The basic equation for 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 or mole fraction C_B/C, Fick’s law is

where ρ = ρ_A + ρ_B and C = C_A + C_B.

The minus sign indicates that the concentration gradient is negative in the direction of diffusion. The proportionality factor D_v is the mass diffusivity or the diffusion coefficient. The total mass flux and molar flux are due to the average velocity of the mixture plus the diffusive flux:

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 forms 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 form using the relation C_B ≡ ρ_B/M_B.

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

when ρ_B << ρ and ρ_A ≈ ρ.

Equation (3) can be used without significant error for water vapor 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 vapor and ρ is the density of moist air (air and water vapor mixture). The error in J_B caused by replacing ρ[d(ρ_B/ρ)/dy] with dρ_B/dy is less than 2%. At temperatures below 60°C where ρ_B < 0.10ρ, Equation (3) can still be used if errors in J_B as great as 10% are tolerable.

Fick’s law can be simplified for cases of dilute mass diffusion in solids, stagnant liquids, or stagnant gases. In these cases, ρ_B << ρ and v ≈ 0, which yields the following approximate result:

For 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,

and when the gradient in T is small, Equation (3) can be written as

or

If v ≈ 0, Equation (4) may be written as

or

The partial pressure gradient formulation for mass transfer analysis has been used extensively; this is unfortunate because the pressure formulation [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 (or concentration) gradient formulation expressed in Equations (1) to (4) is more 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, nor the temperature gradient negligible. Consequently, this chapter emphasizes the density formulation.

For a binary mixture, the diffusion coefficient D_v is a function of temperature, pressure, and composition. Experimental measurements of D_v for most binary mixtures are limited in range and accuracy. Table 1 gives a few experimental values for diffusion of some gases in air. For more detailed tables, see the Bibliography.

In the absence of data, use equations developed from (1) theory or (2) theory with constants adjusted from limited experimental data. For binary gas mixtures at low pressure, D_v is inversely proportional to pressure, increases with increasing temperature, and is almost independent of composition for a given gas pair. Bird et al. (1960) present the following equation, developed from kinetic theory and corresponding states arguments, for estimating D_v at pressures less than 0.1p_{c min}:

where

Dv	=	diffusion coefficient, mm2/s
a	=	constant that depends on units used
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:

For nonpolar gas pairs

For water vapor with a nonpolar gas

In nonpolar gas, intermolecular forces are independent of the relative orientation of molecules, depending only on the separation distance from each other. Air, composed almost entirely of nonpolar gases O₂ and N₂, is nonpolar.

Equation (8) is stated to agree with experimental data at atmospheric pressure to within about 8% (Bird et al. 1960).

Mass diffusivity D_v for binary mixtures at low pressure is predictable within about 10% by kinetic theory (Reid et al. 1987).

where

σAB	=	characteristic molecular diameter, nm
ΩD, AB	=	temperature function, dimensionless

D_v is in mm²/s, p in kPa, and T in K. 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. More realistic models for molecules having intermolecular forces of attraction and repulsion lead to values that are functions of temperature. Bird et al. (1960) and Reid et al. (1987) present tabulations of Ω_D,AB. 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 for Equation (8) using a simplified intermolecular potential field function for water vapor and air (Mason and Monchick 1965). The following empirical equation is for mass diffusivity of water vapor in air up to 1100°C (Sherwood and Pigford 1952):

Figure 1 shows diffusion of one gas through a second, stagnant gas. Water vapor 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 temperature gradient has a negligible effect. Water vapor 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

or

The partial pressure gradient of the water vapor causes a partial pressure gradient of the air such that

or

Air, then, diffuses toward the liquid water interface. Because it cannot be absorbed there, a bulk velocity v of the gas mixture is established in a direction away from the liquid surface, so that the net transport of air is zero (i.e., the air is stagnant):

The bulk velocity v transports not only air but also water vapor away from the interface. Therefore, the total rate of water vapor diffusion is

Substituting for the velocity v from Equation (13) and using Equations (11b) and (12) gives

Integration yields

or

where

P_Am is the logarithmic mean density factor of the stagnant air. The pressure distribution for this type of diffusion is shown in Figure 2. Stagnant refers to the net behavior of the air; it does not move because bulk flow exactly offsets diffusion. The term P_Am in Equation (16b) approximately equals unity for dilute mixtures such as water vapor in air at near-atmospheric conditions. This condition makes it possible to simplify Equations (16) and implies that, for dilute mixtures, the partial pressure distribution curves in Figure 2 are straight lines.

Neglecting the correction factor P_Am for this example gives a difference of less than 1% between the calculated experimental and empirically predicted values of D_v.

Figure 3 shows two large chambers, both containing an ideal gas mixture of two components A and B (e.g., air and water vapor) 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 steady state, the molar flows of A and B must be equal but opposite:

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 (or A):

or

or

Because of the greater density, diffusion is slower in liquids than in gases. No satisfactory molecular theories have been developed for calculating diffusion coefficients. The limited measured values of D_v show that, unlike for gas mixtures at low pressures, the diffusion coefficient for 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

Equation (22) expresses steady-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 Δy = y₂ – y₁. Bird et al. (1960), Eckert and Drake (1972), Hirschfelder et al. (1954), Reid and Sherwood (1966), Sherwood and Pigford (1952), and Treybal (1980) provide equations and tables for evaluating D_v. Hirschfelder et al. (1954) provide comprehensive treatment of the molecular developments.

Diffusion through a solid when the solute is dissolved to form 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 one-dimensional steady-state problems. Values of mass diffusivity are generally lower than they are for liquids and vary with temperature.

Diffusion of a gas mixture through a porous medium is common (e.g., diffusion of an air/vapor mixture through porous insulation or other vapor-permeable building materials). Vapor diffuses through the air along the tortuous narrow passages within the porous medium. Mass flux is a function of the vapor pressure gradient and diffusion coefficient, as indicated in Equation (7a). It is also a function of the structure of the pathways within the porous medium and is therefore called structure-sensitive diffusion. All these factors are taken into account in the following version of Equation (7a):

where μ̅ is called the moisture permeability of the porous medium. For a layer of porous solid L thick, μ̅/L is called the permeance, the unit for which is kg/(s · m² · Pa). Permeance is also known as vapor flow conductance, which is analogous to heat flow conductance in heat transfer. Permeability is the product of permeance and layer thickness [i.e., kg/(s · m · Pa)], but is often given in units of ng/(s · m · Pa).

Consider a slab L = y₂ – y₁ thick with cross-sectional area A_c. The vapor pressure is p_B1 on one side of the slab and p_B2 on the other side. If the permeability μ̅ is constant, integration of Equation (23) yields

where ṁ_B is the vapor flow rate. In Equation (25), p_B1 – p_B2 can be considered a driving potential, and L/(μ̅A_c) a resistance. This is analogous to Equation (1) in Chapter 4, where q is the heat transfer rate, t_s1 – t_s2 is the driving potential, and L/(kA_c) is the resistance.

Chapters 25 and 26 present this topic in more depth.

Convective mass transfer is analogous to convective heat transfer where geometry and boundary conditions are similar. The analogy holds for both laminar and turbulent flows and applies to both external and internal flow problems.

Mass Transfer Coefficients for External Flows. Most external convective mass transfer problems can be solved with an appropriate formulation that relates the mass transfer flux (to or from an interfacial surface) to the concentration difference across the boundary layer illustrated in Figure 5. This formulation gives rise to the convective mass transfer coefficient, defined as

where

hM	=	local external mass transfer coefficient, m/s
	=	mass flux of gas B from surface, kg/(m2 · s)
ρBi	=	density of gas B at interface (saturation density), kg/m3
ρB∞	=	density of component B outside boundary layer, kg/m3

If ρ_Bi and ρ_B∞ are constant over the entire interfacial surface, the mass transfer rate from the surface can be expressed as

where h̅_M is the average mass transfer coefficient, defined as

Mass Transfer Coefficients for Internal Flows. Most internal convective mass transfer problems, such as those that occur in channels or in the cores of dehumidification coils, can be solved if an appropriate expression is available to relate the mass transfer flux (to or from the interfacial surface) to the difference between the concentration at the surface and the bulk concentration in the channel, as shown in Figure 6. This formulation leads to the definition of the mass transfer coefficient for internal flows:

where

hM	=	internal mass transfer coefficient, m/s
	=	mass flux of gas B at interfacial surface, kg/(m2 · s)
ρBi	=	density of gas B at interfacial surface, kg/m3
ρBb ≡ (1/u̅BAcs) ∫Acs uB ρB dAcs)	=	bulk density of gas B at location x
u̅B ≡ (1/Acs) ∫A uB dAcs	=	average velocity of gas B at location x, m/s
Acs	=	cross-sectional area of channel at station x, m2
uB	=	velocity of component B in x direction, m/s
ρB	=	density distribution of component B at station x, kg/m3

Often, it is easier to obtain the bulk density of gas B from

where

ṁBo	=	mass flow rate of component B at station x = 0, kg/s
A	=	interfacial area of channel between station x = 0 and station x = x, m2

Equation (28) can be derived from the preceding definitions. The major problem is determining u̅_B. If, however, analysis is restricted to cases where B is dilute and concentration gradients of B in the x direction are negligibly small, u̅_B ≈ u̅. Component B is swept along in the x direction with an average velocity equal to the average velocity of the dilute mixture.

Most expressions for the convective mass transfer coefficient h_M are determined from expressions for the convective heat transfer coefficient h.

For 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:

and

where f in Equations (31) and (32) 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 (33) and (34). 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 for component B when cast in dimensionless form. Several primary factors prevent the analogy from being perfect. In some cases, the Nusselt number was derived for smooth surfaces. Many mass transfer problems involve wavy, droplet-like, or roughened surfaces. Many Nusselt number relations are obtained for 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 or suction at the surface because of condensation, evaporation, or 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 or v_i/u̅ > 0.01, especially if the Reynolds number is large.

The kind of similarity between heat and mass transfer that results in Equations (31) to (34) can also be shown to exist between heat and momentum transfer. Chilton and Colburn (1934) used this similarity to relate Nusselt number to friction factor by the analogy

where n = 2/3, St = Nu/(Re Pr) is the Stanton number, and j_H is the Chilton-Colburn j-factor for heat transfer. Substituting Sh for Nu and Sc for Pr in Equations (33) and (34) gives the Chilton-Colburn j-factor for mass transfer, j_D:

where St_m = ShP_Am/(Re Sc) is the Stanton number for mass transfer. Equations (35) and (36) are called the Chilton-Colburn j-factor analogy.

The power of the Chilton-Colburn j-factor analogy is represented in Figures 8 to 11. Figure 8 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 portion of the solid line) and Goldstein’s solution for a turbulent boundary layer (right-hand portion). The right-hand part also represents McAdams’ (1954) correlation of turbulent flow heat transfer coefficient for 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 evaporation or absorption with a gas flowing through the tube. Figure 9 illustrates typical data on vaporization 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:

Similarly, McAdams’ (1954) equation for heat transfer in pipes can be expressed as

This is represented by the dash-dot curve in Figure 9, which falls below the mass transfer data. The curve f/2, representing friction in smooth tubes, is the upper, solid curve.

Data for liquid evaporation from single cylinders into gas streams flowing transversely to the cylinders’ axes are shown in Figure 10. Although the dash-dot line in Figure 10 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, normal pressure drag must be subtracted from the measured total drag. At Re = 1000, skin friction is 12.6% of the total drag; at Re = 31 600, it is only 1.9%. Consequently, values of f/2 at a high Reynolds number, obtained by the difference, are subject to considerable error.

In Figure 11, data on evaporation of water into air for single spheres are presented. The solid line, which best represents these data, agrees with the dashed line representing McAdams’ correlation for heat transfer to spheres. These results cannot be compared with friction or momentum transfer because total drag has not been allocated to skin friction and normal 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, vapor condensation occurs. Typically, air dry-bulb temperature and humidity ratio both decrease as air flows through the exchanger. Therefore, 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 transport coefficients h, h_M, and f. It would be convenient if heat transfer and friction data for dry heating coils could be used with the Colburn analogy to obtain the mass transfer coefficients, but this approach is not always reliable, and Guillory and McQuiston (1973) and Helmer (1974) show that the analogy is not consistently true. Figure 12 shows j-factors for a simple parallel-plate exchanger for different surface conditions with sensible heat transfer. Mass transfer j-factors and friction factors exhibit the same behavior. Dry-surface j-factors 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 more of the total flow field is exposed to the droplets. Roughness caused by the droplets induces mixing and larger j-factors.

The data in Figure 12 cannot be applied to all surfaces, because the length of the flow channel is also an important variable. However, water collecting on the surface is mainly responsible for breakdown of the j-factor analogy. The j-factor analogy is approximately true when surface conditions are identical. Under some conditions, it is possible to obtain a film of condensate on the surface instead of droplets. Guillory and McQuiston (1973) and Helmer (1974) related dry sensible j- and f-factors to those for wetted dehumidifying surfaces.

The equality of j_H, j_D, and f/2 for certain streamlined shapes at low mass transfer rates has experimental verification. For 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.

The extensive developments for calculating heat transfer coefficients can be applied to calculate mass transfer coefficients under similar geometrical and flow conditions using the j-factor analogy. For example, Table 8 of Chapter 4 lists equations for calculating heat transfer coefficients for flow inside and normal to pipes. Each equation can be used for mass transfer coefficient calculations by equating j_H and j_D and imposing the same restriction to each stated in Table 8 of Chapter 4. Similarly, mass transfer experiments often replace corresponding heat transfer experiments with complex geometries where exact boundary conditions are difficult to model (Sparrow and Ohadi 1987a, 1987b).

The j-factor analogy is useful only at low mass transfer rates. As the rate increases, the movement of matter normal to the transfer surface increases the convective velocity. For 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 or, at least, insignificant mass transfer rates. Therefore, if such data are to be valid for a mass transfer process, the mass transfer rate (i.e., the blowing) must be low.

The j-factor relationship j_H = j_D can still be valid at high mass transfer rates, but neither j_H nor j_D can be represented by data at zero mass transfer conditions. Chapter 24 of Bird et al. (1960) and Eckert and Drake (1972) have detailed information on high mass transfer rates.

Heat and mass transfer coefficients are satisfactorily related at the same Reynolds number by equating the Chilton-Colburn j-factors. Comparing Equations (35) and (36) gives

Inserting the definitions of St, Pr, St_m, and Sc gives

or

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. For air and water vapor mixtures, the ratio is (0.60/0.71) or 0.845, and (0.845)^2/3 is 0.894. At low diffusion rates, where the heat/mass transfer analogy is valid, P_Am is essentially unity. Therefore, for air and water vapor mixtures,

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 (40)] is usually called the Lewis relation and is nearly true for air and water vapor at low mass transfer rates. It is generally not true for other gas mixtures because the ratio Le of thermal to vapor diffusivity can differ from unity. Agreement between wet-bulb temperature and adiabatic saturation temperature is a direct result of the nearness of the Lewis number to unity for air and water vapor.

The Lewis relation is valid in turbulent flow whether or not α/D_v equals 1 because eddy diffusion in turbulent flow involves the same mixing action for heat exchange as for mass exchange, and this action overwhelms any molecular diffusion. Deviations from the Lewis relation are, therefore, caused by a laminar boundary layer or a laminar sublayer and buffer zone where molecular transport phenomena are the controlling factors.

The water vapor concentration in the air is the humidity ratio W, defined as

A mass transfer coefficient is defined using W as the driving potential:

where the coefficient K_M is in kg /(s · m²). For 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. Therefore,

where ρ_Am = mean density of dry air, kg/m³. Comparing Equation (43) with Equation (26) shows that

The humid specific heat c_pm of the airstream is, by definition (Mason and Monchick 1965),

or

where c_pm is in kJ/(kg_da · K).

Substituting from Equations (44) and (45b) into Equation (40) gives

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 force, the Lewis relation becomes ratio of heat to mass transfer coefficient equals humid specific heat.

For the plate humidifier illustrated in Figure 7, the total heat transfer from liquid to interface is

Using the definitions of the heat and mass transfer coefficients gives

Assuming Equation (46) is valid gives

The enthalpy of the air is approximately

The enthalpy h_s of the water vapor can be expressed by the ideal gas law as

where the base of enthalpy is taken as saturated water at temperature t_o. Combining Equations (50) and (51) gives

If small changes in the latent heat of vaporization of water with temperature are neglected when comparing Equations (50) and (52), the total heat transfer can be written as

Where the driving potential for heat transfer is temperature difference and the driving potential for mass transfer is mass concentration or partial pressure, the driving potential for simultaneous transfer of heat and mass in an air water/vapor mixture is, to a close approximation, enthalpy.

Air-conditioning equipment can be classified by whether the air and water used as a cooling or heating fluid are (1) in direct contact or (2) separated by a solid wall. Examples of the former are air washers and cooling towers; an example of the latter is a direct-expansion refrigerant (or 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 (or heating) fluid. In the dehumidifying coil, contact is direct with condensate 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.

For the direct-contact spray chamber air washer of cross-sectional area A_cs and length l (Figure 13), the steady mass flow rate of dry air per unit cross-sectional area is

and the corresponding mass flux of water flowing parallel with the air is

where

ṁa	=	mass flow rate of air, kg/s
Ga	=	mass flux or flow rate per unit cross-sectional area for air, kg/(s · m2)
ṁL	=	mass flow rate of liquid, kg/s
GL	=	mass flux or flow rate per unit cross-sectional area for liquid, kg/(s · m2)

Because water is evaporating or 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 work on a unit volume basis. If a_H and a_M are the areas of heat transfer and mass transfer surface per unit of chamber volume, respectively, the total surface areas for heat and mass transfer are

The basic equations for the process occurring in the differential length dl can be written for

That is, the water evaporation rate, air moisture content increase, and mass transfer rate are all equal.

Assuming a_H = a_M and Le = 1, and neglecting small variations in h_fg, Equation (59) reduces to

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 or Raschig rings, are used, the two areas may be considerably different because the packing may not be wet uniformly. The validity of the Lewis relation was discussed previously. It is not necessary to account for small changes in latent heat h_fg after making the two previous assumptions.

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. For exact energy balance, the term (c_Lt_LdG_L) should be added to the right side of Equation (61). The percentage change in G_L is quite small in usual applications of air-conditioning equipment and, therefore, can be ignored.

Equations (57) to (62) are the basic relations for solution of simultaneous heat and mass transfer processes in direct-contact air-conditioning equipment.

To facilitate use of these relations in equipment design or performance, three other equations can be extracted from the above set. Combining Equations (60), (61), and (62) gives

Equation (63) relates the enthalpy potential for total heat transfer through the gas film to the temperature potential for this same transfer through the liquid film. Physical reasoning leads to the conclusion that this ratio is proportional to the ratio of gas film resistance (1/K_M) to liquid film resistance (1/h_L). Combining Equations (58), (60), and (46) gives

Similarly, combining Equations (57), (58), and (46) gives

Equation (65) 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 14, state 1 represents the state of air entering the parallel-flow air washer chamber of Figure 13. The washer operates as a heating and humidifying apparatus, so the interface saturation state of the water at air inlet is the state designated 1_i. Therefore, 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. Corresponding air states and interface saturation states are indicated by the letters a, b, c, and d in Figure 14. In each instance, the air path is directed toward the associated interface state. The interface states are derived from Equations (61) and (63). Equation (61) describes how air enthalpy changes with water temperature; Equation (63) describes how the interface saturation state changes to accommodate this change in air and water conditions. The solution for the interface state on the normal psychrometric chart of Figure 14 can be determined either by trial and error from Equations (61) and (63) or by a complex graphical procedure (Kusuda 1957).

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 odors. Adiabatic spray washers, which have no external heating or 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 for water evaporation, are used to humidify and possibly heat air.

This method can also be used to predict performance of existing direct-contact equipment and to determine transfer coefficients when performance 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 to determine by trial and error the proper slope of the tie-line necessary to achieve the measured final air state. The tie-line slope gives the ratio h_La_H/K_Ma_M; K_Ma_M is found from the integral relationship in Example 7 from the known chamber length l.

Additional descriptions of air spray washers and general performance criteria are given in Chapter 41 of the 2020 ASHRAE Handbook—HVAC Systems and Equipment.

A cooling tower is a direct-contact heat exchanger in which waste heat picked up by the cooling water from a refrigerator, air conditioner, or 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 for air, moving by natural or forced convection through the tower, to contact the water. Cooling towers may be counterflow, crossflow, or 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 for design. Therefore, the mass and energy balance equations are based on an overall coefficient K, which is based on (1) the enthalpy driving force from h at the bulk water temperature and (2) neglecting the film resistance. Combining Equations (60) and (61) and using the parameters described previously yields

or

Chapter 40 of the 2020 ASHRAE Handbook—HVAC Systems and Equipment covers cooling tower design in detail.

When water vapor 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 forth for 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. Therefore, if, in Equations (57), (58), and (60), a_Mdl or a_Hdl is replaced by dA/A_cs, these equations become the basic heat, mass, and total energy transfer equations for indirect-contact equipment such as dehumidifying coils. The energy balance shown by Equation (61) 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_rA_r). If this combined resistance is designated as R_i = R_L + R_m + R_r = 1/U_i, Equation (62) becomes, for a coil dehumidifier,

(plus sign for counterflow, minus sign for parallel flow).

The tie-line slope is then

Figure 17 illustrates the graphical solution on a psychrometric chart for 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 corresponding interface and air states are denoted by the same letter symbols, and the solution follows the same procedure as in Example 7.

If the problem is to determine the required coil surface area for a given performance, the area is computed by the following relation:

This graphical solution on the psychrometric chart automatically determines whether any part of the coil is dry. Thus, in the example illustrated in Figure 17, entering air at state 1 initially encounters an interface saturation state 1_i, clearly below its dew-point temperature t_d1, so the coil immediately becomes wet. Had the graphical technique resulted in an initial interface state above the dew-point temperature of the entering air, the coil would be initially dry. The air would then follow a constant humidity ratio line (the sloping W = constant lines on the chart) until the interface state reached the air dew-point temperature.

Mizushina et al. (1959) developed this method not only for water vapor and air, but also for other vapor/gas mixtures. Chapter 23 of the 2020 ASHRAE Handbook—HVAC Systems and Equipment shows another related method, based on AHRI Standard 410, of determining air-cooling and dehumidifying coil performance.