Document qkwgregoGYYjYVEmveRzJBngk

82 CHAPTER 5 1965 Guide And Data Boole =31,600, only 1.9 percent. Consequently, the values of //2 at high Reynolds number, obtained by difference, are subject to considerable error. In fig. 10 some data on the evaporation of water into air for angle spheres are presented. The solid line, which best represents these data, is in quite good, agreement with the dashed line representing McAdams, correlation for* heat transfer to spheres. It is not possible to compare these;results with friction or momentum transfer, since the breakdown of total drag into skin, friction and normal pressure drag has not beenimade. The application of these data to air-water con tacting devices such as air washers and spray, cooling.towers b well substantiated. In summary, the equality of ja, jo, and //2 for certain streamline shapes at low 'mass transfer rates has excellent experimental verification. For flow past bluff objects, ja and jo are much smaller than //2, based on total pressure drag. The heat and mass transfer are, however, still related in a use ful way by equating ja and ]d- Example l. Using solid cylinders of volatile solids (e.g., naphthalene, camphor, dichlorobenzene) with air Sow normal to these cylinders, Bediogfield'and Drew* found that the ratio be tween, the heat and mass transfer coefficients could be closely correlated by the relation h/pho = 0.294(n/pD,)*'** For completely dry air at 70 F flowing at a velocity of 31' fps over a wet-bulb thermometer of diameter d - 0.300 in., detaiy mine the heat and mass transfer coefficients from Fig. 9 and com pare their ratio with the Bedingfield-Drew relation. ' Solution: For dry air at 70 F, p = 0.075 lb per-cu ft, p = 0.0441 lb per (ft)' (hr), k = 0.0150 Btu per (hr) (ft*) (F deg per ft), e, = 0.2398 Btu per (lb) (F deg). Prom Equation 7/ D, -- 0.971 sq ft per hr. Therefore Nr. = pVd/p = 0.075(31H3600)(0.300)/12(0.0441:) - 475 Npr' - - (0.2398)(0.0441)/0.0150 = 0.706 Ns. = it/pD, = 0.0441/0.075(0.971) =' 0.605 From Fig. 9 at Nr.. = 475, ja = ja = 0.026 and A " japCrVi.HNp.)*!* A = (0.025)(0.075)(0.2398)(31)(3600)/(0:706)* ' = 63.3 Btu per (hr)(sq.ft)(F deg) ho - jDVm/(Na.)il* - (0.025)(31)(3600)/(0.605).*/* ' . = 3900 ft.per hr .- h/pKo - 63.3/3900(0.075) - 0.216 Btu per (lb)(F-deg) ' From the Bedihgfield and Drew rebtion hfpn'o - 0;294(0.605)-* = 0.222 Btu per (lb)(F deg) Note further that the Reynolds analogy, Equation 24, suggests that h/phjt -- c. = 0.2396 'Btu- per (lb) (F deg). The close agreement here Is due to the fact that the ratio Ns./Npr ia,0.605 /0.706 or 0.85 , so that the exponent of these numbers has little effect on the ratio of the transfer coefficients. The Lewis' Relation Tire experimental verification of the similarity relations presented in the preceding section suggests that the heat and mass .transfer coefficients are satisfactorily related, at the same.Reynolds number, by equating the Chilton-Colburn jhfactors. This leads to. hppim / p \*ri A tcrp \*/*> VmP, \ pD, ) " pc>Vm\ k ) or A p, r fi/pDj ^*f* pirn / Ct \ hope, " P, L 1 Pt \ D.) (29) The quantity a/D. b called the Lewb number Afrits magnitude expresses, relative rates of propagation of. energy and mass within a system. It b fairly insensitive to tempera ture variation. For air and water vapor mixtures the' ratio is (0.60/0.71) or-0.845, and (0.845)1/1 is 0.894.- Furthermore, at low; diffusion rates, where the heat-mass transfer analogy b valid, the. ratio (pL-/Pf) b essentially unity. Therefore, for air and water vapor- mixtures, we set h hope. (30) That is, the ratio of the heat transfer coefficient to the mass transfer coefficient b equal to the specific heat per unit volume at constant pressure of the mixture. This rebtion, first de rived by W. K. Lewb,1* b usually called the Lewb rebtion. It b important to recognise that this rebtion b fortuitously very nearly.true for air and water vapor at low mass transfer rates. It b not, in general; true for other gas mixtures because the ratio of thermal'to vapor diffusivity,'Ni*,'is usually dif ferent from unity. Another interesting point which deserves attentionhere b that the reading of a wet-bulb thermometer owes its importance to the validity of the Lewb rebtion. That is, agreement between wet-bulb temperature and adiabatic saturation temperature b a direct result of the nearness of Lewb number to unity for air and water vapor. The Lewb relation-is valid in turbulent flow whether or not the ratio of a/D, equals l.'Thb'follows.from the fact that the eddy diffusion process 'in turbulent flow involves the sam* macroscopic mixing action for heat exchange as for masn ex change, and this action completely overwhelms any molecular diffusion process. The dcvbtions from theLewis rebtion are, therefore, due to the presence of' a laminar boundary bycr as in Fig. 2, or A laminar sublayer and buffer zone as in Fig. 6 wherein molecular, transport phenomena are the controlling factor. SIMULTANEOUS HEAT AND MASS TRANSFER BETWEEN WATER-WETTED SURFACES AND AIR As stated in the section Fundamental Principles, a simpli fied method b commonly used for solving problems involving simultaneous heat and mass transfer. This method, which is developed by use of the Lewb rebtion, gives satisfactory results for most air-conditioning processes. It should be noted, however, that extrapobtion to very high mass transfer; rates, where the simple heat-mass transfer analogy b not valid, probably will lead to erroneous results. Enthalpy Potential The quantity usually used to express.the water vapor con centration in the air b the humidity ratio-TF, defined as Wm,/m. = p./p. - It b,.therefore, convenient to define a mass transfer coeffi cient using W as.the driving potential, hence m./A = KD{Wt - Wm) (31). where the. coefficient Kd has .the units pounds per (hour) (square foot)..Using the definition of W and'noting that for dilute.mixtures PmSPw (i-e., the partial mass density of dry air changes, only by a small percentage between interface and free stream conditions), it b permissible to write rn./A " --- (ph -- p.m) Pmm where pom is the.mean density of dry air, pounds per cubic foot. Comparing this equation with Equation 2a shows that hi> " Ko/pam The humid specific heat c^. of the air stream b, by definition Moss Transfer - - .-a + Wm)c, Btu per (F deg)(lb of dry air) p.or e {p/p*m)c. Substituting from these latter expressions into the Lewb rebtion (Equation 30) gives hpam ^ ^ h KpPa.epm KoC*m (32) since due to the small change in dry-air density. Using a rroc^ transfer coefficient with humidity ratio as-the driving force, the Lewb rebtion becomes: ratio ofj heat to mass transfer coefficient .equab humid specific heat. ] For the pan humidifier illustrated in Fig. 1, it has been shown that the total heat transfer from the liquid to the interface b . * q/A - q./A + (m,/A)b/. Utilising the definitions of the transfer coefficients, ' q/A = A(<< -- f) 4- Ko(Wi -- fP)h/ > Aiming the Lewb rebtion (Equation 32)to be valid gives q/A = KnicomiU - u + (HV- WJihf.] ; (33) The enthalpy of the air b, by definition - I h = ctJ 4* IFh, (Btu per.lb dry air) trhere x b enthalpy of air, Btu per pound, h, = enthalpy of water'vapor, Btu per pound. The enthalpy of the water vapor h, may, by the perfect gas law be expressed as h, = *(*.-- 4) 4- hft. where the base of enthalpy b taken as saturated water at.temperature t*. Choosing U =-0 F,to correspond .with the base of the dry-air enthalpy gives h = (Cf. + Wcp.)t + TFh/,, = Cpmt + Whf,, (34) Comparing Equations 33 and 34, it b obvious that if.small changes in the btent heat of vaporisation of water with tem perature are neglected, the total heat transfer can be written q/A = Knthi - h.) ' (35) Thus, whereas the driving potential, for heat transfer b temperature difference and .the driving-potential for, mass transfer b mass concentration or partial: pressure, the driving potential for simultaneous transfer of heat.aind brass in an air water-vapor mixture is enthalpy. Basic Equations for Direct-Contact Equipment Air-conditioning equipment may be classified according to whether there b direct contact between water used as a cool ing or heating fluid and air, or whether the heating or cooling fluid b separated from the air stream by, a-solid walL.Ex amples of the former are air washers, and-cooling towers, whereas the most common example of- the btter b a direct e*Pajiswn refrigerant (or water) cooling and-dehumidifying coiL It should be noted that in both cases the air stream b in contact with a water surface. The term direct contact implies that the contact b directly with, the, cooling (or heating) fluid. In the dehumidifying coil the contact b directly with the condensate removed from the air stream,-1but-indirectly with the refrigerant flowing inside1 the tubes of the coiL-The in ability to evaluate surface areas in directcontact equipment u ffie principal reason for tr^ting these' two cases separately. For the direct-contact spray chamberof/crosa-sectional 83 - - fig. 11 . Air Washer Sproy Chamber- area Am and length l (FigJ ll), the steady mass flow rate of dry air b ' ;. tO./Am , and the mass flow rate of water flowing parallel with the air wl/Aa = Gl ' w.' -- mass flow rate of air, pounds per hour.' ' G. -- mug* velocity, or flow rate per unit croaa occtional' area, . for air, pounds per (horn) (square foot). ! Wl = flow rate of liquid,' pounds per hour. Gl ~ miu velocity, or flow rate per unit cross-sectional area, for liquid, pounds per (hour) (square foot). Since water b evaporating or condensing, Gl changes by an amount dGh in a differential length dl of the chamber. Similar changes occur in temperature, humidity ratio, enthalpy, and other properties. Because it b difficult to evaluate true surface area in direct contact equipment; common practice b to work on a unit volume basis. If-a# and ajr represent the square feet of heat transfer and mmm transfer surface per cubic foot of chamber volume,-respectively, the total surface areas for'heat and macfl transfer are then An = obAc.1 and 'Am v-ovAm! The basic equations for the process occurring in the differ ential length dl may be written for: (1) Mass Transfer -dbL - GadW = KoautWi - W)dl (36) That is,-the water evaporated, the moisture increase of the air, and the mass transfer rate are all equal. (2) Heat Transfer to the Air G^mdU = haOB(U - U)dl (37) (3) Total Energy Transfer to the Air O.ic^di. + b,tJW) = [Kdom(,W{ -- W)hft 4* h&a(U -- ti)]dZ (38a) As shown in the preceding section, aranming an = a*, Nu " 1, and neglecting small variations in h/,, Equation 38a reduces to (7dh TM FrxiwOh -- b)dl (38b) Note: The assumption that the heat'and mass transfer areas are identical (aa = om) is' usually quite'.valid for spray chambers. In equipment where packing materials (e.g.,- wood slats, Raschig rings, etc.) are used, however,-there maybe considerable differ ence in the two areas due to noauniform wetting of the packing^ The validity of the Lewb relation-has been'adequat&y discussed previously. To'attempt toTaccount for the very small changes in