The CRC Handbook of Mechanical Engineering. Chapter 4. Heat and Mass Transfer (776127), страница 7
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For heat transfer withsignificant viscous dissipation see the section on flow over flat plate with zero pressure gradient: Effectof High Speed and Viscous Dissipation. The Eckert number Ec is defined as Ec = U ¥2 / C p (Ts - T¥ ).With a rectangular plate of length L in the direction of the fluid flow the average heat transfer coefficienthL with uniform surface temperature is given byhL =1LLòh0xdxLaminar Boundary Layer (Rex < Recr , ReL < Recr): With heating or cooling starting from the leadingedge the following correlations are recommended. Note: in all equations evaluate fluid properties at thefilm temperature defined as the arithmetic mean of the surface and free-stream temperatures unlessotherwise stated (Figure 4.2.9).FIGURE 4.2.9 Heated flat plate with heating from the leading edge.© 1999 by CRC Press LLC4-28Section 4Local Heat Transfer Coefficient (Uniform Surface Temperature)The Nusselt number based on the local convective heat transfer coefficient is expressed as12(4.2.32)Nu x = fPr Re xThe classical expression for fPr is 0.564 Pr1/2 for liquid metals with very low Prandtl numbers, 0.332Pr1/3for 0.7 < Pr < 50 and 0.339Pr1/3 for very large Prandtl numbers.
Correlations valid for all Prandtl numbersdeveloped by Churchill (1976) and Rose (1979) are given below.12Nu x =Nu x =0.3387Re x Pr 1 3é æ 0.0468 ö 2 3 ùê1 + èúPr ø ûë(4.2.33)14Re1 2 Pr 1 2(27.8 + 75.9Pr0.306+ 657Pr16)(4.2.34)In the range 0.001 < Pr < 2000, Equation (4.2.33) is within 1.4% and Equation (4.2.34) is within 0.4%of the exact numerical solution to the boundary layer energy equation.Average Heat Transfer CoefficientThe average heat transfer coefficient is given by(4.2.35)Nu L = 2 Nu x = LFrom Equation 4.2.35 it is clear that the average heat transfer coefficient over a length L is twice thelocal heat transfer coefficient at x = L.Uniform Heat FluxLocal Heat Transfer CoefficientChurchill and Ozoe (1973) recommend the following single correlation for all Prandtl numbers.12Nu x =0.886Re x Pr 1 2é æ Pr ö 2 3 ùê1 + èú0.0207 ø ûë14(4.2.36)Note that for surfaces with uniform heat flux the local convective heat transfer coefficient is used todetermine the local surface temperature.
The total heat transfer rate being known, an average heat transfercoefficient is not needed and not defined.Turbulent Boundary Layer (Rex > Recr , ReL > Recr): For turbulent boundary layers with heating orcooling starting from the leading edge use the following correlations:Local Heat Transfer CoefficientRecr < Rex < 107:45Nu x = 0.0296Re x Pr 1 3(4.2.37)10 7 < Rex:Nu x = 1.596 Re x (ln Re x )© 1999 by CRC Press LLC-2.584Pr 1 3(4.2.38)4-29Heat and Mass TransferEquation (4.2.38) is obtained by applying Colburn’s j factor in conjunction with the friction factorsuggested by Schlicting (1979).In laminar boundary layers, the convective heat transfer coefficient with uniform heat flux is approximately 36% higher than with uniform surface temperature.
With turbulent boundary layers, the differenceis very small and the correlations for the local convective heat transfer coefficient can be used for bothuniform surface temperature and uniform heat flux.Average Heat Transfer CoefficientIf the boundary layer is initially laminar followed by a turbulent boundary layer at Rex = Recr, thefollowing correlations for 0.7 < Pr < 60 are suggested:Recr < ReL < 107:[(124545Nu L = 0.664 Re L + 0.037 Re L - Re cr)]Pr13(4.2.39)If Recr < ReL < 107 and Recr = 105, Equation 4.2.39 simplifies to()45Nu L = 0.037Re L - 871 Pr 1 3(4.2.40)10 7 < ReL and Recr = 5 ´ 10 5:[Nu L = 1.963Re L (ln Re L )-2.584]- 871 Pr 1 3(4.2.41)Uniform Surface Temperature — Pr > 0.7: Unheated Starting LengthIf heating does not start from the leading edge as shown in Figure 4.2.10, the correlations have to bemodified.
Correlation for the local convective heat transfer coefficient for laminar and turbulent boundarylayers are given by Equations (4.2.42) and (4.2.43) (Kays and Crawford, 1993) — the constants inEquations (4.2.42) and (4.2.43) have been modified to be consistent with the friction factors. Thesecorrelations are also useful as building blocks for finding the heat transfer rates when the surfacetemperature varies in a predefined manner. Equations (4.2.44) and (4.2.45), developed by Thomas (1977),provide the average heat transfer coefficients based on Equations (4.2.42) and (4.2.43).FIGURE 4.2.10 Heated flat plate with unheated starting length.Local Convective Heat Transfer CoefficientRex < Recr :12Nu x =0.332 Re x Pr 1 3é æ xo ö 3 4 ùê1 - ç ÷ úêë è x ø úû13(4.2.42)Rex > Recr :Nu x =© 1999 by CRC Press LLC0.0296Re 4x 5 Pr 3 5é æ x o ö 9 10 ùê1 - ç ÷ úêë è x ø úû19(4.2.43)4-30Section 4Average Heat Transfer Coefficient over the Length (L – xo)ReL < Recr :é æ x ö3 4ù0.664 Re L Pr ê1 - ç o ÷ úêë è L ø úû=L - xo12h L - xo2313k(4.2.44)34æx ö1- ç o ÷è Løh=21 - xo L x = LIn Equation (4.2.44) evaluate hx=L from Equation (4.2.42).Recr = 0:é æ x ö 9 10 ù0.037Re L Pr ê1 - ç o ÷ úêë è L ø úû=L - xo45h L - xo= 1.25351 - ( x o L)89k(4.2.45)9 101 - xo Lhx = LIn Equation (4.2.45) evaluate hx=L from Equation (4.2.43).Flat Plate with Prescribed Nonuniform Surface TemperatureThe linearity of the energy equation permits the use of Equations (4.2.42) through (4.2.45) for uniformsurface temperature with unheated starting length to find the local heat flux and the total heat transferrate by the principle of superposition when the surface temperature is not uniform.
Figure 4.2.11 showsthe arbitrarily prescribed surface temperature with a uniform free-stream temperature of the fluid. If thesurface temperature is a differentiable function of the coordinate x, the local heat flux can be determinedby an expression that involves integration (refer to Kays and Crawford, 1993). If the surface temperaturecan be approximated as a series of step changes in the surface temperature, the resulting expression forthe local heat flux and the total heat transfer rate is the summation of simple algebraic expressions. Herethe method using such an algebraic simplification is presented.FIGURE 4.2.11 Arbitrary surface temperature approximated as a finite number of step changes.© 1999 by CRC Press LLC4-31Heat and Mass TransferThe local convective heat flux at a distance x from the leading edge is given bynq x¢¢ =å h DTxi(4.2.46)si1where hxi denotes the local convective heat transfer coefficient at x due to a single step change in thesurface temperature DTsi at location xi(xi < x).
Referring to Figure 4.2.11, the local convective heat fluxat x (x3 < x < x4) is given byq x¢¢ = hx ( x, 0)DTo + hx ( x, x1 )DT1 + hx ( x, x 2 )DT2 + hx ( x, x3 )DT3where hx(x, x1) is the local convective heat transfer coefficient at x with heating starting from x1; thelocal convective heat transfer is determined from Equation (4.2.42) if the boundary layer is laminar andEquation (4.2.43) if the boundary layer is turbulent from the leading edge.
For example, hx(x, x2) in thethird term is given byRe x < Re crRe cr = 0hx ( x , x 2 ) =hx ( x , x 2 ) =æ rU x ö0.332ç ¥ ÷è m øé æ x2 öê1 - ç ÷êë è x øPr 1 3kx34 13æ rU x ö0.0296ç ¥ ÷è m øé æ x2 öê1 - ç ÷êë è x ø12ùúúû45Pr 3 59 10 1 9ùúúûkxThe procedure for finding the total heat transfer rate from x = 0 to x = L is somewhat similar. Denotingthe width of the plate by W,q=WåhL - xiDTi ( L - xi )(4.2.47)where hL - xi is the average heat transfer coefficient over the length L – xi due to a step change DTi inthe surface temperature at xi. For example, the heat transfer coefficient in the third term in Equation(4.2.47) obtained by replacing xo by x2 in Equation (4.2.44) or (4.2.45) depending on whether ReL <Recr or Recr = 0.Flows with Pressure Gradient and Negligible Viscous DissipationAlthough correlations for flat plates are for a semi-infinite fluid medium adjacent to the plate, mostapplications of practical interest deal with fluid flowing between two plates.
If the spacing between theplates is significantly greater than the maximum boundary layer thickness, the medium can be assumedto approach a semi-infinite medium. In such a case if the plates are parallel to each other and if thepressure drop is negligible compared with the absolute pressure, the pressure gradient can be assumedto be negligible. If the plates are nonparallel and if the boundary layer thickness is very much smallerthan the spacing between the plates at that location, the medium can still be considered as approachinga semi-infinite medium with a non-negligible pressure gradient. In such flows the free-stream velocity(core velocity outside the boundary layer) is related to the pressure variation by the Bernoulli equation:© 1999 by CRC Press LLC4-32Section 42p U¥++ zg = constantr2FIGURE 4.2.12 Flow over a wedge.
bp is the wedge angle.Another situation where the free-stream velocity varies in the direction of flow giving rise to a pressuregradient is flow over a wedge. For the family of flows for which the solutions are applicable, the freestream velocity at the edge of the boundary layer is related to the x-coordinate by a power law, U¥ =cxm. Flows over semi-infinite wedges (Figure 4.2.12) satisfy that condition.
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