_——— w—vvw A STUDY OF A LEAF MODEL USED IN THE SPRENKLING METHOD OF FREEZE PROTECTION OF PLANTS Thesis {or The Degree of DB. D. MICHEGAN STATE UNIVERSETY Jerry Lee Chesness 1965 ABSTRACT A STUDY OF A LEAF MODEL USED IN THE SPRINKLING METHOD OF FREEZE PROTECTION OF PLANTS by Jerry Lee Chesness The need for an effective method of preventing agricultural crops from freezing has led to extensive research into the sprinkling method of freeze protection. Some of this research has been directed toward the derivation of a sprinkling rate prediction equation based on heat transfer theory applied to models of various plant structural parts. In this investigation a theoretical and experimental analysis was made of the commonly accepted leaf model, a thin flat plate, under simulated field conditions . Convective and mass-transfer heat losses from a flat plate were measured in a wind tunnel with controlled air temperatures between 15 and 32 F, air velocities between 50 and 900 ft/min, and relative humidity between 35 and 75 percent. Radiation losses were not considered in this study. Free stream tur- bulence in the wind tunnel was held to a maximum of 4. 0 percent. if Jerry Lee Chesness Phase One of the study was concerned with determining the convective heat loss equation for a thin uniformly heated flat plate with laminar air flow parallel to its surface. The plate was constructed from two 6 in° x 6 in. x .051 in. pieces of aluminum with thermocouples imbeddedin the surface and a heating element sandwiched between them, It was found that the heat loss can be predicted from an equation differing only slightly from the theoreti- cal equation for a plate with a continuously varying surface temperature. In Phase Two the rate ofheat loss due to the mass transfer of water vapor from a stationary water surface into a laminar air stream was determined. Measurements were made of the actual quantity of water removed per unit time from the cloth covered surface of an8 in. x 8 in. x 3/4 in. heated (to prevent the water surface from freezing) insulated plastic tray. Phase Three involved the continuous sprinkling of a 4 in. x 4 in. x 1/8 in. leaf model with laminar air flow occurring over both surfaces (angle of incidence = 0) . The leaf model was fitted with thermocouples to provide a measure of the local surface temperature on the underside, the mean water film temperature on the upper surface, and the mean temperature of the water leaving the surface. The sprinkling rate and mean water drop temperature (at the leaf model surface) was measured by an insulated catchment tray. The equation for the average film heat transfer coefficient Jerry Lee Chesness was derived for the condition of a flowing water film on the leaf model surface. Using this coefficient in the convective and mass-transfer heat loss equations for the upper plate surface, combined with the convective heat loss equation from Phase One for the underside and the heat gained from the sensible heat of the sprinkled water resulted in a theoreticalvalue for the water application rate which correlated with the measured value. The theoretical water application rate can therefore be predicted if the air velocity, air temperature, relative humidity, local surface (un-wetted) temperature, and mean water film temperature are known. 'or Profes sor Approved Department Chairman Date (ti/04 [9.6; fl A STUDY OF A LEAF MODEL USED IN THE SPRINKLING METHOD OF FREEZE PROTECTION OF PLANTS BY Jerry Lee Chesness A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOC TOR OF PHII. OSOPHY Department of Agricultural Engineering 1965 ! 5 T 3 l I ‘E »\. r. 1 it“ \ 22:“ l . _. \x‘: { ACKNOWLEDGMENTS This study was carried out under the guidance and supervision of E. H. Kidder (Agricultural Engineering) whose interest and encouragement has been most gratifying. I am deeply indebted to Dr. A. M. Dhanak (Mechanical Engineering) for his brilliant council and guidance in the application of heat transfer theory to this study. To Dr. C. E. Cutts (Civil Engineering), and Dr. F. H. Buelow (Agricultural Engineering) the author expresses his gratitude for their time, professional interests, and constructive criticisms. This dissertation is dedicated to my wife, Betty, whose unfailing support and endless faith enabled me to attain this goal. .l.fl:a.1‘,- . . TAB LE OF CONTENTS LIST OF FIGURES LIST OF TABLES NOMENCLATURE I. INTRODUCTION . 1.10bjectives.,.......... II. REVIEW OF LITERATURE III. ANALYSIS General Background Convection . . l ,2 .3 Mass—Transfer , , . . . .4 Sprinkling the Leaf Model wwww IV. EXPERIMENTAL EQUIPMENT 4. 1 Wind Tunnel Temperature Velocity Turbulence . . . Relative Humidity . rF-rpt-P-rb r—v—t—v—a I-FU’Nb—I 4. 2 Heated Flat Plate 4.3 Mass-Transfer 4.4 Leaf Model . V. EXPERIMENTAL PROCEDURE 5.1 Convection . . . . . . . . 5.2 Mass-Transfer . . . . . 5. 3 Sprinkling the Leaf Model iii Page viii 12 21 2.3 Z7 Z7 29 31 33 35 40 44 50 50 54 VI. DISCUSSION OF RESULTS 6.1 Convection . . 6.2 Mass-Transfer. 6. 3 Sprinkling the Leaf Model . VII. C ONC LUSIONS SUGGESTIONS FOR FUTURE STUDIES . APPENDICES REFERENCES Page 57 57 61 65 81 84 85 115 Figure LIST OF FIGURES Flow boundary layer and thermal boundary layer on a flat plate heated over its entire surface Approximation of continuously varying wall temperature by straight line segments Wind tunnel layout Inlet section of wind tunnel Temperature recording potentiometers Hot-wire anemometer, root-mean-square voltmeter, and oscilloscope Relative humidity indicator and sensors . Dimensional view of heated flat plate Cutaway view of heated flat plate . Heated flat plate located in test area Equipment set-up for measuring mass-transfer Dimensional view of mass-transfer tray . Cutaway view of mass-transfer tray . . Leaf model in test area of wind tunnel . Tray for measuring water application rate and drop temperature . Page 2 l 28 30 30 34 34 36 38 39 39 4 l 42 45 45 Figure 16 17 18 19 20 21 22 23 24 25 26 27 Wind tunnel with sprinkling equipment attached Sprinkling tower and water control assembly Comparison of measured and calculated average film heat-transfer coefficients in laminar flow over a flat plate Dimensionless mass-transfer coefficient for a flat surface evaporating water into a laminar air flow A comparison of the experimental and theoretical water application rates Average film heat-transfer coefficients in laminar flow over a flat plate A comparison of the experimental and theoretical water application rates with the theoretical water application rate calculated for a flat plate surface covered by a moving water film Dimensionless mass-transfer coefficient for a flat surface evaporating water into a laminar air flow Dimensionless water application rate num- ber for the leaf model in laminar air flow parallel to its surface . Empirical correlation curve for determining ATp' when the air, water film, and local plate temperatures are known Flow boundary layer over a water film on a flat plate Water film on a flat plate vi Page 47 47 60 64 67 76 77 78 79 80 85 91 Figure 28 29 30 31 Page Approximation of continuously varying plate surface temperature by straight line segments for convection test number 8 . . 95 Calibration curve for determining the quantity of water removed from the mass-transfer tray during a test . . . . . . . . . . . 100 Approximation of continuously varying plate surface temperature by straight line seg- ments for sprinkling test 22 , . . . . . . 104 The effect of air velocity on the quantity of water collected over a five minute period for sprinkling rate determinations (tests 21through26)............. 107 LIST OF TAB LES Table 1 Summary of convection tests 2 Summary of mass-transfer tests 3 Summary of sprinkling tests 4 Sprinkling data analyzed on the basis of the theore- tical relationships developed for a flowing water film on a flat plate viii Page 59 63 66 73 Appendix A. LIST OF APPENDICES Theoretical derivation of the average film heat- transfer coefficient over a flowing water film Sample calculations Propagation of errors for the theoretical water application rate equation . ix Page 85 93 110 D“ D“ :r‘| 33‘I NOMENC LAT URE specific heat, Btu/lbm F . . . . Z . diffusmn coeff1c1ent, ft /mm diameter of drops, in. sprinkling frequency, min average convective heat transfer coefficient for a flat plate with a constant surface temperature, Btu /min ft2 F average convective heat transfer coefficient for a flat plate with a continuously varying surface temperature, Btu/min ftz F average mass—transfer coefficient, ft /min latent heat of vaporization, Btu/lb ice load, 1b current to heating wire, milliamperes thermal conductivity, Btu/min ft F length of leaf, ft rate of mass transfer, lb /min m RH SE 2 vapor pressure of the air, lb /ft 2 vapor pressure at the water surface, lb /ft . , 2 total heat removed by convection, Btu/min ft heat removed by convection from a flat plate with a con- , 2 stant surface temperature, Btu/min ft heat removed by convection from a flat plate with a . . . 2 continuously varying surface temperature, Btu/min ft 2 heat removed by mass-transfer, Btu/min ft . . ‘ . 2 heat added through sprinkling, Btu/min ft measured heat loss from the flat plate, defined by equation ( 5.1. 4), Btu/lb universal gas constant, ft/OR radiation characteristics relative humidity, percent standard error of estimate for the y ordinate values absolute temperature of the water film, R local air temperature, F local un-wetted plate surface temperature, F free streamair temperature, F mean temperature of the drops striking the surface of the water film on the leaf model, F xi X, y,Z (TS - Ta) /2, mean temperature for evaluating the fluid properties over the water film, F (average plate surface temperature - Ta) /2, mean temperature for evaluating the fluid properties over the plate surface, F plate surface temperature defined by equation (6. 1. 1) , F local water film temperature, F mean water film temperature, F mean temperature of the water leaving the leaf model surface, F - T Td w velocity at any point in the hydrodynamic boundary layer, ft /min free stream velocity of the air, ft/min voltage to heating wire, volts measured water application rate, in./min theoretical water application rate, in. /min dimensionless water application rate number defined by equation (3 . 4. 4) rectangular coordinates hydrodynamic boundary layer thickness, ft Sc thermal boundary layer thickness , ft ratio of 6t/6 . . . . Z . kinematic Viscosuy, ft /min ratio of y/cSt 2 absolute viscosity, lbm/ft min mass density, lb /ft3 m D‘l W C.‘ m t“ Q Cp Ha 2 , thermal diffusivity, ft /min 9 Nusselt number . dimensionless mass-transfer number , Reynolds number , Prandtl numbe r IE) , Schmidt number (X . *3. LeWis number xiii I. INTRODUCTION Every year partial and in some cases total destruction of agricultural crops by freezing exacts a heavy economic toll from the farmers in this country and abroad. As the crop yields per acre increase along with their increased value the necessity of insuring these yields against loss by freezing increases. A number of methods for preventing freeze damage to agricultural crops have been tried, each meeting with varied degrees of success. The sprinkling of water on agricultural crops as a method of preventing freeze damage has been successfully applied for over 30 years. The success of this method has been somewhat limited with respect to the types of crops it can protect, and the nature and severity of the freeze conditions encountered. The research work leading to the adoption and utilization of the sprinkling method has been primarily of the field experimentation type, directed toward obtaining a broad working knowledge of the method. Precise theoreti- cal descriptions of the sprinkling process in the field have not been obtained owing to the almost inseparable manner in which the effects of plant and environment on the heat and mass transfer process are interwoven. If the limits and accurate application of the sprinkling lht att PE] wil. call: Sent. OVer Phas< method are to be obtained a concise description of the physical pro- cesses leading to a quantitive sprinkling rate prediction equation must be developed. This approach has, in recent years, been undertaken by researchers who have applied heat transfer theory to geometric models representing various plant structural parts. The prediction equations arrived at through these analyses have met with a limited degree of success when applied to the actual sprinkling of plants in the laboratory and field. The limited degree of success may be attributed to the fact that the prediction equations have not been ex- perimentally verified for the models postulated. It is in this area of the sprinkling method of freeze protection research that this study will be conducted. 1. 1 Objectives The overall objective of this study is to examine theoreti- cally and experimentally a leaf model under freeze conditions repre- senting the actual field conditions encountered by the plant. This overall objective can be divided into three separate objectives or phases of study: 1. The convective heat losses from a heated flat plate under freezing conditions will be measured. 2. The mass-transfer from a stationary free water surface subjected to freezing conditions will be measured. 3. The sprinkling rate for a leaf model subjected to freezing conditions will be measured. (11 on 5p Sh] the R01 tior bud size Spra and ling delei freqL tigat in th II. REVIEW OF LITERATURE The first systematic investigations of the sprinkling method of freeze protection for agricultural crops was carried out in 1938 by Kessler and Kaempfert (1949). Investigations were con- ducted by Kidder and Davis ( 1956) and Braud and Hawthorne ( 1965) on the sprinkling rate and frequency required to protect strawberries. Sprinkling rates for the protection of blueberries were determined by Shultz and Parks (1957) in California. Rogers (1952) investigated the physical tolerance of plant tissues to freezing temperatures. Rogers fl“ ( 1954) conducted research in England on the applica- tion of the sprinkling method of freeze protection to deciduous fruit buds. Von Pogrell and Kidder ( 1959) investigated the effect of drop size and distribution, air temperatures, application rates, and spraying frequencies on the frost protection of plant leaves. Gerber and Harrison ( 1963) reported on the results of employing the sprink- ling method of freeze protection on Citrus in Florida. Wheaton (1959) determined experimentally the effect of application rate and spraying frequency on the freeze protection of bean leaves. The research work carried out by these and other inves- tigators has led to an overall understanding of the variables involved in the sprinkling method of freeze protection and their inter- 3 po: aPP tak The (2) Cylh theo Pred tran theh relationships. Strawberries and other low growing crops can be pro — tected from temperatures as low as 20 F (produced by radiation freezes) with sprinkling rates in the neighborhood of 0.1 in. /hr. As the quantity of heat removed from the plant surface by radiation, con— vection, and evaporation increases the sprinkling rate and spraying frequency must be increased in order to prevent freeze damage. Some approximate limits have been established for this method of freeze protection based on the adverse effects of soil saturation accompanying prolonged sprinkling, inability to supply sufficient heat for severe wind borne freezes, and damage to the structural portions of the plant produced by heavy ice loads. A few researchers have attempted a purely theoretical approach to procuring a sprinkling rate prediction equation which takes into account the many variables involved. Geometrical models have been proposed to represent various structural parts of a plant. The three basic models are: ( 1) a flat plate to represent a leaf; (2) a sphere to represent a bud or individual fruit; and (3) a cylinder to represent a shoot or branch. Applying heat transfer theory to these models has resulted in the postulation of theoretical prediction equations for the sprinkling rate. Nieman ( 1958) described the various phases of heat transfer, radiation, convection, and evaporation and how they affect the heat balance of the plant. “~"’-"-"' 1 Beahm ( 1959) calculatEd the theoretical water application rates required to protect various size flat plates, cylinders, and spheres for various convective, evaporative, and radiative heat losses. The author did not attempt to verify these values for the model or the portion of the plant represented by the model. Beahm arrives at the following prediction equation for a flat plate with an angle of incidence equal to zero, laminar air flow, constant surface temperature, constant radiative heat loss, constant plate length, evaporation from top surface only, 100 percent of sprinkled water freezing on plate surface, and 100 percent relative humidity; 28 + h[293l.5-Ta +102(P_S-Pa)] w: 780 In this equation, w = water application rate, in. /hr . . . . 2 h = film coeffic1ent from leaf to air, Btu/hr ft F Ta: free stream air temperature 2 PS 2 saturated vapor pressure at the plate surface, lb/ft Pa = vapor pressure at the plate surface, lb /ft2 Businger ( 1963) made a theoretical study of the heat transfer process taking place on a flat plate and a sphere. Applying the energy balance on the flat plate results in the following sprinkling rate prediction equation fro Com ) ~ 2 = o< +h - W L (hr+h e) (Tm T 1 where w = the water application rate ct = ratio of the actual quantity of water required to the theoretical quantity required ((X > 1) L,1 = latent heat of fusion of water hr = coefficient of radiative heat transfer h = coefficient of convective heat transfer he = coefficient of evaporative heat transfer Tm = minimum tolerable leaf temperature T 1 = leaf temperature The author assumes that both the upper and lower leaf surfaces are covered with a continuous water film and that both surfaces have a con- stant temperature. Comparing experimental values with theoretical values calculated from the above equation results in a value of 1.5 for at . This type of correlation offers no confirmation as to the validity of the theoretical prediction equation since any unaccounted or inaccurately accounted for heat losses will remain hidden in the value of o< . Gates and Benedict ( 1963) observed the free convection from leaves in still air my means of schlieren photographs of broad— leaved trees . A quantitative measure of the rate at which heat was convected away from the leaf was obtained by photographing the size of the convection plume, measuring its rate of flow by means of movie photography, and measuring the temperature of the plume with a fine thermocouple. Their observations of free convection from broad-leaved plants confirmed the values predicted using heat transfer theory for heated plates. III . ANA L YS IS 3. 1 General Background A plant located in its natural environment gains or loses heat from or to its surroundings by three separate phySical processes: ( 1) conduction and convection; (2 ) evaporation and transpiration; and (3) radiation exchange. Conduction is a process by which heat flows from a region of higher temperature to a region of lower temperature within a medium or between different mediums in direct physical contact. For plants this medium can be the solid material forming its physical structure, the water films forming on its surface, and the air surrounding it. Radiation is a process by which heat flows from a high temperature body to a lower temperature body when they are separated in space. Although the term "radiation" is generally applied to all kinds of electromagnetic-wave phenomena, our concern in heat trans- fer is only with those phenomena which are the result of temperature. This energy is called ”radiant heat. " Convection is a process of energy transport by the com- bined action of heat conduction, energy storage, and mixing motion. Convection is the mechanism of energy transfer betWeen the solid surfaces of the plant and the surrounding air mantle. 8 I'( ini Th (fr toh for tur air c0m Watt quér shou Under freezing conditions the plant loses heat to its sur- roundings by these heat transfer processes singularly or in combina— tion. The physiological make up of each plant species dictates what minimum temperature each segment of the plant structure can with- stand before destruction of the cells occurs. In general, however, investigations by Beahm ( 1959 ) , Wheaton (1959 ) , Rogers (1952 ) , and others indicate that the minimum temperature is below 32 F (around 31 F). In the sprinkling method of frost control the heat lost from the plant to its surroundings is replaced by the heat in the water. This heat is in the form of sensible heat and the heat of solidification (freezing) . Since the minimum temperature which the plant can tolerate is approximately 31 F, no freeze damage occurs when ice forms on the plant surface if the ice coat is maintained at a tempera- ture above 31- F. It should be pointed out that when the plant surface is covered with a water film from sprinkling and exposed to a moving air mass an additional heat loss phenomenon occurs. Under these conditions the mass-transfer (evaporation) of water vapor from the water surface into the moving air stream takes place with a subse- quent heat loss termed "latent heat of vaporization. ” The effects of plant and environment on the heat transfer in which both are involved are interwoven to such an extent that they should be described together. However, in order to determine the he WE fol the face basic heat transfer processes involved and give a description of these processes, it is necessary to separate in as much as possible these interwoven effects. To ultimately meet this objective it is necessary to divide the structural parts of the plant into shapes which are geometrically similar and for which a model can be postulated. This investigation will be concerned only with that struc— tural portion of the plant called the leaves. Consider a single leaf exposed to the freezing conditions which can occur in its natural environment. If this leaf is to be kept from freezing the amount of heat it loses to its surroundings must be replaced by the sprinkled water. Thus it is necessary to derive a prediction equation for the required water application rate (termed w) . The development that follows will be directed toward developing a prediction equation for the theoretical application rate. The rate at which water must be applied to the leaf sur- face may be postulated as a function of several independent variables. Ua - velocity of the air Ta - temperature of the air RH - relative humidity of the air f - frequency of water application d — drop size T — average temperature of the leaf surface t — time rat be ind , 9 - angle of incidence qr — rate of heat loss by radiation L - length of the leaf measured in the direction of air flow i - rate of ice formation AT - difference in the mean temperature of the water striking the leaf surface and leaving the leaf surface. Or this may be expressed as: w=w(U,T,RH, f, d,T,t, e,q,L, i, AT ) (3.1.1) a a S r W The following simplifying assumptions will be made. The drop size will be constant or at least restricted to a narrow range of values. The angle of incidence of the leaf surface will be held at zero (leaf surface parallel to the direction of air flow) . Radiant heat exchange is negligible (qr< 10 percent of qtotal) . The frequency of water application will be continuous. Only steady-state conditions will be considered. The ice load will be kept at zero by employing only the sensible heat of the water (admittedly not a practical approach from the field standpoint since this will necessitate very high application rates) . With these conditions imposed, the water application rate becomes a function of only six independent variables. w=w(U , T, T , Rh,AT , L) (3.1.2) a a s w In order to determine the relationship between these six independent variables a heat balance is made on the leaf surface: 12 Heat added through sprinkling = total heat removed by convection + heat removed by mass—transfer, or q =q +q (3.1.3) w ct m The quantity of heat added through sprinkling can be com- puted by the relation qw=wt(5.20) CpATw (3.1.4) where the constant 5.20 converts w from lbm/min to in. /min. To obtain prediction equations for convection and mass- transfer it is necessary to assume that the leaf can be represented geometrically by a thin flat plate. Gates and Benedict (1963 ), found. by the use of schlieren photography that the actual measured values for heat loss by free-convection from a broad leaf compared closely with those values predicted using heat transfer theory for thin heated plates . 3 . 2 Convection The rate of heat transfer from the plate surface to the moving air stream can be computed from the relation qc=hC (TS-Ta) (3.2.1) This equation has been used for many years even though it is a defi- nition of he rather than a phenomenological law of convection. The equation for he in the case of forced convection in laminar flow over a flat plate heated over its entire length and with a constant surface l3 temperature is given by Kreith (1958) as - _ k 1/2 hC — .664 L ReL Pr (3.2.2) or expressed in a dimensionless term called the Nusselt number, the relation becomes Nu = = .664 ReL Pr (3.2.3) Combining these equations the equation 1/2 k 1/3 q - .664 L ReL Pr (T -T) (3.2.4) 5 a represents the convective heat transfer prediction equation for heat loss from one side of the plate. All fluid property values in equation (3. 2. 4) and all sub- sequent heat transfer equations which are dependent on temperature must be introduced at a mean temperature. The mean temperature may be calculated from the equation Many investigators have observed that the leading edge of the leaf freezes first. On the basis of this it would be well to consider the equation for he for the case of a continuously varying plate surface temperature (or constant surface heat flux). The deri- vation of this equation follows: its dir Figure 1. ——Flow boundary layer and thermal boundary layer on a flat plate heated over its entire 'surface The calculations are confined to a steady state, two- dimensional problem, with constant -property fluid, and velocities which are sufficiently small so that temperature increases caused in the boundary layer by internal friction can be neglected. Since symmetry exists about the center line of the plate the derivation will be carried out for the upper plate surface. The hydrodynamic boundary layer and the temperature boundary layer begin at the leading edge of the plate since the plate is heated over its entire length. Both increase their thickness 6 and 6t in the direction of flow. Four boundary conditions can be stated. 1. aty=0 qcz-k% Fr Fr With these four conditions a polynomial with four functions will be used to express the temperature profile in the boundary layer. T=A+By+Cyz+Dy3 Using the four boundary conditions the coefficients A, B, C, and D can be determined. From boundary condition 4. —=2C+6Dy=0 butDy=0.'.C=0 From boundary condition 1. q Cl 8T 2 c 2 . _ _c —y.-B+3Dy-—k butDy—O..B—-k From boundary condition 3. q q 2 2 . ?=--k£+3Dy2=0 buty =6t ..D= C y 3k6 t From boundary condition 2. qc 6t qc 6t 16 The equation for the temperature is From Eckert and Drake (1959) the integral heat—flow equation of the boundary layer is 1 d 8T dxfo (Ta'T) udy‘“(ay)y=o (3.2.5) The integral in this equation can now be evaluated. 6t q 6 q V q Y3 2 flTa'T’ “W '3 It + i "6—? [uldy o o 3k6 , t q a U 6t 2 3 = t a __+ _>'_ __L_ k 3 5 3 o t 36 t 3 3Y l y_ 25 2 6 dy Now the ratios 4., = {St/6 where (St/6 is assumed to. be less than one and r) = y/E)t can be introduced. q 6 U c t a 2 1 3 (Ta—T) udy-—k— [-3 n--n] Since g was assumed to be less than one, the second term in the right hand expression is small compared to the first and can be neglected. Introducing the value of the integral into the heat-flow equation one obtains a C t C X —+ U C a Since the plate is heated over its entire length at x = 0 5 = 0 C = 0 1/3 t — 12" (3.2.6) 18 The integral solution given by Eckert and Drake (1959) for the hydrodynamic boundary layer thickness 6 is 6_4.64x Re 1/2 X also U Xp Re = a (.1 and a: _k_ 9 CP (3.2.7) Introducing these expressions into equation (3. 2. 6) and simplifying yields 1 (1.29) Prl/3 The heat flow from the plate per unit area is given by (3.2.8) (3.2.9) (3.2.10) l9 Equating these two expressions gives (,6 (3.2.11) (3.2.12) h ' = .834?) Re Pr (3.2.13) or expressed in terms of the dimensionless Nusselt number r-"| ‘ L c Nu: k (3.2.14) The prime designation on the coefficient of convection term will be used to distinguish it from the coefficient of convection term for the case of a flat plate with a constant surface temperature. Equation (3. 2. 13) indicates that the rate of heat removal from a plate whose surface temperature is continuously varying is 25. 6 percent greater than that obtained from a plate whose surface temperature is constant. The quantity of heat removed per unit time per unit area for a plate 20 with a continuously varying surface temperature is given by the re— lationship q ' = K' (T -T) (3.2.15) c p a The temperature Tp is obtained by plotting the temperature profile curve (local plate surface temperature versus distance from the leading edge), finding the area. under the curve, and obtaining the temperature which divides this area in half. An alternate solution [Eckert and Drake (1959)] for ob- taining the local heat flux from a flat plate with a continuously varying surface temperature involves integration of the equation q =j;x h(X.§) dtp(§) where dtP is interpreted as a succession of infinitesimally small tem- perature steps occurring at infinitesimally closely spaced locations d5 . In view of the fact that the evaluation of this equation is rather tedious an approximate equation offered by Eckert and Drake (1959) can be used. The approximate equation, which by actual test deviates only a few percent from the exact solution, for the total heat flux with laminar air flow is . = ‘ - - & qC hC Ato +_.969 (Atr1 Ato) .432 L n _ _ _ _ .1 [(2.1 1) Atn mo 2 gain] (3 2 6) Fi 21 where hc is the average coefficient of convection for a plate with a constant surface temperature defined by equation (3.2.2) . At and AL are explained in the following figure; Figure 2. --Approximation of continuously varying wall temperature by straight line segments. Letting ATP' equal the quantity within the outer brackets of equation (3.2.16) the equation can be written as 3. 3 Mass -Transfer Water vapor is transferred from any water film on the leaf surface to the air by the process of evaporation or mass-transfer. For every pound of water removed by this process 1070 Btu's of heat are required (for water at 32 F). To arrive at a prediction equation for this heat loss use is made of the similarity relations between an( 1’at: qui 22 heat and mass-(transfer. The following assumptions are made: 1. The fluid properties are approximately constant. 2. The temperature differences in the field are small when compared to the absolute temperature. 3. The pressure is approximately constant. When these assumptions are met then any of the equations for heat transfer in laminar flow also gives the solutio_n_ for a corresponding h L c mass—transfer problem if the Nusselt number h L is replaced by the dimensionless mass-transfer coefficient 1'3 and the Prandtl number (V/0<) by the Schmidt number (U/D) . The rate of mass transfer can then be given as 5| m =—ri (P mp) (3.3.1) 5 T s a pa where PS is the saturated vapor pressure at the water surface and Pa is the vapor pressure of the air. The validity of the similarity relations has been proven by many investigators . Hartnett and Eckert (1957) determined the temperature and concentration profiles in alaminar boundary layer on a flat plate. Their investigations show that the mass-fraction and the temperature profiles are similar when Pr = Sc or when the ratio Sc/Pr = O+1 1.9.: (”T . moon mmuooq moon. 838 .. emu zmmmom .., membc A mmz<> 3.2352”.me _ uNsaw 2058C $1411.21 .111»:IB.11 LII1111L1||1 05.2 1 ..w lllllllll n .ll: 1' == 2. AIHW. AH”. 29 (Figure 4) . Located in the tunnel entrance was a set of 2 in. di- ameter straightening vanes 6 in. in length. Following this were three screen grids, a second set of straightening vanes 3/16 in. in diameter by 2 1/2 in. in length (constructed from plastic straws), a fourth screen grid, and finally two cotton gauze filters. The top of the tunnel was constructed so that it could be removed in two separate sections to facilitate placement of the straightening vanes, grids, and filters . The cold air supply was obtained from a 650 cu ft cold storage box with an approximate net cooling capacity of 12, 000 Btu/hr. The cold storage refrigeration unit could be set to maintain a desired temperature within i 2.0 degrees Fahrenheit. The air was brought to the fan intake via the velocity control box through a 10 in. diameter metal duct wrapped with a 4 in. thick batting of fiberglass insulation. The cold air leaving the tunnel was returned to the cold storage box by a similarly constructed duct system. 4.1.1 Temperature The air temperature in the test area of the tunnel was measured with two thermocouples located 1 ft upstream from the test area and 5 1/2 in. above the tunnel floor (centered vertically) . The thermocouples were made of 30 gage copper-constantan wire insulated with enamel and glass. These thermocouples have an accuracy of approximately :1: 3/4 percent of the standard emf temperature 3O 1—_ . ,. , .1 1‘. { ./, . 1 I ‘ ‘, ,1. .- 1 -:.'. 4 I. ‘. - '\ .1‘ I y I . - , I I. . -‘ .J a Q ' . . \ o o u o - o o Figure 5. --Temperature recording potentiometers. 31 calibration. A Honeywell two pen strip chart recorder (Figure 5) was used to determine the temperature indicated by the emf output of the thermocouples. The temperature range of the recorder was from —20 to +60 degrees Fahrenheit. It was possible to read the chart scale directly to 1.0 degree and estimate it to 0.1 of a degree. Full scale calibrated accuracy was :1: . 25 percent of instrument span. Although this recorder was calibrated at the factory a check at the 32 degree F reading was made by immersing a thermocouple junction in an agitated ice—water mixture. Each recording pen was connected to a six point manual switch thereby allowing the temperatures sensed by twelve different thermocouples to be measured and recorded. 4.1. 2 Velocity A forward-curved-blade centrifugal fan driven by a single phase 5 hp electric motor was used to provide the required air velocity. The fan housing was wrapped with fiberglass insulation to reduce heat loss. The outlet section of the fan was connected to the inlet section of the tunnel by means of a tapered plywood conduit insulated on the inside by 2 in. of Styrofoam. The connection between the fan outlet and the tapered conduit Was made with a flexible piece of canvas. This reduced the amount of vibration transmitted from the fan to the tunnel. l) ( willfihh..1 Illm»)1111h .1. . k111111.1,1 R11M 1. .11. 11.). . 32 Velocity control was made possible by a sliding door arrangement located in the intake duct. Movement of the door simply increased or decreased the effective size of the fan intake opening thereby increasing or decreasing the air velocity. The time—averaged velocity of air movement in the tunnel was measured with a constant current hot-wire anemometer (Figure 6); model HWB ser. no. 216 by Flow Corporation. The hot-wire probe was a standard 24 in. Flow Corporation probe fitted with a tungsten filament 0.0625 in. long and 0.0005 in. in diameter. The probe sensing element (filament) was located approximately two feet up- stream from the test area and in the center of the tunnel cross section. The hot-wire anemometer was calibrated against a pitot tube installed in the tunnel. The pressure head difference from the pitot tube was sensed by an inclined manometer filled with a fluid of . 797 specific gravity. The scale on the manometer could be read to .005 in. For the pitot tube used, the velocity was determined by the equation 1 Ua = 66.75 h /2 (4.1.2.1) where h is the manometer reading in in. and Ua the velocity in ft/sec. Velocity measurements with the hot—wire anemometer were considered to have an accuracy of t 2.0 percent. 33 4. l. 3 Turbulence The degree of free stream turbulence was investigated by using the hot~wire anemometer described in section 4.1. 2. The signal from the hot—wire anemometer amplifier was fed through a 7KC low pass filter to a true root —mean-square voltmeter (Figure 6); Model No. 320 by Ballantine Laboratories. An oscilloscope was used to obtain the correct square-wave compensation frequency setting for the hot-wire anemometer amplifier. Detailed operating instruc- tions for obtaining the average velocity and degree of turbulence are given in Flow Corporation Bulletin No. 37B. 4.1. 4 Relative humidity The humidity in the closed air system for the tunnel was not controlled. Since a measure of the relative humidity at any time was obtainable it was felt that the difficulties (air temperatures below freezing) and cost accompanying the installation of a humidity control system could not be justified in this study. The relative humidity of the air in the wind tunnel was determined with a Honeywell Model w6llA Portable Relative Humidity Indicator (Figure 7) . Seven lithium chloride humidity sensors were available as plug-in components for the probe assembly, each de- signed for a specific RH range. By a simple change of sensing unit and indicator scale plate relative humidities ranging from 2 to 100 Figure 6. -—Hot —wire anemometer, root-mean—square voltmeter, and oscillosco e. Figure 7. --Relative humidity indicator and sensors. 35 percent could be measured. The over-all instrument accuracy was :1: 3 RH percent. The probe to which the sensors were attached was located just in front of the straightening vanes at the entrance to the tunnel. The access door at this location facilitated the changing of the sensors . 4. 2 Heated Flat Plate The evaluation of the convective (sensible) heat loss under the simulated air temperatures and velocities experienced in the field was made using a heated flat plate. The heated plate used in this study was constructed from two pieces of aluminum each. having a thickness of .051 in. The top section of the 6 in. x 6 in. plate (Figure 8) was provided with .35 in. x .35 in. tabs for mounting on the test stand. The inside surface of each piece of aluminum was sprayed with two coats of Krylon—red insulating varnish forming a thin die -electric layer. Number 32 bare Nickle Chromium wire was laid out over the entire surface a distance of . 25 in. apart and sprayed into place with an additional coat of insulating varnish. This arrangement produced a resistance heating element with a total resistance of 112 ohms. Plate surface temperatures were sensed by means of five thermocouples mounted flush with the surface. The exact locations are shown in Figure 8. The thermocouple junctions were made from number 30 copper- ~ constantan wire. These junctions were forced through a 0.023 in. 653 and woumofl mo 33> Hmcodmcocfiflua .w ousmfm 3m; ozm LI .. ..nm. 3m; sorrow 39>. mo._. a 6 3 no 0 o .o._ «n o noiaouoecuf. w ..d. _ u _ c 11m =QO ..ood :00...» . J.oo._v 3mm. 1.— ‘I In! ..nw.n o o o o A. r :mn : t J . ..no. . . . a .|i.\ , . 7 od . ..oom J :0 37 diameter hole and fitted flush with the top surface. Care was taken to position the thermocouple lead wires between the heating wires on the inside plate surface. Since the boundary layer development for the top of the plate is analogus to that for the bottom, no thermocouples were installed in the bottom surface of the plate. Following installation of the five thermocouple junctions, the two halves of the plate were pressed together and cemented in place with Corona Dope (a high voltage insulating material) . The completed plate (Figure 9) with an overall thickness of 0.125 in. was then sprayed with a single coat of insulating varnish to provide a uniform finish on the upper and lower surfaces. Two variable DC power supply units with a maximum out- put of 500 volts and 200 milliamperes were connected in parallel to provide the current for the heating element in the plate. Two Weston analyzers were employed for measuring the voltage and current supplied to the heating element. These units were calibrated against a standard power supply unit accurate to d: O. 2 percent. The plate was mounted on a test stand (Figure 10) by means of small plastic tabs which were attached to the tabs provided on the top half of the plate. The test stand was constructed so that it was possible to raise or lower any corner of the plate. This pro- vided a means for adjusting the plate surface attitude with respect to the direction of air travel in the tunnel. The plate and the test stand Figure 11. --Equipment set-up for measuring mass-transfer. r J P fi-f 40 so designed, permitted undisturbed development of the hydrodynamic and thermal boundary layers on both plate surfaces. 4. 3 Mass -Transfer Apparatus A special apparatus was designed to measure the mass- -" transfer of water vapor from a free water surface into a moving air stream whose temperature was below 32 degrees Fahrenheit. The . apparatus consisted of a tray constructed from 1/8 in. thick plastic .‘ IL with inside dimensions of 8 in. x 8 in. x 7/8 in, (Figure 12). All four sides and the bottom of the tray were insulated with a 1 in. thickness of Styrofoam. A 1/2 in. hooded water inlet was provided through the center of the tray bottom and attached to three feet of flexible plastic tubing. A plastic grid made of 1/8 in. plastic, 3/10 in. deep was installed flush with the tray surface dividing the surface area into 9 equal parts. The purpose of the grid was to provide sup- port for the porous cotton cloth which was stretched across the tray surface. Thermocouple junctions were threaded into the cloth surface at selected locations (Figure 12) and sewed in place. These thermo- couples measured the temperature of the water surface. The thermo- couple lead wires were located beneath the cloth surface. To keep the water surface from freezing during the test runs, a plastic coated nickel-chromium heating wire was arranged at 1/2 in. intervals over the bottom of the tray (Figure 13) . The resistance of this heating wire was approximately 7 ohms per foot. One end of the heating wire SM: nmwmcmuu: mmme mo 33> Hmcowmcoeflfiuu .N .m mndwwrm . _ ..llllllllllllllnnumuv llfilfilllamlhulhuununlllbl ””“UIHIHIUIHIHIJI—Ill fl... 3m; ozu _ m“ _. _. ... .. 1. IIIII .JHIIIIII: uuuuuuu 1 phi up 41 3w; dog. ..I 2:1: I l I £4 I- , I DI I. iv . .0 1-.. t .l. imimpolhn . , r: t . a) lull”! H I i I 1 till M i. in with it). div-“hm . ”villi. 2.? II" I \ I ll l l l Ir)... |.l|y ) L51... u “it r. i ii i t- .-v. i.-l - H-111»? ' I l - i x I - «m ) . 3 oil 11:! r --liillualilh 43 was wound around the exposed (to the tunnel air) plastic tubing lead- ing from the tray bottom to the outside of the tunnel and insulated to prevent freezing of the water therein. Plastic cement was used to seal the cloth to the top edge of the tray in order to insure that mass- transfer would take place only from the 8 in. x 8 in. tray surface. A similar 8 in. x 8 in. 7/8 in. tray of 1/8 in. plastic was constructed for the purpose of providing a constant-head water reservoir for the test tray in the tunnel (Figure 11) . This tray was covered with the exception of a small access hole, to prevent evapora- tion of the water. The tray was not insulated since it was only subject to the ambient temperature outside the tunnel. The tray bottom was fashioned so that it would rest securely on the head of a balance. This arrangement provided a constant-level support and a means of measur- ing the amount of water removed from the tray (to replenish the water lost by mass—transfer from the tray surface in the tunnel). A Mettler Balance with a capacity of 800 grams which could be read directly to 0.1 gram and estimated to 0.01 gram was used. To provide the required DC current to the heating wire in the test tray three variable DC power supply units were used. Each of these units had a maximum capacity of 500 volts and 200 milli- amperes. When these units were connected in parallel a maximum of 600 milliamperes of DC current was available for the heating wire. i - - hi»:sln.i)lla.ln.)lllhfi.li will) - . i; - t f )I F.‘ ..hl'El 44 4.4 Leaf Model A model of a leaf (flat plate) was constructed so that the validity of equation (3.4. 2) could be examined under actual sprinkling conditions. A 4 in. x 4 in. leaf model (Figure 14) was constructed from two pieces of .051 in. thick aluminum in the same manner as the plate described in section 4. 2. Three thermocouples were located in the bottom plate surface at distances of .168 in., 1. 250 in., g! and 3.000 in. measured from the leading edge along the center line of the plate. The plate was mounted on an adjustable stand. A plastic trough made from 1/2 in. diameter tubing was fastened to each side and the back of the plate to intercept the water leaving the plate surface. The troughs on each side of the plate were closed at the front (leading edge) and open at the back. The trough at the rear of the plate was closed at both ends with an exit located at the center. Two thermocouples were located in each of the three troughs to pro- vide a measure of the temperature of the water as it left the plate. Each trough was covered by a sloping (away from the plate surface) plastic canopy 1/2 in. above the plane of the plate surface. The lead- ing edge of the canopy was located directly above the edge of the plate so that it did not prevent precipitation from falling directly onto the plate itself. It did, however, prevent precipitation from falling directly into the surface water collecting troughs. The canopies were Figure 15. --Tray for measuring water application rate and drop temperature. 46 mounted in such a manner that there was no interference to the free development of the boundary layers over the plate surfaces. The temperature of the water film on the plate surface was measured by three thermocouples laid on top of the plate. The water application rate and the mean temperature of the water drops striking the water surface were measured by means of a 4 in.x 4 in.x 1/2 in plastic tray (Figure 15) . The tray was in- sulated with a one inch thickness of Styrofoam about its exterior and fitted with adjustable support legs. The top edge of the tray and insulation were sloped away from the interior edge of the opening to insure that only the precipitation from a 4 in.x 4 in. area entered the tray. The thermocouples were located approximately 1/4 in. above the inside floor of the tray. The temperature sensed by these thermo- couples was recorded on the two pen strip chart recorder and indi— cated the mean water drop temperature at the surface of the water film. In order to conduct the actual sprinkling tests on the leaf model, it was necessary to make some additions and alterations to the test section of the wind tunnel. Initially a 12 in.x 12 in. roof section over the test area was removed. Fitted to this opening was an in- sulated plywood sprinkling tower 18 1/2 in.high (Figure 16) with a full cone spray nozzle assembly installed at the top. The front section of the tower was removable thereby providing easy access to the spray Figure 16. -—Wind tunnel with sprinkling equipment attached. Figure 17. —-Sprinkling tower and water control assembly. 48 nozzle . The water supply and pressure to the nozzle were controlled by an assembly located adjacent to the tower (Figure 17) . It was comprised of a shut-off value, diaphragm pressure regulator (0-125 . 2 . 2 . lb/in. and a pressure gage (0-100 lb/in. Made the tower (Figure 17) an inverted cone (made from a 6 1/2 in. diameter funnel) with a 3/4 in. diameter opening was located 2 and 3/8 in. below the spray nozzle to control the diameter of the spray pattern in the test area. The inverted cone was placed over a 4 1/2 in. diameter by 2 in. high open cylinder affixed to the center of a 12 in. x 12 in. metal pan which caught and drained-off all water not passing through the cone opening. Attached to the 4 l/2 in. opening on the under side of the pan was a 9 1/2 in. diameter cone which served to intercept and di— vert (away from the center of the test area) any water droplets forming about the periphery of the opening. A 12 in. x 12 in. piece of insulation was removed from the floor of the test area directly beneath the sprinkling tower. A sheet metal pan with drain was installed in its place to collect the sprinkled water. A one inch deep 16 1/2 in. long metal splash pan was installed down wind from the test area to intercept any spray carried past the collector pan. This pan was installed flush with the tunnel floor and drained forward into the col— lector pan in the test area. Thermocouples were attached to the spray nozzle and the drain in the collector pan to measure incoming and outgoing water temperatures respectively. 49 The thermocouples sensing the air temperature approach- ing and leaving the test area and the water temperature at the spray nozzle and in the collector pan were connected to a 16 point Brown Potentiometer (Figure 5) . This recorder had a scale reading from -20 F to 120 F and could be read directly to 0. 5 F. Since the recorder had a 15 second pen speed connecting 4 points in series it provided a recording of each of the four temperatures every one minute. The six thermocouples used to measure the water temperature in the collector troughs were connected to one pen of the two pen strip chart recorder through the six—position manual switch. The three thermo- couples located in the bottom plate surface and the three thermocouples lying on top of the upper plate surface were connected to the second pen through a six-position manual switch. ‘I. - V . EXPERIMENTAL PROCEDURE 5.1 Convection The independent variables and their range of values studied in Phase One of the investigation were: a. air velocity (51 to 961 ft /min) b. air temperature (10.1 to 25.0 F) The heat transferred from both sides of the flat plate was determined for various values of the above variables. A total of 19 tests were made. Prior to the actual test period the cooling coils on the cold storage box refrigeration unit were defrosted, the thermostat set at 19 degrees F, and the temperature in the box allowed to come to equilibrium. The initial step was to properly position the stand sup- porting the plate in the test area of the tunnel. The stand was centered horizontally and the plate adjusted until it was centered vertically making sure the plate surface was parallel with the tunnel floor. After closing the access door to the test area, the velocity control gate was fully opened and the fan started. The strip chart recorder and the DC power supply units were then turned on. The 50 51 output from the power supply units was adjusted until the plate tem— perature indicated by the number one thermocouple was brought to a steady—state value of approximately 31 F. Once assured that steady- state air temperature and plate temperature were reached, the values for the heating element voltage and current along with the current reading from the hot —wire anemometer were recorded. Immediately following these readings, the temperature values indicated by plate surface thermocouples two through five were recorded by manually switching to each thermocouple for a brief time interval. The velocity control gate was then reset to provide a lower air velocity and the above procedure repeated. 5. 2 Mass —Transfer The independent variables and their range of values studied in Phase Two of the investigation were: a. air velocity (161 to 866 ft/min) b. air temperature (15.1 to 22.1 F) c. relative humidity of the air (40 to 63 percent) The mass-transfer from a free water surface was determined for various values of the above variables. A total of 6 tests were run. The mass-transfer tray was positioned in the center of the test area and leveled by adjustment of the support legs. Before each test run the balance supporting the water-reservoir tray was calibrated with respect to water removal from the mass-transfer 52 tray in the tunnel. This was accomplished by bringing the water- reservoir tray to a fixed position using a device on the balance which secured the balance head for transport. Distilled water was added to the water—reservoir tray until the desired level of water was ob- tained in the mass—transfer tray. Following this the clamp located adjacent to the outside tunnel wall on the plastic tubing connecting the trays was closed. The positioning mechanism on the balance was released and a reading taken. The water—reservoir tray was then returned to its initial position the clamp opened and a small quantity of water removed from the mass-transfer tray. After suf- ficient time had elapsed for equalization of the water levels in the two trays, the clamp was closed and a second reading taken. This procedure was repeated five times and each time a different amount of water was removed from the test tray. Upon completion the five samples were weighed and the result plotted against the difference between each corresponding set of balance readings. After calibration of the weighing system, the test area access door and the water line clamp were closed. The access door in the tunnel inlet section was opened and the desired relative humidity sensor plugged into the receptacle whereupon the door was closed. The desired air temperature in the tunnel was obtained by setting the thermostat on the refrigeration unit. The fan was started and the air velocity control gate adjusted to produce the 53 desired air speed. During the initial transient conditions the clamp on the plastic tubing was opened and water was added to the reservoir tray to compensate for the slight static pressure increase acting on the surface of the mass-transfer tray. This was accomplished by observing through the plastic window in the access door, the level of the water surface in the mass-transfer tray. The adjustment was r- n. 1.5.. ._ Lani-1r a.” completed when a thin film of water was visible over the entire cloth surface . When the water surface temperature at location 1 (Figure 12) reached approximately 55 degrees F, the DC power supply units were turned on. The quantity of current flowing to the heating wire was adjusted so that the water surface temperature was maintained at about 51 degrees F. The system was allowed to run for a few minutes without further adjustment to insure that steady-state con- ditions had been attained. The 30 minute duration test run was then started by positioning the event market on a particular time division on the chart, and closing the water line clamp. A balance reading was then made, the tray returned to its starting position and the clamp opened. At 5 minute intervals during the test run velocity, relative humidity, and water surface temperature (at all six locations) readings were taken. A continuous recording of air temperature was made with pen number one of the strip chart recorder. At precisely 30 minutes 54 after the start of the test run the water line clamp was closed and a balance reading (with the fan in operation) obtained. The difference between the initial and final balance reading was calculated and the actual quantity of water removed from the mass-transfer tray ob- tained from the weight calibration curve (Figure 29) . This procedure was repeated for different velocities and air temperatures . 5.3 Sprinkling the Leaf Model The independent variables and their range of values studied in Phase Three of the investigation were: a. air velocity (65 to 703 ft/min) b. air temperature (13. 9 to 27. 7 F) c. relative humidity of the air (36 to 66 percent) (1. change in sprinkled water temperature (2. 5 to 10.0 F) e. local plate surface temperatures (32. 5 to 54. 3 F) The water application rate for a flat plate was measured for various values of the above variables. A total of 4 test series comprising 23 tests were run. The leaf model mounted on its test stand was positioned in the center of the collector pan and adjusted until its surface was level and vertically in the center of the tunnel test area. Once this arrangement was completed, the location of the test stand was marked 55 in the collector pan assuring proper alignment of the leaf model upon removal and replacement. The plastic tray used to determine the sprinkling rate was adjusted and its location marked so that its sur- face intercepted the precipitation at the exact same location in the test area as did the leaf model, E“ The desired nozzle tip size was selected and by means of the access door on the sprinkling tower installed in the nozzle assembly. ) The desired air temperature in the tunnel was obtained Hr by setting the thermostat on the cold storage box refrigeration unit. The fan was started and the desired air velocity obtained by adjusting the air intake gate. The water line shut-off valve was turned on and the water pressure at the spray nozzle set at 10 lb/in. Z by adjusting the pressure regulator. A flood light mounted outside the window of the test area access door was turned on to allow visual observation of the leaf model during sprinkling. The primary reason for the visual observa- tion was to assure that an approximately equal quantity of water was being discharged from each of the three collector trough outlets. Once the air temperature in the test area had reached steady state, recordings of the six water temperatures in the collector trough, the three temperatures of the water film on the plate surface, and the three temperatures of the bottom plate surface were made by switching the two six—point manual switches to each position for a 56 brief time interval. During this same time the temperature of the air approaching the test area along with the water temperature at the spray nozzle and in the collector pan was recorded by the 16 point recorder. The air temperature downwind from the test area was not recorded due to the "icing -up" of the junction and subsequent invalid readings. Immediately following the temperature measure- ments an air velocity measurement was made with the hot-wire anemometer along with a relative humidity measurement. A visual check was then made of the leaf model and the spray nozzle water pressure and the above procedure repeated for a second time, thus affording two replications per test. With the spray shut off the leaf model was removed from the test area and the sprinkling rate cali- bration tray installed in its place. The lead wires from the two thermocouples in the tray were connected to the two pen strip chart recorder. The spray was then turned on for a period of 5 minutes, the time period was observed by stop watch. During this 5 minute period the air velocity and temperature were identical to those occurring during the sprinkling of the leaf model. The quantity of water collected in the tray was weighed to 0.1 of a gram on a Mettler Balance and the calibration procedure repeated a second time. This procedure was then repeated for different air velocities, air temperatures, and sprinkling rates. VI. DISCUSSION OF RESULTS Prior to conducting the actual tests for each phase of the study the percent of free stream turbulence in the test area was determined. The measured free stream turbulence ranged from 2.9 percent at low velocities (100 ft /min) to 4.0 percent at high velocities (1000 ft/min) . 6 .l Convection The experimental data obtained for this phase of the study showed that the temperature of the plate surface increased with dis- tance from the leading edge (Figure 28) . It was observed that during any given test thermocouples 4 and 5 (Figure 8) indicated within 0. 5 F of the same temperature as thermocouple 3. This confirmed the two-dimensionality of the thermal boundary layer development (con- stant in the z direction) . On the basis of equation (3. 2.17) the values for the average Nusselt number for each test can be obtained from the relation Qe L —' : __ .1.1 N“ 2A k ATP' (6 ) 57 58 where Qe is the total measured quantity of heat being removed from the plate per unit time (Btu/min) during any given test. The equation defining Qe is Qe = (0.5692) V I (6.1.2) where V is the voltage to the heating wire and I the amperage. The results for this phase of the investigation are pre- sented in Table 1 and in Figure 18. In Figure 18 the experimentally obtained values for convective heat transfer from a flat plate are compared to the values obtained using the theoretical equations for a constant plate surface temperature and a continuously varying plate surface temperature. This comparison is made by plotting the Nusselt number against the Reynolds number on log-log paper. From a least squares analysis the regression equation for the experi- mental data is 0.494 E: .719 ReL (6.1.3) The standard error of estimate SEy’ which is a measure of the amount of variation from the regression line based on ordinate values is :1: 1.022. The equation for the average coefficient of convection ob- tained from curve (2) , Figure 18, is k l/Z 1/3 . — . .4 757 L ReL Pr (6 l ) 5‘1 o n .Inh 1".IIIII. III-l ll 1...,“ . Am .N .mv do an, posflop onsuduomeg 60.3qu 33m 33980 .m Mom m63d> Hmoflouooea. 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Amy A: A: .mumou dofioo>cou mo >um585mnu .H 3an I00 — in — <2) 30 - (3) 20 I J llllJl I I inf 2 3 4 5 6 7 8 9 IO 20 30 4O 50 60 ReLx IO'3 Figure 18. -—Comparison of measured and calculated average film heat- transfer coefficients in laminar flow over a flat plate. Curve (1) for a continuously varying surface temperature was obtained with equation (3. 2.14) , Curve (2) is obtained from the measured values. Curve (3) for a constant plate surface temperature was obtained with equation ( 3. 2. 3). Regression equation (2); m = 3719 ReL 0.494 Standard error of estimate : :1: 1.022 61 It should be pointed out that at the maximum air velocity encountered of 961 fpm and free stream turbulence of 4.0 percent, the Reynolds number based on the total plate length was 57 x 103. Investigations by Edwards and Furber (1956) show that with 3.8 per— cent free stream turbulence the critical Reynolds number did not occur until a value of 80 x 103 was attained. Thus, during all the tests the air flow regime over the plate surfaces was laminar. This condition can also be verified for the tests conducted in Phase Two and Three. Figure 18 shows excellent correlation among the experi— mental values. The actual measured rates of heat transfer as represented by the Nusselt number was 12.4 percent greater than those predicted by constant surface temperature theory but 5.8 per- cent less than those predicted by continuously varying surface tem- perature theory. 6. 2 Mass -Transfer Modification of equation (3. 3. 2) was necessary before it could be used in analyzing the test data. This equation is appli- cable only for a flat surface with a free water surface starting at the leading edge. The free water surface of the mass—transfer de— vice used in this study starts a distance of 0.188 ft back of the lead- ing edge. To account for this, the following equation for the average 62 coefficient of mass-transfer from Eckert and Drake (1959) is used. — l l h = .664 1:7- Re /2 Sc /3 l L [1_ (Xe/q3/4] l/3. (6.2.1) L in this equation is measured from the leading edge to the inside back edge of the tray where the free water surface ends, a distance of 0. 766 ft. Entering the stated values for X0 and L into equation (6.2.1) results in the relationship — _ E 1/2 1/3 hm — .679 L ReL Sc (6.2.2) The data from the balance calibration run preceding each test was combined into a single calibration curve shown in Figure 29. The regression equation for the line is Actual Weight 2 l. 397 (Balance Reading Difference) - 0.221 with SE = i .1772 Y The results for this phase of the investigation are pre- sented in Table 2 and Figure 19. In .Figure 19 the experimental values, which are denoted by the circles, are compared to the theoretical curve which was calculated from equation (B.l. 2. 3) The measured values, although somewhat scattered, show good correlation with those predicted theoretically. 63 fwd .H 2m: nowadays Eouw #6335030 303.9, Hmucogiomxm .A m .N .H . m3 dodumswo Eon pmudadoado m03d> Houflonoofik . mend; Hduaogfinomxm . A N .N .H .mv Goflmaém 803 63.3.9345 moDHm> Hmofiouoogfi 3; .2 A2 8.3 51$ 2 J. 2 J. 8.: $0. SM. 93 22... 9.8 mam 8 1: a 3:? 3.3. 85 $5 8.2 omo. 3m. 5mm 1on m5 Tom 8. SN m 3.0.: 8.03 8.5 $8 2 .2. :0. 0mm. 93 are... 13 mi. 3 omw e 3.2: cod: 85 $5 5.2. So. mam. v.8 0.8... 1.: e23. 3 33. m 3.8 sod: is. its 8.3. mmo. mam. 8am no.8... mg: 0.2. 3 9:. N 8.42 3&2 £6 2 .s 8...: Bo. SN. mgm imam H .3 «.2. 8 3w 2 Em Em mag: mans... m m a a e. £83 .02 Ez 82 m2 m2 -quum mm mm EH 9 as we mm a: “may A: ANV .mumou Hounds-mods: mo >Hd885muu .N 0138 30" l I I 1 l J_| IO 20 30 4O 50 60 7O 80 ReLx IO‘3 Figure l 9. --Dimensionless mass-transfer coefficient for a flat surface evaporating water into a laminar air flow. The circles indicate the measured values. The curve is calculated from equation (B. 1. 2. 3) . 65 6.3 Sprinkling the Leaf Model Three full—cone spray nozzles with orifice diameters of .024, .027, and ,030 in. were operated at 10 psi to produce sprink- ling rates ranging from .101 to . 220 in. /min with drop sizes varying between 0. 2 and l. 0 mm in diameter... The size of the water droplets at zero air velocity in the wind tunnel was determined by the process described by Engleman (1963 ) . The procedure used was to briefly expose a 11 in. x 9 in. sheet of Ozalid 10 582 paper to the water spray followed by a brief exposure to ammonia fumes. The water spots, after drying, showed- up as yellow spots on a light gray (depending on the ammonia exposure time) background. The true diameter of the drops was calculated according to the empirical relationship D = 0.43 50°74 where D is the drop diameter in mm and S the spot diameter in mm. The results for this phase of the investigation are pre- sented in Table 3 and Figure 20. The theoretical water application rate was calculated from equation (3.4.. 3) where the average film heat-transfer coefficients for qC', qc, and qIn were obtained from Figure 18 [curve (2) and curve (3)] and Figure 19 respectively. In Figure 20 the measured water application rate is compared to the theoretical water application rate in the form of a dimensionless . Juli: BE! Ill om.o wua. ocH. om.H oN.~ ww.m v~.o nvm.e moo. who. 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Boo. ma.m~ om.o~ NNm so m.Nm ~.Hs m.¢m o.o~ w.as ¢.o o mm.o mom. ems. mo.H ms.H oo.~ mm.m ma~.m mmo. coo. mm.ma us.mfi was me m.mm «.ms ~.mm m.aH 0.9m m.m m Ho.H Nos. gas. mq.H sm.H cm.~ mq.m nmm.~ ave. mmo. NH.~H oo.NH mam mo m.mm o.vs N.mm m.aH m.om s.m s NH.H “as. now. so.H mN.H ss.~ sn.s CHH.N mmo. mmo. 0H.o mH.o mes, «m m.om m.ms m.sm o.NN o.mm e.q m OH.H Hwfi. oHN. as.o no.fl 0H.H a¢.m Nmo.H BN0. oNo. No.m oo.m am mm m.ov 0.0m H.Nv a.s~ m.om N.m m “a _E::.cu N: Eccsxw c:£: .mwufisszm .tsfiu s m m m m m m .oz 03 “a 03 no Ev .ow 3v Em um .um $2qu nexqum a: mm .69 .asd Es as as 394 “mos Juno» mcfixfiumn we >ud§§mnu .n 03.3. 67 0353mm m0 .858 pumpnmum 1H 3 mmoé + 0% onoH xwmm n u lol "modemswo Godmwmummm .moumn cofimofimmm Roam? Hmoflouoofi paw 3368:0me 93 mo demmummgoo d7: .om ouswwh .-o. x 5m mm mm «m NN ON 0. w. E . N. o. m -I - _ _ _ 4 _ - W . _ . _ 68 ratio plotted against Reynolds no. If there was perfect correlation, at least within the experimental accuracy of the test system, one would expect the ratio of we /wt to be approximately one for all values of the Reynolds number. As indicated by Figure 20, this is not the case, values of we /Wt range from 0.81 to 1.17 for Reynolds numbers 3 3 . . between 2 x 10 and 27 x 10 .. The regressron equation for the least squares best—fit line is W -6 e = ~9.24x10 Re + 1.093 Q: L with SE = :I: .1072 Y In examining the individual data points plotted in Figure 20 it was noted that primarily all we /Wt values of 1.1 or greater occur at low Reynolds numbers (up to 7 x 103) . The values of we /wt above 7 x 103 lie scattered about an approximate mean value of 0. 90. An attempt will now be made, first of all, to explain the occurrence of large We /wt values at low Reynolds numbers. Visual observations of the plate surface during sprinkling indicated that maximum water film thickness occurs at minimum air velocity. As the air velocity increases the water film thickness decreases due to the drag of the air on the water surface. Also as the air velocity increased, the sprinkling rate decreased (see Table 3) . The com- bined effects of high intensity precipitation striking a relatively thick 3i :x:~ 69 water film surface would tend to promote a very rough or "cratered'l surface. Although exact theory pertaining to laminar air flow over rough surfaces is not available, intuitive thinking leaves little doubt that the coefficient of heat and mass transfer would be greater than that obtained for a smooth surface. This was brought about by the increased surface area and the "disturbed“ (from the drops of falling water) nature of the boundary layer. Since the Wt values are derived for the case of laminar flow over smooth flat surfaces, these values would be lower than those actually occurring during the tests, hence the large values of we /wt. The equation for wt (3.4. 3) also contains the heat loss term for the un—wetted underside of the plate. This term is not affected by the phenomina discussed here or in the fol— lowing paragraph. As air velocity increased and water application rate decreased, this "surface cratering" effect would diminish. In an attempt to explain why such a large portion of the remaining we /wt values were below 0. 90, an examination of the theory used in deriving the heat and mass transfer coefficients was made. These coefficients were derived for the boundary condition of zero air velocity at the plate surface. This condition no longer applies for the case of a moving water film on a flat plate since the air now has some finite velocity at the air -water interface. Appendix A contains an expression for the average coefficient of convective heat transfer derived for this boundary condition. 70 The resulting expression (A.l.ll) is 1/3 /2 _ k l : 0. _ hC 398 L Pr (ZReL + 3R61 ) where Rel, the Reynolds number, is defined in terms of the velocity at the air-water interface (ul) . The equation describing u1 (A.l. 20) is l 2 l 2 (u—u)(2u+3u)/ |.L V l a l a l _ 37 03 w _a_. L 2 ' p. L W 111 a where the a and w subscripts refer to air and water respectively. The derivation of this expression (see Appendix A) for Lil was based on certain assumptions concerning the water film on the plate surface. The validity of these assumptions was further borne out by observa- tions and measurements made of the water film thickness b during sprinkling tests . The measured thickness of the thermocouple junctions lying on the plate surface averaged .040 in. During sprink- ling tests at low air velocities, it was observed that the water film completely covered the thermocouple junctions, however at high velocities a portion of the junction protruded above the water film surface. A check of Table 4 shows that the water film thickness b ranged from a maximum of . 088 in. with an air velocity of 65 fpm to a minimum of .011 in. with an air velocity of 677 fpm. The higher velocities did, however, tend to produce a non-uniform water film 71 thickness by decreasing the depth near the leading edge of the plate and increasing it toward the back. The average film heat ~transfer coefficient over a moving water film [equation A. 1.11)] includes the air -water interface velocity term ul, which was not readily calculable. Since this expression for he differs by an almost constant amount ( 12. 2 percent) from the he term for a flat plate without a water film (Figure 21) an expression was obtained for the moving water film heat—transfer coefficient which did not include ul. This expression was i = .5831‘: Re ”2 Pr1/3 (6.3.1) Similarly, the average coefficient of mass—transfer over a moving water film can be expressed as —- _ D 1/2 1/3 hm — .583 L ReL Sc (6.3.2) The sprinkling data was re-evaluated using equations (6. 3. l) and (6. 3. 2) for calculating the heat and mass transfer (assuming again the validity of the similarity relations proven in Phase Two) coefficients of the water film surface (heat loss ex— pression for the bottom of the plate remaining unchanged) . These results are shown in Table 4 and Figure 22 and 23. The measured values of the water application rate we were used in equation (A. 1. 20) 72 for evaluating u In Figure 21 the values for the average Nusselt 1' number are obtained by using equation (A. 1.11) to define l—lc [curve (4) ] . The regression equation for the experimental points is E; = .620 ReL0°481 with SEY = 11.010 Curve (3) in Figure 21 was obtained from equation (3. 2.3) for the case of a flat plate without a water film on its surface. Comparison of curve (4) with curve (3) indicates that the values for the average Nusselt number are 11., 2 percent (Re = 2. 25 x103) to 13. 2 percent L (ReL = 30.00 x 103) less than those indicated by curve (3) . The average value of 12. 2 percent was used to reduce each of the qc and qm terms in Table 4 and subsequently obtain a second Wt value for each sprinkling test (Table 4) . This value of Wt is compared with we in Figure 22 revealing that the values of we /wt, with the exception of the few values occurring at low Reynolds numbers, are now scattered about an approximate mean value of 0. 98 rather than 0. 90 indicating better agreement between theory and experimentation. The regression equation for the experimental data is W /W = -9.89 x 10-6 Re + 1.173 with SE = :1: .1183 e t L y The dimensionless water application rate number defined by equation (3.4.4) was altered slightly. Since the mean temperature change experienced by the sprinkled water while on the leaf surface 73 Table 4. --Sprink1ing data analyzed on the basis of the theoretical rela- tionships developed for a flowing water film on a flat plate. Test u u1 b ReLx 10‘ KIT. (1) we wt :3 w ft/nihi in. in./rnin Wt 8 97 0.71 .066 3.60 31.80 .210 .168 1.25 9.02 3 163 1.04 .044 6.13 41.05 .207 .165 1.25 7.17 4 318 1.68 .026 12.06 57.03 .194 .178 1.09 5.73 5 408 2.08 .020 15.47 64.54 .186 .194 0.96 5.58 6 522 2.32 .017 19.80 70.58 .178 .187 0.95 4.80 7 657 2.59 .014 24.98 80.46 .157 0148 1.06 3.40 9 98 0.60 .056 3.71 31.83 .151 .121 1.25 6.85 10 163 0.85 .036 6.19 40.80 .139 .149 0.93 6.75 11 285 1.25 .023 10.91 53.72 .129 .148 0.87 5.36 12 412 1.57 .017 15.84 64.36 .118 .134 0.88 4.10 13 514 1.82 .013 19.88 71.91 .110 .115 0.96 3.22 14 677 2.12 .011 26.24 80.99 .101 0103 0.98 2.48 15 65 0.55 .088 2.42 26.27 .218 .173 1.26 11.92 16 109 (3.78 .060 4.09 33.78 .211 .181 1.16 9.71 17 216 1.22 .034 8.18 47.06 .186 .205 0.91 8.22 18 336 1.67 .024 12.85 58.52 .177 .190 0.93 6.31 19 470 2.15 .018 17.97 69.06 .175 .169 1.04 5.08 20 631 2.55 .014 24.26 79.92 .159 .157 1.01 4.07 22 179 1.16 .042 6.74 43.08 .220- .180 1.22 7.61 23 308 1.68 °027 11.62 56.18 .207 .176 1.18 5.95 24 378 1.88 .022 14.30 62.02 .188 .214 0.88 6.16 25 504 2.19 .018 19.05 71.41 .177 .196 0.90 5.05 26 703 2.76 .013 26.63 84.12 .160 .165 0.97 4.19 (1) Calculated from equations (3. 2. 3) and (A. 1. 11) where Rea = Re and the fluid properties are evaluated at the mean value of T for 1. alltests. 7.! 74 ATW was difficult and impractical to measure it was replaced by the temperature difference (T - T ) s a 5.20 wt Cp L (Ts—Ta) m'k'AT'+fik(T—T)+MHD(P-p) (6.3.3) p s a v s a RT The dimensionless numbers El], E1, and Nm are obtained from Figures 18[curve (2)], 21 [curve (4) ], and 23 [curve (6) ] respectively. In Figure 24 the dimensionless water application rate number is plotted against the Reynolds number of a log-log scale. The regression line is w = 1159 ReL‘0°526 with SEY = 41.1% To obtain the bottom plate surface temperature term ATP' defined by equation (6.1. 2) the local surface temperature at three locations (minimum) along the surface in the x direction must be measured and then a temperature profile curve (see Figure 30) plotted. To simplify the acquisition of this term, the following two empirical equations were derived from the test data (Figure 25): For tp at x1 = 0.169 in. (x1 measured from the leading edge) ATp =61.28(ReL/Ta)'0'353, withSEy=:L-1.159 (6.3.4) (TS-tp) 75 For tpatx2 = 1.250in. (xzineasuredfronitheleading edge) AT ' o 379 —-—B—=98.83(Re /T) ' , withSE =41.21o (6.3.5) (Ts-tp) L a y 200 (3) - -1 '°° (4) 90 '- "‘ 80 '- "' 70 - " | :3 60 ‘- —> Z 501- ‘ 4o - - 30— J 20 l I l l I l l I l l I 2 3 4 5 6 7 8 9 IO 20 30 4O ReL x 10" Figure 21. --Average film heat-transfer coefficients in laminar flow over a flat plate. Curve (3) was calculated with equation (3. 2. 3) . Curve (4) was calculated using equation (A.1 .11) for h , which was derived for the case of a moving water film on thg surface of a flat plate. Regression equation (4); K1; : .620 ReL 0.481 Standard error of estimate = i 1.010 77 mwz. H n oumgflmm mo youuo pumpnmuw MNHJ + Jom ouofi x «36; n Isl? ”905.350 codmmoumom .Edm Room? m5>o§ m >9 poum>oo 33m and d .HOH woufldofido mm? 3m.“ coflmofimmm poem? awoflouoofi 69H. .mofip coflmofimmw .533 3039565 USN HMHCoEComxo o5. mo memdhmmgoo <1- .NN madman“ .-o. .33. mm mm cw Nu ON 2 w. .v. N. o. w 0 V N o A l.) _ _ _ . _ _ _ q _ _ _ n._ 200 78 (5) [(6) J 1 1 I l I l I I l l l 3 4 5 6789|O 20 30 40 50607080 ReL 1: IO‘3 Figure 23. --Dimensionless mass-transfer coefficient for a flat surface evaporating water into a laminar air flow. Curve (5) calculated from equation (B. 1 . 2. 3) applies to a stationary water surface. Curve (6) calculated from equations (3. 3. 4) and (6. 3. 2) applies to a moving water film. 79 20 2 l l 1 1 g l n m 2 3 4 5 6 7 8 9 IO 20 3O ReLx l0" Figure 24. --Dimensionless water application rate num- ber for the leaf model in laminar air flow parallel to its surface. Regression equation: w = 427.4 ReL-0.464 Standard error of estimate = i l .1 92 .lF/v. ATp' (TS‘ip) 30 20 - -' IO - '- 9 t- _. 8 b .. 7 " q 6" X2312501n. ‘ 5 r- .. x. = 0.|681n. 4 *- - 3 1 1 1 1 1 1 1 1 I 1 1 I00 200 300 400 500 1000 2000 3000 ReL/To Figure 25. --Empirical correlation curve for determining AT ' when the air, water film, and local plate temperatures are known. . . . p = -0. 353 Regresswn equation (x1). —_T _ t 61. 28 (ReL/Ta) S P AT ' . . . p __ -O.379 RegreSSion equation (x2). _—T _ t - 98.83 (ReL/Ta) S P Standard error of estimate (x ) = :1: 1. 159 : :1: 1.210 Standard error of estimate (x2) VII . C ONC LUSIONS The correlation for the equations. determined experi— mentally in this study is expressed in terms of a plus or minus one standard deviation value. The limits represented by these values are obtained from the standard error of estimate and indicate that 68. 27 percent of all values calculated by the prediction equations will fall within these limits . The convective heat transfer coefficients he and Kc. defined by the equations which follow are expressed in units of Btu/min ft2 . 7 .l Convection The rate of heat flow from a thin uniformly heated flat plate in laminar air flow can be calculated from the relationship 1 = 1 qC hC ATP where RC = 757 — ReLl/Z Pr1/3 a .0007 81 82 AL AT'=A1: + .969(At -At )-.432 — p o n o L n [(2n-1)At -At - 2: At ] (7.1.1) n 0 n=0 n 7. 2 Mass -Transfer The rate of heat flow from a stationary free water sur- face due to water evaporating into a laminar air stream can be calcu- lated from the relationship E Hv m = — P - P qm R T ( s a) where — D 1/2 1/3 hrn — .664 L ReL Sc 7. 3 Sprinkling the Leaf Model The quantity of heat removed from a continuously sprinkled leaf model in laminar air flow can be calculated from the relationships given in the following three paragraphs . l. Convective heat loss from the un-wetted underside of the leaf model can not be accurately predicted by the constant surface temperature heat flow equation (3. 2.4) . The equation describing the heat loss was Ill... .1 83 k 1/2 1/3 ' : . -— :1; _ l qC 757 L ReL Pr 007 ATP AT ' can be readily calculated from either of the empirical equations (6.3.4) or (6. 3. 5) depending on which local surface temperature is spec ified. 2. The coefficient of convective heat loss from the moving water film on the upper surface of the leaf model can best be predicted by equation (A.l.11) which accounts for the velocity at the air -water interface. Adjustment of equation (3. 2.4) to account for this boun- dary condition results in a simplified relationship (elimination of the 1 air -water interface velocity term ul) for the convective heat loss. k 1/2 1/3 : — j; _ q . 583 L ReL Pr .0007 (Ts Ta) 3. The equation describing the heat loss due to mass- transfer from a stationary water surface must be altered to account for the moving water film surface. This equation was H D Re 1/2 SC1/3 (P 'Pa) V qm—'583RTL L s 4. The sprinkling rate can be predicted from a dimen- sionless water application rate number formed by combining the above mentioned heat loss equations with the heat added equation (the sensible heat from the sprinkled water) . . 1. .1, 5|: 1 I .1.1 83 5.20prL(TS-T) w: a qC + qC + qm W can be obtained from the plot of W versus Re (Figure 24) or from L the regres sion equation w = 1159 ReL”O'526 i 1.192 The results obtained from this study show that the constant- surface-temperature heat transfer equations postulated by Beahm (1959) , Businger (1963) , and others do not accurately predict the convective heat losses from the un-wetted under side of the leaf model. The con- stant-surface-temperature heat and mass transfer equations must also be altered slightly when used for predicting the heat loss from the flow- ing water film on the upper surface of the leaf model. The amount of alteration may be reduced to insignificance when the sprinkling rate is reduced to a value approximating that used in actual field practice. This, however, remains to be proven by future studies. SUGGESTIONS FOR FUTURE STUDIES This investigation has revealed the need for additional research in the following areas: 1. The applicability of the flat plate as a leaf model should be verified by sprinkling individual leaves under the same conditions imposed in this study. In addition, the radiation heat loss should be considered. 2. The effect which the angle of incidence has on the convection and mass~transfer terms in the prediction equation should be determined. 3. The effect of intermittent sprinkling, lower appli- cation rates, and drop size on the water application rate prediction equation should be analyzed. 4. The utilization of the latent heat of freezing of the sprinkled water accompanied by the surface ice load should be analyzed with respect to reducing the amount of water applied and any related changes which may occur in the heat transfer theory being applied. 84 APPENDIX A A.l Theoretical Derivation of the Average Film Heat-Transfer Coefficient Over a Flowing Water Film )1 6 __9 L11 ua \x ‘ [/IT/ll/l/f///////r7////}Ib Figure 26. --Flow boundary layer over a water film on a flat plate . Initially, the development of the hydrodynamic boundary layer over the surface of the water film will be considered. Assuming a second order polynomial to express the shape of the velocity profile: 2 u = A + By + Cy From the boundary condition u = 111 at y = 0 one obtains the equation 2 = + + ua 111 By Cy Aty=6, u= ua thus ua = 11.1 + B6 + C62 (A.1.l) 85 86 0 = B + 2C6 (A.l.Z) Solving equations (A. 1. l) and (A. l. 2) simultaneously and rearrang- ing terms results in the following expression for the velocity at any point in the hydrodynamic boundary layer. 1.. N ___)L u-u 6 |‘< (A.l.3) N The integral momentum equation for the case of steady state, two- dimensional fluid flow with constant properties [Eckert and Drake (1959)] is (A.l.4) T _° p 6 d EL (ua-u)udy— In this equation the shear force at the water surface was evaluated as _ d_u To ‘ ** dy y=0 (A.l.S) Substituting for the value of u from equation (A.l.3) 211(ua -u1) 'r =———— 0 6p Substitution of equations (A.l.6) and (A.l.3) into the integral equation produces d 6 2 Y2 2 Z __ _ _ _X__ _ _ _Y__X__ dx [11a (ua ul)( 5 52) ul][(u ul) 6 62 +ul:| dy - 21.1(ua- ul) 6 P Letting r) = '2: one gets 6 u d 2 2 1 2V (ua-ul) 5; 6f [l-2n+n ][2n-n +u _u]dn .—. —6- o a 1 Expanding terms and integrating from 0 to 1 ( , d_6 __2_ . __1_ _ .23: ua ul dx 15 3(u -u ) _ 6 a l Separating variables and integrating 62 z 1801/ x C 6(ua - 111) + 5u1 At the leading edge of the water film x and 6 are both zero, con- sequently C = O. The final expression for the boundary layer thickness is 1 2 5 - Kl 60 / (A 1 7) ’ L2Rea + 3Re1 ‘ ° The local film heat -transfer coefficient for the water film surface is given by equation (3. 2.11) 88 :r ll N101 (A.l.8) _k_ L, 6 For a flat plate with a constant surface temperature Eckert and Drake (1959) give the following expression for t. 1 g : (11.2.9) 1.026 Pr1/3 Substituting for L and 6 in equation (A.l.8) and simplifying one obtains 1/3 1/2 h = 0.199 Pr (2Rea + 3Rel) (A.l.10) 1: x The average film heat ~transfer coefficient is obtained by integrating equation (A.l.lO) with respect to x from 0 to L. 1/3 /2 _ k 1 h = 0.398 — Pr (2Re + 3Re) (A.l.ll) L a l C This expression contains the unknown velocity of the water film sur- face ul, which is a function of the free stream air velocity ua and the water film thickness b. To arrive at an expression for 111 the condition of continuity of shear at the water film air interface is em- ployed. The expression for the shear force on the air at the water surface is 89 where Si;- is obtained from equation (A.l.3) Introducing equation (A.1. 7) for 6 and rearranging terms results in the following expression for shear. 12 0.258 “a (ua ~ u1)(2ua + 3u1) / Ta = 1/2 » (11.1.12) (V X) The average shear over the water surface is obtained by integrating equation (A.l.lZ) with respect to x from 0 to L. _ 1/2 _ L l/Z Ta - 0.516 (7]) “a (ua - 111) (Zua + 3u1) The shearing force within the water film is expressed as du : __ A.1. Tw ”w (dy) y = 0 ( 13) where y is measured from the plate surface. To obtain an expression for the velocity of the water u a linear velocity profile through the water film is assumed. u = Dy + E Assuming the boundary condition that at y = 0 u = 0 gives 90 At y = b u = 111 therefore u = (A.l.l4) By taking the derivative of this equation with respect to y, setting y equal to zero and substituting into equation (A.l.l3) the relation- ship for TW is obtained. .7 lJw U‘1 TW 2 b (A.l.lS) —' L L : : — A.1.16 T L Tw dx 1.1 u b ( ) The two shearing forces, air and water, must be equal at the air- water interface . 1/2 1.1 .516 (17: 11a (ua-ul) (2ua+3u1)1/2 = i—Y— (11.1.17) To arrive at an expression for the water layer thickness b in terms of known parameters it is assumed the water film flows only in the 91 x direction. Visual observation during actual sprinkling tests indicate that the flow pattern is such that approximately 1/3 of the water flows off the back of the plate at x = L with the remainder divided equally between the two sides . Figure 27. ——Water film on a flat plate. The quantity of water flowing out of the shaded element per unit time is 1 b _ = L 3 Qw j; u dy Introducing equation (A.l.l4) to express u beuly Q :30 b dy 3Lu [b b oy Y 92 3 3 . Q = 2 1.11 b L (ft /m1n) (A.l.18) The rate at which the water is leaving the plate must be equal to the rate at which water is applied to the plate by sprinkling. 3 w 2 2 “1“ - F L b : 32L (A.l.l9) 18u1 Introducing this expression for the water film thickness into equation (A.l.l7) and simplifying provides an expression for 1.11 in which all parameters are known . /2 2 1 (u -u) (Zu +3u) p U 1 a” 1 a 1 = 37.03—‘1' (_a) — (A.l.ZO) 2 1.1 L w u a l This equation can now be solved by the method of successive approxi- mations or graphically for the velocity of the water, 111, at the air- water interface. Inspection of equation (A.l.ll) indicates that the value for the average coefficient of convection he will be smaller than that obtained by equation (3. 2. 2) (no water film on plate surface) even though the magnitude of 111, as indicated by equation (A.l. 20) is small compared to ua. )1 ‘ APPENDIX B B.l Sample Calculations Sample calculations will be presented in accordance with the three separate and distinct phases of this investigation. B .1.1 Convection Test 8 will be used to demonstrate the calculations per- formed. For this test the followmg experimental data was obtained: Plate Air Heating H.W. Temperatures Temperature Element Current t1 t2 t3 Ta V I 41 (F) (F) (F) (F) (voltS) (ma) (ma) 31.1 33.7 34.2 13.0 34.3 268 616 From the hot-wire anemometer calibration curve Ua = 288 ft/min The voltage and current to the plate heating element obtained from the calibration curves for the Weston analyzers are V = 34. 7 volts and I = 272 ma The experimental value for the rate of heat transfer is calculated from equation (6.1. 2) 93 94 Q = (.05692) v1 = (.05692) (33.7) (.272) = .522 Btu/min To evaluate ATp', defined by equation (7.1.1) , it is necessary to plot the plate surface temperature profile (Figure 28) . n AT ' 2A1; + .969(At -At ) -(.432)31: (Zn-1)At-At -2: At p o 5 o L n o n-o n =16.6+.969(21.2-16.6) - (.432)—15 5 [(10-1) 21.2-16.6 — 2 2 124.9] 1120 = 26.3 F The mean temperature used to evaluate the fluid properties was taken as l Tm = (average plate surface temperature + air temperature) 2 = 22. 9 F Then - 2 V = 8.44x 10 3 ft /min -5 . 2 k = 22.98x 10 Btu/min ft F Pr: .72 2. The surface area of the plate includes the plate edge area and 3 of the plate tab area. A = .250 + .0208 + .002 = .27lft2 A) = tp-To (F) 95 22 Figure 28. --Approximation 0f continuously varying plate surface temperature by straight line segments for convec- tion test number 8. 96 Evaluating the Reynolds number a _ (22.8) (,5) 8.443.10- = 13.51 x103 The theoretical Nusselt number for the case of a flat plate with a continuously varying surface temperature is given by equation (3. 2.14) where he is defined by equation (3.2.13) 1 2 ‘ .834 ReL / Pr“3 2 c. II 31/2. (.834) (13.507 x10) 86.92 The value of the '1‘? (.72)”3 theoretical Nusselt number for the case of a flat plate with a constant surface temperature is given by equation (3. 2.3) where Kc is defined by equation (3. 2. 2) . l/3 l/2 Pr . 664 ReL 2 C II 12 (.664) (13.407 x103) / I) 69.26 The experimental value of the equation (6.1.1). (.522) (.5) (.72)1 /3 Nusselt number is calculated from = 79.00 (2) (.271) (22.98 x10-5) (26.3) "3 q 97 B. 1. 2 Mass -Transfer Test 5 will be used to demonstrate the calculations per- formed. During the course of this test the following data was obtained: 3.3.3313; 1:: Ta .1 12 13 .5 .6 ‘ F‘ r“ r‘ r‘ F r‘ F‘ 0 613 39 17.5 49.9 56.0 50.3 47.0 47.9 48.0 5 17.4 50.0 55.1 50.3 47.7 47.9 47.7 : 10 17.3 50.1 56.6 50.6 46.8 48.2 47.9 f 15 612 40 17.3 50.4 55.8 50.8 47.0 48.2 48.1 1 20 17.2 50.8 56.1 51.1 47.7 48.3 48.5 25 17.1 51.0 56.2 50.1 47.7 48.9 48.7 30 £43 41 17.0 51.0 55.9 51.0 47.2 48.8 48.7 Balance reading difference = 3.75 gm Averaging water surface temperatures with respect to time and location and air temperatures with respect to time one finds T s 50.1 F T a 17.3 F The average absolute temperature of the water surface is T = 510.1 OR The mean temperature used to evaluate the fluid properties is Ts + Ta 50.1+17.3 n1 2 = 2 33.7 F The vapor pressure at the water surface is the saturated vapor pres- sure of the air evaluated at the average water surface temperature. 98 PS = .364 in./Hg The vapor pressure of the free air stream is evaluated by determining the saturated vapor pressure of the air at the average air temperature and multiplying by the relative humidity. Pa = (.40) (.089) = .036 in./Hg From the hot-wire anemometer calibration curve the average velocity is Ua = 221 ft/min Evaluating the Reynolds number U L ReL = a = M = 19.5ox1o3 V 8.68 x 10 Evaluating the Schmidt number u 8 68 10'3 SC = B = ——"—§—'__3‘ = .532 16.31 x 10 The theoretical rate of mass—transfer is calculated from equations (3.3.1) and (6.2.1) D 1/2 1/3 RTL ReL SC (Ps'Pa) 1 m = .664 (70.70) s [1—(xo/L) 3/4]1/3 (B.1.Z.1) 99 The constant 70.70 converts the values of vapor pressure from in. /Hg 2 to lb /ft . _ (.664) (l6.31xlO-3) (19.5x103)1/2(.532)1/3(.364 -.036) (70.70) 5 (85.74) (510.1) (.766) 1 [1- (.188/.766)3/4]1/3 - Z .870 x 10 3 lb/min ft Converting the mass~transfer rate to the total quantity of water re- moved during the test one obtains M 8 (ms) (time) (area) (453.6) (B.l.2.2) (.870 x10-3) (30) (.444) (453.6) 5.27 gm The theoretical dimensionless mass-transfer number is given by equation (3. 3.4) where Tim is defined by equation (6. 2. 2) —— (.679) ReLl/z Sal/3 (13.1.2.3) z B / 1/3 (.679) (19.5 x103)1 2(.532) 76.88 The measured amount of water lost during the test is obtained from the balance calibration curve (Figure 29) as ‘- $8.4m“ Weight (grams) Actual lOO )- 1 1 I l l O I 2 3 4 0| 0) Balance Reodlng leference (grams) Figure 29. --Calibration curve for determining the quantity of water removed from the mass-transfer tray during a test. Regression equation: Actual Weight = l. 397 (Bal. Reading Diff.) - 0.221 Standard error of estimate = :l: .1772 101 m = 5.01 grams The experimental value for the dimensionless mass-transfer number is defined by the equation m — s R T L run} 2 3.1.2.4) (70.7)(01 (Ps‘Pa’ ( ' :3. Where ms is the experimental mass-transfer rate evaluated as M s (.870 x10-3) (30) (.444) (453.6) 5.01 3 6.04 x 10 .829 x 10'3 lb/min ftZ Substituting into equation (B. l. 2 . 4) one obtains -3 N; ___ (.829x10 )(85.74) (510.1) («766) = 73.21 (70.7) (16.31x 10‘3) (.364- .036) B. l. 3 Sprinkling the Leaf Model Test 22 will be used to illustrate the calculations per- formed. For this test the following experimental data was obtained: Average (from the six thermocouples in the collector troughs, with two replications) bulk temperature of the water leaving the plate T ’49.3F w-off ‘ 102 Average (from the three thermocouples on the plate surface, weighted according to surface area represented, with two replications) temperature of the water film T = 52.7F 5 Average (two replicates) hot-uwire anemometer reading is 677. 5 ma. Average (two replicates) relative humidity of the free -' airstream is 36 percent. 1 Average (two replicates) local plate surface (bottom) tempera- ture: x = .168 in. x = 1.250 in. x = 3.000 in. 46.1 F 47.8 F 52.2F Average (from two replicates) free stream air temperature is 22. 2 F. Average (from two five minute samples) weight of water collected in the application rate tray is 280.0 grams . The change in temperature which the sprinkled water undergoes while on the plate surface was AT = T — T = 53.3 -49.3 = 4.0 F w w-on w-off The mean temperature used for evaluating the fluid properties of the water film surface was T T T : _§+__a_ ___ 52.7+22.2 : 37.51:. m 2 2 103 Evaluation of ATp', defined by equation (6.1.1) requires a plot of the bottom plate surface temperature profile (Figure 30) . ATp' = 23.6 + .969 (29.4 - 23.6) " (.432) .111 2.333 [(6 -1) 29.4 - 23.6 - 20062)] = 42.1 F The mean temperature used for evaluating the fluid properties of the bottom plate surface was 1/2 (At —At)+At +21“ 0 O a 3 m 2 1/2 (29.4 - 23.6) + 23.6 + 2 (22.2) 2 35.5 F The free stream air velocity obtained from the hot —wire anemometer calibration curve was Ua = 179 ft/min Evaluating the Reynolds number for the water film surface Re = ———a = (179) (“333) = 6.74 x103 8.85 x10-3 Evaluating the Reynolds number for the bottom plate surface 1 ReL' = ai = (.179) ”33?; = 6.78 x103 V 8.79 x 10 1r A? = 'P-To (F) 104 29 28 27 '“r Figure 30. --Approximation of continuously varying plate surface temperature by straight line segments for sprinkling test 22. 105 The average coefficient of convection for the bottom plate surface is calculated from equation (3. 2.14) where ET; is obtained from curve (2) , Figure 18 knowing the value of ReL' . — Nu k' ' Z hc L a: _ (56.5) (23.42 x 105) q _ .333 . 2 = .040 Btu/min ft F _ 1‘ The rate of heat removal by convection from the bottom plate surface 1:. was calculated by equation (3. 2.17) 2 qc' = hc ATP': (.040) (42.1) = 1.68 Btu/min ft The average coefficient of convection for the water film surface is calculated from equation (3. 2.2) where ET. is obtained from curve (3) , Figure 18 knowing the value of Re L' — RE k hc '- L -5 (48.5) (23.48 x10 ) .333 .034 Btu/min ftz F The rate of heat removal by convection from the surface of the water film was calculated from equation (3. 2.1) C1 = H (T -T) = .034(52.7-22.2) =1.04Btu/minftZ c c s a 106 The average film heat -transfer coefficient for mass-transfer from the water film surface was calculated from equation (3. 3.4) where Nm is obtained from Figure 19 knowing ReL. a . Kin—D _ (44.9) (16.57 x10-3) m L _ .333 = 2. 214 ft/min The rate of heat removal accompanying the water film mass-transfer process was calculated from equation (3. 3. 5) and (3. 3.1) 1? H m V m RT (PS-Pa) _ (2.214) (1070) ‘ (85.7) (460 + 52.7) (24‘46) = 1.32 Btu/min ft2 The water application rate was determined from the relationship _ (weight of water) (12) e — (453.6) (62.4) (A) (time) where the ”weight of water" term is the adjusted average sample weight obtained from the application rate tray. It was found that when the actual weight of water for any series of tests (low to high velocity) was plotted against the air velocity (see Figure 31) a straight line relationship resulted. This relationship indicated that the weight of water collected and subsequently the application rate decreased as the air velocity increased. On the basis of this a 107 mfiw u” n aofludgfimw mo honno 6835.95 9me + 83 Now.) n Omm “:5 803.930 codmmmuwom . SN :msoufi Hm 3mm: mcofimfigumuov 3m.“ mcfixghmm new 60:an 3.338 63w .m Ho>o 6300200 H395 We 43:93:“. was no .3839, bum we «come 95.... .H m oudmfm can: :4 we 2.02; 005 000 Sn 00¢ con CON oo. o a d g u u u c 2— (stumb) ”WM 10 (qbgaM 108 linear adjustment to the "weight of water" term was made to correct for the "Bernoulli" effect of the air velocity around the application rate tray. Since the tray had a face area (opposing the direction of air flow) that was 17. 3 percent of the tunnel cross-section area, each original "weight of water" increment based on the quantity obtained E at zero air velocity was increased by 17.3 percent. (287.9) (12) *- w = = .220 in/min. e (453.6) (62.4) (.111) (5) 1:1. The rate at which heat was being added to the plate was calculated by equation (3.1.4). 5.20 w C AT W e p W .0 II (5.20) (.220) (1.0) (4.0) 2 4. 62 Btu/min ft The theoretical water application rate was calculated from the relationship _ qc'+qc+qm _1.’68+1.04+1.32w Wt ‘ 5.20 ATW Cp ‘(5.20)(4.0)(1.0) = .194 in. /min The ratio of the experimental to theoretical application rate was then calculated as 109 The dimensionless water application rate number (for a flowing water film on the plate surface) was calculated from the following relationship 5.20 Wt Cp (TS - Ta) w = , (13.1.3.1) C1 +(qc+qm) (.878) C (5.20) (.194) (1.0) (30.7) 2.60 + (1.63+1.73) (.878) II U1 .58 APPENDIX C C.1 Propagation of Errors for the Theoretical Water Application Rate Equation In this study and any related studies which may follow in the same area it is desirable to know what effect the precision of each independent variable measurement has on the accuracy of the dependent variable being predicted. This can be achieved by utilizing the propagation of error method of combining independent errors. According to Mickley, Sherwood, and Reed (1957) the indirectly measured quantity (dependent variable) is a function of one or more directly measured quantities (independent variable) . Q = f(q ..qn) (6.1.1) 1’ q2’ The differential change in Q corresponding to a differential change in each of the q's is 8f 8f 8f : — + —— _— dQ 1 dq1 aqz dq2 + aqn dqn (C.1.2) 3C1 If the differentials dql, dq . , dqn are replaced by small 2’ finite increments 6 qz, . . . , 6qn there results as a good approxi- mation (assuming that the quantities 6 q are small so that all higher order terms in a Taylor's expansion of Q + 6 Q are negligible) for 110 111 6 Q the expression 8f 8f 8f éQ=——- 6q + — 6q +...+ 6q (C.1.3) 1 Z 8q1 aq2 aqn n The quantities éql, éqz, . . . , 6qn may be considered as errors in ql, qz, . . . , qn, and (C.1.3) provides a means of computing the resulting error in the function and the error contribution from each independent term. The theoretical prediction equation for the dependent variable is " + + qc qc qm = ——-———— c.1.4 W 5.20 Cp ATW ( ) Introducing the expression for qc', qC, and q results in the expression m 1 1 1 .757Re1/ZPr/3k'AT '+.664Re 1/21:’r/3k(T -T )+ .664Re (2561/31; D(P-P) L p L s a L v s a RT W : 5.20 Cp 1. AT W (c.1.5) Considering all dimension and fluid property terms as constant 1 1 1 AU ”AT '+BU ”(T -T ) +cu ”(p —p) (c.1.6) a p a s a a s a w: AT W where 12 A _ .757 Prl/3 k' (L/V)/ ' 5.20LCp 1 B_ .664Pr/3k (L/v)1/2 ‘ 5.20ch 13 1 .664SC/ H D(L/1/)/2 C V 5.20 LRCp 112 The saturated vapor pressure in 1b/ft2 canbe expressed in terms of Tas 5 17.25 T sc P :2.78exp 1.43+ ___ s 238+TSC where T is in degrees Centigrade, sc 2 The vapor pressure of the air in 1b /ft canbe expressed in terms of Tas a 17.25 T ac : 1. ————-— Pa (RH)2.78 exp 43 + 238+Tac where Tac is in degrees Centigrade. With these two expressions for the vapor pressure, equation (C. 1.6) becomes 12 12 w=AU/AT'+BU/(T-T) a p a s a AT W 1/2 17.25'1"SC 17.25 Tac CUa 2.78 exp 1.43+ W —exp 13434—2—38.fi_ + SC aC AT W (C.1.9) Taking ATP' and ATW as the difference between two temperature measurements the total derivative of equation (C. 1. 9) is 113 AT ' T -T PS -Pa 6w=A———1/Zp 6Ua+B 172a 6Ua+C 1/2 6U 2U AT J 2U AT 2U (460+T ) AT a w a w a s w 1 Ual/Z(ps -pa) Ua /2 1as + ———Z——— 6TS + (4110) 6T (460+T) AT (460+T )AT (238+T ) SC 5 w s w sc P — Pa 6(RH)- ——3——— (4110) 6Tac (c.1.10) (238 + T ) ac The magnitude of the individual sources of error based on the values obtained for the independent variables in sprinkling test 26 are: 6Ua = :t 14.1 ft/min 6T = i .63 F 5 6T = i .44 C sc 6T = :t .63 F a 6T = :1: .44 C ac 6(RH) = :b 3 percent The maximum error was obtained by determining the absolute value for each term (test 26) in equation (C. l. 10) and combining those terms originating from the same source of error (indicated in parentheses by each value) . |6w| = 117'61(Ua) + 20.32 (Ts) + 2.45 (RH) + 3. 18(Ta)1X10-4 (C. 1. 11) a 114 Thus the maximum (provided all errors occurred simultaneously at their maximum absolute values) possible error due to the pre- cision of the instruments used to measure each independent variable is (6wl = .0044 in./min or, expressed in another manner w = .178 i 2.47 percent, in.,/hr Taking the 6TS error as one in equation (C. 1. 11) the relative order of magnitude of each error term is 6T -- - - 1.00 s 6U - - 4- 0.87 a 6T - — - 0.16 a (HRH) - — 0.12 The mean water film temperature and the air velocity must, there- fore, be measured with a precision which is approximately, 7 times greater than that used to measure air temperature and relative humidity if one expects the same error contribution from each inde- pendent variable in the theoretical prediction equation for w. 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