The CRC Handbook of Mechanical Engineering. Chapter 4. Heat and Mass Transfer (776127), страница 57
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It isconvenient to rewrite Equation (4.7.68) as()m˙ ¢¢ = g *m1 g m1 g *m1 B m1 kg m 2 sec(4.7.69a)g *m1 = lim g m1(4.7.69b)whereB m ®0Now g *m1 is the limit value of gm1 for zero mass transfer (i.e., vs = 0), and Sh* can be obtained fromconventional heat transfer Nusselt number correlations for impermeable surfaces. The ratio ( g m1 / g *m1 )is termed a blowing factor and accounts for the effect of vs on the concentration profiles. Use of Equation(4.7.69) requires appropriate data for the blowing factor. For the constant-property laminar boundarylayer on a flat plate, Figure 4.7.15 shows the effect of the Schmidt number on the blowing factor. Theabscissa is a blowing parameter Bm = m˙ ¢¢ / g *m .The blowing velocity also affects the velocity and temperature profiles, and hence the wall shear stressand heat transfer. The curve for Sc = 1 in Figure 4.7.15 also gives the effect of blowing on shear stressas t s / t *s , and the curve for Sc = 0.7 gives the effect of blowing on heat transfer for air injection intoair as hc / hc* (since Pr = 0.7 for air).Variable Property Effects of High Mass Transfer RatesHigh mass transfer rate situations are usually characterized by large property variations across the flow,and hence property evaluation for calculating gm and hc is not straightforward.
An often-encounteredsituation is transfer of a single species into an inert laminar or turbulent boundary layer flow. The effectof variable properties can be very large as shown in Figure 4.7.16 for laminar boundary layers, andFigure 4.7.17 for turbulent boundary layers.A simple procedure for correlating the effects of flow type and variable properties is to use weightingfactors in the exponential functions suggested by a constant-property Couette-flow model (Mills, 1995).Denoting the injected species as species i, we haveg m1ami Bmi=;g *m1 exp(ami Bmi ) - 1Bmi =m˙ ¢¢g *mi(4.7.70a)org m1 ln(1 + amiB mi );=g *miamiB miB mi =a fi B fts=;*t s exp a fi B f - 1(© 1999 by CRC Press LLC)m˙ ¢¢ mi,e - mi,e=g mimi,s - 1Bf =m˙ ¢¢uet *s(4.7.70b)4-235Heat and Mass TransferFIGURE 4.7.15 Effect of mass transfer on the mass transfer conductance for a laminar boundary layer on a flatplate: g m / g *m vs.
blowing parameter Bm = m˙ ¢¢ / g *m .hcahi Bh=;*hcexp(ahi Bh ) - 1Bh =m˙ ¢¢c pehc*(4.7.70c)Notice that g*mi , t*s , hc* , and cpe are evaluated using properties of the free-stream gas at the mean filmtemperature. The weighting factor a may be found from exact numerical solutions of boundary layerequations or from experimental data. Some results for laminar and turbulent boundary layers follow.1. Laminar Boundary Layers.
We will restrict our attention to low-speed air flows, for which viscousdissipation and compressibility effects are negligible, and use exact numerical solutions of theself-similar laminar boundary layer equations (Wortman, 1969). Least-squares curve fits of thenumerical data were obtained using Equations (4.7.70a) to (4.7.70c). Then, the weighting factorsfor axisymmetric stagnation-point flow with a cold wall (Ts/Te = 0.1) were correlated as10 12ami = 1.65( Mair Mi )© 1999 by CRC Press LLC(4.7.71a)4-236Section 4FIGURE 4.7.16 Numerical results for the effect of pressure gradient and variable properties on blowing factors forlaminar boundary layers: low-speed air flow over a cold wall (Ts/Te = 0.1) with foreign gas injection: (a) mass transferconductance, (b) wall shear stress, (c) heat transfer coefficient.
(From Wortman, A., Ph.D. dissertation, Universityof California, Los Angeles, 1969. With permission.)a fi = 1.38( Mair Mi )ahi = 1.30( Mair Mi )3 125 12(4.7.71b)[c (2.5 R M )]pii(4.7.71c)Notice that cpi /2.5R/Mi) is unity for a monatomic species. For the planar stagnation line and theflat plate, and other values of the temperature ratio Ts/Te, the values of the species weightingfactors are divided by the values given by Equations (4.7.71a,b,c) to give correction factors Gmi,Gfi, and Ghi, respectively. The correction factors are listed in Table 4.7.7.The exponential relation blowing factors cannot accurately represent some of the more anomalouseffects of blowing.
For example, when a light gas such as H2 is injected, Equation (4.7.70c)indicates that the effect of blowing is always to reduce heat transfer, due to both the low densityand high specific heat of hydrogen. However, at very low injection rates, the heat transfer isactually increased, as a result of the high thermal conductivity of H2. For a mixture, k » åxikiwhereas cp = åmicpi. At low rates of injection, the mole fraction of H2 near the wall is much largerthan its mass fraction; thus, there is a substantial increase in the mixture conductivity near thewall, but only a small change in the mixture specific heat.
An increase in heat transfer results. Athigher injection rates, the mass fraction of H2 is also large, and the effect of high mixture specificheat dominates to cause a decrease in heat transfer.2. Turbulent Boundary Layers. Here we restrict our attention to air flow along a flat plate for Machnumbers up to 6, and use numerical solutions of boundary layer equations with a mixing lengthturbulence model (Landis, 1971). Appropriate species weighting factors for 0.2 < Ts /Te < 2 are© 1999 by CRC Press LLC4-237Heat and Mass TransferFIGURE 4.7.17 Numerical results for the effect of variable properties on blowing factors for a low-speed turbulentair boundary layer on a cold flat plate (Ts/Te = 0.2) with foreign gas injection: (a) mass transfer conductance, (b)wall shear stress, (c) heat transfer coefficient.
(From Landis, R.B., Ph.D. dissertation, University of California, LosAngeles, 1971. With permission.)1.33ami = 0.79( Mair Mi )a fi = 0.91( Mair Mi )0.76ahi = 0.86( Mair Mi )0.73(4.7.72a)(4.7.72b)(4.7.72c)In using Equation (4.7.70), the limit values for m˙ ¢¢ = 0 are elevated at the same location alongthe plate. Whether the injection rate is constant along the plate or varies as x–0.2 to give a selfsimilar boundary layer has little effect on the blowing factors.
Thus, Equation (4.7.72) has quitegeneral applicability. Notice that the effects of injectant molecular weight are greater for turbulentboundary layers than for laminar ones, which is due to the effect of fluid density on turbulenttransport. Also, the injectant specific heat does not appear in ahi as it did for laminar flows. Ingeneral, cpi decreases with increasing Mi and is adequately accounted for in the molecular weightratio.Reference State Schemes.
The reference state approach, in which constant-property data are used withproperties evaluated at some reference state, is an alternative method for handling variable-propertyeffects. In principle, the reference state is independent of the precise property data used and of the© 1999 by CRC Press LLC4-238Section 4TABLE 4.7.7Correction Factors for Foreign Gas Injection into Laminar Air Boundary LayersGmi Ts/TeGeometryAxisymmetricstagnation pointPlanar stagnation lineGfi Ts/TeGhi Ts/TeSpecies0.10.50.90.10.50.90.10.50.9HH2HeAirXeCCl4HH2HeCCH4OH2ONeAirACO2XeCCl4I2HeAirXe1.141.031.05—1.211.031.001.001.001.001.001.001.001.00—1.001.001.001.001.000.96—0.921.361.251.18—1.130.951.041.061.041.011.010.981.011.00—0.970.970.980.900.910.98—0.871.471.361.25—1.151.001.091.061.031.001.000.971.000.98—0.940.950.960.830.850.98—0.831.301.191.341.211.381.001.001.001.001.001.001.001.001.001.001.001.001.001.001.000.850.940.901.641.441.491.271.341.030.620.700.660.790.880.790.820.830.870.930.960.961.031.020.530.840.931.791.491.561.271.341.030.450.620.560.690.840.700.730.750.820.910.941.051.071.050.470.810.951.151.561.181.171.191.041.001.001.001.001.001.001.001.001.001.001.001.001.001.000.930.940.931.321.171.321.211.181.040.941.001.000.991.000.981.000.970.990.960.991.060.960.970.910.940.93—1.32————0.541.010.950.871.000.950.990.950.970.950.970.990.930.940.92——Based on numerical data of Wortman (1969).
Correlations developed by Dr. D.W. Hatfield.combination of injectant and free-stream species. A reference state for a boundary layer on a flat platethat can be used in conjunction with Figure 4.7.14 is (Knuth, 1963)m1,r = 1 -ln( Me Ms )M2M2 - M1 ln m2,e Me m2,s Ms()éc p1 - c pr ùTr = 0.5(Te + Ts ) + 0.2r * ue2 2c pr + 0.1ê Bhr + ( Bhr + Bmr )ú (T - Te )c pr úû sêë()(4.7.73)(4.7.74)where species 1 is injected into species 2 and r* is the recovery factor for an impermeable wall. Use ofthe reference state method is impractical for hand calculations: a computer program should be used toevaluate the required mixture properties.ReferencesHirschfelder, J.O., Curtiss, C.F., and Bird, R.B.
1954. Molecular Theory of Gases and Liquids, JohnWiley & Sons, New York.Knuth, E.L. 1963. Use of reference states and constant property solutions in predicting mass-, momentum-, and energy-transfer rates in high speed laminar flows, Int. J. Heat Mass Transfer, 6, 1–22.Landis, R.B. 1972. Numerical solution of variable property turbulent boundary layers with foreign gasinjection, Ph.D. dissertation, School of Engineering and Applied Science, University of California,Los Angeles.Law, C.K. and Williams, F.A. 1972. Kinetics and convection in the combustion of alkane droplets,Combustion and Flame, 19, 393–405.© 1999 by CRC Press LLCHeat and Mass Transfer4-239Mills, A.F.
1995. Heat and Mass Transfer, Richard D. Irwin, Chicago.Wortman, A. 1969. Mass transfer in self-similar boundary-layer flows, Ph.D. dissertation, School ofEngineering and Applied Science, University of California, Los Angeles.Further InformationGeankoplis, C.J. 1993. Transport Processes and Unit Operations, 3rd ed., Prentice-Hall, EnglewoodCliffs, NJ. This text gives a chemical engineering perspective on mass transfer.Mills, A.F.
1995. Heat and Mass Transfer, Richard D. Irwin, Chicago. Chapter 11 treats mass transferequipment relevant to mechanical engineering.Strumillo, C. and Kudra, T. 1986. Drying: Principles, Applications and Design, Gordon and Breach,New York.Mujamdar, A.S.. Ed. 1987. Handbook of Industrial Drying, Marcel Dekker, New York.© 1999 by CRC Press LLC4-240Section 44.8 ApplicationsEnhancementArthur E. BerglesIntroductionEnergy- and materials-saving considerations, as well as economic incentives, have led to efforts toproduce more efficient heat exchange equipment.
Common thermal-hydraulic goals are to reduce thesize of a heat exchanger required for a specified heat duty, to upgrade the capacity of an existing heatexchanger, to reduce the approach temperature difference for the process streams, or to reduce thepumping power.The study of improved heat transfer performance is referred to as heat transfer enhancement, augmentation, or intensification. In general, this means an increase in heat transfer coefficient. Attempts toincrease “normal” heat transfer coefficients have been recorded for more than a century, and there is alarge store of information.
A survey (Bergles et al., 1991) cites 4345 technical publications, excludingpatents and manufacturers’ literature. The literature has expanded rapidly since 1955.Enhancement techniques can be classified either as passive methods, which require no direct application of external power (Figure 4.8.1), or as active methods, which require external power. Theeffectiveness of both types of techniques is strongly dependent on the mode of heat transfer, which mayrange from single-phase free convection to dispersed-flow film boiling. Brief descriptions of thesemethods follow.Treated surfaces involve fine-scale alternation of the surface finish or coating (continuous or discontinuous). They are used for boiling and condensing; the roughness height is below that which affectssingle-phase heat transfer.Rough surfaces are produced in many configurations ranging from random sand-grain-type roughnessto discrete protuberances.
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