The CRC Handbook of Mechanical Engineering. Chapter 4. Heat and Mass Transfer (776127), страница 8
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The exponent m is relatedto the wedge angle bpb=2m1+ mm=b2-bWith laminar boundary layers, the boundary layer thickness, friction factor, and Nusselt numbers aredefined byd=xc1Re xC fx2=tw=rU ¥2c2Re x12Nu x = c3 Re xThe values of c1, c2, and c3 are available in Burmeister (1993). For example, for b = 0.5 (wedge angle= 90°), m = 1/3, c1 = 3.4, c2 = 0.7575, and c3 = 0.384 for Pr = 0.7, and c3 = 0.792 for Pr = 5. Rex isbased on U¥ = cxm; the free-stream velocity is not uniform.Uniform Temperature: Flat Plate with Injection or Suction with External Flows of a FluidParallel to the SurfaceInjection or suction has engineering applications.
When the free-stream temperature of the fluid is high,as in gas turbines, a cooling fluid is introduced into the mainstream to cool the surface. If the coolingfluid is introduced at discrete locations (either perpendicular to the surface or at an angle), it is knownas film cooling. If a fluid is introduced or withdrawn through a porous medium, it is known as transpiration(Figure 4.2.13). An application of suction is to prevent boundary layer separation (Figure 4.2.13).Analytical solutions for a laminar boundary layer with transpiration suction or blowing are availableif the velocity perpendicular to the surface varies in the following manner:v o = constant x ( m -1) 2FIGURE 4.2.13 Flat plate with transpiration injection.© 1999 by CRC Press LLC4-33Heat and Mass TransferSolutions are limited to the cases of the injected fluid being at the same temperature as the surface andthe injected fluid being the same as the free-stream fluid.
Positive values of vo indicate blowing andnegative values indicate suction. Values of Nu x / Re1x 2 for different values of Pr and for different valuesof blowing or suction parameter are given in Kays and Crawford (1993).For example, for a laminar boundary layer over a flat plate with a fluid (Pr = 0.7) the value ofNu x / Re1x 2 is 0.722 for (vo/U¥) rU ¥ x / m = –0.75 (suction) and 0.166 for (vo/U¥) rU ¥ x / m = 0.25(blowing). Heat transfer coefficient increases with suction which leads to a thinning of the boundarylayer.
Blowing increases the boundary layer thickness and decreases the heat transfer coefficient.For turbulent boundary layers Kays and Crawford (1993) suggest the following procedure for findingthe friction factor and convective heat transfer coefficient. Define friction blowing parameter Bf and heattransfer blowing parameter Bh asv o U¥Cf 2(4.2.48)v o U ¥ m˙ ¢¢ G¥=StSt(4.2.49)Bf =Bh =wherevoU¥m˙ ¢¢G¥St=====velocity normal to the platefree-stream velocitymass flux of the injected fluid at the surface (rvo)mass flux in the free stream (rU¥)Stanton number = Nux/RexPr = h/rU¥cpThe friction factors and Stanton number with and without blowing or suction are related byCfC fo=(ln 1 + B f)Bfln(1 + Bh )St=St oBh(4.2.50)(4.2.51)In Equations (4.2.50) and (4.2.51) Cfo and Sto are the friction factor and Stanton number with vo = 0 (noblowing or suction), and Cf and St are the corresponding quantities with blowing or suction at the sameRex(rU¥x/m).For the more general case of variable free-stream velocity, temperature difference, and transpirationrate, refer to Kays and Crawford (1993).Flow over Flat Plate with Zero Pressure Gradient: Effect of High-Speed and ViscousDissipationIn the boundary layer the velocity of the fluid is reduced from U¥ to zero at the plate leading to areduction in the kinetic energy of the fluid.
Within the boundary layer there is also the work done byviscous forces; the magnitude of the such viscous work is related to the velocity of the fluid, the velocitygradient, and the viscosity of the fluid. The effect of such a reduction in the kinetic energy and theviscous work is to increase the internal energy of the fluid in the boundary layer. The increase in theinternal energy may be expected to lead to an increase in the temperature; but because of the heattransfer to the adjacent fluid the actual increase in the internal energy (and the temperature) will be lessthan the sum of the decrease in the kinetic energy and viscous work transfer; the actual temperature© 1999 by CRC Press LLC4-34Section 4increase depends on the decrease in the kinetic energy, the viscous work transfer, and the heat transferfrom the fluid.
The maximum temperature in the fluid with an adiabatic plate is known as the adiabaticwall temperature (which occurs at the wall) and is given byTaw = T¥ + rU ¥22C p(4.2.52)In Equation (4.2.52) r is the recovery factor and is given by Eckert and Drake (1972).Laminar boundary layer0.6 < Pr < 15Turbulent boundary layerr = Pr12r = Pr13Equation (4.2.52) can be recast asTaw - T¥ rU ¥2=2 C p (Ts - T¥ )Ts - T¥(4.2.53)From Equation (4.2.53) the maximum increase in the fluid temperature as a fraction of the differencebetween the plate and free-stream temperatures is given by r Ec/2. With air flowing over a plate at 500m/sec, the increase in the temperature of the air can be as high as 105°C.
With Ts = 40°C and T¥ =20°C, the temperature of the air close to the plate can be higher than the plate temperature. It is thuspossible that although the plate temperature is higher than the free-stream temperature, the heat transferis from the air to the plate. At a Mach number greater than 0.1 for gases, viscous dissipation becomessignificant.The temperature profiles for high-speed flows for different values of Ts are shown in Figure 4.2.14.In high-speed flows, as heat transfer can be to the plate even if the plate temperature is greater than thefluid temperature, the definition of the convective heat transfer coefficient given in Equation (4.2.29) isnot adequate. On the other hand, as the heat transfer is always from the plate if Ts > Taw , the adiabaticwall temperature is more appropriate as the reference temperature.
Thus, in high-speed flows thedefinition of the convective heat transfer coefficient is given byq ¢¢ = h(Ts - Taw )(4.2.54)FIGURE 4.2.14 Temperature profiles for high-speed flows: (a) T¥ < Ts < Taw ; (b) Ts = T¥; (c) Ts ! T¥; (d) Ts > Taw.Equation (4.2.54) is consistent with Equation (4.2.29) as the adiabatic wall temperature equals the freestream temperature if the effects of viscous dissipation and reduced kinetic energy in the boundary layerare neglected. With the adiabatic wall temperature as the fluid reference temperature for the definition© 1999 by CRC Press LLC4-35Heat and Mass Transferof the convective heat transfer coefficient, equations for low speeds can also be used for high-speedflows.
Because of the greater variation in the fluid temperature in the boundary layer, the variation ofproperties due to temperature variation becomes important. It is found that the correlations are bestapproximated if the properties are evaluated at the reference temperature T * defined by Eckert (1956):T * = 0.5(Ts + T¥ ) + 0.22(Taw - T¥ )(4.2.55)With properties evaluated at the reference temperature given by Equation (4.2.55), Equation (4.2.56)through (4.2.61) are applicable to high-speed flows with Prandtl numbers less than 15. It should be notedthat the adiabatic wall temperatures in the laminar and turbulent regions are different affecting both thetemperature at which the properties are evaluated and the temperature difference for determining thelocal heat flux.
Therefore, when the boundary layer is partly laminar and partly turbulent, an averagevalue of the heat transfer coefficient is not defined as the adiabatic wall temperatures in the two regionsare different. In such cases the heat transfer rate in each region is determined separately to find the totalheat transfer rate.Evaluate properties at reference temperature given by Equation (4.2.55):LaminarTurbulent12Local: Re x < Re crNu x = 0.332 Re x Pr 1 3Average: Re L < Re crNu L = 0.664 Re L Pr 1 3Local: 10 7 > Re x > Re crNu x = 0.0296 Re x Pr 1 3Local: 10 7 < Re x < 10 9Nu x = 1.596 Re x (lnRe x )Average: Re cr = 0, Re L < 10 7Nu L = 0.037Re L Pr 1 3Average: Re cr = 0, 10 7 < Re L < 10 9Nu L = 1.967Re L (lnRe L )( 4.2.56)12( 4.2.57)45( 4.2.58)-2.584Pr 1 345( 4.2.59)( 4.2.60)-2.584Pr 1 3( 4.2.61)When the temperature variation in the boundary layer is large, such that the assumption of constantspecific heat is not justified, Eckert (1956) suggests that the properties be evaluated at a referencetemperature corresponding to the specific enthalpy i* given byi * = 0.5(is + i¥ ) + 0.22(is - i¥ )(4.2.62)where i is the specific enthalpy of the fluid evaluated at the temperature corresponding to the subscript.Equation (4.2.62) gives the same values as Equation (4.2.55) if Cp is constant or varies linearly withtemperature.At very high speeds the gas temperature may reach levels of temperatures that are sufficient to causedisassociation and chemical reaction; these and other effects need to be taken into account in those cases.Flow over Cylinders, Spheres, and Other GeometriesFlows over a flat plate and wedges were classified as laminar or turbulent, depending on the Reynoldsnumber, and correlations for the local and average convective heat transfer coefficients were developed.But flows over cylinders (perpendicular to the axis) and spheres are more complex.
In general, the flowover cylinders and spheres may have a laminar boundary layer followed by a turbulent boundary layer© 1999 by CRC Press LLC4-36Section 4FIGURE 4.2.15 A fluid stream in cross flow over a cylinder.and a wake region depending on the Reynolds number with the diameter as the characteristic length.Because of the complexity of the flow patterns, only correlations for the average heat transfer coefficientshave been developed (Figure 4.2.15).Cylinders: Use the following correlation proposed by Churchill and Bernstein (1977): Red Pr > 0.2.Evaluate properties at (Ts + T¥)/2:12Re d > 400, 000:Nu d = 0.3 +0.62 Re d Pr 1 3[1 + (0.4 Pr )23 14]1210, 000 < Re d < 400, 000: Nu d = 0.3 +0.62 Re d Pr 1 323 14[1 + (0.4 Pr) ]é æ Re d ö 5 8 ùê1 + ç÷ úêë è 282, 000 ø úû45é æ Re d ö 1 2 ùê1 + ç÷ úêë è 282, 000 ø úû( 4.2.63)( 4.2.64)12Re d < 10, 000:Nu d = 0.3 +0.62 Re d Pr 1 3[1 + (0.4 Pr )( 4.2.65)23 14]For flow of liquid metals, use the following correlation suggested by Ishiguro et al.
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