Document MJYedxw9OLp4Jja50Qr3wY2m7

76 CHAPTER 5 1965 Guide And Data Book compared to the absolute temperature T, then the mass-,., transfer coefficient ho will have an identical magnitude in" Equations 2a, 2b, and 2c. As the property variations increase, however, the value of the coefficients will diverge. r Since the water vapor transferred to the air stream must-.' be supplied with its latent heat of vaporization (h/a), the heat flux between the liquid *nH the interface (and, consequently, . the heat supplied by the heater) is : ( q/A .-- q,/A .+ (m,/A)h/,. ' or - q/A " h(l( -- tj) + ho(j>n -- P-)h/ (3) Table 1 .... Mass Diffusivities for Gases in Air (Caws at 77 F and t atm) v" , . \ ' Ca* Ammonia Benzene t.hnnl Hydrogen Oxygen Water Vapor . ,V', O, m tq H par hr 1.08 0.34 , 0.64. 0.46 1.60 > 0.80 - 0.99 m.,/A-- - D,(dp./dy) (5a) q/A -- <) (4) per hour. ! h/, - enthalpy of evaporation (latent heat of vaporisation); Btu per pound. > ' '11' hi heat transfer coefficient for the liquid side, Btu per (hour) (square foot) (Fahrenheit degree): . If the flow,conditions are specified so that the heat and mass transfer -coefficients can be predicted, and the air stream properties and pan water temperature are given, Equations 1, 2, 3, and 4 can be solved simultaneously to find: (1) the interfare saturation state, (2) the mass transfer rate, (3) the heat transfer to the air,`and (4) the total heat transfer sup plied to the water by the heater. . . < In the preceding discussion, the temperatures, of.the air stream and the water in the pan remained constant through out the entire process. However, in many, processes^ such as in the counterflow cooling tower,-the air and water temper atures vary throughout the process, with resultant variations in the interface state. Equations 1, 2, 3, and 4 can be'used to determine the total transfer in a step-by-step-procedure, solv ing in each step for the interface state and- the transfer rates; and then summing or integrating over each step to obtain the total transfer. However, a simpler method is available and is more oom- monly used. Before developing this method, it is desirable to Hiw-imh the two means by which mass transfer or diffusion may occur: (1) molecular diffusion, and (2) convective or eddy diffusion. MOLECULAR DIFFUSION Consider a quantity, of air in a.small dosed container, where the air at one end has been slightly heated to cause a tem perature gradient in.the air.. If.the container is isolated, the temperature will soon equalize as a result of the.passing of energy by a.random mixing .of the air molecules. This is,the familiar process of heat conduction, in. a gas. Assume that a amount of water vapor is,introduced at one end of the container which is again isolated. After some time, .the .water vapor will be equally distributed throughout the, air by the e>m> molecular movements. This mass transfer process is known as pwleeular diffusion. The equations'governing this process are analogous to those for,heat conduction.,. Basic Equation ' The bade equation for molecular diffusion is usually re ferred Id as Pick's law and can be derived from the principles of the kinetic theory of gases. Depending upon the parameter chosen* for the driving potential, the one-dimensional dif fusion rate per unit area in~the'y direction (normal to'the transfer surfaceVis cmirsmarf is -- - '-v<5 - 0r...,, or m,/A = -- pD,(dti>,/dy) in,/A..= - (D,/R.T)(.dp./dy). . (5b) (5c) The minus sign appears because the'concentration gradient is' negative in the direction of diffusion. The quantity D, is known as the mass diffusivity or the diffusion coefficient and is a property of the gas mixture. Since the mass flux (th/A) is expressed in pounds per (hour) (square foot) and the partial density (p,) in pounds*per cubic foot, in the English system of units, the,units of Z), for Equation 5a must be ' JD, -r(lb per (hi)(ft*)][(ft* per lb)(ft)] .= ft*/hr. The* Diffusion Coefficient Extensive gas kinetic calculations* show that D, depends only;slightly on the concentration of the.:components in the mixture, that it is inversely proportional-to the pressure, and that-.with temperature its increase is proportional to T14*, where n a. value between 0.5 and-1. The equations for diffusion coefficients in gases, derived by several investigators from-kinetic theory models, may alibe expressed by Equa tion 6.* where ^Ma+ Mb D, = b - Pt(vmaxi* + fw0*)1 ' Ma Mb (6) 6 .TM a constant., , Pt - total pressure, atmospheres. ... Mx,Mb " molecular weights of the two gases in the gas mix ture, pounds per pound moL . vma, *mb =* mnWiilar volumes of the. two gases in the mixture, cubic feet per,.pound moL (See Reference 2 for a table to compute these values.) .< -Upon noting that several theories agreed on this form, Gilli land evaluated' the constant 6 empirically by plotting the re sults-of some 400 experimental determinations. The resulting value was b - 0.0069 The principal weakness of Equation 6 is the. temperature function T*1* since Z). is more nearly proportional to T*. Where experimental data are available for D,, they should be preferred to the semitheoretical Equation 6/A few experi mental1 values for diffusion of some common gases in air am given in Table 1. More extensive tables are available in References 2, 3, and 4. Of particular interest to tire air conditioning engineer is the mftjgt diffusivity of water vapor in air. Analyzing the available date, Spalding found that, up to a,temperature of 2000 F, the mmga diffusivity of water vapor in air may be expressed as Pt ' T +441.. (7) Moss TraiSfer-'' ,J ' where all-quantities are expressed in the same units as in Equation 6. ' ' Analogy to Conduction Heat Transfer Molecular diffusion is directly analogous-to. conduction heat transfer:'Both phenomena are si result of random molecu lar in'a stagnant fluid, or in a fluid in laminar (streamline) flow. In each case the equation governing the process can be expressed in the form Fbuc =* diffusivity X concentration gradient For a two-component gaseous system (e-g., water vapor and air) in which toe density of toe gas p and the specific heat c, are constant," Equations 8 and '9 may be written for the fluxes of and energy in' one-dimensional systems:'. ' .u/a '-- D, -r- (p) (Fick'a law for constant^) (8) dy.. . . < ; .. 77 profiles would all be identical in form. This is true if p and Cp are essentially constant through the fluid field and the dif fusivities are identical. Stated in'terms of dimensionless pa rameters the condition is where Nfr = Ns* -- Schmidt number = --~ dimensionless. ' pD. Where toe Prandtl number (N/,=c^n/k) equals the Schmidt number (AZsc--pfpp*), the temperature and partial pressure profiles are identical; and if, in Addition, both are unity;:to*^e (Fourier's law for constant pe,) (9) where Jc = thermal conductivity, Btu per (hour) (square - foot) (Fahrenheit degree per foot). a thermal diffusivity, square feet per hour. . Cp specific heat at constant pressure, Btu per, (pound) (Fahrenheit degree). Equation 8 states that mass transfer occurs because there is a gradient in mass concentration, and the mam flux equals the mass diffusivity D, times the mass concentration gradient. Equation 9 states that heat transfer occurs because, there is a gradient in energy Concentration, and the heat flux equals the thermal diffusivity a(=k/pcp) times the energy concentra tion gradient. ..--i / - - The equation for the viscous shear force in one-dimensional flow is ud -9 ydy <y.) or mt* rt* = viscous shear force .in the yx plane, pounds force per square foot. 9 m gravitational acceleration = 4.17 X 10* (pounds'.force) ' (feet) per (pound mass) (hour) (hour). P = absolute viscosity, pounds per foot-hour. ; F, " local velocity'of gas in the x direction parallel to the transfer surface, feet per hour. p ** kmematic viscosity, or momentum diffusivity, ' square Test per hour. 10 states that momentum transfer occurs because toere isagradientin momentum concentration, and toe momen- . (jppx) equals the momentum diffusivity;(kinematic ^tsowity) v times the momentum concentration gradient. t is important to note that the mass, thermal, and momen tum diffusivities all have toe same 'units* (ft* per hr) and that the concentration gradients are linear for a uniform medium state. If, under a. fortunate.circumstance, these ^ffusmtws all had the same numerical value, theconcentra- 8r*dients would be identical. That is,.if a fluid were in vea<*y one-dimensional laminar flow o^^ a^surfaCe sucri that ***a=ri toe partial pressure, temperature,' and velocity SmfTortfy ot partial pressure, femperefure, and vetoafy profifei - : > -; fig. 2 .... Laminar Boundary Layer on a fiat Surface profiles are the same as the velocity profile. This condition is illustrated in Fig. 2-for a gas B diffusing from a stationary boundary into a gas A in one-dimensional steady flow past the boundary. For air and water vapor at normal room conditions the Prandtl number is 0.71 and the Schmidt number is 0.60, a condition surprisingly close to that'postulated. Equimoial Counterdiffusion Consider a mixture of air and'water vapor in a closed .con tainer in equilibrium at all points throughout the container so that the total^pressure, (in-petards,'/orce per square foot) is ZYand the absolute-temperature T. A concentration gradient exists through-the mixture so that-toe ^partial pressure p, of toe water vapor is as shown in Fig. 3. It is assumed here that