MICRGENVIRONMENTAL MODIFICATION BY SMALL WATER DROPLET EVAPORATION Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY FRED VERNON NURNBERGER 1972 III/IIIIII/l III III I is. LI BR 4 "’ Y E Michig-lil ‘ 3 3' University 1; This is to certify that the thesis entitled MICROENVIRONMENTAL MODIFICATION BY SMALL WAIER.DROPLET EVAPORATION presented by FRED VERNON NURNBERGER has been accepted towards fulfillment of the requirements for th - degree in Agricnlnuzal Engineering Date ”WW“ Z/77X 0-7639 @367) I Gem/7st Q217+ ABSTRACT MICROENVIRONMENTAL MODIFICATION BY SMALL WATER DROPLET EVAPORATION BY Fred Vernon Nurnberger The relief of heat and moisture stress on actively growing plants is a major concern to agriculturists and horticulturists. To date, sprinkler irrigation has been the primary mode of stress condition relief. The current investigation proposes a different method whereby small water droplets are sprayed into the air and allowed to evaporate before reaching the lower surface. The objectives of this investigation were to: (1) develop a mathematical model for the droplet evapora- tion modification process; (2) experimentally verify the model; and (3) use the model to predict modifications for various atmospheric conditions and spray rates. The model was developed for modification over bare soil condi- tions to facilitate experimental verification over known lower boundary conditions. An evaporation coefficient was developed from the literature to provide liquid water evaporation proportional to the saturated water vapor concentration deficit. The Fred Vernon Nurnberger exponential wind profile law, Swinbank (1964), and similar- ity profiles of temperature and water vapor concentration were used. The numerical solution technique utilized was the miniature control volume integral equation method proposed by Spalding and Patankar (1968). Experimental verification was performed over a bared strip of land at the Michigan State University Experimental Muck Farm. Water droplets were sprayed into the air from a 300 m long elevated line at a height of l m. Measurements of the profiles of wind speed, and dry and wet bulb temperatures were made upstream and downstream from the spray line. Other measurements included net radiation, wind direction, soil heat flux, and soil temper- ature. The Swinbank profile for wind speed was found to be appropriate but the similarity initial profiles for temperature and water vapor concentration exhibited some error. The agreement between the measured and model results was very good for the ratio of turbulent diffusivi- ties suggested by Leichtman and Ponomareva (1969). The maximum predicted cooling for the various atmospheric conditions and spray rates investigated was -l4.5°C. The method proposed does warrent further investi- gation with the influence of a plant canopy included. Fred Vernon Nurnberger References Leichtman, D. L. and Ponomareva, S. M., (1969). On the ratio of the Turbulent Transfer Coefficients for Heat and Momentum in the Surface Layer of the Atmosphere, Izves., Atmos. and Ocean. Phys., Vol. 5, No. 12, 1245-1250. Spalding, D. B. and Patankar, S. V., (1968). Heat and Ma§§ Transfer in Boundary Layers, 138 pp., C. R. C. Press, Cleveland. Swinbank, W. C., (1964). The Exponential Wind Profile, Quart. Jour. Roy. Meteor. Soc., Vol. 90, No. 384, 119-135. Approved / E~ W a3 rofessor fi/AW Department Chairman MICROENVIRONMENTAL MODIFICATION BY SMALL WATER DROPLET EVAPORATION BY Fred Vernon Nurnberger A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering 1972 To Hazle .1 .1 ACKNOWLEDGMENTS The author wishes to thank Dr. James B. Harrington, Jr. under whose guidance this project was conceived and developed. A special thanks to Dr. George E. Merva (Agricultural Engineering) for serving as the guidance committee chairman and for his many hours of invaluable Russian translation. Thanks to Dr. A. M. Dhanak (Mechanical Engineering), Dr. R. C. Hamelink (Mathematics), and Prof. E. H. Kidder (Agricultural Engineering) for serving on the guidance committee. A final thank you to all my fellow graduate students, family and friends who "volunteered" to help with the many tasks of the field experiment. iii TABLE OF CONTENTS DEDICATION . . . . . . . . . ACKNOWLEDGMENTS. . . . .. . . . LIST OF LIST OF LIST OF LIST OF Chapter 1. 2. TABI‘ES C O C O O O O O FIGURES. . . . . . . . APPENDICES. . . . . . . SYMBOLS. . . . . . . . INTRODUCTION . . . . . . LITERATURE REVIEW . . . . 2.1 Scope of Literature Review 2.2 Selection of Technique . 2.3 Relevant Parameters . . 2.4 Atmospheric Diffusion Models 2.5 Evaporation Models. . . THEORETICAL MODEL DEVELOPMENT 3.1 Diffusion Equation. . . 3.2 Initial Conditions. . . 3.3 Source Terms. . . . . 3.4 Boundary Conditions . 3.4.1 Lower Boundary Conditions 3.4.2 Upper Boundary Conditions SOLUTION TECHNIQUE . . . . 4 O l MethOd. I O O O I 4.2 Development of the Finite Difference Equations . . . . . 4.3 Parameter Evaluation . . 4.4 Solution Procedure. . . iv Page ii iii vi viii xi 1 xii .5 FJH WOU'luboh- 22 24 27 31 31 32 33 33 36 55 Chapter Page 5. EXPERIMENTAL PROCEDURE. . . . . . . . 57 5.1 Site Selection . . . . . . . . . 57 5.2 Water Supply System . . . . . . . 58 5.3 Instrumentation. . . . . . . . 59 5.3.1 Profile Measurements . . . . . 59 5.3.2 Other Measurements. . . . . . 66 6. EXPERIMENTAL RESULTS . . . . . . . . 68 6.1 Test Conditions. . . . . . . . . 68 6. 2 Wind Speed and Direction. . . . . 68 6. 3 Net Radiation, Soil Heat Flux, and Sensible Heat Flux . . . . . . . 72 6.4 Temperature and Moisture Measurements . 72 6.4.1 Initial Conditions. . . . . . 72 6.4.2 Modified Conditions . . . . . 76 7. MODEL PREDICTION. . . . . . . . . . 100 8. DISCUSSION. . . . . . . . . . . . 111 9. CONCLUSIONS . . . . . . . . . . . 116 APPENDICES . . . . . . . . . . . . . . 117 LIST OF REFERENCES 0 O O O O O O O O C O O 173 Table 2.5.1. 3.3.1 4.3.1 6.2.1 6.4.2.1 6.4.2.2 7.1 A1 A2 LIST OF TABLES Page Constants for the evaporation kinetics of water drops, (cgs units) . . . . . . . 19 A comparison of values calculated from Equation (3.3.3) and the volume formula for a sphere for various droplet radii, a. . . 29 A summary of the general Spalding-Patankar symbols and the specific symbols of the equations to be solved. . . . . . . . 44 A summary of the variability in the moist air parameters for the temperature and moisture range pertinent to the model being developed and for P=1000 mb . . . . . . 47 The 30 minute average wind direction and mast positions relative to the spray line . 69 Sum of the squared errors using Swinbanks similarity profiles as initial conditions . 87 Sum of the squared errors using linearized initial conditions . . . . . . . . . 98 A summary of the values of L, T-, ui, and the spray rate used in the prediction mOdel O O I O O O O O O O O O O 100 A comparison of saturation times and cloud lifetimes for the Maxwell and Fuchs- Okuyama-Zung relationships with cloud radius, R=10 m, degree of dulution, b/a=20, and T=25°C. . . . . . . . . . . . 122 Cloud lifetimes for the Maxwell relation- ship for various cloud radii and with I a=100 u, b/a=20 and T=25°C . . . . . . 122 vi Table Page A3 A comparison of saturation radii, satura- tion times, and cloud lifetimes for the Maxwell and Fuchs—Okuyama-Zung relation- ships for various degrees of dilution, b/a, with 3:10p, R=l m, and T=25°C . . . . . 123 A4 A comparison of cloud lifetimes for the continuum and cellular models for various degrees of dilution with a=10u. R0: 1.0 m, and T=25°C . . . . . . . . . 126 A5 A comparison of saturation radii, satura- tion times, and cloud lifetimes for various degrees of dilution for Zung's generalized model with a=10 u, R=l m, and T=25°C . . . 127 A6 A comparison of cloud lifetimes for the cellular model for fixed and turbulent clouds for various degrees of dilution with ao=10 u, Ro=l m, T=25°C,vt=l cm/sec, and Maxwellian evaporation assumed . . . . . 129 A7 A comparison of evaporation rates for the continuum model for fixed and turbulent clouds for various degrees of dilution with ao=10 u, Ro=l m, T=25°C, vt=l cm/sec, and Maxwellian evaporation assumed . . . . . 131 A8 A comparison of the evaporation rates for various average cloud lifetimes using Equation (A22) with a0=lO u. Ro=1 m, T=25°C vt=l cm/sec, Cm=0, m2 cs=2.33x"5 g/cm3, D=0.249 cmZ/sec, and b/a=18.0 . . . . . 132 B1 30 min mean wind speed data, m/sec. . . . 136 B2 30 min mean initial conditions (mast l) . . 136 B3 30 min mean temperature and moisture values. 137 B4 A summary of the assigned constants used in the theoretical model . . . . . . . 138 BS Values of the constants used in the prediction model. . . . . . . . . . 139 B6 Prediction model results . . . . . . . 140 vii LIST OF FIGURES Figure Page 3.2.1 Initial liquid water profile. . . . . . 26 4.1.1 Grid nomenclature and control volume to be used in the numerical solution scheme. . . 35 5.2.1 Schematic diagram of the experimental site . 60 5.3.1.1 Psychrometer unit . . . . . . . . . 62 5.3.1.2 Bottom view of the psychrometer's radiation shielding and sensing unit . . . . . . 62 5.3.1.3 Cross sectional schematic diagram of the psychrometer sensing unit and radiation shielding . . . . . . . . . . . . 63 5.3.1.4 Instrumentation arrangement on masts l and 2 with the elevated spray line in the background. . . . . . . . . . . . 64 5.3.1.5 A close up view of the anemometer and psychrometer arrangement . . . . . . . 64 5.3.1.6 Data acquisition system . . . . . . . 65 5.3.2.1 Schematic diagram of the instrumentation and spray line . . . . . . . . . . 67 6.2.1 Wind speed profiles. . . . . . . . . 70 6.4.1.1 Initial temperature profiles. . . . . . 74 6.4.1.2 Initial moisture concentration profiles . . 75 6.4.2.1 Operational spray line for small water droplets . . . . . . . . . . . . 77 6.4.2.2 Droplet dispersion and disappearance between masts 2 and 3. . . . . . . . . . . 77 viii Figure 6.4.2.3 6.4.2.4 6.4.2.5 6.4.2.6 6.4.2.7 6.4.2.8 6.4.2.9 6.4.2.10 6.4.2.11 6.4.2.12 6.4.2.13 6.4.2.14 6.4.2.15 6.4.2.16 6.4.2.17 6.4.2.18 6.4.2.19 Temperature profiles at mast 2 with Swinbank initial conditions . . . . . Temperature profiles at mast 3 with Swinbank initial conditions . . . . . Temperature profiles at mast 4 with Swinbank initial conditions . . . . . Temperature profiles at mast 5 with Swinbank initial conditions . . . . . Moisture concentration profiles at mast 2 with Swinbank initial conditions . . . DJ Moisture concentration profiles at mast with Swinbank initial conditions . . . Moisture concentration profiles at mast with Swinbank initial conditions . . 0 :5 Moisture concentration profiles at mast with Swinbank initial conditions . . 0 U" Initial linearized temperature profile . Temperature profiles at mast 2 with the linearized initial conditions . . . . Temperature profiles at mast 3 with the linearized initial conditions . . . . Temperature profiles at mast 4 with the linearized initial conditions . . . . Temperature profiles at mast 5 with the linearized initial conditions . . . . Initial linearized moisture concentration prOfile. O O O I O O O O O 0 Moisture concentration profiles at mast 2 with the linearized initial conditions . Moisture concentration profiles at mast 3 W1th the linearized initial conditions . Moisture concentration profiles at mast 4 w1th the linearized initial conditions . ix Page 79 80 81 82 83 84 85 86 88 89 90 91 92 93 94 95 96 Figure Page 6.4.2.20 Moisture concentration profiles at mast 5 with the linearized initial conditions . . 97 7.1 Predicted temperature profiles with Ti= BOOOOC O O O O O O O O O O O O O 103 7.2 Predicted moisture concentration profiles for Ti=30.0°C and R.H.i=25% . . . . . . 104 7.3 Predicted temperature profiles with Ti=40 00°C 0 O O O O O O O O O O O 105 7.4 Predicted moisture concentration profiles for Ti=40.0°C and R.H.i=25% . . . . . . 106 7.5 Maximum cooling with ui=0.5 m/sec and three spray rates . . . . . . . . . . . 107 7.6 Maximum cooling with ui=l.0 m/sec and three spray rates . . . . . . . . . . . 108 7.7 Maximum cooling with ui=2.0 m/sec and three spray rates . . . . . . . . . . . 109 7.8 Maximum cooling with ui=4.0 m/sec and three spray rates . . . . . . . . . . . 110 LIST OF APPENDICES Appendix Page A. Detailed Review of Cloud Droplet Evaporation Models . . . . . . . . . . 118 B. Field Data and Prediction Model Results . . . 135 C. FORTRAN IV Program for Theoretical Model. . . 142 xi LIST OF SYMBOLS Chapter 2 and Appendix A Symbol Meaning Droplet radius Initial droplet radius 0 aS Droplet radius at cellular saturation a(a) Evaporation-condensation coefficient a' 4n D (3/41rpw)l/3 aH Ratio of turbulent diffusivity of sensible heat to the turbulent diffusivity of momentum do aH at the ground surface 9-x dH at large C b Cell radius B(t) Average cell radius at time t B(a) Gas phase resistance for the droplet 8 d'[c (r,t) - COJm2 c Water vapor concentration not near the drop's surface cO Water vapor concentration at the drop's surface cp Isobaric specific heat C“, Water vapor concentration far from the drop cS Saturated vapor concentration at the surface of the drops C(a) Surface resistance of the droplet xii Symbol c(t) DI FOZ H z N 7“ WU- w :12 3 X I." t" 7% 7: Meaning Average liquid molecule concentration Diffusivity of water vapor through air New diffusion coefficient ED/(l-E) 2 3 ~3—T KV Differential operator, Partial differential Free angle ratio Fuchs' concentration jump distance Laplacian Operator Void fraction Integrated error function Exponential function z/L Acceleration of gravity Sensible heat flux Fuchs'-Okuyama-Zung evaporation rate per drop Maxwellian evaporation rate per drop Moisture flux Von Karman's constant Boltzman constant Turbulent diffusivity for sensible heat Turbulent diffusivity for momentum Turbulent diffusivity for moisture vapor Monin-Obukhov scale length Modified Monin-Obukhov scale length xiii Symbol 1n 0' Q cloud n d we Meaning Natural logarithm Droplet mass Molecular mass of the evaporating liquid Mass of a vapor molecule Total cloud mass Micron Droplet density in the cloud Kinematic visosity of air 3.14159 Steady state continuum model evaporation rate in a turbulent cloud Nonstationary rate of evaporation in the cloud Steady state rate of evaporation in the fixed cloud Position in a spherical cloud Initial cloud radius Cloud radius at time t Ambient air density Liquid water density Dimensionless coefficient in the KEYPS equation Modified 0 Liquid surface tension Time Total cloud lifetime Cloud lifetime after saturation xiv 0-9- FOz X* Meaning Lifetime of outer droplets after cellular saturation Lifetime of outer droplets before cellular saturation Time required for inner cloud cells to become saturated Cloud lifetime for the unsaturated case Ambient temperature Absolute temperature Functional relationship of t used in Zung's continuum model Temperature at the height 20 Similarity temperature profile coefficient Shear stress Shear stress at the ground surface Wind speed in the direction of mean flow Friction velocity Volume of a single liquid molecule Average turbulent relative velocity Droplet evaporation size coefficient ¢ (a0) Fuchs'-Okuyama-Zung evaporation rate per unit droplet surface area Maxwellian evaporation rate per unit droplet surface area Ambient water vapor concentration Water vapor concentration at the height 20 Similarity water vapor concentration profile coefficient xv Symbol Chapter 3 Symbol K(x) K(y) K(z) Meaning Dimensionless wind speed gradient used in the KEYPS equation Height Surface roughness height 3 v oS/kBTA Dimensionless height ratio, z/L Meaning Contaminant (liquid water) concentration Initial liquid water concentration profile Average initial liquid water concentration Evaporation source term coefficient Total differential Turbulent diffusivity for liquid water drops Turbulent diffusivity in the x-direction Turbulent diffusivity in the y-direction Turbulent diffusivity in the Z-direction Latent heat of vaporization Average droplet concentration Latent heat flux Soil heat flux Sensible atmospheric heat flux Net all wavelength radiation Spray rate per meter of line xvi Symbol Meaning 3 = Mean wind speed in the x-direction 5 = Mean wind speed in the y-direction W = Mean wind speed in the z-direction ¢ = Contaminant source term $1 = Liquid water concentration source term @2 = Sensible heat source term ¢3 = Water vapor concentration source term x = Direction along the mean wind y = Direction transverse to the mean wind 2 = Vertical direction 22 = Bottom vertex of initial liquid water concentration profile 23 = Top vertex of initial liquid water concentrat1on prof1le zmax = Maximum height of solution zMS = Location of the maximum initial liquid water concentration Other symbols as previously defined Chapter 4 Symbols Meaning A,A' = Spalding-Patankar solution coefficients A1,A2,A3 = Intermediate source term Parameters B,B' = Spalding-Patankar solution coefficients C = Spalding—Patankar solution coefficient Cf = Compressibility factor xvii Symbol C(w) Ac P Meaning General turbulent diffusivity Isobaric specific heat residual. Generalized source term Downstream solution node Midpoint between vertically adjacent down- stream nodes Saturated vapor pressure Ratio of the molecular weight of water to the molecular weight of dry air Intermediate solution parameters Simpsons integration Ambient atmospheric pressure = Intermediate solution parameters Dry air gas constant Water vapor gas constant Relative humidity Virtual temperature Upstream solution node Midpoint between vertically adjacent upstream nodes Mixing ratio Saturated mixing ratio Solution increment in the x-direction General diffusing property Saturated water vapor concentration Dimensionless vertical axis = -C Solution increment in the w-direction xviii Subscripts + Chapter 8 Symbols Adjacent node above the solution point Adjacent node below the solution point Others as previously defined Meaning Dynamic viscosity of air Terminal velocity of falling water droplets in still air The solution point closest to the ground surface xix 1 . INTRODUCTION The injurious effects of the commonly known problems of heat and moisture stress on plants have been of major concern to agriculturists and horticulturists for many years. The results of these stresses are to reduce the crop yield in quantity and/or quality and may even prove fatal to the plants under extreme conditions. The two stresses are not independent. Moisture stress can occur in the presence of a low soil moisture content. The root system cannot obtain the water required for the plant's normal development. Moisture stress can also occur during periods of high evapotranspiration rates. Under such conditions water is lost from the above ground portions of the plant, primarily the leaves, faster than the below ground root system can supply the water. Evaporation is the principal means of cooling the plant during the day. When evaporation is restricted, leaf temperatures rise and the plant is subjected to heat stress. It is obvious that the most severe conditions are low soil moisture and high potential evapotranspiration rates. The widely accepted practices of supplemental and total irrigation have been used many years to reduce the stress caused by low soil moisture. Methods for the reduction of high evapotranspiration rate induced moisture and heat stress are not as well developed. Some of the factors affecting the evapotranspira- tion rates are: (1) wind speed resulting in transport of the water vapor from the plant canopy; (2) low ambient moisture conditions which results in an increase in the water vapor diffusion rates from the leaf stomates; and (3) high plant temperatures, due to high insolation rates, which increase the evaporative cooling demands. The wind speed is largely uncontrollable, except where wind breaks are used e.g. Geiger (1965), Brown and Rosenberg (1971). Reduced wind speed can have a reverse affect, though, if an ample supply of soil moisture is available. The reduced wind flow lowers the evaporation rate thereby decreasing the evaporative cooling and increasing the heat stress. Ambient temperature and moisture conditions, however, can be modified. Carolus, Erickson, Kidder, and Wheaten (1965), Carolus and Van Den Brink (1965), and Carolus (1965,1969), have investigated and demonstrated the affects of low rate sprinkler irrigation to provide a source of water for evaporative cooling exterior to the plant. A dis- advantage of the sprinkling approach is that the plant environment remains nearly saturated thus possibly increasing disease susceptability. A different approach would be to modify and cool the air before it reaches the plants. Thus the plant canopy would not be continuously wet. The method proposed herein for modifying and cooling the air is the complete evaporation of small water droplets. The cooling will be provided by the latent heat energy required for evaporation of the droplets. Before any practical engineering applications of this technique can be designed more knowledge of the relevant parameters is.needed so that a mathematical model of the problem can be developed. The current investigation will be limited to the development and testing of a suitable model for describing the affects of the evaporation of small water droplets on the downwind microclimatic temperature, humidity, and wind profiles. The model will be restricted to a two- dimensional problem for an elevated line source of water droplets above a bare soil surface. 2. LITERATURE REVIEW 2.1 Sc0pe of Literature Review The review of the pertinent literature included in this chapter will be limited to the establishment of sufficient background information for the subsequent development of the model. Literature citations directly pertinent to the model itself will be deferred until the apprOpriate section. A more detailed review of evaporation models is included in Appendix A. 2.2 Selection of Technique The region of the atmosphere to be considered in this model is within the surface boundary layer of the earth. The environmental study of this region involves a study of the microclimate and is commonly referred to as micro- meteorology. Many investigators have studied the various aspects of the microclimate. The investigations have been either in the form of a statistical analysis of the turbulent characteristics or in the form of profile gradients. Excellent reviews of the statistical and profile gradient relationships are given by Sutton (1953), Pasquill (1962), Lumley and Panofsky (1964), Harrington (1965), Waggoner (1965), Monin and Yaglom (1971), and others. The profile gradient techniques were the ones chosen for this investiga- tion. This choice implicitly neglects the turbulent kinetic energy exchange discussed by Lumley and Panofsky (1964), Zilitinkevich, Leichtmann and Monin (1967), and others. 2.3 Relevant Parameters The atmospheric parameters of importance in this investigation are: the shear stress; the wind speed, temperature and humidity profiles; the net radiation; the surface heat and moisture fluxes; and the stability. Many investigators, e.g. Calder (1939), Sutton (1953), Monin and Obukhov (1954) have shown that for normal atmospheric conditions within the region up to an average height of 50 meters, the turbulent shear stress, I, is constant and equal to that at the ground surface, To. Additional sub- stantiation was provided in the work reported by Lettau and Davidson (1957) for the O'Neal, Nebraska Project Prairie Grass. The various expressions for the wind speed profile have been thoroughly reviewed by Harrington (1965). Harrington (1965, p. 123) concluded that the Swinbank (1964) exponential-law profile is "the most apprOpriate expression for the wind profile near the ground". Swinbank's model for the wind profile is: u* exp(%)-l u(z) = E"“n{"‘75“‘} . (2.3.1) exp(i9)-l The corresponding momentum diffusivity is: KM(z) = k u*L[1-exP(-§)J, (2.3.2) where u* = the friction velocity, (m/sec), k = Von Karman's constant 20.4, 20 = roughness height, (m) L = Monin—Obukhov (1954) scale height, (m), 3 ———_l k H c T g /p p with g acceleration of gravity, (m/secz), H = sensible surface heat flux, (cal/mz—sec), p = ambient air density, (g/m3), c = specific heat of the air, (cal/g-OK) P and T absolute temperature, (0K). The friction velocity, u*, as defined by Sutton (1953) is: Since the shear stress, T=TO as noted above, u*=/|r;7pl, and is customarily assumed constant for given flow and stability conditions. The roughness height, 20, is the dynamic roughness parameter at which the velocity is zero. The Monin-Obukhov scale height is in fact a stability para— meter arrived at through dimensional analysis. The sign of L is chosen such that for a positive sensible heat flux, i.e. H>0 and unstable conditions, L<0 and for a negative sensible heat flux, i.e. H<0 and stable conditions, L>0. The theory of similarity develOped by Monin and Obukhov (1954) and used by Lumley & Panofsky (1964), Swinbank (1964), Harrington (1965), Monin and Yaglom (1971) and others is assumed to hold. Under the similarity theory, the initial temperature and moisture vapor profiles have a form mathematically similar to the wind velocity profile. The initial temperature profile thus becomes: = _ exp(z[L)—l T To T*[£n(exp(zo/L)-l]’ (2.3.3) where To = temperature at zo, (0C), _ _ l H o and T* - ku* EE;-, ( C). The initial moisture concentration profile is: _ _ exp(z/L)-l X "Xo x*[£n(eXp(zo/L)—lj’ (203-4) where x0 = moisture concentration at 20, (g/m3), . 3 x. = -J/ku*. (g/m ) j = moisture flux, (g/mZ-sec), and the others as previously defined. The relationships of the turbulent diffusivities for momentum, heat and moisture have not been well established. The most common assumption is where C = z/L: KM = momentum diffusivity, Ki = diffusivity for parameter 1. Sutton (1953, p. 319) deduced that: KM = KXKM 1n stable cond1t1ons, where KH = diffusivity of heat KX = diffusivity of any contaminant Swinbank (1968) suggested 0.24 dH = 2.7ICI . Stewart and Lemon (1969) proposed Leichtmann & Ponomareva (1969) indicated that F’ 008 -003KM in unstable conditions, i.e. L<0, KHzKX0, which is almost completely Opposite to Sutton's earlier suggestion. Panofsky, Blackadar, and McVehil (1960), Webb (1960), Panofsky (1963), Lumley and Panofsky (1964) and Monin and Yaglom (1971) discuss an alternate approach by defining a new scale factor L'=aHL. 10 The various derivations by Kazanski and Monin (1956), Ellison (1957), Xamamoto (1959), Panofsky (1961) and Sellers (1962), satisfies an equation of the form: W4 — 013—11. W3 = 1 (2.3.5) 2 _ kz Bu Where II) (fr) — E;- '5';- Equation (2.3.5) is known as the.KEYPS equation. The value of c is not definitely determined. Monin and Yaglom (1971) report values ranging in size from 4 to 14 that were proposed by Panofsky, Blackadar and McVehil (1960), and Charnock (1967) respectively. Likewise the value of o'=a_mo has not been definitely determined. Monin and Yaglom (1971) report a range of values between 10 and 20. A comparison of the values for o and 0', however, indicates that a_w>l, and is consistant with the previously stated conclusions. 2.4 Atmospheric Diffusion Models The diffusion of substances in the atmosphere near the ground have been of interest to micrometeorologists for many years. Sutton (1953) presents the solution to various diffusion models for both instantaneous and continuous point, line, and plane sources. He first used the Fickian diffusion equation which required the diffusivities to be constant and assumed the wind speed was also a constant value. For 11 the diffusion of heat, Sutton (1953, p. 145) states: "As yet, there is no generally accepted formulation of the problems of heat transfer by atmospheric turbulence." Sutton reviewed the theories presented by G. I. Taylor, Brunt, L. F. Richardson, Calder, and Priestley & Swinbank, and concluded in agreement with Deacon that the wind speed and diffusivity profiles were simple power functions of height under diabatic conditions. He resolved the diffusion equation using the power law profiles. The results have proved to be valid only in the limited near neutral stability situation but were an important first step in understanding the atmospheric diffusion process. For an elevated source,Sutton (1953, p. 139) introduced the method of images to conveniently handle the assumed inpervious boundary condition. This method utilizes a mirror image technique whereby a virtual source, corres- ponding to the actual source, is located below the zero plane. Thus, no net flux occurs across the boundary. Philip (1959) included advection into the diffusion model but retained the power law profiles, and aH=l. Rider, Philip and Bradley (1964) reviewed the work done by Timofeev (1954), deVries (1959), and Philip (1959) to develop a model for a freely evaporating soil surface. They retained the power law profile and aH=l. 12 Yordanov (1966) developed a two layer analytical model for continuous diffusion from an elevated point source, in which he used the Lagrangian correlation coefficients. The wind speed was assumed constant with height while the turbulent diffusivity was that of Monin and Obukhov (1954) i.e. K(z)=ku*Lf(§). An implicit assumption, of dH=l was made. The two layers employed were those indicated by the results of Priestley (1955), Deacon (1959) and Gurvich (1965) to be the thermal sublayer and the dynamic sublayer. The transition region from the dynamic to the thermal sublayer was indicated to be for C in the range from -0.03 to -0.05. (Seealso Waggoner (1959) and Monin and Yaglom (1971)). Yordanov (1968) extended his model to an infinite elevated line source. His results, though agreeing with other researcher's data, were much to complicated mathematically to be of practical use in the current investigation. Jaffe (1967) used the results of Monin and Obukhov (1954), Priestly (1959) and Lumley and Panofsky (1964) to develOp a three layer diffusion model. The diffusivities for the layers were: (1) "log + linear" of Monin & Obukhov for ICI<0.03 (2) KM¢(z)4/3 of Priestly for 0.031 M 13 The common assumption of dH=1 was retained, "because of the confusion surrounding the KH/KM ratio-—and for convenience." [Jaffe (1967) p. 302]. The numerical solution of the two- dimensional steady state diffusion equation provided results consistent with the Project Prairie Grass data reported by Lettau and Davidson (1957). The adoption of Swinbank's exponential law profile in Section 2.3 above and the fact that the thermal sublayer exists where|§|>0.05 leads to the adoption of a single layer model as the current diffusion model. 2.5 Evaporation Models Models for the atmospheric evaporation of water droplets have not been widely developed. This is in contrast to the evaporation studies conducted in enclosed chambers for combustion and food drying processes. Milburn (1957) develOped a model for non-turbulent, homogeneous cloud evaporation restricted by the following assumptions: (1) "The individual cloud droplets are sufficiently far apart for the average vapor pressure and temperature in the immediately surrounding medium into which they evaporate to be described by simple scalar 'field' functions of space and time." (2) "The individual drOplets will be able to reach a steady-state condition with respect to their immediate surroundings in a time short 14 compared with the duration of processes affecting the cloud as a whole.“ (3) "The individual water drOplets remain fixed in space so that the number of such drops in a unit volume is constant." The last assumption is to presume non-turbulent idealized conditions. Assumption (2) was verified by demonstrating that the evaporation of about 2% of the droplet's mass would reduce the cell temperature to the wet-bulb value. The resulting equation for drOplet mass transfer was: 5% m(r,t) = 47TD(3m/4'npw)1/3[c(r,t)-co(r,t)], (2.5.1) where; m(r,t) = drOplet mass at (r,t), (g), D = diffusivity of water vapor through air, (cmz/sec). m = initial mass of individual droplets, (g), pw = density of water, (g/cm3), c(r,t) = water vapor concentration at (r,t), not near drop surface, (g/cm ), and co(r,t) = same as c above but at the drop surface, (g/cm3). The inclusion of the bulk vapor and heat diffusion equations, and the droplet heat transfer equation yielded the relation- ship: 15 am 3 4 —n —— 3 3t _ (8nD)( —— 8t 2 - DV )m 5?, ":w)1/3 (2.5.2) [with the incorporation of a correction noted by Zung (1967, b, p. 3579)], where n = droplet density in the cloud, (No./cm3), and V2 = the Laplacian Operator. Equation (2.5.2) was not solved until a linearizing assumption was applied to the dimensionless form and its applicability was reduced to the very early stages of evaporation i.e. small t and evaporation of less than 20% of the mass of the drOplets. Milburn (1958) extended the previous model to turbulent clouds, thereby relaxing assumption (3) above, but retaining the first two. In addition, he assumed that the evaporation of individual drOplets was describable by "quasistatic" or equilibrium-flow equations. The quasi- static assumption implied that the transient terms had been damped out, and the drOplets were able to attain equilibrium temperatures before saturation was reached. The inidividual drOplet evaporation was assumed to be governed by Equation (2.5.1) which neglects the effect of turbulence since the laminar boundary layer around the droplet was estimated to be more than 2 orders of magnitude larger than the droplet radius. "It may be argued . . . that diffusion in a tempera- ture gradient is more accurately represented by an equation in partial vapor pressures than by one in vapor concentra- tion. At practical temperatures there is but little difference, 16 however." (Milburn, 1958, p. 116). The diffusivities of heat, momentum, and vapor were assumed to be the same. The resulting unsolved equation was: J’nUnnnt)=(3a/2){§%-ml/3n(m,r,t) }-{£dm-m-n(m,r,t) + C (rct)-‘I’(rrt)}t (205-3) where; _ _§ _ 2 .8— 3t Kv , n(m,r,t) = number density of drOplets in a unit mass interval at (r,t), K = turbulent diffusivity, (cmZ/sec), 0‘ = (17312) (1%3;>1/3. (1mm). 00 c(r,t) + [mm m n(m,r,t), (g/cm3). and w(r,t) Okuyama and Zung (1967) noted that Maxwell's derivation for the stationary evaporation of a spherical drop in a motionless media has the form: C’C _ S ” QM "" W, (2.504) where; OM = Maxwell rate of evaporation per drOp per unit surface, (g/sec-cmZ-drOp), GS = saturated vapor concentration at the drOp surface, (g/cm3), cco = vapor concentration at an infinite distance from the drOp, (g/cm3), 17 D = molecular diffusion coefficient of the vapor in air, (cmz/sec), and a drOplet radius, (cm). "Equation (2.5.4) is valid only for drops larger than 10—2 cm radius (200p diameter), becomes less accurate for smaller drOps, and includes only prOperties of the vapor phase." Okuyamam and Zung demonstrated that the formulations of Fuchs, and of Monchick and Reiss were essentially the same, but did include both liquid and vapor prOperties. The derivation of the evaporation-condensation coefficient for very small drOplets was made in the form u(a) = 6:¢(a); where; O free-angle ratio, and O size coefficient. The coefficient was found to be: c(a) = 6 exp(-3vo/akBTA) (2.5.5) where; v = volume of a single liquid molecule, (cm3), GS = surface tension of the liquid, (ergs/cmz), k3 = the Boltzman constant, (ergs/OK), and TA = absolute temperature, (OK). The inclusion of Equation (2.5.5) into Fuchs' equation leads to the more general rate equation: FOZ where; (DFOZ A and v m v Equation (2.5.6) and liquid-vapor Okuyama and Zung 18 (co-cm) (a/D)[a/(a+A)]+(l/v5¢) (2.5.6) the generalized Fuchs-Okuyama-Zung evaporation rate par drOp per unit surface, (g/sec-cm -dr0p), Fuchs' concentration jump distance, and is the distance from the drop surface at which the steady concentration is main- tained, (cm), 1(8kBTA)1/2 4 nmv ’ mass of a vapor molecule, (g). (cm/sec), includes prOperties of the liquid, vapor, interface. The values reported by (1967, p. 1582) for the above parameters are given in Table 2.5.1. The denominator of Equation (2.5.6) can be redefined as B(a) + C(a), where; B(a) and C(a) It can be seen (a/D) [a/ Ia+A)] gas phase resistance, surface resistance. l/v6¢ that for large a, 4 reduces to the F02 Maxwell equation and is diffusion controlled. When a+0, the process is governed by ¢(a), the size coefficient, and A is completely eliminated. By examining B(a) and C(a), Okuyama & Zung discovered that B increases with increasing radius while C increases with decreasing radius for 19 TABLE 2.5.1.--Constants for the evaporation kinetics of water drOps, (cgs units). T (°K) parameter 373 273 C93 un1ts prOperty of the _23 _23 3 liquid phase, v 3.11 x 10 2.98 x 10 cm prOperties of the gas-liquid inter— 2 face, OS 58.8 75.6 ergs/cm a¢ = .5 1.5 x 10_7 3 x 10—7 cm 6 0.042 0.039 prOperties of the _4 —6 3 gas phase, Cs 5.98 x 10 4.87 x 10 g/cm v 1.65 x 104 1.42 x 104 cm/sec D 3.60 x 10.1 1.98 x 10-1 cmZ/sec A 1.09 x 10‘5 7.0 x 10'6 cm a<10-4om(i.e.].u). This is due to ¢(a) decreasing very rapidly with decreasing radius below about a=10-5cm. Thus ¢F02(a) has a maximum value at some value of a. For . o ' ' C 4 as shown to be for evaporat1on 1n air at 0 , FOZ,MAX(a) w a lo-Scm(i.e. 0.1u). The range of maximum values was for radii between 1 and 0.01m. 20 The above derivations are valid only for individual drOplets. Zung (l967,a) extended the results to drOplet assemblages in air by use of a modified cellular model. The cellular model has been used primarily for enclosed systems for combustion and spray drying. As such the model is not directly applicable to cloud evaporation. For the modified cellular model, Zung limited his considerations to monodisperse systems evaporating into a motionless medium. The cells are assumed spherical with radius b containing one drOplet with radius a per cell. The inner cell's drOplets will cease to evaporate after cellular saturation has occurred while the outer cell layers will continue to evaporate. The Maxwell expression, Equation (2.5.4) which was I I 2 0 for evaporat1on 1n g/sec-cm , can be rewr1tten as, IM = 4naDm2(cS-cm), - (2.5.7) where; IM = Maxwell evaporation, (g/sec-drOp), m = molecular mass of the evaporating liquid, (9) . and the other parameters are as previously noted. In a similar manner, the general exPression of Fuchs- Okuyama-Zung, (i.e. Equation 2.5.6) which is valid for allO-ch, is 4na D m2(cS-cm) I = , (2.5.8) FOZ (D/ava¢)+ a/(a+A) 21 where IFOZ = evaporation rate per drop, (g/sec-drop), -3vo ¢ = exP(-——-§-) . a kBTA v = (kBTA/211m2)1/2 A = D/2v a = evaporation coefficient =6, and the other parameters as noted previously. For a more detailed review of the modified cellular and continuum models develOped by Zung for both still and turbulent clouds, see Appendix A. For the problem in question in this thesis the models reviewed fall short of the ideal in several respects. (1) The diffusivity was assumed constant with position and only a function of time. The models for fixed clouds assumed the diffusivity was constant and equal to the mole- cular diffusivity of water vapor in air. (2) In the continuum model develOpment, the assumption was made in the derivation of the new diffusion coefficient that "The radius, a, of the drOplet does not vary much". This is not true when it is permitted to go from a=ao to a=0. It would be true only during the initial stages of evaporation. (3) The continuum model is valid only in the saturated and near saturated cases while the cellular model is valid in the unsaturated case. (4) None of the models have been experimentally verified and can be therefore used only as a guide to the evaporation process. 3. THEORETICAL MODEL DEVELOPMENT 3.1 Diffusion Equation The problem under investigation is an active contam— inant diffusion problem. The mathematical description of the general three dimensional, unsteady, incompressible, diffusion problem [Sutton (1953), Pasquill (1962), and Harrington (1965)] is: 3% = a—E2(z,x), (3.1.4) where; T = temperature, (0C), p = ambient air density, (g/m3), cp = isobaric specific heat of moist air, (cal/g-OC), and 92 = energy source term, (cal/m3-sec). For evaporation it will be a negative source, i.e. a sink. The coupling of Equations (3.1.2), (3.1.3), and (3.1.4) is through the source term. With the general problem defined above, the forms of the wind speed profile, diffusivities, initial conditions, source terms and boundary conditions must now be specified. 3.2 Initial Conditions The selection of the Swinbank similarity profiles in Section 2.3 specifies the wind speed profile by Equation 25 (2.3.1), the momentum diffusivity by Equation (2.3.2), the initial temperature profile by Equation (2.3.3) and the initial moisture vapor concentration by Equation (2.3.4). The initial solutions of the present problem will be made for aH=l. Other OH relationships will be tested for comparison. The profiles of wind and diffusivities will be assumed to remain unchanged by the introduction of the evaporating mist. The initial liquid water droplet concentration at the spray line will be assumed to have'a triangular profile. The location of the verticies will be adjusted to the observed spray pattern. The average uniform spray concen- tration is determined by the following formula: _ 5 co - z , (3.2.1) 3 u(z)dz I22 where; c = mean liquid water concentration, (g/m3), (D II mean rate of liquid sprayed into the air, (g/my-sec), fz3u(z)dz = total wind flow past the spray line 2 between the heights 22 & 23, (mzmx/sec), = lower and upper bounds respectively of the drOplet distribution, (m). 26 For a triangular profile, the same concept can be utilized, but restricted to each node in question and multiplied by an apprOpriate weighting function so that a continuity check will yield the same amount of water per unit area for the initial profile, i.e. I::Ci(z)dz = constant. Ci(z) is the initial liquid water concentration profile and is equal to zero below 22 and above z3. For the average uniform con- centration the continuity value would be co(23-zz). The initial triangular profile is shown in Figure 3.2.1. where z = z for maximum spray, and 2212 MS M512 3 ' FIGURE 3.2.l.--Initia1 liquid water profile. 2 The weightingfunction must be such that [23wt. f(z)dz=l. 2 . It can be shown that the following function satisfies the above conditions: 27 2(Az)(z-zz) for zz HOw onetmm m How OHDEHOM oEdHO> on» one Am.m.mv cowumoom Eoum OODmHsono mosam> mo comwummEOo «1|.H.m.m momma 30 a mean radius in the 10 to lOOu range. From the comparison of droplet volumes1 in Table 3.3.1, it can be easily seen that the evaporation of drOplets larger than lOu radius would affect the environmental cooling more than those less than lOu radius, by several orders of magnitude. Hence, the value of El/N to be chosen should be between 0.0022 and 0.03. The average value will be several orders of magnitude too large when the drOplets become very small, but their negligible contribution to the cooling process will not introduce any significant error. The sink term for energy removal will be the corresponding latent energy required for the evaporation source term. Thus, L ‘ _ v P where; Lv = latent heat of vaporization, (cal/g), and the other symbols are as previously defined. The increase in water vapor will be the negative of the evaporation source term so that the total mass of liquid plus water vapor remains constant. Thus, 4 = -4 (3.3.5) lSince p is very nearly = l g/cm3 at ambient temperatures, the numercial values of the drOplet mass and volume are the same. 31 3.4 Boundary Conditions 3.4.1 Lower Boundary Conditions The problem being considered is one for a bare vegetationless surface. As demonstrated by Sutton (1953), pp. 139-140), the method of images can be used to establish a lower boundary condition of zero flux across the surface. For the liquid water drOplets and the vapor concentration, the technique of images will be employed. It must be modified when a crOp canopy is introduced, since the zero flux assumption is not applicable in the presence of evapotranspiration. The lower boundary condition for heat flux will be determined by the energy balance at the surface. From Munn (1966), ON = QG + QH + QE, (3.4.1.1) where; QN = net all wavelength radiation, ++'energy gained by the surface, QG = heat transfer through the ground,-++ downward flow, OH = turbulent transfer of sensible heat to the atmosphere, +-+ upward flow, and QE = latent heat flow, +-+ upward flow. QE is identical to H in Section 2.3. 32 For zero flux of moisture across the surface, QE = 0. Therefore, after rearrangement Equation (3.4.1) becomes: QH = QN - QG, (3.4.1.2) 3.4.2 Upper Boundary Conditions The upper boundary conditions for all three profiles will be ones of constancy. Since above a given level the modification process will have no affect, the values for temperature, moisture concentration, and liquid water con- centration will remain equal to the initial conditions at that level. T (x'zmax) = T (o’zmax) C (x’zmax) = C (o’zmax) (3.4.2.1) x (X.zmax) = x (0.2max) 4. SOLUTION TECHNIQUE 4.1 Method The solution of Equations (3.1.2), (3.1.3) and (3.1.4) with the initial conditions, Equations (2.3.1), (2.3.3), (2.3.4) and (3.2.3), and boundary conditions, Equations (3.4.1.2) and (3.4.2.1), must be by numerical means. The method prOposed by Richtmyer and Morton (1967, pp. 185-201) for the solution of equations with variable coefficients was rejected because the trans- formation of variables led to inconsistent boundary condi- tions. The methods reviewed by Harrington (1965), and the basic Crank-Nicolson method prOposed therein would have required large amounts of computer space and time, since matrix inversions would have been necessary. The much faster and simpler form proposed by Spalding and Patankar (1968) has been adOpted herein. The solution technique is a marching-integration procedure whereby the values for the unknown variables will .be evaluated in a stepwise manner downstream for all ‘Vertical nodes at each step. Thus no matrix inversions are required. The general Spalding-Patankar method has the additional advantages that uniform grid spacing is not required, 33 34 variable coefficients may be included, variable forward step increments are permitted and all forms of boundary conditions are permitted, i.e. constant values, constant gradients or entrainments along free boundaries. The equations to be solved have the general form 3?;- = (TIE flew) 3%) + d, (4.1.1) where; O = dependent variable, x = distance along the mean horizontal direction of flow, w = vertical axis, u = vertical wind profile, c = turbulent diffusivity, and d = source term not containing terms of the form BO/Bw. The variables used in the development of the solution technique correspond to those of Spalding and Patankar. The symbols previously used in this paper will be reintroduced upon actual solution. The finite-difference equation to be derived from Equation (4.1.1) will not be by the use of Taylor-series expansion as is commonly done, e.g. Harrington (1965), Smith (1965), Richtmyer and Morton (1967), or any standard numerical analysis text. Instead, a miniature integral equation over a chosen control volume is used coupled with 35 an assumption regarding the nature of the variation of O between the grid points. "In other words, the finite- difference equation is obtained by expressing each term in the parent differential equation as an integrated average over a small control volume. The advantage of this procedure is that, unlike the conventional method, it ensures that the conservation equation will be satisfied over any part of the boundary layer" (Spalding & Patankar, 1968, p. 35). The grid and control volume used is shown in Figure 4.1.1, UU DD+ . ZW/é/fl/AT . U. W/Wfl/fl FIGURE 4.l.l.--Grid nomenclature and control volume to be used in the numerical solution scheme. where; U & D represent the upstream and downstream nodes respectively at a given w, U+,U_ ,D+,D_ represent corresponding adjacent points to U & D for adjacent m values, 36 and UU+, UU_, DD , DD_ represent the midpoints that define the control volume between the indicated U's & D's. The evaluation of the 30/3w terms can be at xv or x As noted by Spalding and Patankar, since it can be D' shown that the use of the downstream station values are more accurate for large values of the forward step, yield stable solutions and are more convenient, the downstream evaluation will be the procedure adOpted. To obtain equations linear in O, any coefficients will always be evaluated from the known upstream values of ¢. It is assumed that, in the w direction, ¢ varies linearly with w between grid points while in the x direction, the variation is considered to be stepwise. The values of ¢ for the interval xU+P2(¢D ¢U)+P3(¢D- ¢U_)I (4.2.2) where; : (wD+-wD) _ 3 D+ D- D U (U ’0.) and P33 D D- 4(wD+-wD-)(XD_XU) (Note: a typographical error in the expression for P1 reported by Spalding and Patankar has been corrected). The expression for the flux term is: 1 a 3 ~ XD “00+ 1 a a¢ m '37:“ (95%“ {fo (mm. m '57)“: (”5'6”“de (4.2.3) {(xD-XU)(wDD+-wDD-)}' Since u(w)#f(x) and a/aw are to be evaluated at xD, Equation (4.2.3) reduces immediately to: W '37:“: (”’34) m) :9: 53133: —(c (mg—gmwv (4.2.4) (wDD+-wDD-)' 38 The integration must be performed by parts, fudv = uv - fvdu. Let u = l/u(m) and dv = 534C(w)g ¢)dw, 1 Eu so then du = - ‘—— dw and v = c(w) ——. u2(w) 8m 3w Equation (4.2.4) becomes: wDD+ wDD+ 3¢ 18¢-13u '7—7'3w(6(w)fi M) ~{-7—74c(w)g %)I WD-‘fwn [C(w)— —][;§?;;’3 adew}/ (wDD+-wDD-) wDD+ 13¢ 1 3¢ = L————C (a —) ( ) [C(w) J uDD+ cDD+ DD+ uDD_ cDD- 5"DD- wDD 5i" 1 Bu [u2(w) 55de}/(wDD+_wDD—) (4.2.5) Since in this problem, functional forms of u(w) and c(w) are known, the midpoint calculations can be made. The evaluation of the partial derivatives must be made in con- junction with the assumed linearity of o with w between nodes. Thus (ii) = 32::E2.. (.i) = ¢ D-¢D' 3w DD+ wD+-wD DD- wD-wD_ w -w and ,(m ) = (21—92). (4.2.6) 39 The integration is performed using Simpson's rule: x +2Ax 0 ~Ax IS = f y(x)dx~-3—-(y0+4yl+y2). (4.2.7) x 0 This technique requires that Ax be constant. Thus the general Spalding-Patankar technique has become modified by requiring Aw to be uniform or at least a very slowly changing value between nodes. For this case x0 = wDD-' xo+2Ax = wDD+' and therefore Ax = %(wD+-wD_). Since the w increment is now required to be constant, let Now Ax = %—(2Aw) = —— and Equation (4.2.7) is: I =I HDD+(c< >-$)< S— w _2-1—_3w>dw~TY[(c(w)'§-t%)(T-)(g—_w—)I DD- (w ) (w ) DD- +4 (C (w)g%) (m)(%— 3&1)” +(c(— M2( ))<§—)Iw J. D... (4.2.8) If one uses the results of Equations (4.2.6) in Equation (4.2.8) and notes that (ii) = ( ¢D+-¢D' 3w *wD+-wD_ ). (4.2.9) 40 the integral becomes: ¢ -¢ ~Aw D D- 1 Eu Is”2(3){cDD-(wD-wD_)(u§ )(§E)DD- DD- ¢ -¢. (EL—Lulngg) +4[c _ J D ”0+ wD- u D D ¢ -¢ D+ D 1 Eu +CDD+(mD+-wD)(u2 )(35)DD+}' (4'2'10) DD+ After substituting the above expressions into Equation (4.2.5), rearranging and collecting like terms, the general expression for the flux term is found to be: 5%67'534°(w’%%°:(P4+P5)¢D++(P6‘P4)¢D"P5+P6)¢D-' (4.2.ll) where; 2 1 CDD+ Aw Bu p =( _ fl__ _ :{l+-———v-—-%-0 J} 4 wD+ wD_ wD+ wD uDD+ 2(3)uDD+ 3w DD+ ' P =( 1 >( 4“” HEM-33) . 5 wD+-wD_ 3uD(wD+-wD_) “0 3w D 2 1 CDD- Aw an and P =( _ ){ _ (. )[( )(-—) _-l]}. 6 wD+ wD_ (wD wD-YluDD- 2(3)uDD_ 3w DD Under the restriction of the uniform grid spacing, the expressions for the P's become: 41 1 3 (4.2.12) P2 = 3/4Ax, P4 = __l_7424 [1 + 6A —(a u)DD+] (Aw) u DD+ “ DD+ “ _ l c Bu P5 ‘ 3Aqu(E°D(§E)D' and P = -———§%-) [— (a —) -l]. 6 (Am) DD— 63 DD 3“ DD The source term must now be included in the finite difference equation. The value of d will be assumed uniform over the control volume and equal to that at D. Since d may not be linear with ¢, it can be approximated by dD 2dU + (3¢) U(¢D-¢U). (4.2.13) The expression to be evaluated becomes d 2 I, _(d) _ dw/(w -w _). (4.2.14) DDD_ x—xD DD+ DD If one assumes d is linear with w between grid points, Equation (4.2.14) can be broken into two parts and evaluated from w to w and from w to w Since d is assumed DD- D D DD+° linear in these intervals, it can be easily shown that the average value of d for the interval is d = i—(3dD + dDi)' where i is - or + for the respective interval. By perform- ing the indicated operations with equations (4.2.13) and 42 and (4.2.14) and employing the definitions of the P's, one obtains d z G1¢D+ + G2¢D + G3¢D- + G4, (4.2.15) where; Gl = P1(%%(U+(XD_XU)’ <32 = Pgfibu (xD-xU). G3 = P3(§$)U_ (xd- XU), and G4 = {P1Edu+ (a¢) U+¢ ++P2 dU( a4 HU¢U +p3th_-<§%)U_¢:-J} (xD-xU). After substituting Equations (4.2.2), (4.2.11) and(4.2.15) into Equation (4.1.1) and collecting like terms, one obtains the finite difference equations: ¢D = A¢D+ + B¢D_ + c (4.2.16) where; A = (P4 + P5 - P1 + Gl)/(P2 - P6 + P4 - GZ)’ B = -(P3 + P5 + P6 - G3)/(P2 - P6 + P4 - G2), and C - (qu)u+ + P2¢U + P3¢U_ + G4)/(P2 - P6 + P4 - G2). 43 It can be shown by using the Gaussian elimination method that the solution of Equation (4.2.16) can be obtained from: 4i = Ai¢i+l + Bi (4.2.17) where; Ai & Bi are the recursion formulae: A2 = A2 I = __ I - A1 Ai/(l BiAi-l)’ 1>2' ' = 32 B2 ¢l + C2. I = I __ I ' and Bi (BiBi-l + Ci)/(1 BiAi-l)’ 1>2. The general form of the solution is thus established by Equation (4.2.17) and the preceding parametric defini- tions. 4.3 Parameter Evaluation The conversion from the general Spalding-Patankar symbols to the symbols of the specific problem to be solved are summarized in Table 4.3.1. The temperature range of applicability of this model will be from 20°C to 50°C, since this range extends beyond the active plant growing temperature extremes for which cooling would be required and/or desirable. The ambient moisture conditions will range from very dry at the high temperatures to nearly saturated at the lower temperatures. 44 TABLE 4.3.l.—-A summary of the general Spalding-Patankar symbols and the specific symbols of the equations to be solved. Specific Problem Symbols General Spalding-Patankar Liquid Water Water Vapor Symbols Concentration Energy Concentration m C C C c(w) KC(C) KH(C) KX(C) ¢(w) C (C) T(C) X(C) d Pl/u(c) PZ/U(C) @3/u(c) The variation in several atmospheric parameters of interest within this temperature and moisture range are namely: (1) mixing ratio, (2) compressibility factor i.e. deviation from the ideal gas law, (3) moist air density, (4) moisture concentration, (5) isobaric specific heat, (6) moist air heat capacity, (7) latent heat of varporiza- tion, and.(8) evaporative cooling rate. The following is a summary of the definitions and calculation procedures for the above parameters. Mixing ratio, W = R.H. x WS ,(g/kg), (4.3.1) where: R.H. = relative humidity in decimal form, and W = saturated mixing ratio, (g/kg). The moist air density and compressibility factor, are defined by the gas law, [List (1966, p. 295)], 45 P p = -——————V (4.3.2) CfRdaTv where; p = ambient density of moist air, P = ambient atmospheric pressure, Cf = compressibility factor representing deviation from ideal gas, Rda = dry air gas constant, (2.8704x106erg/g-OK), Tv = virtual temperature, (OK), = 1+W/e T 1+W A e = ratio of the molecular weight of water vapor to the molecular weight of dry air, (0.62197), [List,(1966, p. 332)]. TA = absolute temperature, (0K), and W = actual mixing ratio, (g/g). Moisture concentration, x, is found by: W x = ———l+w p, (4.303) where W and p are as defined above. The isobaric specific heat for moist air is, [List (1966, p. 339)] 0 II 0.2399 + 0.4409 + Acp, (4.3.4) where; Ac isobaric specific heat residual, and W is as defined above. 46 The moist air heat capacity and evaporative cooling coefficient are given by (p op) and(Lv/p cp) respectively, where; LV = the latent heat of vaporization for water. The results of these calculations are given in Table 4.3.2 for an atmospheric pressure of 1000 mb. A close examination of various sections of Table 4.3.2 leads one to the following conclusions: SECTION EXAMINED CONCLUSION l. Compressibility Factor The deviation of moist air from an ideal gas is negligible. 2. Evaporative Cooling The evaporative cooling rate will Parameter be on the order of 2°C per gram of water evaporated per cubic meter. 3. Moisture Concentrations The maximum cooling under normal and Evaporative Cooling conditions will be less than Parameter 20°C. 4. Moist Air Density, Iso- All can be assumed constant and baric Specific Heat, evaluated at an intermediate Latent Heat of Vapor- value within the range of condi- ization, Heat Capacity, tions applicable to a specific and Evaporative Cooling problem. The maximum errors so Rate incurred will be about 3% for the extreme conditions of initially 50°C air cooled to 30°C. The implied assumption in the development of Equation (3.1.4) that pcp=constant is completely justified by conclusion (4) above. For the evaluation of the solution coefficients Gl, G2, GB, and G4, not only the values of d are required but 47 .mmm .a .4m magma .Ammaae Dmeqm .4om .mom .ma .mh magma .Amomav Dmflgm .mm .a .4 magma .Ammaav Dmaqfl maaa. maaa. maaa. maaa. ma.¢a mm.aa av.a vh.m am on maaa. maaa. naaa. naaa. am.hm an.om ma.ma ~a.m mm om maaa. maaa. naaa. naaa. Hm.av mm.>m oa.am mv.ma «ca ov maaa. maaa. ~H.aa ao.mm mo.vv. mo.~m «NH om ooa mm on mm «OOH m» cm mm Hmo 00 w ..m.m w ..m.m e :3}: s Houomm mafiaflnflmmoumaou oflumm msaxflz ouspmummfioa .QE oooaum How can pmmoao>mp mswmn Hopos 0:» Op ugosaumm magma muzumwofi was mnnumummfimu on“ How mnmumamumm Ham umHoE on» Ca mpwaflnmanm> on» no MHDEEsm dun.~.m.v mqm4a 48 .mam .a .Hn magma .Ammaae Dmaa m pmuomammc was muflawnflmmoumEou onam .Homnmmm .ma .mh magma .Ammaae Dmflqa mmH.H m.mm o.mm m.HN h.om mm.m na.a mm.H mm. om awa.a o.mm m.mm m.mm m.am na.v m>.m mm.~ nm.a om NHH.H o.av a.mv m.¢¢ m.mv mo.a ma.m mm.v vm.~ ow ano.H m.mm N.vm a~.m ¢N.¢ om ooH mm om mm ooH mb om mm 00 Ams\axv m ..m.m w ..m.m A s\axv.moa Aoov . emu n .e Loo. . Ba m auamsoo mousumummama vucoeouocH B owed who Hmsuufl> poumsnpm musumummfims Hmsuna> .vaGHuQOUII.N.m.¢ mandB 49 .amm .mam .mm .Hn magma Lemmas Dana b m.hH H.MH om.m H¢.v th.H omH.H mmH.H mmH.H 0N mm.om o.mm mm.ma mm.h oma.H mma.H ovH.H va.H om ¢.Hm N.am o.mm m.mH Hmo.H mmo.H mmo.H voa.H ow m.vv m.NN Hmo.H «mo.H om OOH mm om mm ooa mh om mm 00 w somom W somom E m .x m ~a 1m \ 1 $5 5 a. coaumnucmocoo musumaoz a muamsmo Had umfloz .pmscwusooll.m.m.¢ mqmda .amm .m .mm magma .Ammmao Dmaq 50 a mom. mom. vow. mom. mooo. mooo. wooo. oooo. om mmm. omN. mom. vow. aooo. aooo. oooo. mooo. om mom. mmm. mmm. mom. vaoo. Naoo. aooo. wooo. ow How. omm. maoo. aooo. om ooH mm on om ooH mo om mm 00 W somom w sum-Ham A00 I m\Hmov .mo mo< .mamspflmmm B Doom oamaommm UHHDQOmH “mom oamwommm oaumnomH .GTSGHDCOUII.N.M.¢ mqmfia 51 .mvm .a .NO magma .AOOOHO “was a Ho.~ mo.m mo.~ vo.m o.omm Ham oam amm ham om mo.m vo.m oo.m no.m v.omm mom vow Nam omm om mo.m vo.m ao.m HH.N n.oom «am How mam mum ov mo.~ va.m o.aom ohm mom om OOH mO Om mm Aa\HmoO OOH ma Om mm oo w sow-Ham > w somom A acoflumn m > Iflnomm> m :OEEBOO . o E O to Dam Go-ma>8O . o O O umumfimumm mafiaoou m>Humuomm>m ucmumq Had unwoz mo huflommmu umwm .GTDGHDEOUII.N.M.¢ mqmda 52 also the values of 3d/3¢. For the liquid water droplet equation this becomes Bdl a -cl(XS‘X) 15$.1..= 5E£ fiTE)’ ] , (4.3.5) Since c1 and u(z) are assumed constant during the modifica— tion process, Equation (4.3.5) becomes: -c13xs 3 1 _ ‘Cl 3 _ A, _ 31 __ r. S'E‘Xs -x) fir; 3T 5%(3C). (4.3.6) If one uses an energy balance, it can be easily seen that the change in Tchxeto the evaporation of liquid water is Lv/p cp and therefore, 3d -cl L 8x 1 v s 3 3‘5; ‘ u(C) (p cp)[3—T— " 5%3- ”'3'” If one uses the moist air relationships, List (1966, p. 347), assumes the water vapor behaves as an ideal gas and uses the Clausius-Clapeyron equation, Hess (1959, p. 48), it can be shown that 3x3 - xSPLv 4 a _ 2, ( .3.3) R (P-(l-e)e xT+273.16) V S and %= .‘P ‘ , (4.3.9) (l-e)Rv(T+273.16)2 where: and the others are as previously defined. 53 P R v ambient atmospheric pressure, (mb), specific gas constant for water vapor, (0.110226 cal/g-OC) [List,(l966, p. 289)]. If one substitutes Equations (4.3.8) and (4.3.9) into Equation (4.3.7), converts all units for homogeneity and simplifies the expression, one obtains; Bdl -clLvP 5.?¢_._= 2 X 1 p chvu(c)(T+273.16) (41.8684) l:0.0418684 vas + 1 ] P-(l-e)eS (I-e) ° Let: A1 = -c1 Lv P/(4l.8684 p chv), A2 = 0.0418684 L , v and A3 = (l-e), then; Bdl = A1 Azxs + £_] 3¢l u(;)(T+273.16)2 P-ABes A3 The 3d/3¢ term for the energy equation is: (4.3.10) (4.3.11) 54 8d 3 -L c ¢2 3 0 Cp 11(5) (xS-x))]. (4.3.12) The term (Lv/p cp) has been assumed constant, thus Equation (4.3.12) reduces to Equation (4.3.7) and 3d2 Bdl -——-= -—— . (4.3.13) a¢2 a¢1 The 8d/8¢ term for the water vapor equation is 3d c 3 _ §_ 1 _ 3E; - 3X [u(;)(xs X)]l (4°3°l4 which can be rewritten as: 3d c 3x c 3x 3 _ 1 s _ _ 1 S. 21 - 57); - uTCT [3x 1] ' mtaw ax 13' ‘4'“5’ Since %§-= 1/(8x/3T), if one substitutes Equations (4.3.8) and (4.3.9) into Equation (4.3.15) it yields 3d3 -cl 0.0418684 vas 1 = (l-e)[ _ _ -+.f:—]. (4.3.16) 353 u(z) P (l e)eS 1 e If one compares Equations (4.3.10), (4.3.11), and (4.3.16), the latter is easily seen to be: 3d -c A A x 1 3 2 S + —l]. (4.3.17) 3- — 3¢3 - u(z) I:P-A3es A3 55 4.4 Solution Procedure The solution procedure is to: (l) diffuse the evaporating liquid water droplets from x = xU to x = xD and determine the new profile; (2) solve the energy diffusion equation for the same step; (3) solve the water vapor diffusion equation for the same step; (4) check the vapor concentration values for misapportionment with respect to the saturated vapor concentration values at the computed temperatures in step 2; (5) if inconsistent values are found, reapportion the error by a halving iteration in proportion to the concentration at the nodes, adjust the temperature and liquid water profiles accordingly, and print out the adjustments required; (6) sum the integrated liquid and water vapor profiles to verify the continuity of total moisture; and (7) allow this procedure to continue until an arbitrarily small amount of liquid water remains as determined by the integration of the liquid water profile. For this problem, the limiting value will be 0.05 g/my-sec. Beyond this point the diffusion of energy and water vapor will continue with the source terms equated to zero. A computer program was written in FORTRAN IV language to solve the problem, (see Appendix C). A second program was written utilizing a graphing subroutine, 56 GRAPHL, [See Breasbois and Nurnberger (1970)] developed for this project to plot field data and model results on the CALCOMP x-y plotter. All programs were executed on the CDC-3600 computer in the Computer Center at Michigan State University, East Lansing, Michigan. 5. EXPERIMENTAL PROCEDURE 5.1 Site Selection The experimental site requirements for this investigation are: (1) level uniform ground; (2) clear upwind fetch; (3) dry vegetation free strip, and (4) an ample clean water supply. Various values for the ratio of the fetch to measurement height have been suggested for uniform re-establishment of the boundary layer. Some of the investigators and their suggested values are: Inoue, gt_§l. (1958), 100:1; Priestly (1959), 20:1; Brooks (1961), 50 tree heights for an accuracy of 3%; Dyer (1962), a range of 140 to 330 for measurements between .5 and 10m respectively under neutral stability; and Panofsky and Townsend (1964), 10:1. The site selected for this investigation was a 40 acre (16.2 hectare), square, grass covered field at the Michigan State University Experimental Muck Farm. The minimum fetch to tree height ratio was approximately 20:1. The fetch over the bare strip to the spray line compared to the upwind grass height was maintained at a minimum of 200:1. A strip 300 m long and 120 m wide was established with a NW-SE orientation across the northeast corner of 57 58 the field to take maximum advantage of the normal southerly to westerly wind direction. The strip was plowed, tilled, leveled, and sprayed with herbicides to establish a vegetation free surface. 5.2 Water Supply_System The spray line was positioned along the center of the bared strip. The construction of the line consisted of a base feeder pipe of 3 inch irrigation pipe with welded couplings and 3/4 inch galvanized steel pipe risers supplying an elevated 3/4 inch line. The elevated line was positioned at a height of l m with Bette fog nozzles pointed downward spaced at intervals of 2/3 m. The nozzle selection and spacing was based upon preliminary laboratory studies by Schissler (1968). The nozzles were calibrated by collecting their discharge in graduated cylinders for one minute time periods. Water was supplied to the center of the line through a 3 inch feeder line by a high pressure, 125 psi, low volume pump located off of the bared strip on the downwind grass. Two 4 ft high by 18 ft diameter above ground swimming pools were utilized as water reserviors for a total capacity of approximately 18,000 gallons. The. large capacity assured constant water temperature and a slowly changing supply head during any test run. The 59 water from each pool was filtered by two 100 mesh well point screens on the pump intake. Figure 5.2.1 illustrates the schematic arrangement of the experimental site. 5.3 Instrumentation 5.3.1 Profile Measurements Profile measurement masts were constructed from 30 foot sections of 3 inch aluminum irrigation pipe mounted on tilt up bases which were staked to the ground. Instru- ment arms were constructed of 1/2 inch diameter galvanized steel pipe and mounted on the masts with saddle-T risers. The instruments were mounted 2 feet outward from the main mast support. The mast array consisted of five masts aligned normal to the spray line. For convenience in identification, the masts will be referred to by number. Mast number 1 was 10 m upwind from the spray line to monitor initial conditions. Masts 2 through 5 were at 10 m intervals downstream from the spray line starting with mast 2, 10 m downstream. Wind speed profiles were measured on Masts l and 3 with Climet 3-cup anemometers mounted at the l m, 2 m, 4 m and 8 m levels. Temperature and moisture profiles were made on all masts at the l m, 2 m, 4 m, 6 m, and 8 m levels with aspirated constant water level radiation shielded psychro- meters. The psychrometers were constructed as shown in .mpfim HousmeHmmxm on» no Emummwp oaumamnomnl.a.m.m mmDon 60 e e e O . ..K _ _ a e r a .m. mmmuo .m mmono .m u u m. w D D .1 /Q// P I _. pmom Dawn 1 w e a . . .m O m .m D m pom pom mmouo a D . a m ”u couwn ommswmnn _ Ema Honucmw magma Houmz We Z Trees mwmhfi 61 Figures 5.3.1.1, 5.3.1.2 and 5.3.1.3 using dual bakelite tubes for radiation shielding, a squirrel cage fan for aspiration, plexiglass water reservoir and constant water level well, and Yellow Springs Instruments thermo-linear thermistors as the dry and wet bulb sensing elements. The mast instrumentation is shown in Figures 5.3.1.4 and 5.3.1.5. The thermistors were calibrated in the lab by testing in a constant temperature water bath at tempera- tures of 20°C, 30°C, 40°C and 50°C. From an initial supply of 100 thermistors, 60 were selected that were the most closely matched and incorporated into 30 psychometers, thus providing 5 spare psychrometers. The digital data signals from the eight light chopper type anemometers and the millivolt analog data from the psychrometers were transmitted via wire to a data acquisition system designed for this project by Information Instruments Inc. of Ann Arbor, Michigan, Figure 5.3.1.6. The anemometer data pulses were counted, the psychrometers scanned sequentially, the analog signals converted to digital data, and all data punched onto paper tape in binary coded decimal at the rate of two cycles per minute. A FORTRAN IV computer program was written to interpret the paper tape, write the data onto magnetic tape, and analyze the data. 62 .UHan mchcmm can mchHoHnm :oHuMHuMH N m.uouoaouso>mm on» no 30H> souuomll H m m MMDUHM .uHas mansom can msHpHOHnm :OHumHomu on» mH GOHuuom souuon on» can .uHus can map mH :oHuowm OHpoHE .HHo>Homou nouns on» mH QOHuoom mos .uHcs Houosouno>mmII.H.H.n.m mmouHm 63 To Water Supply -To Fan fl Plexiglass Tubing Bakelite Radiation Shielding Wet Bulb Thermister Dry Bulb Thermister Wick Signal Wires U'IIBwNH Constant Level Water Well Plug in Sensing Unit Brass Support Screws Air Flow FIGURE 5.3.1.3.--Cross sectional schematic diagram of the psychrometer sensing unit and radiation shielding. 64 FIGURE 5.3.l.4.--Instrumentation arrangement on Masts 1 and 2 with the elevated spray line in the background. FIGURE 5.3.1.5.--A close up view of the anemometer and a psychrometer arrangement. 65 FIGURE 5.3.1.6.-—Data acquisition system. 66 5.3.2 Other Measurements Wind direction was measured with a Climet anemovane and Esterline-Angus sprip chart recorder. The calibration was by distant range pole alignment at 0,15,30, and 450 from the mast array. Net solar radiation was measured by a Beckman & Whitley net radiomemter and recorded on a Brown single pen strip chart recorder modified for millivolt input. Calibration was by comparison to a shaded and unshaded pyroheliometer and evaluated by the calibration factor supplied by the manufacturer. The soil temperature profile was measured at 2.5 cm intervals to a depth of 60 cm by copper-constantan thermocouples and a Leeds and Northrup 24-point recorder. The top thermocouple was immediately below the soil surface with a thin layer of soil to cover it. Soil heat flux was measured by a Thornthwaite heat flux plate and a Leeds and Northrup pen recorder. The heat flux plate was placed immediately below the soil surface with a thin layer of soil to cover the sensor. The calibration chart was supplied by the manufacturer. A schematic diagram of the instrumentation arrange- ment is shown in Figure 5.3.2.1. 67 Recording Center <::>>19 Pools ' O '2 \Pump and Mbtor Heat & Soil Flux Temperature Net Radiometer4,p. Anemo vane Spray Line FIGURE 5.3.2.l.--Schematic diagram of the instru- mentation and spray line. 6. EXPERIMENTAL RESULTS 6.1 Test Conditions The data herein reported was collected during an experimental test on 12 September, 1970, under a cloudless sky. The reported data were the results of an average of the data recorded for each sensing element for a 30 minute time period. The test run was terminated when a dense altostratus cloud cover approached. The ambient atmOSpheric pressure was 984.8 mb. The average spray rate was 8 g/m-sec. 6.2 Wind Speed and Direction The average wind direction was at an angle of 25 degrees from the normal to the spray line. The deviation from perpendiCularity to the spray line resulted in the masts being positioned at different effective distances. The mast positions are summarized in Table 6.2.1. The wind speed data is given in Appendix B, Table B1. The anemometer on mast 3 at the 8 m level was inoperative. The wind speed profiles are shown in Figure 6.2.1 as the upwind mast, A, and the downwind mast, B. The solid Curve is the Swinbank profile for this time period. The values of the profile parameters were found to be: 68 69 H.vv ov o.mm om o.- om o.HH oH 0mm ooom Hmsuod Hmfinoz Hmouofi Hmfiuoz Hospoe HmEHoz Hmsuom Hmfinoz Hmenoz m .02 v .02 m .02 chzczoo N.oz Eoum mummfiou Dom: poo: poo: o woman: H . oz oHafiH osH3C3oo “EV ucmfimomHm Mums . coHuomHHn OGHB .DGHH woman on» on m>HumHmH mCOHDHmom DmmE tam COHuowHHo UGH3 omnum>m mpscHE om OSBII.H.~.o mqmma F L 7.0 .. 5.0 [ A = Upwind B = Downwind 5.0 ,. H L. I: ,_ L..o , I D H b M I 3.0 ,. L. 2.0 ,. L 1.0 . L l 4 1 l 1 1 1 .l. 00 2.00 00 £4.00 N Ith SFDELEIJ ( M/’ SETC FIGURE 6.2.l.--Wind speed profiles. 71 L = —1.66 m, u* = 0.28 m/sec, and 20 = 0.0016 m. The very close agreement between the upwind profile and the profile within the active evaporation region justifies the previous constant wind speed profile assumption. The value of 20 is well within the range of values given by Sutton (1953, p. 233) which are: Surface 20, m Very smooth mud flats 0.00001 Lawn grass up to 1 cm 0.001 Downland, thin grass up to 10 cm 0.007 The value of u* is also well within the comparable range of values calculated from the values for surface shear stress over 1-5 cm high grass reported by Sutton (1953, p. 259). The values reported by Sutton and the 3 values of u* computed using a moist air density of 1.15 kg/m are: Wind Speed To u* at 1m 2 EZSGC dynes/cm m/sec 4.03 0.90 0.280 4.78 1.44 0.354 72 6.3 Net Radiation, Soil Heat Flux and Sensible Heat Flux The average net radiation for the period was 0.27 ly/min. The soil heat flux as measured by the sur- face heat flux plate was 0.14 ly/min and thus the sensible heat flux as found by Equation (3.4.1.2) was 0.13 ly/min. 6.4 Temperature and Moisture Measurements 6.4.1 Initial Conditions The uppermost soil thermocouple recorded an average temperature of 24.50C (76.1 0F). The psychrometric data was analyzed by using a field calibration factor. This factor was developed for each dry bulb and wet bulb temperature sensor by initializing in such a manner that the readings for no spray conditions for all psychrometers at a given level were equal. The technique used was to average the temperatures for all psychrometers at a given height for a preliminary time period. The deviation of each sensing element from its corresponding mean value during the initializing period was then used as each element's calibration factor. The resulting dry bulb and wet bulb temperature data for the test run is presented in Appendix B, Table B2. The comparison of the dry bulb temperature and the Swinbank similarity profile, Equation (2.3.3), is given in Figure 6.4.1.1. The computed value of T* was 73 -0.869 0C. The test for goodness of fit to the data was by the method of minimizing the sum of the squared errors. [See Himmelblau (1969).] For the initial conditions, the best agreement occurred with a temperature of 22.50C at a height of l m. The computed value of T0 was 27.8 0C. The deviation of the measured temperature from the Swinbank profile was consistant for all time periods tested, i.e. measured data was less than the Swinbank profile for 2 less than or equal to 4 m and greater than the Swinbank profile for z greater than 4 m. This is believed by the author to be the influence of the upwind grass area and its modifying influence on the boundary layer. Measurements seemed to have been made both within and above the reestablishing boundary layer. It is a commonly accepted fact that wet bulb temperature measurements are much more difficult to make and the accuracy greatly reduced compared to dry bulb temperature measurements. Thus it is expected that the initial moisture concentration data would not agree with Swinbanks similarity profile as well as the dry bulb temperature profile. This is illustrated in Figure 6.4.1.2. The initial moisture conditions at 20 were assumed to be 80% R.H. at the To temperature and ambient pressure. The minimum sum of squared errors was achieved for a moisture concentration at the 1 m level of 11.5 g/m3. L 7.0 _ L ‘0.0 _ a Solid Line = Swinbank Similarity Profile B = Measured Values L 5.0 .. (0 u: L k1 +— U 0.0 . a z '_ L I o ,_. 3.0 _ u) I L 2.0 _ B 1.0 . a L L 1 L L L L 1 L _l 22.00 2LI.00 2b.00 28.00 30.00 TEZMF’EFRA'TUFRE (C) FIGURE 6.4.l.l.--Initia1 temperature profiles. 75 7.0 . L 5.0 _ a Solid Line = Swinbank's Similarity Profile B = Measured Values L 5.0 . (D m . u) I- uJ 0.0 L e t .- L I D ..... 3.0r -u I. L 2.0 L a 1.0 . B 10.00 10.00 15.00 22.00 25.00 MIDI Sl'UFRE (IOIVCIEIJTFQA‘TIIJN (G)/ ClJ M) FIGURE 6.4.1.2.--Initia1 moisture concentration profiles. 76 The values of x0 and x* computed for these conditions were 21.7 g/m3 and -1.66 g/m3 respectively. The sum of squared errors for the temperature and moisture concentration for the five levels measured were 0.871 and 13.1 respectively. The comparison of these two values further illustrates the reduced accuracy of the moisture profile measure- ments. The moisture profile measurements exhibit a peculiar bulge at the 4 m level. This anomally was consistant for all measurements made and is believed to be the direct result of the evaptranspiration of the upwind grass area mentioned above. 6.4.2 Modified Conditions The observed dry and wet bulb temperatures are included in Appendix B, Table B3. Figure 6.4.2.1 illustrates the small water droplet supply at the spray line and Figure 6.4.2.2 the optical disappearance of the mist between the first and second downwind masts, i.e. masts number 2 and 3. The effective evaporation has therefore been assumed to be completed in the neighborhood of mast number 3. The choice of OH = 1 did not yield satisfactory model results. The active evaporation distance required was between 30 and 40 m if the evaporation source term coefficient, cl, was chosen to yield reasonable cooling 77 spray line for small water droplets. FIGURE 6.4.2.1.--Operational . >-- ~ .- can‘t.- an... I '--...: ,rv-A. - ‘ * ‘. noo?)k¥i§*4*f¢¥1+ffi‘f£g¥¥i¥¥l*f15“,7 . My£+§fiLNA~INW . mas‘trrmvurv‘nwhfi‘k‘ ‘ FIGURE 6.4.2.2.--Drop1et dispersion and disappearance between masts and 3. 78 profiles. If larger values of cl were used to achieve complete evaporation in the neighborhood of the second mast, then the predicted temperature profiles were much cooler than the observed data. The choice of “H to be that of Leichtman and Ponomareva (1969) and a value of cl = 0.02/sec yielded improved results. The active evaporation distance was 21.0 m with the Swinbank similarity profiles used as initial conditions. The values of the constants used in the model are summarized in Appendix B, Table B4. Figures 6.4.2.3 thru 6.4.2.6 compare the measured dry bulb temperature values and the model profiles for masts 2 thru 5 respectively. Since the model solution step did not always coincide with the exact placement of the masts relative to the wind flow distances from the spray line due to changing wind direction, the solution step closest to the actual distance was used. This did not introduce a significant error because the maximum misalign- ment distance would be one half of the downwind solution step, i.e. in this case 0.5 m. Figures 6.4.2.7 thru 6.4.2.10 compare the observed and predicted moisture concentration profiles. The sum of squared errors for the five observation levels are given in Table 6.4.2.1. The maximum cooling computed by the model was -1.02 0C at 19.0 m downstream and a height of 0.67 m. 8.0 r' I 79 L. 7.0 , 5,0 F a Solid Line = Model Prediction B = Measured Values 5.0 n F to m L LI 3.. U Lo.0 F B t F I ‘I u H 3.0 , u: I. 2.0 , e L 1.0 F L 25.00 30.00 TEZHF’EFQA'IUIWE (C) FIGURE 6.4.2.3.--Temperature profiles at mast 2 with Swinbank initial conditions. L 7.0 . bun . a Solid Line = Model Prediction B = Measured Values L 5.0 F £0 m. . kl i. U Lo.0 . 3 2 P b I I: H 3.0 . m: I. L. 2.0 r I. 1.0 P 25.CD so CD TEIHF‘EFlA'fUFWE. (C) FIGURE 6.4.2.4.--Temperature profiles at mast 3 with Swinbank initial conditions. BJDF. a 81 b “7.0 , buo . a Solid Line = Model Prediction B = Measured Values 5.0 [. In K L U I- u 0.0 , a I P L I U ,_. 3.0 F LI 1 2.0 r 1.0 . TEHPERATURE (C) FIGURE 6.4.2.5.--Temperature profiles at mast 4 with Swinbank initial conditions. 82 L 7.0 .. 500 p * Solid Line = Model Prediction B = Measured Values L 5.0 . to c. L U '- 1 u 0.0. l I: F b I D ,. 3.0 . u: I. 2uo . a 1.0 D T'EI1F'EF1A'TLJRIE (II) FIGURE 6.4.2.6.--Temperature profiles at mast 5 with Swinbank initial conditions. F 7.0 .. L b~° r 3 Solid Line = Model Prediction B = Measured Values L 5.0 .. to m . kl .- U h.0 ' B a: F .- L I u H 3.0 r kl I. L 2.0 F L 1.0 ,. L .18 (I) 22.CD 2!: CD Hill S'TLIHIE CfOlinElil'HlkT ICJN (04/ (TU r4) FIGURE 6.4.2.7.--Moisture concentration profiles at mast 2 with Swinbank initial conditions. F L 1.0 . L 5.0 . a Solid Line = Model Prediction L B = Measured Values 5.0 . m 8'. L u .— h] 0.0 . I I ._ L I U H 3.0 . ii I. L 2.0 . 1.0 . a L .10 CI) 1L: 00 1.8.00 22 00 Z: 00 van 1531 UIQEI CEJNICEIN'TFiA'II 0!“ (ISI' Cl) H) FIGURE 6.4.2.8.--Moisture concentration profiles at mast 3 with Swinbank initial conditions. L 7.0 ,. L 5,0 _ 3 Solid Line = Model Prediction B = Measured Values L 5.0 . m I. L Ll .— El 0 O a: F ,_ L I U ... 3.0 . U .I; L 2.0 D , 1.0 L I L 10.00 lh.00 13.00 2?.(13 25.00 HOJSTURE CONCENTRATION iR/CU M) FIGURE 6.4.2.9.--Moisture concentration profiles at mast 4 with Swinbank initial conditions. L 7.0 { L 5,o { . Solid Line = Model Prediction B = Measured Values 5.0 . u I. L t] p. U “DO . t P L I u H 3.0 r t] I. L 2.0 L a L lOO I a L .1000 10.00 13.00 22.CD 25.00 P10.1£iTIJFIE (IOIUCIEIUT'RIKT IIJN 1C3! (3U r1) FIGURE 6.4.2.10.--Moisture concentration profiles at mast S with Swinbank initial conditions. 87 TABLE 6.4.2.l.--Sum of the squared errors using Swinbanks similarity profiles as initial conditions. Profile Mast Number ‘ 1 2 3 4 5 Temperature 0.871 .684 .311 .071 .218 Moisture Concentration 13.1 8.51 8.86 17.9 8.82 The moisture concentration profiles are much less in agreement than are the dry bulb temperatures. Mast 4 has the lowest sum of squared errors for temperature but the highest value for moisture concentration. Since the sum of squared errors is computed for the observation points only, the inexact initial Swinbank profiles could contribute to the downstream errors. To test the influence of the initial profiles, the measured data points were assumed to be connected linearly. The results for the temperature profiles are shown in Figure 6.4.2.11 through Figure 6.4.2.15 and for the moisture concentration profiles in Figure 6.4.2.16 through Figure 6.4.2.20. The solution procedure retained the Swinbank wind speed profile, diffusivities, and lower boundary sensible heat flux relationships. A11 constants were the same. The sum of squared errors are given in Table 6.4.2.2. 8.0 r s 88 0.0 . 5.0 . Solid Line = Model Prediction B = Measured Values 5.0 F m r u p. u 0.0 r E ,_ L I u ,_. 3.0 u I. L 2.0 . 1.0 L L J l l L l 4 l l l 22.00 2h.00 23.00 28.00 30.00 T EPGP EFiA‘TtJR E ICJ FIGURE 6.4.2.ll.--Initia1 linearized temperature profile} 5.0 r I 89 L 7.0 ,. 5.0 F a Solid Line = Model Prediction B = Measured Values L 5.0 . u: I . u p. U (4.0 L s I .- L- I D H 3'0 P u: I. L 2.0 . a 1.0L TEiHF’EFlA'FUFREZ ((1 FIGURE 6.4.2.12.--Temperature profiles at mast 2 with the linearized initial conditions. .00 90 L 7.0 .. L b-0 , % Solid Line = Model Prediction L B = Measured Values 5.0 L m I L Ll '- U Lo.0 ,; F F L I. u H 3.0 p u I. L 2.0 ,. 8 1.0L L 22 00 2h 00 28.00 25 00 30 1'Ef1PlERJA1’UFiE (CJ FIGURE 6.4.2.13.--Temperature profiles at mast 3 with the linearized initial conditions. .00 (P1C'TCIHSEJ HIE] GriT L 7.0 . L Solid Line = Model Prediction 8.0 . B B = Measured Values 5.0 . L 0.0 r L 3.0 . L 2.0 r L 1.0 .. r 22.00 24.00 23.00 28.00 30.00 T EP1P EFiA'lLJH E IC) FIGURE 6.4.2.14.--Temperature profiles at mast 4 with the linearized initial conditions. 7.0 .. 8L0 . a Solid Line = Model Prediction B = Measured Values L 5.0 . w I L L1 '— U 0.0 . B I |_ L I u H 5.0 P L: I f" O In T EP1P EF‘A'TLifiii ((3) FIGURE 6.4.2.15.--Temperature profiles at mast 5 with the linearized initial conditions. 5‘0 r 8 93 L 7.0 L L 5.0 . Solid Line = Model Prediction B = Measured Values 5.0 L m I L u p. U 0.0 I F ._ L I u H 3.0 u I L 2.0 D 1.0 L L L l l Mil] S'FUIQE: CEJNiiEIN'FHFAl’JKJN (04/ ClJ MJ FIGURE 6.4.2.16.--Initial linearized moisture concentration profile. L 7.0 L L b 0 a Solid Line = Model Prediction B = Measured Values 5.0 L m 0: L u }_ t1 0.0 L z .— L I u H 3.0 . u I. L 2.0 L L .100 D L- 1 L L 1 J .18 00 22.00 2|: 00 HID] S'IUFQE (IOFQCTEhllflA'11l3hl IG.’ Cl) M} FIGURE 6.4.2.17.--Moisture concentration profiles at mast 2 with the linearized initial conditions. 7.0 L 5.0 _ a Solid Line = Model Prediction B = Measured Values L 5.0 L m m b u p. U 0.0 L I +- I- I o H 3.0 L u I. 2.0 L 1.0 L 22.00 2|: 00 M13] S'TLIREI CC)N(IEIVT'RI\T ICJN (0" ClJ Ml FIGURE 6.4.2.18.--Moisture concentration profiles at mast 3 with the linearized initial conditions. L 7.0 L L Solid Line = Model Prediction 8.0 L a B = Measured Values L 5.0 L tn I L t: .— U 8.0 L I .- L- ‘I u H 3.0 . ll I. 2.0 F .0 1 r H'DJ S'IUFQE (2010C EthFih'lJIJN lficf CLJ H) FIGURE 6.4.2.19.--Moisture concentration profiles at mast 4 with the linearized initial conditions. 7.0 L L 8.0 L a Solid Line = Model Prediction B = Measured Values L 5.0 m I. L u p. L: 0.0 L I F L I u H 3.0 L L! I. L 2.0 L L 1.0 P L H13] S'1LJR E CFOIUC'EEGT'RIKW JEJN lti/ (TU r41 FIGURE 6.4.2.20.--Moisture concentration profiles at mast 5 with the linearized initial conditions. 98 TABLE 6.4.2.2.—-Sum of squared errors using linearized initial conditions. Mast Number Profile 1 2 3 4 5 Temperature 0.002 0.329 0.029 .0.158 0.565 Moisture Concentration 0.008 5.68 6.27 20.0 5.86 The nonzero sum of squared errors for the initial conditions is due to the noncoincidence of the measured heights and the solution heights. The errors thus incurred are negligible. The linearized initial conditions improved the model results compared to the results obtained with the Swinbank initial conditions, for temperature profiles at masts 1, 2, and 3, but the results were slightly poorer for masts 4 and 5. The moisture concentration profile results were improved for all masts except number 4. Mast 4 consistently exhibited the highest error throughout the test runs. A change of psychrometers did not significantly affect the anomally. The cause of the deviation of mast 4 is unknown to the author. The active evaporation distance for the linearized initial conditions was 23.0 m. This is about 10% farther than with the Swinbank initial conditions. The maximum cooling computed by the model was -2.90°C at 23.0 m down- stream and a height of 0.34 m. 99 Neither of the two forms of the initial conditions caused saturation in the active lower layers. 7. MODEL PREDICTION The basic model was used to predict the temperature modification under various atmospheric conditions. The parameters that were allowed to vary were: (1) the Monin- Obukhov stability factor, L; (2) the initial temperature at 1 m; (3) the initial wind speed at l m; and (4) the spray rate. The values used for the four selected variables are summarized in Table 7.1. Not all combinations of all variables were utilized. TABLE 7.l.--A summary of the values of L, T., u., and the spray rate used in the prediction model. Variable Range of Values L, m -001 “1.0 -1000 Ti(@ z=1 m), °c 30.0 35.0 40.0 ui (@ z=l m), m/sec 0.5 1.0 2.0 4.0 Spray rate, g/m-sec 10.0 15.0 20.0 The combinations for uiz2.0 m/sec and L = -0.1, for example, produces an impossible requirement for the sensible heat flux. The consistancy between L, u*, T*, and H for the temperature range of 30 to 40 oC would require H to be nearly three times the solar constant. These combinations were therefore 100 101 omitted. The constants used in the prediction model and selected results are tabulated in Appendix B, Table B5 and Table B6 respectively. The results for two different initial temperatures have been selected for graphical presentation. The two temperatures are 30.0 and 40.0 0C. The parameters used were: L = -l.0, u1 = 2.0 m/sec, spray = 20.0 g/m-sec, and all the other constants are as reported above in Table B5. Figures 7.1 and 7.2 are the temperature and moisture concentration profiles for 30 oC initial temperature and 25% initial relative humidity at the 1 m level. Curve A is the initial condition and curve B is the profile at the point downstream where maximum cooling occurred. The maximum cooling for these conditions was -8.4 0C at 21.0 m downstream and at a height of 0.15 m. Figures 7.3 and 7.4 are the temperature and moisture concentration profiles for the 40 oC initial temperature and 25% initial relative humidity. The maximum cooling was -9.5 0C at 19.0 m downstream and at a height of 0.15 m. Figures 7.5 through 7.8 illustrate the maximum temperature change with respect to the initial 1 m temperature for the three spray rates at each selected wind speed. The maximum cooling occurs with the lowest wind speed and highest initial temperature and spray rate. 102 This is due to the higher wind speeds diffusing and trans- porting the water much more than the lower wind speeds. Thus the lower wind speeds provide greater localized cooling. The maximum cooling for all Conditions tested was -l4.6 0C at 41.0 m downstream and at a height of 1.25 m for Ti = 40.0 0C, ui = 0.5 m/sec, spray rate = 20.0 g/m-sec and L = -10.0 m. The maximum cooling value is within the expected range of values concluded from Table 4.3.2. A comparison of Figures 7.5 and 7.8 reveals that the rate of change in the maximum cooling with respect to the initial temperature is reduced with increasing wind speed. The variation between spray rates is also reduced in a similar manner. 8.0 103 L —. .. W L 8.0 L A = Initial Conditions B = Maximum Cooling L 5.0 L M I L u .. U “.0 D C ._ L 1 u L. 3.0 . u I. L A 2.0 L L .l.0 L 20 00 25.00 30 00 35 00 80.“) . 1 Er1P'EFiALTlJFiE |CIJ FIGURE 7.l.--Predicted temperature profiles with Ti=30.0 O C. 8.0 L 104 7.0 L 8.0 L A = Initial Conditions B = Maximum Cooling L 5.0 L to I L I! .— U Lo.0 L I F L I ID L. 3.0 L n: I. L 2.0 [ L .l.0 L L 1 1 A 1 L 1 J 1 1 I 10 00 20100 30L00 00:00 50'00 MID] S'IUIQEI (IOIQCIEFITFRA'TI 0f! ((3/'CLJ M) FIGURE 7.2.--Predicted moisture concentration profiles for Ti=30.0°C and R.H.i = 25%. (NCTCHSJ HEIGHT 8.0 L 105 Initial Conditions 3’ ll 00 ll Maximum Cooling 33.00 00.00 145.00 30.00 33.00 1 Er1P EFIA'TLJR E LC.) 0 FIGURE 7.3.--Predicted temperature profiles with Ti=40.0 C. 8.0 106 F L 7.0 L L 8.0 F A = Initial Conditions L B = Maximum Cooling 5.0 L (0 I L m: .— U 8.0 a F H F '— L 'I 0 L.. 3.0 L m I. L 2.0 L 1.0 F L 1 l l L l l L L 1 _l 10.00 20.00 30.CD L00.CD SIC!) MID] S'TLJRIi C(lNIZEZN'TFIA'TI 01¢ ((3/'ClJ N) FIGURE 7.4.--Predicted moisture concentration profiles for Ti=40.0°C and R.H.i=25%. 18.0 L L, 12.0 L D i- Z A 10 o / L. 8.0 b — . g m sec 4 B = 15.0 g/m-sec c3 0 L C = 20.0 g/m-sec u E 0.0 D F I H x L 4 I 30.00 35.00 80.00 85.00 T EP1P EFlA‘TLJH E (C ) FIGURE 7.5.--Maximum cooling with ui=0.5 m/sec and three spray rates. 18.0 L L U 12.0 L D +- L ._A Z / H 8.0 L. 5 C) C) L A = 10.0 g/m-sec U B = 15.0 g/m-sec I C = 20.0 g/m-sec 0.0 L 3 I x I- <. .I 30.00 35.00 (40.00 05.00 T Er1P EFlA TLJR E (C ) FIGURE 7.6.--Maximum cooling with ui=l.0 m/sec and three spray rates. 109 18.0 L D b c J 2 / "'" 8.0 L 4 o a e s 0 u I A’ A ¢_A ‘ 0.0 L j Z A = 10.0 g/m-sec H x L B = 15.0 g/m-sec < C = 20.0 g/m-sec I. 30.00 35.00 80.00 85.00 T EF1P EFQA'TLJH E 'IC) FIGURE 7.7.--Maximum cooling with ui=2.0 m/sec and three spray rates. 110 18.0 L U 12.0 . A = 10.0 g/m-sec U B = 15.0 g/m-sec _ C = 20.0 g/m-sec u 2 H 3.0 L 4 c: D L 0 :1 0 0 L 3 c_fl_fl_fl,,.4;.—-~'*‘”L Z 8_________3.—~——————3 x b < Z J 30 00 33 00 80.00 '43 00 T'EP1PIER.A1'UF?E (C) FIGURE 7.8.--Maximum cooling with ui=4.0 m/sec and three spray rates. 8 . DISCUSSION The solution of the differential equations in the form of Equation (4.2.17) was required to be above the roughness height 20. By definition, at the height 20 the wind speed is zero. The solution coefficients have terms involving division by u(z), therefore for the coefficients to remain finite, u(z) must be greater than zero. The lowest solution height chosen was 21 = 0.005 m, which introduces no appreciable error since 20 was 0.0016 m. Early preliminary results disclosed that the selection of the verticies of the initial triangular liquid water droplet concentration profile was not critical. The points chosen below the l m spray line were: bottom, zz=0.4 m; middle, maximum spray concentration, zMS=0.6 m; and the top, z3=0.8 m. Turbulent diffusion was sufficient to mask variations about these values. An implied assumption in the develOpment of the source term for Equation (3.1.2) was that gravitational settling of the drops was negligible. The meteorlogical definition of a cloud drop is one whose diameter is less than 200 u, [Huschke (1959), p. 111]. The mean diameter of the spray drops are well within the cloud definition. 111 112 Stokes Law, [Huschke (1959), p. 543], can be used to estimate the terminal velocity of the drops in still air, Equation (8.1). _ 2 2 _ VT - g a 9(Dw o)/n (3-1) where: VT = terminal velocity of spherical drops in still air, (cm/sec), a = droplet radius, (cm), 9 = acceleration of gravity, (cm/secz), pw = water droplet density, (g/cm3), p = density of medium, (g/cm3), and n = dynamic viscosity, (g/cm-sec). For this case the air density is much less than the droplet density so it can be neglected. The dynamic viscosity for air at 30°C is 1.866 x 10:4 g/cm-sec, [List (1966), p. 395]. The terminal velocities for 10 and 50 u radius drOps become 0.012 and 0.291 m/sec respectively. Since the droplet radius is constantly decreasing due to evaporation, its terminal velocity will also be decreasing. Thus the fall velocity is much less than the turbulent velocity fluctuations and the assumption is justified. The use of the method of images for the liquid water and water vapor concentration profiles to establish a zero flux lower boundary condition caused a point of discontinuity in the wind speed profile. The diffusivity profile was 113 likewise affected at the lowest solution point since it is a function of the wind speed gradient. The value of the wind speed gradient at the lowest node was therefore assumed zero. Zero provided reasonable results in contrast to using either of the gradients from above or below this point. The choice of zero for the velocity gradient caused the calculation of the intermediate solution coefficient, P4, at that node to produce erroneous solutions near the boundary if the downwind solution step was too small. The size of the solution step, DX, was therefore chosen to be greater than 0.125 / P4IZl, where P4IZl is the value of P4 at the node IZl closest to the ground surface. In all cases the minimum value set for DX was 1.0 m to reduce computer time. This caused no problem since for the Leichtman- Ponomareva form of a the computed minimum for Dx was H' 0.25 m whereas the use of aH=l required Dx to be 1.25 m. The continuity check of the three profiles at each solution step revealed.an error of less than 0.5% of the total integrated value for each profile.i.e. liquid water concentration, energy, and water vapor concentration, at that solution step. The low value of the error demonstrates the accuracy of the solution procedure used. The small amount of water vapor added to and sensible heat removed from the total system with respect to the water vapor and energy 114 initially present was severely masked by the compounding of the relatively small percent error at each step. A scheme was therefore developed for reapportioning the error at each solution step to each vertical node in proportion to that nodes contribution to the total sum of profile values for all nodes at that solution step. The change was very minor at each node but the elimination of the compounding error problem was achieved. The exact cause of the error was never determined. The solution for water vapor concentration at each downstream step was checked for saturation at the lowest active nodes. None of the runs reported in this thesis achieved saturation. If an excessive amount of water was sprayed into the air, e.g. 40 g/m-sec, the combined small evaporation source term and vapor diffusion caused local saturation. These conditions were beyond the applicability of the model, however, since large amounts of water would require higher droplet concentrations and/or larger drops which in turn affects the evaporation coefficient, cl, and the assumed negligible settling rate of the drops. For solutions of the problem using drops larger than 200 u (0.2 mm diameter» e.g. evaporation in sprinkler irrigation, the source term must be rewritten to include the settling rate. The droplet deposition process could be included at the same time the model is redeveloped for 115 a plant canopy. With a plant canopy at the lower boundary and droplet deposition at the surface, all three profiles would have flux rates as the lower boundary condition. The method of images could then be replaced thereby reducing the number of solution nodes by a factor of two for the liquid and vapor profiles. Model wise, the redevelopment is not believed particularly difficult. The complexity of the problem would instead lie in the experimental verifica- tion of the flux terms at the "surface" of the plant canopy. 9. CONCLUSIONS The following conclusions have been drawn from this investigation. (1) (2) (3) (4) (5) (6) Significant cooling can be achieved by mist evaporation. Maximum cooling occurs with light winds when the liquid water is not diffused out of the region. Minimum distances required for complete evapora- tion occur with high winds creating more turbulent diffusion and dispersion of the liquid water drops. More precise humidity measurement systems are required. The Leichtman—Ponomareva approximate relation- ship for a was adequate for this investigation H and much more realistic than aH=l. The Swinbank profile for wind speed was very good, but the similarity profiles for temperature and moisture concentration were much poorer for this investigation. 116 APPENDICES 117 APPENDIX A DETAILED REVIEW OF CLOUD DROPLET EVAPORATION MODELS 118 The following is a more detailed review of the models developed by Zung (1967,a,b, and 1968) for cloud droplet evaporation. Zung (l967,a) develOped the modified cellular method. In the modified cellular method, the mass balance equation for a cell was used to determine the average concentration of molecules per unit volume: at) = [Ib c(r,t)4nr2er/g-Mb3—a3). (A1) a A It was then assumed that co = c = 0, i.e. the initial 00 ,vapor concentration was zero, to obtain c(t) = pw(ag-a3)/m2(b3-a3). (A2) By equating 5(t) to cs, the drOplet radius at cellular saturation, as, was determined to be 3 _ 3_ 3 _ aS — (pwao mzcsb )/(pw mzcs). (A3) The value of a: was then used to ascertain if.the cloud would become saturated internally during its lifetime. If a:>0 the cloud would become saturated and if a230, it would not. Through the use of Equation (2.5.7), the relationship of drOplet mass to radius, and Equation (A2), the equation for the steady-state evaporation time was found by integration to be: 119 120 ——!-[b3- 3f__393——» = t + const. (A4) where; W = p-c m For the unsaturated case, i.e. asio, Equation (A4) was integrated to: tu = [cw/B(ow-csmz)](-%a§+(b3-a3)(%as){£n(ao-as)- l2 ( 2+ 2 3 15 -1 la )/ 2 n aO aoas+as)+( ) tan [(ao+2 s %(3)*5as]-%-(3>’5n}) (A5) which is the lifetime of the cloud when its interior remains unsaturated under initially dry ambient conditions. The time required to reach saturation was found in a similar manner to be: 3 3 2 t _ D‘ p T{%-(az-a2 +(b3-a8) ré‘floi“ 23as 2) s p-c m s o a L s 2 s ao+aoas+aS _ k -l ao+15aS - % +9.n(ao as)+(3) tan (m) (3) TTJ}. (A6) 5 Equation (A6) would be the final result for an enclosed system of droplets. In the atmosphere, however, the droplets around the periphery of the cloud will continue to evaporate since they will not reach an equilibrium state of saturation. 121 The cellular model must be modified to account for this. The outer droplets were assumed to evaporate at one half the rate of a free drOp, since they were evaporating into a semi-infinite medium. Thus the droplet lifetimes would be twice that of a free drOplet. The lifetime of outer drOplets before saturation were found to be: _ 2 _ touter — aopw/[m2D(co cm)] , Maxwell, (A7) p Da = W r1 2+__°(¢‘1-Z—E. (Ln szTco-cm) 260 va 0 a0 1 a0 2 ao+A -aOA + A £n( A‘ )], F-O-Z, (A8) where; Z = 3 vos/kBTA ¢O = ¢(ao)l x and Ei(x) = ft-letdt. The lifetimes of the outer droplets after saturation, tj, has the same form as equations (A7) and (A8) with aS substituted for ac. The cloud lifetime after saturation was found to be: 122 t % s t = {[R(1-E) /2a ]-(——————)}t., (A9) e O touter 3 where; E = void fraction, and the other symbols are as previously defined. The term tS/t is the number of layers evaporated outer prior to saturation and R(l-E)!5/2aO is the total number of layers in the cloud. The total lifetime of the cloud in the saturated case would be: tcloud = t5 + te' (A10) A few of the resultsreported by Zung using the above equations are given in Tables Al, A2 and A3. 'TABLE Al.--A comparison of saturation times and cloud life- times for the Maxwell and Fuchs-Okuyama-Zung relationships with cloud radius, 30:10m, degree of dilution, b/a=20, and T=25°C. a,(u) ts,(sec) tcloud, (sec) Maxwell FOZ 100 8.40 ‘ 38,036 41,392 10 .11 3,804 7,162 TABLE A2.-—Cloud lifetimes for the Maxwell relationship for various cloud radii and with a=100u, b/a=20, and T=25°C. Cloud Radius tcloud' (36C) 1 3.3 x 103 10 3.8 x 104 5 100 3.8 x 10 123 TABLE 3A.--A comparison of saturation radii, saturation times, and cloud lifetimes for the Maxwell and Fuchs-Okuyama—Zung relationships for various degrees of dilution, b/a, with a=10p, R=1m, and T=25°C. tcloud,(sec) b/a as,(u) ts, (sec) Maxwell FOZ 2 9.9994 1.57 x 10'4 4356 7951 10 9.9226 1.59 x 10'2 857 1571 30 7.2292 .41 152 325 40 -7.8016 .165 50 -12.343 .112 It is also reported that for fairly high concentrations, b/a S .4(b/a) 1, aS is nearly the same as a and tS critica o is much less than t The critical value of b/a is cloud' the degree of dilution where the cloud changes from one that remains unsaturated to one that becomes saturated. From the results for the saturated cloud conditions evaporating into initially dry ambient air, the above results indicate that the cloud lifetimes predicted by the Fuchs-Okuyama-Zung relationship are approximately twice that predicted by Maxwell for ab=10u but about the same for ab=100u. Zung (l967,b) develOped an alternate model for the cloud evaporation problem, using the continuum approach develOped by Milburn (1957, 1958). The resulting highly non- linear model for a drOplet system in a fixed atmosphere, 124 for a dilute cloud or one in the initial phase of evaporation where c(r,t)Zc(r), was found to be: g?-c(r,t)=DV2c(r.t)+nd'[mg/3+%Bt]%[cs'c(rrt)Jr (All) where; D = diffusivity of vapor in air, (cmZ/sec), V2 = Laplacian Operator, n = number of drOplets/unit volume, mb = initial mass of a single drOplet, (g), c = vapor concentration, (molecules/cm3), a' E 47rD(3/47rpw)l/3 B E d[c(r,t)-cs]m2. and m2 = mass of a water molecule. Equation (All) is essentially the same as that proposed by Milburn. To linearize the cloud equation Zung assumed the vapor concentration c(r,t) is approximately linear with the drOplet mass. Under steady-state conditions this leads to: dM st - dt 4an2D (CS Cm): (A12) where; ss = rate of evaporation, (g/seC): M = total cloud mass, (9): D' = new diffusion coefficient, D/(1”E)I 125 E = void fraction = l-(a/b)3, i.e. the fraction of the total volume of the cloud occupied by the gas phase, c = droplet surface saturated concentration, (molecules/cm3), and c = water vapor concentration outside the sphere of influence, (molecules/cm3). note: D' = D/(a/b)3. For nonstationary conditions the rate of evaporation would be: Q = 4nRD'm2(cS—cw)[l + n(nn't)‘3]. (A13) NS The resulting cloud lifetime equation for the steady-state condition is: _ _ 2 _ 2 tcloud — (l E) [ow/mzmcS cm)](Ro/2). (A14) where; pw = liquid density, (g/cm3), Eb = initial cloud radius, (cm), and the others are as previously noted. An extraction of the results of calculations using equation (A14) and those using the cellular model, as reported by Zung (l967,b, Table I), are given in Table A4. Tflmabelow results illustrate the conclusions from Zung (1967,b, p. 3580) that "The cellular model gives good results for an unsaturated 126 TABLE 4A.--A comparison of cloud lifetimes for the continuum and cellular models for various degrees of dilution with a=10u1, RO=1.0m, and T=25°C. tcloud, (sec) b/a Continuum Cellular Model Model 1 8.71 x 108 7.00 x 103 2 1.36 x 107 4.36 x 103 6 1.87 x 104 1.45 x 103 10 8.71 x 102 8.58 x 102 30 1.20 1.52 x 102 -1 -1 40 2.13 x 10 1.65 x 10 50 5.58 x 10"2 1.12 x 10‘1 {lFor a free single 10p radius drOplet, t=8.714 x 10"2 sec. cloud but fails to converge to the upper limit where (b/a)=1.0, where the cloud should evaporate as a giant drOp of radius R according to the Maxwell equation. The continuum model . . . yields the desired lifetime for (b/a)=l.0.and also shows a small transition from a saturated cloud to an unsaturated one. However, the lifetimes for an unsaturated cloud do not approach exponentially the lower limit as (b/a)+0° as seen in the cellular model." Zung's final con— clusion was to combine the two models and use the cellular method to compute cloud lifetimes for an unsaturated cloud and for calculating the time required to achieve saturation. 127 For saturated clouds and for the elapsed time, te’ after saturation, Equation (A14) should be used with some modifica— tions, namely: (1) RC should be replaced by Rs=Eo-(tS/touter)2b where touter is computed from Equation (A7), (2) E = 1-(aS/b)3, and (3) tcloud = tS + te' Some of the results of this more general model are reported in Table A5. TABLE A5.--A comparison of saturation radii, saturation times, and cloud lifetimes for various degrees of dilution for Zung's generalized model with a=lOu, Ro=l m, and T=25°C. b/a as, (p) ts, (sec) tcloud' (sec) saturated 8 1 10.0000 0 8.71 x 10 2 9.9994 1.57 x 10'4 1.36 x 107 10 9.92 1.59 x 10’2 8.32 x 102 30 7.23 .41 .581 unsaturated 40 -7.80 .165 50 -12.3 .112 All ofthe models discussed to this point have been for evaporation into still air. Zung (1968) extended his model to include the effects of turbulent diffusion and cloud 128 expansion. He also investigated the effects of Brownian Motion and Electric Charge and found that they were much less significant than the turbulence. The current review will, therefore, be limited to the turbulent portion of the paper. The basic assumptions employed in this model were: (1) a monodisperse droplet system; (2) uniform spreading throughout the cloud; and (3) Maxwellian single droplet evaporation. The cellular and continuum models were included for comparison. The semi-empirical formula for b(t) for small values of t, i.e. lifetimes of small droplets in dilute clouds, given by Batchelor and Townsend was used: B(t) = (bO + 0.25 Gt2t2)*, (A15) where; average turbulent relative velocity, <3 II and b initial cell radius. 0 For the cloud radius, Zung obtained: Gtt 2 5 R(t) = Ro[1 + (53;) J . (A16) The cellular model yields the equation; 3 3 Dm c p (a -a (t) _ _ 2 (aT:T .. all _0 3 T ), (A17) pw 2 (a(t)b (t)-a (t)) 9.10: rrm I which Zung solved numerically using a combination of the second-order Runge-Kutta and the Predictor-Corrector methods. 129 A portion of the reported results computed from the above equation is given in Table A6. TABLE A6.--A comparison of cloud lifetimes for the cellular model for fixed and turbulent clouds for various degrees of dilution with a =10 u, R =1 m, T=25°C, vt=l cm/sec, and Maxwellian evap8ration agsumed. Cellular Model 35135.? _ gamma (.1, 2 b/a Fixed Turbulent F/T 18 440 .140 3.14 x 103 30 152 .130 1.18 x 103 40 .165 .115 1.43 50 .112 .110 1.02 lfrom Zung (l956,a), p. 2069, Table III. 2from Zung (1968), p. 5185, Table V. The column labeled F/T contains the ratio of cloud lifetimes for the fixed and the turbulent clouds. The less dilute clouds differ by an order of 103. The more dilute cloud ratios approach unity since the turbulence has a minimal dilution effect on an already diluted system. The continuum model now contains a diffusion equation that is a function of time, (see the note following Equation (A12).) 130 D'(t) = D b3(t)/ao3 (A18) Solution of the equation 3c _ , 2 —t--D(t)VC gave for steady-state conditions, QC = 4nR(t)D'(t)m2(cS-cm),(g/sec), (A19) and for non steady-state conditions, _ % 0NS - octl + R(t)/(nTC) J. (g/sec). (A20) where; TC = (D/ao 3){0. 25t(b§ +0. 25V it 2)3/2 + 23.16: uh: +0. 25v :13)!“ 4 % +(3 bO/4vt )£n[0. 5vt t+(b: +0. 25v2 tt 2) J} A portion of the results reported for the stationary continuum model using the same conditions as the cellular model above are given in Table A7. The below results cannot be compared directly with those given for the cellular model in Table A6 since the latter is for cloud lifetimes, not evaporation rates. To convert the below results to cloud lifetimes, Zung (1968, p. 5186) suggests using the results for the fixed cloud lifetimes reported in Zung (l967,b) and the ratio 131 TABLE A7.--A comparison of evaporation rates for the continuum model for fixed and turbulent clouds for various degrees of dilution with =10 u, 30:1 m, T=25°C, vt=l cm/sec, and Maxwellian evaporation assumed. Continuum Model Degree of dilution Evaporation rates, (g/sec) b/a Fixed Turbulent 18 4.20 x 10I 6.33 x 1010 30 1.94 x 102 3.80 x 1010 40 4.61 x 102 2.85 x 1010 50 9.01 x 102 2.28 x 1010 of the above evaporation rates. This does not give a comparable result, however. The values above would indicate 9 less for the turbulent a cloud lifetime of the order of 10- case versus the fixed case which is inconsistent with the cellular model's results of 0(103). The values reported above were "averages over the entire lifetimes of the cloud--". No indication of the actual times used was reported. To check this, Equations (A15) and (A16) were rewritten to be: b(t) bo (A21) _ v t b(t)=[b§+(—§-—)2]k and R(t)=Ro By substituting equations (A18) and (A21) into equation (A19) one obtains: 132 411DRDm2 4 QC= —————§— b (t)(cS-cm). (A22) bbao If one solves Equation (A22) for b(t) and utilizes the results with the expression for B(t), a value of t can be found. For the following conditions: a0 = 100 = 10'3cm, T=25°c, 100 cm, v = l cm/sec, 6" II t coo = O (the most severe condition, i.e. maximum evaporation), mzcs = 2.33 x 10‘5 g/cm3§T=25°c, (List(l966)), D = 0.249 cmZ/sec, (Zung(l967,a)), b/a = 18, and Qc = 6.333 x 1010 g/sec, (Table A7), t was found to be 7.06 sec, clearly outside the lifetime of the cloud as indicated by the cellular model. Thexariation of QC with t for the above conditions is given in Table A8. TABLE A8.--A comparison of the evaporation rates for various average cloud lifetimes using Equation (A22) with _a5;0=103 u, Ro =1 m, T=250C, vt= l cm/sec, 2cw= —0, m c =2. 33 x 10 g/Cm3 , D=O. 249 cm 2/sec and g/a=18. t, (sec) .Qc' (g/sec) 0 42.7 .01 49.1 .1 3.23 x 103 1.0 2.55 x 107 10.0 2.55 x 1011 133 For a cloud lifetime of the order of 0.1 sec as indicated by the cellular model the average evaporation rate during its lifetime would be of the order of 103 g/sec not 1010 g/sec as reported. The mass of the cloud can be found by: volume/cloud)(l drop)(mass) volume/cell cell drop ' A cloud mass = ( (A23) By substitution of the formulae for the respective spherical volumes, the cloud mass becomes: 4n 3a03 cloud mass = —§ RD(E;) ow (A24) where; p = water density, (g/cm3). W For the above conditions the cloud mass becomes 7.23 x 102 g/cloud. Thus the "average" evaporation rate of()(103) would yield a cloud lifetime of (3(10-1) as in the purely cellular model. Clearly the evaporation rates reported for the continuum model with turbulent diffusion are incon- sistent with the cellular model and the conclusions derived therefrom suspect. The evaporation rates in the expanding cloud are_greater than those in a fixed cloud by a factor more nearly 103 rather than the reported value of 107 or 108 134 None of the above models have been experimentally verified and can be therefore used only as a guide to the evaporation process. The use of the Maxwell formulation in the diffusing cloud would not be expected to be accurate since Zung (1967,b) pointed out that it was valid only for droplets with ao>1000. The models do serve the purpose of demonstrating the complexity and incomplete solution of the problem. APPENDIX B FIELD DATA AND PREDICTION MODEL RESULTS 135 TABLE Bl.--30 min mean wind speed data, m/sec. Mast l Mast 3 Height Mean Standard Mean Standard . m Deviation Deviation 8 4.88 1.34 4 4.28 1.24 4.83 1.19 2 4.53 1.20 4.56 1.08 l 4.45 1.08 4.32 1.03 TABLE B2.--30 min mean initial conditions (mast l). Moisture Concentration Ratio Humidity Height Dry Bulb Wet Bulb m o o Mixing Relative C C 9/m3 g/kg % 8 22.3 15.7 9.81 8.56 50.7 6 22.2 15.0 8.94 7.79 46.2 4 21.7 16.9 11.66 10.18 62.3 2 21.5 14.6 8.78 7.63 47.4 1 22.2 18.7 13,35 12.17 71.8 136 137 m.mn mp.HH m4.mH H.4H m.H~ H m.wm Hm.m 4m.oH m.mH H.H~ m 4.4m m~.m Ho.oH o.4H m.H~ 4 4.44 4m.n 04.4 4.4H o.Hm m 4.44 04.4 44.4 4.4H R.Hm m m m.Hm am.m mm.HH m.wH m.H~ H m.m4 mm.p m4.m m.4H 4.Hm m m.wm Hm.m mm.oH o.mH 4.H~ 4 H.44 m~.n 4m.m 4.4H m.H~ m H.H4 mn.m 44.4 m.mH m.Hm m 4 m.mm HH.HH -.~H m.~H m.H~ H H.om mm.m HH.HH 4.4H m.HN m m.Hm 44.m 44.4 4.mH m.Hm 4 «.44 km.h 44.4 4.4H H.~m 4 m.M4 -.4 om.m 4.4H o.~m m m m.mm om.HH 4~.mH H.mH m.Hm H m.wm eh.m mo.oH m.mH H.mm m m.~m mh.m mo.oH m.mH H.m~ 4 «.44 m4.n mm.m 4.4H m.wm m 0.44 no.4 44.x m.4H o.m~ m m w mx\m mE\m muHuHssm oHumm coHumuucmocoo oo oo 8 .oz 0>H08H0m monHs musumHos anm 003 anm and panmm umms .mO5HM> 0H5HMflOE 680 OHSHMHOQEOD. G608 CHE omllomm mafia... 138 TABLE B4.--A summary of the assigned constants used in the theoretical model. L = -l.66 m u* = 0.28 m/sec z = 0.0016 m H = 0.13 ly/min T* = -o.869 °c T = 27.8 °c x* = -l.66 g/m3 x0 = 21.7 g/m3 0 = 1.15 kg/m3 L = 585.0 cal/g v cp = 0.242 cal/g-OC Ax = 1.0 m Aw = 0.1 SPRAY = 8 g/m-sec Cl = 0.02 sec”1 22 = 0.4 m 2M8 = 0.6 m 2 = 0.8 m 139 TABLE B5.--Values of the constants used in the Prediction model. R.H.. = 0.25 p = 1.12 kg/m3 Lv = 580.0 cal/g cp = 0.250 cal/g-OC C1 = 0.02 /sec P = 1000 mb 20 = 0.01 m 22 = 0.4 m zMS = 0.6 m 23 = 0.8 m Aw = 0.1 Ax = 1.0 m 140 mo.mI mh.mm Hm.oI mm.04 oa.o No.o mm.o o.mH mo.~HI o.a I 04 om.hI m4.o4 ma.oI nn.mm oa.o ac.o mm.o o.Hm 45.0HI o.H I mm 4n.4I m~.m~ No.0I mo.om mo.o oo.ov mm.o o.~4 mm.m I o.oHI om om mm.mI mh.mm Hm.oI mm.ov oH.o No.0 mmuo o.hH 44.HHI o.H I 04 om.nI m4.o4 mH.OI nn.mm oH.o Ho.o m4.o o.mH hm.oaI o.a I mm m4.mI -.om ma.0I mm.om oa.o Ho.o mm.o o.o~ 4o.m I o.H I om ma mo.mI mh.mm Hm.oI mm.o4 oH.o No.o mH.o o.ma nm.m I o.H I O4 om.nI m4.o4 mH.¢I hh.mm oa.o Ho.o mH.o o.mH ~4.m I o.H I mm m4.mI N~.om mH.on mm.om oa.o Ho.o m~.o c.5H 54.x I o.H I om oa H oom\E o.H .5 mom >~.mI Hm.Hm Ho.OI no.04 4o.o oo.ov m~.H o.H4 mm.4HI o.oaI 04 h~.mI om.mm Ho.oI mo.mm 40.0 oo.ov mm.H o.m4 m4.mHI o.oaI mm ~5.4I NH.m~ oo.OIv No.om 40.o oo.ov mm.a o.mm hN.~HI o.oaI om om m~.mI mm.am mo.0I NN.O4 mo.o oo.ov mm.o o.mH mm.maI o.H I O4 nm.mI om.mm Ho.OI No.mm 4o.o oo.ov mm.o o.mm m.NHI o.oHI mm mh.4I NH.mm oo.OIA No.om 4o.o oo.ov mo.H o.m4 m.HHI o.oHI om ma mm.mI mm.Hm mo.0I mm.ov mo.o oo.ov mm.o o.ma 4.NHI o.H I o4 mm.oI ha.mm mo.oI mH.mm mo.o oo.ov mH.o o.mH m.HHI o.H I mm m~.mI nm.m~ 40.0I ma.om no.0 oo.ov mh.o o.mH o.oaI o.H I on ca owm\E m.o u as you m5\m mE\m 00 oo omm\E .nHE\>H E E 00 E 00 ommIE\m «x ox 4e as 4s m xmsu xmex x6894 H 49 mmumm .mflflgmmH HOUOE GOHHUHUOHmllomm Waugh. 141 04.0 I 00.00 0m.oI 00.44 00.0 00.0 00.0 0.04 00.4I 0.04I 04 40.0 I mm.~4 4m.oI 00.00 00.0 00.0 00.0 0.40 00.0I 0.04I 00 00.0 I 44.40 0~.0I 40.40 00.0 00.0 00.0 0.00 00.0I 0.04I om om 04.0 I 00.00 0m.oI 00.44 00.0 00.0 04.0 0.04 0m.mI 0.04I 04 40.0 I mm.~4 4m.oI 00.00 00.0 00.0 04.0 0.04 00.0I 0.04I 00 N~.0 I 44.40 0m.oI 40.40 00.0 00.0 04.0 0.00 00.0I 0.04I 00 04 04.0 I 00.00 0m.oI 00.44 00.0 00.0 04.0 0.44 00.NI 0.04I 04 40.0 I mm.~4 4m.oI 00.00 00.0 00.0 04.0 0.04 N4.NI 0.04I mm N~.0 I 44.40 0m.oI 40.40 00.0 00.0 04.0 0.04 04.~I 0.04I on 04 omm\fi 0.4 .9 Mom 00.44I 0o.m0 40.0I 40.04 04.0 04.0 04.0 0.04 40.0I 0.4 I 04 40.0 I 00.04 40.0I 00.00 04.0 44.0 04.0 0.00 40.0I 0.4 I mm 00.0 I 00.00 m0.oI m0.mm 04.0 00.0 04.0 0.4m N4.0I 0.4 I om om 00.44I 00.00 40.0I 40.04 04.0 «4.0 04.0 0.04 04.0I 0.4 I 04 40.0 I 00.04 40.0I 00.00 04.0 44.0 04.0 0.04 00.0I 0.4 I 00 00.0 I 00.00 m0.oI m0.~m 04.0 00.0 04.0 0.00 00.0I 0.4 I om 04 00.0 I 40.00 04.0I 04.04 04.0 40.0 00.0 0.04 00.4I 0.04I 04 40.0 I 00.04 40.0I 00.00 04.0 44.0 04.0 0.04 00.4I 0.4 I 00 00.0 I 00.00 m0.oI m0.Nm 04.0 00.0 04.0 0.04 00.4I 0.4 I om o4 owm\E 0.~ .5.Hom mE\0 0E.0 oo 00 omm\E GHE\>4 E E 00 E 00 ommIE\0 8x 0x 4B 09 «s m x880 xmex x0890 4 49 mmumm .vmsaHucoouu.0m mHmme APPENDIX C FORTRAN IV PROGRAM FOR THEORETICAL MODEL ‘ 142 The following FORTRAN IV program is for the theoretical solution of the environmental modification problem with Swinbank initial conditions. The total program is composed of the main program DIFSN, subroutines CONST, BEGIN, TOTAPE, I C, B C, SATURAT, CORRECT, ODDEVEN, OTPT, SOURCE, COEFF, SOLVE and INTGRL, and functions ALPHA, FRCTN, and FSUM. Subroutine CONST initializes all the constants that must be set to start the solution. The integer node numbers are calculated for several points of interest, eg. IZl for Z 122 for IZ for z , etc. z 2' max Subroutine BEGIN computes the nodal and bi-nodal ll midpoint values for: wind speed, U (J) and UM (J), wind speed gradient, DUDZ (J) and DUDZM (J), diffusivities, AK (J) and AKM (J), and diffusivities divided by wind speed, AKBU (J) and AKBUM (J). Subroutine TOTAPE writes the array 0 (J) onto magnetic tape for later access for graphing purposes. The array 0 (J) acts as a buffer for intermediate handling of other arrays or portions of arrays. The first call to TOTAPE also writes the vertical node heights. All sub- sequent calls write only 0 (J) plus identification information. 143 144 Subroutine I C calculates the initial profiles for liquid water concentration, temperature, and water vapor concentration. Subroutine B C is an entry within I C and establishes the fixed boundary conditions. Subroutine SATURAT computes the saturated vapor pressure, saturated mixing ratio, and saturated water vapor concentration. Subroutine CORRECT provides the continuity error reapportionment. Subroutine ODDEVEN determines if the number of intervals between two specified nodes is ood or even because the Simpson's integration procedure has been modified to include an odd number of spaces. Subroutine OTPT is used to write model profiles. Any single dimension array can be written with the appropriate call to OTPT. Subroutine SOURCE calculates the modification source term for all three solutions. Some of the intermediate parameters are also computed. Subroutine COEFF calculates the remaining intermediate parameters and the solution coefficients. Subroutine SOLVE physically solves the diffusion equations. I Subroutine INTGRL uses Simpson's approximation to integrate any array between specified nodal limits. If the limits are out of sequence an error diagnostic is printed. 145 Function FRCTN calculates the weight function for the initial liquid water concentration profile. Function SUM computes the sum of the elements in any one dimensional array. 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