M is the transport of one component of a mixture relative to the

Transcription

1 Related Commercial Resources CHAPTER 5 MASS TRANSFER Molecular Diffusion Convection of Mass Simultaneous Heat and Mass Transfer Between Water-Wetted Surfaces and Air Symbols ASS transfer by either molecular diffusion convection M is the transpt of one component of a mixture relative to the motion of the mixture and is the result of a concentration gradient. In air conditioning, water vap is added removed from the air by simultaneous transfer of heat and mass (water vap) between the airstream and a wetted surface. The wetted surface can be water droplets in an air washer, wetted slats of a cooling tower, condensate on the surface of a dehumidifying coil, a spray of liquid absbent, wetted surfaces of an evapative condenser. Equipment perfmance with these phenomena must be calculated carefully because of the simultaneous heat and mass transfer. This chapter addresses the principles of mass transfer and provides methods of solving a simultaneous heat and mass transfer problem involving air and water vap, with emphasis on airconditioning processes. The fmulations presented can help analyze perfmance of specific equipment. F discussion of perfmance of air washers, cooling coils, evapative condensers, and cooling towers, see Chapters 19, 21, 35, and 36, respectively, of the 2004 ASHRAE Handbook HVAC Systems and Equipment. MOLECULAR DIFFUSION Most mass transfer problems can be analyzed by considering diffusion of a gas into a second gas, a liquid, a solid. In this chapter, the diffusing dilute component is designated as component B, and the other component as component A. F example, when water vap diffuses into air, the water vap is component B and dry air is component A. Properties with subscripts A B are local properties of that component. Properties without subscripts are local properties of the mixture. The primary mechanism of mass diffusion at dinary temperature and pressure conditions is molecular diffusion, a result of density gradient. In a binary gas mixture, the presence of a concentration gradient causes transpt of matter by molecular diffusion; that is, because of random molecular motion, gas B diffuses through the mixture of gases A and B in a direction that reduces the concentration gradient. Fick s Law The basic equation f molecular diffusion is Fick s law. Expressing the concentration of component B of a binary mixture of components A and B in terms of the mass fraction ρ B /ρ mole fraction C B /C, Fick s law is J B ρd B v ( ρ) J A J* B CD dc B v ( C) J* A where ρ ρ A + ρ B and C C A + C B. (1a) (1b) The preparation of this chapter is assigned to TC 1.3, Heat Transfer and Fluid Flow. The minus sign indicates that the concentration gradient is negative in the direction of diffusion. The proptionality fact D v is the mass diffusivity the diffusion coefficient. The total mass flux m B and molar flux m B * are due to the average velocity of the mixture plus the diffusive flux: m B d( ρ B ρ) ρ B v ρd v m B * C B v * dc ( B C) CD v (2a) (2b) where v is the mixture s mass average velocity and v * is the molar average velocity. Bird et al. (1960) present an analysis of Equations (1a) and (1b). Equations (1a) and (1b) are equivalent fms of Fick s law. The equation used depends on the problem and individual preference. This chapter emphasizes mass analysis rather than molar analysis. However, all results can be converted to the molar fm using the relation C B ρ B /M B. Fick s Law f Dilute Mixtures In many mass diffusion problems, component B is dilute, with a density much smaller than the mixture s. In this case, Equation (1a) can be written as J B D dρ B v when ρ B << ρ and ρ A ρ. Equation (3) can be used without significant err f water vap diffusing through air at atmospheric pressure and a temperature less than 27 C. In this case, ρ B < 0.02ρ, where ρ B is the density of water vap and ρ is the density of moist air (air and water vap mixture). The err in J B caused by replacing ρ[d(ρ B /ρ)/] with dρ B / is less than 2%. At temperatures below 60 C where ρ B < 0.10ρ, Equation (3) can still be used if errs in J B as great as 10% are tolerable. Fick s Law f Mass Diffusion Through Solids Stagnant Fluids (Stationary Media) Fick s law can be simplified f cases of dilute mass diffusion in solids, stagnant liquids, stagnant gases. In these cases, ρ B << ρ and v 0, which yields the following approximate result: m B dρ B J B D v Fick s Law f Ideal Gases with Negligible Temperature Gradient F dilute mass diffusion, Fick s law can be written in terms of partial pressure gradient instead of concentration gradient. When gas B can be approximated as ideal, (3) (4) Copyright 2005, ASHRAE 5.1

2 ASHRAE Handbook Fundamentals (SI) and when the gradient in T is small, Equation (3) can be written as M B D v dp J B B R u T D J* v dp B B R u T If v 0, Equation (4) may be written as ρ p B R u T B C M B R u T B m B M B D v dp J B B R u T m B * * D v dp J B B R u T (5) (6a) (6b) (7a) (7b) The partial pressure gradient fmulation f mass transfer analysis has been used extensively; this is unftunate because the pressure fmulation [Equations (6) and (7)] applies only when one component is dilute, the fluid closely approximates an ideal gas, and the temperature gradient has a negligible effect. The density ( concentration) gradient fmulation expressed in Equations (1) through (4) is me general and can be applied to a wider range of mass transfer problems, including cases where neither component is dilute [Equation (1)]. The gases need not be ideal, n the temperature gradient negligible. Consequently, this chapter emphasizes the density fmulation. Diffusion Coefficient F a binary mixture, the diffusion coefficient D v is a function of temperature, pressure, and composition. Experimental measurements of D v f most binary mixtures are limited in range and accuracy. Table 1 gives a few experimental values f diffusion of some gases in air. F me detailed tables, see the Bibliography. In the absence of data, use equations developed from (1) they (2) they with constants adjusted from limited experimental data. F binary gas mixtures at low pressure, D v is inversely proptional to pressure, increases with increasing temperature, and is almost independent of composition f a given gas pair. Bird et al. (1960) present the following equation, developed from kinetic they and cresponding states arguments, f estimating D v at pressures less than 0.1p c min : T b 1 1 D v a T ca T cb M A M B ( p ca p cb ) 1 3 ( T ca T cb ) p where D v diffusion coefficient, mm 2 /s a constant, dimensionless b constant, dimensionless T absolute temperature, K p pressure, kpa M relative molecular mass, kg/kg mol Subscripts ca and cb refer to the critical states of the two gases. Analysis of experimental data gives the following values of the constants a and b: F nonpolar gas pairs, a and b (8) F water vap with a nonpolar gas, a and b In nonpolar gas, intermolecular fces are independent of the relative ientation of molecules, depending only on the separation distance from each other. Air, composed almost entirely of nonpolar gases O 2 and N 2, is nonpolar. Equation (8) is stated to agree with experimental data at atmospheric pressure to within about 8% (Bird et al. 1960). The mass diffusivity D v f binary mixtures at low pressure is predictable within about 10% by kinetic they (Reid et al. 1987). where σ AB characteristic molecular diameter, nm Ω D, AB temperature function, dimensionless D v is in mm 2 /s, p in kpa, and T in kelvins. If the gas molecules of A and B are considered rigid spheres having diameters σ A and σ B [and σ AB (σ A /2) + (σ B /2)], all expressed in nanometres, the dimensionless function Ω D, AB equals unity. Me realistic models f molecules having intermolecular fces of attraction and repulsion lead to values of Ω D, AB that are functions of temperature. Reid et al. (1987) present tabulations of this quantity. These results show that D v increases as the 2.0 power of T at low temperatures and as the 1.65 power of T at very high temperatures. The diffusion coefficient of moist air has been calculated f Equation (8) using a simplified intermolecular potential field function f water vap and air (Mason and Monchick 1965). The following empirical equation is f mass diffusivity of water vap in air up to 1100 C (Sherwood and Pigfd 1952): where D v is in mm 2 /s, p in kpa, and T in kelvins. (9) (10) Diffusion of One Gas Through a Second Stagnant Gas Figure 1 shows diffusion of one gas through a second, stagnant gas. Water vap diffuses from the liquid surface into surrounding stationary air. It is assumed that local equilibrium exists through the gas mixture, that the gases are ideal, and that the Gibbs-Dalton law is valid, which implies that the temperature gradient has a negligible effect. Water vap diffuses because of concentration gradient, as given by Equation (6a). There is a continuous gas phase, so the mixture pressure p is constant, and the Gibbs-Dalton law yields Table 1 Mass Diffusivities f Gases in Air* Gas D v, mm 2 /s Ammonia 27.9 Benzene 8.8 Carbon dioxide 16.5 Ethanol 11.9 Hydrogen 41.3 Oxygen 20.6 Water vap 25.5 *Gases at 25 C and kpa. T 1.5 D v p( σ AB ) 2 Ω DAB, T 2.5 D v p T p A + p B p constant ρ A ρ B p constant M A M B R u T M A M B (11a) (11b)

3 Mass Transfer 5.3 The partial pressure gradient of the water vap causes a partial pressure gradient of the air such that dρ A dρ B (12) M A M B Air, then, diffuses toward the liquid water interface. Because it cannot be absbed there, a bulk velocity v of the gas mixture is established in a direction away from the liquid surface, so that the net transpt of air is zero (i.e., the air is stagnant): (13) The bulk velocity v transpts not only air but also water vap away from the interface. Therefe, the total rate of water vap diffusion is (14) Substituting f the velocity v from Equation (13) and using Equations (11b) and (12) gives Integration yields dp dp A B m A D dρ A v ρ A v 0 m B D dρ B v ρ B v m B D v M B p dρ A ρ A R u T m B D v M B p R u T ln( ρ AL ρ A0 ) y L y 0 m B Dv P ρ BL ρ B0 Am y L y 0 (15) (16a) (16b) p ln( ρ where P Am ρ AL ρ A0 ) (17) p AL AL ρ AL ρ A0 P Am is the logarithmic mean density fact of the stagnant air. The pressure distribution f this type of diffusion is illustrated in Figure 2. Stagnant refers to the net behavi of the air; it does not move because the bulk flow exactly offsets diffusion. The term P Am in Equation (16b) approximately equals unity f dilute mixtures such as water vap in air at near atmospheric conditions. This condition makes it possible to simplify Equations (16) and implies that, f dilute mixtures, the partial pressure distribution curves in Figure 2 are straight lines. Example 1. A vertical tube of 25 mm diameter is partially filled with water so that the distance from the water surface to the open end of the tube is 60 mm, as shown in Figure 1. Perfectly dried air is blown over the open tube end, and the complete system is at a constant temperature of 15 C. In 200 h of stea operation, 2.15 g of water evapates from the tube. The total pressure of the system is kpa. Using these data, (1) calculate the mass diffusivity of water vap in air, and (2) compare this experimental result with that from Equation (10). Solution: (1) The mass diffusion flux of water vap from the water surface is m B g/h The cross-sectional area of a 25 mm diameter tube is π(12.5) mm 2. Therefe, m g/(m 2 s). B The partial densities are determined with the aid of the psychrometric tables. ρ BL 0; ρ B g/m 3 ρ AL kg/m 3 ; ρ A kg/m 3 Because p p AL kpa, the logarithmic mean density fact [Equation (17)] is P Am ln( ) The mass diffusivity is now computed from Equation (16b) as m B ( y D L y 0 ) v ( ) ( 0.060) ( ) P Am ( ρ BL ρ B0 ) ( 1.009) ( ) 28.2 mm 2 s (2) By Equation (10), with p kpa and T K, D v mm 2 s Fig. 2 Pressure Profiles f Diffusion of Water Vap Through Stagnant Air Fig. 1 Diffusion of Water Vap Through Stagnant Air Fig. 1 Diffusion of Water Vap Through Stagnant Air Fig. 2 Pressure Profiles f Diffusion of Water Vap Through Stagnant Air

4 ASHRAE Handbook Fundamentals (SI) Neglecting the crection fact P Am f this example gives a difference of less than 1% between the calculated experimental and empirically predicted values of D v. Equimolar Counterdiffusion Figure 3 shows two large chambers, both containing an ideal gas mixture of two components A and B (e.g., air and water vap) at the same total pressure p and temperature T. The two chambers are connected by a duct of length L and cross-sectional area A cs. Partial pressure p B is higher in the left chamber, and partial pressure p A is higher in the right chamber. The partial pressure differences cause component B to migrate to the right and component A to migrate to the left. At stea state, the molar flows of A and B must be equal but opposite: (18) because the total molar concentration C must stay the same in both chambers if p and T remain constant. Because molar fluxes are the same in both directions, the molar average velocity v* 0. Thus, Equation (7b) can be used to calculate the molar flux of B ( A): (19) * A cs D v m p B0 p BL B R u T L (20) (21) Example 2. One large room is maintained at 22 C (295 K), kpa, 80% rh. A 20 m long duct with cross-sectional area of 0.15 m 2 connects the room to another large room at 22 C, kpa, 10% rh. What is the rate of water vap diffusion between the two rooms? Solution: Let air be component A and water vap be component B. Equation (21) can be used to calculate the mass flow of water vap B. Equation (10) can be used to calculate the diffusivity. From a psychrometric table (Table 3, Chapter 6), the saturated vap pressure at 22 C is kpa. The vap pressure difference p B 0 p BL is Fig. 3 p B0 m B Equimolar Counterdiffusion Fig. 3 * * m A + m B 0 * D v m dp B B R u T M B A cs D v p B0 p BL R u T L D v mm 2 /h ( )2.645 kpa 1.85 kpa p BL Equimolar Counterdiffusion Then, Equation (21) gives ( m 6 ) b kg/s Molecular Diffusion in Liquids and Solids Because of the greater density, diffusion is slower in liquids than in gases. No satisfacty molecular theies have been developed f calculating diffusion coefficients. The limited measured values of D v show that, unlike f gas mixtures at low pressures, the diffusion coefficient f liquids varies appreciably with concentration. Reasoning largely from analogy to the case of one-dimensional diffusion in gases and using Fick s law as expressed by Equation (4) gives m B ρ B1 ρ B2 D v y 2 y 1 (22) Equation (22) expresses stea-state diffusion of solute B through solvent A in terms of the molal concentration difference of the solute at two locations separated by the distance y y 2 y 1. Bird et al. (1960), Eckert and Drake (1972), Hirschfelder et al. (1954), Reid and Sherwood (1966), Sherwood and Pigfd (1952), and Treybal (1980) provide equations and tables f evaluating D v. Hirschfelder et al. (1954) provide comprehensive treatment of the molecular developments. Diffusion through a solid when the solute is dissolved to fm a homogeneous solid solution is known as structure-insensitive diffusion (Treybal 1980). This solid diffusion closely parallels diffusion through fluids, and Equation (22) can be applied to onedimensional stea-state problems. Values of mass diffusivity are generally lower than they are f liquids and vary with temperature. Diffusion of a gas mixture through a pous medium is common (e.g., diffusion of an air/vap mixture through pous insulation). Vap diffuses through the air along the ttuous narrow passages within the pous medium. Mass flux is a function of the vap pressure gradient and diffusivity, as indicated in Equation (7a). It is also a function of the structure of the pathways within the pous medium and is therefe called structure-sensitive diffusion. All these facts are taken into account in the following version of Equation (7a): m B µ dp B (23) where µ is called the permeability of the pous medium. Chapter 23 presents this topic in me depth. CONVECTION OF MASS Convection of mass involves the mass transfer mechanisms of molecular diffusion and bulk fluid motion. Fluid motion in the region adjacent to a mass transfer surface may be laminar turbulent, depending on geometry and flow conditions. Mass Transfer Coefficient Convective mass transfer is analogous to convective heat transfer where geometry and boundary conditions are similar. The analogy holds f both laminar and turbulent flows and applies to both external and internal flow problems. Mass Transfer Coefficients f External Flows. Most external convective mass transfer problems can be solved with an appropriate fmulation that relates the mass transfer flux (to from an interfacial surface) to the concentration difference across the boundary layer illustrated in Figure 4. This fmulation gives rise to the convective mass transfer coefficient, defined as

5 Mass Transfer 5.5 m B h M ρ Bi ρ B (24) where h M local external mass transfer coefficient, m/s m mass flux of gas B from surface, kg/(m 2 s) B ρ Bi density of gas B at interface (saturation density), kg/m 3 ρ B density of component B outside boundary layer, kg/m 3 If ρ Bi and ρ B are constant over the entire interfacial surface, the mass transfer rate from the surface can be expressed as m B h M ( ρ Bi ρ B ) where h M is the average mass transfer coefficient, defined as 1 h M -- h A m da A (25) (26) Mass Transfer Coefficients f Internal Flows. Most internal convective mass transfer problems, such as those that occur in channels in the ces of dehumidification coils, can be solved if an appropriate expression is available to relate the mass transfer flux (to from the interfacial surface) to the difference between the concentration at the surface and the bulk concentration in the channel, as shown in Figure 5. This fmulation leads to the definition of the mass transfer coefficient f internal flows: m B h M ρ Bi ρ Bb where h M internal mass transfer coefficient, m/s m mass flux of gas B at interfacial surface, kg/(m 2 s) B ρ Bi density of gas B at interfacial surface, kg/m 3 Fig. 4 Nomenclature f Convective Mass Transfer from External Surface at Location x Where Surface is Impermeable to Gas A (27) ρ Bb ( 1 u B A cs ) bulk density of gas B at location x u B ρ B da cs A cs u B ( 1 A cs ) average velocity of gas B at location x, u B da cs A m/s A cs cross-sectional area of channel at station x, m 2 u B velocity of component B in x direction, m/s ρ B density distribution of component B at station x, kg/m 3 Often, it is easier to obtain the bulk density of gas B from m Bo + m B da A ρ Bb u B A cs where m mass flow rate of component B at station x 0, kg/s Bo A interfacial area of channel between station x 0 and station x x, m 2 (28) Equation (28) can be derived from the preceding definitions. The maj problem is the determination of u B. If, however, analysis is restricted to cases where B is dilute and concentration gradients of B in the x direction are negligibly small, u B u. Component B is swept along in the x direction with an average velocity equal to the average velocity of the dilute mixture. Analogy Between Convective Heat and Mass Transfer Most expressions f the convective mass transfer coefficient h M are determined from expressions f the convective heat transfer coefficient h. F problems in internal and external flow where mass transfer occurs at the convective surface and where component B is dilute, Bird et al. (1960) and Incropera and DeWitt (1996) found that the Nusselt and Sherwood numbers are defined as follows: Nu Sh f ( X, Y, Z, Pr, Re) f ( X, Y, Z, Sc, Re) (29) (30) and Nu g( Pr, Re) (31) Sh g( Sc, Re) (32) Fig. 4 Nomenclature f Convective Mass Transfer from External Surface at Location x Where Surface is Impermeable to Gas A Fig. 5 Nomenclature f Convective Mass Transfer from Internal Surface Impermeable to Gas A Fig. 5 Nomenclature f Convective Mass Transfer from Internal Surface Impermeable to Gas A where f in Equations (29) and (30) indicates a functional relationship among the dimensionless groups shown. The function f is the same in both equations. Similarly, g indicates a functional relationship that is the same in Equations (31) and (32). Pr and Sc are dimensionless Prandtl and Schmidt numbers, respectively, as defined in the Symbols section. The primary restrictions on the analogy are that the surface shapes are the same and that the temperature boundary conditions are analogous to the density distribution boundary conditions f component B when cast in dimensionless fm. Several primary facts prevent the analogy from being perfect. In some cases, the Nusselt number was derived f smooth surfaces. Many mass transfer problems involve wavy, droplet-like, roughened surfaces. Many Nusselt number relations are obtained f constant temperature surfaces. Sometimes ρ Bi is not constant over the entire surface because of varying saturation conditions and the possibility of surface dryout. In all mass transfer problems, there is some blowing suction at the surface because of the condensation, evapation, transpiration of component B. In most cases, this blowing/suction has little effect on the Sherwood number, but the analogy should be examined closely if v i /u > 0.01 v i u > 0.01, especially if the Reynolds number is large.

6 ASHRAE Handbook Fundamentals (SI) Example 3. Air at 25 C, 100 kpa, and 60% rh flows at 10 m/s, as shown in Figure 6. Find the rate of evapation, rate of heat transfer to the water, and water surface temperature. Solution: Heat transfer to the water from the air supplies the energy required to evapate the water. where h convective heat transfer coefficient h m convective mass transfer coefficient A m 2 surface area (both sides) m evapation rate t s, ρ s temperature and vap density at water surface t, ρ temperature and vap density of airstream This energy balance can be rearranged to give The heat transfer coefficient h is found by first calculating the Nusselt number: f laminar flow f turbulent flow The mass transfer coefficient h m requires calculation of Sherwood number Sh, obtained using the analogy expressed in Equations (31) and (32) With Nu and Sh known, f laminar flow f turbulent flow This result is valid f both laminar and turbulent flow. Using this result in the preceding energy balance gives This equation must be solved f ρ s. Then, water surface temperature t s is the saturation temperature cresponding to ρ s. Air properties Sc, Pr, D v, and k are evaluated at film temperature t f (t + t s )/2, and h fg is evaluated at t s. Because t s appears in the right side and all the air properties also vary somewhat with t s, iteration is required. Start by guessing t s 14 C (the dew point of the airstream), giving t f 19.5 C. At these temperatures, the values on the right side are found in property tables calculated as k W/(m K) Pr Fig. 6 Fig. 6 q ha( t t s ) m h fg h m A( ρ s ρ )h fg ρ s ρ h h m Nu 0.664Re 1/2 Pr 1/3 Nu 0.037Re 4/5 Pr 1/3 Sh 0.664Re 1/2 Sc 1/3 Sh 0.037Re 4/5 Sc 1/3 ρ s h m h h m ( ) t t s h fg Sh D v h Nu k L L ρ Nu k Pr /3 k Sh D v Sc D v Pr /3 k ( ) Sc Water-Saturated Flat Plate in Flowing Airstream D v t t s h fg Water-Saturated Flat Plate in Flowing Airstream D v m 2 /s [from Equation (10)] ρ kg/m 3 µ kg/(m s) Sc µ/ρd v h fg J/kg (at 14 C) ρ kg/m 3 (from psychrometric chart at 25 C, 60% rh) t s 14 C (initial guess) Solving yields ρ s kg/m 3. The cresponding value of t s 21.6 C. Repeat the process using t s 21.6 C as the initial guess. The result is ρ s kg/m 3 and t s 18.3 C. Continue iterations until ρ s converges to kg/m 3 and t s 19.4 C. To solve f the rates of evapation and heat transfer, first calculate the Reynolds number using air properties at t f ( )/ C. ρu Re L ( 1.191) ( 10) ( 0.1) L µ where L 0.1 m, the length of the plate in the direction of flow. Because Re L < , the flow is laminar over the entire length of the plate; therefe, m Sh 0.664Re 1/2 Sc 1/3 144 h m ShD v m/s L h m A( ρ s ρ ) kg/s q m h fg 65.8 W The same value f q would be obtained by calculating the Nusselt number and heat transfer coefficient h and setting q ha(t t s ). The kind of similarity between heat and mass transfer that results in Equations (29) to (32) can also be shown to exist between heat and momentum transfer. Chilton and Colburn (1934) used this similarity to relate Nusselt number to friction fact by the analogy Nu j H Re Pr ( 1 n) St Pr n f (33) where n 2/3, St Nu/(Re Pr) is the Stanton number, and j H is the Chilton-Colburn j-fact f heat transfer. Substituting Sh f Nu and Sc f Pr in Equations (31) and (32) gives the Chilton-Colburn j-fact f mass transfer, j D : Sh j D Re Sc ( 1 n) St m Sc n f (34) where St m ShP AM /(Re Sc) is the Stanton number f mass transfer. Equations (33) and (34) are called the Chilton-Colburn j-fact analogy. The power of the Chilton-Colburn j-fact analogy is represented in Figures 7 to 10. Figure 7 plots various experimental values of j D from a flat plate with flow parallel to the plate surface. The solid line, which represents the data to near perfection, is actually f /2 from Blasius solution of laminar flow on a flat plate (left-hand ption of the solid line) and Goldstein s solution f a turbulent boundary layer (right-hand ption). The righthand part also represents McAdams (1954) crelation of turbulent flow heat transfer coefficient f a flat plate. A wetted-wall column is a vertical tube in which a thin liquid film adheres to the tube surface and exchanges mass by evapation absption with a gas flowing through the tube. Figure 8 illustrates typical data on vapization in wetted-wall columns, plotted as j D versus Re. The point spread with variation in µ /ρd v results from Gilliland s finding of an exponent of 0.56, not 2/3, representing the effect of the Schmidt number. Gilliland s equation can be written as follows:

7 Mass Transfer 5.7 (35) Similarly, McAdams (1954) equation f heat transfer in pipes can be expressed as (36) This is represented by the dash-dot curve in Figure 8, which falls below the mass transfer data. The curve f /2, representing friction in smooth tubes, is the upper, solid curve. Data f liquid evapation from single cylinders into gas streams flowing transversely to the cylinders axes are shown in Fig. 7 j D 0.023Re µ 0.56 ρd v j H 0.023Re c µ p 0.7 k Mass Transfer from Flat Plate Figure 9. Although the dash-dot line in Figure 9 represents the data, it is actually taken from McAdams (1954) as representative of a large collection of data on heat transfer to single cylinders placed transverse to airstreams. To compare these data with friction, it is necessary to distinguish between total drag and skin friction. Because the analogies are based on skin friction, nmal pressure drag must be subtracted from the measured total drag. At Re 1000, skin friction is 12.6% of the total drag; at Re , it is only 1.9%. Consequently, values of f /2 at a high Reynolds number, obtained by the difference, are subject to considerable err. In Figure 10, data on evapation of water into air f single spheres are presented. The solid line, which best represents these data, agrees with the dashed line representing McAdams crelation f heat transfer to spheres. These results cannot be compared with friction momentum transfer because total drag has not been allocated to skin friction and nmal pressure drag. Application of these data to air/water-contacting devices such as air washers and spray cooling towers is well substantiated. When the temperature of the heat exchanger surface in contact with moist air is below the air s dew-point temperature, vap condensation occurs. Typically, air dry-bulb temperature and humidity ratio both decrease as air flows through the exchanger. Therefe, sensible and latent heat transfer occur simultaneously. This process is similar to one that occurs in a spray dehumidifier and can be analyzed using the same procedure; however, this is not generally done. Cooling coil analysis and design are complicated by the problem of determining transpt coefficients h, h M, and f. It would be convenient if heat transfer and friction data f dry heating coils could be used with the Colburn analogy to obtain the mass transfer Fig. 9 Mass Transfer from Single Cylinders in Crossflow Fig. 7 Mass Transfer from Flat Plate Fig. 8 Vapization and Absption in Wetted-Wall Column Fig. 9 Fig. 10 Mass Transfer from Single Cylinders in Crossflow Mass Transfer from Single Spheres Fig. 8 Vapization and Absption in Wetted-Wall Column Fig. 10 Mass Transfer from Single Spheres

8 ASHRAE Handbook Fundamentals (SI) Fig. 11 Sensible Heat Transfer j-facts f Parallel Plate Exchanger From Figure 9 at Re da 4700, read j H , j D From Equations (33) and (34), h j H ρc p u ( Pr) ( 0.709) W ( m 2 K ) 2 3 h M j D u ( Sc) ( 0.605) m/s h ρh M 128 ( ) 803 J ( kg K) From the Bedingfield-Drew relation, h ρh M 1230( 0.605) J ( kg K) Fig. 11 Sensible Heat Transfer j-facts f Parallel Plate Exchanger coefficients, but this approach is not always reliable, and Guilly and McQuiston (1973) and Helmer (1974) show that the analogy is not consistently true. Figure 11 shows j-facts f a simple parallel-plate exchanger f different surface conditions with sensible heat transfer. Mass transfer j-facts and friction facts exhibit the same behavi. Dry-surface j-facts fall below those obtained under dehumidifying conditions with the surface wet. At low Reynolds numbers, the boundary layer grows quickly; the droplets are soon covered and have little effect on the flow field. As the Reynolds number increases, the boundary layer becomes thin and me of the total flow field is exposed to the droplets. Roughness caused by the droplets induces mixing and larger j-facts. The data in Figure 11 cannot be applied to all surfaces because the length of the flow channel is also an imptant variable. However, water collecting on the surface is mainly responsible f breakdown of the j-fact analogy. The j-fact analogy is approximately true when the surface conditions are identical. Under some conditions, it is possible to obtain a film of condensate on the surface instead of droplets. Guilly and McQuiston (1973) and Helmer (1974) related dry sensible j- and f-facts to those f wetted dehumidifying surfaces. The equality of j H, j D, and f /2 f certain streamline shapes at low mass transfer rates has experimental verification. F flow past bluff objects, j H and j D are much smaller than f /2, based on total pressure drag. The heat and mass transfer, however, still relate in a useful way by equating j H and j D. Example 4. Using solid cylinders of volatile solids (e.g., naphthalene, camph, dichlobenzene) with airflow nmal to these cylinders, Bedingfield and Drew (1950) found that the ratio between the heat and mass transfer coefficients could be closely crelated by the following relation: Equations (34) and (35) are called the Reynolds analogy when Pr Sc 1. This suggests that h/ρh M c p 1006 J/(kg K). This close agreement is because the ratio Sc/Pr is 0.605/ , so that the exponent of these numbers has little effect on the ratio of the transfer coefficients. The extensive developments f calculating heat transfer coefficients can be applied to calculate mass transfer coefficients under similar geometrical and flow conditions using the j-fact analogy. F example, Table 9 of Chapter 3 lists equations f calculating heat transfer coefficients f flow inside and nmal to pipes. Each equation can be used f mass transfer coefficient calculations by equating j H and j D and imposing the same restriction to each stated in Table 9 of Chapter 3. Similarly, mass transfer experiments often replace cresponding heat transfer experiments with complex geometries where exact boundary conditions are difficult to model (Sparrow and Ohadi 1987a, 1987b). The j-fact analogy is useful only at low mass transfer rates. As the rate increases, the movement of matter nmal to the transfer surface increases the convective velocity. F example, if a gas is blown from many small holes in a flat plate placed parallel to an airstream, the boundary layer thickens, and resistance to both mass and heat transfer increases with increasing blowing rate. Heat transfer data are usually collected at zero, at least, insignificant mass transfer rates. Therefe, if such data are to be valid f a mass transfer process, the mass transfer rate (i.e., the blowing) must be low. The j-fact relationship j H j D can still be valid at high mass transfer rates, but neither j H n j D can be represented by data at zero mass transfer conditions. Eckert and Drake (1972) and Chapter 24 of Bird et al. (1960) have detailed infmation on high mass transfer rates. Lewis Relation Heat and mass transfer coefficients are satisfactily related at the same Reynolds number by equating the Chilton-Colburn j- facts. Comparing Equations (33) and (34) gives h µ [ 1230 J ( kg K) ] ρh M ρd v F completely dry air at 21 C flowing at a velocity of 9.5 m/s over a wet-bulb thermometer of diameter d 7.5 mm, determine the heat and mass transfer coefficients from Figure 9 and compare their ratio with the Bedingfield-Drew relation. Solution: F dry air at 21 C and standard pressure, ρ kg/m 3, µ kg/(s m), k W/(m K), and c p kj/ (kg K). From Equation (10), D v mm 2 /s. Therefe, Re da ρu d µ ( ) 4690 Pr c p µ k Sc µ ρd v ( ) St Pr n f ---- St m Sc n 2 Inserting the definitions of St, Pr, St m, and Sc gives h c p µ h M P Am µ ρc p u k u ρd v h ( µ ρd P v ) h M ρc Am p ( c p µ k) P Am ( α D v ) (37)

9 Mass Transfer 5.9 The quantity α /D v is the Lewis number Le. Its magnitude expresses relative rates of propagation of energy and mass within a system. It is fairly insensitive to temperature variation. F air and water vap mixtures, the ratio is (0.60/0.71) 0.845, and (0.845) 2/3 is At low diffusion rates, where the heat/mass transfer analogy is valid, P Am is essentially unity. Therefe, f air and water vap mixtures, h h M ρc p (38) The ratio of the heat transfer coefficient to the mass transfer coefficient equals the specific heat per unit volume of the mixture at constant pressure. This relation [Equation (38)] is usually called the Lewis relation and is nearly true f air and water vap at low mass transfer rates. It is generally not true f other gas mixtures because the ratio Le of thermal to vap diffusivity can differ from unity. The agreement between wet-bulb temperature and adiabatic saturation temperature is a direct result of the nearness of the Lewis number to unity f air and water vap. The Lewis relation is valid in turbulent flow whether not α/d v equals 1 because ed diffusion in turbulent flow involves the same mixing action f heat exchange as f mass exchange, and this action overwhelms any molecular diffusion. Deviations from the Lewis relation are, therefe, due to a laminar boundary layer a laminar sublayer and buffer zone where molecular transpt phenomena are the controlling facts. SIMULTANEOUS HEAT AND MASS TRANSFER BETWEEN WATER-WETTED SURFACES AND AIR A simplified method used to solve simultaneous heat and mass transfer problems was developed using the Lewis relation, and it gives satisfacty results f most air-conditioning processes. Extrapolation to very high mass transfer rates, where the simple heatmass transfer analogy is not valid, will lead to erroneous results. Enthalpy Potential The water vap concentration in the air is the humidity ratio W, defined as W ρ B (39) ρ A A mass transfer coefficient is defined using W as the driving potential: m B K M ( W i W ) (40) where the coefficient K M is in kg/(s m 2 ). F dilute mixtures, ρ Ai ρ A ; that is, the partial mass density of dry air changes by only a small percentage between interface and free stream conditions. Therefe, m B K M ( ρ ρ Bi ρ ) Am (41) where ρ Am mean density of dry air, kg/m3. Comparing Equation (41) with Equation (24) shows that K h M M ρ Am (42) The humid specific heat c pm of the airstream is, by definition (Mason and Monchick 1965), c pm ( 1 + W )c p (43a) ρ c pm cp (43b) ρ A where c pm is in kj/(kg da K). Substituting from Equations (42) and (43b) into Equation (38) gives h ρ Am h K M ρ A c K pm M c pm (44) because ρ Am ρ A because of the small change in dry-air density. Using a mass transfer coefficient with the humidity ratio as the driving fce, the Lewis relation becomes ratio of heat to mass transfer coefficient equals humid specific heat. F the plate humidifier illustrated in Figure 6, the total heat transfer from liquid to interface is q q A + m B h fg (45) Using the definitions of the heat and mass transfer coefficients gives q ht ( i t ) + K M ( W i W )h fg Assuming Equation (44) is valid gives q K M c pm ( t i t ) + ( W i W )h fg The enthalpy of the air is approximately h c pa t + Wh s (46) (47) (48) The enthalpy h s of the water vap can be expressed by the ideal gas law as h s c ps ( t t o ) + h fgo (49) where the base of enthalpy is taken as saturated water at temperature t o. Choosing t o 0 C to crespond with the base of the dry-air enthalpy gives h ( c pa + Wc ps )t + Wh fgo c pm t+ Wh fgo (50) If small changes in the latent heat of vapization of water with temperature are neglected when comparing Equations (48) and (50), the total heat transfer can be written as q K M ( h i h ) (51) Where the driving potential f heat transfer is temperature difference and the driving potential f mass transfer is mass concentration partial pressure, the driving potential f simultaneous transfer of heat and mass in an air water/vap mixture is, to a close approximation, enthalpy. Basic Equations f Direct-Contact Equipment Air-conditioning equipment can be classified by whether the air and water used as a cooling heating fluid are (1) in direct contact (2) separated by a solid wall. Examples of the fmer are air washers and cooling towers; an example of the latter is a directexpansion refrigerant ( water) cooling and dehumidifying coil. In both cases, the airstream is in contact with a water surface. Direct contact implies contact directly with the cooling ( heating) fluid. In the dehumidifying coil, contact is direct with condensate

10 ASHRAE Handbook Fundamentals (SI) Fig. 12 Air Washer Spray Chamber Total energy transfer to air G a ( c pm dt a + h fgo dw) [ K M a M ( W i W )h fg + h a a H ( t i t a )]dl (57) Assuming a H a M and Le 1, and neglecting small variations in h fg, Equation (57) reduces to G a dh K M a M ( h i h)dl (58) removed from the airstream, but is indirect with refrigerant flowing inside the coil tubes. These two cases are treated separately because the surface areas of direct-contact equipment cannot be evaluated. F the direct-contact spray chamber air washer of crosssectional area A cs and length l (Figure 12), the stea mass flow rate of dry air per unit cross-sectional area is (52) and the cresponding mass flux of water flowing parallel with the air is where m a mass flow rate of air, kg/s G a mass flux flow rate per unit cross-sectional area f air, kg/(s m 2 ) m L mass flow rate of liquid, kg/s G L mass flux flow rate per unit cross-sectional area f liquid, kg/(s m 2 ) (53) Because water is evapating condensing, G L changes by an amount dg L in a differential length dl of the chamber. Similar changes occur in temperature, humidity ratio, enthalpy, and other properties. Because evaluating the true surface area in direct-contact equipment is difficult, it is common to wk on a unit volume basis. If a H and a M are the area of heat transfer and mass transfer surface per unit of chamber volume, respectively, the total surface areas f heat and mass transfer are (54) The basic equations f the process occurring in the differential length dl can be written f Mass transfer Fig. 12 (55) That is, the water evapated, moisture increase of the air, and mass transfer rate are all equal. Heat transfer to air Air Washer Spray Chamber m a A cs G a m L A cs G L A H a H A cs l and A M a M A cs l dg L G a dw K M a M ( W i W )dl G a c pm dt a h a a H ( t i t a )dl (56) The heat and mass transfer areas of spray chambers are assumed to be identical (a H a M ). Where packing materials, such as wood slats Raschig rings, are used, the two areas may be considerably different because the packing may not be wet unifmly. The validity of the Lewis relation was discussed previously. It is not necessary to account f the small changes in latent heat h fg after making the two previous assumptions. Energy balance G a dh ± G L c L dt L (59) A minus sign refers to parallel flow of air and water; a plus refers to counterflow (water flow in the opposite direction from airflow). The water flow rate changes between inlet and outlet as a result of the mass transfer. F exact energy balance, the term (c L t L dg L ) should be added to the right side of Equation (59). The percentage change in G L is quite small in usual applications of air-conditioning equipment and, therefe, can be igned. Heat transfer to water ± G L c L dt L h L a H ( t L t i )dl (60) Equations (55) to (60) are the basic relations f solution of simultaneous heat and mass transfer processes in direct-contact airconditioning equipment. To facilitate the use of these relations in equipment design perfmance, three other equations can be extracted from the above set. Combining Equations (58), (59), and (60) gives h h h i L a H h L t L t i K M a M K M (61) Equation (61) relates the enthalpy potential f total heat transfer through the gas film to the temperature potential f this same transfer through the liquid film. Physical reasoning leads to the conclusion that this ratio is proptional to the ratio of gas film resistance (1/K M ) to liquid film resistance (1/h L ). Combining Equations (56), (58), and (44) gives dh dt a h h i t a t i Similarly, combining Equations (55), (56), and (44) gives dw dt a W W i t a t i (62) (63) Equation (63) indicates that, at any cross section in the spray chamber, the instantaneous slope of the air path dw/dt a on a psychrometric chart is determined by a straight line connecting the air state with the interface saturation state at that cross section. In Figure 13, state 1 represents the state of the air entering the parallel-flow air washer chamber of Figure 12. The washer is operating as a heating

11 Mass Transfer 5.11 and humidifying apparatus, so the interface saturation state of the water at air inlet is the state designated 1 i. Therefe, the initial slope of the air path is along a line directed from state 1 to state 1 i. As the air is heated, the water cools and the interface temperature drops. Cresponding air states and interface saturation states are indicated by the letters a, b, c, and d in Figure 13. In each instance, the air path is directed toward the associated interface state. The interface states are derived from Equations (59) and (61). Equation (59) describes how air enthalpy changes with water temperature; Equation (61) describes how the interface saturation state changes to accommodate this change in air and water conditions. The solution f the interface state on the nmal psychrometric chart of Figure 13 can be determined either by trial and err from Equations (59) and (61) by a complex graphical procedure (Kusuda 1957). Air Washers Air washers are direct-contact apparatus used to (1) simultaneously change the temperature and humidity content of air passing through the chamber and (2) remove air contaminants such as dust and ods. Adiabatic spray washers, which have no external heating chilling source, are used to cool and humidify air. Chilled-spray air washers have an external chiller to cool and dehumidify air. Heated-spray air washers, with an external heating source that provides additional energy f water evapation, are used to humidify and possibly heat air. Example 5. A parallel-flow air washer with the following design conditions is to be designed (see Figure 12). Water temperature at inlet t L1 35 C Water temperature at outlet t L C Air temperature at inlet t a C Air wet-bulb at inlet t a1 7.2 C Air mass flow rate per unit area G a kg/(s m 2 ) Spray ratio G L /G a 0.70 Air heat transfer coefficient per cubic metre of chamber volume h a a H 1.34 kw/(m 3 K) Liquid heat transfer coefficient per cubic metre of chamber volume h L a H kw/(m 3 K) Air volumetric flow rate Q 3.07 m 3 /s Solution: The air mass flow rate m a kg/s; the required spray chamber cross-sectional area is, then, A cs m /G a 3.68/ m 2 a. The mass transfer coefficient is given by the Lewis relation [Equation (44)] as K M a M ( h a a H ) c pm kg ( m 3 s) Figure 14 shows the enthalpy/temperature psychrometric chart with the graphical solution f the interface states and the air path through the washer spray chamber. 1. Enter bottom of chart with t a1 of 7.2 C, and follow up to saturation curve to establish air enthalpy h 1 of 41.1 kj/kg. Extend this enthalpy line to intersect initial air temperature t a1 of 18.3 C (state 1 of air) and initial water temperature t L1 of 35 C at point A. (Note that the temperature scale is used f both air and water temperatures.) 2. Through point A, construct the energy balance line A-B with a slope of dh c L G L dt L G a Point B is determined by intersection with the leaving water temperature t L C. The negative slope here is a consequence of the parallel flow, which results in the air/water mixture s approaching, but not reaching, the common saturation state s. (Line A-B has no physical significance in representing any air state on the psychrometric chart. It is merely a construction line in the graphical solution.) 3. Through point A, construct the tie-line A-1 i having a slope of h h h i L a H K M a M 1.33 t L t i The intersection of this line with the saturation curve gives the initial interface state 1 i at the chamber inlet. [Note how the energy balance line and tie-line, representing Equations (59) and (61), combine f a simple graphical solution on Figure 14 f the interface state.] Fig. 14 Graphical Solution f Air-State Path in Parallel Flow Air Washer Fig. 13 Chart Air Washer Humidification Process on Psychrometric Fig. 13 Air Washer Humidification Process on Psychrometric Chart Fig. 14 Graphical Solution f Air-State Path in Parallel Flow Air Washer

12 ASHRAE Handbook Fundamentals (SI) 4. The initial slope of the air path can now be constructed, accding to Equation (62), drawing line 1-a toward the initial interface state 1 i. (The length of line 1-a depends on the degree of accuracy required in the solution and the rate at which the slope of the air path changes.) 5. Construct the hizontal line a-m, locating point M on the energybalance line. Draw a new tie-line (slope of 12.6 as befe) from M to a i locating interface state a i. Continue the air path from a to b by directing it toward the new interface state a i. (Note that the change in slope of the air path from 1-a to a-b is quite small, justifying the path incremental lengths used.) 6. Continue in the manner of step 5 until point 2, the final state of air leaving the chamber, is reached. In this example, six steps are used in the graphical construction, with the following results: State 1 a b c d 2 t L h t i h i t a The final state of air leaving the washer is t a C and h kj/kg (wet-bulb temperature t a C). 7. The final step involves calculating the required length of the spray chamber. From Equation (59), G a l K M a M dh ( h) The integral is evaluated graphically by plotting 1/(h i h) versus h, as shown in Figure 15. Any satisfacty graphical method can be used to evaluate the area under the curve. Simpson s rule with four equal increments of h equal to 8.2 gives N 2 1 Therefe, the design length is l (1.628/1.33)(0.975) 1.19 m. This method can also be used to predict perfmance of existing direct-contact equipment and to determine transfer coefficients when perfmance data from test runs are available. By knowing the water and air temperatures entering and leaving the chamber and the spray ratio, it is possible, by trial and err, to determine the proper Fig. 15 Graphical Solution of dh/(h i h) dh ( h 3) ( y ( h) 1 + 4y 2 + 2y 3 + 4y 4 + y 5 ) h i N ( 8.2 3)[ ( ) + ( ) h i + ( ) ] slope of the tie-line necessary to achieve the measured final air state. The tie-line slope gives the ratio h L a H /K M a M ; K M a M is found from the integral relationship in Example 5 from the known chamber length l. Additional descriptions of air spray washers and general perfmance criteria are given in Chapter 19 of the 2004 ASHRAE Handbook HVAC Systems and Equipment. Cooling Towers A cooling tower is a direct-contact heat exchanger in which waste heat picked up by the cooling water from a refrigerat, air conditioner, industrial process is transferred to atmospheric air by cooling the water. Cooling is achieved by breaking up the water flow to provide a large water surface f air, moving by natural fced convection through the tower, to contact the water. Cooling towers may be counterflow, crossflow, a combination of both. The temperature of water leaving the tower and the packing depth needed to achieve the desired leaving water temperature are of primary interest f design. Therefe, the mass and energy balance equations are based on an overall coefficient K, which is based on (1) the enthalpy driving fce due to h at the bulk water temperature and (2) neglecting the film resistance. Combining Equations (58) and (59) and using the parameters described previously yields G L c L dt K M a M ( h i h)dl G a dh K a dv( h h a ) A cs K a V m L t 2 t 1 c L dt ( h h a ) (64) (65) Chapter 36 of the 2004 ASHRAE Handbook HVAC Systems and Equipment covers cooling tower design in detail. Cooling and Dehumidifying Coils When water vap is condensed out of an airstream onto an extended-surface (finned) cooling coil, the simultaneous heat and mass transfer problem can be solved by the same procedure set fth f direct-contact equipment. The basic equations are the same, except that the true surface area of coil A is known and the problem does not have to be solved on a unit volume basis. Therefe, if, in Equations (55), (56), and (58), a M dl a H dl is replaced by da/a cs, these equations become the basic heat, mass, and total energy transfer equations f indirect-contact equipment such as dehumidifying coils. The energy balance shown by Equation (59) remains unchanged. The heat transfer from the interface to the refrigerant now encounters the combined resistances of the condensate film (R L 1/h L ); the metal wall and fins, if any (R m ); and the refrigerant film (R r A/h r A r ). If this combined resistance is designated as R i R L + R m + R r 1/U i, Equation (60) becomes, f a coil dehumidifier, ± m L c L dt L U i ( t L t i )da (66) (plus sign f counterflow, minus sign f parallel flow). The tie-line slope is then h h U i i t L t i K M (67) Fig. 15 Graphical Solution of dh/(h i h) Figure 16 illustrates the graphical solution on a psychrometric chart f the air path through a dehumidifying coil with a constant refrigerant temperature. Because the tie-line slope is infinite in this case, the energy balance line is vertical. The cresponding interface