting air. A long thermocouple rod with a diameter of 1/4 inch was inserted into  Chapter 3. Experimental  Apparatus  . '5b 3  a o  E O  o  O  3 4> 3  oo  (0 C  • *  s ^  D< O o "--1  a -2 •si <3  n  • *v o « «i  . 4) v  JS  -a T3  o w (-i 4>  £ W  O  3  S3 •  Ou  * <° s 4)  0  1-1  — ° C  £ s £ 4)  D.  H ^  Chapter 3. Experimental  Apparatus  25  the spouted bed from the top to measure the temperature at different vertical positions in the bed. The average value of the temperature measurement along the bed height was taken as the average bed temperature. Two open U-tube manometers containing water were alternately connected to the two pressure taps before and after the rotameter and to the two ports below and above the inlet orifice to the spouted bed. They were used to determine the absolute pressure inside the rotameter and the absolute pressure in the spouted bed, respectively. These values were used for calculating the gas flowrate and the minimum spouting velocity at bed conditions. The absolute pressure inside the rotameter was obtained from the average of the two manometer readings at the ports before and after the rotameter. The pressure port below the orifice in the conical section was used to measure the overall pressure drop of the spouted bed, —AP S , from which the average absolute pressure in the bed was determined. A stainless steel screen was placed under the orifice to prevent sand particles from falling into the gas inlet tube. But the screen also caused blockage by the entrained small broken pieces of ceramic packing from the heating section and by the sand particles as well. This made measurement of the bed pressure drop unreproducible. To solve this problem, an alternative pressure tap was located 38 mm above the orifice in the actual experimental runs. A calibration was obtained by correlating — APS under no-screen conditions with the measured bed pressure drop, —AP a , using the pressure tap above the orifice, where the latter term was obtained from Equation (4.27) [22]. The equation in Figure 3.6, obtained by Wu [22], is  -APa  = 0.171 + 0 . 9 7 6 ( - A P a )  (3.24)  The pressure profile along the bed was measured using pressure transducers through the other seven pressure ports at the back of the column at intervals of 100 mm.  A  Chapter 3. Experimental  Apparatus  26  6.0-j  -£  4.0-  „  J2T  3.0-  / *  < 1  / /  2.0i.o-  y^  o.o4 0.0  i 1.0  1 2.0  1 3.0  1 4.0  5.0  Figure 3.6: Calibration curve for — APS versus — APa. 2 meter long stainless steel tube with a diameter of 1/4 inch was used to connect the transducers to the pressure ports to ensure that the transducers were not exposed to the high temperature air. The signals from the transducers were logged in to the computer through a cable with a 37-pin female connector. A dash-8 board interface and Labtech software were installed in the computer. Photographic slides were taken using a camera to record each run, from which maxim u m spoutable bed height, spout diameter and spout shape could be determined.  3.3  Bed Material  Target sand, supplied by Target Products Ltd., was used as bed material in this study. The sand, with a sphericity only a little below unity, was screened to a relatively narrow size range before particle sizes and particle density were measured. Five different mean sizes were prepared in this study. The mean particle diameter of each size fraction was  Chapter 3. Experimental  Apparatus  27  Table 3.1: Typical measurement of sand particles. mesh -7 + 9 - 9 + 12 - 1 2 + 14 - 1 4 + 16 - 1 6 + 20 - 2 0 + 24 -24 Total  dia.(mm) 2.80/2.00 2.00/1.40 1.40/1.18 1.18/1.00 1.00/0.85 0,85/0.71 0.71/0  net weight(g) 0.2 44.7 905.6 234.4 69.0 9.8 14.0 1277.7 x da - 1 .1942 mm P> ~ £(*./d P l ) - X  avg.dia., dPi{mm) 2.40 1.70 1.29 1.09 0.925 0.78 0.355  Xi  0.00016 0.03498 0.70877 0.18345 0.05400 0.00767 0.01096  Xi/dPi  0.00007 0.02058 0.54944 0.16831 0.05838 0.00983 0.03087 0.83747  determined from a U.S. sieve analysis using the following equation:  "'• - W  (3 25)  '  where Xi is the weight fraction of particles with an average adjacent screen aperture size of dPi. Several measurements were taken for each size to yield an average diameter. Table 3.1 is a typical measurement of sand particles. In order to determine the difference in the particle diameter of cold and heated sand particles, the cold sand size was first measured at room temperature. Then, the sand was heated at 300 °C for five hours so as to remove the moisture in the particles. It was found that at 300 °C, the color of the sand changed appreciably. After the heated sand was cooled down to room temperature, it was then screened to measure its mean particle diameter. T h e results are listed in Table 3.2. The heated sand values were the actual particle diameters used in the present experiments. All the sands were first heat-treated in this manner. The density of heat-treated sand particles was obtained by measuring the volume of water displaced by a known weight of particles. Because the sand particles could be permeable to water, the particles were first coated with a water seal (Thomson's  Chapter 3. Experimental  Apparatus  28  Table 3.2: Mean diameters of sand particles  Heated sand Cold sand avg. dia. (mm) avg. dia. (mm) 2.216 2.025 1.646 1.630 1.216 1.200 1.027 1.010 0.919 0.915  % diff. 9.43 0.98 1.33 1.68 0.44  Seal) before the measurement. In the density measurement, a 100 cm3 volumetric flask and a high accuracy (0.05 mg) balance were used. The volume occupied by the sand was calculated from volume difference, from which the density of the sand particles was determined. It was found that the density of the uncoated sand was higher than that of the coated sand by about 10 %. The latter value was 2547 kg/m3  for all particle sizes.  The bulk density of loosely packed sand was measured using the procedure of Oman and Watson [61]. First, a 250 cm3 graduate cylinder was partially filled with a known weight of sand. Then this cylinder was inverted with its open end covered and quickly reinverted to its original position. The volume of sand was then recorded and the bulk density thus determined. The loosely packed solids voidage was determined from the particle density and the bulk density.  Chapter 4 E x p e r i m e n t a l P r o c e d u r e s and Conditions  4.1 4.1.1  O p e r a t i n g procedure Operation  Before running the experiment, the large heating unit with three electric heaters was turned on for about 20 minutes to preheat the heating section and the ceramic packings inside them. The air flow was not turned on during this heating period. Then the heater controllers were set to the appropriate temperature level so as to reach the first desired temperature in the spouted bed. With a small flow rate of air, sand was added to the bed from the top of the column through the sand input system, which consisted of a funnel and a ball valve. The valve could control the amount of sand being put into the system. After the column was fed with a certain amount of sand, the valve was closed and the funnel still contained some sand for later use. The height of the bed was adjusted either by adding more sand from the hopper or by releasing some sand through the discharge line. The air flow rate was increased and adjusted to maintain a steady spouting condition. The long thermocouple was inserted from the top of the column to different levels of the bed for measuring the bed temperature. When the bed reached the desired temperature within ± 5 ° C , measurements were taken as described in the next subsection. When all the measurements were completed, the heaters were turned off and the outlet valve was opened to discharge the hot sand particles into a container. The sands were drained either by gravity or by maintaining a high flowrate, which yielded a  29  Chapter 4. Experimental  Procedures and  30  Conditions  spout fountain to accelerate the discharge of the sands. The column could be emptied in about 20 minutes. Air flow was kept on for an additional 60 minutes to cool off the whole apparatus.  4.1.2  Measurement  Hm was determined by increasing the bed height until stable spouting could not be obtained for any gas flowrate. The corresponding loosely-packed bed height was then taken as Hm. The minimum spouting velocity was measured by observing the bed through the transparent front panel. The gas flowrate was first increased to a value above the minimum spouting condition and then decreased slowly until spouting ceased. The gas flowrate at which the fountain just collapsed was taken as the minimum spouting flowrate. The calculation of the minimum spouting velocity is given in the next subsection. Measurement of spout diameter was performed in two steps.  The first step was  effected during an experimental run by holding a stainless steel rule horizontally against the transparent front panel and measuring the local spout diameter at several bed levels to yield a full spout shape. The more accurate second step involved making a photographic slide of the spouted bed for each run and, after the experiment, projecting the slide and measuring the spout diameter at 10 cm intervals along the bed height. The area-average spout diameter was calculated as follows, always at Us/Ums = 1.05: 1  H fHa °  2  .  dz  (4.26)  where Ds{z) was the measured spout diameter at bed level, z. The numerical integration was done with "QINT4P", a routine described by Tom Nicol [63]. The routine is shown in Appendix E.2.  Chapter 4. Experimental  Procedures and  31  Conditions  The pressure drop due to the bed according to Mathur and Epstein [1] should be determined as follows: - APa = ^Pl  - PI + P\TM  - PATM  (4.27)  where PB is the measured absolute upstream pressure for the bed and PE is the corresponding value at the same flowrate for an empty column. The calibration of PE versus rotameter reading was obtained in the form of a polynomial equation as follows: PE = 3.50 x 1(T 3 + 1 . 7 3 x lO~3R+2.35  x l ( T 4 i ? 2 - 3.63 x l O " 6 ^ + 3.21 x K T 8 # 4 (4.28)  where R is the large rotameter reading. The pressure profile was measured by connecting a set of manometers to the corresponding ports along the bed height. A set of pressure transducers was also installed and connected to a data-logging computer.  4.1.3  C a l c u l a t i o n of Ums  F l o w r a t e in t h e S p o u t e d B e d Figure (4.7) is a simplified flow diagram of the experimental apparatus. Applying the ideal gas law, VS = VR From the rotameter reading, VSTD  wa  PRTS  .PsTR\  (4.29)  s determined from one of the two calibration curves  in Appendix A. This value is not equal to the actual volumetric flowrate VR through the rotameter. However, it has been shown via Equations (A.85) and (A.87) in Appendix A that for a rotameter, VR = - ~ \/PR  (4.30)  Chapter 4. Experimental  Procedures and  32  Conditions  "*  EXHAUST  Ps T5  S P O U T E D BED  ROTAMETER  VALVE Figure 4.7: Simplified flow diagram of the apparatus. Therefore VR_  IPSTD  VsTD  PR  (4.31)  and T/ T/ jPSTD VR = VSTD\ = .PR  jPsTD  T/  VsTD\ V  p  (4.32)  R  Substituting Equation (4.32) into Equation (4.29) gives the flowrate in the spouted bed, PRTSI  Vs — VsTD\ PR  IPSTRI  V,STD  Tc l  VPSTDPR  (4.33)  R  Minimum Spouting Velocity From Equation (4.33), we can proceed with the detailed calculation of Ums as follows:  Chapter 4. Experimental  (1)  Procedures and  33  Conditions  Determine the temperature both in the rotameter, TR, and in the spouted bed, Ts-  Note that Ts is an average value of all the tem-  perature values along the bed height. (2)  Determine the flowrate of the air, VSTDVSTD VSTD  = 0.4800 + 0.2945 x R  (large  rotameter)  (4.34)  = 0.2693 -f 0.0212 x R  (small  rotameter)  (4.35)  where R is the rotameter reading. (3)  Determine the average absolute pressure of the rotameter, PR. PR = P9-T^  + Patm  where Pg is the gauge pressure upstream of the rotameter and AR\  (4.36) is  the pressure difference across the rotameter. (4)  Determine the average absolute pressure of the spouted bed, Psa.  Calculate absolute pressure at the port above the orifice, PRPB=Patm+AR2  (4.37)  where AR2 is the gauge pressure at the port above the orifice. b.  From Equation (4.28), calculate the corresponding value at the same flow rate for an empty column, PE-  c.  From Equation (4.27), calculate — APa.  d.  From Equation (3.24), calculate — APS.  e.  Then PS = Patm + ^ ^ ^  (4-38)  Chapter 4. Experimental  (5)  Procedures and  Conditions  34  From Equation (4.33), calculate the volumetric flow rate through the spouted bed, Vs-  (6)  Calculate Ums: Urns = ^  4.2 4.2.1  (4.39)  Experimental Conditions Range  For t h e experimental work, three orifice diameters, five particles sizes and six temperature settings were used. The scheduled number of runs for the experimental program thus came to 3 x 5 x 6 = 90. The operating conditions of the experiments are listed in Appendix C. The range encompassed was  4.2.2  Us/ Ums  1.0-1.1  dp(mm)  0.915 - 2.025  Diimm)  12.70 - 26.64  H{m)  0.10-1.00  T(°C)  20°C - 580°C  E x p e r i m e n t a l Error Calculation  In this thesis, the following definitions are used for the comparison of the experimental values with predicted values: . CAL-EXP % dev = —— M  innn. x 100%  RMS % ERROR = J[J2(% dev)2}/M AVG  ERROR  = [£  | % dev \ ]/M  (4.40)  (4.41) (4.42)  Chapter 4. Experimental  Procedures and Conditions  where EXP = experimental value CAL = predicted value M  = number of data points  35  Chapter 5  Results: Minimum Spouting Velocity  5.1  M e a s u r e m e n t difficulties  The minimum spouting velocity, Ums, was calculated using the procedure described in Section 4.1.3. Generally the Ums value was more difficult to obtain at high temperature than at room temperature, partly because spouting became less stable at high temperature but mainly because of the spouting equipment itself. At high temperature, the fluid density is low and thus a very small change in the flowmeter setting could result in a large flowrate change. The smaller flowmeter was occasionally used when required air flowrates at high temperature were relatively low. The ceramic rings inside the heaters easily broke into small pieces because of high bed temperatures, thus blocking the screen under the orifice and thereby changing the measured value of Ums. The screen was therefore cleaned before each run and measurement of Ums usually performed several times to ensure a certain level of reproducibility. Another factor which made the measurement of Ums  a  t high temperature more difficult was that to reach the required high temperature,  the rate of heating affected the approach to the set point value of the temperature controller. It was relatively difficult to maintain a high temperature at the desired value in the bed because the signal to which the controller responded was not from a point inside the spouted bed, but rather, from a point at the heater outlet. In these experimental runs, all the elevated bed temperatures could therefore only be maintained within ± 5 ° C of their desired values.  36  Chapter 5. Results: Minimum  5.2  Spouting  Velocity  37  Effect of Particle D i a m e t e r  Although the Target sand employed in this work was almost spherical, its exact particle shape factor remained uncertain. The mean particle size was narrowed down by screening and the average particle diameter calculated using Equation (3.25). The effect of particle diameter is shown in Figures 5.8 and 5.9. In all case, minimum spouting velocity Uma increases with particle diameter for a fixed orifice diameter, at any given bed height.  This observation is consistent with the empirical equation of  Mathur and Gishler, Equation (2.1). Only four particle sizes are shown in Figure 5.8, because the smallest size could not be spouted with this intermediate size orifice. The effect of temperature can also been seen in the two graphs. Generally, Ums increases with increasing temperature. For the intermediate size orifice this temperature effect is consistent, but for the small size orifice this generalization only applies unambiguously to the largest particles.  5.3  Effect of Orifice D i a m e t e r  According to the Mathur - Gishler equation, when other conditions are fixed, Ums increases with orifice diameter. Figure 5.10 and 5.11 show the effect of orifice diameter on Ums- In Figure 5.10 at room temperature, Ums oi the middle orifice has the smallest value at the given bed height of 0.3 m when dp = 2.025 mm; while at the temperature of 300° C, the same orifice shows the largest value. At the high temperature of 580° C, when orifice diameter becomes larger, the Ums value also increases. For dp = 1.010 mm at the lower bed height of 0.2 m, maxima are observed in Figure 5.11 at both elevated temperatures but not at room temperature. At 580°C, Ums became smaller than at 300°C for the large orifice. In both figures, there was no consistent trend of Ums with orifice diameter. The difference between the two figures could be attributed to the differences in particle  38  Chapter 5. Results: Minimum Spouting Velocity  Dc=156 mm Di=19.05 m m H=0J20 m OS  00  b-  1  ©  03  A3. b-  b-  © = 300#"C • = 580 C  OP  02  b. O.B  T— 1.0  1.2  1.4  1.6  1.8  2.0  2.2  Particle diameter, dp [mm]  Figure 5.8: Effect of particle diameter on Um,. (Z?c=156 mm, A-19.05 mm, H=0.2 m)  Chapter 5. Results: Minimum Spouting Velocity  Oi  Dc=156 mm Di=12.70 m m H=0.20 m  OS  of 00  .dl  1. 6  d' d'  Ts = 20°C = 300 °C = 580 *C d' d.  o.e  1.0  1.2  1.4  1.6  1.8  2.0  2.2  Particle diameter, dp [mm]  Figure 5.9: Effect of particle diameter on Um>- (.Dc=156 mm, Z?i = 12.70 mm, H=0.2  40  Chapter 5. Results: Minimum Spouting Velocity  Dc=156 mm d p =2.025 mm H=0.30 m  CO  CM  JO"  6 05  00  Tg o = 20 °C © •  d"  = 300 °C = 580 °C  to  d_L  — I  10,0 12.0  1  1  I  1  1  1  1  1  14.0  16.0  18.0  20.0  22.0  24.0  26.0  28.0  1  30.0  Orifice diameter, Di [mm] Figure 5.10: Effect of orifice diameter on Unt. (P c = 156 mm, dp=2.025 mm, H=0.3 m)  41  Chapter 5. Results: Minimum Spouting Velocity  CO  Dc=156 m m dp=1.01 mm H=0.20m to -  d  I  d  B  d-  Tg  o © •  = 20°C = 300 8C = 580 'C  02  d. 10,0 12.0  14.0  16.0  18.0  20.0  22.0  24.0  26.0  28.0  30.0  Orifice diameter, Di [mm]  Figure 5.11: Effect of orifice diameter on Um3. (£>c = 156 mm, c£p = 1.010 mm, H=0.2 m)  Chapter 5. Results: Minimum  Spouting  Velocity  42  size and bed height. This result shows that the Mathur - Gishler equation might not be suitable for predicting Ums at all temperature levels for different particles. The increase of Ums with temperature is again illustrated for most cases plotted in Figures 5.10 and 5.11.  5.4  Effect of B e d H e i g h t  Figure 5.12 shows the effect of bed height H on Ums at different temperatures. It is seen that Ums always increases with H and that the previously mentioned temperature effect on Ums increases as H increases.  5.5  Effect of T e m p e r a t u r e  The effect of temperature and particle size for a given bed height at three different orifice diameters, is illustrated in Figures 5.13 - 5.15. The curves in these figures, as well as in Figures 5.8 - 5.12, were fitted to the data by the method of cubic splines assuming in most cases that Ums was reproducible to ± 5 % . In Figure 5.13 for the intermediate size orifice, it is observed that the higher the temperature, the larger the value of Ums.  This trend is consistent with Equation (2.1)  of Mathur and Gishler. Thus, when the temperature of the air is high, the air density becomes smaller, which results in a higher value of Ums. A similar effect of temperature is shown in Figures 5.14 and 5.15, but mainly for the larger particles. The data for the 1.63 m m particles in Figure 5.15 display more erratic behaviour than the rest. As already illustrated by Figures 5.8 and 5.9, Ums in Figures 5.13 - 5.15 always increases with dp. It is basically known that for small particles at high temperature, viscous forces are dominant. For large particles especially at low temperature, kinetic forces are dominant. Considerations such as these, which might explain some of the apparent anomalies or  43  Chapter 5. Results: Minimum Spouting Velocity  oo  Dc=156 mm Di=26.64 mm d p =2.025 mm  to  02  J, 00  co  dTg  a  = 20°C = 300 °C • = 580 'C  A  CO  dl  d. 0.0  1—  0.1  0.2  ~I—  0.3  0.4  0.5  Bed height [m]  0.6  n— 0.7  0.8  Figure 5.12: Effect of bed height on Ums. (Dc=lo6 mm, D{=26.64 mm, dp=2.025 mm)  Chapter 5. Results: Minimum Spouting Velocity  D c =156 m m Di=19.05 m m H=0.2 m «  CO  d ©  t» to  ID O'  00  d-  dp, mm o = 2.025 n = 1.630 A = 1.200 o = 1.010  02  ©©  d. 0.0  —1  100.0  200.0  300.0  400.0  — i —  500.0  600.0  Bed temperature, °C  Figure 5.13: Effect of temperature on Umt. (Z?c=156 mm, A=19.05 mm)  45  Chapter 5. Results: Minimum Spouting Velodty  Dc-156 nun Di=26.64 mm H=0.2m o  mto SdH  A A  o  o  o  O'  s  $  CO  d"  dp o= a= A= o=  02  d ©-  o  d. 0.0  mm 2.025 1.630 1.200 1.010  — I  100.0  200.0  300.0  400.0  500.0  Bed temperature, °C  Figure 5.14: Effect of temperature on Umt. (Z>c=156 mm, D,=26.64 mm)  600.0  Chapter 5. Results: Minimum Spouting Velocity  46  Dc=156 m m Di=12.70 m m H=0.2 m o doo  d" ©  oa to  d  m to  O"  0~ CO  d"  ffl—  Lp m m  d-  o = 2.025 • = 1-630 A = 1.200 o = 1.010 ffl = 0.915  oq d. 0.0  100.0  200.0  300.0  400.0  —i  500.0  600.0  Bed temperature, "C Figure 5.15: Effect of temperature on Umt. (DC=156 mm, L>;=12.70 mm)  Chapter 5. Results: Minimum  Spouting  47  Velocity  irregularities in Figures 5.8 and 5.15, are best approached by dimensional analysis.  5.6  D a t a Correlation  5.6.1  First O p t i o n  Ignoring p and particle shape, after Mathur and Gishler [24] and Wu et al. [39], Um, = f{dp,pp-  pf,pf,Dc,Di,H,g)  (5.43)  By dimensional analysis,  Ums ^^fd^DjJLPr-pA  (544)  The Mathur - Gishler relation, Equation (2.1), can be expressed as follows  Ums yfl^E  ( dp\ (DiY ( H\° fpp- p/Y \DCJ \DJ \DJ V Pf  (5.45)  The equation of Wu et al. is 77 Ums  y/2pl  1 0 5  ( A \  =10.6(#)  ' \DCJ  / n \ 0-266 , U \ -0.095 / „  (%)  \Dj  (%)  \DJ  n  \ 0-256  (eJLZU)  V  Pi  (5.46)  The simple power relation based on Equation (5.44), of which Equation (5.45) and (5.46) are particular examples, is  y/2jE  \DCJ \DJ \DJ \  P}  with dp in the present study evaluated as the reciprocal mean diameter by screen analysis. The five constants based on a least squares correlation of all the present data, K, a, r, u and £, were 28.4, 1.17, 0.127, -0.0452 and 0.151, respectively. These constants, together with those of Mathur and Gishler and of Wu et al. are summarized in Table 5.3, which also contains the corresponding RMS errors on Ums when applying the corresponding  Chapter 5. Results: Minimum  Spouting  48  Velocity  Table 5.3: Constants in Equation (5.47) and root mean square errors for three correlations  parameter K Mathur-Gishler eq. 1.0 Wu et al. eq. 10.6 This work 28.4 * **  a 1.0 1.05 1.17  T  UJ  i  0.333 0.266 0.127  0 -0.095 -0.0452  0.5 0.256 0.151  RMS, 17.4* 16.5* 8.10*  % 18.7** 7.82** 13.1**  RMS using present data RMS using Wu's data  empirical equations both to the present data and to the data of Wu [22]. It is seen in the table that the RMS  error for the present equation applied to the present data is  less than half that of the other two equations, and that even for Wu's data, the present equation does significantly better than the Mathur - Gishler equation.  5.6.2  S e c o n d Option  Ignoring particle shape but including //, Ums = f(dp,(pP-  pf),pf,fl,Dc,Di,H,g)  (5.48)  By dimensional analysis, dpUmspf _ P  (<%(pP ~ Pf)pf9 A H_ A PP~ pj ^ \ p2 ' D' rc: A dj ^c ' "p PS  (5.49)  i.e.  \  Dc Dc  dp  pf  (5.50) j  If one ignores the last group on the assumption that particle and fluid densities are adequately accounted for by the Archimedes number, then Rems = tp  [Ar,  Di  H  DA  ZV ZV dp  (5.51)  Chapter 5. Results: Minimum  Spouting  Velocity  49  By forcing a direct proportionality between Rems and Ar in Equation (5.50), thereby effectively eliminating / t a s a variable and therefore making the result just another form of Equation (5.44),  JJ^ = A^Wg*£,iZ££)  (5.52)  Correlating the present data by simple power relationships based on Equation (5.50), (5.51) and (5.52), the resulting empirical equations were , n \ 0.0364 ,  ^=4, 5X1 0-M^(|)  u  \ 0-464 / n \ ° - 0 9 4 3 / „  (|)  g)  „ \ °-258  (^M)  (,53,  / n \ °-346 / u \ °-459 / n \ -°- 2178 He„,=40.05^-(|) (-£) g )  (5.54)  and / r> \ 0-795 / 17 N 0.457 / n \ -0.665 /  \ -0.346  "—^"(f) (£) (£) ( ^ )  («•«)  respectively. Note that Equation (5.55) is equivalent to f A \ 1-165 / n \ 0.130 , IT \ -0.043 /  TT  ^  V^F  - 26.92 ( i ) (#) (£) ' v^J \DJ Vzv  „ \ 0.154  (^^) V />/  (5.56)  which is very similar to Equation (5.47) with the empirical constants as listed previously. The RMS errors were 8.22%, 8.17% and 8.10% for Equation (5.53), (5.54) and (5.55), respectively. The differences between these values are insignificant, and t h e absolute match between the RMS errors obtained by Equations (5.55) and (5.47) is attributable to correlating the same variables by different but inter-convertible dimensionless groups. 5.6.3  Third Option  If we assume that the effects of fluid and particle properties are fully accounted for in the minimum fluidization velocity, Umf, for the given fluid-particle system, then Ums = fctn(Umf,  dp, A , £>c, H)  (5.57)  50  Chapter 5. Results: Minimum Spouting Velocity  By dimensional analysis,  E=L=*^ >mS  = i,  n-emJ  *,£*) \ L>C uc  (5.58)  ap j  or  If one includes the additional ratio (pp — Pf)/pj in the correlation, then  Two well tested functional relationships, f(Ar),  from the literature are that of Wen  and Yu [43], Remf = f(Ar)  = ^(33.7)2 + 0.0408 Ar - 33.7 = 33.7(^1 + 3.59 x 10~5Ar - 1] (5.62)  and that of Grace [64], Remf = f(Ar)  = ^/(27.2)2 + 0.0408 Ar - 27.2 = 27.2[\/l + 5.51 x 10- 5 Ar - 1] (5.63)  Simple power relationships based on Equations (5.60) and (5.61), each combined with either Equation (5.62) or (5.63), were used to correlate the present data. The resulting equations and their root mean square errors are: From Equation (5.60) plus (5.62), , T) v 0.0296 / LJ \ 0.311 / n \ 0.0604  Rems = 2 4 . 6 [ N / l + 3 . 5 9 x l 0 - M r - 1] (£)  (-g-J  '  '  dp  ±  12.6% (5.64)  From Equation (5.60) plus (5.63), Dt\0U8 Rems = 25.7[Vl+5.51 x 10~Mr - 1] (jj-"j  fH\0-350 (—) c'  fDt' \ Up  ±  10.5% (5.65)  Chapter 5. Results: Minimum Spouting Velocity  51  From Equation (5.61) plus (5.62), -0.0177 / IT \ 0.438 / »-. \  ^ - l ^ l  + SJWxlO-Ar-l]^)--  0 1 3 2  /  N \ °-272  ( I ) ' ( £ ) ' ( a ^ a l ) ' W V Pf  ± 8.28%  (5.66) From Equation (5.61) plus (5.63), ,  / n \ 00847 / r r v 0.439 /  ^.-4.1,(^1 +SJ lxI^Ar-.](^)  (£)  n  UJU \ U0.0302 * // .  g )  u i o 9 . N, \\ 0.1897 '  (*jfd)  ±8.17* (5.67)  The inclusion of (pp — p/)/pf  thus gives better correlation than its exclusion, and the use  of the Grace f(Ar) is then marginally better than that of Wen and Yu. 5.6.4  Fourth Option  Alternately, if we assume that fluid and particle properties are best accounted for by the free setting velocity, Ut, of the particles, which is related to the minimum inlet jet velocity, Umi, then  m \2 Umi = (jf)  Ums = fctn(Ut, Dc, Dh H, dp)  (5.68)  By dimensional analysis, Ums  =  Rems  =,(  (Di H DA  7r isr 'U , D;'^j  (5 69)  -  But Ret = ^  ^ = <f> (Ar) f1  (5.70)  Therefore  «e„,^(^(f,f,g  (5.71)  If, as before, one includes the additional ratio (pp — pj)/pf in the correlation, then  52  Chapter 5. Results: Minimum Spouting Velocity  A correlation for Ret as a function of Ar, i.e. <f> (Ar), over a wide range of Ret was ontained from Table (5.3) of Clift et al. [62]: logw Ret = -1.81391 + 1.34671 W - 0.1242W2 + 0.006344W3  (5.73)  12.2 < Ret < 6.35 x 103 where W = logwND and Np = 4Ar/3. Based on simple power relationships amongst the remaining non-dimensional ratios in Equations (5.71) and (5.72), the resulting empirical correlations and their root mean square errors are: , r) \ 0.515 , IT v 0.521 / n \ -0-374  Rems = 4(Ar) x 0.391 ^ j  f^-j  Uf\  ± 9.01%  (5.74)  and , n \ 0.541 , u \ 0-452 / n \ ~ 0 - 4 1 4 / „  „ \ -0-149  (5.75) Note that in the correlations all the data were used which satisfied the condition H > 0.2 m. The Fortran program for the Ums correlations is listed in Appendix E. A parity plot for Equation (5.75), the best fit correlation of all those generated in the present work, is presented in Figure 5.16. The goodness of fit of all the present data for Equation (5.75) is compared in Table 5.4 with that of Mathur and Gishler [24], Equation (2.1); Wu et al. [39], Equation (2.6); and Grbavcic et al. [32], Equation (2.4). It is seen that, while Equation (5.75) shows considerably smaller average and RMS errors than the others, the Grbavcic equation gives a better overall fit than that of Wu et al, which in turn is slightly better than that of Mathur and Gishler. Percentage deviations for individual runs are listed in Appendix F.  53  Chapter 5. Results: Minimum Spouting Velocity  0.0  1  I  1  I  1  1  1  I  0.2  0.4  0.6  0.8  1.0  1.2  1.4  1.6  I  1.8  2.0  Predicted U m s [m/s] Figure 5.16: Experimental values of Umt vs. values predicted by Equation (5.75).  Chapter 5. Results: Minimum  Spouting  Velocity  54  Table 5.4: Comparison of average and root mean square errors of Ums by equations of Mathur and Gishler, Wu et al, Grbavcic et al. and best fit by present work  AVG ERR, % RMS ERR, %  M-GEq. 14.2 17.4  WuEq. 13.3 16.5  Grbavcic Eq. This work 5.82 11.2 13.6 7.43  A comparison of the experimental data with the above four correlations for the two largest particle sizes is shown in Figures 5.17 and 5.18 for the two smaller orifice sizes, at both room temperature and 580° C.  For these particular particles it appears that  the Mathur - Gishler equation actually gives better predictability than t h e equation of Wu et al. at high temperature and vice versa at room temperature, while t h e equation of Grbavcic et al. gives its best agreement for both temperatures at low bed height. Equation (5.75) gives somewhat more consistent agreement with the experimental data than the others, irrespective of temperature or bed height. Although data for H = 0.1 m were ignored in arriving at this empirical equation (as well as at all the others generated in this thesis), data points for H = 0.1 rn are shown in Figures 5.17 and 5.18 for comparison purposes. Applied to the experimental data of Wu [22], Equation (5.75) shows an RMS error of 10.38 %. The same method of correlating Wu's data yields the empirical equation ,  n  N  * „ - * . , > * 2.03(f)  0.632 , IT s 0.381 / n \ - ° - 3 7 7 / „  ( £ )  ( g )  „ \ -0-145  ( < ^ )  (5,6,  with an RMS error of 6.62 %. This value is smaller than 7.82 %, t h e RMS error obtained for the same data by Equation (2.6) of Wu et al. [39], which ignores viscosity as a parameter, and supports the choice of free-settling terminal velocity of the particles is a key parameter in the correlation of Ums. Correlating the 305 data points of the present  Chapter 5. Results: Minimum Spouting Velocity  55  study along with the 112 data points of Wu [22] by the same scheme yields , n \ 0.555 , IT \ 0-467 /  ^  = ^)xU.(g)  (|)  n  (f)  \ -0.388 /  (**)  \ -0.126  (5.77,  with an RMS error of 8.21 %. That this value exceeds the RMS error for both Equation (5.75) and (5.76) could be due to a global difference in the way the respective data sets are clustered.  Chapter 5. Results: Minimum Spouting Velocity  56  CO  Dc=156 m m / Di=19.05 m m • dp=2.025mm/ /  02  1 LEGEND • 20 #C • 580 °C This work M-G_Ecpiation Wu'Equation " Grbavcic Eq.  CO  CO  d.  — i  0.0  0.1  1  0.2  0.3  1  1  1  1  1  1  0.4  0.5  0.6  0.7  0.8  0.9  1.0  Bed height [m] Figure 5.17: Comparison of correlations for Um, with experimental data. (Z?c=156 mm, £,'=19.05 mm, dp=2.025 mm)  Chapter 5. Results: Minimum Spouting Velocity  57  W  Dc=156 m m Di=12.70 m m d p =1.630 m m  /  • •  0.0  T  T  0.1  0.2  LEGEND 20 «C 580 °C This work M—G jEojiatioii WiTEq'uatlon Grbavcic Eg.  " I —  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1.0  Bed height [m] Figure 5.18: Comparison of correlations for Um, with experimental data. (J9C=156 mm, A-12.70 mm, dp=1.630 mm)  Chapter 6  Results: Maximum Spoutable B e d Height  The maximum spoutable bed height , Hm, is the maximum bed height at which steady or stable spouting can be obtained. Above such a height, spouting can not be effected for any gas flow, so it is a transition point on a regime m a p . The measurement of Hm was approached from bed heights above Hm, so that solids were intermittently discharged from t h e column until stable spouting could just be achieved, for which H was taken as Hm. Further measurements were then made for values of H below Hm by progressively discharging more solids.  6.1  Spoutability  In the present study, four of the five particle sizes could spout at all temperatures. The smallest particles, with a mean diameter of 0.915 mm, could only spout when the smallest orifice was used at both room temperature and high temperature and when the intermediate size orifice was used at room temperature. Table 6.5 lists all the spoutability trials for the sand particles. Chandnani [23] developed a criterion, based on experiments at room temperature, which states that stable spouting can only occur if Di/dp  < 25.4.  However, this criterion failed for two situations. For the intermediate size orifice with Di/dp  = 20.82, spouting only occurred at room temperature. For the large size orifice  with Di/dp  = 26.38, spouting was obtainable for all temperature levels. These results  suggest that temperature has some effect on the criterion. Similar results were obtained by Wu et al [39]. 58  Chapter 6. Results: Maximum Spoutable Bed Height  59  Table 6.5: Spoutability of sand particles Run 1 -6 7-12 13-18 19-24 25-30 31-36 37-42 43-48 49-54 55-60 61-66 67-72 73-78 79-84 85-90  Di (mm) 19.05 19.05 19.05 19.05 19.05 26.64 26.64 26.64 26.64 26.64 12.70 12.70 12.70 12.70 12.70  dp (mm) 2.025 1.630 1.200 1.010 0.915 2.025 1.630 1.200 1.010 0.915 2.025 1.630 1.200 1.010 0.915  Di/dp 20°C 9.407 11.69 15.88 18.86 20.82 13.16 16.34 22.20 26.38 29.11 6.272 7.791 10.58 12.57 13.88  V V V V V V V V V  Spoutability 170°c 300°C 420° C  500°C  580° C  V V V V  V V V V  V V V V  V V V V  V V V V  X  X  X  X  X  V V V V  V V V V  V V V V  V V V V  V V V V  X  X  X  X  X  X  V V V V V  V V V V V  V V V V V  V V V V V  V V V V V  V V V V V  Chapter 6. Results: Maximum  6.2  Spoutable  Bed Height  60  Maximum Spoutable Bed Height  A frequently used equation in predicting Hm is that of McNab and Bridgwater [44], Equation (2.10). With &i = 1.11 to best fit their existing experimental data, it becomes Hm = \ D l \ dv  \Dc] LAJ  2/3  [-7001 (Vl+35.9 xlO-Mr-1)2 .Ar.  (2.10a)  In the present study, using the experimental data obtained, two graphs were composed based on the McNab - Bridgwater equation. In Figure 6.19, Hm/Dc  was plotted against  [Dddp] [ A / A ] 2 / 3 [700/Ar] (y/l + 35.9 x 10~6 Ar - l ) 2 . The predicted values show fair agreement with the experimental values (RMS=2A.7  %).  However, with bx = 1.11,  Equation (2.10a) is not the best fit. By applying a least squares analysis, a best fit straight line through the origin for the experimental data in the present work has a slope 0.881 (RMS=22.1%).  Therefore a new value of &i, 1.04, was obtained. This suggested  value predicts a lower bed height than the McNab - Bridgwater equation. Both equations are plotted in Figure 6.19. Another graph, which plots [HmdPlD2c] [ A / A ]  against Ar,  is presented in Figure 6.20. Along with the experimental data, it also shows the McNab - Bridgwater equation plotted with both the old and the new value of 6X. In both graphs, solid lines represent the McNab - Bridgwater equation and dashed lines represent the newly fitted equation.  6.2.1  Effect of P a r t i c l e d i a m e t e r on Hm  If the expression for Ar is substituted into Equation (2.10a), the latter becomes Hm =  C2 d  $  y/l + C3<% - 1  61  Chapter 6. Results: Maximum Spoutable Bed Height  o  q o  /  o  / o S O  £1 « o "*1  O /CD o / cv'  O  q  ^O °9,  o o © o  o o<  Dc=156 mm o  d. 0.0  1.0  2.0  i  3.0  4.0  5.0  6.0  7.0  8.0  Dc \Dc]2/3 [700' ( \ / l + 35.9 x 1 0 - 6 4 r - l) 5  UP\  .Ar.  Figure 6.19: Comparison between experimental data (points), prediction by Equation 2.10a (solid line) and prediction by modified equation (broken line).  Chapter 6. Results: Maximum Spoutable Bed Height  62  Figure 6.20: Comparison between experimental data (points), prediction by Equation 2.10a (solid line) and prediction by modified equation (broken line).  Chapter 6. Results: Maximum  Spoutable  where C2 = 700D8J3D^/3p,2/(pp  -  Ps)P}g  63  Bed Height  and C3 = 35.9 x l O " 6 ^ -  2  Pf)Pfg/p  .  (dp)crit  is found by setting d(Hm) / d(dp) equal to zero. The solution is (dp)3crit =  (6.79)  8 C  / 3  or  1/3  F  {dp)crit = 60.6  (6.80)  .(pp-Pf)9Pf. Because d?Hmld(dp)2  from Equation (6.78) is negative at dp — (dp)crit, this critical value  of dp represents the particle diameter at which Hm achieves a maximum as dp is increased for a fixed column geometry and fixed fluid and particle properties. Equation (6.80) states that the critical value of dp depends on particle density, gas density and gas viscosity. In this thesis, the particle density in all the experiments is the same, so only the gas properties, which depend on temperature, could change the value of (dp)crit- For air spouting of sand particles at atmospheric pressure, the critical values of dp as given by Equation (6.80) are listed in Table 6.6. Table 6.6: Change of critical value of dp with temperature  temperature T, (°C) density pf , (kg/m3) viscosity p x 105 , (kg/m-s) critical value dp , (mm)  20 170 1.205 0.797 1.84 2.48 1.358 1.902  300 0.616 3.00 2.353  420 0.509 3.43 2.741  500 0.457 3.65 2.962  580 0.414 3.79 3.139  T h e experimental data showing the effect of particle diameter for different orifice sizes and temperatures, and the same effect calculated by the McNab - Bridgwater relation, Equation (2.10a), are plotted in Figures 6.21, 6.22 and 6.23. Generally, Equation (2.10a) overpredicted Hm substantially at room temperatures and underpredicted Hm slightly at high temperatures. Considering the fact that the least squares fitted equation whereby bx equals 1.04 instead of 1.11 gives lower bed height prediction over all temperature levels,  Chapter 6. Results: Maximum  Spoutable Bed Height  64  the modified McNab - Bridgwater equation with bi = 1.04 would strike a better balance between its predictions at low and high temperatures. From Table 6.6 and the discussion above about the critical value of particle diameter for Hm, it is noted that Hm increases with increasing dp below the critical value and decreases with increasing dp above it. This trend is demonstrated in figures 6.21 - 6.23 at room temperature. The trend towards a maximum is also exhibited at the two higher temperatures, but since the values of (dp)^  listed for these two temperatures in Table  6.6 exceed the largest particle size studied, the corresponding maxima are not achieved within the range of the plots. It should also be noted that the use of the approximate Wen - Yu [43] constant, 35.9 x 10~ 6 , in the derivation of Equation (2.10) may be a source of error in the prediction of {dp)CTit by that equation.  6.2.2  Effect of Orifice D i a m e t e r on Hm  The effect of orifice diameter on Hm for three different temperatures, both experimentally and by the McNab - Bridgwater Equation (2.10a), are shown in Figures 6.24, 6.25, 6.26 and 6.27 for the four sand diameters of 2.025 mm, 1.630 mm, 1.200 mm and 1.010  mm,  respectively. If all other conditions are fixed, then Hm decreases with increasing value of the orifice diameter. The observed trends were pretty much consistent with that predicted by Equation (2.10a).  6.2.3  Effect of T e m p e r a t u r e on Hm  Equation (2.10) shows that Hm is a function of Ar, which incorporates the entire effect of fluid properties. Therefore, provided that other conditions remain the same, the effect of temperature on Hm is given by the effect of Ar on Hm. When temperature increases, air density decreases while air viscosity increases, which results in a lower value of Ar. If the McNab - Bridgwater Equation (2.10) is written as a relation between Hm and Ar, it  Chapter 6. Results: Maximum Spoutable Bed Height  65  Dc=156 mm Di=26.64 mm  1.2-  LEGEND D 2 0 °C 2 0 *C B 300 «C 300 #C 580-C 5 8 0 «C  1.1-  B 0.9-  <u &  0.7-  -d  a  r^  0.6  3«J  0.5 H  o A  m  m 0.4H  m  CO  M d  0.3-1 0.20.10.00.8  1.0  1.2  ~i— 1.4  i  1.6  1.8  2.0  2.2  Particle diameter, d p [mm]  Figure 6.21: Effect of particle diameter on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. (Dc = 156 mm, D{ = 26.64 mm)  Chapter 6. Results: Maximum Spoutable  66  Bed Height  1.2-  LEGEND  1.1-  a 20 #*C 20 C ffl  580 °C 580 *C  1 ^ B  300'C 300 *C  0.90.80.7-  a 0.6GB  0.5O  P<  ffl  0.4 H 0.30.2-  Dc=156 mm Dj=19.05 mm  0.10.0-  0.8  1.0  1.2  1.4  1.6  1.8  2.0  2.2  Particle diameter, d p [nun] Figure 6.22: Effect of particle diameter on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. (Dc = 156 mm, D{ = 19.05 mm)  67  Chapter 6. Results: Maximum Spoutable Bed Height  1.21.11.0-  0.90.80.70.6-  m  ea 0.50.4 /  LEGEND  a 20 'C 20 *C a 300 'C 300 "C 580'C 580 CC  0.30.2-  Dc=156 m m Di=12.70 m m  0.1 0.0-  i  0.8  1.0  1.2  1.4  1.6  i  1.8  2.0  2.2  Particle diameter, d p [mm] Figure 6.23: Effect of particle diameter on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. (Dc - 156 mm, £>,- = 12.70 mm)  Chapter 6. Results: Maximum Spoutable  1.2-  I  i  i« JO  «  3o a  n X es  2  68  Bed Height  LEGEND a 20 »c JIPc  1.11.0-  •  0.9  3 0 0 *C 680'C  0.80.7O.S 0.6 0.40.30.2  O.H 0.0-  Dc-156 mm dp=S.025 mm  —I 10.0 12.0  1 14.0  1 16.0  1 18.0  1 1 20.0 22.0  1 1 1— 24.0 26.0 28.0 30.0  Orifice diameter. Di [mm]  Figure 6.24: Effect of orifice diameter on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. (Dc=156 mm, <fp=2.025 mm) L21.1  B  i  L0 0.9 0.8  0)  si  •a ©  •8  1  0.7-| 0.6 0.6H 0.4  a,  0.3  m X  0.2-  3  0.10.0-  Dc=156 mm dp=1.630 mm —i 10.0 12.0  1 1 1 1 1 1 — - I — —I 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0  Orifice diameter. Dj [mm]  Figure 6.25: Effect of orifice diameter on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. (£> c =156 mm, dp=1.630 mm)  Chapter 6. Results: Maximum  Spoutable  Bed Height  1.2-  LEGEND a  1.1-  J.0  69  20 *C SO _*c  olPc  1.0-  3 0 0 *C  X  EBIT'C  0.9-  i  5 8 0 *Q  0.8  A  0.7  •a v  0.6H 0.6  •8 o a  0.3  a  0.2  0.-H  CO  2  D c =156 m m d p =1.200 m m  0.1 0.0-  —I 10.0 12.0  1 14.0  1 16.0  1 16.0  1 20.0  1 22.0  1 1 1— 24.0 26.0 28.0 30.0  Orifice diameter, Di [mm]  Figure 6.26: Effect of orifice diameter on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. (£> c =156 mm, dp=1.200 mm) L2-  LEGEND a 20 *C 20 *C  1.1  a a  4 0) J3 •O V  B  1.0  •  0.9  0.70.6 0.6H  §  0.4  2  580'C ._58_O_;Q  0.8-  •8 S3  300'C  IgO'C  0.3 H 0.2 0.1 0.0-  Dc=156mm d p =1.010 m m —i 10.0 12.0  1 14.0  1 16.0  1 16.0  1 20.0  1 22.0  1 24.0  Orifice diameter, Di [mm]  1 26.0  1— 28.0 30.0  Figure 6.27: Effect of orifice diameter on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. (Dc=156 mm, rfp=1.010 mm)  Chapter 6. Results: Maximum  Spoutable  Bed Height  70  has the form  Hm = Ci  yj-h+35-9  x 10 Mr  ~ - )f£j  (6 81)  -  Differentiating both sides of Equation (6.81) with respect to Ar, while other variables included in C\ are kept constant, leads to (see Appendix B): dH„, --P- > 0 for dAr  Ar>0  v(6.82)  '  A derivative greater than zero for all values of Ar implies that Hm increases with increasing Ar.  Thus, Hm should increase with decreasing temperature. The prediction  was well supported by the experimental results plotted in Figures 6.28, 6.29 and 6.30 for two particle sizes. The Hm data using the intermediate size orifice (Figure 6.29) are reasonably well predicted by Equation (2.10a); and data from the other two orifices (Figure 6.28 and Figure 6.30) were qualitatively in agreement with this equation. The existence of a critical diameter, as discussed in Section 6.2.1 and illustrated in Figures 6.21 - 6.23, helps to explain why the smaller particles, for which Hm always fall to the left of the maximum (i.e. on the Hm - rising side of the curve) on these figures, show a greater temperature effect in Figures 6.28 - 6.30 than the larger particles, which fall to the left of the maximum at the high temperatures but to the right of the maximum (i.e. on the Hm - falling side of the curve) at room temperature.  71  Chapter 6. Results: Maximum Spoutable Bed Height  1.1d  • PJ5S*2.025 2.025 o 1.010 1.010  1.0-  a  0.9-  i  0.8-  • i-t  &  0.70.6-  • O  t—t  \  \  •  .  0.5* O OH  • \  0.4-  cd  •  ^ s  CO  M  O \  N  N  0.3-  N  \  o  N X  o o  0.2*""—'-*•*.  Dc=156 m m Di=26.64 m m  0.10.0-  i  0.0  100.0  i  200.0  o  1  i  300.0  400.0  i  500.0  600.0  Temperature, "C  Figure 6.28: Effect of temperature on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. (Dc=156 mm, D{ =26.64 mm)  Chapter 6. Results: Maximum Spoutable Bed Height  72  1.1-  dp, mm • 2.025 2.025 o 1.010 1.010  1.0-  J, a  0.9\  i  0.8-  <D X>  0.6-  o  \  \  0.7-  \ i—i  ^3  o ft M  2  0.5-  s  \  \  \  o\ 0.4-  \\  • s  V  N N  0.3-  o  0.2-  Dc=156 m m Di=19.05 mm  0.1-  o.o-  i  0.0  100.0  i  200.0  "  1  1  300.0  400.0  •  •  —  -  1  500.0  600.0  Temperature, °C  Figure 6.29: Effect of temperature on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. (Z?c=156 mm, Z)t=19.05 mm)  Chapter 6. Results: Maximum Spoutable  x> ex  1.0-  \  <L)  0.7-  o .a  0.6-  r—1  •s  o a. CO  dp, mm • 2.025 2.025 o 1.010 1.010  \X \ X \ X \ X \ X \ X \ X \ X  0.9-  i  73  >  1.1-  0.8-  Bed Height  \ \ \ \ \ \  °•  x.  X  X X  \  \ X X  N  •\ \  \  •  \  ° \\  0.50.4-  N  •  ^v^^ - ^ B  \  \  N  %  % N  O  N  o  0.3-  M  "•**••  0.2-  O ^m  """—--  Dc=156 m m Di=12.70 m m  0.10.0-  i  0.0  100.0  i  200.0  1  1  1  300.0  400.0  500.0  600.0  Temperature, °C  Figure 6.30: Effect of temperature on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. (Z) c =156 mm, D,=12.70 mm)  Chapter 7 Results: Average Spout Diameter  From visual observations in the experiments, the spout diameter expanded and then converged slightly in the conical region . Above the conical region, the spout diameter remained constant but diverged near the bed surface. The average spout diameter was determined using Eq.(4.26). As mentioned earlier, all Ds values were obtained at the condition U/Ums = 1.05.  7.1  Effect of B e d T e m p e r a t u r e o n Ds  Basically the temperature had an almost negligible effect on the average spout diameter, as shown in Figures 7.31 and 7.32. This result was generally in agreement with the prediction of Wu et al. [39], Equation (2.23), but contradicted that of McNab [50], Equation (2.22), which predicts that Ds decreases with increasing temperature. The Wu et al. equation also gave better absolute prediction than the McNab equation, especially at elevated bed temperatures, where the latter consistently underpredicted Ds.  7.2  Effect of B e d H e i g h t on Ds  Average spout diameter was found to change with bed height. As shown in Figure 7.33, bed height had a large effect on the value of Ds. At the high bed temperatures, the spout diameter increased with increase of bed level, while at some of the lower temperatures, the spout diameter first decreased slightly with increasing bed level and then increased.  74  Chapter 7. Results: Average Spout Diameter  75  4.5  4.0-  S .CJ.  3.5 -i  m Q  3.0-  <D -P (1)  a as  2.5-  •3  2.0-  O  ft CQ  <D 00  1.5-  fi  5  1.0-  •  0.5-  0.00.0  LEGEND Experimental McNab Equation Wu.Equation  — i — 100.0  200.0  300.0  Di=19.05 m m d p =1.630 m m H=0.3 m 400.0  500.0  600.0  Bed temperature, °C Figure 7.31: Effect of temperature on D.. (D c =156 mm, Z><=19.05 mm, ^=1.630 mm)  76  Chapter 7. Results: Average Spout Diameter  4.5  4.0-  3.5  3.0-  2.5-  2.0-  1.5-  LEGEND • Experimental McNab Equation Wu Equation  1.0-  0.5-  Di=26.64 m m d p =2.025 m m H=0.3 m  0.0 0.0  100.0  200.0  300.0  400.0  500.0  600.0  Bed temperature, °C Figure 7.32: Effect of temperature on D,. (D c =156 mm, Z>,=26.64 mm, <fp=2.025 mm)  Chapter 7. Results: Average Spout  Diameter  77  At the high bed temperatures the spout diameter diverged continuously, as per spout shape (a) of Mathur and Epstein [1], while at the lower temperatures the spout diameter followed their undulating spout shape (e), except that it necked only once rather than twice. The spout shape and change of spout diameter in the vicinity of the gas inlet is a matter of importance since it directly affects the longitudinal profile of gas velocity in the spout, and consequently also influences particle velocity and voidage profiles.  7.3  C o m p a r i s o n w i t h E x i s t i n g Correlations  Two empirical equations for Ds were compared with all the experimental data. One was that of McNab [50], Equation (2.22), while the other was that of Wu et al. [39], Equation (2.23). Parity plots comparing the experimental values with the calculated values are given in Figures 7.34 and 7.35.  In Figure 7.34, the predicted values by the McNab  equation almost matched the experimental values in the temperature range 20 - 170°C, but at the higher temperatures, the experimental values were consistently underpredicted by this equation (overall RMS  — 24.4%). Figure 7.35, in contrast, shows that calculated  spout diameters using the Wu equation were very close to the experimental values at all temperature levels (RMS  = 10.4%). Both the experimental values and the calculated  values, together with their percentage errors, are listed in Appendix G. The superiority of the Wu et al equation over that of McNab apparently arises from the fact that the former, unlike the latter, explicitly includes the effect of gas density and gas viscosity, hence of gas temperature.  Chapter 7. Results: Average Spout Diameter  78  4.50  4.25-  I 09  Q  4.00-  T S /C • = 20 o = 170 o = 300 A = 420 x = 500 + = 580  •  u -p  .1-1  3.75-  o CO  So  3.50-  3.25-  Di=26.64 m m dp=2.025 mm  3.000.2  0.3  0.4  0.5  I— 0.6  0.7  Bed height H [cm] Figure 7.33: Effect of bed height on Dt. (Dc=156 mm,  JD,-=26.64  mm, ^=2.025 mm)  79  Chapter 7. Results: Average Spout Diameter  3.5-  1  .1  D  TS.°C o = 20 o = 170 o = 300 A = 420 X = 500 + = 580  4.0-  3.0-  /  n/ B D  u/  By 0  9  n/  A  £y o  6  *4 J r V  A  X  2.5D  T3  <u i—i  / 2.0-  3  o  S  /§ ^  /  1—f  1.5-  /  *  A  0  +  X 1  +  ^ +  Z o * ? X ^ ° +AA A  *  o  A +>$<  x  +  1.0-  0.5-  0.0-  i  0.0  0.5  i  1.0  T 1.5  1 2.0  1 2.5  1 3.0  —1  3.5  1  4.0  4.5  Experimental Ds [cm] Figure 7.34: Comparison of D, measured experimentally with D, predicted by McNab equation. (Dc=156 mm)  Chapter 7. Results: Average Spout  Diameter  80  Experimental Ds [cm] Figure 7.35: Comparison of D, measured experimentally with D, predicted by Wu et al. equation. (D c =156 mm)  Chapter 8  Conclusions  1. Generally the value of Ums is more difficult to obtain at high temperature than at room temperature, partly because spouting becomes less stable at high temperature but also because of increased measurement difficulties at elevated temperatures. 2. Minimum spouting velocity Ums increases with particle diameter for a fixed orifice diameter, at any given bed height.  This observation is consistent with the empirical  equation of Mathur and Gishler, Equation (2.1). 3. There is no consistent trend of Ums with orifice diameter, showing that the Mathur - Gishler equation might not be suitable for predicting Ums at all temperature levels for different particles. 4.  When the bed temperature is raised, Ums increases, primarily because of the  corresponding decrease in spouting gas density. Temperature has a larger effect on the Ums of large particles than on that of small particles, possibly because viscous as opposed to inertial forces become more dominant for the latter. 5. Ums always increases with H and the temperature effect on Ums increases as H increases. 6. A best fit Ums correlation is obtained by including the free settling velocity, Ut, of the particles, which largely accounts for fluid and particle properties. Ut is found to be better than Umf as a correlating parameter for Ums.  81  Chapter 8.  82  Conclusions  7. The best fit equation for Ums and its root mean square error is: /  n  x o.54i / IT N o.452 /  Rem, = ,iM), 1.63(|i)  (I)  n  \ -°-414 / .  (*J  . \ -°  ( ^ )  1 4 9  ±7.43% (5.7.)  where <f>(Ar) = i?e t . Equation (5.75) shows considerably smaller average and RMS  errors  than the Grbavcic equation, which gives a better overall fit than t h a t of Wu et al, which in turn is slightly better than that of Mathur and Gishler. 8. The McNab - Bridgwater equation with b\ = 1.11 overpredicts Hm significantly at room temperatures and underpredicts Hm slightly at high temperatures. The same equation with b\ = 1.04 gives better overall agreement with the experimental data. 9.  There exists a critical value of dp at which Hm achieves a maximum as dp is  increased for a fixed column geometry and fixed fluid and particle properties. The higher the temperature, the larger this value is. 10. Temperature has an almost negligible effect on the average spout diameter. At high bed temperatures, the spout diameter increases with increase of bed height, while at lower temperatures, the spout diameter sometimes first decreases slightly with increasing bed height before it increases. 11. The Wu et al  equation gives better absolute prediction of Ds than does the  McNab equation, especially at elevated bed temperatures, where the latter consistently underpredicts Ds. This is attributed to the fact that the Wu et al equation explicitly includes the effect of gas density and gas viscosity, hence of gas temperature.  Notation  Notation  A  Ratio given by Equation (2.15)  Ai  Cross-sectional area of the rotameter tube  A2  Area of annulus between the float and tube  Ac  Cross-sectional area of the column  Ap  Maximum cross-sectional area of the float  Ar  Archimedes number,  a5  Ratio of spout area to column area  b  Value of exponent on Hm in equation for U„.  bi  Um/Um/  Co  Drag coefficient  CAL  Predicted value  Dc  Inside diameter of column  Dj-  Diameter of inlet orifice  D5  Mean spout diameter  D s (z)  Local spout diameter  dp  Reciprocal mean diameter of particles  (dp)crtt  Value of d p at which H m is a maximum  EXP  Experimental value  f  Friction factor  fx  150(1 - o V/<£<£  f2  1.75(1 -  G  Mass flowrate of gas  g  Acceleration due to gravity  p  ~f  ea)Pf/dpe3a  /g  Notation  H  Static bed height  (m)  Hm  Maximum spoutable bed depth  (m)  h  H/Hm  (- )  k  Constant in Equation (2.2)  (- )  M  Number of data points  (- )  ND  Best number, | Ar  (- )  n  Number of particles accelerated per unit time  (- )  Pat™  Atmospheric pressure  (Pa)  PB  Absolute pressure measured just below inlet orifice with solids in the bed  PE  (Pa)  Absolute pressure measured just below inlet orifice without solids in the bed  (Pa)  Pg  Gauge pressure upstream of rotameter  (Pa)  PM  Absolute pressure of the gas meter  (Pa)  PR  Absolute pressure of the rotameter  (Pa)  Ps  Absolute pressure in the bed  (Pa)  PSTD  1 atm  (Pa)  Qs  Volumetric flowrate in the spout  (m3/  R  Rotameter reading  (- )  TR  Temperature of the rotameter  (° C)  Ts  Temperature of the spouted bed  (° C)  Ua  Superficial gas velocity in the annulus  (m/s  Um  Minimum superficial spouting velocity at Hm  (m/s  UTO/  Minimum superficial fluidization velocity  (m/s  Umi  Minimum gas inlet velocity for spouting  (m/s  Notation  UTOS  Minimum superficial spouting velocity  (m/s)  Us  Superficial gas velocity  (m/s)  U*  Free settling terminal velocity of the particles  (m/s)  VF  Volume of the float  (m 3 )  VM  Measurement volumetric flowrate of the gas meter  (m3/s)  Vs  Volumetric flowrate through the spouted bed  (m3/s)  VSTD  Volumetric flowrate taken from the calibration curves  (m3/s)  W  logw  (-)  Xi  Weight fraction of particles  (-)  z  Vertical distance from inlet orifice  (m)  APa  Measured pressure drop above the orifice  (Pa)  AP/  Pressure drop across bed of particles  ND  at minimum fluidization  (Pa)  APms  Overall pressure drop at minimum spouting condition  (Pa)  AP.,  Overall spouting pressure drop  (Pa)  7  Angle of repose of solids  (-)  e  Overall voidage of the bed  (-)  emj  Voidage at minimum  (-)  A  Reciprocal of sphericity  (")  H  Fluid viscosity  (kg/m-s)  pb  Bulk density of particles  (kg/m 3 )  PF  Density of the rotameter float  (kg/m 3 )  Pf  Fluid density  (kg/m 3 )  fluidization  Notation  86  pp  Particle density  (kg/m 3 )  <f>  Particle sphericity  (- )  ift  Net downward force of solids per unit volume  (kg/m2  • s2)  Bibliography  [1] Mathur, K. 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[26] Charlton, B. G., Morris, J. B., and Williams, G. H., " An Experimental Study of Spouting Beds of Spheres", Rep. Mo. AERE-R4852, U. K. Atomic Energy Authority, Harwell (1965). [27] Lim, C. J. and Grace, J. R., "Spouted Bed Hydrodynamics in a 0.91 m Diameter Vessel", Can. J. Chem. Eng., 6 5 , 366 (1987).  Bibliography  89  [28] He, Yanlong, "Spouted Bed and Spout-fluid Bed Hydrodynamics in a 0.91 m Diameter Vessel", M. A. Sc. Thesis, University of British Columbia, Vancouver, Canada (1990). [29] Ghosh, B., "A Study on the Spouted Bed, Part I: A Theoretical Analysis", Indian Chem. Eng., 56, 353 (1978). [30] Green, M. C. and Bridgwater, J., "An Experimental Study of Spouting in Large Sector Beds", Can. J. Chem. Eng., 6 1 , 281 (1983). [31] Mamuro, T. and Hattori, H., "Flow Pattern of Fluid in Spouted Beds", J. Eng. Japan, 1, 1 (1968); correction, J. Chem. Eng. Japan, 3, 119 (1970).  Chem.  [32] Grbavcic, Z. B., Vukovic D. V., Zdanski, F. K. and Littman, H., "Fluid Flow Pattern, Minimum Spouting Velocity and Pressure Drop in Spouted Beds", Can. J. Chem. Eng., 54, 33 (1976). [33] Thorley, B., Saunby, J. B., Mathur, K. B. and Osberg, G. L., "An Analysis of Air and Solid Flow in A Spouted Wheat Bed", Can. J. Chem. Eng., 37, 184 (1959). [34] Buchanan, R. H. and Manurung, F., "Spouted Bed Low-temperature Carbonization of Coal", Brit. Chem. Eng., 6, 402 (1961). [35] Smith, J. W. and Reddy, K. V. S., "Spouting of Mixed Particle-size Beds", Can. J. Chem. Eng., 42, 206 (1964). [36] Manurung, F., "Studies in the Spouted Bed Technique with Particular Reference to Low Temperture Coal Carbonization", Ph.D. Thesis, University of New South Wales, Kensington, Australia (1964). [37] Cowan, C. B., Peterson, W. S. and Osberg, G. L., "Spouting of Large Particles", Eng. Journal (Canada), 4 1 , 60 (1958). [38] Madonna, L. A., Lama, R. F. and Brisson, W. L., "Solids-air Jets", Brit. Eng., 6, 524 (1961).  Chem.  [39] Wu, S. W. M., Lim, C. J. and Epstein, N., "Hydrodynamics of Spouted Beds at Elevated Temperatures", Chem. Eng. Comm., 62, 251 (1987). [40] Nakamura, M., Hamada, Y. and Toyama, S., "An Experimental Investigation of Minimum Fluidization Velocity at Elevated Temperatures and Pressures", Can.J.Chem.Eng, 6 3 , 8 (1985). [41] Becker, H. A., "An Investigation of Laws Governing the Spouting of Coarse Particles", Chem. Eng. Sci., 13, 245 (1961).  Bibliography  90  [42] Ergun, S., "Fluid Flow Through Packed Columns", Chem. Eng. Progr., 48, No.2, 89 (1952). [43] Wen, C. Y. and Yu, Y. H., "A General Method for Predicting the Minimum Fluidization Velocity", AIChE J., 12, 610 (1966). [44] McNab, G. S. and Bridgwater, J., "Spouted Beds—Estimation of Spouting Pressure Drop and the Particle Size for Deepest Bed", Proc. European Congress on Particle Technology, Nuremberg (1977). [45] Thorley, B., Saunby, J. B., Mathur, K. B. and Osberg, G. L.,"An Analysis of Air and Solid Flow in a Spouted Wheat Bed", Can. J. Chem. Eng., 37, 184 (1959). [46] Malek, M. A. and Lu, B. C. Y., "Pressure Drop and Spouted Bed Height in Spouted Beds", Ind. Eng. Chem. Process Des. Develop, 4, 123 (1965). [47] Lefroy, G. A. and Davidson, J. F., "The Mechanics of Spouted Beds", Trans. Inst. Chem. Eng., 47, T120 (1969). [48] Littman, H., Morgan III, M. H., Vukovic, D. V., Zdanski, F. K. and Grbavcic, Z. B., "A Theory for Predicting the Maximum Spouting Height in a Spouted Bed", Can. J. Chem. Eng., 55, 497 (1977). [49] Littman, H., Morgan III, M. H., Vukovic, D. V., Zdanski, F. K. and Grbavcic, Z. B., "Prediction of the Maximum Spoutable Height and the Average Spout to Inlet Tube Diameter Ratio in Spouted Beds of Spherical Particles", Can. J. Chem. Eng., 57, 684 (1979). [50] McNab, G. S.,"Prediction of Spout Diameter", Brit. Chem. Eng. and Proc. Technol, 17, 532 (1972). [51] Bridgwater, J. and Mathur, K. B., "Prediction of Spout Diameter in a Spouted Bed—A Theoretical Model", Powder Technol, 6, 183 (1972). [52] McNab, G. S. and Bridgwater, J., "The Application of Soil Mechanics to Spouted Bed Design", Can. J. Chem. Eng., 52, 162 (1974). [53] Littman, H., "The Measurement and Prediction of the Maximum Spoutable Height, Spout Diameter, Minimum Spouting Velocity and Pressure Drop at Minimum Spouting In Spouted Beds", Lecture Notes for C.S.Ch.E. Continuing Education Course on Spouted Beds, Vancouver, 1982. [54] Krzywanski, R. S., Epstein, N. and Bowen, B. D., "Spout Diameter Variation in Two-dimensional and Cylindrical Spouted Beds: A Theoretical Model and its Verification", Chem. Eng. Sci., 44, 1617 (1989).  Bibliography  91  [55] Epstein, N. and Levine, S., "Non-Darcy Flow and Pressure Distribution in a Spouted Bed", Fluidization, Proc. Second Engineering Foundation Conference on Fluidization, Davidson, J. F. and Keairns, D. L., eds., p.98, Cambridge Univ. Press (April 1978). [56] Morgan, M. H. and Littman, H., "General Relationships for the Minimum Spouting Pressure Drop Ratio, APms/APmF, and the Spout-Annular Interfacial Condition in a Spouted Bed", in 'Fluidization', Proc. 3rd Engng. Found. Conf., Henniker, Grace, J. R. and Matsen, J. M., eds., p.287, Plenum Press, New York (1980). [57] Pattipati, R. R. and Wen, C. Y., "Minimum Fluidization Velocity at High Temperatures", Ind. Eng. Chem. Process Des. Dcv., 20, 705 (1981). [58] Whiting, K. J. and Geldart, D., "A Comparison of Cylindrical and Semi-cylindrical Spouted Beds of Coarse Particles", Chem. Eng. Sci., 3 5 , 1499 (1980). [59] Geldart, D., Hemsworth, A., Sundavadra, R. and Whiting, K. J., "A Comparison of Spouting and Jetting in Round and Half-round Fluidized Beds", Can. J. Chem. Eng., 5 9 , 638 (1981). [60] Lim, C. J., "Gas Residence Time Distribution and Related Flow Patterns in Spouted Beds", Ph.D. Thesis, University of British Columbia, Vancouver, Canada (1975). [61] Oman, A. 0 . and Watson, K. M., "Pressure Drop in Granular Beds", Management and Petroleum Chem. Tech., 36, R-795 (November 1, 1944).  Refinery  [62] Clift, R., Grace, J. R. and Weber, W. E., "Bubbles, Drops, and Particles", Academic Press, N.Y. (1978). [63] Nicol, T., "Integration of Unequally Spaced Data Points", UBC QINT4P, of British Columbia, Vancouver, Canada (1976).  University  [64] Grace, J. R., "Fluidized-bed Hydrodynamics", Chapter 8.1 in 'Handbook of Multiphase Systems', Hetsroni, M., ed., Hemisphere - McGraw - Hill, New York (1982).  Appendix A  Calibration of R o t a m e t e r s  For a rotameter, the governing equation is:  G = CDA2  2gVF(pF ~ Pf)pJ AF[l (A2/A1)^]\  (A.83)  The coefficient C D depends on the shape of the float and the Reynolds number for flow through the annular space of area A2. If the float is kept at a fixed vertical position, C D can be assumed constant. For a specific rotameter, the only independent variable is then the fluid density. Equation (A.83) then becomes in the case of a gas flow,  G = Bly/pJ  (A.84)  Figure A.36 is a simple flow sheet of the rotameter calibration set-up. If the ideal gas law is assumed, and TM — TR = 20°C, then  VR  =  VM  (A.85) VPR\  and G = GR = PRVR  — PRV;\M  PM^ iPj R  (A.86)  where the subscripts M and R refers to gas meter and rotameter, respectively, and PR — Pf-, the fluid density in the rotameter. Combining Equations (A.84) and (A.86) yields B1 = VIM  M PR  92  PR  (A.87)  Appendix A. Calibration of Rotameters  93  ATMOSPHERE  GAS METER  ROTAMETER  PM,VM,TM  Pi  VALVE  PM M = -  24^  2  Figure A.36: Schematic set-up for rotameter calibration. A standard condition of P = 1 atm and T = 20°C was chosen. Substituting Eq. (A.87) into Eq. (A.84) gives GsTD —  ViM  [PMm  y/PSTD  PR\  (A.88)  and VsTD =  ^STD PSTD  = VM  M  I PR  PR\  PSTD  (A.89)  For an ideal gas, PR PSTD  PR  (A.90)  PsTD  Substituting this relation into Eq. (A.89) gives VsTD = V}M  PM  (A.91)  .VPRPSTD.  Using Equation (A.91), the two calibration curves which follow were produced by Wu [22]. These curves were checked against a gas meter and found to be accurate.  CO  CO  &  •(-4  4  -  50  Pressure — 1 o t m .  100  150  200  V = 0.2692 + 0.0212 X R  Rotameter reading  Temperature = 20 deg C  250  4*.  X  Appendix  A. Calibration of  Rotameters  95  o o  in  -<t en  o  CO  C>J  o  +  bO  o o  00  o II  8 *d o u u  >  Tj<  CO  O  o  01  §  2  Figure A.38: Calibration curve (large rotameter).  Appendix B  Derivation of the Expression for  dAr  The McNab and Bridgwater Equation for predicting Hm is: Hm =  Dl  \ ]  Dc  2/3  \ ]  . dp .  '5686?' (Vl + 35.9 x 10- 6 Ar - \f Ar  (2.10)  The above equation can be rewritten as l — + 35.9 x 10- 6 - \ — Ar V Ar  Hm = C\  (B.92)  whence dHm = 2Ci dAr  Ar  + 35.9 x 10- 6  1 x  2 ^  + 3 5 . 9 x l 0 - 6 V Ar2)  ^ 2 Ar  Ar.2  — +35.9 x 1 0 - 6 - i / - — Ar V Ar r  x  Ar Ar2  1 Ar 2 Ar y/Ar + 35.9 x 10- 6 Ar 2  -J- + 35.9 x 10-6 - J - J -  = Ci  x  1 ^ArVAr  Ar  1 Ar\A4r + 35.9 x lQ-Mr 2  96  V Ar  > 0  /or  Ar > 0  (B.93)  Appendix C Experimental Conditions  Run No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26* 27* 28* 29* 30* 31 32 33 34  D, (mm)  dp (mm)  19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 26.64 26.64 26.64 26.64  2.025 2.025 2.025 2.025 2.025 2.025 1.630 1.630 1.630 1.630 1.630 1.630 1.200 1.200 1.200 1.200 1.200 1.200 1.010 1.010 1.010 1.010 1.010 1.010 0.915 0.915 0.915 0.915 0.915 0.915 2.025 2.025 2.025 2.025  T(°C) 20 170 300 420 500 580 20 170 300 420 500 580 20 170 300 420 500 580 20 170 300 420 500 580 20 170 300 420 500 580 20 170 300 420  Appendix  C. Experimental  35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55* 56* 57* 58* 59* 60* 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81  Conditions  26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70  2.025 2.025 1.630 1.630 1.630 1.630 1.630 1.630 1.200 1.200 1.200 1.200 1.200 1.200 1.010 1.010 1.010 1.010 1.010 1.010 0.915 0.915 0.915 0.915 0.915 0.915 2.025 2.025 2.025 2.025 2.025 2.025 1.630 1.630 1.630 1.630 1.630 1.630 1.200 1.200 1.200 1.200 1.200 1.200 1.010 1.010 1.010  500 580 20 170 300 420 500 580 20 170 300 420 500 580 20 170 300 420 500 580 20 170 300 420 500 580 20 170 300 420 500 580 20 170 300 420 500 580 20 170 300 420 500 580 20 170 300  to O  -*^  ' »»i  R  -c  3  •—•*  o <$  c  -*J  a  X  W O .X  Hs  CNO00OJN-OOJO00  o o o o o o o o o  OOOlDlOWBlfllfi  H r t r l H H H H H r l  O O O O i O O O O O ) • ^ I T H T H O O O O O O  o o o o o o o o o  KN-N-l>-KI*-l--r-NCNCNCJCNCNCNCMCSOJ H r l H H r l r l r l H r l  cNfO^wtosmroo cooococococococoo  o aO C O  -*J  fl  2 "3 o <J £ CO  S 3  (-^ -a  3  T3 CD r^A  CO  u  CJ Oi  *  CX  a <  o o  (XI  cn  cn  4=>  CO •^1  to  *  *  *  *  #  #  o o o o  o o o o  o O O o o O o o O o o o  4*  !-»•  OJ CO CO CO (0 00 - 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' i - ' ^ i o w W i i i t J i m s p K i o t i K D f f l m h ' K H i o i o i Oi M O o u o o K K t J i o K i - ' i o w ^ ^ c n m aio^ioOTU)i^ooustnococriwco^mcoK(DMcnoiucn(»MowNh *ioOTK!DMrocott)0)WM.oo(ocn  C 3 > m O ) C 0 C » 0 0 0 0 0 0 W N J I O W t O I O W M W ^ ^ ^ ^ ^ ^ ^ C n W m w a c n O 5 O ) O ) C » ( » 0 0 0 0 0 0 0 0 I O ( O N 3 I O I O I O I O ( O  O O O O O O O O M - l -  #  o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o u^^h>iou*cnKWO)^tno)SOo«)PWMHS)UWpMWCJMMWJi^i-'iotoif>cncni-'ioWi(»oioiSOO O O > - ' O O O O i - ' O O O O O O O O O O O - > J O O O t O O O O - s I O O O O - - J O O O O O 0 0 O O O O O O O O o o e n o o o o o o o o o o o o o o o o o o o o o o o o o o o o o e n o o o o o e n o o o o o o o o  (OtOtD(0(D(D(0(0(0(OtDtO(OOtO(DtDtD(0(OtDtD(D(DlOtOli)(D(0(D(0(D(0(DtD(0(D(DtOlD(0(0(0(OlD(D(0 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o CTicncTiaicricTicricricTiCTicncnaicTiCTicncTioiaimwmmuicJimcTiaic^  o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o  M N 3 M S 3 I O I O W M S 3 M M W M W M M S 3 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) ( J ) 0 ) 0 ) C J ) ( J ) 0 ) C n O ) C I ) 0 ) 0 1 0 ) f f l ( J ) a 0 1 0 ) 0 ) O l O l O ) 0 1 OOOOOOOOOOOOOOOOOtOGOGOCOCOCOGOGOCAJGOtoGOGOGOGOGOOOGOOOGOGOCOGOGOGOGOCOGOOOGO  C0O1CO0lSOl^O)C0CnS(0SIOSK(»O0l)Sl- WUtO(0tnS(|!.00^(»S^C»IOO00-JOOOS01l-'t0-^if» OOOTO)h'MmO)WlOMHtO^M*i^lOWtjOCI)0--Ji^i^(0000>KtOWailOW>^WOO^SIOI-'0(OOWO)IOl-1  c n o ) 0 ) M ^ c n o ) S i O i | i t n m o ) - j - < ) o o o o ^ o ) o o *l s o o o o w o v o o i - ' O o ( i ) > i o o o u o ) S S O O o u o ) a N ( » o o ( O o  O 0 0 O O O O 0 O O O O O 0 O O O O O O O O O 0 O O O h - ' - O O O 0 > - » - O O 0 0 O l - * 0 0 0 0 0 0 0 l - '  CJl  en  o o o o  o o CO o o <tf CO CO co o o o o o o o o  CO  sf co o o o o o o o o  o o o o  o o o o  o o o o  o o o o  co o o o o  o o o o  •* ^  o o o o  o o o o  «*  o o o o  o o o o  o o o o  o o o o  o o o o  ^  o o o o  o o o o  ^  o o o o  o o o o  <*  o o o o  CO LO LO LO o> <J> O) <tf <tf <tf <tf 00 CO 00 00 CO "tf CD CO CO K K h- 00 00 00 00 CO 00 CO 00 •tf <tf t-i T-l f l 1H t H T-l T H t-l CN CN CN CN CN CO CO CO CO CO  <* CO o o o o  o o o o  o o o o  o o o o  o o o CO CO o o o <tf <tf CO CO CO CO co o o o o o o o o  •* o o o o  o o o o  o o o o  co o o o o  o o o o  o o o o  <*  o o o o  o o o o  ^  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  ^ ^ ^ <* ^ ^ <*  CO LO to LO 05 o en <tf <? CD CD CD f~ K h- CO CO CO 00 00 CO CO 00 CO 00 00 rH T-l 1-1 i-H T H T H CO CO CO CO CO CO • r H t-l T H i H t-l  o o o o  o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o  *  *  *  *  *  *  *  *  CN O O CD LO if  O O CO CN  * o o o o o  O O O O O O O O O O O O O O O O O r H - . - l ' r H i - l - r H ' r H v H i H T - l  C^mNlDC>l^r(010)(10t0in^t0NCNinN(0intN00'*HNr>IClNlflCS(»)CN|0)Cn00(DO00O<tN(0a)O(NU)N (NrlHr(H(D(flli3r<r(HmiO^(l)(l)WrliHTKrl000010lrlHHTH!0(OlOHrHrllflC)WNHW^^I<)tNH (O^WlDW^**^^^CSnCNCNCNO)CNC^000000(O00<OCO«)U)inU3^^*^^^CNCNCSCNCNCNCNC^CN(N(N  *  o o o o o o o o o o o  *  O O L O O O L O O O L O O O O O O O O O O O O O O O O O O O L O O O O O O O O O O O O O O O O O O T O O L O O O C N O O C O O O O O O O O C O O O O O O O O L O O O s ^ O O C N O O L O O O O O O  o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o  0)HCNCNHtNCNHC^CNHSKlflin^flCNW^^(«)(Nr((<)CNHO)OIHC<JCN'riCSCNH(Oin*nO)TH  W L O L O L O U ^ L O L O L O L O L O L O l O L O L O L O W L O L O L O l O l O L O L O L O L O L O l O L O L O L O L O t n L O L O L O L O L O L O L O L O L O L O ^ ^ ^ ^ ^ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O C D C O C D C O C O  O)O5O5O^O)cj)o^o>a5a)O)C)o^a)OioiOTa)O)O50)oi<j)o^o)O^O)Oicno^o>c^OT I H T H I H , H , H , H T H T H I H T H T H I H T H I H T H I H T H V M T H V H I H Y H I H T H ^ T H I M T H , - I I H I H I H T M , H ^ , H ^  C N C N C N C N C N C N C N C N C N C N C N O O O O O O O O O O O O O O O O O O O O O O O O O O ) 0 > C n a ) 0 ) 0 ) 0 0 0 0 0  O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O L O L O L O L O L O L O L O L O L O L O L O OOOOOOOOOOO'^iHlHiH^iHiHlHiHl-lTH'iHTH-^THiMTMi-liHxHvHiMiHiHi-liHiHrHiH-tH-rHCNCNCNCNCN I H T H T H T H I H ^ ^ I H I H T H T H T H T H * H I H T H I H T H I H I H T H T - I V H I H T H I H T H I H T H T H T H T H V H ^ I H ^  H O O K O H N H O S P I f f l H O N H ^ t D N l O O O ) ( 0 ( D O ) 0 ) N n m U 3 0 ( O i n ( J ) W l O O l f l ! O n o f f l * ( D ( J ) ( 0 ( 0 ( f l O  O  O  O  O  CO  O  O  O  O  O  O  O  O  O  O  O  O  O  LO  O  00  o  O  CO  CJ>  CN  00  O  O  O  O  O  O  O  CO  O  CO  CN CN  O  CN  O  O  O  CN  co  CO  O  O  O  O  CO  O  CO  LO CN  O  CN  O  O  O  I  H  I  -  I  CO  I  H  T  H  O  otNnioiow^inwoiioooocsNcsNtoooininiflocicNtomTHPjcofDncoN^ocNoococsoicitnoiomn incMSU5csu5ifln<ou)CNS©«3inu3<,(OH^in^<*c>iifl'*iHin*H^^'H^'5fcN(oin'j,*(0'HwcNHOo) O  CO  ^> CO  Appendix  32(7)  33(6)  34(5)  35(5)  36(4)  37(6)  38(6)  39(5)  40(4)  D. Experimental  0.450 1.772 1.619 1.343 1.197 1.049 0.921 0.496 1.696 1.376 1.236 1.100 0.975 0.486 1.768 1.350 1.200 1.024 0.606 1.639 1.385 1.191 1.025 0.556 1.612 1.212 1.014 0.531 1.152 0.941 0.845 0.778 0.689 0.365 1.315 1.157 1.041 0.881 0.755 0.367 1.222 1.121 0.895 0.737 0.378 1.152 0.996  Data  2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630  103  26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64  0.100 0.615* 0.600 0.500 0.400 0.300 0.200 0.100 0.590* 0.500 0.400 0.300 0.200 0.100 0.540* 0.400 0.300 0.200 0.100 0.500* 0.400 0.300 0.200 0.100 0.440* 0.300 0.200 0.100 0.630* 0.500 0.400 0.300 0.200 0.100 0.585* 0.500 0.400 0.300 0.200 0.100 0.440* 0.400 0.300 0.200 0.100 0.350* 0.300  1.213 0.838 0.835 0.826 0.820 0.814 0.809 0.804 0.644 0.638 0.633 0.628 0.625 0.621 0.528 0.523 0.520 0.516 0.513 0.472 0.468 0.465 0.462 0.460 0.424 0.420 0.418 0.417 1.247 1.239 1.230 1.223 1.217 1.213 0.827 0.824 0.819 0.814 0.805 0.802 0.635 0.633 0.628 0.622 0.620 0.522 0.519  0.0000184 0.0000248 0.0000248 0.0000248 0.0000248 0.0000248 0.0000248 0.0000248 0.0000300 0.0000300 0.0000300 0.0000300 0.0000300 0.0000300 0.0000343 0.0000343 0.0000343 0.0000343 0.0000343 0.0000365 0.0000365 0.0000365 0.0000365 0.0000365 0.0000379 0.0000379 0.0000379 0.0000379 0.0000184 0.0000184 0.0000184 0.0000184 0.0000184 0.0000184 0.0000248 0.0000248 0.0000248 0.0000248 0.0000248 0.0000248 0.0000300 0.0000300 0.0000300 0.0000300 0.0000300 0.0000343 0.0000343  CO «* CO  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  CO LO LOix) o> en en ^ ^ ^ ^ «* "sF ^ CO 0000 <tf 10 to( O N N S 00 00 00 00 CO 00 00 «# <tf<* CO CO COCO CO CO COH H iH H H ^ H M CN CN o o o o  o o o o  o o COCO oo oo oo oo  00 00 o O  ^ <* CN CN o o oo oo oo o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  O O CO CO CO LO LO ID en en to r- N- 0 0 0 0 0 0 0 0 0 0 0 0 ^ ^ ^ ^ ^ 0 0 0 CO CO CO T - I V H - T - I T - I V H T H C N C N C N C N C N C O C O C O  o o ^ <& ^ to CO CO CO CO CO CO COCO o o o o o oo o o o o o oo o o o o o oo o o o o o oo  CO  C O  o o o o  o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o  #  T  H  T  H  T  -  •&  I  I  H  O  O  O  O  -X-  O  O  O  O  O  *  O  O  O  O  O  O  O  O  -  >  -  1  o o o o o o o o o  H T H ^ ( 0 i n C ^ H r t i n ^ ^ 0 0 C J - H H H H H O O ( N O l C N C S T H T H r l « ) ( D l f l H H ^ ^ ( r | C N H r H T - I H T H O O C ^ O I ( N ' H  mc^u)rtoiooowMoiH^scono)U)oincNO)(otNO<oio(N^'HO)0)(Dnon(OKnin(Ootnois(NO(£)  #  O O O O O O O O T - I T - I  #•  O O L O O O L O O O O O O O O O O L O O O O O O O O O O O O L O O O L O O L O O O O O O O O O O O O O O O O O O O O t O O O L O O O O O O O K O O O O C n o O O C O O O C O O O C N O ^ O O O O O ^ O O O O O O O L n (NTHClOCNvHCNCNTHtOtDLO^COCN-rH^^COCNTHCOCOCN-rHCNCN-rH CNCN^CN-rHLOLO^COCNTH^'^COCNT-tCOCN'rHCN  O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O  ^* ^* ^* w* ^* ^* ^* w* ^t* ^* ^* w* ^* ^* ^* ^5* **}* ^^ ^^ ^^ ^* ^r ^* ^^ w* ^7* ^* ^* w* *^r ^* ^^ ^* ^r w* ^* ^* *5* ^* ^* ^* ^* *^ ^* ^* ^r ^*  (D(0<O<OlD(D(D(0tD(D(O(0(D(O(D(0(O(O(D(Dt0<O<D(0(OfO(D(O<O(0(0lD(O(O(DlD(0(0(D(Dt0lD(O(0(D(Ot0 (DtOU)(D(0<0(0<0(DU)tO(0<0(OtO<0(0(0(OU)(0(OU><0(0<OU)tO(0<0<0(0<0(0(0<0(0(OU>(OU}(0(OU>!OQ(C CNCNCNCNCN<NCNCNCNCNCNCNCNC^CNCNCNCNCNCSCNCNCNCNCNCNCNCNCNCNCNCNCNCNC^  O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O C 0 C 0 C 0 C 0 C 0 C 0 C 0 C 0 O O O O O O O O O O O O O O O O O O O O O O O O ' « H i H T H ' « H T - l T - l - r H ' r H T H T H ' r H T H T H - r H T H tOtOtOlOU3tOtDlOCNCNCNCNCNCNCNCNCNCNCNC>ICNCNCNCNCNCNCNCNCNCNCNCNOOOOOOOOOOOOOOO  ocono)(S'HO(<)rioio^oo)Wrt^Lnono(<)nocii>-(0(o<oH(OTHitscoiDO)Oooi)nHri(oooioOH  ^ l f i ( O N O r t m O N O W N « l O S O H 0 ) H ( < ) r l 0 0 < J l O H C S U ) T < C N 0 ) 0 ) H « t i S 0 0 C I l U ) O ) O l t < ) O ) C N ^ U N O < t  CO CO  LO  ^  CD  CO  h<tf  CO  ^  00  CN  ^  O)  CO  o  LO LO  <tf  LO  CO  LO  CO CN LO  00*OS^ffiCO*COSS(OO^OIStDin*CNKlOlO<N(Din(NinU)CNlOCNlO(DLO^<}l(N(0!DU5^CNin'!t,rHin O O - r H O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O CO  CN  Appendix D. Experimental Data  53(3)  54(3)  61(7)  62(6)  63(6)  64(5)  65(5)  66(5)  67(9)  0.437 0.178 0.447 0.410 0.190 0.419 0.408 0.191 1.373 1.233 1.138 1.042 0.953 0.816 0.409 1.447 1.314 1.228 1.115 0.975 0.447 1.476 1.382 1.254 1.078 0.936 0.490 1.459 1.279 1.200 0.986 0.501 1.496 1.372 1.169 1.010 0.497 1.381 1.267 1.156 1.048 0.510 1.101 1.058 1.019 0.974 0.925  1.010 1.010 1.010 1.010 1.010 1.010 1.010 1.010 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 2.025 1.630 1.630 1.630 1.630 1.630  105  26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70  0.200 0.100 0.220* 0.200 0.100 0.210* 0.200 0.100 0.735* 0.600 0.500 0.400 0.300 0.200 0.100 0.610* 0.500 0.400 0.300 0.200 0.100 0.555* 0.500 0.400 0.300 0.200 0.100 0.530* 0.400 0.300 0.200 0.100 0.500* 0.400 0.300 0.200 0.100 0.485* 0.400 0.300 0.200 0.100 0.880* 0.800 0.700 0.600 0.500  0.515 0.512 0.463 0.462 0.459 0.418 0.418 0.416 1.265 1.239 1.247 1.243 1.233 1.219 1.213 0.829 0.823 0.821 0.809 0.804 0.801 0.638 0.635 0.632 0.625 0.622 0.619 0.526 0.522 0.517 0.515 0.512 0.473 0.468 0.466 0.462 0.459 0.425 0.423 0.420 0.418 0.416 1.274 1.272 1.266 1.257 1.248  0.0000343 0.0000343 0.0000365 0.0000365 0.0000365 0.0000379 0.0000379 0.0000379 0.0000184 0.0000184 0.0000184 0.0000184 0.0000184 0.0000184 0.0000184 0.0000248 0.0000248 0.0000248 0.0000248 0.0000248 0.0000248 0.0000300 0.0000300 0.0000300 0.0000300 0.0000300 0.0000300 0.0000343 0.0000343 0.0000343 0.0000343 0.0000343 0.0000365 0.0000365 0.0000365 0.0000365 0.0000365 0.0000379 0.0000379 0.0000379 0.0000379 0.0000379 0.0000184 0.0000184 0.0000184 0.0000184 0.0000184  Appendix  68(6)  69(5)  70(4)  71(4)  72(4)  73(9)  74(6)  75(5)  D. Experimental  0.856 0.774 0.655 0.230 1.101 0.988 0.855 0.748 0.637 0.303 1.287 1.071 0.939 0.849 0.375 1.117 0.982 0.895 0.379 0.997 0.820 0.751 0.380 0.960 0.872 0.753 0.371 0.919 0.874 0.819 0.767 0.704 0.636 0.563 0.474 0.239 0.718 0.674 0.605 0.529 0.437 0.236 0.608 0.560 0.512 0.478 0.204  106  Data  1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.630 1.200 1.200 1.200 1.200 1.200 1.200 1.200 1.200 1.200 1.200 1.200 1.200 1.200 1.200 1.200 1.200 1.200 1.200 1.200 1.200  12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70 12.70  0.400 0.300 0.200 0.100 0.635* 0.500 0.400 0.300 0.200 0.100 0.545* 0.400 0.300 0.200 0.100 0.435* 0.300 0.200 0.100 0.415* 0.300 0.200 0.100 0.395* 0.300 0.200 0.100 0.940* 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100 0.595* 0.500 0.400 0.300 0.200 0.100 0.490* 0.400 0.300 0.200 0.100  1.244 1.230 1.216 1.212 0.826 0.823 0.819 0.812 0.805 0.802 0.633 0.630 0.624 0.622 0.619 0.523 0.518 0.515 0.512 0.464 0.464 0.462 0.459 0.421 0.419 0.418 0.416 1.273 1.263 1.261 1.254 1.246 1.235 1.226 1.218 1.213 0.826 0.821 0.816 0.810 0.804 0.801 0.634 0.630 0.625 0.622 0.619  0.0000184 0.0000184 0.0000184 0.0000184 0.0000248 0.0000248 0.0000248 0.0000248 0.0000248 0.0000248 0.0000300 0.0000300 0.0000300 0.0000300 0.0000300 0.0000343 0.0000343 0.0000343 0.0000343 0.0000365 0.0000365 0.0000365 0.0000365 0.0000379 0.0000379 0.0000379 0.0000379 0.0000184 0.0000184 0.0000184 0.0000184 0.0000184 0.0000184 0.0000184 0.0000184 0.0000184 0.0000248 0.0000248 0.0000248 0.0000248 0.0000248 0.0000248 0.0000300 0.0000300 0.0000300 0.0000300 0.0000300  o o o o  oo oo oo oo  o o o o  o o o o  o o o o  o o o o  o o o o  CO CO CO CO CO CO CO 00 CO CO CO  o o o o  o o o o  CO CO CO CO LO lO LO LO CD CD CD  ^ ^ ^ ^  o o o o  T H  ^ 00 o o o o  o o o o o o  ^ ^ ^  CO CO CO CO • *  o o o o  o o o o  o o o o  co o o o o  o o o o  CO CO N - N 00 CO  co o o o o  >-  co o o o o  to LO LO CD CD CD  co o o o o  T H r l CM CN CN O) CN CN CO CO CO CO CO CO CO CO CO  o o o o  rl  o o o o  T-i  o o o o  T-l  o o o o  vH  •tf <* <tf ^ ^ ^f 00 00 00 CO CO CO o 00 00 00 00 CO 00 <tf «tf «tf ^ ^ •tf o  o o o o  o o o o o o o o o o o o o o o o o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  o o o o  <tf vH  T-l  o o o o  T-l  ^  ^ <* CN CM  CO  T-l  o o o o o o o o  00  o o o o  CO CO T-l  o o o o  T-l  1-1  o o o o o o o o o o o o  ^ ^ ^ <* ^ CO CO CO CO CO o o o o  o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o  Jt  J/.  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Newton's Method will be used. This method requires that the initial values of bl,b2,b3,...bn are very close to the solution in order to obtain convergence. The method of Multiple Linear Regression is used to provide this set of initial values of bi's. To apply the latter method, the above equation has to be rewitten as  Nl=bl*(N2**b2)*(N3**b3)*...*(Nn**bn)  ie  Y=f(X1,X2,X3,...Xn)=al*Xl+a2*X2+a3*X3+...+an*Xn lnNl=bl+b2*lnN2+b3*lnN3+...+bn*lnNn  This program is set to handle a maximum of 10 unknowns (bl,b2,b3,...bn) and 305 sets of data points. To accomodate more unknowns and/or sets of date points, change the dimensions of the arrays accordingly and of course, the DATA statement in the beginning of the MAIN program. The data file should be in the form of N11,N21,N31,...Nnl  Nlm,N2m,N3m,...Mnm where m=number of sets of data points n=number of unknowns sought  DATA M,N,FACTOR,DB/305,5,1.,0.0001/ DIMENSION X(10,384),XN(10,384),Y(384),YN1(384),A(384),B(384) DIMENSION PR0D(384),DIFF1(384),DIFF2(384),DELB(10),C0E(384), * UMS(384),DP(384),DI(384),DC(384),SUM1(384),SUM2(384),  109  Appendix  E. Fortran  * * C C C  Programs  H(384),RH0P(384),RH0(384),VISC(384),UIF(384),AR(384), W(384),RET(384)  Read input data READ(4,10)(UMS(I),DP(I),DI(I),DC(I),H(I),RH0P(I),RH0(I),VISC(I), * 1=1,M) 10 F0RMAT(F11.3,F8.3,F8.2,F8.1,F8.3,F8.0,F8.3,F11.7) NP=N+1  C C C  Give value to elements of the non-linear equations DO 20 K=1,M AR(K)=(DP(K)*1.E-3)**3*(RH0P(K)-RH0(K))*9.8067*RH0(K)/VISC(K)**2 W(K)=AL0G10(4./3.*AR(K)) RET(K)=10.**(-!.81391+1.34671*W(K)-0.12427*W(K)**2 * +0.006344*W(K)**3) YN1(K)=(RHO(K)*UMS(K)*DP(K)*1.E-3)/VISC(K)/RET(K) XN(2,K)=DI(K)/DC(K) XN(3,K)=H(K)/(DC(K)*l.E-3) XN(4,K)=DI(K)/DP(K) XN (5, K) = (RHOP (K) -RHO (K) ) /RHO (K) 20 CONTINUE  C C C  Transform input data into x-values & y-values  30 40  DO 40 MK=1,M Y(MK)=AL0G(YN1(MK)) X(1,MK)=1 DO 30 NK=2,N X(NK,MK)=AL0G(XN(NK,MK)) CONTINUE CONTINUE  C CALL LSqM(X,Y,M,N,A,VAR) C C C  Transform A to B B(1)=EXP(A(1))  50 C C C  DO 50 IN=2,N B(IN)=A(IN) CONTINUE Calculate the variance of the fit SSUMT1=0.  Appendix  90  100  120 130 140 150 160 170  E. Fortran  Programs  SSUM1=0. RMST=0. RMS=0. DO 100 K=1,M SUM1(K)=B(1) DO 90 1=2,N SUM1(K)=SUM1(K)*XN(I,K)**B(I) CONTINUE SSUMT1=SSUMT1+(YN1(K)-SUM1(K))*(YN1(K)-SUM1(K)) RMST=RMST+((SUM!(K)-YN1(K))*100./YN1(K))**2 UIF(K)=SUM1(K)*VISC(K)/(DP(K)*1.E-3)/RHO(K)*RET(K) SSUM1=SSUM1+(UIF(K)-UMS(K))*(UIF(K)-UMS(K)) RMS=RMS+((UIF(K)-UMS(K))*100./UMS(K))**2 CONTINUE VART1=SSUMT1/(M-N) VAR1=SSUM1/(M-N) RMST=SQRT(RMST/M) RMS=SQRT(RMS/M) WRITE(6,120) F0RMAT(5X,'INDIRECT APPROACH-Multiple Linear Regression WRITE(6,130) FORMAT(/,15X,'The fitting parameters are:-') WRITE(6,140) F0RMAT(15X, ' ') WRITE(6,150)(I,B(I),I=1,N) F0RMAT(l5X,'b',Il,'=',F10.4) WRITE(6,160) VART1,VAR1 FORMAT(/,15X,'Variance =',F9.5,5X,'V (Urns) =',F9.5) WRITE(6,170) RMST,RMS FORMAT(/,15X,'RMST =' ,F9.5,5X,'RMS(ums) =',F9.5) B(1)=B(1)/FACT0R  180  250  DO 180 1=1,M YN1(I)=YN1(I)/FACT0R CONTINUE MM=0 IFLAG=0 MM=MM+1 IF(MM.GT.5000) GO TO 500 DO 270 1=1,M PR0D(I)=1. DO 260 J=2,N PROD(I)=PROD(I)*XN(J,I)**B(J)  Appendix  260  270  E. Fortran  Programs  CONTINUE DIFFl(I)=YNl(I)-l.*B(l)*PROD(I) DIFF2(I)=YNl(I)-2.*B(l)*PR0D(I) CONTINUE CALL LSQM2(X,XN,YN1,M,N,PROD,DIFF1,DIFF2,B(1),DELB)  280  290  300 C C C  310  320 330 360 370 390  DO 280 1=1,N B(I)=B(I)+DELB(I) IF(ABS(DELB(I)).GT.DB) IFLAG=1 CONTINUE IF(IFLAG.EQ.O) GO TO 290 GO TO 250 B(1)=B(1)*FACT0R DO 300 1=1,M YN1(I)=YN1(I)*FACT0R CONTINUE Calculate the variance of the fit SSUMT2=0. SSUM2=0. RMST=0. RMS=0. DO 310 K=1,M SUM2(K)=B(1)*PR0D(K) SSUMT2=SSUMT2+(YN1(K)-SUM2(K))*(YN1(K)-SUM2(K)) RMST=RMST+((SUM2(K)-YN1(K))*100./YN1(K))**2 UIF(K)=SUM2(K)*VISC(K)/(DP(K)*l.E-3)/RH0(K)*RET(K) SSUM2=SSUM2+(UIF(K)-UMS(K))*(UIF(K)-UMS(K)) RMS=RMS+((UIF(K)-UMS(K))*100./UMS(K))**2 CONTINUE VART2=SSUMT2/(M-N) VAR2=SSUM2/(M-N) RMST=SQRT(RMST/M) RMS=SQRT(RMS/M) WRITE(6,320) FORMAT(/,5X,'DIRECT APPROACH - Newton s Method') WRITE(6,330) MM,DB FORMAT(/,15X,'No of Iterations =',15,': Epsilon =',F7.6) WRITE(6,360) F0RMAT(/,15X,'The fitting parameters are :-') WRITE(6,370) F0RMAT(15X, ' ') WRITE(6,390)(I,B(I),I=1,N) F0RMAT(15X,'B',I1,'=',F10.4)  Appendix  400 410 470  C C C 500 600 650  E. Fortran  Programs  WRITE(6,400) VART2,VAR2 FORMAT(/,15X,'Variance = ' , F 9 . 5 , 5 X , ' V (Urns) =',F9.5) WRITE(6,410) RMST,RMS F0RMAT(/,15X,'RMST = ',F9.5,5X,'RMS(ums) =',F9.5) WRITE(7,470) ( B ( I ) , I = 1 , N ) F0RMAT(F9.5) CALL COMPAR(UIF,UMS,M) STOP P r i n t warning message! WRITE(6,600) F0RMAT('***************WARNING**************') WRITE(6,650) F0RMAT(12X,'Convergence n o t a c h i v e d a f t e r 5000 i t e r a t i o n s ' ) STOP END  SUBROUTINE LSqM(X,Y,M,N,A,VAR) C C C C C C C C C C C C  Arguement: X Y M N A VAR  real array of independent x-value real array of dependent y-value interger number of pairs of (x,y) points interger number of terms in fitting equation fitting coefficients variance of fit  DIMENSION X(10,112),Y(M),A(N),C0EFF(10,11) NP=N+1 Form the arguement coefficient matrix  C C  DO 80 1=1,N C0EFF(I,NP)=0  50  DO 50 K=1,M C0EFF(I,NP)=C0EFF(I,NP)+X(I,K)*Y(K) CONTINUE DO 70 J=1,N C0EFF(I,J)=0  Appendix  E. Fortran  Programs  60  DO 60 K=1,M C0EFF(I,J)=C0EFF(I,J)+X(I,K)*X(J,K) CONTINUE  70 80  IF(I.Eq.J) GO TO 70 COEFF(J,I)=COEFF(I,J) CONTINUE CONTINUE  C C C  Solve for the unknown A coefficients CALL GAUSS(COEFF,N,10,11,A,RNORM,IERROR)  C C C  Calculate variance of multiple linear regression  130 140  SSUM=0 DO 140 K=1,M SUM=A(1) DO 130 J=2,N SUM=SUM+A(J)*X(J,K) CONTINUE SSUM=SSUM+(Y(K)-SUM)**2 CONTINUE VAR=SSUM/(M-N) RETURN END  SUBROUTINE LSQM2(X,XN,YN,M,N,PR0D,DF1,DF2,B1,DELB)  C  c c c c c c c c c c c c c c c  Arguement:  X XN YN M N PROD  DF1 DF2 Bl DELB  real array of independent LN(XN) values real array of independent Ni values (i=2,3....n) real array of dependent Nl values interger number of pairs of (X,y) points interger number of terms in fitting equation (N2**b2)*(N3**b3)*...*(Nn**bn) Nl-1.0*(bl*PR0D) Nl-2.0*(bl*PR0D) bl real array of unknowns to be sought  Appendix  E. Fortran  Programs  DIMENSION X(10,112),XN(10,112),YN(112),PR0D(112),DF1(112) DIMENSION DF2(112),DELB(112),C0EFF(10,11) NP=N+1 C C C  Form the arguement coefficient matrix DO 80 1=1,N C0EFF(I,NP)=0  50  DO 50 K=1,M C0EFF(I,NP)=C0EFF(I,NP)+DF1(K)*B1*X(I,K)*PR0D(K) CONTINUE DO 70 J=1,N C0EFF(I,J)=0  60  DO 60 K=1,M C0EFF(I,J)=C0EFF(I,J)+DF2(K)*(-PR0D(K)*X(I,K)*X(J,K)) CONTINUE  70 80  IF(I.EQ.J) GO TO 70 COEFF(J,I)=COEFF(I,J) CONTINUE CONTINUE  90  C0EFF(1,1)=0 DO 90 K=1,M C0EFF(1,1)=C0EFF(1,1)+PR0D(K)*PR0D(K) CONTINUE  100  C0EFF(1,NP)=0 DO 100 K=1,M C0EFF(1,NP)=C0EFF(1,NP)+DF1(K)*PR0D(K) CONTINUE  C  C C  Call subroutine GAUSS to solve for DELB's  CALL GAUSS(C0EFF,N,10,11,DELB,RNORM,IERROR) RETURN END SUBROUTINE GAUSS(A,N,NDR,NDC,X,RNORM,IERROR) C  Appendix  C C C C C C C C C C C C C C C C  E. Fortran  Programs  Purpose: Uses Gauss elimination with partial pivot selection to solve simultaneous linear equations of form [A]*{X}={C}• Arguments: A Augmented coefficient matrix containing all coefficients and r.h.s. constants of equations to be solved. N Number of equations to be solved. NDR First (row) dimension of A in calling program. NDC Second (column) dimension of A in calling program. X Solution vector. RNORM Measure of size of residual vector {C}-[A]*{X}. IERROR Error flag. =1 Successful Gauss elimination. =2 Zero diagonal entry after pivot selection. DIMENSION A(NDR,NDC),X(N),B(10,11),BIG(10) NM=N-1 NP=N+1  C C C  Set up working matrix B  10 20 C C C  DO 20 1=1,N DO 10 J=1,NP B(I,J)=A(I,J) CONTINUE CONTINUE Carry out elimination process N-l times DO 80 K=1,NM  C C C C  Search for largest coefficient in column K, rows K through N IPIV0T is the row index of the largest coefficient  25 22 C C C  DO 22 I=K,N BIG(I)=ABS(B(I,1)) DO 25 J=K,N AB=ABS(B(I,J)) IF(AB.LE.BIG(I)) GOTO 25 BIG(I)=AB CONTINUE CONTINUE KP=K+1 Search for the largest Si value in column K, rows K through N IPIV0T is the row index of the largest Si  Appendix  E. Fortran  Programs  C  30 C C C  40 50 C C C  60 70 80 C  SK=(ABS(B(K,K)))/BIG(K) IPIVOT=K DO 30 I=KP,N SI=(ABS(B(I,K)))/BIG(I) IF(SI.LE.SK) GO TO 30 SK=SI IPIV0T=I CONTINUE Interchange rows K and IPIVOT if IPIVOT.NE.K IF(IPIVOT.EQ.K) GO TO 50 DO 40 J=K,NP TEMP=B(IPIVOT,J) B(IPIVOT,J)=B(K,J) B(K,J)=TEMP CONTINUE IF(B(K,K).EQ.O) GO TO 130 Eliminate B(I,K) from rows K+1 through N DO 70 I=KP,N QUOT=B(I,K)/B(K,K) B(I,K)=0. DO 60 J=KP,NP B(I,J)=B(I,J)-QUOT*B(K,J) CONTINUE CONTINUE CONTINUE IF(B(N,N).EQ.O.) GO TO 130  C C C  Back substitute to find solution vector  90 100 C C  X(N)=B(N,NP)/B(N,N) DO 100 11=1,NM SUM=0. I=N-II IP=I+1 DO 90 J=IP,N SUM=SUM+B(I,J)*X(J) CONTINUE X(I)=(B(I,NP)-SUM)/B(I,I) CONTINUE Calculate norm of residual vector, C-A*X  Appendix  C C  110 120  C C C C  E. Fortran  Programs  Normal return with IERR0R=1 RSQ=0. DO 120 1=1,N SUM=0. DO 110 J=1,N SUM=SUM+A(I,J)*X(J) CONTINUE RSq=RSQ+(ABS(A(I,NP)-SUM))**2 CONTINUE RNORM=SQRT(RSQ) IERR0R=1 RETURN Abnormal return because of zero entry on diagonal IEER0R=2  130  IERR0R=2 RETURN END SUBROUTINE C0MPAR(X,Y,M) IMPLICIT REAL*4(A-H,0-Z) DIMENSION X(M),Y(M),X0(2),Y0(2) DATA X0/0.,2./,Y0/0.,2./ CALL DSPDEV('PLOT') CALL NOBRDR CALL COMPLX CALL PAGE(8.5,11.0) CALL AREA2D(4.5,5.0) CALL HEADIN('Ums (pred) vs Urns (exp)$',100,1.2,1) CALL XNAMEOPredicted Urns (m/s)$M00) CALL YNAMEC Experimental Urns (m/s)$\100) CALL GRAF(0.,0.2,2.,0.,0.2,2.) CALL THKFRM(.02) CALL FRAME CALL MARKER(15) CALL CURVE(X0,Y0,2,0) CALL CURVE(X,Y,M,-1) CALL ALNLEG(1.0,0.0) CALL ENDPL(O) CALL DONEPL RETURN END  Appendix E. Fortran Programs  E.2  Program to calculate average spout diameter  10  C C C C C C C  C  C C C  IMPLICIT REAL*4(A-H,0-Z) DIMENSION Z(9),DS(9) DATA Z/0.,5.,10.,20.,30.,40.,50.,60.,70/ DATA DS/2.34,3.48,3.24,3.00,3.14,3.30,3.48,3.72,4.14/ N=9 AREA=QINT4P(Z,DS,N,1,N) ADS=SQRT(AREA/DS(N)) WRITE(6,10)ADS F0RMAT(1X,F5.2) STOP END FUNCTION QINT4P(X,Y,N,IA,IB) DIMENSION X(N),Y(N) WHERE: QINT4P = THE RESULTING INTEGRAL X = AN ARRAY CONTAINING THE "N" ABSCISSAE Y = AN ARRAY CONTAINING THE CORRESPONDING ORDINATES N = THE NUMBER OF POINTS IA = X(IA) IS THE FIRST POINT OF INTEGRATION IB = X(IB) IS THE LAST POINT OF INTEGRATION REAL*8 AC(64) DATA HALF,SIXTH,TWLVTH,TWO 1/0.5,Z402AAAAB,Z40155555,2.0/ 1/2 , 1/6 , 1/12 , 2 DUM=ACSUM(AC,0.0,0) IF (N.LT.4.0R.IA.GE.IB.0R.IA.LT.1.0R.IB.GT.N) GO TO 60 I1=IA IF (IA.LT.3) 11=3 IF (IA.EQ.(N-1).AND.N.GT.4) Il=N-2 I2=IB+1 IF (IB.GT.(N-2)) I2=N-1 IF (IB.EQ.2.AND.N.GT.4) 12=4 DO 50 1=11,12 IF (I.NE.I1) GO TO 10  INITIALIZATION H2=X(I-l)-X(I-2) D3=(Y(I-l)-Y(I-2))/H2 H3=X(I)-X(I-1) D1=(Y(I)-Y(I-1))/H3 Hl=X(I)-X(I-2) D2=(D1-D3)/H1 H4=X(I+l)-X(I)  Appendix  E. Fortran  Programs  R1=(Y(I+1)-Y(I))/H4 R2=(R1-D1)/(X(I+1)-X(I-1)) Hl=X(I+l)-X(I-2) R3=(R2-D2)/H1 IF (IA.NE.l) GO TO 20 C C C  10  HANDLE THE FIRST SEGMENT WITH FORWARD DIFFERENCE FORMULA DUM=ACSUM(AC, 1 H2*(Y(1)+H2*(D3*HALF-H2*(D2*SIXTH-(H2+TW0*H3)*R3*TWLVTH)))) GO TO 20 H4=X(I+1)-X(I) R1=(Y(I+1)-Y(I))/H4 R2=(R1-D1)/(X(I+1)-X(I-1)) R3=(R2-D2)/(X(I+l)-X(I-2)) IF (I.LE.IA.OR.I.GT.IB) GO TO 30  20 C C HANDLE MOST WITH CENTRED DIFFERENCE FORMULA C DUM=ACSUM(AC, 1 H3*((Y(I)+Y(I-1))*HALF-H3*H3*(D2+R2+(H2-H4)*R3)*TWLVTH)) 30 IF (I.NE.I2) GO TO 40 IF (IB.NE.N) GO TO 50 C C HANDLE THE LAST SEGMENT WITH BACKWARD DIFFERENCE FORMULA C DUM=ACSUM(AC, 1 H4*(Y(N)-H4*(R1*HALF+H4*(R2*SIXTH+(TW0*H3+H4)*R3*TWLVTH)))) GO TO 50 40 H1=H2 H2=H3 H3=H4 D1=R1 D2=R2 D3=R3 50 CONTINUE C 60 QINT4P=ACSUM(AC) C RETURN END  Appendix F Error % for t h e Ums Values P r e d i c t e d by Four E q u a t i o n s  s  in rn/s  Ums U, ns % dev expt. Eq.(5.75)  Uma % dev Eq.(2.1)  Ums % dev Eq.(2.6)  Ums % dev Eq.(2.4)  1.399 1.164 1.036 0.950 0.886 0.794 1.503 1.361 1.190 1.085 0.989 0.848 1.464 1.374 1.261 1.129 0.971 1.453 1.408 1.280 1.141 0.997 1.427 1.309 1.133 0.967 1.344 1.282 1.180 0.998 1.117 1.041 0.972 0.896  1.069 0.999 0.912 0.820 0.712 0.583 1.241 1.224 1.122 1.007 0.875 0.717 1.331 1.277 1.146 0.996 0.816 1.430 1.405 1.261 1.095 0.898 1.447 1.332 1.157 0.948 1.466 1.400 1.216 0.996 0.948 0.921 0.864 0.803  1.425 1.345 1.249 1.144 1.020 0.867 1.510 1.492 1.389 1.271 1.133 0.963 1.536 1.484 1.358 1.211 1.029 1.582 1.558 1.426 1.271 1.081 1.570 1.467 1.308 1.111 1.563 1.504 1.341 1.140 1.227 1.198 1.137 1.070  1.056 1.061 1.039 0.984 0.871 0.683 1.153 1.156 1.150 1.110 1.004 0.804 1.179 1.179 1.160 1.077 0.886 1.172 1.173 1.161 1.088 0.905 1.169 1.166 1.114 0.946 1.172 1.172 1.137 0.985 0.882 0.883 0.880 0.865  1.393 1.307 1.204 1.092 0.961 0.801 1.458 1.440 1.329 1.204 1.059 0.884 1.456 1.402 1.270 1.117 0.932 1.476 1.452 1.315 1.157 0.965 1.448 1.342 1.181 0.985 1.429 1.370 1.205 1.005 1.212 1.181 1.113 1.041  -0.404 12.263 16.192 14.937 8.417 0.941 -2.961 5.778 11.658 10.965 7.100 4.237 -0.528 2.034 0.701 -1.048 -3.979 1.597 3.112 2.767 1.402 -3.163 1.483 2.547 4.223 1.860 6.341 6.854 2.139 0.686 8.527 13.414 14.522 16.190  -23.573 -14.210 -11.973 -13.687 -19.622 -26.527 -17.441 -10.040 -5.735 -7.189 -11.550 -15.460 -9.069 -7.052 -9.129 -11.822 -15.952 -1.606 -0.244 -1.477 -4.005 -9.952 1.395 1.759 2.144 -1.965 9.091 9.173 3.086 -0.243 -15.129 -11.512 -11.140 -10.361 121  1.838 15.507 20.566 20.442 15.107 9.176 0.467 9.654 16.701 17.154 14.571 13.601 4.889 8.013 7.697 7.234 6.018 8.850 10.666 11.432 11.423 8.421 10.022 12.059 15.418 14.943 16.277 17.355 13.684 14.200 9.847 15.101 16.925 19.434  -24.508 -8.884 0.310 3.628 -1.670 -14.038 -23.261 -15.079 -3.332 2.287 1.530 -5.146 -19.495 -14.189 -8.035 -4.640 -8.763 -19.343 -16.687 -9.297 -4.641 -9.261 -18.099 -10.933 -1.707 -2.146 -12.764 -8.566 -3.640 -1.276 -21.012 -15.163 -9.461 -3.436  Appendix F. Error % for the Ums Values Predicted by Four Equations  7 - 5 7 - 6 7 - 7 7 - 8 8 - 1 8 - 2 8 - 3 8 - 4 8 - 5 9 - 1 9 - 2 9 - 3 9 - 4 10 - 1 10 - 2 10 - 3 11 - 1 11 - 2 11 - 3 12 - 1 12 - 2 13 - 1 13 - 2 13 - 3 13 - 4 13 - 5 13 - 6 13 - 7 13 - 8 14 - 1 14 - 2 14 - 3 14 - 4 15 - 1 15 - 2 15 - 3 15 - 4 16 - 1 16 - 2 17 - 1 17 - 2 18 - 1 18 - 2 19 - 1 19 - 2 19 - 3 19 - 4  0.813 0.750 0.679 0.600 1.002 0.877 0.784 0.708 0.623 1.043 0.872 0.785 0.643 1.141 0.876 0.658 0.894 0.834 0.737 0.876 0.683 0.882 0.814 0.774 0.722 0.679 0.622 0.571 0.452 0.765 0.646 0.556 0.472 0.696 0.655 0.588 0.501 0.732 0.563 0.592 0.521 0.657 0.592 0.701 0.680 0.627 0.571  0 .960 18.132 0 .870 16.007 0 .765 12.720 0 .638 6.401 1 .116 11.392 1 .049 19.650 0 .950 21.139 0 .836 18.131 0 .698 12.022 2.845 1 .073 0 .994 13.974 0 .874 11.372 0 .729 13.413 0 .986 -13.571 2.606 0 .899 0 .750 13.915 0 .940 5.155 0 .913 9.482 0 .762 3.330 0 .886 1.110 0..775 13.444 0 .893 1.292 0 .849 4.329 0,.801 3.489 0..749 3.720 0..691 1.736 0..626 0.601 0..550 -3.652 0.,459 1.538 0..748 -2.247 0..671 3.865 0..590 6.147 0..492 4.252 0..702 0.838 0..691 5.425 0.607 3.290 0..507 1.100 0.602 -17.702 0.514 -8.733 0.578 -2.370 0.519 -0.451 0.553 •-15.841 0.525 •-11.394 0.673 -3.966 0.661 -2.837 0.618 -1.514 0.570 -0.225  0.735 0.660 0.573 0.469 0.967 0.903 0.810 0.705 0.577 1.003 0.922 0.801 0.656 0.975 0.882 0.722 0.961 0.931 0.762 0.928 0.801 0.717 0.679 0.637 0.592 0.542 0.486 0.422 0.346 0.673 0.598 0.519 0.425 0.692 0.680 0.590 0.484 0.633 0.532 0.632 0.562 0.625 0.590 0.547 0.536 0.498 0.456  -9.563 -12.032 -15.611 -21.801 -3.539 3.020 3.263 -0.484 -7.316 -3.859 5.776 2.082 2.082 -14.517 0.645 9.720 7.502 11.576 3.426 5.912 17.331 -18.736 -16.623 -17.716 -18.005 -20.185 -21.816 -26.092 -23.517 -11.973 -7.455 -6.593 -9.938 -0.624 3.753 0.411 -3.469 -13.491 -5.595 6.804 7.826 -4.869 -0.343 -21.911 -21.139 -20.531 -20.084  0 .995 0 .911 0 .812 0 .690 1 .169 1 .106 1 .012 0 .903 0 .767 1 .158 1 .081 0 .964 0 .819 1 .100 1 .012 0 .860 1 .068 1 .041 0 .885 1 .023 0..907 0 .909 0,.869 0..825 0,.776 0..722 0..661 0..589 0.,500 0..809 0..734 0..655 0..556 0..796 0.,784 0.699 0.,594 0.719 0.624 0.707 0.642 0.690 0.658 0.700 0.688 0.648 0.603  22 .434 21 .451 19 .571 14 .991 16 .677 26 .127 29 .018 27 .475 23 .163 11 .015 24 .009 22 .802 27 .426 -3 .604 15 .560 30 .742 19 .489 24 .781 20 .019 16 .770 32 .870 3 .118 6,.764 6..544 7,.525 6..349 6,.240 3..108 10..714 5..807 13..700 17..761 17..861 14..335 19..742 18.,910 18.,619 -1. 749 10.,784 19. 396 23.,158 4. 973 11. 139 -0. 154 1.216 3.318 5.549  122  0 .828 0 .759 0 .652 0 .495 0 .927 0 .926 0 .903 0 .830 0 .675 0 .912 0 .910 0 .869 0 .738 0 .883 0 .878 0 .800 0 .868 0 .867 0 .823 0 .860 0 .846 0 .654 0 .655 0,.650 0 .634 0..602 0,.548 0..466 0..351 0..638 0,.632 0..595 0..496 0..593 0..593 0.,581 0.,512 0..552 0..538 0.,532 0..527 0..521 0.520 0.540 0.540 0.538 0.525  1 .790 1 .264 -4 .038 -17 .483 -7 .525 5 .575 15 .150 17 .199 8 .305 -12 .525 4 .336 10 .709 14 .840 -22 .602 0 .280 21 .515 -2 .944 4,.013 11..648 -1..799 23..895 -25 .824 -19 .522 -16,.013 -12,.121 -11..327 -11,.906 -18..394 -22..275 -16..602 -2..127 6..990 5..172 -14..761 -9..407 -1..131 2..175 -24..545 -4..455 -10..145 1.,139 -20..757 -12..159 -23..021 -20. 632 -14.,242 -8. 103  Appendix  19 19 19 20 20 20 20 21 21 22 22 23 23 24 24 25 25 25 25 25 31 31 31 31 31 32 32 32 32 32 32 33 33 33 33 33 34 34 34 34 35 35 35 35 36 36 36  - 5 - 6 - 7 - 1 - 2 - 3 - 4 - 1 - 2 - 1 - 2 - 1 - 2 - 1 - 2 - 1 - 2 - 3 - 4 - 5 - 1 - 2 - 3 - 4 - 5 - 1 - 2 - 3 - 4 - 5 - 6 - 1 - 2 - 3 - 4 - 5 - 1 - 2 - 3 - 4 - 1 - 2 - 3 - 4 - 1 - 2 - 3  F. Error % for the Ums Values Predicted by Four  0.524 0.478 0.367 0.650 0.559 0.456 0.408 0.529 0.437 0.519 0.435 0.468 0.435 0.472 0.445 0.625 0.538 0.483 0.420 0.329 1.556 1.229 1.103 1.036 0.936 1.772 1.619 1.343 1.197 1.049 0.921 1.696 1.376 1.236 1.100 0.975 1.768 1.350 1.200 1.024 1.639 1.385 1.191 1.025 1.612 1.212 1.014  0 .516 -1 .588 0 .454 -5 .112 0 .378 3 .094 0 .581 -10 .572 0 .546 -2 .285 0 .480 5 .321 0 .400 -1 .863 0 .489 -7 .596 0 .408 -6 .690 0 .457 -11 .867 0 .410 -5 .698 0 .447 -4 .457 0 .412 -5 .292 0 .433 -8 .168 0 .415 -6 .656 0 .571 -8 .635 0 .509 -5 .474 0 .461 -4 .638 0 .405 -3 .568 0 .338 2 .664 1..388 -10,.814 1 .258 2 .321 1,.139 3,.287 1 .002 -3,.260 0..836 -10,.659 1..520 -14..245 1,.499 -7..421 1..385 3..116 1..255 4..830 1..104 5..276 0..921 0..027 1..570 -7..434 1..461 6..161 1..324 7.,097 1.,165 5.,916 0.,971 -0.,363 1.,566 -11.,431 1.371 1.557 1.206 0.491 1.006 -1. 733 1.545 -5. 760 1.400 1.064 1.231 3.387 1.027 0.204 1.491 -7. 494 1.258 3.755 1.048 3.395  0 .409 0 .355 0 .291 0 .539 0 .504 0 .437 0 .358 0 .497 0 .407 0 .504 0 .447 0 .517 0 .472 0 .520 0 .497 0 .469 0 .413 0 .371 0 .322 0 .264 1 .138 1 .022 0 .917 0 .796 0,.652 1,.384 1,.364 1,.252 1..124 0..977 0..800 1..540 1..425 1..279 1..112 0.,910 1..628 1.,408 1.222 1.002 1.657 1.488 1.293 1.059 1.640 1.360 1. 113  -21 .951 -25 .692 -20 .717 -17 .035 -9 .874 -4 .082 -12 .252 -6 .137 -6 .854 -2 .862 2 .839 10 .435 8 .578 10 .206 11 .586 -24 .977 -23 .191 -23 .228 -23 .353 -19 .879 -26,.840 -16 .853 -16,.867 -23 .130 -30,.302 -21..879 -15..736 -6..766 -6..095 -6.,861 -13..116 -9.,170 3.545 3.510 1. 126 -6. 622 -7. 938 4. 263 1.873 -2. 147 1.070 7.435 8.545 3.314 1.712 12. 234 9.795  0 .551 0 .491 0 .418 0 .648 0 .613 0 .547 0 .464 0 .583 0 .496 0 .574 0 .520 0 .576 0 .535 0 .570 0 .549 0 .603 0 .543 0 .497 0 .443 0 .376 1 .494 1 .367 1,.251 1 .115 0,.948 1..649 1,.629 1..517 1..388 1..238 1..052 1..729 1..621 1..484 1..323 1..124 1..755 1.,558 1.389 1.,181 1.751 1.,603 1.429 1.215 1.709 1.467 1.246  Equations  5 .187 2 .778 13 .782 -0 .248 9 .711 19 .852 13 .812 10 .245 13 .477 10 .539 19 .644 23 .038 23 .018 20 .822 23 .375 -3 .584 0 .966 2 .916 5 .469 14 .419 -4,.014 11,.229 13..414 7,.626 1..254 -6..948 0..595 12..949 15..992 18..022 14..247 1..935 17..777 20.,029 20..280 15.,292 -0.,739 15.,398 15.716 15. 296 6.810 15. 728 19. 975 18.490 5.991 21. 010 22. 885  123  0 .492 -6 .078 0 .432 -9 .611 0 .336 -8 .502 0 .499 -23 .265 0 .498 -10 .917 4 .928 0 .478 0 .399 0 .410 0 .450 -14 .901 0 .434 -0 .714 0 .411 -20 .791 0 .407 -6 .415 0 .393 -16 .056 0 .391 -10 .093 0 .382 -18 .986 0 .382 -14 .124 0 .477 -23 .653 0 .472 -12 .199 0 .452 -6 .476 0 .405 -3 .654 0 .321 -2 .483 1..063 -31 .706 1 .057 -13 .975 1 .019 -7..641 0 .920 -11 .153 0,.736 -21,.343 1..151 -35,.020 1,.153 -28 .792 1,.149 -14,.454 1..108 -7..402 1..003 -4..359 0..803 -12..777 1..175 -30,.736 1..174 -14..691 1..141 -7.,655 1..043 -5.,175 0..843 -13..541 1..171 -33..746 1..154 -14.,533 1.,073 -10. 602 0.,884 -13.,648 1. 168 -28.,765 1.,161 -16.,206 1.,097 -7. 918 0.920 -10. 243 1. 172 -27. 267 1. 137 -6. 184 0.985 -2. 834  Appendix  37 - 1 37 - 2 37 - 3 37 •- 4 37 - 5 38 - 1 38 -- 2 38 -- 3 38 -- 4 38 -- 5 39 -- 1 39 -- 2 39 -- 3 39 -- 4 40 -- 1 40 -- 2 40 -- 3 41 -- 1 41 -- 2 42 -- 1 42 -- 2 43 -- 1 43 -- 2 43 -- 3 43 -- 4 43 -- 5 43 -- 6 44 -- 1 44 -- 2 44 -- 3 44 -- 4 45 -- 1 45 -- 2 45 -- 3 46 -- 1 46 -- 2 47 -• 1 47 -• 2 48 -• 1 49 -• 1 49 -• 2 49 -- 3 49 -- 4 49 -• 5 50 -• 1 50 -• 2 50 -• 3  F. Error % for the Ums Values Predicted by Four  1.152 0.941 0.845 0.778 0.689 1.315 1.157 1.041 0.881 0.755 1.222 1.121 0.895 0.737 1.152 0.996 0.804 1.036 0.797 0.911 0.805 0.817 0.790 0.705 0.647 0.602 0.495 0.710 0.641 0.559 0.491 0.731 0.618 0.509 0.671 0.532 0.561 0.512 0.519 0.674 0.637 0.558 0.499 0.405 0.682 0.633 0.519  1 .111 1 .003 0 .908 0 .799 0 .666 1 .173 1 .094 0 .991 0 .872 0 .728 1 .081 1 .036 0 .912 0 .761 1 .003 0 .937 0..782 0,.960 0..795 0..917 0..808 0..803 0..781 0..720 0..652 0..574 0..479 0..755 0..700 0..616 0..513 0,,713 0..634 0..528 0..623 0.,536 0.,576 0.,541 0.,565 0.,617 0.594 0.538 0.473 0.395 0.594 0.570 0.501  -3 .586 6 .542 7 .510 2 .719 -3 .276 -10 .780 -5 .438 -4 .811 -1 .054 -3 .554 -11 .563 -7 .577 1 .877 3 .284 -12 .897 -5 .889 -2 .777 -7,.354 -0,.251 0,.648 0,.426 -1,.695 -1,.188 2,.163 0,.813 -4..695 -3..283 6..386 9..179 10..121 4..564 -2..522 2..539 3..827 -7..113 0.774 2. 598 5.693 8.898 -8. 430 -6. 747 -3. 599 -5. 185 -2. 528 -12. 839 -9. 994 -3. 509  0 .921 -20 .077 0 .823 -12 .553 0 .739 -12 .581 0 .642 -17 .538 0 .525 -23 .786 1 .090 -17 .145 1 .009 -12 .782 0 .905 -13 .032 0 .786 -10 .732 0 .646 -14 .475 1 .078 -11 .752 1 .030 -8 .133 0 .895 0 .045 0 .735 -0 .324 1 .061 -7 .914 0 .985 -1 .107 0 .807 0,.318 1 .049 1..278 0 .853 7,.067 1,.029 12,.963 0 .896 11..323 0 .682 -16,.547 0,.661 -16 .335 0,.605 -14,.141 0,.543 -16,.084 0..472 -21..671 0..386 -21..933 0..726 2..296 0.,668 4.,233 0..580 3.,830 0..475 -3.,183 0,.751 2..735 0..660 6.,835 0..541 6.,251 0..701 4.,504 0.,594 11. 722 0.,672 19. 698 0.,628 22. 698 0.683 31. 642 0.531 -21. 146 0.510 -19. 988 0.457 -18. 072 0.397 -20. 433 0.325 -19. 659 0.590 -13. 520 0.563 -11. 052 0.489 -5. 875  1 .194 1 .089 0 .996 0 .888 0 .755 1 .287 1 .209 1 .106 0 .986 0 .839 1 .227 1 .181 1 .054 0 .896 1 .176 1 .106 0,.940 1,.145 0,.968 1,.111 0,.992 0,.870 0..848 0,.789 0,.722 0..643 0..547 0..860 0..803 0..716 0..608 0..849 0..764 0..650 0..780 0.682 0.,741 0..702 0.,740 0.,681 0.659 0.602 0.537 0.457 0.696 0.670 0.597  124  Equations  3 .608 15 .697 17 .928 14 .164 9 .527 -2 .149 4 .459 6 .232 11 .895 11 .112 0 .392 5 .378 17 .711 21 .597 2 .061 11 .065 16,.926 10,.561 21,.406 21,.909 23,.251 6,.486 7,.350 11,.915 11..571 6..879 10..506 21..065 25..238 28..017 23..871 16..154 23..694 27..648 16..289 28.,178 32.,112 37.,016 42.,632 1.,109 3.377 7. 972 7. 616 12. 726 2. 087 5.892 15. 056  0 .888 0 .882 0 .849 0 .766 0 .611 0 .926 0 .924 0 .900 0 .823 0 .668 0 .912 0 .912 0 .885 0 .768 0 .883 0 .882 0 .815 0 .868 0 .834 0 .861 0 .849 0 .658 0 .658 0 .653 0 .626 0,.563 0,.448 0,.638 0..636 0,.607 0,.516 0,.594 0,.587 0,.526 0..552 0..540 0..532 0..531 0..521 0..541 0..541 0..532 0..494 0..406 0.,499 0.,499 0.,483  -22 .911 -6 .230 0 .495 -1 .549 -11 .292 -29 .561 -20 .101 -13 .584 -6 .577 -11 .577 -25 .367 -18 .642 -1 .117 4 .182 -23 .341 -11 .493 1,.423 -16 .247 4,.583 -5,.537 5,.404 -19 .491 -16 .698 -7,.427 -3,.269 -6..543 -9..539 -10,.165 -0..782 8..653 5..083 -18,.802 -5..066 3..305 -17,.704 1.,445 -5..180 3..729 0.,313 -19..735 -15..067 -4..674 -1..080 0.,244 -26.,865 -21.,231 -6. 897  Appendix  50 51 51 52 52 53 53 54 54 61 61 61 61 61 61 62 62 62 62 62 63 63 63 63 63 64 64 64 64 65 65 65 65 66 66 66 66 -  67 67 67 67 67 67 67 67 68 68  -  4 1 2 1 2 1 2 1 2 1 2 3 4 5 6 1 2 3 4 5 1 2 3 4 5 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 5 6 7 8 1 2  F. Error % for the Ums Values Predicted by Four Equations  0.412 0.505 0.427 0.514 0.437 0.447 0.410 0.419 0.408 1.373 1.233 1.138 1.042 0.953 0.816 1.447 1.314 1.228 1.115 0.975 1.476 1.382 1.254 1.078 0.936 1.459 1.279 1.200 0.986 1.496 1.372 1.169 1.010 1.381 1.267 1.156 1.048 1.101 1.058 1.019 0.974 0.925 0.856 0.774 0.655 1.101 0.988  0.418 0.510 0.425 0.473 0.428 0.458 0.430 0.448 0.433 1.354 1.244 1.143 1.034 0.911 0.761 1.378 1.262 1.142 1.007 0.840 1.394 1.332 1.206 1.062 0.886 1.415 1.249 1.100 0.917 1.406 1.274 1.120 0.935 1.418 1.301 1.145 0.955 1.168 1.120 1.056 0.987 0.911 0.824 0.726 0.607 1.109 0.996  1.367 0.993 -0.363 -7.939 -2.059 2.383 4.841 6.869 6.224 -1.375 0.854 0.435 -0.728 -4.442 -6.743 -4.778 -3.939 -6.998 -9.656 -13.813 -5.543 -3.633 -3.847 -1.463 -5.373 -3.001 -2.350 -8.359 -7.032 -6.049 -7.111 -4.153 -7.411 2.669 2.709 -0.957 -8.915 6.120 5.830 3.599 1.316 -1.535 -3.704 -6.156 -7.341 0.718 0.858  0 .400 0 .555 0 .455 0 .559 0 .500 0 .566 0 .528 0 .576 0 .555 0 .958 0 .875 0 .796 0 .713 0 .620 0 .509 1 .078 0 .980 0 .878 0 .766 0..627 1 .173 1..116 1 .000 0 .871 0 .713 1..262 1..101 0..958 0,.784 1,.293 1,.162 1..009 0..827 1..343 1..223 1..063 0..870 0..841 0..802 0..752 0..699 0..640 0..574 0..500 0..410 0.,887 0.,789  -2 .887 9 .953 6 .601 8 .708 14 .475 26 .576 28 .823 37,.406 36,.099 •30 .208 •29 .049 •30 .050 •31 .561 •34..932 •37,.597 •25,.470 25,.423 28,.538 31,.336 35..687 20 .553 19 .273 20,.236 •19..196 23,.831 13,.497 13,.947 20,.187 20,.535 13..589 15..278 13..703 18..094 -2..741 -3.,499 -8.,077 17..012 23..615 24..150 26..159 28..222 30,.757 32..967 35..433 37..346 19..409 20.,163  0 .508 0 .638 0 .542 0 .623 0 .569 0 .619 0 .585 0 .618 0 .600 1 .305 1 .209 1 .121 1,.025 0,.914 0,.778 1..349 1,.247 1..140 1,.018 0..865 1,.388 1,.332 1,.219 1,.088 0,.924 1,.432 1,.280 1,.142 0..970 1..437 1..316 1..173 0..997 1..459 1..351 1..204 1..023 1..116 1..074 1..019 0..959 0,.892 0..816 0..728 0..620 1..093 0..993  23..183 26 .260 26 .970 21 .113 30 .209 38,.431 42,.697 47..514 47..118 -4..926 -1,.964 -1..503 -1..643 -4..087 -4..669 -6..786 -5..117 -7..187 -8..680 -11..241 -5,.948 -3 .591 -2..814 0..907 -1..262 -1..880 0..068 -4..840 -1..628 -3..962 -4..070 0..316 -1..257 5..615 6..604 4..180 -2..367 1..363 1..529 -0..013 -1,.545 -3,.532 -4,.685 -5,.908 -5..374 -0..761 0..479  125  0 .419 0 .450 0 .434 0 .411 0 .408 0 .393 0 .392 0 .382 0 .382 1 .057 1 .060 1 .029 0 .964 0 .847 0,.659 1,.156 1,.152 1..112 1,.013 0,.813 1 .178 1 .179 1,.156 1 .070 0,.876 1,.173 1,.158 1,.082 0,.894 1,.167 1..161 1..096 0..920 1..172 1..167 1..110 0..937 0..881 0..881 0,.876 0,.857 0,.816 0,.745 0..637 0..483 0..927 0..919  1 .622 -10 .857 1 .611 -20 .008 -6 .652 -12 .112 -4 .380 -8 .727 -6 .298 -23 .004 -14 .070 -9 .615 -7 .494 -11 .146 -19 .202 -20 .123 -12 .332 -9,.412 -9..174 -16 .564 -20 .187 -14 .721 -7,.843 -0,.709 -6,.369 -19 .635 -9..498 -9..873 -9..292 -21,.995 -15,.412 -6..232 -8..910 -15..145 -7..913 -4..006 -10..546 -19..982 -16..744 -14,.081 -12..022 -11..750 -13,.012 -17..694 -26..289 -15..840 -7.,014  Appendix  68 •- 3 68 •- 4 68 •- 5 69 •- 1 69 •- 2 69 •- 3 69 •- 4 70 •- 1 70 •- 2 70 •- 3 71 •- 1 71 •- 2 71 •- 3 72 •- 1 72 •- 2 72 -- 3 73 -- 1 73 •- 2 73 -- 3 73 -- 4 73 -- 5 73 -- 6 73 -- 7 73 -- 8 74 •- 1 74 -- 2 74 -- 3 74 -- 4 74 -- 5 75 -- 1 75 -- 2 75 -- 3 75 -- 4 76 -- 1 76 -- 2 76 -- 3 77 -- 1 77 -- 2 77 -- 3 78 -- 1 78 -- 2 79 -• 1 79 -• 2 79 -- 3 79 -• 4 79 -• 5 79 -- 6  F. Error % for the Ums Values Predicted  0.855 0.748 0.637 1.287 1.071 0.939 0.849 1.117 0.982 0.895 0.997 0.820 0.751 0.960 0.872 0.753 0.919 0.874 0.819 0.767 0.704 0.636 0.563 0.474 0.718 0.674 0.605 0.529 0.437 0.608 0.560 0.512 0.478 0.604 0.565 0.467 0.603 0.583 0.476 0.570 0.467 0.655 0.615 0.583 0.510 0.439 0.356  0 .902 0 .794 0 .663 1 .085 0 .945 0 .832 0 .693 1 .007 0 .854 0 .712 1 .005 0 .868 0 .723 0 .999 0 .883 0 .736 0 .866 0 .807 0 .760 0 .710 0 .655 0..594 0 .522 0..436 0..760 0..703 0..637 0..560 0..468 0..718 0..656 0..577 0..481 0..666 0..585 0..488 0..625 0.,591 0.492 0.593 0.498 0.627 0.586 0.541 0.490 0.431 0.359  5 .522 6 .181 4 .078 -15 .716 -11 .809 -11 .437 -18 .367 -9 .816 -13 .049 -20 .441 0 .786 5 .837 -3 .671 4 .077 1 .313 -2 .251 -5 .798 -7 .695 -7 .219 -7 .443 -6 .959 -6,.646 -7 .195 -8..041 5..801 4..360 5..286 5..948 7..003 18..063 17..139 12..731 0.,663 10..275 3.,617 4.,523 3. 636 1.349 3.459 4. 089 6.703 -4. 324 -4. 744 -7. 189 -3. 968 -1. 875 0.961  0 .707 -17 .282 0 .615 -17 .764 0 .504 -20 .813 0 .939 -27 .036 0 .806 -24 .705 0 .702 -25 .270 0 .574 -32 .406 0 .923 -17 .369 0 .770 -21 .569 0 .631 -29 .532 0 .957 -3 .999 0 .814 -0 .758 0 .666 -11 .333 0 .980 2 .117 0 .856 -1 .792 0 .700 -7..030 0 .640 -30 .343 0 .593 -32 .163 0 .555 -32 .229 0 .515 -32 .816 0 .472 -32 .967 0 .424 -33,.338 0 .369 -34 .545 0,.302 -36,.313 0,.632 -11..933 0,.581 -13,.737 0..522 -13..781 0..453 -14..290 0..372 -14..969 0..655 7.,730 0..594 6.,013 0.,818 0..516 0..422 -11..615 0.,653 8. 190 0.,567 0.,356 0.,464 -0. 576 0.637 5.657 0.,599 2.762 0.490 2. 988 0.624 9.551 0.515 10.361 0.467 -28. 678 0.434 -29. 393 0.398 -31. 651 0.357 -29. 975 0.310 -29. 348 0.254 -28. 600  by Four  0 .908 0 .810 0 .689 1 .099 0 .971 0 .867 0 .736 1 .054 0 .909 0 .772 1 .066 0 .935 0 .794 1 .071 0 .960 0 .815 0 .831 0 .780 0 .739 0 .696 0,.647 0,.593 0 .528 0,.449 0..772 0,.720 0..659 0..588 0..500 0..763 0..704 0..628 0..534 0.,740 0.,659 0.,560 0.713 0.,678 0.576 0.691 0.591 0.617 0.581 0.541 0.495 0.441 0.375  Equations  6 .209 8 .288 8 .140 -14 .571 -9 .319 -7 .721 -13 .324 -5 .658 -7 .453 -13 .706 6 .929 13 .999 5 .740 11 .595 10 .038 8 .196 -9 .556 -10 .732 -9,.715 -9 .300 -8 .066 -6..819 -6 .138 -5..238 7..459 6..853 8..924 11..082 14..321 25..527 25.,736 22..649 11.,615 22.,448 16.,618 19.,902 18. 172 16.,249 20. 952 21. 145 26.483 -5. 799 -5. 552 -7. 212 -2. 995 0.446 5.305  126  0 .882 0 .795 0 .634 0 .913 0 .897 0 .833 0 .684 0 .883 0 .858 0 .746 0 .868 0 .850 0 .748 0 .860 0 .849 0 .758 0 .654 0 .654 0 .645 0 .626 0,.591 0,.535 0 .453 0 .340 0,.637 0..635 0..616 0..561 0..453 0..593 0..590 0..559 0..471 0..552 0..544 0..484 0.,532 0.,531 0.,495 0.,521 0.,503 0.,539 0.539 0. 529 0.499 0.442 0.346  3 .144 6 .230 -0 .537 -29 .082 -16 .288 -11 .325 -19 .427 -20 .969 -12 .627 -16 .665 -12 .938 3 .602 -0 .417 -10 .392 -2 .648 0 .610 -28 .793 -25 .192 -21 .198 -18 .341 -16 .027 -15 .833 -19 .526 -28 .332 -11,.328 -5..801 1..796 6..129 3..662 -2..495 5..309 9..277 -1.,398 -8.,615 -3..791 3..534 -11.,803 -8.,906 3.994 -8. 680 7.782 -17.,718 -12. 439 -9. 275 -2. 068 0.599 -2. 884  Appendix  80 80 80 80 80 81 81 81 82 82 82 83 83 84 84 85 85 85 85 85 85 86 86 86 86 87 87 87 87 88 88 88 89 89 90 90 -  1 2 3 4 5 1 2 3 1 2 3 1 2 1 2 1 2 3 4 5 6 1 2 3 4 1 2 3 4 1 2 3 1 2 1 2  F. Error % for the Ums Values Predicted by Four  0.568 0.541 0.501 0.429 0.355 0.593 0.447 0.397 0.483 0.465 0.407 0.449 0.399 0.439 0.396 0.606 0.558 0.519 0.465 0.389 0.308 0.484 0.405 0.356 0.283 0.494 0.464 0.419 0.338 0.433 0.391 0.339 0.379 0.297 0.341 0.287  0 .590 0 .573 0 .518 0 .456 0 .380 0 .549 0 .465 0 .387 0 .504 0 .467 0 .390 0 .477 0 .391 0 .463 0 .395 0 .553 0 .524 0 .483 0..437 0 .385 0..321 0..514 0..459 0,.404 0..337 0..479 0..464 0..409 0.,341 0.,438 0..409 0.,341 0.412 0.341 0.400 0.343  3 .859 5 .825 3 .477 6 .283 7 .145 -7 .487 3 .921 -2 .464 4 .377 0 .481 -4 .254 6 .149 -1 .913 5..378 -0,.355 -8,.780 -6..155 -6..896 -5..923 -1..093 4..177 6..128 13..349 13..441 18.,931 -3.,001 0.,089 -2.,463 0.749 1.180 4. 553 0.528 8.823 14. 958 17. 159 19. 630  0 .506 0 .490 0 .439 0 .382 0 .313 0 .521 0 .434 0 .355 0 .519 0 .477 0 .391 0 .513 0 .413 0 .517 0 .434 0 .418 0,.394 0 .361 0 .324 0 .281 0..230 0,.450 0..398 0,.346 0..283 0..467 0..452 0..394 0..322 0..466 0..432 0.,354 0.,460 0.374 0.464 0.393  -10 .936 -9 .491 -12 .315 -11 .045 -11 .901 -12 .129 -2 .883 -10 .503 7 .379 2 .532 -3 .982 14 .161 3 .410 17 .673 9 .541 -31 .057 -29 .359 -30 .416 -30 .366 -27 .707 -25,.235 -6..983 -1..673 -2..766 0..056 -5..366 -2..518 -6..064 -4.,768 7.,642 10.,468 4.,334 21. 271 25. 857 36. 033 36. 764  0 .617 0 .601 0 .550 0 .490 0 .417 0 .608 0 .524 0 .445 0 .588 0 .550 0 .467 0 .573 0 .480 0 .568 0 .493 0 .550 0,.524 0,.488 0,.446 0 .398 0..338 0,.548 0..496 0,.442 0..376 0..544 0..529 0..472 0..401 0..527 0..495 0..421 0.,513 0.433 0.,509 0.,444  Equations  8 .708 11 .118 9 .793 14 .298 17 .430 2 .505 17 .178 12 .093 21 .803 18 .180 14 .801 27 .637 20 .405 29 .434 24 .466 -9 .211 -6..064 -6 .019 -4,.050 2 .231 9..724 13..315 22..475 24..244 32..750 10..183 14..105 12..738 18..689 21..721 26..700 24..188 35. 269 45. 820 49.,142 54.,723  127  0 .498 0 .498 0 .491 0 .457 0 .377 0 .450 0 .437 0 .379 0 .411 0 .409 0 .377 0 .393 0 .375 0 .382 0 .372 0 .477 0,.477 0,.469 0,.445 0,.396 0..311 0..428 0,.423 0..397 0..331 0..380 0..380 0..370 0..322 0..344 0..343 0..317 0.,328 0.,315 0.,318 0..311  -12 .311 -7 .923 -2 .043 6 .520 6 .244 -24 .126 -2 .284 -4 .418 -14 .913 -11 .948 -7 .373 -12 .515 -5 .925 -12 .908 -5,.988 -21 .362 -14,.561 -9..544 -4..205 1..757 1..063 -11,.617 4..553 11..646 16..865 -23..066 -18..073 -11,.676 -4.,603 -20..479 -12..184 -6..379 -13..490 5.,938 -6.,611 8.,384  Appendix G Error % for t h e S p o u t D i a m e t e r  T °C 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 170 170 170 170 170 170 170 170 170 170 170 170 170  dp (mm)  D; (mm)  Hm (m)  2 2 2 1 1 1 1 1 1 1 1. 1 1 1.  19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 26.64 26.64 26.64 26.64 26.64 26.64 26.64 26.64 12.70 12.70 19.05 19.05 19.05 19.05 19.05 19.05 19.05 19.05 26.64 26.64 26.64 26.64 26.64  0.700 0.400 0.200 0.850 0.700 0.500 0.300 0.900 0.700 0.500 0.300 0.730 0.500 0.300 0.620 0.500 0.300 0.630 0.650 0.500 0.545 0.400 0.680 0.400 0.585 0.400 0.300 0.510 0.300 0.460 0.400 0.300 0.615 0.500 0.300 0.500 0.475  025 025 025 630 630 630 630 200 200 200 200 010 010 010  2.025  2 .025 2 .025 1 .630 1 .200 1 .200 1 .010 1..010 0 .915 0 .915 1 .630 1 .630 1 .630 1.200 1.200 1.010 1.010 1.010 2.025 2.025 2.025 1.630 1.200  Ums (m/s)  d s -exp (cm)  McNab's (cm)  1.399 0.950 0.794 1.117 0.972 0.813 0.679 0.882 0..774 0..679 0..571 0 .701 0.571 0.478 .556 .229 .036 .152 0.817 0.705 0.674 0.558 0.606 0.465 1.002 0.784 0.708 0.765 0.556 0.650 0.559 0.456 1.772 1.343 1.049 1.157 0.710  3.370 3.100 3.010 3.580 3.320 3.020 2.680 3.100 2.930 2.690 2.620 2.970 2.820 2.500 4.330 3.720 3.750 3.970 2.690 2.890 2.920 2.580 2.250 1.940 2.830 2.650 2.510 2.710 2.410 2.380 2.220 1.930 3.960 3.790 3.570 3.200 2.890  4.089 3.336 3.037 3.664 3.409 3.102 2.823 3.270 3.044 2.836 2.592 2.900 2.604 2.376 4.276 3.796 3.471 3.688 3.123 2.892 2.832 2.571 2.703 2.349 2.815 2.485 2.353 2.454 2.088 2.262 2.097 1.893 3.748 3.249 2.858 3.017 2.368  128  %dev  21.349 7.600 0.885 2.340 2.688 2.701 5.325 5.472 3.906 5.435 -1.071 -2.351 -7.647 -4.971 -1.236 2.047 -7.451 -7.107 16.078 0.081 -3.030 -0.347 20.113 21.106 -0.540 -6.208 -6.254 -9.435 -13.373 -4.960 -5.540 -1.903 -5.348 -14.271 -19.942 -5.730 -18.075  Wu's (cm)  %dev  3.726 3.137 2.897 3.380 3.178 2.935 2.710 3.053 2.878 2.714 2.514 2.757 2.517 2.327 3.893 3.511 3.254 3.416 2.946 2.760 2.707 2.491 2.589 2.301 3.142 2.821 2.695 2.791 2.427 2.599 2.434 2.226 4.031 3.567 3.198 3.343 2.703  10.556 1.191 -3.754 -5.590 -4.262 -2.801 1.123 -1.520 -1.761 0.895 -4.053 -7.156 -10.727 -6.902 -10.101 -5.630 -13.222 -13.944 9.501 -4.512 -7.294 -3.438 15.087 18.634 11.016 6.464 7.386 2.987 0.685 9.214 9.618 15.354 1.788 -5.887 -10.427 4.455 -6.483  Appendix  170 170 170 170 300 300 300 300 300 300 300 300 300 300 300 300 420 420 420 420 420 420 420 420 420 420 420 500 500 500 500 500 500 500 500 580 580 580 580 580 580 580 580 580  G. Error % for the Spout  1 .200 1 .010 1 .010 0 .915 1 .630 1 .630 1 .200 1 .200 1 .010 2 .025 2 .025 2 .025 1 .630 1 .200 1 .010 0 .915 2 .025 2 .025 1 .630 1..200 1 .010 2..025 2..025 2..025 1..630 1..200 0..915 1..630 1..200 1..010 2..025 2..025 1..200 1..010 0..915 2..025 2.,025 1.630 1..200 1.,010 2. 025 2. 025 1.200 0.915  26 .64 26 .64 26 .64 12 .70 19 .05 19 .05 19 .05 19 .05 19 .05 26 .64 26 .64 26 .64 26 .64 26 .64 26 .64 12 .70 19 .05 19 .05 19 .05 19 .05 19 .05 26,.64 26 .64 26..64 26,.64 26..64 12..70 19..05 19..05 19..05 26..64 26.,64 26..64 26..64 12..70 19.,05 19..05 19.,05 19.,05 19.,05 26. 64 26. 64 26. 64 12. 70  0 .300 0 .440 0 .300 0 .515 0 .475 0 .300 0 .415 0 .300 0 .300 0 .590 0 .500 0 .300 0 .400 0 .390 0 .300 0 .430 0 .520 0 .300 0,.370 0,.285 0 .255 0,.540 0,.400 0..300 0,.350 0..280 0..350 0..320 0..255 0..240 0..500 0..300 0..235 0.,220 0..305 0.,440 0.,300 0.270 0.225 0.,220 0.440 0.300 0.225 0.280  129  Diameter  0.559 0.682 0.519 0.484 1.043 0.785 0.696 0.588 0.529 1.696 1.376 1.100 1.121 0.731 0.505 0.494 1.453 1.141 1.141 0.732 0.519 1.768 1.350 1.200 1.152 0.671 0.433 0.894 0.592 0.468 1.639 1.191 0.561 0.447 0.379 1.344 1.180 0.876 0.657 0.472 1.612 1.212 0.519 0.341  2 .640 2 .510 3 .100 2 .190 2 .860 2 .560 2 .660 2 .360 2 .190 3 .790 3 .560 3 .700 3 .530 2 .840 2 .940 2 .160 3,.370 3 .070 2 .660 3,.090 2 .450 3,.940 3,.710 3.,500 3,.500 2..790 2..060 2..660 2..810 2..880 4..030 3..640 2..880 2..840 2..010 3..240 3..110 2.,700 2.,720 2.,900 3.790 3.680 2. 960 2. 050  2 .094 2 .316 2 .020 1 .964 2 .522 2 .182 2 .063 1 .892 1 .798 3 .224 2 .897 2 .576 2 .610 2 .110 1 .758 1 .750 2,.709 2 .387 2..396 1..918 1,.621 2,.986 2,.604 2..451 2..407 1..840 1..487 2..009 1..640 1..460 2..723 2..312 1..597 1..427 1..320 2..344 2..189 1.,894 1.,641 1..396 2.,563 2.,218 1.462 1. 192  -20 .663 -7 .735 -34 .850 -10 .342 -11 .833 -14 .769 -22 .429 -19 .815 -17 .890 -14 .921 -18,.619 -30,.374 -26 .055 -25 .693 -40,.212 -18,.994 -19,.600 -22 .261 -9.,939 -37,.920 -33,.845 -24..221 -29,.815 -29..974 -31..231 -34.,052 -27..799 -24..488 -41..654 -49..318 -32..429 -36..491 -44..553 -49..748 -34..305 -27.,645 -29..605 -29.,846 -39.,659 -51..871 -32.,381 -39.,723 -50. 602 -41.,873  2 .433 2 .654 2 .356 2 .290 3 .165 2 .794 2 .655 2 .465 2 .355 3 .917 3 .573 3 .235 3 .265 2 .710 2,.308 2 .290 3 .614 3 .247 3,.250 2 .678 2 .307 3,.936 3 .497 3,.320 3,.264 2..579 2 .135 2 .896 2..422 2,.186 3..774 3..279 2,.366 2..143 1,.997 3..421 3..228 2..838 2..504 2..170 3..701 3,.266 2..261 1..885  -7,.855 5 .737 -24 .013 4,.549 10..677 9,.140 -0 .204 4,.432 7..524 3..345 0,.351 -12,.579 -7,.505 -4,.566 -21 .501 6 .033 7..250 5,.750 22..191 -13 .343 -5,.836 -0..097 -5..741 -5..139 -6..748 -7..549 3..626 8..871 -13,.825 -24,.085 -6..346 -9..911 -17 .856 -24,.532 -0..631 5..572 3..807 5.,128 -7..940 -25 .182 -2..350 -11,.248 -23..620 -8..033  

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