rOL-rD TA1 C-b Cee-b~ .. ffi-35 Clp•Z.. 'Technical Report EVAPORATION FROM SMALL WIND WAVES by Juey-rong Lai and Erich J. Plate Prepared under National Science Foundation Grant No. GK Fluid Dynamics and Diffusion Laboratory Civil Engineering Department Colorado State University Fort Collins, Colorado June 1969 CER68-69JRL35 Also CED68-69RL27 .. , Technical Report EVAPORATION FROM SMALL WIND WAVES .. by Juey-rong Lai and Erich J. Plate Prepared under National Science Foundation Grant No. GK - Fluid Dynamics and Diffusion Laboratory Civil Engineering Department Colorado State University Fort Collins, Colorado June 1969 CER68-69JRL35 ABSTRACT The evaporation rates from small wind-waves by forced convection in a range where the spray of water by strong wind act i on is not important has been studied in the laboratory. The effects of free stream velocity, wave conditions, and temperature difference between air and water (e i ther inversion conditions or lapse conditions) on evaporat i on were investigated, and the results were compared with previous work. The experimental data were correlated in terms of dimensionless groups, which were based on well­ known theories for exchange processe s in forced convection over sol i d surface. The transitiona l phenomenon was ana­ lyzed for evaporation as the wind blew over the solid sur­ face onto the water surface. For the lapse condition, the temperature difference was found to cause larg~r growth rates of the waves as I well as increased evaporation rates. The stratification . of air velocity atove the water surface was calculated, based on the Richardson criterion. No significant change was detected based on this criterion in this study. iii ACKNOWLEDGMENTS The author wishe s to express his deep gratitude to his major professo~ , Dr. Erich J. Plate and to Dr. George M. Hidy for their advice and encouragement at all stages of the work. The author is also indebted to other members of his committee , Dr. R. N. Meroney and Dr. R. D. Haberstroh, who reviewed the rough draft and gave many valuable comments. Finally, the author would like to thank Mr. K. s. Su and Mr. Aziz F. Eloubaidy for helping in the data collection. The work was sponsored by the National Science Founda­ tion through its contract with the National Center for Atmospheric Research, and through the grant to Colorado State University, No. GK. ' iv Chapter I II III IV TABLE OF CONTENTS LIST OF TABLES ... LIST OF FIGURES . NOMENCLATURE . INTRODUCTION . . REVIEW OF LITERATURE . 2.1 Wind over Sma ll Water Wav es . 2.1.1 Wind Profiles ....... . 2.1.2 Movi ng and Flexibl e Boundary 2.2 Analyti c a l Method to Evaluate Evapora- tion Rates .......•.•... 2.2.1 Direct Mass Balance .. 2.2.2 Turbulence Di ff u sion . 2.2.3 Profile Method •••. 2.2.4 Dime n s ionless Corre l ation . 2.2.5 So lution of Transitional Boundary Layer ' 2.3 Evaporation from Field a nd Experimental Data. 2.3.1 Semi-Empirica l Method . 2.3. 2 Evaporation by Forced Convection THEORETICAL CONSIDERATIONS . .•... EXPERIMENTAL EQUIPMENT AND PROCEDURE. 4.1 Wind-Water Channel . 14.2 Instrumentation 4. 2.1 Wave Records .. V vi i vi ii X 1 5 5 5 7 8 9 9 11 13 18 20 20 21 25 30 30 32 3 2 Chapter · V VI VII 4.2. 2 4. 2 . 3 4. 2 . 4 Air Flow .. • •• Mean Temperature . Specific Humidity . 4 .3 Experimental Procedure .• RESULTS . . . . . . . . . . . . . . . . . . . 5.1 Properties of the Water Surface . 5.2 Air Flow •. • 5.3 Mean Temperature . 5.4 Humidity Profiles and Evaporation Rates • . • . . . • . • . • . • • • DISCUSSION OF RESULTS . 6.1 Nature of t he Water Surface 6.2 Air Flow . . . . . . . . . . . . . . 6.3 Humidity and .Evaporation Rates •• 6.3. 1 Universal Concentration .• 6.3 .2 Methods of Obtaining the Ev aporation Rates .. . • 33 35 36 41 45 45 50 65 66 73 73 76 79 79 83 6. 4 ' Roughness Ef.fect on Evaporatio n Rates 8 9 6. 5 Evaporation Rates innimensionless Gr oups . • • • ••.••.••• 6.6 Stepwise Change .and-£vaporation . CONCLUSIONS. REFERENCES . vi 94 103 107 111 Table 1 2 3 4 LIST OF TABLES Boundar y conditions of flow systems . Characteristic parameters of flow sys tern . . . . • . . . . • • . . . • • • • Evaporation rates by different methods Numerical values of dimensionless groups . vii 71 · 72 86 106 Figure 1 2 LIST OF FIGURES Distribution of the diffused substance in turbulent stream (from Levich, 1962) .•• Sketch of air and water motion associated with evaporation and growing waves .•••• 15 26 3 Schematic drawing of wind-water channel · •••• . 31 4,5 Calibration curves of waves - gauge for cold and warm water cases. • • • • . • • • ••• 34 6 Calibration curves of water surface temperature . 37 7 Schematics of arrangement for measuring humidity. . . . . . . . . . . . . . . . 38 8 Reading of ppm water vapor on moisture monitor. 40 9 10 11 Variation of o, f and u against fetches m oo Variation of o and f with u* at constant free stream velocity~ ••••••••••• Therma l effect on o and fro. 12,13, Spectra diagram of water waves at different 14,15 fetch, different air velocities and air- 46 47 49 water tempe rature difference .•.••••••• 51-54 16,17, Measured velocity, temperature and humidity 18 profiles against fetch for cold a~d warm water cases .•••••.•.••••••••• 55-57 19,20 Vertical velocity profiles of air at different downs tream positions of cold and warm water cases .•.••••••• .••.• 59-60 21,22 Variation of u* and q* at different 23 24,25 downstream position of cold and warm water cases . . • • • • • • • • • • • • •• 61-62 Variation in aerodynamic roughness length with dimensionless ratio u*a/v . •..•• Vertical s pecific humidity distribution for cold and warm water cases .•.•• viii 64 67 Figure 26 Mass flux against the variation of Z •.•••• 69 27,28 29 30 31,32 33 34 35 36 37 Dimensionless specific-humidity d istribution over small water waves ...•..•.•• u*z om Correlation between Peclet number D and dimensionless roughness length .. Variation of evaporation coefficients against U •••••.•.••••••• C0 Evaporation rates against fe tch Evaporation coefficient against wave parameter . . . . . . . . . . . . . . Correlation in dimensionless groups analogous to the heat transfer problem ...•••.•. Comparison of results with Levich theoretical mode 1 . • • . • • • • • • • . • • Comparison of the experimental data with modified Levich model ..•••••••• with modified roughness 80 82 84 87 93 96 98 • • 100 Variation in coefficient. zom • • . . 102 38 Comparison of the experimental data with Spalding's numerical solution •.•.•••••• 104 ix NOMENCLATURE Symbol Definition A Constant a Wave heigh~ B Constant o r mass transfer p arameter C C p D e E f g H Ht Phase speed of significant waves Specific heat Molecular diffusivity of water vapor Vapor pressure Evaporation flux Wave frequency Frequency of spectral peak Acceleration of gravity Heat flux at wall Heat flux due to evaporation [= ~hxE] cP Dimension [cm] [cm/sec] [Joules/gm°C] [crn2/sec] [gm/cm2 ] [gm/cm2-sec] [Hz] [Hz] [cm/sec2 ] [c~l/cni2-sec] He k K Heat flux due to temperature gradient [= a(!~) 5 ] m ie L von Karman's constant [= 0.4] Eddy diffusivity for momentum Eddy diffusivity for heat Eddy diffusivity for mass Evaporation parameter [= K/Z] Length scale of turbulence eddy Concentration thickness Latent hea t of evaporation m Net amplification rate X [crn2/sec] [crn2/sec] [cm2 /sec] [cm/sec] [cm] [cm] [ca:I./gm] Symbol m w p p q s T u V X Damping factor Growth factor Pressure Definition Wave period Specific humidity Friction specific humidity Source strength Temperature Velocity along x-axis Friction velocity [= -rr-] Velocity along i-axis Horizontal axis (fetch) x+ Dimensionless x-axis Z Vertical axis z 0 Roughness coefficient z Saturation length om (l Thermal diffusivity Boundary layer thickness Dimension [gm/en/] [cps] [gm of water v~p or1 gm of dry air [gm of water v a !)Or] gm o f dry air · [gm/cm2-sec] [°C] [cm/sec ] [cm/sec] [cm/sec] [cm] [cm] [cm] [cm] 0 o* e A Boundary layer thickness for mass transfer Wave length [cm2 /sec] [cm] [cm] [cm] V p 0 "[ Kinematic viscosity Density Standard d eviation of water waves Shearing stress xi [cm2 /sec] [grn/cm3 ] [cm] [gm/en/] Symbol Definition ~ Amplitude spectra r 0 Evaporation coefficient ru Velocity coefficient w Stability function of Miles Subscripts 1 Water vapor 2 Dry air a Inner layer b Outer layer m Momentum rgn Roughness surface s Water surface t Turbulence CD Free stream WI Water Dimensionleis numbers B Bowen ratio [= He/He] Drag coefficient [= T/½pu 2 ] 00 Pr Re Gukhman number Prandtl number [=v/a] Reynolds number [= ~] . V Dime nsion [cm2/sec] Richardson number in flux form [ =- gH/c T T(au)] po az xii Symbol Definition Dimensionless numbers continued Ri Rgn Sc Sh Richardson number in gradient form [ = g(ae)/T (au)- 2 ] az o az Modified roughness coefficient [ = A (1 - ~)-l] Fi u* Schmidt number [ = v/D] Sherwood number [ = Ex/pD(q -q )] s CD Stanton number [ = Sh/Sc•Re] xiii Chapter I INTRODUCTION The problem of evaporation by forced convection from a free surface has interested many scientists because of its numerous applications to technology and water conserva­ tion. For example, many industrial processes depend on simultaneous heat and mass transfer with evaporation or con­ densation. Evaporation from the ocean plays an important rol e in controlling the humidity and temperature distribu­ tion near the sea surface. Meteorologists consider the microscale convect i ve transport across the air-sea inter­ fac e an essential process in affecting the general circula­ tion of the atmosphere (Roll, 1965). The prediction of evaporation rates i s also critical to the design and deve lopment of water resources systems to reduce water los se s from lakes and reservoirs. The phenomenon of evaporation takes place when the vapor pressure above a free surface is less than the satu­ rated vapor pressure at that surface. When a vapor pressure dif fe rence exists, the kinetic theory of gases shows that a net flux of molecules must be directe d away from the water surface. When the liquid is in contact with its saturated vapor, the rate of evaporation of molecules is equal to the rate of condensation (i.e., the evaporation and condensation are in d ynamic equ ilibrium) . There is no mass loss due to 2 the evaporation at this condition. The higher the water temperature, the higher the observed saturated vapor pres­ sure ; thus, the total amount of evaporation will be increased by raising the water temperature . When wind blows over a free water surface, the evapora­ tion mechanism becomes more complex . The water vapor near the water surface is carried away by the wind. Thus, a gradient of vapor concentration is established , which combined with the wind field provides the driving force to decrease the vapor p ressure at the interface and eventually increase the evaporation rate. The complexity arises from ·the interrelationship among the velocity, water surface, surface films, say of organic hydrophobic materials, and temperature difference s between water and air. The wind velocity distribution is affected by the surface waves (Miles, 1962; Kinsman , 1965; Plate and Hidy, 1967; and Chang , 1968). The water surface waves are affected by the temperature differe nce (Fleagle, 1956; and Hidy and Plate, 1968) and the surface film (Le Mer and Schaefer, 1965; and Hidy and Plate, 1968). These four factors are interrelated to some extent and do not independently contribute to the evaporation. It is, therefore, very difficult to describe the whole evaporation me chanism by a simple relationship. Clean water has be en assumed in most laboratory studies or field obse rvations, so the effect of surface film is 3 presumed small. Yet, the three otner major factors which strongly affect the evaporation rates have not been studied simultaneously. In previous analytical or experimental work , as will be described briefly in Chapter II, one or two fac tors were emphasized, but the temperature difference be tween air and water was left out. Furthermore, some experimental results contradicted each other . and Hayami, 1959; and Easterbrook, 1968 ) (See Okuda The purpose of this study was to provide experimental data which can be used for developing a practical method to predic t the evaporation r ates from small wind waves . The situation of this study is indicated in Figure 2, (see Chapter III, p. 2). A turbulent boundary layer develops first over a solid boundary and continues onto the water surface. When waves are generated, the approaching boundary laye~ flow become s the outer layer while a new inner layer ~eve~ops over the waves. The boundary layer for mass trans­ fer (inner layer for mass transfer) develops from the . leading edge (x = O), while the inner layer for momentum transfer deve lops somewhere at~ position downstream of the region where the water surface changes from smooth to rough. The inner layer for momentum transfer is assoc iated with a change in shear stress at the surface. For the flow far enough downstream, the inner layer has grown to encompass the whole shear layer , and this layer is in essential 4 equilibrium with the rough surface underneath. The portion of flow in this zone is define d as a fully developed turbu­ lent flow. The experiments were conducted in a laboratory channel in a range where a spray of water from breaking waves as would be cause d by a strong wind was unimportant. _The specific objectives of the study were: (1) to bring together the previous results of other investigations on predicting evaporation rates by using the ana lysis of the aut hor's experiment; (2) to inves t igate the effect of free stream velocity, fetch , and temperat ure difference on evaporation rates; (3) to analyze the transitional phenomenon of evapora­ tion when the wind blew over the solid plate onto the water surface; and (4) to correlate the aerodynamic and thermodynamic factors in the form of dimensionless equqtions. 5 Chapter II REVIEW OF LITERATURE As wind blows over a water surface, the evaporation rate is affected by velocity, wave shapes, and temperature differences. This chapter will review the available litera­ ture concerning the velocity profile over the small water waves and concentrate on the evaporation problem on the basis of analytical methods and -experimental data. 2.1 Wind .Over Small Water Waves 2.1.1 Wind Profiles The mean velocity profile .of .air above the mean water surface is needed to understand .the ,exchange of energy between air and water during wave generation by wind. Exami­ nation of Miles' (1957) inviscid shear f low model showed that a logarithmic velocity profile of air had been assumed to calculate the total energy growth of water waves. In marine ' physics, the logarithmic profile has been used to describe the air flow in the atmospheric boundary over water. In laboratory studies, Hidy and Plate (1966), Plate and Hidy (1967), Shemdin and Hsu (1966), Hess (1968), and Chang (1968) all used the logar ithmic law to corre late the mean velocity data of their experiments. Therefore, the ''law of wall" has also been used throughout the author's experiment to describe the velocity profiles, as follows: where u u* u = Q,n z k z 0 is velocity, u* 6 is friction velocity, 2 T (u = -* p (2 -1) , is shear stress at wall ), z is the roughness length, 0 and k is von Karman's constant and assumed to be 0.4. To illustrate the wind action over the solid surface onto the liquid sur face with different roughness, a t wo­ layer model of shear flow was proposed by Plate and Hidy (1967). The upstream flow and outer layer was referred to as conditions over the smooth solid surface . The inner layer was referred to as conditions over the wavy liquid surface . Both layers were assumed to follow the logarithmic law of the wall with different friction velocity and rough­ nes s length. The i nner layer will grow in depth downstream and eventually coincide with the outer layer . Plate and Hidy investigated the values of friction velocity in tran­ sition with the given velocity profile in the outer layer ups tream by qpplying the momentum balance, mass conservation, and the conditions of velocity con tinuity . The agreement between the two-layer model and the experiments was quite satisfactory except very close to the leading edge. Although Equation (2-1) has been verified by many lab­ oratory studies and field observations, the relation between z and u* are not unique (see for example, Karaki and 0 Hsu, 1967). This leads to further studies in measuring the wind profile over the wavy surface. Shemdin (1967) 7 pointed out that the velocity profile, which was given by a fixed probe, needed correction for the effects o f shifting the streamline and of wave-induced perturbation . Chang (1 968 ), using the technique of moving probe, suggested the possible existence of separation ·near the peak of wave which affected the wind profile near the water surface . Thus, the wind profile above a wave height from mean water surface deviated from the logarithmic law . Dynamical relations between z and u* have been 0 reported in some laboratory studies. Kunishi (19 63 ) sug- gested that the aerodynamic roughnes s of wavy surfaces is related to a characteristic wave height for a s mal l wave , and for the condition that the wave speed i s much less than the mean air speed. Hidy and Plate (19 66) found that a Reynolds number which is based on u* and a, (a is the standard deviation of water waves), correlated quite wel l wi th 1 z 0 on log-log paper . 2.1.2 ~oving and Flexible Boundary The concept of turbulent flow and boundary layer theory over the rigid wa ll has been studied quite extensively both theoretically and exper imentally. For the first approx­ imation, the theoretical approach of turbulent flow over rigid wall could be used to describe the wind over wavy surface , which was moving and flexible, and the experimental results could also be compared. Gupta and Mollo-Christenson (1 966 ) measured pressure s and constant speed lines in the 8 air flow over a boundary of solid waves to compare with Benjamin's (1959) theoretical prediction of phase-shift in wind-wave theory. The resulting s h ifts were only one-tenth of Benjamin's values . The reason for this is at present not understood. Zagustin et al. (196 6) carried out experiments in a laboratory flume with a moving belt in sine wave motion. Their results were compared to Miles' (1959) theoretical estimation of pressure component in shear flow model of wind-wave theory. It only agreed qualitatively, because the coupling of the flow in the two fluids cannot be neglected. The flexible and moving boundary can induce fluctua tion and turbulence, which are not considered in studying the turbulent flow over solid bounda ry . . Thus, applications o f the results of turbulent flow over solid boundaries are limited in the study o f air-sea interactions. 2.2 Analytical Method to Evaluate Evaporation Rates The purpose of this section is to present and compare various form~las to estimate the evaporation rates from liquid surface. In reviewing the mechanism for the water vapor transport from interface to the gas stream, the rela­ tively straightforward theories for direct mass balance and turbulent transport are considered first as opposed to more sophiiticated methods attempting to account for a c hanging surface structure. The dimensionless correlations, which are based on analytical equation, are also included in this 9 section . The effect of roughness changes and other transitional phenoillenon on evaporation are discussed later . 2. 2.1 Direct Mass Balance By u sing the concept of direct mass balance in control volume , an integral boundary-layer equation of mass transfer can be expressed as (see Eckert and Drake , 1959) : 6* d e E = dx J p [q( z) - q<.) u ( z ) dz ( 2 -2 ) 0 where q is the concentration of water vapor. In deriving Equation (2-2) , one basic assumption was made; that is, neg­ l ecting the vertical velocity at the interface . For a given velocity profile and c oncentration displacement, the evapora­ t ion r ate is c alculated from Equation (2- 2). This procedure was adopted in this thesis for obtaining the "measured evaporation rate E." 2 .2.2 Turbulence Diffusion A ge neral eddy diffusion equation for evaluating t he evaporation rate in t urbulent flow' is derived from a erodynamic theory, and may be written as (see for example , - Bird , Stewart , and Lightfoot , 1960): E = - pK e (2- 3) where E i s the evaporation rate , which is assumed constant everywhere, K the eddy diffusivity for mass, and q the e specific humidity . Knowledge of K e is still a challenge to the investigators of this field . In fact , Ke is affected b~ v elocity, temperature, concentration, surface condition , and 10 position. Different expressions for K have been derived e analytically in terms of many f a ctors at different boundary conditions (Brutsaert, 1965). Yet~ only under very limited conditions has even the most simple equation for verified. K been e By using the Equation (2-3), evaporation rates can be determined by averaging across the boundary layer. One inte­ gral technique approach has been described by Sheppard (1958). He considered the existence of diffusion sublayer near the water surface and applied Equation (2-3) in the form: E = - p (D + K ) ~ e az (2-4) where D is the molecular diffusivity of water vapor in air. The molecular and turbulent exchanges were supposed to occ ur simultaneously. He further assumed that K increased e linearly with height, z, or, (2-5) By inserting the relationship (2-5) into (2-4), Equation (2-4) was integrated, yielding the following relationship: ( 2-6) where the subscripts s and a refer to heights, z = 0, and z = a. Since there are few field or experimental data J.l available to confirm this equation , the extent of applicabil­ ity for this equation remains to be .verified {Roll, 1965). 2.2.3 Profile Method To derive a logarithmic law of mass transfer, t he fol lowing classical assumptions may be made: (1) t he change in scale of the eddy motion is a function of distance from the surface; and (2) t he air is saturated at the f ree sur­ fac e at the surface t emperature . Based on assumption 1, the coeffi c ient of turbulent d iffusion h as the relationship (Levich, 1965): K e '\., Q,2 au '\., z 2 au a z a z ( 2-7) where Q, is the length scale of important eddies. With Equations (2-3) and (2-7), the evaporation rate , E, can be ·expr;e ssed as: au an E = pf3z 2 _:_:;t az az ( 2-8) where f3 is a constant . Using Equations (2-1) and (2-8), - the logarithmi c l aw of mas s transfer is obtained a s: (2-9a) (2-9b) where t he value of k' is normally set equal to Karman's 12 constant as a first approximation , and in analogy to u* , q* can be defined as a friction humidity. The length scale , z , may be considered a hypothetical length or d istance om across a "layer '' of saturated air near the surface. Equation (2-9) also can be derived from the concept of Reynolds a nalogy which is d emonstrated in Schlichting ' s book (19 62 ), or by other methods describe d by Roll (19 65 ). The concept o f Reynolds analogy is based on the assumption, i.e.: T m H - = - = H , T m w w w to express the analogy between momentum, heat and mass as: and T ~ TW H H w = (1 K du + + ~) \) dy + I m E = (!_+Ke) dq+ S v dy+ c~ (2-9c) where T is shear stress; m is mass flux; H is heat flux; + + + u = u/u*' t = (t -t ) pC u*/H , q - (qw-q ) pku*/E and w p w + Y = y/yo . The evaporation rates de termined by this method depend on knowledge of suitable values of q* and z 0 m, which vary with u, T, q surface conditions and height z Unfortunately, our knowledge is still rather limited on this aspect of the problem. At present , the only way to determine 13 these values is empirically from the profiles taken by experiments for different boundary cond i tions. 2.2.4 Dimensionless Correlation Correlations in terms of dimensionless number were derived based on di fferent approaches, such as dimensional arguments,Reynolds analogy, numerical solution, and boundary layer theory. For the heat transfer p roblem , these correla­ tions have been studied extensively. Based on the analogy between heat and mass transfer, dimensionless mass transfer rate s can be obtained from heat transfer problems by replacing the Nusselt number with the Sherwood number and the Prandtl number with t he Schmidt number. Based on Reynolds analogy, Chi lton and Colburn (1934) derived the following dimensionless correlation for a smooth fla t plate: for laminar flow: Sh = const. (Sc) l/3 (Re) 112 for turbulent flow: Sh~ const. (Sc) 1/ 3 (Re)O.S where · Sh= pEx D (q - q ), s co (= Sherwood number) Sc= \)/D , (= Schmidt number ) (2-10) (2-11) The above relation, Equation 2-10, was derived numerically by Lighthill (1950). Equations were also shown in Schlichting's book (1962), which were derived from the boundary layer theory . Reynolds et al. (1958) carried out a s er ies of experiments to arrive at a similar result. They 14 considered the effect of drag force and found the following relation for turbulent flow: Sh= const. C 1/2 ( _f_) (SC ) 1 / 3 (Re) 0 . 8 2 (2-12) where 1 Cf = ,/2 u~ , (= drag coefficient). If the drag coefficient is indepe ndent of the Reynolds number , then Equations (2-11) and (2-12) are seen to be identical. Either of the equations is commonly used to express the Sherwood number. When wind blows over the water waves, the air flow near the interface is affected by the wavy surface. For the first approximation , this effect can be considered as the roughness effect on a flow system over solid boundary. Papers citing the effect of surface roughness on heat (or mass) transfer are limited and contradict each other. For example, Smith and Epstein (1957), after examining commercial pipes of different roughness, indicated that the . roughness increased the pressure drop, and thus substantially increased the heat (or mass) transfer rate. According to Kolar's r esults (19 65 ), the rough pipe has low efficiency in heat (or mass) transfer. The efficiency is defined as the ratio of the amount of energy transferred as heat per unit temperature di f ference with the amount of energy needed to passing the fluid through the tube . The efficiency of the rough tube decrease s with increasing velocity more I oin Turbul nt Stream Tur u nt Bound ry Loyer q - ill Vi cous Su oy r - rJJ= Const. * e z Fig. 1. Distribution of the diffused sub stance in turbulent stream (from Levich, 1962). X 16 rapidly than that for smooth tube . Thus , beyond some limit, fo r any combination of Reynolds number a nd Prandtl number , a n increase of roughness will no longer increase t he heat transfer coefficient . Levich (1965) proposed an explanation of the effect o f sur face roughne ss on mass transfer in turbulent flow. His model is s h own in Figure 1 . For a fully developed turbulent flow over the plate ,the c oncentration r emained constant * some d istance away from the wall . At o < z < o , there o e is a turbulent boundary layer in which both mean velocity and average concentration decreas e according to logarithmic law. In this zone, both momentum and matter are transferred by turbulent eddies . In the zone of the viscous sublayer ( o < z < o ) , turbulent eddi e s became so weak that the 0 momentum transferred by the molecular process exc eeds t ha t transferred by turbulent eddies~ In the diffusion sublayer (z < o), the molecular mechanism domi nates over t he . turbulent mechanism. The thickness of the diffusion sublayer, which forms the main resistance to mass transfer, is related to the thickness of the viscous sublayer . A viscous sublaye r u s ually will d evelop around the roughness ·peak. However, a s eparation in the flow over individual protrusions will al so occur in the case of solid roughness where t he roughness height h is greater than o but smaller than the boundary layer 0 17 thickne ss of momentum transfer o , and where the Reynolds m hu* number based on h and u (Re = --) is much larger * rgn v t han unity . The motion in the r egion , z = h , therefore must be turbulent , or at least highly agitated. The nature o f the motion in t h is zone, then, is likely not to be a function of viscosity ; it should be the function principally of height of roughness , h So the sca l e of characteristic turbulent eddies must be proportional to h , o r : 1 ~ h . (2-13) Due to the a bove hypothesis , the velocity distribution of a turbulent flow in t he zone z = h is obtained as u = (2-14) Levich suggested that this velocity distribution is valid on ly when the corresponding Reynolds numbe r uo / v 0 is greater than or equal to unity . Thus, t ~e viscous sublayer develops near the roughness peak. The t hickness o of the ' 0 vi scous sublayer is determined by the condi tion or uo 0 V 0 0 u o2 * 0 = hv ~ 1 = h/(Re ) 1/ 2 . rgn (2 -15 ) The evaporation rate can also be determined by expres­ sing the thickness of the diffus ion sublayer as 18 E:::: pD(q - q ) o . X oo (2-16) In the ca se of Sc == l, where o = o , from Equations (2-15) 0 and (2-16), the evaporation rate is given by: 1/2 ( _ q )/(vh)l/2 E :::: pDu* q s 00 and using the relation C 1/4 E == pD (-f) 1/2 u 00 (-) vh (2-17a) (2-17b) In dimensionless form, the above equatio~ c an be written as: Sh == Ex/pD(qs - q 00 ) , rgn or Sh = const . rgn (2-18) where (~) is the roughness coefficient, A is width of protiusion and rg~ refers to roughness surface. 1 For the case Sc >> 1, such as for salt in water, . Levich derived the following correlation of dimensionless groups: Sh = const. rgn C 1/4 A 1/2 (-~) (Re) 1/2 (Sc) 1/4 (-) 2 h (2-19) This has been partially verified in the case of the pipe flow for dis solved oxygen in water by Mahato and Shemitt (1967) in the range where 5 x 10 3 - 'qb = V a from ( 3-3) For an air-water mixture, E and V s can also be expressed, following Eckert and Drake (1959): E = - pls D (~) + Pls V s az s ( 3-4a) z=o D ( aq) E = s 1 qs az z=o (3-4b) Combining Equations (3-3), (3-4a), and (3-4b), the final expression for E is given by: 28 x' o* o* -(l+B) J Edx J e p(q-qs)ub dz-p(q -q ) J e dz = ub b s (3-5) 0 0 0 - where B = (q -q )/(q -1) (= mass transfer parameter). a s s In deriving Equation (3-5), the vapor concentration in outer layer was assumed small and constant. So the effect of outer layer on evaporation is only shown in ub and o* e in Equation (3-5). The values of B have been used to express the mass transfer rate in similar solution by Dono­ van, Hanna, and Yeragunis (1967). They obtained a closed form similar solution of the probl em of turbulent boundary layer mass transfer with a finite interfacial veloci ty . Their results showed that the Stanton number is a function of Schmidt number, drag coeff icie nt, mass transfer para­ meter B and Spalding function (see Spalding, 1963) with zero interfacial velocity. For the velocity of Equation (2-1) and the humidity profile of Equation (2-5a) and assuming that the value of (qb - qs) is •constant in downstream direction (which is true in the laboratory when the wind-water channel reaches steady state, see also next chapter), the evaporation rat~ is obtained from Equation (3-5) as: z z om (1 9-n om)] -~ - z e 0 z z + om (1 9-n om) ~ - - z e 0 This equation was 29 - p (qw -qs ) 1] } che cked for u a* 9- * e z om o* o* + 9- n ~)-1 z 0 * e [ ( 9-n ~) -k- z 0 ( 3-6) consistency thr ough the experimental data of this study, and the results are dis­ cussed in Chapter VI. Equation (3- 6 ) a pplies both to tran­ sitional boundary and to a ful ly developed turbulent bound­ ary. .. 30 Chapter IV EXPERIMENTAL EQUIPMENT AND PROCEDURE Most of the laboratory instrume nts and facilities used in this experiment have been described in previous reports by Plate and Hidy (1967), Hess (19 68 ), and Chang (1968), exce pt for the temperatu~e and humidity m~asurements. The 1 latter equipment will be discussed in detail in the following section, but the facilities and instruments for wind measure­ ments will be summarized only briefly. 4.1 Wind-Water Channe l The wind-water channel in the Fluid Mechanic s and Diffusion Laboratory at Colorado State University has been used for this experiment. The channe l (Figure 3) consists of a water channel 0.92 m wide and 11.2 cm deep. At the ups t ream end of the tunnel, a smooth aluminum plate 3.7 m long was installed at approximately the same height as the • water surfac e . Over the aluminum plate and water surface is a wind tunnel 1.09 m high. The channel has a plexiglass test section 13.7 m long. For a referenc e coordinate, the downstream edge of the plate was considered to be at x = 0, and along the water tunnel was a positive direction of x-coordinate (Figure 2). The air velocity was controlled by an axial fan at the outlet of the tunnel . The air flow was made uniform at the inle t and outlet sections through mesh screen and honeycomb ADJUSTABLE TURNING VANES NO.2 HEATER (IOOOW) WATER SURFACE\ NO.I HEATER (IOOOW) RUBBER SECTION TEST SECTION (44ft x 24in x 5Oin) NO.3 HEATER (2OOOW} VALVE WATER STORAGE TANK :· .. ~--- -· •.'·:· ::_.-~ .. _. ·. •···. •:.: SCHEMATIC VIEW OF WIND-WATER CHANNEL Figure 3. PITCH FAN w FAN STAND I-' SLOPE ADJUSTMENT MOTOR WITH . DRIVE SCREW 32 grids, as de scribed in previous works. To hea t the water, imme rsion heaters of a total capacity of 5,000 volts were installed at four different positions in t he test section at the bottom of the channel (Figure 3). These heaters were controlled by powerstats located out s ide the channel. An instrument carriage capab le of manual horizontal positioning and automatic vertical position ing was used to hold the sampling tube , t hermocouple and the Pitot-static tube during measurements. The movement of the carriage was remotely controlled by a counter in th~ control panel, located beside the channel. Because previous work has demonstrated that the properties of fluid flow in the channel are approximately uniform in a cross-stream direction hori­ zontally, measurements in the study were made only at dif­ ferent distances, x, along the centerline. 4.2 Instrume ntat ion 4.2.1 Wave Records A capacitance probe whose sensor was a 34-gauge magnet wire, was installed to continuously measure the water surface displacement at a given distance, x, as a function of time. The gauge was constructed so that the vertically stretched wire and the water surface formed two "plates'' of a condenser, and the wire insulation (Nyclad) provided the dielectric medium. The difference in capaci­ tance, due to the water depth, was measured by a capacitance bridge d eveloped in the Engineering Research Laboratory at 33 CSU. The circuit diagram of the bridge is shown in Figure 4 in Chang's thesis (1968). The output signal of the CSU capacitance bridge was fed to an oscillograph recorder. The capacitance gauge-oscillograph combination was calibrated against water depth after each series of experi­ ments . The calibrations proceeded as follows: t he water in the channel was discharged to the sump very slowly, and the water depth and the output of the r e corder was simultaneously read for each period of 0.5 cm of water depth. Typical c ali ­ bration curves, which indicated a linear proportionality between water depth and recorde d elevation of the water sur­ face, are shown in Figures 4 and 5. For further statistical analysis , the wave record s were dig i tiz ed at equal time intervals of ~t = 0.02 seconds. Cal­ culations were made to obtain for each run the values of standard deviation, cr , amplitude spectra, ~ (f), and fre­ quency of maximum spectral density, f . The statistical com- m puting metho~ used for these properties was that of Blackman and Tukey (1958), as discussed by Hidy and Plate (1966). (See also Chang, 1968). 4.2.2 Air Flow The mean air velocity was measured by a Pitot­ static tube , using a 0.325 cm OD probe manufactured by the United Sensor Co. The probe was placed on the instrument carrier which ro se or descended step-by-step to give the mean air velocity profiles. In addition to the traveling 34 8000 ..-~-..--.-----,-,..--,----,,--,--,-,--,-,--,-,--,-.--,-.--,-.-, 7000 6000 .,,; E 5000 4000 3000 2 ooo 2L_5_1-_..1.._..1,__.L_3...1._o_...1___1 _ _J___J_~3_L:-5-L----L-.._____._--:;4:':_o::-__.___._....,__~-:4 _5 Water Depth (in) Fig. 4. Calibration curves of waves - gauge for cold and warm water cases . 9000 ,--.--~--.-.--.--,---,---,-,--,---,r--.-.--.---,--,----.-,,--,--, 8000 7000 ~ 6000 5000 4000 3o oo3L_o_.L,C____._ _ _,__,__3~_~5_.__....___._....,_-:-4_~0-~~-.._~-4~_5=--......____.._....___..__-::'50 Water Depth (in) Fig. 5. Calibration curves of waves - gauge for cold and warm water cases . 35 probe , a fixed Pitot-static tube of 0.65 CB OD , which yielded the reference velocity for every run, was located a bove t he aluminum plate and outside of the boundary layer. To calculate the air flow, t he Pitot tubes were con­ nected to an electronic micromanometer (Transonic Equibar Type 12 0) . The manometer measured the difference between total pressure and the stat ic press r e , from which the mean air speed is calculate d by means of t he fo l lowing relation­ ship : u(x, z ) = ✓~p x 16.5 (m/sec) . (4-1) The pressure dif ference r ead off the micromanometer instru­ ment was c alibrated against a water manometer (Flow Corpora­ tion , Type MM2 ). 4.2.3 Mean Temperature The fr ee stream temperature , T , was measured by a 00 merc ury-in-glass thermometer placed on t he upper frame of carriage whioh was located 10 cm below t he top of the channel at each run. The local air tempera t ure, T, in the boundary layer was measured with a 40-gauge Copper-Constantan thermo­ coup l e (Thermo-Electric Co.) us ing he f ree stream t empera­ ture a s a r eference. The output of the thermocouple was read with a potentiometer (Leeds & Northrup Co., Mode l 868 6) . Vol tag e differences were converted to temperature differences (T-T u sing standard calibrations , as given in the National 00 Bureau of Standards Circular 561). . 36 The surface temperature of the water, T , was measured s with an infrared radiometer (Branes Engineering Co., Mode l IT-3). The radiometer was calibrated against a known, stable black body source (Mar latt and Grassman, 1968). To obtain T, the water t emperature T, which was measured by a ther-s w mometer immersed 2 . 5 cm b eneath the mean water surface, was also determined . With the same free stream velocity and temperature, the water surface t emperature was found to be a function of water temperature. This relationship is plotted in Figure 6, covering the range 10°C to 32°C. This curve is used as a calibration for obtaining the local surface temper­ ature throughout the wind-water tu _ne l. 4.2.4 Specific Humidity The specific humidity of the air-vapor mixture was measured by sampling the gas stream through a static pressure I tube connected to a Consolidated Electrodynamic Co. moisture I monitor (Model 26-303) at a constant flow rate maintained by a vacuum pump. The sketch of the arrangement is shown in Figure 7. The sampling tube was made of a 0.163 cm OD brass tube. The Pitot-~ube, thermocouple and the sampling tube were set 2 cm apa~t on the instrument carriage. The only inlet for sampling the gas was a side hole in the tube to avoid the pressure drop in the tube. The brass tube was connected to the moisture monitor by a 5 mm OD Teflon Tube. The brass and Teflon tubes were recommended by Consolidated Electro-dynamic Co. because they absorbed only very little -u 0 30 20 10 0 sl_ _ ___j ___ l~2---1---~,6----l..--~20L-..--'----2~4---'---2~8--........ --='32 T5 (°C) Fig . 6. Calibration curves of water surface temperature. w -.J By Pass Valve Flow Meter Liter Ballast Volume Sampling Probe Control Valve Bulb-0-Meter Moisture Monitor Fig. 7. Schematics of arrangement for measuring humidity. Vacuum Pump w co 39 moisture . The bypass value was use d to maintain a large · flow rate to insure stable operation , while only small s amples of gas were fass e d into the moisture moni tor. A soap fi l m type flowme ter (a " bubble - o-:neter ") and a valve were used to ma intain a constant flow rate through the moisture monitor . Two one-liter ballast volumes in the outlet line smoothed the pulsating motion caused by the vacuum pump . The moisture mo~itor offered a practical measure of trace water in the gaseou s mixture , s ~nce the instrument ' s electrolytic c e ll is used spe cifically to measure water mo isture. The electrolytic c e ll contained t wo platinum wire s , the space between was coated with phosphorous pentoxide (P2o5 ) which is a strong desiccant . When water vapo r wetted the P2 o5 , a potential afplied to the wires change d , ~reducing a me asurable electrolysis current. This current is directly proportional to the mass flow rate of wate~ vapor into the cell. Electrolysis o f the water absorbed in t~e P2o5 continuously regenerated the cell, thus permitting it to continuously measure all the moisture in the sample stream . The CEC moisture monitor u sed to detect low moisture contents was designed to operate at a constant flow rate (i. e . V=lOO cm3/min .). The instrument ' s range of applica­ t ion was extended to higher moisture contents by reducing t he flow rate . Thi s was done by by-passing the pressure 39a regulator of t he monitor and r epl acing it by t he bubble-o­ meter flowmeter c ircuit described above . The instrument was c a librated for u se at different flow rates by t he following procedu re . A flow of constant humid i ty was conducted through t he moisture monitor and the flow rate was d e creased in ste?S - The relationship between flow rate and apparent moistur e content obtained in t his manner is shown in Fig. 8. It is seen that the relation v1;v 2 =(ppm) 1 /(ppm) 2 holds for whole t esting range . The true moisture contents (ppm) 2 c an be expressed as : (ppm) 2 = (ppm ) 1 x 100/Vl where (ppm ) 1 is t he meter reading at flow rate v1 = 100 cm3/ min. Through most of experiments of this study, the flow rates were reduced to 10-15 c m3/min. Only in two c ases , in whic h the water temperature and wave height were larges t, the I flow rates were reduced to 3 cm3/min. The humidity data us ed ♦ were di fferences between local humidity in t he boundary layer and that of the fr ee stream gas . In thi s manner some of the systematic error due to the errors i n the low-flow-rate mea surements are cancelled . The r esponse time of t he mois­ ture monitor was found to be l ess t han 30 seconds (approxi ­ mately 63 % in 30 s e cond s to a step change in either dire ction ). A sta ~ l e r eading was obtai ned by tak ing the reading at each point at least t wo minutes after the probe had been positioned and t he flow thr ough the meter adjusted . 100 q = Constant Humidity 80 . - I C E I ,,., 60 E u - (I) -0 ct: 4 0 ~ .i::.. 0 LI.. 0 20 Oi:c;.. ___ _.,_ _ _ _ ...1., ___ -J. ____ ,__ ___ ..i.,.. _ __ ...1., ___ --1, ___ .....,1..._ ___ ..__ _ _ ___, 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Scale Reading ,. ppm Water Vapor Fig . 8 . Reading of ppm water vapor on moisture monitor . 41 The output reading of the moisture monitor was in part per million (ppm) of water particles by volume . For convenience , p pm was transforme d to specific humidity by t~e relation : 18 10-6 q = (ppm) X 2§" X gm of water [gm _of dry air] ( 4-2) The above equation came from the definition of q and ppm . 4 . 3 Experimental Procedure Local measurements of mean velocity , temperature, and specific humidity of air were simultaneously taken by a Pitot-tube , thermocouple , and sampling tube for conditions o f steady flow. To achieve such conditions for a case where the water was cooler than air , the following p rocedure was used . The tunnel was allowed to operate at a normal air flow for two or three hours . During this period, the water t emperature would stop decreasing as a result of evaporative cooling and stay at a nearly constant value , deviating between+ 0. 1°C from an average value. The deviation was c aused by room temperature fluctuation . This case was defined as the "colj-water condition, '' where T = 12° ~ w 15 °C and T = 2 0 ° - 25° C. Under such a c ondition, aT/az 00 i s positive , and stratification in air flow is stable . When the air temperature was lower than the water tempera­ ture, it was difficult to achieve stable temperatures, because of the limited heating capacity of the water channel . 42 The following procedure was used to slow the temperature d rop during the experimental time period . First , the water was heated to about 38°C, and the fan was started to blow a ir over the water surface for five to ten minutes . The f an was stopped, and the water was heated again . After r epeating this process several times , the temperature of t he water throughout the channel was very uniform . o ne run of the expe=iment , it would drop 1° - 2 °C. During This c ase was defined as the warm-water condition , where aT/az was negative , ~nd the air flow unstably stratified . The warm-water cases were c hosen as T = 27 ° - 34°C and T = W 00 20 ° - 25°C . During all operations , the water level decreased gradually due . t o t~e evaporation , and a gap between the a luminum plate and the water surface developed . 'I'o maintain a condition of steady flow , it was then necessary to add water (cold or war~ ) continuously i nto the channel to keep t he water surface depth at x = 0, within 0. 2 mm of the same d epth . This would assure the same boundary condition at a d iscontinuity l ine between the solid and the l i qu id , as well a s a satisfactory smooth transition from the plate onto t he water surface . The velocity , humidity , and temperature distribution profiles were obtained for three different series of experi­ ments . First , the data were collected simultaneously f o r T , u and q at different downst~eam positions (x = 56 cm, 43 112 cm, 214 cm, 460 cm, 778 cm) with different reference velocities (vf = 4.17 m/sec. 5.64 ~/sec, 6.89 m/sec) for the cold-water case. For the warm-water case, the data were col lected simultaneously for T , u and q at different downstream positions (x = 116 cm, 214 c m, 460 cm, 610 cm) with a moderate temperature difference of T - T = 5° ~ 8°C s 00 and then with a small temperature difference of T ~ T = s 00 1° ~ 3°C. For the cold-water case, a third series of experi- ments was carried out at a fixed fetch of x = 610 cm with a variation of air velocity corresponding to v = 8 m/sec, 00 8.8 m/sec, 9.9 m/sec , 12.1 m/sec, and 13.3 m/sec. The three meas~ring instruments were set parallel on the carriage 2 cm apart . The carriage was lowered close to the mean water surface, yet far enough away to avoid the instrument being hit by splash of the highest waves. The measured vertical distance from the mean water surface was set equal to z for all calculations. The carriage was raised step-by-step, and measurements were taken after each step until all instruments indicated an output which was independent of height . Then, it was lowered again, step-by­ step to reach the initial vertical position . The lowest values of z were measured before and after each run to the vertical displacement . The step-by-step method with a sampling period of two minutes would give enough time for the moisture monitor and the thermocouple to respond . The measurements close to the water surface were taken with 0.5 44 c m pe riod at different heights. The distance was increased a way from the water surface . Eight points per r un were taken when the carriage neared the l eading ·edge and increased to fi fteen points in downstream position. Finally, after each s eries of exper iments , a recording of the waves was t aken on a strip chart r ecorder . 45 Chapter V . RESUL'I'S 5.1 Properties of t he Water Surface The propert i es of the water surface are importa nt fac­ tor s affecting the evaporation by increasing the apparent sur face area, and by changing the air flow near the surface . The c harac teristics of the surface waves c an be determined experimentally from the wave records. The re sults of typi­ cal statistical c omputation for the water surface displace­ ment were the standard deviation , o , wave energy spectrum , ¢(f), and frequency , f , o f t he spectral peak . m . The values o f o and f r epresent the geome tric properties of signif­ m icant wave s . The wave energy spectrum ¢ (f) is t he Fourier transform of the autocorrelation function of the water s ur - face displacement . Tabl e I. The values o f o and f are given in m To compare cold- and warm-water cases , the variations of o , f , and u with f etch are shown in Figure 9, and m co the effect of u* on f and o m at a fi xed position , x, is s hown in Figure 10 . For the cold-water case, t he stan­ dard deviation o was found to incr2ase linearly at a fixed downstream position with friction v e locity u*, and al so with fetch x. The peak fr equency f d i d c hange m rapidly at s mall values of x and u*, but its variation was sma l ler for the larger values of x and u*. 12 14 II 12 lo ~ Cold-Water Case fm.... ~) ~ Hot -Water Case r ~ Cold- V ater Case o- .... A) - Hot - ater Case 0.3 Ua, l □ ~ Cold -Water Case ~ Hot-Water Case 10 10 - 0.2 0 9 8 Q) (/) ....... q E E -- - 0 8 3 ~ ::, 8 6 - O"I --~-.,--,. 7 4 ,,,rf - 0 .1 ,,, --------cr -.-o- ----------- 6 2 5 0 .__ __ __._ ___ _.__ __ ___. ___ ....._ ___ ,__ __ __._ ___ ...__ __ ~---- --- 0.0 0 2 3 4 5 6 7 8 9 X { eters) Fig. 9. Variation of o , f and u against fetches . m oo 4 0.6 q -- 3 0.4 (") N 3 .i:. :c - -.J -E - 2 0 .2 12~0---'---4...L.0----'----'6'-0--..,_ __ 8...._0 _____ IOO0.0 u* (cm-sec- 1) Fig. 10. Variation of a and fm with u* at constant free stream velocity. 4 8 The effect of x and u* on o and f has been m shown in Figures 9 and 10. The effect of temperature dif­ f erence between air and water on wave geometry is shown in Figure 11. However , due to the limited capacity of the heating facility of the wind-water channel , the equilibrium fo r · the warm-water cases was hard to reach . So the data o f cold-water cases were adjusted to compare with warm-water c ases . In Figure 11, the values of u* of both cases are identical . This was accomplished by using the relation of Figure 10 to r ~duce the values of o and f of c old-water m cases until both cases had the same value of u*. The data in Figure 11 showed that in warmer water, a lower frequ ency f is found at x > 3 m. Also , the standard d eviation , o , m ii larger with the exception of t wo stat:ons , x = 460 cm and x = 778 cm. The wave energy spectra of wind waves , as described in Chapter IV , are shown in Figures 1 2 , 13, 14, and 15 for dif ferent conditions . In the spectral diagram the difference between cold- and warm-water cases is difficult to dis- tinguish . The peak spectra f , o f course, decreases with m increasing fri c tion velocity u*, as determined from other work (for example , Hidy and Plate , 1966 ). An f -S law for ~(f), which was suggested by Phillips (1 966 ) to show the equilibrium range of the gravity waves , can be applied to t he data to describe the slope of the envelope to all the spectral curves . However , beyond the peak , the value of -N ::I: - E - 12 o-, Warm-Water Case 0 .3 10 A f m, Cold-Water Case 8 0 .2 6 a; Cold-Water Case f m, Warm-Water Case 4 0 .1 2 0 .._ ___ o..__ __ __,__ ___ _..._2 ___ ...... 3 ___ ....1.4 ___ _.5 ___ _.6 ___ ....J7------s...._ __ ....J9 °·0 Fig . 11. X (Meters) Thermal effect on a and f m q -(") ~ I.O 3 - 50 ¢ ( f) decreased following a -7 f law u p to f = 2 f . m For large values of f (f > 2 f ) , m

13 Hz by Hicks (as quoted by Phillips , 1966 ) as an equilibrium range for pure c apillary waves. All the r esults of the amplitude spectra of wind waves indicate the consistent agreement o f results of this e xper i ment with thos e of previous works such as Plate et al . (19 68 ), and Chang (1 968 ) . The data further suggest that there is no detectable thermal effect on frequency spectra . In wave e~ergy spectra of c old- and warm-water c ases for f > 13 Hz, the effect of capillary becomes a primary influ ­ ence on wave behavior . The temperatur e showed no effect o n the break frequency from gravity wave behavior to c apillary wave behavior. 5 . 2 Air Flow . The measured velocity distributions above aluminum plate and water surface are shown in Figures 16 , 17, and 18 fo r cold- and warm-water cases . At the s ame reference v elocity, the free stream velocity u 00 inc reased in the downstream direction . This was due to a favorab le pressure gradient along the positive x-direction . This was due to a favo rable pressure gradient along the positive x-direction in the c hannel . The boundary conditions at the water sur­ fac e are given in Table I. The dimensionless velocity ♦ 51 0 o~ :t: 38(200Log) X=l.12m Cold- o ~ ~ 44 (200 Log ) X =4.32m Water 6 ~ # 4 7 ( 200 Log ) X = 6 . Im 108 L___....L.____.___,__..L...L...L.L..LJ._ _ _.____,L...-L-..L...L....L.J.....L.L __ ,___,__..,_,LJ-l ........... 10-I 10° f0 I 102 f(cps } Fig. 1 2 . Spectra diagram of water waves at di fferent fetch , different air velocities and air-water temperature difference . 52 103 - 0 D 48~ 6 49 COLD·-V!ATE R AT 0 52 wm1 0IFFEr:[rH ◊ 53 .. J . 10° 10 1 f(c ps) Fig . 13. Spectra diagram of water waves at different fetch , d~f ferent air velocities and air-water tempera ture difference . Hot­ Water 53 o ~ # 5 7 ( 200 Log ) o~ # 61 (200Lag )) 6 ~ # 67(66) ( 200 Log ) • ~ # 91 f(cps ) High­ Temp. Diff. Fig. 14. Spectra di agram of water waves at different fetch, different air velocities and air-water temperature differenc e . '9t 162 103 10~ 10 5 ♦ Hot ­ Water 54 •• 11 □ rs ll. •□ L!I:, I I R.' 0. ·1 e !fr Ir. ·j ib • \ o/ ~ If • ~ ~I Ji□ r A A/ J 0 m o ~,:, 56 (200 Log} X=l.l6 □ ~ #64 (65}( 200Log } X=6.lm }~~~~­ m Diff. A~ .'J,60 (100 Log} X=4.32 • ~#90 (200 Log ) X=7.78 m I08 L__..1,__J,_..J._L.LJ...J....l.L-_...L.,_J__J,_..L..J.....L.LJ'--'--. _ _.___.__~_._._._._.,_, 101 10° 101 102 f(c ps ) Fig . 15 . Spectra diagram of water waves at different fetch , d ifferent air velocities and air-water temperature d ifference . 20 E u 1 • 22.5 fl/sec I re I u"'=22.6ft~sec\ ,I u"'=22.6f t/sec I I I I '-~-HE~i . I u"'•21.6ft/sec u 10 N q"' -q' = - 0 .0051 0 - U } a - q- q, COLD-WATER, HIGH VELOCITY 6 - T-T, CASE u"'•26ft/sec QCl)-q, = - 0.00468 s· • u"' =26.I ft/sec QCl) -q, = -0.00659 9 T"' -T1 =972°C u"' =28.2 ft/sec I qCl)-q,=-0.00486 \ I ✓ ✓ \/_,/ ,,." 0 b===777~~'L-.-.=.«~~=-='¢-~=---~~....::=-----..1.._-------i;.,_T.~,"-. ::10:-:_1::0 -=c-----i;~T.~,'-. 1:--:4-:_5::0-=c-----4 "-:T.~,.-:1-=o-=.1::-·c=---7 0 Fig . 16. Water q,=0.0082 q,=0.0104 q,=0.082 Tw• l2°C Tw = l5.4°C T•• 12°C 3 4 5 6 7 8 X ( Meters) Measured ve l oci ty, temperature and humidity profiles against fetch for cold and warm water cases. 9 LT1 Vl 20 u,.,•20ft/sec 10 0 Fig. 17. T1 = 29°C q,=0.0275 r.=30°C 2 T1 a33.2°C q1 =0.0328 Water T,.=34°C 3 4 X ( Meters) T1 =29°C q, =0.00256 r .=30°C 5 uco=25.2ft/sec T1 =3l.2°C q,=0.0294 T.=32.6°C 6 7 o ~u Velocity □ -q -q, A~T-T1 8 Measured velocity, temperature, and humidity profiles against fetch for cold and warm water cases. 9 tn O"I 20 E 0 10 N 0 ua,= 20ft - sec-1 ua,= 20ft/sec z Lx 0 ua,=20.9 ft/sec I ua,=20.4 ft/sec qco-q,= - 0.0213 Ts=305°C T,.=31 .a 0 c Qs=0.028 2 a: 3 X (Meters) um= 24 ft/sec \ / Air ' ,, .,,., T1 =24.3°C T .. =2s.2°c Water q. =0.0 191 4 5 6 7 Fig. 18. Measured velocity, temperature, and humidity profiles against fetch f or co l d and warm wat er cases. 8 LT1 --.J f · 1 (u-) pro i es u (X) 58 for different fetches are shown in Figures 19 a nd 20. Like· other data taken in the CSU tunnel, all o f t hese profiles also can be correlated satisfactorily by the law of t he wal l (Eq . 2-1). In this study , the values of u* were obtained from the velocity profiles by the following method using two l evels of an assumed logarithmi c distribution: (5-1) where and are velocitie s at and which lie in the logarithmic part of the distribution curve . The v alues of u* found in thi s manner are given in Table II. For the fully d eve_oped rough flow defined by t he r egime b eyond a> 0.15 c o , the values of u* a re also given by an empirical formula given by Hidy and Plate (1 967 ) who com­ bined their findings with earlier results of other workers : = 0.0 185 u 1 · 5 (cm/sec). (X) ( 5-2) Values of u* d etermined by use of Equations (5-1) and (5 -2) agreed within+ 5%. The variation of friction v e locity with fetch is plotted in Figures 21 and 22 , the curves showing that the friction velocity changes ma inly with the free stream velocity and only to a small d egree with the temperature difference (T _ T Compare d with the previous work of Plate and Hidy s (X)) • 59 101 Cold Wote, I 0 - #35 □ - ¢38 A- ¢4 1 ◊ - #44 v - #47 E u N u/u co Fig . 19 . Vertical velocity profiles of air at d ifferent d ownstream positions o f c old and warm water c ases . E u N 101 ( o ~ #57 □ ~ #59 Hot Water t::.~ :fl:G I ◊~ #67 60 10° .._ __ .,___ __ .,___ _ __.,__.....L.__,C--_--1 __ __. __ -J 0.4 0 .6 0 .8 1.0 u /u a:, Fig. 20. Vertical velocity profiles of air at different I downstream positions o f cold and warm water cases. ---------------~---~-----,------.-----r-----,-4 60 0 40 a, -!!? E (.) * :, 20 / -----□--------□ 6/ □ 0 / { A ~ High Vel. / q- o' * - Med. Vel. / Cold ater u ~ * { 6 ~ High Vel. □ ~ Med . Vel. 0'-----.__ ___ ..__ ___ ..__ ___ ..__ ___ ..__ ___ ,__ __ ___, ___ __. ___ __, 0 2 4 6 8 X (Meters) Fig. 21. Variation of u* and q* at different downstream position of cold and warm water cases. -6 -12 -14 -16 - -0 O E E O> O> V 0 -u (l) V) ' E u * :, ,-----~ ---,~- ----.--- -.-----r-----.----.----,-2 60 40 20 Hot Water { ~ High Tem . Deff. q ~ * o ~ Low Temp. Deff. u. ~ ( □ ~ High Temp. Deft. o ~ Low Temp. Deff. 0 L----L-------'------'-----'-----'-------'--------'------- 0 2 4 6 X ( eters) Fig . 22 . Variation of u * and q* at different downstream position of cold and warm water cases . 8 - 4 - 6 - 8 -10 -1 2 -1 4 * CT I O'\ N 63 (1967), u* showed a more gradual change from aluminum plate to the wavy surface , probably because the free stream v eloci ty in this experiment was smaller than in t he previ ­ ou s one. An increased free stream velocity tended to i ncrease the sharp change of u* at the transitional zone near x = 0. The values of z were obtained from Equation (2 - 1) 0 a fte r the values of u* had been obtained from Equation (5-1). Numerical values of z 0 for e ach run are given in Table II. According to Hidy and Plate (1966), the values of roughness length over small water waves, z02 , c an be correlated empirically with a Reynolds number based on friction velocity and the standard deviation of the waves . Hidy and Plate reported an empirical formula for I at short fetch e s -4 z 02 = 1.4 x 10 in a wind-water c hannel as : (5-3) Such a correlation for t his experiment is shown in Figu re 23. It indicated satisfactory consistency between the experimental data at high and low free stream veloc ities of air, and earlier data o f Plate and Hidy (1967). The corre lation also held r easonably well for both warm- and co l d-water cases . However, t here may be a s ma ll d ifference related to a thermal effect on momentum transfer, which will b e discussed in the next sect ion. -E u -0 N 64 ,o-' 0/ o/ 0/ / 3 HIDY AND 102 /~ PLATE (1 966) fJ 0 / 163 / 0 ~ Cold - Voter Case Hot - Water Cose 0 J: (High Temp.Diff.) Hot- Voter Case ( Low Temp. Diff. ) 10-4...__ _ __._ _ _,_____.__,__..__._..L...L....__ _ ___, _ __.___,____._..L..J.~-'------l--L--'-.-L...L..I-L.J...J I 10 100 1000 Fig. i3 . Variation in aerodynamic roughne ss length with dimensionless ratio u* o/v . 65 5.3 Mean Temperature The vertical temperature distributions are also shown in Figures 16, 17 , and 18 over the different fetches. The temperature profiles have been modified by assuming that the thermal boundary layer had the same thickness as the concentration boundary layer , thus avoiding the t emperatu re gradient caused by elevation in t he building. During summer time, the temperature in the building increased about l°C per meter in elevation . There was no constant temperature zone. In other words , the t emperatu re , T , was selected as the temperature at 00 z > 8* , where 8* indicated the t hickness of concentra-e · e tion boundary layer. In t he cold-water case , t he ten- dency of increasing the gradient of temperature along the downstream direction was similar to the gradient of velocity. Thus , the t endency of increasing shear stress ; i r downstream position was similar to the heat transfer . In the warm-water case , the temperature profiles near the x = 0 showed a small bump, the shape of which enlarged, then d isappeared along the downstream direction. This may have been due to the effect on an outer boundary layer formed along the cold aluminum plate . The thermal condi tions at the bou ndary are given in Table I. It is difficult to construct a dimens ionless temperature profile in this experiment, since the temperature difference was small, and there were so many factors to influence such 66 a small temperature dif ference . For example , the fr ee stream velocity , the wavy surface , evaporation r ate , radi­ a tion and c onduction all contributed to this transfer mechanism . For better results , further study i s recommended. 5. 4 Humidity Profiles and Evaporation Rates The spec ific humidity at the water surface c an be de termined u nder the assumption that it is equal to the saturated humidity qs surface temperature . at T, where T s s With the value i s the water at z = O, the measured humidity profiles at the different fet c hes are shown i n Figures 16, 17, and 18. In the cold water case, the gradient of the humidity profile , which showed the amount of mass transfer rate , increased slowly with x when x was small, but the g radient started to increase fa ster when x was large , indicating the effect of wind waves . In the warm-water c ase , t he tendency was similar , except at x·= 610 c m. At t hat particular point, t he humidi ty difference between different height seemed linearly increasing with height . Several vertical distributions of humidity a t dif­ ferent heights are shown in semi-logarithmic form in Figures 24 and 25. Most of t he data points form a straight line o n th i s type of plot , indicating that a logarithmic profi l e seems to be a u seful approximation for many of the humidity profi l es . The application of t he logarithmic p rofi l es will be discussed in Chapter VI. E <.) N Fig . 2 4 . 10 1 N 67 10 1 o ~ ""35 o ~ #38 l 6 ~ #41 Cold-WoterCose 0 ~ #44 Q ~ #47 io-' oL.0------0 .L..2- °'_ b_-"b--~o .L..4------=-o .6 °' b--=--ti Vertical specific humidity distribution fo r c old and warm water c ases . o ~ # 57} o ~ ..;,59 6 ~ # 6 1 Hot-Water Cose ◊ ~ #67 10..J '----'--- -'-:---'----'---- L----L-----' 0 .0 0 .2 0 .4 Fig. 25 . q - Qco q, - Qco 0 .6 Vertical specific humjdity distribution fo r cold and warm water cases . 68 The evaporation rat e , E , in gm/cm 2 /sec c an be evalu­ a ted from the experimental data for q(z) and u( z ) by considering a mass balance on a contro l volume in the air (F i gure 2 ). By considering the mass flux in terms of such a volume and n e glec t ing V s at the surface (see Chapter II) , co E = p~x J u(z) 0 [q (z) - q ] dz . co (5-4 ) To find the experimental values of the evaporation rate from Equation (5-4 ), the value s of u(q - q) were plotted co against z on linear paper. An optical planimeter (Mi lano Co., Type 236) was u sed to obtain the value of the integral. When z was small , it was not possible to obtain experi­ mentally the data for q and u, so they were extrapolated according to logarithmic law to give q and u at small z. Some typical curves of this linear plot are shown in I Figure 26. The values o f E calculated by this method ♦ a re given i n Table III . The values of E decreased along the downstream direction until x is a pproximately 3 m, then the values of E increased . This indicates that the waves s eemed to increase the evaporation rate in this experiment once thei r amplitude exceeded a= 0.l cm . The thickness of the conc2ntration boundary layer, c:, was defined as the value of z, where the local spe­ ci fi c humidity difference was equal to 0 . 01 x (q - q ). s co -8 O"' I O"' -::, 1.2 ,----,.---r--~--,----r----y----,----.---.---r---.---.-.---~-ic--~--.---~-.----i 6 \ 6 1.0 0 .8 0.6 0.4 0.2 0 0 0 .4 • 0.8 1.2 1.6 0 ~ # 33 Cold - { 0 ~ # 36 oter 6 ~ #42 Unit = 0 .852 2.0 Z ( in .) 2.4 2.8 Fig. 26. Mass flux against the variation of z. 3.2 3.6 4.0 70 The boundary layers of vapor concentra tion are shown in Figures 16, 17 and 18. The effect of thermal stability in raising the internal boundary layer was predicted by Elliott (1968). The tendency of his results are consis­ tent with the experime ntal data of the present study. Th~ strong effect of surface temperature on evaporation can be seen in Figure 18 at the ups t ream position or transition zone, while the values o f 8* showed a sharp e increase due to the increasing of positive t emperature dif­ ference between air and water . For the fully developed turbu lent flow, the boundary layer thickness of mass trans­ fer eventually became the same for both cold- and warm-water cases . The values o f o* , which are obtained from experi-e mental data, are given in Table II. Another concentration thickness , le , has been defined by Kays (1966) as: 1 00 le= (5-5) The concentration thickness of the boundary layer in mass trans fer problems is analogous to the momentum thickness of the flow field. The values of £e are connected with the flow of mass through an area normal to the surface. The values of le are listed in Table II. In solving the boundary layer equation of mass flow, important characteristic l ength. le should be an Table I . Boundary c onditions o f flow systems . q x103 q x103 f s 00 Run T T T gm of water gm of water X u a m w s 00 00 (gm of dry air) (gm of dry air ) No . (cm ) (m/sec ) (cm ) • (Hz ) ( oc) ( oc ) ( oc) 34 56 5 . 64 -- -- 13 11. 9 25. 6 8 . 8 4 . 28 37 112 5 . 67 -- - - 12 10 . 7 25 . 0 8.1 4.14 40 214 5 . 95 -- -- 12 10 . 7 22 . 6 8 . 0 2 . 98 43 460 6 . 40 - - -- 12 10 . 7 20 . 8 8 . 1 3.7 46 778 6 . 95 - - -- 12 10 . 7 20 . 2 8 . 1 3 . 31 35 56 6 . 89 -- -- 13 11 . 9 24 . 2 8 . 8 3.72 38 112 7 . 24 0 . 0245 10 .5 12 10 . 7 25 . 0 8 . 2 3 . 72 41 214 7 . 61 0 . 0701 7 . 02 12 10 . 7 22 . 6 8 . 2 3 . 36 --.i 44 460 8 . 54 0 . 165 4 . 25 12 10 . 7 20 . 8 8 . 1 3 . 52 I-' 47 778 9 . 25 0 . 323 3. 2 12 10 . 7 20 . 2 8 . 0 3.14 48 610 7. 98 0 . 253 3. 9 15 .4 14 . 5 27 1 0 . 3 5 3 . 76 49 610 8 . 78 0 . 309 3. 3 11.2 9 . 7 20.8 7 . 6 4 . 24 51 610 9.88 0 . 37 3 . 0 11. 0 9. 6 21. 6 7 . 5 3 . 86 52 610 12 . 1 0 . 449 2 . 8 12 . 2 10 . 9 21 8 . 2 3 . 59 53 610 13 . 3 0 . 523 2 . 5 11 . 7 10 . 4 21 7. 95 4 . 42 57 116 6 . 70 0 . 0248 1 0 . 5 30 29 22 . 6 27 . 5 7.05 59 214 6 . 83 0 . 0659 7 . 3 34 33 . 2 27 33 . 2 6 . 74 61 460 7 . 90 0 . 101 4 . 65 30 29 22 24 . 1 5 . 03 67 610 8 . 05 0 . 227 3. 6 33 .8 32 . 1 26 .8 30 5 . 25 56 116 6 . 69 0 . 01 7 3 1 1 26 . 6 25 . 6 21 . 6 20 . 9 6 . 44 58 214 6 . 86 0 . 0559 7 . 4 30.5 29 . 3 24 28 . 2 6 . 92 60 46 0 7 . 87 0 . 0897 4 . 6 25 . 9 25 . 2 23 19 . 1 4 . 6 64 610 8 . 04 0 . 0237 3 . 7 27 . 1 26 . 4 25 . 4 22 5 . 17 Table II. Characteristic parameters of flow system. u* 4 -q*x10 4 xl0 3 cS * ExlOS z xl0 z Run* X cm 0 (gm o f wa~er) om t e e gm No . (cm) (sec) (cm) gm of air (cm) (cm) (cm) (cm~-sec) 34 56 17 2.54 6 .3 8 2.54 0.283 3.5 1.3 37 112 21 5. 85' 6 . 8 12.7 0.52 4.31 1.05 40 21 4 26 .3 17.7 8 .2 0.522 5.1 0 .7 37 43 460 30.2 30.2 9 . 32 81. 2 1. 403 10.2 0 .8 45 46 77 8 34 45 .7 9 . 97 127 1 . 96 12.7 0 . 8G5 35 56 25.5 2.44 5 . 83 0.36 0.284 2. 8 8 1.79 38 112 29 .5 6.22 7.51 7.86 0 . 488 4 . 82 1. 43 41 214 36 .3 25.0 8 . 66 18.8 0.781 7.64 1. 36 44 460 46 . 0 82.0 I 9 .3 0 57 1 .3 5 11. 4 1.16 47 778 52 . 0 175 9.84 132 2 . 33 16.2 1 . 36 "-.l 48 610 42 115 12.0 76 .3 2.1 14.6 18.3 N 49 610 51. 9 150 5 . 85 122 2 . 96 15.2 14.5 51 610 57 .2 228 6.80 183 3.12 16 . 5 18.6 52 610 79 381 10.9 330 3 . 43 17 . 4 31.7 53 610 90 485 9 . 48 510 3.53 17 . 8 27.4 57 116 27.4 5.08 26.4 0.813 0.531 7.36 6.24 59 214 33.1 25 . 5 50.8 54 1.11 9. 8 9.35 61 460 37 .7 61 65 234 2.15 13.7 7.0 67 610 42 . 6 192 105 1430 4.11 15 . 6 13.4 56 116 23.5 1. 78 23 6.35 0.484 6.5 4.02 58 214 31.4 15.5 44 .8 44.5 1.16 8.39 7. 88 60 460 37.6 45.3 50 .5 144 1. 93 12. 4 4 .77 64 610 41 . 6 152 66 965 3 . 56 15.7 7.85 * Run no. 34-46 and 35-47 are cold water cases. Run no. 48-53 are cold water cases at fixed position. Run no. 57-67 and 56-64 are warm water cases. 7 3 Chapter VI DISCUSSION OF RESULTS The experimental results of this study have been pre­ s ented in Chapter v. To check the data of this study , both t he momentum and mass transfer phenomena will be compared with experimental or analytical results of earlier authors. To better understand the transport mechanism, the physical b ackground of different parameters was considered and is d iscussed to some extent in the light of the experimental r esul ts . An empirical relationship between momentum and mass transfer has been worked out using the experimental data . Finally , a direct and practical method to evaluate t he evaporation rate is proposed . 6. 1 Nature o f the Water Surface The temperatu~e difference between water and air has s ome effect on structure of the surface waves (Fleagle , 1 956 ). One•way to estimate the thermal effect on wave s tructure is to evaluate the net amplification rate of waves r esulting from the temperature difference . A theoretic al approach which includes physical parameters that vary with t emperature has been suggested by Miles (1962 ) and Benjamin (1 9 59 ) . Miles showed that (see also Hidy and Plate, 1968) t he net amplification rate for sma l l waves is given by : m = m + m a w (6-1 ) 74 where m is the growth factor predicted by Miles ' mathe­ a matical model , and m w is the d amping factor : where c is the phase s peed of wave , -2 kd e w ( 6-2) is the kinematic viscosity of water , k is wave number , and d is water depth . The growth factor has the form: m a ::: 1/2 u' s 0 322 2 343 1/3 [ . (~) (~) + 0. (uk *v ) ] '(6 -3) w2 k va u* w a where w is a comp lex stability function which is given in t erms of the variable z = c/u' o* s where u' is the slope of air velocity profile at the s water surface. ( 6-4) Some calculations of m(k ) have been shown in Table II o f Hidy and Plate (1968). They assumed t ~a t -r = l dyne/ s cm2 and 6T = + 10°C, and they us e d Miles ' estimated value o f w adjusted for the change in the physical properties with temperature. They found an approximately 10 % change in net amplification rate for this range of temperature differe nce. Experimental data of this study (Figures 9, 10 a nd 11) show that the values of f decrease and the m values of a increase as the temperature difference , Ts - T 00 , changes from positive to negative. By using 75 Figure 10 to account for the tempera ture effect in the ~hange of f and m a , one obtains for a temperature d ifference of 6T = + 12°C to 6T = - 6°C and the same free stream velocity, that the standard deviation increased about 25 %. Since the standard deviation also indicates the wave h e ight; the increased standard deviation shows the dif ference in amplification rate for such a temperature difference . It was found from the experimental data and the thermal conditions of this study, that the net ampli­ fication rate , which was calculated froill Equations (6-2) a nd (6-3) , showed a 30 % increase from 6T = + 12°C to 6T = - 6°C. The d ecrease of the damping factor (Eq. (6-2)) was largely due to the t emperature difference . However , the increase of growth factor (Eq . (6-3)) due to the temperature d ifference was small . Thus , while the Benjamin-Miles theory - - also shows that m increases with increase of water tempera- tur~ , it does not predict the l arge increases observed in the experim~ntal results . Roll (as quoted by Fleagle, 1956) d evised a statistical method to study the temperature effect on wave generation. He found that for the same winds , the mean wave height increased 22% as the air and sea temperature difference increased from 0° to 6 . 7°C. His results were based on measurements taken by North Atlantic weather ships . Fleagle (1956) later us ed the mean tabulate d data of wind speed , wave h e ight and air temperatures , based on measurements by 76 Atlantic and Pacific weather ships, to plot a relationship between air-sea temperature difference and wave height. His results showed that higher waves are generated on warm water (relative to air tempe rature ) than on cold water . The di fference amounted to an increase in wave height of roughly ten percent per d egree centigrade . Rolls' results agree with the experimental data of this study . Yet Fleagles ' results showed a stronger effect on t emperature difference . This may have been partially due to the deficiency in measuring sea surface temperature, and the way he s elected the field data and excluded the c ases o f a randomly agitated sea . 6.2 Air Flow In the warm-water c ase , an unstable stratification developed in t he flow system, which may have had some effect on :t he air fiow n ear the water . A criterion for the magni­ tude of this effect is based on the flux form of the Richardson number, which is defined by : where = - gH/c T , ( au) p o a z H is the heat flux at the surface and T 0 (6-5) is the mean absol~te temperature . The values of H are contribu- ted by: H =H t + He (6 -6 ) where Ht = latent heat due to evaporation ( = p EL/C ) , w p 77 aT He= sensible heat due to t emperature gradient (=Cp (32)s). The s e nsible heat was contributed by air and water . Here~ o nly sensible heat of air was cons idered and denoted by He. The ra tio between He and H£ is the Bowen ratio (B = Hc/H£ ) . In this exper iment , the values of B were equal to (+ ) 0.2 - 0.3 for the cold-wa ter case and equal to (-) 0.003 - 0.00 4 for the warm-water case . This indicated that the sensible heat of air was a n important source for evaporation of the cold-water ca se , while the sensible heat o f water was an important source for evaporation of the warm-water case. For the first approx imation in the warm­ water c a se , which had the grea ter hea t flux at the surface than the cool surface , the following as s umptions were made : K K au m T = p m az = p k u* (6 -7a) H = H£ - p EL/c w p (6-7b) and • K Rf h R ' Ri = - l "' K (6-7c) m where Ri is t he Richardson number in gradient form. The d ifference between Ri and Rf depends on the value of Kh/~ . Here , Kh/~ is assumed equal to unity . The values of Ri can be calculated by: u* 2 Rf "' Ri = gx lixE/T (-) s k ( 6-8) 78 In the experiments of this study, Riis the order of 10-3 . For small Ri , the flow system can be considered as near­ adiabatic, which leads to the following equation to describe the velocity distribution (Roll , 1965): au=~ 1 az K 2 1 - aRi ' (6-9) where a is a constant and approximately equal to 10. On the basis of Equation (6-9), the velocity profile in the warm-water case would involve 2% error by neglecting the thermal effect of the Richardson criterion. Therefore, the effect of the unstable density stratification on mome ntum e xchange was considered negligible. The values and relation- s hips between u* and z 0 in warm-water case, such as ex- pres sed in Equations (5 -1) and (5-2), were similar for the s ame free stream conditions in the cold-water c ase . There­ fore, it was assumed that temperature difference between the air and the water did not modify the momentum transfer in this experime nt. The correlation between z and u*cr/v for both cold­o ' a nd warm-water cases c an also be used to check the thermal e ffect on momentum transfer. As shown in Figure 23, the correlation held reasonably well for both cases, but there was a small difference which may not be accounted for by applying the Richardson criterion. Since virtually no i nformation is available on the combine d effects of waves and temperature difference on shear ing flow of air overhead , 79 this question should be investigated further in later studies . 6.3 Humidity and Evaporation Rates 6.3.1 Universal Concentration The "logarithmic law" for mean humidity profiles in turbulent boundary layers over a flat or a wavy surface has been d erived in the previous section and is given by Equation (2-9). The curves ·of dime nsionless humidity pro­ files placed in the form o f (q - q 8 )/q* with dimensionless height (z/z ) are shown in Figures 27 and 28 . Most of the om experimental data are well correlated along the line which is give by Equation (2-9) , except at large values of z/z . om Large values of z/z for each run indicate large distances om from the mean water sur face , where the logarithmic law did not hold, as is also the c ase for the velocity profiles. The results of this study suggested that the law of wall for the humidity distribution was a satisfactory approxima­ tion near the water surface for both the cold-water case (inversion condition) and warm-water case (lapse c onditions). Using Equation (2-9), the values of q* were evaluated from the humidity profiles by the relationship: (6 -10) where and designated the specific humidity at 10 8 6 tJo- 4 10 8 • 6 f:f ..... • u I f:f 4 2 80 o ~ # 34 V ~ 1(;}45 o ~ -# 37 C,~ 43 co ld-water 6 ~ ,:, 40 Cl~ #' 33 0...- 0--- • ~# 4 4 ....&j60 6 ,¥)Q; , CJ""" ◊ - ~ 46 ~CJCJCJ Z I Z0 m Fig. 27. Dimensionl ess spec ific-humidity di stribution over small water waves . o ~ ¢56 c, ~ <> 67 Hot- Ylote, l o ~ C.-58 v~ C:61 6 ~ ,P 60 ◊ ~ # 59 ~~a,O •~ 64 ◊ ..p-c1'5oo ~a~ · ~ o.,,vt'.~ ~(J· q -q. = 2. 3 Log~ '6 q* Zom . 0 Z I Z0,. Fig. 28. Dimension l ess specific-humidity distribution over small water wave s . 81 a nd and lay in the curve of the logarithmi c part . After de termining t he values of z was evaluated om from Equation (2-9) . The numerical values of q* and z for this study are given in Table II. om The trends of q * were similar to that of u*. The variation of q* with fetc h is shown in Figures 21 and 22. The values of q* appeared to be mainly affected by ~he t emperature dif ference between air and water . The effect o f air velocity on q* was smaller than the effect of t emperature difference. To compare the properties of q* wi th u*, t he effect o f temperature difference between air a nd water on q* was similar to the effect of free stream velocity on u*. The l ength scale , z , is analogous to z , and is om o introduced as a characteristic l ength for the (logarithmic) humidity profiles. Es timation o f the values of z from om the momentum field is a us efu l and practical way for predicting t~e universal profile of humidity. A reason­ a ble correlation was found for the set of experiments . The dimensionless roughness lengths ( z / o ) 0 are shown in Figure 29 as a function of the Peclet number Pe= u*z /D, om which is based on the length z om The standard d evi- at ion o f z /o from a straigh t line was+ 15 %, which is 0 considered to be a satisfactory fit for both the cold- and warm-water c ases studied. From this corre lation, the values of evaporation can be estimated from knowledge of the b ' 0 N o Cold Water Low Velocity • Worm Water Low Temperature Difference r:l Warm Water Medium Temperature Difference 161 162 '----'---L-L..L...L...L..Ju..L--L-....L-_,_a......i.....,_,_u-_--t.._L..-<....__._ ................... __ _._ _____ ......... ___ ,0- 1 10° 10 1 102 103 Fig. 29. u*Zom D Correlation between Peclet number roughness length. and dimensionless (X) r:v 83 velocity field only, provided the thermal conditions at the water surface and free stream are given, or measured. 6.3.2 Methods of Obtaining the Evaporation Rate In Chapter II, several semi-empirical methods for estimating the evaporation rate were outlined. In this section, these results are compared for consistency with those of previous investigators and with each other . For the latter comparison, the data of E calculated by the mass balance method, as shown in Table II, were taken as the standard. The other approximate methods were each shown to be u seful for estimating E over a c ertain range of boundary layer development. From the measured humidity distribution, the profile coefficient, r 0 , can be calculated by means of Equation (2-22). One set of the results at a fixed position, x = 6.10 m, is shown in Figure 30, together with Okuda and Hay 1 ami 's results ta.ken in a wind-water tunnel under identi- • cal conditions. When the wind speed, · u, was less than CX) 10 m/sec, the values r 0 were nearly independent of u . 00 Okuda and Hayami observed that above u = 10 m/sec, the CX) surface became sufficiently agitated to produce spray from breaking waves . This caused a marked increase in evapora- tion rate. The data taken in this study showed a similar sharp increase in r D at about u = 10 m/sec, which evi- oo dently was related to spray formation. Droplets of spray •. 0:3 -----....--,----,lr------,,----,,----,,----Tl----r--,----,1---7 0.2- 0.1 - 0 I 0 Fig. • o - By Okuda and Hoyomi □ - This Experiment I I I I 4 8 u00 ( m/sec) I 12 30. Variation of evaporation coefficients against - 0 - I I 16 u 00 85 were actually observed above the wavy surface around or above this wind spe ed . From experimental knowledge of the profile parameters of the fields of velocity, temperature, and vapor concen­ tration, the evaporation rate can be c alculated by the approximate methods outlined in Chapter II as: El = - pk u*q* (2 -9b ) E2 = pk u* (q - q )/ tn [ (D + ku*o* )/D] s CX) e ( 2-6) and d a* E3 = p dx J eu(q - q (X) ) dz, 0 (2 -2) where the values of E1 , E 2 , and E 3 refer to the profile method, integral method, and experimental data, respectively. The values of E's are listed in Table III and plotted against . the x-coordinate in Figures 31 and 32. E 3 is the I . • . d measured evaporation rate which is taken as the standar . for comparison . In the cold-water case, and for x > 3 meters, the difference between E2 and E3 was small, so that Equation (2-6) gave a satisfactory simplified method to evaluate the evaporation rate in a well-developed turbu­ lent boundary layer . However, t he difference between E1 and E 3 was l arge and systematic over x > 3 meters , so the constant of Equation (2-9b) had to be adjusted from Karman 's constant of 0.4. As pointed out earlier, the Karman con­ stant, K, was used only for a first approximation. A 86 Table III. Evaporation rates Run No. 34 37 40 43 46 35 38 41 44 47 4 8 49 51 52 53 57 59 61 67 5 6 58 60 6 4 * * E 1 xio 5 X (cm) ** 56 0.438 112 0.56 214 0.871 460 1.14 77 8 1.37 56 0.60 1 112 0.8 59 214 1. 27 460 1.56 778 1.70 610 2.04 610 1 . 33 610 1. 57 610 3.48 610 3.45 116 3.03 214 6.36 460 8.21 610 14.2 116 . 2.15 214 5 .6 1 460 6.45 610 9.27 (gm of water ) c m2-sec E2xl0 5 ** 0.605 0.631 0.97 0.935 1.0 0.9 95 1.06 1.14 1. 26 1.41 1.68 1.03 1.19 1.97 1. 68 3.63 5.39 4. 5 6.75 2 . 43 4.26 4 . 52 5 . 49 by different methods . 5 E*xl0 5 5 E 3 xl0 E 4 xl0 1 ** ** ** 1.30 1.55 1.05 1. 23 0.745 0.706 0.8 85 0.91 0. 911 1.0 4 0.99 5 1.09 1. 27 1 .7 9 1. 82 1 . 43 1.48 1.3 6 1.0 2 1.51 1.16 1.25 1. 35 1 .3 6 1. 36 1. 76 1. 83 1.63 1.25 1.06 1. 5 6 1.26 3.17 2.79 2.74 2.77 6.24 7.21 9.35 9.54 7.0 6.58 5 .75 13.4 11.40. 17.10 4.02 4.2 7.88 8.07 4.77 5.15 4.96 7.85 7.42 9.75 .,Q )( w ., 0 )( w 87 2.0 ~--~--~----.-----,.----r---.----r---,----, 1.6 1.2 0 .8 0 .4 'a Cold­ Water = Cose o ~ E3 , Measured Value 6 ~ E 1 , Profile 1c thod □ ~ E 2 , In tegral Method v ~ E;, Profile Method With Empcricol Const. 0 L-----'----'-----'----~--~--~--------~ 0 2 4 6 8 X (Meters) Fig. 31. Evaporation rates against fetch. o ~ E 3 , Meosured Valu~ 14 Worm- 6 ~ E 1 , Profile Method Wa ter Cose □ ~ E z , Integral Method v ~ E~ , Profilo Method With Emp~ricol Const. 10 . -- 6 2:----~---::----~---:----....L....----L.--