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

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SPOUTED BED HYDRODYNAMICS AT TEMPERATURES UP 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 OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED 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 this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada & Date faiytj DE-6 (2/88) A b s t r a c t A study of the hydrodynamics of spouted beds at temperatures ranging from room tem-perature 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 diame-ters of 0.915 mm, 1.010 mm, 1.200 mm, 1.630 mm and 2.025 mm, 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. A similar equa-tion with a slightly smaller value of Um/Umf 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 Contents Abstract ii List of Tables vii List of Figures viii Acknowledgement xi 1 Introduct ion 1 1.1 Rationale for the Present Work 1 1.2 Objectives of the Present Work 3 2 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 Exper imenta l 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 4 Exper imenta l Procedures and Condit ions 29 4.1 Operating procedure 29 4.1.1 Operation 29 4.1.2 Measurement 30 4.1.3 Calculation of Ums 31 4.2 Experimental Conditions 34 4.2.1 Range 34 4.2.2 Experimental Error Calculation 34 5 Resul t s : M i n i m u m Spout ing Veloci ty 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 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 8 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 109 E.l Program on Ums correlation 109 E.2 Program to calculate average spout diameter 119 v F Error % for the Ums Values Predic ted by Four Equat ions 121 G Error % for the Spout Diameter 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 corre-lations 48 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 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 2.2 Typical pressure drop versus velocity curve for a spouted bed of coarse particles 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 26 4.7 Simplified flow diagram of the apparatus 32 5.8 Effect of particle diameter on Ums. (DC=156 mm, Z),=19.05 mm, H=0.2 m) 38 5.9 Effect of particle diameter on Ums. (Z)c=156 mm, Z?,=12.70 mm, H=0.2 m) 39 5.10 Effect of orifice diameter on Ums. (Dc=156 mm, dp=2.025 mm, H=0.3 m) 40 5.11 Effect of orifice diameter on Ums. (£)c=156 mm, dp=1.010 mm, H=0.2 m) 41 5.12 Effect of bed height on Ums. (£>c=156 mm, A - 2 6 . 6 4 mm, dp=2.025 mm) 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 mm, A = 19.05 mm, <fp=2.025 mm) 56 viii 5.18 Comparison of correlations for Ums with experimental data. (Dc=156 mm, A - 1 2 . 7 0 mm, dp=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). . . . 62 6.21 Effect of particle diameter on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. ( J D C = 1 5 6 mm, A=26 .64 mm) 65 6.22 Effect of particle diameter on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. (7JC=156 mm, Z?j=19.05 mm) 66 6.23 Effect of particle diameter on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. (Dc=156 mm, Ds=12.70 mm) 67 6.24 Effect of orifice diameter on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. (Z?c=156 mm, dp=2.025 mm) . 68 6.25 Effect of orifice diameter on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. (DC=156 mm, <ip=1.630 mm) . 68 6.26 Effect of orifice diameter on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. (DC=156 mm, dp=1.200 mm) . 69 6.27 Effect of orifice diameter on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. (Z)c=156 mm, dp=1.0W mm) . 69 6.28 Effect of temperature on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. (Z)c=156 mm, 7J,=26.64 mm) . 71 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) . 72 ix 6.30 Effect of temperature on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. (Dc=156 mm, Z),=12.70 mm) . 73 7.31 Effect of temperature on Ds. (Dc=156 mm, D,=19.05 mm, dp=1.630 mm) 75 7.32 Effect of temperature on Ds. (Dc=156 mm, £>t=26.64 mm, dp=2.025 mm) 76 7.33 Effect of bed height on Ds. (Z>c=156 mm, A - 2 6 . 6 4 mm, dp=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 Work-shop and Stores for their invaluable assistance. Finally, to my parents for their encouragement and support. XI Chapter 1 Introduction 1.1 Rat ionale for the Present 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 parti-cles 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. Introduction 2 P« « C e 0 0 ill'. b0r FOUNTAIN BED SURFACE SPOUT ANNULUS SPOUT-ANNULUS INTERFACE CONICAL BASE FLUID INLET Figure 1.1: Schematic diagram of a spouted bed. Chapter 1. Introduction 3 Bridgwater [3]. As pointed out by Lim et al. [11], spouted beds exhibit some advantages over conven-tional fluidized beds. They have been used for various physical and chemical processes and have achieved increasing recognition. Recently, high temperature spouting has at-tracted 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 Object ives of the Present Work 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 as-pects 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 pub-lished 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] stud-ied 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 signifi-cantly 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 Spout ing Veloci ty 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 experimen-tally by reducing the fluid flow rate to a point at which a further decrease of flowrate will Chapter 2. Literature Review 6 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. Fig-ure 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 Mathur and Gishler equat ion 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: 1/3 Urns — dp [DC\ \Dt] [DJ N 2gH(pp-pfJ 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 Review 8 between the entering fluid and the entrained particles fan Ums dp Dc \Di 2gH(Pp - PI) ( 2 2 ) \.Del\ 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 close-fractions and mixed sizes, obtained Ums a e^62 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 Review 9 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 Grbavc ic 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. 3 1 a =l~ l-JT (2-3) 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 a ms -. l -H Hm. 3 (2.4) Umf 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 h = 1-75 (H)2£/ Pfi1 ~ ernf) Since the Grbavcic equation was verified only for water spouted beds at room tempera-ture, its application to high temperature air spouted beds has yet to be examined. Chapter 2. Literature Review 10 2.3.3 Wu 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 [A] Wc\ 0.266 r/n LAJ -0 .095 Pp~ Pf . PS . 0.256 (2.6) The most significant difference from the original Mathur and Gishler equation is that the exponent on (pp — Pf)/pj is 0.256 instead of 0.5. Unlike Mathur and Gishler [24], (pp — Pf)/pf 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 temper-ature 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. The 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 Review 11 2.3.4 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. Experimental data by previous workers show that Um often exceeds Umf. 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 B e d 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 Hm. 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 Chapter 2. Literature Review 12 U„ dp [A] [DC\ 1/3 i 2gHm(pp-pf) Pf (2.1a) As mentioned above, Um has a close relationship with Umf. In general, U„ u, = h = 1.0 - 1.5 mf (2.7) On the other hand, Umf c a 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 P where Rtmf = dpUyPf = 33.7(Vl + 35.9 x 1 0 - M r - 1) dl(pp-pf)gpf Ar = _ vy V-1 (2.8) (2.9) Equations (2.1a), (2.7) and (2.8) are combined to eliminate Um and Umf, the result being (2.10) Hm = dp \DC1 [Dil 2/3 "5686?" Ar {Vl + 35.9 x lO"6Ar - 1): 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 varia-tions 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„ Dc = 0.105 'De' dp 0.75 [•Del 0.4 A 2 " k2J (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: -tawy = 0.36 (2.12) 8DsHm 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) Lit tman et al. [48, 49] developed two models using monodispersed spherical particles. The first of these models states that »~D- 0.215 + 5 ^ / o r / t > 0 . 0 2 Di where A is defined by A pUmfUt (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 Ret = 1.745ArU5; Ret > 1000 (2.14) (2.15) (2.16) Chapter 2. Literature Review 14 The second model states that HmDs 0.345 DA DJ -0 .384 (2.17) Dl - D] 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 ' -?- + 35.9 x 10-6 - W-J-Ar V Ar 1 1 Ar3 Ar3 + 35.9 x 10~6Ar4 > 0 for Ar > 0 (2.18) The above equation shows that for all value of Ar, dHm/dAr > 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. This phenomenon was verified in both Wu's [22] and Zhao's [20] experiments. 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 f 1/3 (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 D i a m e t e r 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, at tempts 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 de-rived from a force balance analysis. Their theoretical equation is WO'® = 1 ( 2 2 0 ) 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 Ds = 0.384 £0.500.75 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: Da = 2.0 '0.49 n0.68 £ 0 . 4 9 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 (2.23) ( , W ) 0 ' 2 8 3 4 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 Pressure D r o p and Pressure Distr ibut ion 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 Lit tman [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 Choice and Descr ipt ion of Equipment 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 II Side View * Front View 1. Spouting flow line. 2. Pressure port. 3. Conical base. 4. Solids discharge lines. 5. Measuring port. 6. Half-column. 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 22 Size S M L Dimension (mm) A 50.8 50.8 50.8 B 41.0 41.0 41.0 C 12.70 19.05 26.64 D 3.2 3.2 3.2 E 3.2 3.2 3.2 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 Ins trumentat ion 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 thermo-couples, 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 a o o 3 3 E O O oo • * 4> 3 (0 C D< o "--1 s ^ O a -2 •si <3 • *v o n « «i -a T3 . o w 4) (-i 4> v £ 3 JS W S3 O • 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, —APS, 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, —APa, 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 .976(-AP 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 2.0- / i . o - y ^ o . o 4 i 1 1 1 0.0 1.0 2.0 3.0 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 maxi-mum spoutable bed height, spout diameter and spout shape could be determined. 3.3 B e d M a t e r i a l 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 -12 + 14 - 1 4 + 16 -16 + 20 -20 + 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 avg.dia., dPi{mm) 2.40 1.70 1.29 1.09 0.925 0.78 0.355 d - x - 1 aP> ~ £(*./dPl) - X net weight(g) 0.2 44.7 905.6 234.4 69.0 9.8 14.0 1277.7 .1942 mm 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. The 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 Cold sand avg. dia. (mm) 2.216 1.646 1.216 1.027 0.919 Heated sand avg. dia. (mm) 2.025 1.630 1.200 1.010 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 Experimental Procedures and Condit ions 4.1 Operat ing procedure 4.1.1 Operat ion 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 Conditions 30 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 M e a s u r e m e n t 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 trans-parent 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 fH° Ha . dz 2 (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 Conditions 31 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 corre-sponding 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(T3 +1.73 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 corre-sponding ports along the bed height. A set of pressure transducers was also installed and connected to a data-logging computer. 4.1.3 Calculat ion of Ums Flowrate in the Spouted B e d Figure (4.7) is a simplified flow diagram of the experimental apparatus. Applying the ideal gas law, VS = VR PRTS (4.29) .PsTR\ From the rotameter reading, VSTD w a 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 = - ~ (4.30) \/PR Chapter 4. Experimental Procedures and Conditions 32 Therefore and ROTAMETER VALVE "* EXHAUST P s T 5 SPOUTED BED Figure 4.7: Simplified flow diagram of the apparatus. VR_ VsTD IPSTD PR T/ T/ jPSTD T / jPsTD VR = VSTD\ = VsTD\ (4.31) (4.32) .PR V pR Substituting Equation (4.32) into Equation (4.29) gives the flowrate in the spouted bed, Vs — VsTD\ PR PRTSI IPSTRI V, STD Tc lR VPSTDPR (4.33) M i n i m u m Spout ing Veloc i ty From Equation (4.33), we can proceed with the detailed calculation of Ums as follows: Chapter 4. Experimental Procedures and Conditions 33 (1) 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, VSTD-VSTD = 0.4800 + 0.2945 x R (large rotameter) (4.34) VSTD = 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 (4.36) where Pg is the gauge pressure upstream of the rotameter and AR\ is the pressure difference across the rotameter. (4) Determine the average absolute pressure of the spouted bed, Ps-a. Calculate absolute pressure at the port above the orifice, PR-PB=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 + ^ ^ ^ (4-3 8) Chapter 4. Experimental Procedures and Conditions 34 (5) From Equation (4.33), calculate the volumetric flow rate through the spouted bed, Vs-(6) Calculate Ums: Urns = ^ (4.39) 4.2 Exper imenta l Condit ions 4.2.1 Range For the 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 Us/ Ums dp(mm) Diimm) H{m) T(°C) 1.0-1.1 0.915 - 2.025 12.70 - 26.64 0.10-1.00 20°C - 580°C 4.2.2 Exper imenta l Error Calculation In this thesis, the following definitions are used for the comparison of the experimental values with predicted values: M . CAL-EXP innn. % dev = — — x 100% (4.40) RMS % ERROR = J[J2(% dev)2}/M (4.41) AVG ERROR = [ £ | % dev \ ]/M (4.42) Chapter 4. Experimental Procedures and Conditions 35 where EXP = experimental value CAL = predicted value M = number of data points Chapter 5 Resul ts : M i n i m u m Spout ing Veloci ty 5.1 Measurement 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 tempera-ture 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 there-fore 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 con-troller. 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 Spouting Velocity 37 5.2 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 iameter According to the Mathur - Gishler equation, when other conditions are fixed, Ums in-creases 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 tempera-tures 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 Chapter 5. Results: Minimum Spouting Velocity 38 OS 00 b-© 1 03 A3. b-b -OP 02 b. Dc=156 mm Di=19.05 mm H=0J20 m © = 300 "C • = 580 #C 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 OS o f 00 .dl 1. 6 d' d' d' d. Dc=156 mm Di=12.70 mm H=0.20 m Ts = 20°C = 300 °C = 580 *C o.e 1.0 1.2 1.4 1.6 1.8 Particle diameter, dp [mm] 2.0 2.2 Figure 5.9: Effect of particle diameter on Um>- (.Dc=156 mm, Z?i = 12.70 mm, H=0.2 Chapter 5. Results: Minimum Spouting Velocity 40 CO CM JO" 6 05 00 d" to d_L Dc=156 mm dp=2.025 mm H=0.30 m o © • Tg = 20 °C = 300 °C = 580 °C — I 1 1 I 1 1 1 1 1 1 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 5.10: Effect of orifice diameter on Unt. (P c = 156 mm, dp=2.025 mm, H=0.3 m) Chapter 5. Results: Minimum Spouting Velocity 41 CO to d-I B d d -02 d. Dc=156 mm dp=1.01 mm H=0.20m o © • Tg = 20°C = 300 8C = 580 'C 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 Height 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 Temperature 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 mm 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 Chapter 5. Results: Minimum Spouting Velocity 43 oo to Dc=156 mm Di=26.64 mm dp=2.025 mm 02 J, 00 co d-CO d l d. a A • Tg = 20°C = 300 °C = 580 'C 0.0 1 — 0.1 0.2 ~ I — 0.4 0.3 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 « CO d © Dc=156 m m Di=19.05 m m H=0.2 m t» to ID O ' 00 d-02 © -© d . 0.0 — 1 100.0 dp, mm o = 2.025 n = 1.630 A = 1.200 o = 1.010 200.0 300.0 400.0 Bed temperature, °C — i — 500.0 600.0 Figure 5.13: Effect of temperature on Umt. (Z?c=156 mm, A=19.05 mm) Chapter 5. Results: Minimum Spouting Velodty 45 o Dc-156 nun Di=26.64 mm H=0.2m m to SdH O ' CO d" 02 d © -o d. A A o o o s $ 0.0 — I 100.0 dp mm o = 2.025 a = 1.630 A = 1.200 o = 1.010 200.0 300.0 400.0 Bed temperature, °C 500.0 600.0 Figure 5.14: Effect of temperature on Umt. (Z>c=156 mm, D,=26.64 mm) Chapter 5. Results: Minimum Spouting Velocity 46 d-oo d" © Dc=156 m m Di=12.70 m m H=0.2 m o oa to d to m O " CO d" d -o-q d. 0 ~ ffl— Lp m m o = 2.025 • = 1-630 A = 1.200 o = 1.010 ffl = 0.915 — i 500.0 0.0 100.0 200.0 300.0 400.0 Bed temperature, "C 600.0 Figure 5.15: Effect of temperature on Umt. (DC=156 mm, L>;=12.70 mm) Chapter 5. Results: Minimum Spouting Velocity 47 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 Option 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 ( dp\ (DiY ( H\° fpp- p/Y yfl^E \DCJ \DJ \DJ V Pf The equation of Wu et al. is (5.45) 77 ( A \ 1 0 5 / n \ 0-266 , U \ -0 .095 / „ n \ 0-256 Ums =10.6(#) (%) (%) (eJLZU) (5.46) y/2pl ' \DCJ \Dj \DJ V Pi 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 Velocity 48 Table 5.3: Constants in Equation (5.47) and root mean square errors for three correlations parameter Mathur-Gishler eq. Wu et al. eq. This work K 1.0 10.6 28.4 a 1.0 1.05 1.17 T 0.333 0.266 0.127 UJ 0 -0.095 -0.0452 i 0.5 0.256 0.151 RMS, % 17.4* 18.7** 16.5* 7.82** 8.10* 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 Second Option Ignoring particle shape but including //, Ums = f(dp,(pP- pf),pf,fl,Dc,Di,H,g) (5.48) By dimensional analysis, dpUmspf _ (<%(pP ~ Pf)pf9 A H_ A PP~ pj ^ \ p2 ' Dr: A ' dj P ' c ^c " p PS i.e. \ Dc Dc dp pf j (5.49) (5.50) If one ignores the last group on the assumption that particle and fluid densities are adequately accounted for by the Archimedes number, then Di H DA Rems = tp [Ar, 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 , u \ 0-464 / n \ ° - 0 9 4 3 / „ „ \ ° - 2 5 8 ^=4,5X10-M^(|) (|) g ) (^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 TT f A \ 1-165 / n \ 0.130 , IT \ -0.043 / „ \ 0.154 ^ - 26.92 ( i ) (#) (£) ( ^ ^ ) (5.56) V ^ F ' v^J \DJ Vzv V />/ 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 the 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) Chapter 5. Results: Minimum Spouting Velocity 50 By dimensional analysis, E = L = * ^ = i, *,£*) (5.58) >mS n-emJ \ L>C uc 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 10-5Ar - 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 = 24.6[N/ l+3.59xl0-Mr - 1] (£) (-g-J ' ' From Equation (5.60) plus (5.63), Dt\0U8 fH\0-350 fDt' dp Rems = 25.7[Vl+5.51 x 10~Mr - 1] (jj-"j ( — ) c' \ Up ± ± 12.6% (5.64) 10.5% (5.65) Chapter 5. Results: Minimum Spouting Velocity 51 From Equation (5.61) plus (5.62), -0.0177 / IT \ 0.438 / »-. \ 0 1 3 2 / N \ ° - 2 7 2 W V Pf (5.66) 0.0302 / , \ 0.1897 ^ - l ^ l + S J W x l O - A r - l ] ^ ) - - ( I ) ' ( £ ) ' ( a ^ a l ) ' ±8.28% From Equation (5.61) plus (5.63), , / n \ 00847 / rr v 0.439 / n \ U U J U * / . . N \ u i o 9 ' ^.-4.1,(^1+SJlxI^Ar-.](^) (£) g ) (*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 Therefore Ret = ^ ^ = <f> (Ar) (5.70) f1 «e„,^(^(f,f,g (5.71) If, as before, one includes the additional ratio (pp — pj)/pf in the correlation, then Chapter 5. Results: Minimum Spouting Velocity 52 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. Chapter 5. Results: Minimum Spouting Velocity 53 1 I 1 I 1 1 1 I I 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Predicted Ums [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. 11.2 13.6 This work 5.82 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 the equation of Wu et al. at high temperature and vice versa at room temperature, while the 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 0.632 , IT s 0.381 / n \ - ° - 3 7 7 / „ „ \ -0-145 * „ - * . , > * 2 . 0 3 ( f ) ( £ ) ( g ) ( < ^ ) (5 ,6 , with an RMS error of 6.62 %. This value is smaller than 7.82 %, the 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 / n \ -0.388 / \ -0.126 ^ = ^)xU.(g) (|) (f) (**) (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 02 1 CO CO d. Dc=156 m m / Di=19.05 m m • d p = 2 . 0 2 5 m m / / LEGEND • 20 #C • 580 °C This work M-G_Ecpiation Wu'Equation " Grbavcic Eq. — i 1 1 1 1 1 1 1 0.0 0.1 0.2 0.3 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 dp=1.630 m m / LEGEND • 20 «C • 580 °C This work M—G jEojiatioii WiTEq'uatlon Grbavcic Eg. T T " I — 0.3 0.0 0.1 0.2 0.4 0.5 0.6 0.7 Bed height [m] 0.8 0.9 1.0 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 : M a x i m u m 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 map. The measurement of Hm was approached from bed heights above Hm, so that solids were intermittently discharged from the 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 Spoutabi l i ty 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 - 1 2 1 3 - 1 8 1 9 - 2 4 2 5 - 3 0 3 1 - 3 6 3 7 - 4 2 4 3 - 4 8 4 9 - 5 4 5 5 - 6 0 6 1 - 6 6 6 7 - 7 2 7 3 - 7 8 7 9 - 8 4 8 5 - 9 0 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 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 Spoutability 20°C V V V V V V V V V X V V V V V 170°c V V V V X V V V V X V V V V V 300°C V V V V X V V V V X V V V V V 420° C V V V V X V V V V X V V V V V 500°C V V V V X V V V V X V V V V V 580° C V V V V X V V V V X V V V V V Chapter 6. Results: Maximum Spoutable Bed Height 60 6.2 M a x i m u m 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 .Ar. ( V l + 3 5 . 9 x l O - M r - 1 ) 2 (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 Part ic le d iameter on Hm If the expression for Ar is substituted into Equation (2.10a), the latter becomes C2 Hm = d$ y/l + C3<% - 1 Chapter 6. Results: Maximum Spoutable Bed Height 61 £1 « o q o o "*1 q o o / / o S O O o / cv' O O / C D o o © o ^ °9, o< Dc=156 mm o d. 1.0 2.0 0.0 i 3.0 4.0 5.0 6.0 7.0 8.0 Dc UP\ \Dc] 2/3 [700' .Ar. ( \ / l + 35.9 x 10 -6 4r - l )5 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 Bed Height 63 where C2 = 700D8J3D^/3p,2/(pp - Ps)P}g and C3 = 35.9 x l O " 6 ^ - Pf)Pfg/p2. (dp)crit is found by setting d(Hm) / d(dp) equal to zero. The solution is (dp)3crit = 8/C3 (6.79) or {dp)crit = 60.6 F 1/3 (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 1.205 1.84 1.358 170 0.797 2.48 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 The 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 iameter 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 Temperature 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 1.2-1.1-B 0.9-<u & 0 .7--d r^ 0.6 3 «J 0.5 H o A 0.4H CO M d 0.3-1 0.2-0 . 1 -0.0-Dc=156 mm Di=26.64 mm 0.8 LEGEND D 20 °C 20 *C B 300 «C 300 #C 5 8 0 - C 5 8 0 «C a m m m 1.2 ~i— 1.4 i 1.6 1.0 .  1.8 Particle diameter, dp [mm] 2.0 2.2 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 Bed Height 66 1.2-1.1-1 ^ B 0.9-0.8-0.7-0.6-0.5-O P< 0.4 H 0.3-0.2-0.1-0.0-LEGEND a 20 *C 20 #C ffl 300'C 300 *C 580 °C 580 *C a GB ffl Dc=156 mm Dj=19.05 mm 0.8 1.0 1.2 1.4 1.6 1.8 Particle diameter, d p [nun] 2.0 2.2 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) Chapter 6. Results: Maximum Spoutable Bed Height 67 1.2-1.1-1.0-0.9-0.8-0.7-0.6-0.5-0.4 0.3-0.2-0.1 0.0-ea m / Dc=156 mm Di=12.70 mm LEGEND a 20 'C 20 *C a 300 'C 300 "C 580'C 580 CC 1.4 0.8 1.0 1.2 i 1.6 i 1.8 2.0 Particle diameter, dp [mm] 2.2 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 Bed Height 68 I i i « JO « 3 o a n X es 2 1.2-1.1-1.0-0.9 0.8-0.7-O.S 0.6 0.4-0.3-0.2 O.H 0.0-Dc-156 mm dp=S.025 mm LEGEND a 20 »c JIPc 3 0 0 *C • 680'C — I 1 1 1 1 1 1 1 1 — 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 6.24: Effect of orifice diameter on Hm. Points represent experimental data, lines represent McNab - Bridgwater equation. (Dc=156 mm, <fp=2.025 mm) B i 0) si •a © •8 1 a, m X 3 L2-1.1 L0 0.9 0.8 0.7-| 0.6 0.6H 0.4 0.3 0.2-0.1-0.0-Dc=156 mm dp=1.630 mm — i 1— 10.0 12.0 14.0 - I — 16.0 —I 1 1 1 1 1 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 69 J. 0 X i A •a v •8 o a a CO 2 1.2-1.1-1.0-0.9-0.8 0.7 0.6H 0.6 0.-H 0.3 0.2 0.1 0.0-LEGEND a 20 *C SO _*c olPc 3 0 0 *C EBIT'C 5 8 0 *Q Dc=156 m m dp=1.200 m m —I 1 1 1 1 1 1 1 1 — 10.0 12.0 14.0 16.0 16.0 20.0 22.0 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) a a 4 0) J3 •O V •8 § S3 2 L2-1.1 1.0 0.9 0.8-0.7-0.6 0.6H 0.4 0.3 H 0.2 0.1 0.0-LEGEND a 20 *C 20 *C B 300 'C IgO'C • 5 8 0 ' C ._58_O_;Q D c = 1 5 6 m m dp=1.010 m m — i 1 1 1 1 1 1 1 1 — 10.0 12.0 14.0 16.0 16.0 20.0 22.0 24.0 26.0 28.0 30.0 Orifice diameter, Di [mm] 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 Ar>0 (6.82) dAr v ' A derivative greater than zero for all values of Ar implies that Hm increases with in-creasing 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. Chapter 6. Results: Maximum Spoutable Bed Height 71 a i • i-t & t—t * O O H CO M cd 1.1-1.0-0.9-0.8-0.7-0.6-0.5-0.4-0 .3-0.2-0.1-0.0-O \ \ . \ O ^ N \ s N N \ N X Dc=156 mm Di=26.64 mm i i • o 1 dPJ5S-• *2.025 2.025 o 1.010 1.010 • • o o *""—'-*•*. i i • o 0.0 100.0 200.0 300.0 400.0 500.0 600.0 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 J, a i <D X> i—i ^3 o ft M 2 1.1-1.0-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1-o.o-\ o \ \ \ s \ \ \ o\ \ \ s V N Dc=156 mm Di=19.05 mm i i dp, mm • 2.025 2.025 o 1.010 1.010 N 1 1 1 • o " • • — -0.0 100.0 200.0 300.0 400.0 500.0 600.0 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 Bed Height 73 i <L) o .a r—1 •s o a. CO M 1.1-1.0-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1-0.0-> x> ex \ X \ X \ X \ X \ X \ X \ X \ X \ x. \ X \ X \ X \ \ \ X \ X • \ N ° •\ \ \ \ ° \ \ N \ \ N % Dc=156 mm Di=12.70 mm i i • % N N 1 dp, mm • 2.025 2.025 o 1.010 1.010 • ^ v ^ ^ O o "•**•• ^m 1 1 - ^ B O """—--0.0 100.0 200.0 300.0 400.0 Temperature, °C 500.0 600.0 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 D i a m e t e r 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 Temperature on 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 Height 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 S . C J . m Q <D - P (1) a as •3 O ft CQ <D 00 fi 5 4.5 4.0-3.5 -i 3.0-2.5-2.0-1.5-1.0-0.5-0.0-0.0 LEGEND • Experimental McNab Equation Wu.Equation Di=19.05 m m dp=1.630 m m H=0.3 m — i — 100.0 200.0 300.0 400.0 Bed temperature, °C 500.0 600.0 Figure 7.31: Effect of temperature on D.. (Dc=156 mm, Z><=19.05 mm, ^=1.630 mm) Chapter 7. Results: Average Spout Diameter 76 4.5 4.0-3.5 3.0-2.5-2.0-1.5-1.0-0.5-0.0 LEGEND • Experimental McNab Equation Wu Equation Di=26.64 m m dp=2.025 m m H=0.3 m 0.0 100.0 200.0 300.0 400.0 Bed temperature, °C 500.0 600.0 Figure 7.32: Effect of temperature on D,. (Dc=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 mat ter 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 Comparison w i th Exist ing 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 4.25-4.00-3.75-I 09 Q u -p .1-1 o CO So 3.50-3.25-3.00-0.2 TS/C • = 20 o = 170 o = 300 A = 420 x = 500 + = 580 • Di=26.64 mm dp=2.025 mm 0.3 0.4 0.5 Bed height H [cm] I — 0.6 0.7 Figure 7.33: Effect of bed height on Dt. (Dc=156 mm, JD,-=26.64 mm, ^=2.025 mm) Chapter 7. Results: Average Spout Diameter 79 1 .1 T3 <u i—i 3 o 1 — f S 4.0-3.5-3.0-2.5-2.0-1.5-1.0-0.5-0.0-o o o A X + i TS.°C = 20 = 170 = 300 = 420 = 500 = 580 i By 0 n/ £y o *4 J r V D / § ^ 0 A + / * ^ + Z o * ? X ^ ° +A A / * Ao / A +>$< x + T 1 1 1 D / n / B u/ D 9 A 6 A X X 1 + —1 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Experimental Ds [cm] 4.0 4.5 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. (Dc=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. Conclusions 82 7. The best fit equation for Ums and its root mean square error is: / n x o.54i / IT N o.452 / n \ - ° - 4 1 4 / . . \ - ° 1 4 9 Rem, = ,iM), 1.63(|i) ( I ) (*J ( ^ ) ±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 that 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 Nota t ion 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, p ~f / g a5 Ratio of spout area to column area b Value of exponent on Hm in equation for U„. bi U m / U m / Co Drag coefficient CAL Predicted value D c 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 dp at which H m is a maximum EXP Experimental value f Friction factor fx 150(1 - o V/<£<£ f2 1.75(1 - ea)Pf/dpe3a G Mass flowrate of gas g Acceleration due to gravity Notation H Static bed height (m) Hm Maximum spoutable bed depth (m) h H / H m ( - ) k Constant in Equation (2.2) ( - ) M Number of data points ( - ) N D Best number, | Ar ( - ) n Number of particles accelerated per unit t ime ( - ) Pat™ Atmospheric pressure (Pa) P B Absolute pressure measured just below inlet orifice with solids in the bed (Pa) PE Absolute pressure measured just below inlet orifice without solids in the bed (Pa) Pg Gauge pressure upstream of rotameter (Pa) P M Absolute pressure of the gas meter (Pa) PR Absolute pressure of the rotameter (Pa) P s Absolute pressure in the bed (Pa) PSTD 1 atm (Pa) Q s Volumetric flowrate in the spout ( m 3 / 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 U T O S Minimum superficial spouting velocity U s Superficial gas velocity U* Free settling terminal velocity of the particles VF Volume of the float VM Measurement volumetric flowrate of the gas meter Vs Volumetric flowrate through the spouted bed VSTD Volumetric flowrate taken from the calibration curves W logw ND Xi Weight fraction of particles z Vertical distance from inlet orifice (m/s) (m/s) (m/s) (m3) (m3/s) (m3/s) (m3/s) ( - ) ( - ) (m) A P a Measured pressure drop above the orifice A P / Pressure drop across bed of particles at minimum fluidization A P m s Overall pressure drop at minimum spouting condition AP., Overall spouting pressure drop 7 Angle of repose of solids e Overall voidage of the bed emj Voidage at minimum fluidization A Reciprocal of sphericity H Fluid viscosity pb Bulk density of particles PF Density of the rotameter float Pf Fluid density (Pa) (Pa) (Pa) (Pa) ( - ) ( - ) ( - ) ( " ) (kg/m-s) (kg/m3) (kg/m3) (kg/m3) Notation 86 pp Particle density (kg/m3) <f> Particle sphericity ( - ) ift Net downward force of solids per unit volume (kg/m2 • s2) Bibliography [1] Mathur, K. 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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 Spout-ing 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 Veri-fication", 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 Fluidiza-tion, 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] Patt ipati , R. R. and Wen, C. Y., "Minimum Fluidization Velocity at High Temper-atures", 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., 35 , 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., 59, 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", Refinery Management and Petroleum Chem. Tech., 36, R-795 (November 1, 1944). [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, University of British Columbia, Vancouver, Canada (1976). [64] Grace, J. R., "Fluidized-bed Hydrodynamics", Chapter 8.1 in 'Handbook of Multi-phase Systems', Hetsroni, M., ed., Hemisphere - McGraw - Hill, New York (1982). A p p e n d i x A Calibration of R o t a m e t e r s For a rotameter, the governing equation is: 2gVF(pF ~ Pf)pJ G = CDA2 (A.83) AF[l - (A2/A1)^]\ 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 VPR\ and G = GR = PRVR — PRV;\ M PM^ iPj R (A.85) (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 = VI M M PR PR (A.87) 92 Appendix A. Calibration of Rotameters 93 ATMOSPHERE ROTAMETER P i VALVE GAS METER P M , V M , T M PM = 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 [PMm GsTD — Vi M PR\ y/PSTD and For an ideal gas, ^ S T D VsTD = = VM PSTD M PR\ I PR PSTD PR PR PSTD PsTD Substituting this relation into Eq. (A.89) gives PM VsTD = V} M (A.88) (A.89) (A.90) (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. •(-4 & CO Temperature = 20 deg C Pressure — 1 otm. 4 -CO V = 0.2692 + 0.0212 X R 50 100 150 Rotameter reading 200 250 X 4*. Appendix A. Calibration of Rotameters 95 in -<t en C>J o + o o 00 o II > o CO bO 8 *d o u u o o 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] . dp . \Dc] 2/3 '5686?' Ar (Vl + 35.9 x 10-6Ar - \f The above equation can be rewritten as Hm = C\ whence l— + 35.9 x 10-6 - \ — Ar V Ar x dHm dAr 1 = 2Ci + 35.9 x 10-6 Ar Ar 2 ^ + 3 5 . 9 x l 0 -6 V Ar2) 2 ^ Ar .2 — +35.9 x 1 0 - 6 - i / - — Ar V Ar rAr 1 Ar x Ar2 Ar2 y/Ar + 35.9 x 10-6Ar2 = Ci -J- + 35.9 x 10-6 - J - J -Ar V Ar 1 1 x ^ArVAr Ar\A4r + 35.9 x lQ-Mr2 > 0 /or Ar > 0 (2.10) (B.92) (B.93) 96 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) 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 dp (mm) 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 Conditions 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 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 ooooooooo CNO00OJN-OOJO00 OOOlDlOWBlfllfi HrtrlHHHHHrl OOOOiOOOOO) to O -*^ ' »»i -c R 3 o •—•* <$ -*J c a X W O .X Hs CJ Oi CX a < •^ITHTHOOOOOO ooooooooo KN-N-l>-KI*-l--r-N-CNCNCJCNCNCNCMCSOJ HrlHHrlrlrlHrl cNfO^wtosmroo cooococococococoo o a CO -*J O fl 2 "3 o <J £ CO S 3 (-^ -a 3 T3 CD r^A u CO * CD (XI cn cn 4=> CO to •^1 i-*-OOi-kl-i l->-00 h-^l-LOOl-J-l-t l-J-l->-00 l-'-HJ-OOOM-l-»-l-'-l-*-OOOOl-kl-»-l-*-KC7)CDHIOtj001U)KWi>O1C0KlO^iJiCn«)l-'IOUI>^00«)Ol-'UCnJ!>S00(0Ol-'W MOtooooo^iotDuoio^tD^ooocnwsiooi-jm^^coooiomootooocnwoiio S(OOOOM*>CDSU<OS(DSKO(»C))IO)-'(OKji^^(10tD010KC>)00^(J)0(JI*tO KlOIOMIO(OIOIOMIOION)[0(OIOIO(OIOMIOMIOMIOIOIOION3lO(ON3IOIOIOIOIOIO OlOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO COIOIOIOtOIOIO(OIOIOK3IOIOIOIOIOIO(OIO(OIOK3(OIO(OIOIO(OIOIOIOIOIOM(OIOIO owcncnoioioimcnmtntiicncnOTcnaicnoiaicnoioiaioicncrimuiaiWOTcnmuiOTUi o o (OlD(D(OtDl0(D(OlD(D(OlO(O!O(i3(D(O(O<O(D(D(O(O(DtOtO(OlDtDtO(DlDtOtD(OtO(O ooooooooooooooooooooooooooooooooooooo wwmoiaicntJimcncnOTOicTicnwmcncncnmcricncnOTWcncnaicnuicncrioioicncntfl ooooooooooooooooooooooooooooooooooooo COHMCo^^^MW^^i-'(OCo*'CntJih'iou^oioii-'ioto^cno)mi-'ioWif>tnms cnoooo4*oooo^oooootoooooo4*ooooootoooooooo ooooooooooaiooooooooooocnoooooooooooooo * * * * * # # h-'OOOOOOOOOOOOOOOOOOOOOOOOOOOOOM-l-'-l-^M-l-'l-^l-' ^^^^^^^^^ji^oiaiaicncnuiammoioimooajooroooooajioioiofOKJioio 0)Hl-'10MIOU10)0)01Sl-'l-'l-'M(OIOM10MWWWOOKKlOW(jOHK»IOW*i(i>0) tD01COOC01fi(DK^>IOIO^COKCnSOIOSl-'fflSIOO)IO>IW(D*>CO-<Cn(OCnO>00 ooooooooooooooooooooooooooooooooooooo o o o o !-»• 00 4* o o o o OJ -J CD o o o o CO •s CD o o o o CO -vl CD O o o o CO ->I CD O O O o (0 •s CD O O O O CO a> cn o o o o CO 01 cn 0 0 0 0 CO at cn 0 0 0 0 CJO CT> cn 0 0 0 0 CO O) cn 0 0 0 0 CO 4^ CO 0 0 0 0 CO 4* CO 0 0 0 0 CJO 4* co 0 0 0 0 CO 4* CO 0 0 0 0 CO 4* CO 0 0 0 0 CO 4* CO 0 0 0 0 CO 0 0 0 0 0 0 GO 0 0 0 0 0 0 CO 0 0 0 0 0 0 w 0 0 0 0 0 0 Co 0 0 0 0 0 0 Co 0 0 0 0 0 0 to 4* 00 0 0 0 0 to 4* 00 0 0 0 0 to 4* 00 0 0 0 0 to 4* 00 0 0 0 0 to 4* CO 0 0 0 0 to 4* CO 0 0 0 0 to 4* 00 0 0 0 0 I-1 00 4* 0 0 0 0 H» CO 4* 0 0 0 0 l-k 00 4* 0 0 0 0 M-00 4* 0 0 0 0 l-J-00 4*. 0 0 0 0 M-CO 4* 0 0 0 0 !-»• 00 4^ en CJl en to GO ^ o 4* CO C71 00 O00OOOO0OOOOO0OOOOOOOOO0OOOh-'-OOO0>-»-OO00Ol-*0000000l-' cno)0)M^cno)SiOi|itnmo)-j-<)oooo^o)oo*soooowovooi-'Oo(i)>iooouo)SSOOouo)aN(»oo(Oo C0O1CO0lSOl^O)C0CnS(0SIOSK(»O0l)Sl-lWUtO(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 MN3MS3IOIOWMS3MMWMWMMS30)0)0)0)0)0)0)0)(J)0)0)CJ)(J)0)CnO)CI)0)010)ffl(J)a010)0)OlOlO)01 OOOOOOOOOOOOOOOOOtOGOGOCOCOCOGOGOCAJGOtoGOGOGOGOGOOOGOOOGOGOCOGOGOGOGOCOGOOOGO ooooooooooooooooooooooooooooooooooooooooooooooo (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 ooooooooooooooooooooooooooooooooooooooooooooooo CTicncTiaicricTicricricTiCTicncnaicTiCTicncTioiaimwmmuicJimcTiaic^ ooooooooooooooooooooooooooooooooooooooooooooooo u^^h>iou*cnKWO)^tno)SOo«)PWMHS)UWpMWCJMMWJi^i-'iotoif>cncni-'ioWi(»oioiSOO OO>-'OOOOi-'OOOOOOOOOOO->JOOOtOOOO-sIOOOO--JOOOOO00OOOOOOOO ooenoooooooooooooooooooooooooooooenoooooenoooooooo # * * * * * * * OOOOOOOOM-l-Al-kl-kl-kM-l->-i-»-OOOOOOOOOOOOOOOOOOOOOOM-M-t-J-i-ke-»-l-kl-J-f-' C3>mO)C0C»000000WNJIOWtOIOWMW^^^^^^^CnWmwacnO5O)O)C»(»00000000IO(ON3IOIOIOIO(O louuoooKi-'i-'^iowWiiitJimspKiotiKDfflmh'KHioioiOMOouooKKtJioKi-'iow^^cnm aio^ioOTU)i^ooustnococriwco^mcoK(DMcnoiucn(»MowNhi*ioOTK!DMrocott)0)WM.oo(ocn ooooooooooooooooooooooooooooooooooooooooooooooo o o o o o o o o o o o o o o o o o o o o oo Go GO to to O O O >£• 4* O O O 00 00 00 o o o o o o o o o o o o to to to 4* 4* 4* 00 00 00 4* o o o o o o o o o o o o o o o o K* \-L !-»• M-00 00 00 00 o o o o o o o o o o o o o o o o M- t-* !->• »-»• 00 00 00 00 4* 4=> •£» if* 00 GO GO GO —4 -~J —4 cr> to CD to en o o o o o o o o o o o o GO GO GO GO Oi O) O) 45> en en en GO o o o o ooooooooo ooooooooo ooooooooo ooooooooo COOOGOOOGOCOGOOOtO 4*4*>t*00000*> GJG0G0OOOOO00 o o o o o o o o o o o o o o o o to to to to tf* £» >f» 4* 00 00 00 00 o o o o o o o o o o o o tO I-1 - M- >->• 4* 00 00 00 00 4* 4* *> o o o o o o o o o o o o o o o o o o o o o o o o I-1  h* I-1  I-* I-' 00 00 00 00 00 »£» 4=> »£» i£> 4^ o o CO o o o o o o CO o o o o CO CO <tf sf co co o o o o o o o o CO <* CO o o o o LO CD CO o o o o LO CO CO o o o o LO CO CO o o o o o> K CO o o o o <J> K CO o o o o O) h-co o o o o •* 00 T-l o o o o ^ 00 fl o o o o "tf 00 t-i o o o o «* 00 1H o o o o <tf CO tH o o o o <tf 00 T-l o o o o <tf CO TH o o o o <tf 00 t-l o o o o 00 ^ CN o o o o CO <tf CN o o o o 00 ^ CN o o o o 00 •tf CN o o o o CO <* CN o o o o o o CO o o o o o o CO o o o o o o CO o o o o CO <tf CO o o o o CO <tf co o o o o CO •* CO o o o o LO CD CO o o o o to CD CO o o o o LO CD CO o o o o 05 f~ co o o o o o K CO o o o o en h-CO o o o o <* CO •rH o o o o <tf CO t-l o o o o ^ CO TH o o o o <? 00 iH o o o o ^ 00 t-l o o o o ^ CO rH o o o o ^ CO T-l o o o o <* 00 1-1 o o o o ^ CO i-H o o o o ^ 00 TH o o o o <* 00 TH o o o o ooooooooooooooooooooooooooooooooooooooooooooooo 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 ooooooooooo OOOOOOOOOOOOOOOOOrH-.-l'rHi-l-rH'rHvHiHT-l * * * * * * * * * * OOLOOOLOOOLOOOOOOOOOOOOOOOOOOOLOOOOOOOOOOOOOOO OOOTOOLOOOCNOOCOOOOOOOOCOOOOOOOOLOOOs^OOCNOOLOOOOOO 0)HCNCNHtNCNHC^CNHSKlflin^flCNW^^(«)(Nr((<)CNHO)OIHC<JCN'riCSCNH(Oin*nO)TH * o o o o o CN O O O O CD LO if CO CN ooooooooooooooooooooooooooooooooooooooooooooooo WLOLOLOU^LOLOLOLOLOLOlOLOLOLOWLOLOLOlOlOLOLOLOLOLOlOLOLOLOLOtnLOLOLOLOLOLOLOLOLOLO^^^^^ OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOCDCOCDCOCO O)O5O5O^O)cj)o^o>a5a)O)C)o^a)OioiOTa)O)O50)oi<j)o^o)O^O)Oicno^o>c^OT IHTHIH,H,H,HTHTHIHTHTHIHTHIHTHIHTHVMTHVHIHYHIHTH^THIMTH,-IIHIHIHTM,H^,H^ OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOLOLOLOLOLOLOLOLOLOLOLO OOOOOOOOOOO'^iHlHiH^iHiHlHiHl-lTH'iHTH-^THiMTMi-liHxHvHiMiHiHi-liHiHrHiH-tH-rHCNCNCNCNCN CNCNCNCNCNCNCNCNCNCNCNOOOOOOOOOOOOOOOOOOOOOOOOOO)0>Cna)0)0)00000 IHTHTHTHIH^^IHIHTHTHTHTH*HIHTHIHTHIHIHTHT-IVHIHTHIHTHIHTHTHTHTHVH^IH^ HOOKOHNHOSPIfflHONH^tDNlOOO)(0(DO)0)NnmU30(Oin(J)WlOOlfl!Onoffl*(D(J)(0(0(fl otNnioiow^inwoiioooocsNcsNtoooininiflocicNtomTHPjcofDncoN^ocNoococsoicitnoiomn incMSU5csu5ifln<ou)CNS©«3inu3<,(OH^in^<*c>iifl'*iHin*H^^'H^'5fcN(oin'j,*(0'HwcNHOo) OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOIHI-IIHTHO CO ^> CO CO CO 00 00 CJ> LO o CN CO CN CO CN CN CO co CN CO CN CO LO CN CO Appendix D. Experimental Data 103 0.450 2.025 26.64 0.100 1.213 0.0000184 32(7) 1.772 2.025 26.64 0.615* 0.838 0.0000248 1.619 2.025 26.64 0.600 0.835 0.0000248 1.343 2.025 26.64 0.500 0.826 0.0000248 1.197 2.025 26.64 0.400 0.820 0.0000248 1.049 2.025 26.64 0.300 0.814 0.0000248 0.921 2.025 26.64 0.200 0.809 0.0000248 0.496 2.025 26.64 0.100 0.804 0.0000248 33(6) 1.696 2.025 26.64 0.590* 0.644 0.0000300 1.376 2.025 26.64 0.500 0.638 0.0000300 1.236 2.025 26.64 0.400 0.633 0.0000300 1.100 2.025 26.64 0.300 0.628 0.0000300 0.975 2.025 26.64 0.200 0.625 0.0000300 0.486 2.025 26.64 0.100 0.621 0.0000300 34(5) 1.768 2.025 26.64 0.540* 0.528 0.0000343 1.350 2.025 26.64 0.400 0.523 0.0000343 1.200 2.025 26.64 0.300 0.520 0.0000343 1.024 2.025 26.64 0.200 0.516 0.0000343 0.606 2.025 26.64 0.100 0.513 0.0000343 35(5) 1.639 2.025 26.64 0.500* 0.472 0.0000365 1.385 2.025 26.64 0.400 0.468 0.0000365 1.191 2.025 26.64 0.300 0.465 0.0000365 1.025 2.025 26.64 0.200 0.462 0.0000365 0.556 2.025 26.64 0.100 0.460 0.0000365 36(4) 1.612 2.025 26.64 0.440* 0.424 0.0000379 1.212 2.025 26.64 0.300 0.420 0.0000379 1.014 2.025 26.64 0.200 0.418 0.0000379 0.531 2.025 26.64 0.100 0.417 0.0000379 37(6) 1.152 1.630 26.64 0.630* 1.247 0.0000184 0.941 1.630 26.64 0.500 1.239 0.0000184 0.845 1.630 26.64 0.400 1.230 0.0000184 0.778 1.630 26.64 0.300 1.223 0.0000184 0.689 1.630 26.64 0.200 1.217 0.0000184 0.365 1.630 26.64 0.100 1.213 0.0000184 38(6) 1.315 1.630 26.64 0.585* 0.827 0.0000248 1.157 1.630 26.64 0.500 0.824 0.0000248 1.041 1.630 26.64 0.400 0.819 0.0000248 0.881 1.630 26.64 0.300 0.814 0.0000248 0.755 1.630 26.64 0.200 0.805 0.0000248 0.367 1.630 26.64 0.100 0.802 0.0000248 39(5) 1.222 1.630 26.64 0.440* 0.635 0.0000300 1.121 1.630 26.64 0.400 0.633 0.0000300 0.895 1.630 26.64 0.300 0.628 0.0000300 0.737 1.630 26.64 0.200 0.622 0.0000300 0.378 1.630 26.64 0.100 0.620 0.0000300 40(4) 1.152 1.630 26.64 0.350* 0.522 0.0000343 0.996 1.630 26.64 0.300 0.519 0.0000343 CO CO LO LO «* <tf 10 to CO CO CO CO o o o o o o o o o o o o o o o o ix) o> en en (ONNS CO CO CO CO o o o o o o o o o o o o o o o o ^ ^ ^ ^ 00 00 00 00 H H iH H o o o o o o o o o o o o o o o o «* "sF ^ CO 00 CO 00 00 «# <tf H ^ H M CN o o o o o o o o o o o o o o o o o o o o 00 00 00 o <* ^ <* o CN CN CN CO o o o o o o o o o o o o o o o o O O O CO CO CO LO o o o ^ <& ^ to CO 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 o o o o o o o o o LO ID en en CO to r- N-CO CO CO CO o o o o o o o o o o o o o o o o 000000000000^^^^^000 T-IVH-T-IT-IVHTHCNCNCNCNCNCOCOCOCO ooooooooooooooo oooooooooooooo oooooooooooooo oooooooooooooo CO o o o ooooooooooooooooooooooooooooooooooooooooooooooo mc^u)rtoiooowMoiH^scono)U)oincNO)(otNO<oio(N^'HO)0)(Dnon(OKnin(Ootnois(NO(£) HTH^(0inC^Hrtin^^00CJ-HHHHHOO(NOlCNCSTHTHrl«)(DlflHH^^(r|CNHrHT-IHTHOOC^OI(N'H OOOOOOOOT-IT-I THTHT-IIHOOOOOOOOOOOOOOOOO->-1 ooooooooo #• # # •& -X- * OOLOOOLOOOOOOOOOOLOOOOOOOOOOOO OOOOOtOOOLOOOOOOOKOOOOCnoOOCOOO (NTHClOCNvHCNCNTHtOtDLO^COCN-rH^^COCNTHCOCOCN-rHCNCN-rH LOOOLOOLOOOOOOOOOOOOOOO COOOCNO^OOOOO^OOOOOOOLn CNCN^CN-rHLOLO^COCNTH^'^COCNT-tCOCN'rHCN OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO ^* ^* ^* 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^ OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO C0C0C0C0C0C0C0C0OOOOOOOOOOOOOOOOOOOOOOOO'«HiHTH'«HT-lT-l-rH'rHTHTH'rHTHTH-rHTH tOtOtOlOU3tOtDlOCNCNCNCNCNCNCNCNCNCNCNC>ICNCNCNCNCNCNCNCNCNCNCNCNOOOOOOOOOOOOOOO ^lfi(ONOrtmONOWN«lOSOH0)H(<)rl00<JlOHCSU)T<CN0)0)H«tiS00CIlU)O)Olt<)O)CN^UNO<t ocono)(S'HO(<)rioio^oo)Wrt^Lnono(<)nocii>-(0(o<oH(OTHitscoiDO)Oooi)nHri(oooioOH 00*OS^ffiCO*COSS(OO^OIStDin*CNKlOlO<N(Din(NinU)CNlOCNlO(DLO^<}l(N(0!DU5^CNin'!t,rHin OO-rHOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO CO CO CN CO LO CO CO CN CO LO <tf CD ^ h-<tf 00 ^ O) ^ LO o LO CO LO CO CN LO Appendix D. Experimental Data 105 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 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 D. Experimental Data 106 0.856 1.630 12.70 0.400 1.244 0.0000184 0.774 1.630 12.70 0.300 1.230 0.0000184 0.655 1.630 12.70 0.200 1.216 0.0000184 0.230 1.630 12.70 0.100 1.212 0.0000184 68(6) 1.101 1.630 12.70 0.635* 0.826 0.0000248 0.988 1.630 12.70 0.500 0.823 0.0000248 0.855 1.630 12.70 0.400 0.819 0.0000248 0.748 1.630 12.70 0.300 0.812 0.0000248 0.637 1.630 12.70 0.200 0.805 0.0000248 0.303 1.630 12.70 0.100 0.802 0.0000248 69(5) 1.287 1.630 12.70 0.545* 0.633 0.0000300 1.071 1.630 12.70 0.400 0.630 0.0000300 0.939 1.630 12.70 0.300 0.624 0.0000300 0.849 1.630 12.70 0.200 0.622 0.0000300 0.375 1.630 12.70 0.100 0.619 0.0000300 70(4) 1.117 1.630 12.70 0.435* 0.523 0.0000343 0.982 1.630 12.70 0.300 0.518 0.0000343 0.895 1.630 12.70 0.200 0.515 0.0000343 0.379 1.630 12.70 0.100 0.512 0.0000343 71(4) 0.997 1.630 12.70 0.415* 0.464 0.0000365 0.820 1.630 12.70 0.300 0.464 0.0000365 0.751 1.630 12.70 0.200 0.462 0.0000365 0.380 1.630 12.70 0.100 0.459 0.0000365 72(4) 0.960 1.630 12.70 0.395* 0.421 0.0000379 0.872 1.630 12.70 0.300 0.419 0.0000379 0.753 1.630 12.70 0.200 0.418 0.0000379 0.371 1.630 12.70 0.100 0.416 0.0000379 73(9) 0.919 1.200 12.70 0.940* 1.273 0.0000184 0.874 1.200 12.70 0.800 1.263 0.0000184 0.819 1.200 12.70 0.700 1.261 0.0000184 0.767 1.200 12.70 0.600 1.254 0.0000184 0.704 1.200 12.70 0.500 1.246 0.0000184 0.636 1.200 12.70 0.400 1.235 0.0000184 0.563 1.200 12.70 0.300 1.226 0.0000184 0.474 1.200 12.70 0.200 1.218 0.0000184 0.239 1.200 12.70 0.100 1.213 0.0000184 74(6) 0.718 1.200 12.70 0.595* 0.826 0.0000248 0.674 1.200 12.70 0.500 0.821 0.0000248 0.605 1.200 12.70 0.400 0.816 0.0000248 0.529 1.200 12.70 0.300 0.810 0.0000248 0.437 1.200 12.70 0.200 0.804 0.0000248 0.236 1.200 12.70 0.100 0.801 0.0000248 75(5) 0.608 1.200 12.70 0.490* 0.634 0.0000300 0.560 1.200 12.70 0.400 0.630 0.0000300 0.512 1.200 12.70 0.300 0.625 0.0000300 0.478 1.200 12.70 0.200 0.622 0.0000300 0.204 1.200 12.70 0.100 0.619 0.0000300 CO CO CO CO ^ ^ ^ ^ CO CO CO CO o o o o o o o o o o o o o o o o LO lO LO LO CD CD CD CO CO CO 00 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 ^ 00 TH o o o o •tf 00 vH o o o o <* 00 T-l o o o o <tf 00 T-i o o o o ^ ^ f^ 00 CO 00 rl TH rl o o o o o o o o o o o o 00 <tf CM o o o o 00 «tf CN o o o o 00 «tf CN o o o o CO ^ O) o o o o CO ^ CN o o o o CO •tf CN o o o o o o CO o o o o o o CO o o o o o o CO o o o o o o CO o o o o CO •* CO o o o o CO ^ CO o o o o CO ^ CO o o o o CO ^ CO o o o o to co CO o o o o LO CO co o o o o LO CO CO o o o o CD N-co o o o o CD N-00 o o o o CD >-co o o o o ^ CO 1-1 o o o o ^ CO T-l o o o o ^ CO T-l o o o o <* CO vH o o o o ^ CO T-l o o o o <tf CO T-l o o o o ^ CO T-l o o o o 00 ^ CN o o o o CO <* CM o o o o ooooooooooooooooooooooooooooooooooooooooooooooo OCOlOtNlfl^CNfflOtOfflHHfflrttONmofOlCO^i-IOWtOOWCDWM^CJOlOtOWNtDNTi^Mrjo^ NHHHU(fltDiriNrlH©W(0(0WrHrlWNrlTHOOnWN0lTlt-lrlrl(0(0l0«r(HlC^f0(r)(NT<HOIH U)lOU)U)^^^**^^CSCNWCNCNNnC00000CO00a)^^lOlOWWlfilfl^^^^^^OICNCNCN(NCVICvlC0(O OOOOOOOOOOOTHTHT-IT-II-I T-»OOOOOOOOOOOOOOOOOOOOT-IT-(T-IT-IT-ITHT-<OO JL Jt J/. J(. Jt .If. Jf. J£. 4f. Jt -¥-ooooooooinoooooooooLooooooinooomoooooomoooooooooino OOOO^OOOCDOOOOOOOOOCOOOOOOOOOOOLOOOOTHOOOOOOOOOOOOOOTHO ooooooooooooooooooooooooooooooooooooooooooooooo ooooooooooooooooooooooooooooooooooooooooooooooo CNCNCSCNCNCN<N<NCNCNCvlCNCNCN<NCNCN<NCNCNCNCNCNCNCNCNCNC^ ooooooooooooooooooooooooooooooooooooooiominmioinLouoLo OOOOOOOOOOOTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTH CSCMCMCMC^CNCN<NC^CNCNOOOOOOOOOOOOOOOOOOOOOOOOOOO05C^C7)C^CnCDCDCT)CD THTHTHTHT-IT-IT-IOOOOOOOOO <j,ioNtNnn(osoKSinifinoo)(o(<)ooTHHO)inoooss(D(iiflNKO)0)om(Dioiooomwffl<io(»'tw 0!0(OrlO!DSfl)S(OOOWH(OTHC)lON(fl^OO)WO)0)^01^<D(00(0^mN(OfflNOWH(0(100SCOO wio^N©iJ)^THin^HUi(ou)in<i |cOHiniflin^(<:riu)^(OH^'t^H^(«)H^(0'HU3inin'*to(<5H^^ ooooooooooooooooooooooooooooooooooooooooooooooo co CO CO 1^ CD CO o CO CO CN CO CO 00 CO CO CO LO CO LT3 co CO tr o> f> o •-+> h-1  • (W CO !"+> O P- (T fc3 <-»- cr o> 10 tr CO g- 0> p o tr tO £ Z p oJ i-i P 1—» &- P> c-t- J" b ii i—i Ol 05 a 8 "a II to Ol >r-ce> 3 H tr ft £ 2 cr a> >~i CO O CO 00 to Co 00 00 00 en oooooooooooooooooo t-ktOCOM-tOCOi-kCOCO>£»t-'-CO^^i^H*tOCO W00i|iU«3SWW(OCOO)WHO)tOai00Ol (OSl-i<OStOm«)i-kUH00(O^^WWO> oooooooooooooooooo (D(C(D(0(DU)(0(OtD(OlO(OtO(0<OtC(DID l-»-l-*l-kh-''l-*l-*-l-»-l-»-l-»-l-il-kl-»t-»-l-kl-»-l-kl-»-l-k cnoiaioitnoimoicnoioicncnoicnaioioi totototototototototototototototototo oooooooooooooooooo oooooooooooooooooo l-'IOIOKIOWKIO(ji)ti)l-'IOWifii^l-kIOO) oooooooooocnoooocoooo oooooaioooooooooooo * * # * oooooooooooooooooo ^^^^.tfatficnuicncnoicncnOTOiOooooo h'KMUIOlfflKH'l-'lOHMlOtoUOOO m(oo(Dioo)(oro(DOtDWtJii-'aiMoioo oooooooooooooooooo o a-o 01 o o o o o o o o CO CO -J -J CD CD o o o o o o o o o o o o o o o o CO CO CO CO -4 <J> oi <y> ID Ol Ol Ol o o o o o o o o o o o o o o o o CO CO CO CO »t» 4* 4=» 4=» CO CO CO CO o o o o CO o o o o o o CO o o o o o o CO 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 Co to to to O if» (^ >P» O 00 00 00 Appendix E Fortran Programs E . l P rogram on Ums correlation C This program i s wr i t ten to f i t experimental data to a dimensionless C equation of the form C C Nl=bl*(N2**b2)*(N3**b3)*...*(Nn**bn) C C bl,b2,b3,...bn are constants to be determined. Newton's Method C will be used. This method requires that the initial values of C bl,b2,b3,...bn are very close to the solution in order to C obtain convergence. The method of Multiple Linear Regression C is used to provide this set of initial values of bi's. To apply C the latter method, the above equation has to be rewitten as C C Y=f(X1,X2,X3,...Xn)=al*Xl+a2*X2+a3*X3+...+an*Xn C ie lnNl=bl+b2*lnN2+b3*lnN3+...+bn*lnNn C C This program is set to handle a maximum of 10 unknowns C (bl,b2,b3,...bn) and 305 sets of data points. To accomodate C more unknowns and/or sets of date points, change the dimensions C of the arrays accordingly and of course, the DATA statement in the C beginning of the MAIN program. C C The data file should be in the form of C C N11,N21,N31,...Nnl C C C C Nlm,N2m,N3m,...Mnm C where C m=number of sets of data points C n=number of unknowns sought C C 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 Programs * H(384),RH0P(384),RH0(384),VISC(384),UIF(384),AR(384), * W(384),RET(384) C C Read input data C 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 Give value to elements of the non-linear equations C 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 Transform input data into x-values & y-values C 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)) 30 CONTINUE 40 CONTINUE C CALL LSqM(X,Y,M,N,A,VAR) C C Transform A to B C B(1)=EXP(A(1)) DO 50 IN=2,N B(IN)=A(IN) 50 CONTINUE C C Calculate the variance of the fit C SSUMT1=0. Appendix 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) 90 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))**2 100 CONTINUE VART1=SSUMT1/(M-N) VAR1=SSUM1/(M-N) RMST=SQRT(RMST/M) RMS=SQRT(RMS/M) WRITE(6,120) 120 F0RMAT(5X,'INDIRECT APPROACH-Multiple Linear Regression WRITE(6,130) 130 FORMAT(/,15X,'The fitting parameters are:-') WRITE(6,140) 140 F0RMAT(15X, ' ') WRITE(6,150)(I,B(I),I=1,N) 150 F0RMAT(l5X,'b',Il,'=',F10.4) WRITE(6,160) VART1,VAR1 160 FORMAT(/,15X,'Variance =',F9.5,5X,'V (Urns) =',F9.5) WRITE(6,170) RMST,RMS 170 FORMAT(/,15X,'RMST =' ,F9.5,5X,'RMS(ums) =',F9.5) B(1)=B(1)/FACT0R DO 180 1=1,M YN1(I)=YN1(I)/FACT0R 180 CONTINUE MM=0 250 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 E. Fortran Programs 260 CONTINUE DIFFl(I)=YNl(I)-l.*B(l)*PROD(I) DIFF2(I)=YNl(I)-2.*B(l)*PR0D(I) 270 CONTINUE CALL LSQM2(X,XN,YN1,M,N,PROD,DIFF1,DIFF2,B(1),DELB) DO 280 1=1,N B(I)=B(I)+DELB(I) IF(ABS(DELB(I)).GT.DB) IFLAG=1 280 CONTINUE IF(IFLAG.EQ.O) GO TO 290 GO TO 250 290 B(1)=B(1)*FACT0R DO 300 1=1,M YN1(I)=YN1(I)*FACT0R 300 CONTINUE C C Calculate the variance of the fit C 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))**2 310 CONTINUE VART2=SSUMT2/(M-N) VAR2=SSUM2/(M-N) RMST=SQRT(RMST/M) RMS=SQRT(RMS/M) WRITE(6,320) 320 FORMAT(/,5X,'DIRECT APPROACH - Newton s Method') WRITE(6,330) MM,DB 330 FORMAT(/,15X,'No of Iterations =',15,': Epsilon =',F7.6) WRITE(6,360) 360 F0RMAT(/,15X,'The fitting parameters are :-') WRITE(6,370) 370 F0RMAT(15X, ' ') WRITE(6,390)(I,B(I),I=1,N) 390 F0RMAT(15X,'B',I1,'=',F10.4) Appendix E. Fortran Programs WRITE(6,400) VART2,VAR2 400 FORMAT(/,15X,'Variance = ' , F 9 . 5 , 5 X , ' V (Urns) WRITE(6,410) RMST,RMS 410 F0RMAT(/,15X,'RMST = ' ,F9.5,5X, 'RMS(ums) WRITE(7,470) (B( I ) , I=1 ,N) 470 F0RMAT(F9.5) CALL COMPAR(UIF,UMS,M) STOP C C P r i n t warning message! C 500 WRITE(6,600) 600 F0RMAT('***************WARNING**************') WRITE(6,650) 650 F0RMAT(12X,'Convergence no t ach ived 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 Arguement: C C X real array of independent x-value C Y real array of dependent y-value C M interger number of pairs of (x,y) points C N interger number of terms in fitting equation C A fitting coefficients C VAR variance of fit C C DIMENSION X(10,112),Y(M),A(N),C0EFF(10,11) NP=N+1 C Form the arguement coefficient matrix C DO 80 1=1,N C0EFF(I,NP)=0 DO 50 K=1,M C0EFF(I,NP)=C0EFF(I,NP)+X(I,K)*Y(K) 50 CONTINUE DO 70 J=1,N C0EFF(I,J)=0 =',F9.5) =',F9.5) Appendix E. Fortran Programs DO 60 K=1,M C0EFF(I,J)=C0EFF(I,J)+X(I,K)*X(J,K) 60 CONTINUE IF(I.Eq.J) GO TO 70 COEFF(J,I)=COEFF(I,J) 70 CONTINUE 80 CONTINUE C C Solve for the unknown A coefficients C CALL GAUSS(COEFF,N,10,11,A,RNORM,IERROR) C C Calculate variance of multiple linear regression C SSUM=0 DO 140 K=1,M SUM=A(1) DO 130 J=2,N SUM=SUM+A(J)*X(J,K) 130 CONTINUE SSUM=SSUM+(Y(K)-SUM)**2 140 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 Form the arguement coefficient matrix C DO 80 1=1,N C0EFF(I,NP)=0 DO 50 K=1,M C0EFF(I,NP)=C0EFF(I,NP)+DF1(K)*B1*X(I,K)*PR0D(K) 50 CONTINUE DO 70 J=1,N C0EFF(I,J)=0 DO 60 K=1,M C0EFF(I,J)=C0EFF(I,J)+DF2(K)*(-PR0D(K)*X(I,K)*X(J,K)) 60 CONTINUE IF(I.EQ.J) GO TO 70 COEFF(J,I)=COEFF(I,J) 70 CONTINUE 80 CONTINUE C C0EFF(1,1)=0 DO 90 K=1,M C0EFF(1,1)=C0EFF(1,1)+PR0D(K)*PR0D(K) 90 CONTINUE C0EFF(1,NP)=0 DO 100 K=1,M C0EFF(1,NP)=C0EFF(1,NP)+DF1(K)*PR0D(K) 100 CONTINUE 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 E. Fortran Programs C Purpose: C Uses Gauss elimination with partial pivot selection to C solve simultaneous linear equations of form [A]*{X}={C}• C C Arguments: C A Augmented coefficient matrix containing all coefficients C and r.h.s. constants of equations to be solved. C N Number of equations to be solved. C NDR First (row) dimension of A in calling program. C NDC Second (column) dimension of A in calling program. C X Solution vector. C RNORM Measure of size of residual vector {C}-[A]*{X}. C IERROR Error flag. C =1 Successful Gauss elimination. C =2 Zero diagonal entry after pivot selection. C DIMENSION A(NDR,NDC),X(N),B(10,11),BIG(10) NM=N-1 NP=N+1 C C Set up working matrix B C DO 20 1=1,N DO 10 J=1,NP B(I,J)=A(I,J) 10 CONTINUE 20 CONTINUE C C Carry out elimination process N-l times C DO 80 K=1,NM C C Search for largest coefficient in column K, rows K through N C IPIV0T is the row index of the largest coefficient 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 25 CONTINUE 22 CONTINUE KP=K+1 C C C 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 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 30 CONTINUE C C Interchange rows K and IPIVOT if IPIVOT.NE.K C 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 40 CONTINUE 50 IF(B(K,K).EQ.O) GO TO 130 C C Eliminate B(I,K) from rows K+1 through N C 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) 60 CONTINUE 70 CONTINUE 80 CONTINUE C IF(B(N,N).EQ.O.) GO TO 130 C C Back substitute to find solution vector 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) 90 CONTINUE X(I)=(B(I,NP)-SUM)/B(I,I) 100 CONTINUE C C Calculate norm of residual vector, C-A*X Appendix E. Fortran Programs C Normal return with IERR0R=1 C RSQ=0. DO 120 1=1,N SUM=0. DO 110 J=1,N SUM=SUM+A(I,J)*X(J) 110 CONTINUE RSq=RSQ+(ABS(A(I,NP)-SUM))**2 120 CONTINUE RNORM=SQRT(RSQ) IERR0R=1 RETURN C C Abnormal return because of zero entry on diagonal C IEER0R=2 C 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 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 10 F0RMAT(1X,F5.2) STOP END FUNCTION QINT4P(X,Y,N,IA,IB) DIMENSION X(N),Y(N) C WHERE: C QINT4P = THE RESULTING INTEGRAL C X = AN ARRAY CONTAINING THE "N" ABSCISSAE C Y = AN ARRAY CONTAINING THE CORRESPONDING ORDINATES C N = THE NUMBER OF POINTS C IA = X(IA) IS THE FIRST POINT OF INTEGRATION C 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/ C 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 C C INITIALIZATION C 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 HANDLE THE FIRST SEGMENT WITH FORWARD DIFFERENCE FORMULA C DUM=ACSUM(AC, 1 H2*(Y(1)+H2*(D3*HALF-H2*(D2*SIXTH-(H2+TW0*H3)*R3*TWLVTH)))) GO TO 20 10 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)) 20 IF (I.LE.IA.OR.I.GT.IB) GO TO 30 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 the Ums Values Predicted by Four Equations s in rn/s U m s U,ns % dev Uma % dev Ums % dev Um s % dev expt. Eq.(5.75) Eq.(2.1) Eq.(2.6) Eq.(2.4) 1.399 1.393 - 0 . 4 0 4 1.069 -23 .573 1.425 1.838 1.056 -24 .508 1.164 1.307 12.263 0.999 -14 .210 1.345 15.507 1.061 -8 .884 1.036 1.204 16.192 0.912 -11 .973 1.249 20.566 1.039 0.310 0.950 1.092 14.937 0.820 -13 .687 1.144 20.442 0.984 3.628 0.886 0 .961 8.417 0.712 -19 .622 1.020 15.107 0.871 -1 .670 0.794 0 .801 0.941 0.583 -26 .527 0.867 9.176 0.683 -14 .038 1.503 1.458 - 2 . 9 6 1 1.241 -17 .441 1.510 0.467 1.153 -23 .261 1.361 1.440 5.778 1.224 -10 .040 1.492 9.654 1.156 -15 .079 1.190 1.329 11.658 1.122 -5 .735 1.389 16.701 1.150 -3 .332 1.085 1.204 10.965 1.007 -7 .189 1.271 17.154 1.110 2.287 0.989 1.059 7.100 0.875 -11 .550 1.133 14.571 1.004 1.530 0.848 0.884 4 .237 0.717 -15 .460 0.963 13.601 0.804 -5 .146 1.464 1.456 - 0 . 5 2 8 1.331 -9 .069 1.536 4 .889 1.179 -19 .495 1.374 1.402 2.034 1.277 -7 .052 1.484 8.013 1.179 -14 .189 1.261 1.270 0 .701 1.146 -9 .129 1.358 7 .697 1.160 - 8 . 0 3 5 1.129 1.117 - 1 . 0 4 8 0.996 -11 .822 1.211 7.234 1.077 -4 .640 0.971 0.932 -3 .979 0.816 -15 .952 1.029 6.018 0.886 - 8 . 7 6 3 1.453 1.476 1.597 1.430 -1 .606 1.582 8.850 1.172 -19 .343 1.408 1.452 3.112 1.405 -0 .244 1.558 10.666 1.173 -16 .687 1.280 1.315 2 .767 1.261 - 1 . 4 7 7 1.426 11.432 1.161 - 9 . 2 9 7 1.141 1.157 1.402 1.095 -4 .005 1.271 11.423 1.088 - 4 . 6 4 1 0.997 0.965 - 3 . 1 6 3 0.898 -9 .952 1.081 8.421 0.905 - 9 . 2 6 1 1.427 1.448 1.483 1.447 1.395 1.570 10.022 1.169 -18 .099 1.309 1.342 2.547 1.332 1.759 1.467 12.059 1.166 -10 .933 1.133 1.181 4 .223 1.157 2.144 1.308 15.418 1.114 -1 .707 0.967 0.985 1.860 0.948 - 1 . 9 6 5 1.111 14.943 0.946 -2 .146 1.344 1.429 6.341 1.466 9.091 1.563 16.277 1.172 -12 .764 1.282 1.370 6.854 1.400 9.173 1.504 17.355 1.172 -8 .566 1.180 1.205 2.139 1.216 3.086 1.341 13.684 1.137 -3 .640 0.998 1.005 0.686 0.996 - 0 . 2 4 3 1.140 14.200 0.985 -1 .276 1.117 1.212 8.527 0.948 -15 .129 1.227 9.847 0.882 -21 .012 1.041 1.181 13.414 0.921 -11 .512 1.198 15.101 0 .883 -15 .163 0.972 1.113 14.522 0.864 -11 .140 1.137 16.925 0.880 - 9 . 4 6 1 0.896 1.041 16.190 0.803 -10 .361 1.070 19.434 0.865 -3 .436 121 Appendix F. Error % for the Ums Values Predicted by Four Equations 122 7 -7 -7 -7 -8 -8 -8 -8 -8 -9 -9 -9 -9 -10 -10 -10 -11 -11 -11 -12 -12 -13 -13 -13 -13 -13 -13 -13 -13 -14 -14 -14 -14 -15 -15 -15 -15 -16 -16 -17 -17 -18 -18 -19 -19 -19 -19 -5 6 7 8 1 2 3 4 5 1 2 3 4 1 2 3 1 2 3 1 2 1 2 3 4 5 6 7 8 1 2 3 4 1 2 3 4 1 2 1 2 1 2 1 2 3 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 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0. 0 0 0, 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. .960 .870 .765 .638 .116 .049 .950 .836 .698 .073 .994 .874 .729 .986 .899 .750 .940 .913 .762 .886 .775 .893 .849 .801 .749 .691 .626 .550 ,459 .748 .671 .590 .492 .702 .691 607 .507 602 514 578 519 553 • 525 • 673 661 618 570 18.132 16.007 12.720 6.401 11.392 19.650 21.139 18.131 12.022 2.845 13.974 11.372 13.413 -13.571 2.606 13.915 5.155 9.482 3.330 1.110 13.444 1.292 4.329 3.489 3.720 1.736 0.601 -3.652 1.538 -2.247 3.865 6.147 4.252 0.838 5.425 3.290 1.100 -17.702 -8.733 -2.370 -0.451 -15.841 -11.394 -3.966 -2.837 -1.514 -0.225 0.735 -9.563 0.660 -12.032 0.573 -15.611 0.469 -21.801 0.967 -3.539 0.903 3.020 0.810 3.263 0.705 -0.484 0.577 -7.316 1.003 -3.859 0.922 5.776 0.801 2.082 0.656 2.082 0.975 -14.517 0.882 0.645 0.722 9.720 0.961 7.502 0.931 11.576 0.762 3.426 0.928 5.912 0.801 17.331 0.717 -18.736 0.679 -16.623 0.637 -17.716 0.592 -18.005 0.542 -20.185 0.486 -21.816 0.422 -26.092 0.346 -23.517 0.673 -11.973 0.598 -7.455 0.519 -6.593 0.425 -9.938 0.692 -0.624 0.680 3.753 0.590 0.411 0.484 -3.469 0.633 -13.491 0.532 -5.595 0.632 6.804 0.562 7.826 0.625 -4.869 0.590 -0.343 0.547 -21.911 0.536 -21.139 0.498 -20.531 0.456 -20.084 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 1 1 0 1 0. 0 0, 0. 0, 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. .995 .911 .812 .690 .169 .106 .012 .903 .767 .158 .081 .964 .819 .100 .012 .860 .068 .041 .885 .023 .907 .909 .869 .825 .776 .722 .661 .589 ,500 .809 .734 .655 .556 .796 ,784 699 ,594 719 624 707 642 690 658 700 688 648 603 22 21 19 14 16 26 29 27 23 11 24 22 27 -3 15 30 19 24 20 16 32 3 6, 6. 7, 6. 6, 3. 10. 5. 13. 17. 17. 14. 19. 18. 18. -1. 10. 19. 23. 4. 11. -0. 1. 3. 5. .434 .451 .571 .991 .677 .127 .018 .475 .163 .015 .009 .802 .426 .604 .560 .742 .489 .781 .019 .770 .870 .118 .764 .544 .525 .349 .240 .108 .714 .807 .700 .761 .861 .335 .742 ,910 ,619 749 ,784 396 ,158 973 139 154 216 318 549 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0, 0 0. 0, 0. 0. 0. 0, 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. .828 .759 .652 .495 .927 .926 .903 .830 .675 .912 .910 .869 .738 .883 .878 .800 .868 .867 .823 .860 .846 .654 .655 .650 .634 .602 .548 .466 .351 .638 .632 .595 .496 .593 .593 ,581 ,512 .552 .538 ,532 .527 .521 520 540 540 538 525 1 1 -4 -17 -7 5 15 17 8 -12 4 10 14 -22 0 21 -2 4, 11. -1. 23. -25 -19 -16, -12, -11. -11, -18. -22. -16. -2. 6. 5. -14. -9. -1. 2. -24. -4. -10. 1. -20. -12. -23. -20. -14. -8. .790 .264 .038 .483 .525 .575 .150 .199 .305 .525 .336 .709 .840 .602 .280 .515 .944 .013 .648 .799 .895 .824 .522 .013 .121 .327 .906 .394 .275 .602 .127 .990 .172 .761 .407 .131 .175 .545 .455 .145 ,139 .757 .159 .021 632 ,242 103 Appendix F. Error % for the Ums Values Predicted by Four Equations 123 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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1. 1 1, 1 0. 1. 1, 1. 1. 1. 0. 1. 1. 1. 1. 0. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. .516 .454 .378 .581 .546 .480 .400 .489 .408 .457 .410 .447 .412 .433 .415 .571 .509 .461 .405 .338 .388 .258 .139 .002 .836 .520 .499 .385 .255 .104 .921 .570 .461 .324 ,165 ,971 ,566 371 206 006 545 400 231 027 491 258 048 -1 -5 3 -10 -2 5 -1 -7 -6 -11 -5 -4 -5 -8 -6 -8 -5 -4 -3 2 -10, 2 3, -3, -10, -14. -7. 3. 4. 5. 0. -7. 6. 7. 5. -0. -11. 1. 0. -1. -5. 1. 3. 0. -7. 3. 3. .588 .112 .094 .572 .285 .321 .863 .596 .690 .867 .698 .457 .292 .168 .656 .635 .474 .638 .568 .664 .814 .321 .287 .260 .659 .245 .421 .116 .830 .276 .027 .434 .161 ,097 ,916 ,363 ,431 557 491 733 760 064 387 204 494 755 395 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0, 1, 1, 1, 1. 0. 0. 1. 1. 1. 1. 0. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. .409 .355 .291 .539 .504 .437 .358 .497 .407 .504 .447 .517 .472 .520 .497 .469 .413 .371 .322 .264 .138 .022 .917 .796 .652 .384 .364 .252 .124 .977 .800 .540 .425 .279 .112 ,910 .628 ,408 222 002 657 488 293 059 640 360 113 -21 -25 -20 -17 -9 -4 -12 -6 -6 -2 2 10 8 10 11 -24 -23 -23 -23 -19 -26, -16 -16, -23 -30, -21. -15. -6. -6. -6. -13. -9. 3. 3. 1. -6. -7. 4. 1. -2. 1. 7. 8. 3. 1. 12. 9. .951 .692 .717 .035 .874 .082 .252 .137 .854 .862 .839 .435 .578 .206 .586 .977 .191 .228 .353 .879 .840 .853 .867 .130 .302 .879 .736 .766 .095 ,861 .116 ,170 545 510 126 622 938 263 873 147 070 435 545 314 712 234 795 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1, 1 0, 1. 1, 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. .551 .491 .418 .648 .613 .547 .464 .583 .496 .574 .520 .576 .535 .570 .549 .603 .543 .497 .443 .376 .494 .367 .251 .115 .948 .649 .629 .517 .388 .238 .052 .729 .621 .484 .323 .124 .755 ,558 389 ,181 751 ,603 429 215 709 467 246 5 2 13 -0 9 19 13 10 13 10 19 23 23 20 23 -3 0 2 5 14 -4, 11, 13. 7, 1. -6. 0. 12. 15. 18. 14. 1. 17. 20. 20. 15. -0. 15. 15. 15. 6. 15. 19. 18. 5. 21. 22. .187 .778 .782 .248 .711 .852 .812 .245 .477 .539 .644 .038 .018 .822 .375 .584 .966 .916 .469 .419 .014 .229 .414 .626 .254 .948 .595 .949 .992 .022 .247 .935 .777 ,029 .280 ,292 ,739 ,398 716 296 810 728 975 490 991 010 885 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1. 1 1 0 0, 1. 1, 1, 1. 1. 0. 1. 1. 1. 1. 0. 1. 1. 1. 0. 1. 1. 1. 0. 1. 1. 0. .492 .432 .336 .499 .498 .478 .410 .450 .434 .411 .407 .393 .391 .382 .382 .477 .472 .452 .405 .321 .063 .057 .019 .920 .736 .151 .153 .149 .108 .003 .803 .175 .174 .141 .043 .843 .171 .154 ,073 ,884 168 ,161 ,097 920 172 137 985 -6 -9 -8 -23 -10 4 0 -14 -0 -20 -6 -16 -10 -18 -14 -23 -12 -6 -3 -2 -31 -13 -7. -11 -21, -35, -28 -14, -7. -4. -12. -30, -14. -7. -5. -13. -33. -14. -10. -13. -28. -16. -7. -10. -27. -6. -2. .078 .611 .502 .265 .917 .928 .399 .901 .714 .791 .415 .056 .093 .986 .124 .653 .199 .476 .654 .483 .706 .975 .641 .153 .343 .020 .792 .454 .402 .359 .777 .736 .691 ,655 ,175 .541 .746 ,533 602 ,648 ,765 ,206 918 243 267 184 834 Appendix F. Error % for the Ums Values Predicted by Four Equations 124 37 -37 -37 -37 • 37 -38 -38 -38 -38 -38 -39 -39 -39 -39 -40 -40 -40 -41 -41 -42 -42 -43 -43 -43 -43 -43 -43 -44 -44 -44 -44 -45 -45 -45 -46 -46 -47 -47 -48 -49 -49 -49 -49 -49 -50 -50 -50 - 1  2  3 - 4  5  1 - 2 - 3 - 4 - 5 - 1 - 2 - 3 - 4 - 1 - 2 - 3 - 1 - 2 - 1 - 2 - 1 - 2 - 3 - 4 - 5 - 6 - 1 - 2 - 3 - 4 - 1 - 2 - 3 - 1 - 2 • 1 • 2 • 1 • 1 • 2 - 3 - 4 • 5 • 1 • 2 • 3 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 1 0 0 0 1 1 0 0 0 1 1 0 0 1 0 0. 0, 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0, 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. .111 .003 .908 .799 .666 .173 .094 .991 .872 .728 .081 .036 .912 .761 .003 .937 .782 .960 .795 .917 .808 .803 .781 .720 .652 .574 .479 .755 .700 .616 .513 ,713 .634 .528 .623 ,536 ,576 ,541 ,565 ,617 594 538 473 395 594 570 501 -3 6 7 2 -3 -10 -5 -4 -1 -3 -11 -7 1 3 -12 -5 -2 -7, -0, 0, 0, -1, -1, 2, 0, -4. -3. 6. 9. 10. 4. -2. 2. 3. -7. 0. 2. 5. 8. -8. -6. -3. -5. -2. -12. -9. -3. .586 .542 .510 .719 .276 .780 .438 .811 .054 .554 .563 .577 .877 .284 .897 .889 .777 .354 .251 .648 .426 .695 .188 .163 .813 .695 .283 .386 .179 .121 .564 .522 .539 .827 .113 774 598 693 898 430 747 599 185 528 839 994 509 0 0 0 0 0 1 1 0 0 0 1 1 0 0 1 0 0 1 0 1, 0 0 0, 0, 0, 0. 0. 0. 0. 0. 0. 0, 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. .921 .823 .739 .642 .525 .090 .009 .905 .786 .646 .078 .030 .895 .735 .061 .985 .807 .049 .853 .029 .896 .682 .661 .605 .543 .472 .386 .726 ,668 .580 .475 .751 .660 .541 .701 ,594 ,672 ,628 683 531 510 457 397 325 590 563 489 -20 -12 -12 -17 -23 -17 -12 -13 -10 -14 -11 -8 0 -0 -7 -1 0, 1. 7, 12, 11. -16, -16 -14, -16, -21. -21. 2. 4. 3. -3. 2. 6. 6. 4. 11. 19. 22. 31. -21. -19. -18. -20. -19. -13. -11. -5. .077 .553 .581 .538 .786 .145 .782 .032 .732 .475 .752 .133 .045 .324 .914 .107 .318 .278 .067 .963 .323 .547 .335 .141 .084 .671 .933 .296 ,233 ,830 ,183 .735 ,835 ,251 ,504 722 698 698 642 146 988 072 433 659 520 052 875 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0, 1, 0, 1, 0, 0, 0. 0, 0, 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. .194 .089 .996 .888 .755 .287 .209 .106 .986 .839 .227 .181 .054 .896 .176 .106 .940 .145 .968 .111 .992 .870 .848 .789 .722 .643 .547 .860 .803 .716 .608 .849 .764 .650 .780 682 ,741 .702 ,740 ,681 659 602 537 457 696 670 597 3 15 17 14 9 -2 4 6 11 11 0 5 17 21 2 11 16, 10, 21, 21, 23, 6, 7, 11, 11. 6. 10. 21. 25. 28. 23. 16. 23. 27. 16. 28. 32. 37. 42. 1. 3. 7. 7. 12. 2. 5. 15. .608 .697 .928 .164 .527 .149 .459 .232 .895 .112 .392 .378 .711 .597 .061 .065 .926 .561 .406 .909 .251 .486 .350 .915 .571 .879 .506 .065 .238 .017 .871 .154 .694 .648 .289 ,178 ,112 ,016 ,632 ,109 377 972 616 726 087 892 056 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0, 0, 0, 0. 0, 0, 0, 0, 0, 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. .888 .882 .849 .766 .611 .926 .924 .900 .823 .668 .912 .912 .885 .768 .883 .882 .815 .868 .834 .861 .849 .658 .658 .653 .626 .563 .448 .638 .636 .607 .516 .594 .587 .526 .552 .540 .532 .531 .521 .541 .541 .532 .494 .406 ,499 ,499 ,483 -22 -6 0 -1 -11 -29 -20 -13 -6 -11 -25 -18 -1 4 -23 -11 1, -16 4, -5, 5, -19 -16 -7, -3, -6. -9. -10, -0. 8. 5. -18, -5. 3. -17, 1. -5. 3. 0. -19. -15. -4. -1. 0. -26. -21. -6. .911 .230 .495 .549 .292 .561 .101 .584 .577 .577 .367 .642 .117 .182 .341 .493 .423 .247 .583 .537 .404 .491 .698 .427 .269 .543 .539 .165 .782 .653 .083 .802 .066 .305 .704 ,445 .180 .729 ,313 .735 .067 .674 .080 ,244 ,865 ,231 897 Appendix F. Error % for the Ums Values 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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0. 1 1. 1 0 0 1. 1. 0. 0, 1, 1, 1. 0. 1. 1. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. .400 .555 .455 .559 .500 .566 .528 .576 .555 .958 .875 .796 .713 .620 .509 .078 .980 .878 .766 .627 .173 .116 .000 .871 .713 .262 .101 .958 .784 .293 .162 .009 .827 .343 .223 .063 .870 .841 .802 .752 .699 .640 .574 .500 .410 ,887 ,789 Predicted by Four Equations 125 -2 9 6 8 14 26 28 37, 36, •30 •29 •30 •31 •34. •37, •25, 25, 28, 31, 35. 20 19 20, •19. 23, 13, 13, 20, 20, 13. 15. 13. 18. -2. -3. -8. 17. 23. 24. 26. 28. 30, 32. 35. 37. 19. 20. .887 .953 .601 .708 .475 .576 .823 .406 .099 .208 .049 .050 .561 .932 .597 .470 .423 .538 .336 .687 .553 .273 .236 .196 .831 .497 .947 .187 .535 .589 .278 .703 .094 .741 ,499 ,077 .012 .615 .150 .159 .222 .757 .967 .433 .346 .409 ,163 0 0 0 0 0 0 0 0 0 1 1 1 1, 0, 0, 1. 1, 1. 1, 0. 1, 1, 1, 1, 0, 1, 1, 1, 0. 1. 1. 1. 0. 1. 1. 1. 1. 1. 1. 1. 0. 0, 0. 0. 0. 1. 0. .508 .638 .542 .623 .569 .619 .585 .618 .600 .305 .209 .121 .025 .914 .778 .349 .247 .140 .018 .865 .388 .332 .219 .088 .924 .432 .280 .142 .970 .437 .316 .173 .997 .459 .351 .204 .023 .116 .074 .019 .959 .892 .816 .728 .620 .093 .993 23. 26 26 21 30 38, 42, 47. 47. -4. -1, -1. -1. -4. -4. -6. -5. -7. -8. -11. -5, -3 -2. 0. -1. -1. 0. -4. -1. -3. -4. 0. -1. 5. 6. 4. -2. 1. 1. -0. -1, -3, -4, -5, -5. -0. 0. .183 .260 .970 .113 .209 .431 .697 .514 .118 .926 .964 .503 .643 .087 .669 .786 .117 .187 .680 .241 .948 .591 .814 .907 .262 .880 .068 .840 .628 .962 .070 .316 .257 .615 .604 .180 .367 .363 .529 .013 .545 .532 .685 .908 .374 .761 .479 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0, 1, 1, 1. 1, 0, 1 1 1, 1 0, 1, 1, 1, 0, 1, 1. 1. 0. 1. 1. 1. 0. 0. 0. 0, 0, 0, 0, 0. 0. 0. 0. .419 .450 .434 .411 .408 .393 .392 .382 .382 .057 .060 .029 .964 .847 .659 .156 .152 .112 .013 .813 .178 .179 .156 .070 .876 .173 .158 .082 .894 .167 .161 .096 .920 .172 .167 .110 .937 .881 .881 .876 .857 .816 .745 .637 .483 .927 .919 1 -10 1 -20 -6 -12 -4 -8 -6 -23 -14 -9 -7 -11 -19 -20 -12 -9, -9. -16 -20 -14 -7, -0, -6, -19 -9. -9. -9. -21, -15, -6. -8. -15. -7. -4. -10. -19. -16. -14, -12. -11. -13, -17. -26. -15. -7. .622 .857 .611 .008 .652 .112 .380 .727 .298 .004 .070 .615 .494 .146 .202 .123 .332 .412 .174 .564 .187 .721 .843 .709 .369 .635 .498 .873 .292 .995 .412 .232 .910 .145 .913 .006 .546 .982 .744 .081 .022 .750 .012 .694 .289 .840 ,014 Appendix F. Error % for the Ums Values Predicted by Four Equations 126 68 • 68 • 68 • 69 • 69 • 69 • 69 • 70 • 70 • 70 • 71 • 71 • 71 • 72 • 72 • 72 -73 -73 • 73 -73 -73 -73 -73 -73 -74 • 74 -74 -74 -74 -75 -75 -75 -75 -76 -76 -76 -77 -77 -77 -78 -78 -79 -79 -79 -79 -79 -79 -- 3 - 4 - 5 - 1 - 2 - 3 - 4 - 1 - 2 - 3 - 1 - 2 - 3 - 1 - 2 - 3 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 1 - 2 - 3 - 4 - 5 - 1 - 2 - 3 - 4 - 1 - 2 - 3 - 1 - 2 - 3 - 1 - 2 • 1 • 2 - 3 • 4 • 5 - 6 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 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0. 0 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. .902 .794 .663 .085 .945 .832 .693 .007 .854 .712 .005 .868 .723 .999 .883 .736 .866 .807 .760 .710 .655 .594 .522 .436 .760 .703 .637 .560 .468 .718 .656 .577 .481 .666 .585 .488 .625 ,591 492 593 498 627 586 541 490 431 359 5 6 4 -15 -11 -11 -18 -9 -13 -20 0 5 -3 4 1 -2 -5 -7 -7 -7 -6 -6, -7 -8. 5. 4. 5. 5. 7. 18. 17. 12. 0. 10. 3. 4. 3. 1. 3. 4. 6. -4. -4. -7. -3. -1. 0. .522 .181 .078 .716 .809 .437 .367 .816 .049 .441 .786 .837 .671 .077 .313 .251 .798 .695 .219 .443 .959 .646 .195 .041 .801 .360 .286 .948 .003 .063 .139 .731 ,663 .275 ,617 ,523 636 349 459 089 703 324 744 189 968 875 961 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0, 0, 0, 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. .707 .615 .504 .939 .806 .702 .574 .923 .770 .631 .957 .814 .666 .980 .856 .700 .640 .593 .555 .515 .472 .424 .369 .302 .632 .581 .522 .453 .372 .655 .594 .516 .422 ,653 ,567 ,464 637 ,599 490 624 515 467 434 398 357 310 254 -17 -17 -20 -27 -24 -25 -32 -17 -21 -29 -3 -0 -11 2 -1 -7. -30 -32 -32 -32 -32 -33, -34 -36, -11. -13, -13. -14. -14. 7. 6. 0. -11. 8. 0. -0. 5. 2. 2. 9. 10. -28. -29. -31. -29. -29. -28. .282 .764 .813 .036 .705 .270 .406 .369 .569 .532 .999 .758 .333 .117 .792 .030 .343 .163 .229 .816 .967 .338 .545 .313 .933 .737 .781 .290 .969 ,730 ,013 ,818 .615 190 ,356 576 657 762 988 551 361 678 393 651 975 348 600 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0, 0, 0 0, 0. 0, 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. .908 .810 .689 .099 .971 .867 .736 .054 .909 .772 .066 .935 .794 .071 .960 .815 .831 .780 .739 .696 .647 .593 .528 .449 .772 .720 .659 .588 .500 .763 .704 .628 .534 ,740 ,659 ,560 713 ,678 576 691 591 617 581 541 495 441 375 6 8 8 -14 -9 -7 -13 -5 -7 -13 6 13 5 11 10 8 -9 -10 -9, -9 -8 -6. -6 -5. 7. 6. 8. 11. 14. 25. 25. 22. 11. 22. 16. 19. 18. 16. 20. 21. 26. -5. -5. -7. -2. 0. 5. .209 .288 .140 .571 .319 .721 .324 .658 .453 .706 .929 .999 .740 .595 .038 .196 .556 .732 .715 .300 .066 .819 .138 .238 .459 .853 .924 .082 .321 .527 ,736 .649 ,615 ,448 ,618 ,902 172 ,249 952 145 483 799 552 212 995 446 305 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0, 0, 0 0 0, 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. .882 .795 .634 .913 .897 .833 .684 .883 .858 .746 .868 .850 .748 .860 .849 .758 .654 .654 .645 .626 .591 .535 .453 .340 .637 .635 .616 .561 .453 .593 .590 .559 .471 .552 .544 .484 ,532 ,531 ,495 ,521 ,503 ,539 539 529 499 442 346 3 6 -0 -29 -16 -11 -19 -20 -12 -16 -12 3 -0 -10 -2 0 -28 -25 -21 -18 -16 -15 -19 -28 -11, -5. 1. 6. 3. -2. 5. 9. -1. -8. -3. 3. -11. -8. 3. -8. 7. -17. -12. -9. -2. 0. -2. .144 .230 .537 .082 .288 .325 .427 .969 .627 .665 .938 .602 .417 .392 .648 .610 .793 .192 .198 .341 .027 .833 .526 .332 .328 .801 .796 .129 .662 .495 .309 .277 ,398 ,615 .791 .534 ,803 ,906 994 680 782 ,718 439 275 068 599 884 Appendix F. Error % for the Ums Values Predicted by Four Equations 127 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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0. 0 0. 0. 0. 0, 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. .590 .573 .518 .456 .380 .549 .465 .387 .504 .467 .390 .477 .391 .463 .395 .553 .524 .483 .437 .385 .321 .514 .459 .404 .337 .479 .464 .409 ,341 ,438 .409 ,341 412 341 400 343 3 5 3 6 7 -7 3 -2 4 0 -4 6 -1 5. -0, -8, -6. -6. -5. -1. 4. 6. 13. 13. 18. -3. 0. -2. 0. 1. 4. 0. 8. 14. 17. 19. .859 .825 .477 .283 .145 .487 .921 .464 .377 .481 .254 .149 .913 .378 .355 .780 .155 .896 .923 .093 .177 .128 .349 .441 ,931 ,001 ,089 ,463 749 180 553 528 823 958 159 630 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0, 0 0 0 0. 0, 0. 0, 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. .506 .490 .439 .382 .313 .521 .434 .355 .519 .477 .391 .513 .413 .517 .434 .418 .394 .361 .324 .281 .230 .450 .398 .346 .283 .467 .452 .394 .322 .466 .432 ,354 ,460 374 464 393 -10 -9 -12 -11 -11 -12 -2 -10 7 2 -3 14 3 17 9 -31 -29 -30 -30 -27 -25, -6. -1. -2. 0. -5. -2. -6. -4. 7. 10. 4. 21. 25. 36. 36. .936 .491 .315 .045 .901 .129 .883 .503 .379 .532 .982 .161 .410 .673 .541 .057 .359 .416 .366 .707 .235 .983 .673 .766 .056 .366 .518 .064 ,768 ,642 ,468 ,334 271 857 033 764 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0, 0, 0, 0 0. 0, 0. 0, 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. .617 .601 .550 .490 .417 .608 .524 .445 .588 .550 .467 .573 .480 .568 .493 .550 .524 .488 .446 .398 .338 .548 .496 .442 .376 .544 .529 .472 .401 .527 .495 .421 ,513 433 ,509 ,444 8 11 9 14 17 2 17 12 21 18 14 27 20 29 24 -9 -6. -6 -4, 2 9. 13. 22. 24. 32. 10. 14. 12. 18. 21. 26. 24. 35. 45. 49. 54. .708 .118 .793 .298 .430 .505 .178 .093 .803 .180 .801 .637 .405 .434 .466 .211 .064 .019 .050 .231 .724 .315 .475 .244 .750 .183 .105 .738 .689 .721 .700 .188 269 820 ,142 ,723 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0, 0, 0, 0, 0. 0. 0, 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. .498 .498 .491 .457 .377 .450 .437 .379 .411 .409 .377 .393 .375 .382 .372 .477 .477 .469 .445 .396 .311 .428 .423 .397 .331 .380 .380 .370 .322 .344 .343 .317 ,328 ,315 ,318 .311 -12 -7 -2 6 6 -24 -2 -4 -14 -11 -7 -12 -5 -12 -5, -21 -14, -9. -4. 1. 1. -11, 4. 11. 16. -23. -18. -11, -4. -20. -12. -6. -13. 5. -6. 8. .311 .923 .043 .520 .244 .126 .284 .418 .913 .948 .373 .515 .925 .908 .988 .362 .561 .544 .205 .757 .063 .617 .553 .646 .865 .066 .073 .676 ,603 .479 .184 .379 .490 ,938 ,611 ,384 Append ix G Error % for the Spout D i a m e t e r T dp D; Hm Ums °C (mm) (mm) (m) ( m / s ) ds-exp McNab's (cm) (cm) %dev Wu's (cm) %dev 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 2 2 2 1 1 1 1 1 1 1 1. 1 1 1. 025 025 025 630 630 630 630 200 200 200 200 010 010 010 2.025 .025 .025 .630 .200 .200 .010 .010 .915 .915 .630 .630 .630 1.200 1.200 1.010 1.010 1.010 2.025 2 .025 2.025 1.630 2 2 1 1 1 1 1. 0 0 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 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 1.399 0.950 0.794 1.117 0.972 0.813 0.679 0.882 .774 .679 .571 .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 0. 0. 0. 0 1.157 170 1.200 26.64 0.475 0.710 3.370 4 .089 21.349 3.100 3.336 7.600 3.010 3.037 0 .885 3.580 3 .664 2.340 3.320 3.409 2 .688 3.020 3.102 2 .701 2.680 2 .823 5.325 3.100 3.270 5.472 2.930 3 .044 3.906 2.690 2.836 5.435 2.620 2.592 - 1 . 0 7 1 2.970 2 .900 - 2 . 3 5 1 2.820 2 .604 - 7 . 6 4 7 2.500 2.376 - 4 . 9 7 1 4 .330 4 .276 - 1 . 2 3 6 3.720 3.796 2 .047 3.750 3 .471 - 7 . 4 5 1 3.970 3 .688 - 7 . 1 0 7 2.690 3 .123 16.078 2.890 2.892 0 .081 2.920 2.832 - 3 . 0 3 0 2.580 2 .571 - 0 . 3 4 7 2.250 2 .703 20 .113 1.940 2.349 21.106 2.830 2 .815 -0 .540 2.650 2 .485 - 6 . 2 0 8 2.510 2 .353 - 6 . 2 5 4 2.710 2 .454 - 9 . 4 3 5 2.410 2 .088 - 1 3 . 3 7 3 2.380 2.262 -4 .960 2.220 2 .097 -5 .540 1.930 1.893 - 1 . 9 0 3 3.960 3 .748 - 5 . 3 4 8 3.790 3 .249 - 1 4 . 2 7 1 3.570 2 .858 -19 .942 3.200 3 .017 -5 .730 2.890 2 .368 -18 .075 3.726 10.556 3.137 1.191 2 .897 -3 .754 3.380 -5 .590 3 .178 -4 .262 2.935 - 2 . 8 0 1 2.710 1.123 3 .053 -1 .520 2.878 - 1 . 7 6 1 2 .714 0.895 2 .514 - 4 . 0 5 3 2 .757 -7 .156 2 .517 -10 .727 2 .327 -6 .902 3 .893 -10 .101 3 .511 -5 .630 3 .254 -13 .222 3.416 -13 .944 2.946 9.501 2.760 -4 .512 2 .707 -7 .294 2 .491 -3 .438 2 .589 15.087 2 .301 18.634 3.142 11.016 2 .821 6.464 2 .695 7.386 2 .791 2.987 2 .427 0.685 2.599 9.214 2 .434 9.618 2.226 15.354 4 .031 1.788 3 .567 -5 .887 3 .198 -10 .427 3 .343 4 .455 2 .703 - 6 . 4 8 3 128 Appendix G. Error % for the Spout Diameter 129 170 170 170 170 300 300 300 300 300 300 300 300 300 300 300 300 420 420 420 420 420 420 420 420 420 420 420 500 500 500 500 500 500 500 500 580 580 580 580 580 580 580 580 580 1 1 1 0 1 1 1 1 1 2 2 2 1 1 1 0 2 2 1 1. 1 2. 2. 2. 1. 1. 0. 1. 1. 1. 2. 2. 1. 1. 0. 2. 2. 1. 1. 1. 2. 2. 1. 0. .200 .010 .010 .915 .630 .630 .200 .200 .010 .025 .025 .025 .630 .200 .010 .915 .025 .025 .630 .200 .010 .025 .025 .025 .630 .200 .915 .630 .200 .010 .025 .025 .200 .010 .915 .025 ,025 630 .200 ,010 025 025 200 915 26 26 26 12 19 19 19 19 19 26 26 26 26 26 26 12 19 19 19 19 19 26, 26 26. 26, 26. 12. 19. 19. 19. 26. 26. 26. 26. 12. 19. 19. 19. 19. 19. 26. 26. 26. 12. .64 .64 .64 .70 .05 .05 .05 .05 .05 .64 .64 .64 .64 .64 .64 .70 .05 .05 .05 .05 .05 .64 .64 .64 .64 .64 .70 .05 .05 .05 .64 ,64 .64 .64 .70 ,05 .05 ,05 ,05 ,05 64 64 64 70 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0, 0, 0 0, 0, 0. 0, 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. .300 .440 .300 .515 .475 .300 .415 .300 .300 .590 .500 .300 .400 .390 .300 .430 .520 .300 .370 .285 .255 .540 .400 .300 .350 .280 .350 .320 .255 .240 .500 .300 .235 ,220 .305 ,440 ,300 270 225 ,220 440 300 225 280 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 2 3 2 2 2 2 2 2 3 3 3 3 2 2 2 3, 3 2 3, 2 3, 3, 3. 3, 2. 2. 2. 2. 2. 4. 3. 2. 2. 2. 3. 3. 2. 2. 2. 3. 3. 2. 2. .640 .510 .100 .190 .860 .560 .660 .360 .190 .790 .560 .700 .530 .840 .940 .160 .370 .070 .660 .090 .450 .940 .710 ,500 .500 .790 .060 .660 .810 .880 .030 .640 .880 .840 .010 .240 .110 ,700 ,720 ,900 790 680 960 050 2 2 2 1 2 2 2 1 1 3 2 2 2 2 1 1 2, 2 2. 1. 1, 2, 2, 2. 2. 1. 1. 2. 1. 1. 2. 2. 1. 1. 1. 2. 2. 1. 1. 1. 2. 2. 1. 1. .094 .316 .020 .964 .522 .182 .063 .892 .798 .224 .897 .576 .610 .110 .758 .750 .709 .387 .396 .918 .621 .986 .604 .451 .407 .840 .487 .009 .640 .460 .723 .312 .597 .427 .320 .344 .189 ,894 ,641 .396 ,563 ,218 462 192 -20 -7 -34 -10 -11 -14 -22 -19 -17 -14 -18, -30, -26 -25 -40, -18, -19, -22 -9. -37, -33, -24. -29, -29. -31. -34. -27. -24. -41. -49. -32. -36. -44. -49. -34. -27. -29. -29. -39. -51. -32. -39. -50. -41. .663 .735 .850 .342 .833 .769 .429 .815 .890 .921 .619 .374 .055 .693 .212 .994 .600 .261 ,939 .920 .845 .221 .815 .974 .231 ,052 .799 .488 .654 .318 .429 .491 .553 .748 .305 ,645 .605 ,846 ,659 .871 ,381 ,723 602 ,873 2 2 2 2 3 2 2 2 2 3 3 3 3 2 2, 2 3 3 3, 2 2 3, 3 3, 3, 2. 2 2 2. 2, 3. 3. 2, 2. 1, 3. 3. 2. 2. 2. 3. 3, 2. 1. .433 .654 .356 .290 .165 .794 .655 .465 .355 .917 .573 .235 .265 .710 .308 .290 .614 .247 .250 .678 .307 .936 .497 .320 .264 .579 .135 .896 .422 .186 .774 .279 .366 .143 .997 .421 .228 .838 .504 .170 .701 .266 .261 .885 -7, 5 -24 4, 10. 9, -0 4, 7. 3. 0, -12, -7, -4, -21 6 7. 5, 22. -13 -5, -0. -5. -5. -6. -7. 3. 8. -13, -24, -6. -9. -17 -24, -0. 5. 3. 5. -7. -25 -2. -11, -23. -8. .855 .737 .013 .549 .677 .140 .204 .432 .524 .345 .351 .579 .505 .566 .501 .033 .250 .750 .191 .343 .836 .097 .741 .139 .748 .549 .626 .871 .825 .085 .346 .911 .856 .532 .631 .572 .807 ,128 .940 .182 .350 .248 .620 .033 

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