The CRC Handbook of Mechanical Engineering. Chapter 4. Heat and Mass Transfer (776127), страница 56
Текст из файла (страница 56)
TheChilton-Colburn form is of adequate accuracy for most external forced flows but is inappropriate forfully developed laminar duct flows.For example, air at 1 atm and 300 K flows inside a 3-cm-inside-diameter tube at 10 m/sec. Usingpure-air properties the Reynolds number is VD/n = (10)(0.03)/15.7 ´ 10–6 = 1.911 ´ 104. The flow isturbulent. Using Equation (4.7.60b) with Sc12 = 0.61 for small concentrations of H2O in air,(Sh D = (0.023) 1.911 ´ 10 4© 1999 by CRC Press LLC)0.8(0.61)0.4 = 50.24-230Section 4g m1 = rD12 Sh D = rnSh Sc12 D =(1.177) (15.7 ´ 10 -6 ) (50.2)= 5.07 ´ 10 -2 kg m 2 sec(0.61)(0.03)Further insight into this analogy between convective heat and mass transfer can be seen bywriting out Equations (4.7.60a) and (4.7.60b) as, respectively,(h c )D = 0.023Recpk cp0.8Dæ m öçk c ÷èpøæ m ögmD= 0.023Re 0D.8 ç÷rD12è rD12 ø0.4(4.7.63a)0.4(4.7.63b)When cast in this form, the correlations show that the property combinations k/cp and rD12 play analogousroles; these are exchange coefficients for heat and mass, respectively, both having units kg/m sec, whichare the same as those for dynamic viscosity m.
Also, it is seen that the ratio of heat transfer coefficientto specific heat plays an analogous role to the mass transfer conductance, and has the same units (kg/m2sec). Thus, it is appropriate to refer to the ratio hc/cp as the heat transfer conductance, gh, and for thisreason we should not refer to gm as the mass transfer coefficient.Simultaneous Heat and Mass TransferOften problems involve simultaneous convective heat and mass transfer, for which the surface energybalance must be carefully formulated. Consider, for example, evaporation of water into air, as shown inFigure 4.7.13.
With H2O denoted as species 1, the steady-flow energy equation applied to a controlvolume located between the u- and s-surfaces requires that()m˙ h1,s - h1,u = A(qcond¢¢ - qconv¢¢ - q rad¢¢ ) W(4.7.64)where it has been recognized that only species 1 crosses the u- and s-surfaces. Also, the water has beenassumed to be perfectly opaque so that all radiation is emitted or absorbed between the u-surface andthe interface.If we restrict our attention to conditions for which low mass transfer rate theory is valid, we can writem˙ / A . j1,s = gm1 (m1,s – m1,e).
Also, we can then calculate the convective heat transfer as if there wereno mass transfer, and write qconv = hc(Ts – Te). Substituting in Equation (4.7.64) with qconv = – k¶T/¶y|u,h1,s – h1,u = hfg, and rearranging, gives-k¶T¶y()= hc (Ts - Te ) + g m1 m1,s - m1,e h fg + qrad¢¢ W m 2(4.7.65)uIt is common practice to refer to the convective heat flux hc(Ts – Te) as the sensible heat flux, whereasthe term gm1 (m1,s – m1,e) hfg is called the evaporative or latent heat flux. Each of the terms in Equation4.7.65 can be positive or negative, depending on the particular situation. Also, the evaluation of theconduction heat flux at the u-surface, –k¶T/¶y|u, depends on the particular situation.
Four examples areshown in Figure 4.7.13. For a water film flowing down a packing in a cooling tower (Figure 4.7.13b),this heat flux can be expressed in terms of convective heat transfer from the bulk water at temperatureTL to the surface of the film, –k¶T/¶y|u = hcL (TL – Ts). If the liquid-side heat transfer coefficient hcL islarge enough, we can simply set Ts . TL, which eliminates the need to estimate hcL.
The evaporationprocess is then gas-side controlled. Figure 4.7.13c shows film condensation from a steam-air mixtureon the outside of a vertical tube. In this case we can write k¶T/¶y|u = U(Ts – Tc), where Tc is the coolant© 1999 by CRC Press LLCHeat and Mass TransferFIGURE 4.7.13 The surface energy balance for evaporation of water into an air stream.© 1999 by CRC Press LLC4-2314-232Section 4bulk temperature. The overall heat transfer coefficient U includes the resistances of the condensate film,the tube wall, and the coolant. Sweat cooling is shown in Figure 4.7.13d, with water from a reservoir(or plenum chamber) injected through a porous wall at a rate just sufficient to keep the wall surface wet.In this case, the conduction across the u-surface can be related to the reservoir conditions by applicationof the steady-flow energy equation to a control volume located between the o- and u-surfaces.
Finally,Figure 4.7.13e shows drying of a wet porous material (e.g., a textile or wood). During the constant-rateperiod of the process, evaporation takes place from the surface with negligible heat conduction into thesolid; then –k¶T/¶y|u . 0. The term adiabatic vaporization is used to describe evaporation when qcond= 0; constant-rate drying is one example, and the wet-bulb psychrometer is another.Consider a 1-m-square wet towel on a washline on a day when there is a low overcast and no wind.The ambient air is at 21°C, 1 atm, and 50.5% RH.
In the constant-rate drying period the towel temperatureis constant, and qcond = 0. An iterative calculation is required to obtain the towel temperature usingcorrelations for natural convection on a vertical surface to obtain hc and gm1; qrad is obtained as qrad =se(Ts4 - Te4 ) with e = 0.90. The results are Ts = 17.8°C, hc = 1.69 W/m2K, gm1 = 1.82 ´ 10–3 kg/m2sec,and the energy balance is()qcond = hc (Ts - Te ) + g m1 m1,s - m1,e h fg + qrad0 = -5.4 + 21.7 - 16.3 W m 2Evaluation of composition-dependent properties, in particular the mixture specific heat and Prandtlnumber, poses a problem. In general, low mass transfer rates imply small composition variations acrossa boundary layer, and properties can be evaluated for a mixture of the free-stream composition at themean film temperature. In fact, when dealing with evaporation of water into air, use of the propertiesof dry air at the mean film temperature gives results of adequate engineering accuracy.
If there are largecomposition variations across the boundary layer, as can occur in some catalysis problems, propertiesshould be evaluated at the mean film composition and temperature.The Wet- and Dry-Bulb PsychrometerThe wet- and dry-bulb psychrometer is used to measure the moisture content of air. In its simplest form,the air is made to flow over a pair of thermometers, one of which has its bulb covered by a wick whoseother end is immersed in a small water reservoir. Evaporation of water from the wick causes the wetbulb to cool and its steady-state temperature is a function of the air temperature measured by the drybulb and the air humidity. The wet bulb is shown in Figure 4.7.14. In order to determine the water vapormass fraction m1,e, the surface energy balance Equation (4.7.66) is used with conduction into the wick¢¢ set equal to zero.
The result isand qradm1,e = m1,s -c p æ Pr öh fg çè Sc12 ÷ø-2 3(Te- Ts )(4.7.66)Usually m1,e is small and we can approximate cp = cp air and (Pr/Sc12)–2/3 = 1/1.08. Temperatures Ts andTe are the known measured wet- and dry-bulb temperatures. With Ts known, m1,s can be obtained usingsteam tables in the usual way. For example, consider an air flow at 1000 mbar with measured wet- anddry-bulb temperatures of 305.0 and 310.0 K, respectively. Then P1,s = Psat (Ts) = Psat(305.0 K) = 4714Pa from steam tables. Hence, x1,s = P1,s/P = 4714/105 = 0.04714, andm1,s =© 1999 by CRC Press LLC0.04714= 0.029790.04714 + (29 18) (1 - 0.04714)4-233Heat and Mass TransferFIGURE 4.7.14 Wet bulb of a wet- and dry-bulb psychrometer.Also, hfg (305 K) = 2.425 ´ 106 J/kg, and cpm1,e = 0.02979 -x1,e =air= 1005 J/kg K; thus1005(1.08) (2.425 ´ 10 6 )(310 - 305) = 0.027870.02787= 0.044150.02787 + (18 29) (1 - 0.02787)( )P1,e = x1,e P = (0.04415) 10 5 = 4412 PaBy definition, the relative humidity is RH = P1,e/Psat(Te); RH = 4415/6224 = 70.9%.In the case of other adiabatic vaporization processes, such as constant-rate drying or evaporation ofa water droplet, m1,e and Te are usually known and Equation (4.7.66) must be solved for Ts.
However,the thermodynamic wet-bulb temperature obtained from psychrometric charts or software is accurateenough for engineering purposes.High Mass Transfer Rate TheoryWhen there is net mass transfer across a phase interface, there is a convective component of the absoluteflux of a species across the s-surface. From Equation (4.7.23a) for species 1,n1,s = r1,s n s + j1,s kg m 2 sec(4.7.67)During evaporation the convection is directed in the gas phase, with a velocity normal to the surface vs.When the convective component cannot be neglected, we say that the mass transfer rate is high.
Thereare two issues to consider when mass transfer rates are high. First, the rate at which species 1 is transferredacross the s-surface is not simply the diffusive component j1,s as assumed in low mass transfer ratetheory, but is the sum of the convective and diffusive components shown in Equation 4.7.67. Second,the normal velocity component vs has a blowing effect on the concentration profiles, and hence on theSherwood number.
The Sherwood number is no longer analogous to the Nusselt number of conventionalheat transfer correlations, because those Nusselt numbers are for situations involving impermeablesurfaces, e.g., a metal wall, for which vs = 0.© 1999 by CRC Press LLC4-234Section 4Substituting for j1,s from Equation (4.7.57) into Equation (4.7.67) givesm1,e - m1,s= g m1B m1m1,s - n1,s m˙ ¢¢m˙ ¢¢ = g m1(4.7.68)where m˙ ¢¢ = ns is the mass transfer rate introduced in the section on heterogeneous combustion andBm1 is the mass transfer driving force. In the special case where only species 1 is transferred, n1,s / m˙ ¢¢= 1, for example, when water evaporates into air, and dissolution of air in the water is neglected.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
