Heat Transfer from Arbitrary Nonisothermal Surfaces in a Laminar Flow

Transcription

1 3 Heat Transfer from Arbitrary Nonisothermal Surfaces in a Laminar Flo In essence, the conjugate heat transfer problem consiers the thermal interaction beteen a boy an a flui floing over or insie it. As a result of such interaction, a particular temperature istribution establishes on the boy flui interface. This temperature fiel etermines the heat flu istribution on the interface an virtually the intensity of heat transfer. Hence, the properties of heat transfer of any conjugate problem are actually the same as those of some nonisothermal surface ith the same temperature fiel, no matter ho this fiel is establishe, as a result of conjugate heat transfer or given a priori. Thus, in general, a theory of conjugate heat transfer is in fact a theory of an arbitrary nonisothermal surface, because the temperature istribution on the interface in a conjugate problem is unnon a priori. Such theory applicable to both arbitrary nonisothermal an conjugate convective heat transfer presente in this an the net to chapters as evelope by the author as a result of stuying this subject since 97 []. The main part of this or as performe together ith grauate stuents an colleagues from the Urainian Acaemy of Science an then by the author uring his time as a visiting professor at the University of Michigan since 996. Different parts of this theory have been publishe in many articles an in a boo [] that as the author s octoral thesis. Although many of these publications ere originally publishe in Russian, almost all are available in English. 3. The Eact Solution of the Thermal Bounary Layer Equation for an Arbitrary Surface Temperature Distribution [3] There are only a fe eact solutions of the thermal bounary layer equation. Most of them are erive for a specific surface temperature istribution. The first eact solution of the steay-state bounary layer equation as given by Pohlhausen [4] for a plate ith constant surface temperature an free stream velocity. The same problem for a plate ith a polynomial surface temperature istribution as solve by Chapman an Rubesin [5]. Levy [6] gave the eact solution for the case of a poer la istribution of both surface temperature an free stream velocity. Solution in the form of multiple series in by Taylor & Francis Group, LLC 55

2 56 Conjugate Problems in Convective Heat Transfer terms of ynamic an thermal shape parameters as given by Bubnov an Grishmanovsaya [7] an Oa [8] (Section.6). Here is given an eact solution to the thermal bounary layer equation for an arbitrary temperature istribution an a poer la of the free stream velocity. The solution is constructe in the form of series containing consecutive erivatives of the temperature hea istribution. Such series converge much more rapily than multiple series of shape parameters. The thermal bounary layer equation an the bounary conitions are use in the Prantl-Mises form (Equation.3), hich in Görtler variables (Equation.), ϕ becomes θ ϕ θ u θ ϕ Pr ϕ U ϕ = U c p u u U ϕ U (3.) ϕ =, θ = θ ( ) ϕ, θ = (3.) ψ = Uζ ϕ = θ = T T ν ν (3.3) The solution of Equation (3.) is sought in the form θ θ = G U ϕ ( ) + G ϕ ( ) c = p (3.4) Substitution of Equation (3.4) into Equation (3.) an replacement of the ine + by in one of the sums yiels the folloing to equations: θ u G + G ϕg ϕ U G Pr = = (3.5) U U G u G U G u u ϕ Pr ϕ U ϕ U = (3.6) Here, prime enotes a erivative ith respect to variable ϕ. If the free stream velocity has poer la epenence U = C m, the velocity istribution in the layer is self-similar (Section.). In this case, the ratio of velocities in the layer an in the outer flo u/ U = ϕ ( η, β) epens only on variable h an parameter b, hile the coefficient in the first term of the secon equation equals β. The epressions in bracets in Equation (3.5) also turn out to epen only on ϕ an β. Equating these epressions to zero leas to a system of orinary equations: ( / Pr)[ ωϕ (, β) G ] + ϕg G = G ( =,,, ) (3.7) by Taylor & Francis Group, LLC

3 Heat Transfer from Arbitrary Nonisothermal Surfaces in a Laminar Flo 57 etermining the coefficients of series (3.4), here ω( ϕ, β) enotes the ratio u/ U in self-similar flos. The bounary conitions for these equations follo from conitions (3.): ϕ =, G =, G = ( =, 3, ) ϕ, G = ( =,, ) (3.8) The equation an the bounary conitions for G ( ϕ ) are foun from Equation (3.6): ( / Pr)[ ωϕ (, β) G ] + ϕg βg = ω( ϕ, β)[ ω ( ϕ, β) ] ϕ =, G = ϕ, G = (3.9) The heat flu at the all is efine via shear stress assuming that close to the all τ τ : q θ u = u y = λ θ λ τ = µ ν ϕ y = y= ϕ= ρ u u ϕ (3.) Integrating both sies of the last equation, one fins that close to the all (ecept near the separation point here τ ), the velocity u ϕ is etermine via shear stress as u = 3/ 4 / 4 ( τ ϕ/ ρ ) / (3.) Using this result an etermining the erivative θ/ ϕ from Equation (3.4), one gets θ q h g g U = + θ c = p g C f St = Pr / / 4 (3.) Here, g = / 4( ϕ / G ) ϕ =, g = ( G / G ) ϕ=, g = ( G / G ) ϕ =, h is the heat transfer coefficient for an isothermal surface in the case of negligible issipation. In that case, the sum in series (Equation 3.) becomes zero, an this equation reuces to the bounary conition of the thir in, an Equation (3.) for the Stanton number gives for Pr = an β = (Figure 3.) St =. 576 Re / 4( C / ) / = C / (3.3) f f as it shoul be accoring to the Reynols analogy. by Taylor & Francis Group, LLC

4 58 Conjugate Problems in Convective Heat Transfer g g β =.6 4. β = β = β = Pr Figure 3. Depenence of coefficient g on Pr an β for a laminar bounary layer. Series (3.) is an eact solution for a poer la free stream velocity. For graientless flo, =U. /ν, an this series becomes q h g g U = θ + c = θ p (3.4) It is seen that in this case, the heat flu is governe by the erivatives of temperature hea ith respect to the coorinate. It follos from Equation (3.) that hen there is a pressure graient, the role of the longituinal coorinate plays the variable, an the heat flu is etermine by the erivatives of the temperature hea ith respect to. Thus, the function θ ( ) plays the same role in the case of a pressure graient flo as the function θ ( ) oes for graientless flo. This situation is plausible physically an reflects the fact that the flo characteristics at a given point are governe not only by local quantities but also by the prehistory of the flo, hich is taen into account by the variable. The coefficients G are etermine by inhomogeneous Equation (3.7). Presenting these coefficients by sum gives for G an G the folloing epressions: G G i= + i = ( ) Fi, ( i)! i= i= i= i= ( ) + i ( ) + i ( ) + i = ifi = Fi + F i F ( i)! ( i)! ( i)! ( ) i i= i= i= (3.5) by Taylor & Francis Group, LLC

5 Heat Transfer from Arbitrary Nonisothermal Surfaces in a Laminar Flo 59 Substituting the sums (Equation 3.5) into Equations (3.7) an (3.8) transforms these into homogeneous equations an the folloing bounary conitions: ( / Pr)[ ωϕ (, β) F ] i + ϕfi ifi = ( i =,,... ) (3.6) i= i= i= + + F ( ) i ( ) =, Fi ( ) =, ( i)! i= ( ) i Fi ( i)! ( ) = (3.7) From the last sum, one gets F i ( ) =. Because F ( ), the other values of F i ( ) in the secon sum cannot be equal to zero. In this case, one shoul tae F ( ) i = / i!. Then, the secon sum becomes zero as a sum of binomial coefficients ith alternate signs. Thus, the bounary conitions for the functions F i ( ϕ ) are ϕ =, F = / i!, ϕ, F (3.8) i Equation (3.6) at the all (ϕ = ) in the case hen i has a singularity as ell as the Prantl-Mises equation (Equation.3). This singularity can be remove by using a ne variable z = ϕ /, hich transforms Equation (3.6) an bounary conitions (Equation 3.8) into the form ( / Pr)[ ω( z, β) F ] z [ z4 ω( z, β)] F 8iz3 + F =, F( ) = / i!, F ( ) i i i i i i (3.9) The solution of this linear problem can be presente as a sum of to others: Vi () F z V z i W W z V i() = i() i(), i( ) =, Wi ( ) =, Vi ( ) =, Wi ( ) =! i() (3.) Using the Runge-Kutta metho for numerical integration, one gets Vi () z an W () i z an then F ( ϕ ), coefficients G ( ) i ϕ from sum (Equation 3.4), an finally, coefficients of the series (3.): g i= g = i= V ( ) = W ( ) ( ) + i Vi ( )/ V ( ) i!( i)! W( ) / W ( ) i (3.) Equation (3.6) simplifies in the limiting cases Pr an Pr. In the first case, a slug velocity profile may be use (Section.3). Then, Equations (3.6) an (3.8) become F + Pr( ϕ F if ) =, ϕ =, F = / i! ϕ, F (3.) i i i i i by Taylor & Francis Group, LLC

6 6 Conjugate Problems in Convective Heat Transfer The functions giving the solution of this problem an its first erivative at ϕ = are [9] i ξ Fi = i i!( i )! ep( ξ ) ξ ep( ξ ) e Pr p( ) = / ζ ζ ξ ϕ i π (3.3) ( ) i+ F ( ) =, Fi ( ) = π πi! + ( i + ) ( ) i + m= i m= Calculation shos that the folloing simple formulae are vali: ( ) mi! ( i m+ ) m!( m)! (3.4) π i Fi ( ) = = g = 35 ( i ) ( i )!!, ( ) +!( ) (3.5) The coefficients ecrease rapily ith increasing number: g =, g = / 6, g = / 3, g = / 68, g = / 8, g6 = / 79 (3.6) In the other limiting case, hen Pr, the linear velocity profile cϕ / may be use (Section.), an Equation (3.6) an bounary conitions (Equation 3.8) tae the form ( cϕ / F ) + Pr( ϕf if) =, ϕ =, F = / i! ϕ, F i (3.7) i i i i The ne variables Hi = ep( z)( z/3) 3 / Fi an z = ( Pr / 3c) ϕ / transform Equation (3.7) into a confluence hypogeometric equation: zh + [( 4/ 3 z)] H [( 4i/ 3) + ] H = (3.8) i i i To satisfy the corresponing bounary conitions, one shoul use the asymptotic epression for the confluence hypogeometric function H( abz,, ). Then, the function Fi () z an the coefficients of series (3.) are obtaine as follos: Fi = z H i+ z i 3! ep( ) ( ), 3, g Γ( / 3) Γ[( 4i/ 3) + ] z Γ( 4/ 3) Γ[( 4/ 3) i + / 3] 3 / H[ ( 4i/ 3) + 43, /, z] (3.9) i= ( ) + iγ( 4i/ 3) = Γ 3 3 (3.3) ( i)!( i )! Γ[( 4i/ 3) + / 3] i= by Taylor & Francis Group, LLC

7 Heat Transfer from Arbitrary Nonisothermal Surfaces in a Laminar Flo 6 g. Pr.9.8 β = β = Pr Pr Figure 3. Depenence of coefficient g on Pr an β for a laminar bounary layer. The calculation gives g =. 63, g =. 345, g =. 98, g =. 57 (3.3) 3 4 Coefficients g calculate by Equation (3.6) for ifferent free stream velocity graients (ifferent β) an various Prantl numbers are plotte in Figures 3., 3., an 3.3. These are compare ith limiting values (Equations 3.6 an 3.3). It follos from Figures 3. an 3.3 that. Coefficient g epens on the velocity graient an on the Prantl number; this epenence is more significant for small Prantl numbers (Pr < 5;. ) for Pr an Pr, the values of g for all β ten to the greatest g = an to the loest g =. 63 values, respectively. g. g g. 3 g 3 4 g Pr Figure 3.3 Depenence of coefficients g on Pr an β for a laminar bounary layer. Numerical solution of Equation (3.6), Pr, 34,, Pr. by Taylor & Francis Group, LLC

8 6 Conjugate Problems in Convective Heat Transfer. Coefficient g is practically inepenent of the velocity graient an epens slightly on the Prantl number in the region of small Prantl numbers; for Pr an Pr, the values of g also ten to the greatest absolute value g = 6 / an to the loest absolute value g =. 345, respectively. 3. Coefficients g 3 an g 4 are inepenent of both the velocity graient an the Prantl number. Therefore, the values obtaine by numerical integration of Equation (3.6) practically coincie ith the limiting values g 3 = 3 / an g 4 = / All subsequent coefficients ill apparently be more inepenent of the velocity graient an of the Prantl. This follos from Equation (3.7) because the role of the first term, hich epens only on Pr an β, reuces as the number increases. The inepenence of the coefficients of both β an Pr for large maes it possible to use for all Prantl numbers an for arbitrary pressure graient the limiting values of g (3.6) for Pr. 3. Generalization for an Arbitrary Velocity Graient in a Free Stream Flo The results obtaine above, in particular, series (Equations 3. an 3.4), are eact solutions for a poer la velocity in a free stream flo. These epressions are also a highly accurate approimate solution for an arbitrary free stream velocity. The folloing consierations sho this fact. If the coefficients g ere inepenent of the pressure graient (i.e., of parameter β), Equations (3.) an (3.4) oul be eact solutions for an arbitrary pressure graient. Actually, all coefficients g are practically inepenent of β, ecept g. So, the eactness of series (Equations 3. an 3.4) for an arbitrary pressure graient epens only on variation of coefficients g via parameter β. Accoring to Figure 3., coefficient g slightly epens on β in the range of mean Prantl numbers, but for limiting cases Pr an Pr, this epenence eteriorates. The greatest effect of the pressure graient on g is in the range of Prantl numbers close to Pr =,. here the eviation from the average value of the first coefficient g =. 675 reaches ±% hen the velocity graient in the self-similar flo changes from β = (accelerate flo) to β =6. (close to separation flo). Thus, the relations (Equations 3. an 3.4) q = h θ + g = θ, q = h θ + g θ = (3.3) by Taylor & Francis Group, LLC

9 Heat Transfer from Arbitrary Nonisothermal Surfaces in a Laminar Flo 63 give the high-accuracy epression for heat flu from nonisothermal surfaces in the case of an arbitrary free stream velocity graient. Diviing both sies of these equations by q = hθ, one obtains corresponing relations for the coefficient of nonisothermicity: = q θ q χt = = + g χ θ t = = + q, q = g θ, (3.33) θ hich shos ho much the heat transfer intensity from a particular nonisothermal surface is more or less than that from an isothermal surface. The value of the parameter β in a general case shoul be etermine satisfying an integral equation that can be obtaine by integrating Equation (3.) across the bounary layer. Omitting the issipative term, one gets θ ϕ θ u θ ϕ = (3.34) ϕ Pr ϕ U ϕ Substituting the self-similar solution ωϕ (, β) for the velocity ratio u/ U, yiels θ ϕ θ ϕ ϕ ωϕ β θ (, ) ϕ = Pr ϕ (3.35) The parameter β shoul be etermine from this equation. Hoever, Equation (3.35) is satisfie by any value of β. This is clearly seen by subtracting Equations (3.34) an (3.35): u θ u θ ωϕ (, β) ϕ = ω(ϕ β ϕ U ϕ, ) U ϕ = (3.36) This equation is satisfie by any value of β because both eact an selfsimilar velocity profiles satisfy the same bounary conitions at the en points of interval (, ). This means that solutions (Equation 3.3) satisfy the eact integral equation (Equation 3.34), in other ors it means that series (Equation 3.3) in the case of an arbitrary free stream velocity are on the average (across the bounary layer) eact solutions. Because the integral equation is satisfie, one can increase the eactness of solutions (Equation 3.3) by using some aitional conition to estimate the parameter β. The simplest such conition is obtaine by equating the average values of the given U ( ) an poer la C m velocity istributions on each interval (, ): Cζm ζ = U( ζ) ζ, β = Re (3.37) by Taylor & Francis Group, LLC

10 64 Conjugate Problems in Convective Heat Transfer As the coefficient g slightly epens on parameter β, the metho of etermining the latter is not important, an so β may be estimate in some other ay, for instance, by approimating U( ) by the poer function U = C m, as in []. In fact, ifferent estimations of β lea to practically the same final values of coefficient g. The heat transfer coefficient for isothermal surface h in Equation (3.3) is etermine using Equation (3.) for the Stanton number an Figure 3.. For laminar flo, the heat transfer coefficient can also be calculate by methos such as those reviee by Spaling an Pun []. 3.3 General Form of the Influence Function of the Unheate Zone: Convergence of the Series [] The superposition principle leas to Equation (.38) for the heat flu on a surface ith an arbitrary temperature istribution. In the case of a continuous temperature hea istribution ithout breas for graientless flo over a plate, this equation has the form q h f θ = + θ ( ) ( ξ/ ) ξ (3.38) ξ The influence function of an unheate zone f( ξ/ ) is calculate for some simple cases using approimate methos. All of these functions have a form (Section.5): f C C ( ξ/ ) = ( ξ/ ) (3.39) Because series (3.3) is an eact solution for the case of poer la free stream velocity, one can estimate eactness of the function (Equation 3.39) by comparing to forms of solution efining heat flu: ifferential (Equation 3.3) an integral (Equation 3.38). This can be one by etermining the relation beteen these to epressions []. It is shon above that in the case of the graient free stream flo, the variable in Equation (3.4) shoul be substitute for by. To compare both ifferential an integral forms for heat flu in the general case, the same substitution shoul be one in Equation (3.38) to get θ q = h + f θ ( ) ( ξ/) ξ ξ (3.4) This equation is transforme using integration of parts assuming for the -transform: by Taylor & Francis Group, LLC

11 Heat Transfer from Arbitrary Nonisothermal Surfaces in a Laminar Flo 65 u ζ θ ( = v v = + ) ξ, γ, ζ = ξ! (3.4) Taing v = f ( ζ ), one obtains for =, v = f( γ) γ an then q θ = h f f + θ ζ ( ζ) ζ θ ( γ) γ ξ+ θ ( ) ξ = (3.4) Repeating integration by parts yiels q θ θ = h f f + + () () ( ) + + θ θ!! = = = ζ θ θ + + ( )! f () + θ ( ) + + θ f( ξ/) ξ (3.43) f ( ζ) = ζ ζ f( ζ) ζ + ζ ζ ζ n= n= ( ) nζ n n!( n)! (3.44) In the last epression, the integral is repeate times. The secon sum in Equation (3.43) beginning from θ ( ) represents an epansion of function θ ( ) as a Taylor series at =. Therefore, one conclues that the epression (Equation 3.43) coincies ith Equation (3.3) if the remainer in the form of the integral in Equation (3.43) goes to zero hen an if it is set Then, because + ξ θ lim f g ( ) + ξ = ξ + f () (3.45) ζ ζ ζ f( ζ) ζ = ( ζ) f( ζ) ζ ( )! ζ n= ( ζ ) n n = ζ ζ!( )!! [( ) n n ] n= (3.46) by Taylor & Francis Group, LLC

12 66 Conjugate Problems in Convective Heat Transfer one obtains, accoring to Equations (3.45) an (3.46), the relation beteen the coefficients g an the influence function of an unheate zone in the form g = ( ) + ( ζ) f( ζ) ζ (3.47)! Thus, epressions (Equations 3.38 an 3.4) etermining the heat flu are integral sums of series (Equation 3.3), hich present the same heat flu in equivalent ifferential forms. Corresponing integral forms for a nonisothermicity coefficient equivalent to (Equation 3.33) are χt = θ ξ θ f ξ θ χt + ξ θ ( ) = f ξ θ ξ θ + ( ) (3.48) Results just obtaine mae it possible to fin the relation beteen eponents C an C in the influence function (Equation 3.39) an coefficients g. Substituting Equation (3.39) into Equation (3.47), epaning ( ζ) via a C binomial formula, an introucing a ne variable r = ζ, one reuces the integral to the sum of beta functions, an Equation (3.47) becomes g ( = ) + ( )! C n= Bi (, j) = ri( r) j r n ( )! B n +, n!( n )! C C (3.49) Accoring to Equation (.4), the eponents in Equation (3.39) in the case of the graientless free stream flo are C = 3/ 4 an C = 3 /. The values of coefficients g =. 6, g =. 3, g3 =. 3, g4 =. 56 obtaine from Equation (3.49) using these eponents iffer only little from the results (Equation 3.3) obtaine from the eact solution for Pr. Thus, although Equation (3.39) is foun by an approimate integral metho, it is reasonably accurate, so that the calculation by Equation (3.38) ith function (Equation 3.39) an C = 3/ 4an C = 3 / gives virtually the same result as that obtaine by the eact solution (Equation 3.3). Another result erive from comparing the solutions in to forms, of the series an of the integral, is the influence function of the unheate zone in the general case. A comparison of Equations (3.3) an (3.38) shos that if one neglects the slight epenence g ( β ), Equation (3.39) can be use for the influence function in the general case after substituting by Taylor & Francis Group, LLC

13 Heat Transfer from Arbitrary Nonisothermal Surfaces in a Laminar Flo 67 variable ξ/ for ξ/: C C f ( ξ/ ) = [ ( ξ/ ) ] (3.5) In such a case, the eponents C an C shoul be etermine so that the values of g an g obtaine by applying Equation (3.5) oul be the same as those given by Figures 3. an 3.3. First, it shoul be consiere ho formula (Equation 3.5) conforms to non cases. In particular, for the case of self-similar graient flos an for Pr > 5,. for hich the coefficients g are practically inepenent on Pr (Figure 3.), Equation (3.5) shoul yiel formula (.44) erive by Lighthill. Because for the self-similar flos U = C m an, consequently, = [ C/ν ( m+ + )] m, it is evient that in this case Equation (3.5) gives Lighthill s formula. It is also reaily seen that the result obtaine by Equation (3.5) ith eponents C = an C = / correspons to the eact solution in the limiting case Pr. This follos from the fact that for C = an C = /, the beta function an the epression (3.49) for g can be presente in the folloing form: Γ( n + ) Γ( / ) n! Bn ( +, / ) = = Γ [ n + + ( / )] [ n + ( / )][ n( / )][ n ( 3/ )] ( / ) g = ( ) + n= ( ) nn ( n+ )!!( n )! (3.5) (3.5) Direct calculations sho that g given by this formula are in agreement ith the values (Equation 3.6) obtaine from the eact solution for Pr. Thus, the influence function (Equation 3.5) escribes quite accurately the effect of an unheate zone in the non cases for self-similar flos ith pressure graient an for practically all relevant values of Prantl numbers. In these cases, the eponents C an C vary relatively little from 3/4 an /3 at meium an high Prantl numbers to an / for Pr. Figure 3.4 shos the eponents C an C for the general case as the functions of Pr for various β. These are foun by solving a system of to equations obtaine by substituting g an g into Equation (3.49). To estimate β for etermining eponents C an C in the general case, Equation (3.37) can be use in the same ay as for estimating coefficients g an g. Consiering function (Equation 3.5) an epression (Equation 3.45), the convergence of series (Equation 3.3) is stuie. Accoring to the theorem of the mean value, one gets J = f + ξ θ ξ ξ + f ξ + θ ξ = ξ ξ θ = f ( ζ) ζ (3.53) by Taylor & Francis Group, LLC

14 68 Conjugate Problems in Convective Heat Transfer C..8 Pr β = β = Pr C.5 Pr.4 β=.3 β = Pr Figure 3.4 Depenence of eponents C an C on Pr an β for a laminar bounary layer. here is a suitable value of from segment [, ]. Applying Equation (3.47) gives an estimation for the integral in Equation (3.53): f( ζ) ζ ( ζ) f( ζ) ζ + ( ζ) ζ ζ! ( ζ) f( ζ) ζ + ( ζ) + ζ ζ! ζ f ζ ζ! + = ( ) ( ) + (3.54) The influence function (Equation 3.5) increases as the eponent C ecreases an as the eponent C increases. Because the range of the values by Taylor & Francis Group, LLC

15 Heat Transfer from Arbitrary Nonisothermal Surfaces in a Laminar Flo 69 of these eponents is 3/ 4 C / an / 3 C /, the maimum value of the function (Equation 3.5) is hen C = C = /. Hence, the integran in Equation (3.54) can be estimate using these values of the eponents: ( ζ) ( ζ) < ( ζ)( ζ / ) / < 8 3/ 9 < / 9 (3.55) f here 8 3/ 9 is the maimum value of function (Equation 3.55) in interval [,]. Thus, f ( ) ζ ζ +! 9 + = ( + )! < 5 ( + )! (3.56) an finally, estimation of the remainer (3.45) is obtaine as follos: J = f + ξ θ 5 ξ < ξ + ( + )! + + θ (3.57) + = This epression iffers from the estimation of the Taylor series remainer only by factor 5. Therefore, the non results for Taylor series convergence are vali for series (3.3) as ell. In particular, if the function θ ( ) has erivatives of all orers in an interval [, ], the series (3.3) converges to integral (3.4). 3.4 The Eact Solution of the Thermal Bounary Layer Equation for an Arbitrary Surface Heat Flu Distribution [3] This section consiers the inverse problem hen surface heat flu istribution is specifie an the corresponing temperature hea istribution nees to be establishe. Accoringly, Equation (3.3) is solve for the heat flu to obtain θ θ q + g = = θ h = ( ), (3.58) here θ ( ) etermines the temperature hea on an isothermal surface ith the same heat flu istribution an is a non function of an, hence, of, because q ( ) is given. Equation (3.58) can be consiere as a ifferential by Taylor & Francis Group, LLC

16 7 Conjugate Problems in Convective Heat Transfer equation efining the unnon function θ ( ). The solution of this equation is sought in a form similar to that of series (Equation 3.3): = θ + n n= θ nθ h n (3.59) n here h are coefficients similar to g. Substituting Equation (3.59) into Equation (3.58), one gets n= = n θ h n n g g + θ n + nθ n h n = n = n= (3.6) Performing the ifferentiation in the last term an assembling terms containing lie epressions of ( θ / ) an n ( n θ n ) / yiels the relation h + g ( h + h ) + g [ ( ) h + ( ) h + h ] + g3[ ( )( ) h + 3( )( ) h + 3( ) h + h + g [ ( )( )( ) h + ( )( )( 3) h ( )( 3) h + 4( 3) h + h ],,, = = 3 h = (3.6) Then, the epressions etermining coefficients h via non g are obtaine: h + g ( h + ) =, h + g ( h + h ) + g ( h + h + ) =, h + g ( 3h + h ) + g ( 6h + 4h + h ) + g ( 6h + 6h + 3h + ) =, h + g ( 4h + h ) + g ( h + 6h + h ) + g ( 4h + 8h + 6h g ( 4h + 4h + h + 4h + ) = + (3.6) ] + h ) Figure 3.5 presents the values of the first four coefficients h as functions of the Prantl number an β. For the limiting cases Pr an Pr, the coefficients h are h = /, h = 3/ 6, h = 5/ 96, h = 35/ h = 38., h = 35., h =. 37, h =. 795 (3.63) 3 4 Lie the coefficients g, the first fe coefficients h are ea functions of β, an the rest are practically inepenent of β an Pr. The ea epenence of the coefficients h on β means that Equation (3.59), lie Equation (3.3), can be use ith high accuracy for arbitrary free stream flos. In this case, β can be foun as before using Equation (3.37). by Taylor & Francis Group, LLC

17 Heat Transfer from Arbitrary Nonisothermal Surfaces in a Laminar Flo 7 h.5 Pr β =.4.3 β =.6 h. Pr h β =. β =.6 Pr h 3 4 Pr Pr h 4 Figure 3.5 Depenence of coefficients h on Pr an β for a laminar bounary layer. It as shon in the preceing section that ifferential form (3.3) for the heat flu correspons to the integral presentation (Equation 3.4). To obtain a solution of the inverse problem in an integral form corresponing to ifferential form (Equation 3.59), the temperature hea from Equation (3.4) shoul be foun, taing q ( ) as a given function. For the case of graientless free stream flo, such a problem as consiere in Chapter, an the solution for the temperature hea as obtaine in the form (Equation.5). Because the epressions for graient an graientless flos iffer only by variables an, the solution of the inverse problem for graient flos is foun by substituting variable for in Equation (.5) to get θ C ξ = Γ( C C ) Γ( ) C C ξ ( ) q ( ξ) h ( ξξ ) ξ C C (3.64) In the simplest case of graientless flo an Pr for hich = C, = 3/ 4, C = 3 /, an h =. 33 λ( U/ ν) / Pr / 3, Equation (3.64) reuces to the folloing ell-non relation [4]: θ = ξ. 63 Re / Pr 3 / / 3 3/ 4 (3.65) by Taylor & Francis Group, LLC

18 7 Conjugate Problems in Convective Heat Transfer For the case of a constant heat flu, Equations (3.59) an (3.64) can be use to obtain a relation beteen heat transfer coefficients for an isothermal (T W = const.) an for the surface ith q = const. Diviing each term of Equations (3.59) an (3.64) by q, one gets h C h = + h h ( / ) = Γ( C ) Γ( C ) q = ξ ( ) ξ h ( ξξ ) C C C C ξ (3.66) If the customary poer la relation is use for the isothermal heat transfer coefficient, h = C n, the ratio h / h q is inepenent of an is etermine by one of the relations h h q h h q = = + nn ( )( n) ( n + ), n n = Γ C + C C + /Γ( ) Γ C (3.67) These epressions give the ratio h / h q for arbitrary graient flo an arbitrary Prantl number. For graientless flo, the ell-non results follo from Equation (3.67). For Pr an Pr, one gets h / h q =. 74 an h / h q = / π, respectively. For the stagnation point, the isothermal heat transfer coefficient is inepenent of, so n =, an h / h q =. To erive the influence function of an unheate zone for the case of the heat flu jump, Equation (3.64) is integrate by parts setting C ζ u( ξ) = q, v( ξ, ) = Γ ( C ) Γ ( C ) ξ C C ζ C( C) ζ h ( ζζ ) (3.68) Because accoring to Equations (3.66) an (3.68) v( =, ) an v(, ) = / h q, one has θ q( ) q = + v( ξ, ) ξ h ξ q (3.69) Because ratio q( / ) hq etermines the temperature hea ( θ ) q on the surface ith q = const, Equation (3.69) has the same form as Equation (3.4). by Taylor & Francis Group, LLC

19 Heat Transfer from Arbitrary Nonisothermal Surfaces in a Laminar Flo 73 This implies that quantity v( ξ, ) in front of the erivative in the integran, lie f ( ξ/ ) in Equation (3.4), gives the influence function of the unheate zone. The integran in Equation (3.69) is the increment of the temperature hea, an the value ( q / ξ ) ξ is the corresponing increment of the heat flu. Consequently, the function v( ξ, ) is equal to the reciprocal of the heat transfer coefficient ( h q ) ξ after the heat flu jump at the point =ξ, an the influence function f q ( ξ/ ) in this case is etermine as follos: h Ch = = f ( ξ, ) ( h ) Γ( C ) Γ( C ) q C ( ζ ) C q ξ ξ/ ζ ζ h ( ζ) C( C) (3.7) Because in the problem at han the heat flu is given an ( hq) ξ = q/ θ, Equation (3.7) in fact etermines the istribution of the temperature hea after the heat flu jump for the arbitrary graient flos. To obtain the ellnon epression for the graientless flo from Equation (3.7), consier that in this case, = Re, h ~ Re /, an hence, h( ζ) = h( ) ζ /. Then, C using variable σ = ζ an beta function (3.49), Equation (3.7) is transforme to the folloing form: f q here B ξ hq ξ BC C = ( ) h = (, ) B { C,[ C ( C ) ] }, (, ) = + ( ) B i j ri j σ r r / /C σ σ (, i j) is an incomplete beta function. The ell-non formula θ = q σ σ (3.7). 76( / h ) B ( 34 /, / 3) (3.7) etermining the temperature hea after the heat flu jump for graientless flo follos from Equation (3.7) for Pr an C = 3/ 4 an C = 3 /. It is evient from Equation (3.7) that influence function f q ( ξ, ) for the case of non heat flu, unlie the influence function f ( ξ, ) for the case of non temperature, epens not only on the ratio ξ/ but also on the function h ( ). Therefore, the function f q ( ξ, ) in contrast to the function f ( ξ/ ) epens on each of the variables ξ an. 3.5 Temperature Distribution on an Aiabatic Surface in an Impingent Flo This problem is consiere as an eample hen the heat flu naturally is prescribe an temperature hea istribution nees to be foun. by Taylor & Francis Group, LLC

20 74 Conjugate Problems in Convective Heat Transfer Let a thermally insulate section of a surface be precee by an isothermal section. At the entrance to the aiabatic section, the heat flu rops abruptly to zero, hereas the temperature hea ecreases graually, becoming practically equal to zero only at a certain istance from entrance point. To etermine the variation of the temperature hea in this case, Equation (3.64) is again integrate by parts putting q u( ξ) = = θ, h ξ C v ( ζ) = C C Γ( C ) Γ( C ) ( ζ) ξ ζ ζ C C ( ) (3.73) Because in this case v() =, v( ) =, an q( ) / h ( ) = θ ( ), Equation (3.64) becomes θ = θ ( ) + ξ θ v ξ ξ (3.74) here θ ( ) is the temperature hea on the isothermal section at the entrance to the aiabatic section. It is seen that the function v( ξ/ ) of the integran in Equation (3.74) is sought. Reasoning as in the erivation of Equation (3.7), one arrives at the conclusion that the function v( ξ/ ) escribes the variation of the temperature hea after the heat flu jump, referring to the corresponing variation of the temperature hea that oul occur on an isothermal surface for the same heat flu. In the stuie case of flo impingent on the aiabatic section, the temperature hea changes from the temperature hea θ ( ) on the preceing isothermal surface at the entrance to the aiabatic section to some value θ after the heat flu jump (i.e., on θ( ) θ ). Hence, in the problem in question, the function in Equation (3.74) is v( ξ/ ) = [ θ θ] / θ. Solving this equation for θ an using Equations (3.73) an (3.49), one fins the temperature hea variation on the aiabatic section in an impingement flo in the general case: C Bσ C C θ = θ ( ) (, ) ξ, σ = BC (, C) (3.75) For graientless flo an Pr, the familiar relation follos from Equation (3.75): Bσ θ = θ ( ) ( 3 /, / 3) ξ, σ = B( 3 /, / 3) 3/ 4 (3.76) by Taylor & Francis Group, LLC

21 Heat Transfer from Arbitrary Nonisothermal Surfaces in a Laminar Flo The Eact Solution of an Unsteay Thermal Bounary Layer Equation for Arbitrary Surface Temperature Distribution Consier an incompressible laminar steay-state flo ith the free stream velocity U( ) an temperature T past a boy ith surface temperature T ( ). At the moment t =, the surface temperature starts to change ith time accoring to a function T (, t ). The problem is to etermine the temperature fiel an the surface heat flu istribution for t [5]. The unsteay bounary layer Equation (.3) in imensionless variables (Equation 3.3) becomes ν U ν U θ θ z ϕ θ U u U + z z + u ϕ Pr ϕ U θ = z ϕ = Ut (3.77) The initial an bounary conitions are as follos: z, ϕ =, θ = θ (, z) ϕ, θ (3.78) In the case of poer la free stream velocity U = C m, the terms ν/u an ( ν/ U)( U/ ) epen only on eponent m, an the term ( θ/ ) is equal to ( θ/ ) /( m + ). Then, Equation (3.77) taes the folloing form: U m z m + + u ( ) θ z + θ m+ ϕ θ u ϕ Pr ϕ U θ ϕ = (3.79) The solution of this equation subjecte to initial an bounary conitions (3.78) can be presente in a series similar to series (Equation 3.4): + + θ θ = G z i i θ i(, ϕ) = G θ Ui ti G G θ + + U t = i= + G θ + G U θ θ + G + (3.8) U t Substituting this series into Equations (3.78) an (3.79), one obtains a set of equations an initial an bounary conitions that etermine the coefficients by Taylor & Francis Group, LLC

22 76 Conjugate Problems in Convective Heat Transfer of the series (3.8): m + U u G + ( m) z z i + Gi + Gi ( ) ( + i) Gi + G i m + ( ) ϕ ϕ u G i Pr ϕ U ϕ = (3.8) z, ϕ =, G =, Gi =,( i =, >, =, i >, i >., > ), ϕ, G i (3.8) For a poer la free stream velocity, the ratio u/ U epens only on variable ϕ. Hence, the coefficients G i are a function only of variables z, ϕ an parameters m an Pr. The surface heat flu is etermine by ifferentiating Equation (3.8) to obtain q θ =λ y = G i y= = ϕ i= ϕ = θ U t + i + i h i i θ θ = + g + + g θ θ θ g + g + g U t U t U θ t + (3.83) gi = [( Gi/ ϕ)( / G / ϕ)] ϕ = ( i =, >, =, i > i >, > ) (3.84) Equation (3.8) subjecte to conitions (Equation 3.8) are solve numerically for a plate an Pr = using the finite ifference metho. The coefficients of the first four terms containing the erivatives ith respect to time ( i ) are given in Figure 3.6. If the surface temperature hea epens on the coorinate only, one puts i =, an Equations (3.8) an (3.83) become the proper form of the steay-state solutions (3.4) an (3.3) ith coefficients g (Equation 3.3). The coefficients gi () z graually gro ith time an finally attain the values of ( g ) i t that coincie ith those obtaine by Sparro ithout initial conitions [6]. To obtain a satisfactory result, one can use only several terms of the series (Equation 3.83), because coefficients g i ecrease rapily ith groing the value ( i ). The ratio gi()( z / gi) t is about.99 hen z = Ut/ = 4.. Hence, for z > 4., coefficients g i are practically inepenent of time an become the values ( g i ) t. Applying the same technique of repeate integration by part as in the case of steay-state heat transfer (Section 3.3), one can sho that ifferential form by Taylor & Francis Group, LLC

23 Heat Transfer from Arbitrary Nonisothermal Surfaces in a Laminar Flo 77 g i tu/ Figure 3.6 Depenence of coefficients g i on imensionless time z = tu/ for an unsteay graientless laminar bounary layer. Pr =. g, g, 3 g, 4 g. (Equation 3.83) for heat flu is ientical to the folloing integral epression: t q = h θ θ θ(, t ) + f( ξ/,, z) ξ+ f(, η/ tz, ) ξ η η + t θ η f( ξ/, η/ t, z) ξ (3.85) ξη Here f( ξ/, η/ tz, ) is an influence function of an unheate zone for the unsteay heat transfer that epens on the integration variables ξ/ an η/t an imensionless time z. The relation beteen the function of the unheate zone an the coefficients of series is g i ( ) ++ i () z = ( )!( i)! z ξ η ( z σ) i ( ζ) f ( ζσ, / zz, ) σ ζ ζ =, σ = t (3.86) The epression for the nonisothermicity coefficient follos from Equation (3.83): χ = + g t θ θ g Uθ θ + + t + g Uθ θ + g t Uθ g θ θ θ t + (3.87) by Taylor & Francis Group, LLC

24 78 Conjugate Problems in Convective Heat Transfer 3.7 The Eact Solution of a Thermal Bounary Layer Equation for a Surface ith Arbitrary Temperature in a Compressible Flo In this section, the Doronizin (or Illingorth-Steartson) inepenent variables [7] (or [8]) are use to sho that solution (Equation 3.4) for incompressible flui is vali in the case of graientless compressible flo past the plate []. Substituting a Doronizin s variable η for the variable y an transforming energy equation (Equation.6) for compressible flui to the variables an ϕ, y U ρ η ν ρ ξ ϕ ψ =, =, (3.88) C ρ Cν U one arrives at the folloing thermal bounary equation of Prantl-Mises- Görtler s type: i i u i u U u ϕ ( )M ϕ Pr ϕ ϕ U = (3.89) ϕ U Here, C is a coefficient in Chapman-Rubesin s la (Equation.63) for viscosity, i = ( J J )/ J is the imensionless ifference of a gas enthalpy, an is the specific heat ratio. This equation is in agreement ith Equation (3.) if one rites the latter for the graientless flo an taes into account that in the case of graientless incompressible flo =Re, an terms ( ) Μ an enthalpy ifference i become U / c p an θ, respectively. Consequently, the corresponing changes in epression (3.4) transform it into a solution of Equation (3.89): i i = G ( ϕ) + G M ( ϕ)( ) (3.9) = The heat flu an shear stress are etermine in the case of compressible flui as q ρ T C U u i T λ ρ µ =, τ = ρ ν ϕ ρ C U u u ν (3.9) ϕ= ϕ ϕ = Integrating both sies of the secon equation, substituting the result obtaine for u in the first equation, an using the relation for shear stress [] by Taylor & Francis Group, LLC

25 Heat Transfer from Arbitrary Nonisothermal Surfaces in a Laminar Flo 79 τ. 33ρ C U3ν / C yiels the epression for heat flu in to forms: = U q g TC i g i =. 576 λ + g C ( ) ν = Μ C = T T / T + S T + S (3.9) q q C = ia + C = a g i ia = J J J a r = i ( ) M The first part of Equation (3.9) is ritten using the imensionless gas enthalpy i, but in the secon form, the imensionless stagnation gas enthalpy i is employe, here J a. is an aiabatic all enthalpy an r is a recovery factor (Section 3.9.). Coefficient C = λρ/ λρ = µ ρ / µ ρ in contrast to coefficient C is etermine using the local all temperature (see Equation.63), an q is the heat flu on an isothermal surface ith the average temperature hea of the stuie nonisothermal surface. The coefficients g an g are the same as in the case of incompressible flui given in Section 3.. Equations (3.9) an (3.9) are the eact solutions of the thermal bounary layer equation for an arbitrary plate temperature istribution. Chapman- Rubesin s solution [5] for a polynomial plate temperature istribution follos from Equations (3.9) an (3.9). 3.8 The Eact Solution of a Thermal Bounary Layer Equation for a Moving Continuous Surface ith Arbitrary Temperature Distribution A number of inustrial processes, lie a forming of synthetic films an fibers, the rolling of metals, glass prouction, an so forth, are base on the systems, in hich a continuous material goes out of a slot an moves through a surrouning coolant ith a constant velocity U. As a result of a coolant viscosity, a bounary layer is forme on such a surface (Figure 3.7). Although this bounary layer is similar to that on a stationary or moving plate, it iffers. In this case, the bounary layer gros in the irection of the motion, as oppose to flo over the plate, on hich it gros in the opposite irection of that in hich it is moving. It can be shon that in a coorinate system attache to the moving surface, the bounary layer equations iffer from the equations for the case of flo over a plate, but the bounary by Taylor & Francis Group, LLC

26 8 Conjugate Problems in Convective Heat Transfer y U Pr T T T T U CR T U Pr T Figure 3.7 Schematic of a bounary layer on a moving plate for symmetrical an asymmetrical (Eample 8.) flos. conitions are ientical. These equations of a moving surface in the moving frame are unsteay, but if the coorinate system is fie an attache to the slot, the problem becomes steay, an both bounary layer equations coincie; hoever, the bounary conitions iffer because the flo velocity on the moving surface is not zero. Eact solutions of the ynamic an thermal bounary layer problems analogous to Blasius an Pohlhausen solutions for a streamline semi-infinite plate are given in References [9] an []. The friction coefficient on the moving surface is greater by 34%, an the heat transfer coefficient for isothermal surface an Pr = 7. is greater by % than for a plate. The eact solution for a nonisothermal surface is obtaine in Reference [] for stationary an moving coolants ith ifferent ratios ε = U / U of g /Pr / U /U = U /U = Pr Figure 3.8 Depenency of g Pr / on the Prantl number an ratio of velocities ε = U / U for a plate moving through surrouning meium. by Taylor & Francis Group, LLC

27 Heat Transfer from Arbitrary Nonisothermal Surfaces in a Laminar Flo 8 g (a) Pr g (ε ) g (ε = ), g (b) Pr Figure 3.9 Depenence of coefficients g () a an g () b on the Prantl number an ratio of velocities ε = U / U for a plate moving through surrouning meium. ε =,., 33., 4. 5, 58., 6 streamline plate, 7() b g3. the velocities of a surface U an flui U. The epressions for the heat flu an nonisothermicity coefficient are the same Equations (3.3) an (3.4) an Equations (3.33) an (3.48), respectively, but only in variable, because the coolant flo is graientless. The heat transfer coefficient for an isothermal surface is Nu = g Re / / or h = g U / ν (3.93) here Nu, Re, an imensionless in the first formula may be efine by any length. Coefficients g an g are given in Figures 3.8 an 3.9. The corresponing eponents C an C are plotte in Figure 3.. As in the case of a streamline plate, the coefficients g for 3 are practically inepenent of the Prantl number an the parameter ε an by Taylor & Francis Group, LLC

28 8 Conjugate Problems in Convective Heat Transfer C (ε = ) C (ε ) (a) Pr C (b) Pr Figure 3. Depenence of eponent C (a) an C (b) on Prantl number an ratio of velocities ε = U / for a plate moving through surrouning meium. ε =, 3., 3., 4. 5, 58.. U can be calculate by formula (Equation 3.5). The coefficient g for flo over a stationary plate in the intervals of large an meium values of Pr is consierably smaller than in the case of a continuous moving surface. For Pr =.7, for eample, the coefficient g is.7 times for ε = 8. an tice for ε = as great as in the case of a flo over a fie surface. This means that the influence of nonisothermicity is substantially greater for a continuous moving surface than for the case of a flo over a stationary plate. by Taylor & Francis Group, LLC

29 Heat Transfer from Arbitrary Nonisothermal Surfaces in a Laminar Flo The Other Solution of a Thermal Bounary Layer Equation for an Arbitrary Surface Temperature Distribution Some other solutions of the thermal bounary layer equation are obtaine for arbitrary nonisothermal surfaces. Because the approach use in all cases presente here is the same as in the above-iscusse problems, these solutions are escribe briefly, inicating only the istinctions of each problem Non-Netonian Flui ith a Poer La Rheology [] The poer rheology means that flui obeys basic poer las []: n ˆ τ = Kτ ˆ,, I e q = Kq I grat I S u = + v + u + v 4 4 y y (3.94) Here, ˆτ an ê are the stress an rate of eformation tensors, q is the heat flu vector, an I is the secon invariant of a rate of eformation tensor. Poer las (Equation 3.94) aequately escribe the behavior of such atypical fluis as suspensions, polymer solutions an melts, starch pastes, clay mortars, an so forth. Neton s friction an Fourier s heat conuction las follo from Equation (3.94) hen n = an s =. Therefore, the eviation of n an s from these values can be a measure of flui anomaly. The bounary layer equations for poer la fluis are as follos: u + v = + u u v u U U Kτ u, y y ρ y y + u T v T Kq u y ρcp y y s T K y ρc τ p u y n+ n = = (3.95) Self-similar solutions of system (Equation 3.95) eist [] in the same cases as for Netonian fluis that is, hen the eponents in las (Equation 3.94) are equal ( s= n ) an free stream velocity an temperature hea istributions obey the poer las (.5). The equality s= n means that viscosity an heat conuctivity efine by epression [( / ) I ] in las (Equation 3.94) are proportional to each other. In such a case, the thermal bounary layer in Equation (3.95) has an eact solution for the case of poer la free stream istribution. by Taylor & Francis Group, LLC

30 84 Conjugate Problems in Convective Heat Transfer Transforming the thir part of Equation (3.95) to Prantl-Mises-Görtler s variables, one gets = n ρ L ψ Un ( ξ) ξ, ϕ = U 3 [ nn ( + )( Kτ / ρ)( U3/ L) n ] n K τ + (3.96) n θ u u nn ( + ) n ϕ θ n θ ϕ Pr ϕ U ϕ U ϕ U c p u U n u ϕ U n+ = (3.97) Substituting solution (Equation 3.4) into Equation (3.97) leas to orinary ifferential equations similar to Equations (3.7) an (3.9): ( / Pr){[ ωϕ (, β, n)] n ω ( ϕ, β, n) n G } + nϕg nn ( + ) G = nn ( + ) G ( / Pr){[ ωϕ (, β, n)] n ω ( ϕ, β, n) n G } + nϕg (3.98) nn ( + ) βg = [ ω( ϕ, β, n)] n ω ( ϕ, β, n) n (3.99) The graient pressure parameter is epenent in this case not only on eponent m as in the case of Netonian fluis, but also on eponent n: ( n+ ) m β = ( n ) m+ (3.) Because of that, the pressure graient is characterize not using β, hich in this case epens also on n, but by using eponent m. Bounary conitions for Equations (3.98) an (3.99) remain the same conitions (3.8) an (3.9) as ell as Equations (3.3) an (3.33) for the heat flu an nonisothermicity coefficient. For an isothermal surface, one obtains Nu Re n n+ C f = g Re n+ n n Re n+ ( ), [ ( )] ( ) g = / n n+ ( n+ ) ϕ G ϕ = (3.) here Nu an Re are generalize numbers as given in the Nomenclature. by Taylor & Francis Group, LLC

31 Heat Transfer from Arbitrary Nonisothermal Surfaces in a Laminar Flo 85 g /Pr / , n Figure 3. Depenence of g 3 Pr / on Prantl number an eponents n an m for non-netonian flui s = n, m=, Pr >, m= 3 /, Pr >, 3 m =, Pr =, 4 m=,pr =, 5 m=, Pr =. Calculations are performe for large Prantl numbers Pr =,,, typical for non-netonian fluis, eponents n from. to.8 an m = 3, / an. The results are given in Figures 3. an 3.. It is seen from Figure 3. that for large Pr, the value g 3 /Pr / slightly epens on the Prantl number. This inicates that the heat transfer coefficient for an isothermal surface for non-netonian fluis is proportional to Pr 3 / as ell as for usual Netonian fluis. The epenencies of coefficients g on Prantl number an pressure graient (Figure 3.) are similar to those for Netonian fluis. In particular, for Pr > an small pressure graients m = an /3, the coefficients g are practically inepenent on Pr. As the pressure graient increases, this epenence becomes more mare. Functions g ( n ) an g ( n) for m= 3 /, m= an Pr = practically merge in one curve. Because in the case uner consieration s= n, the basic relation (3.3) remains the same as for Netonian fluis, all other formulae erive above remain vali as ell, in particular, integral form (Equation 3.4) an Equation (3.48). Of course, the eponents C an C of the influence function in integral form, coefficients h in formula (3.58) an others shoul be etermine accoring to coefficients g given in Figures 3. an 3.. This can be one similar to that performe in previous sections. As in the case of Netonian fluis, the relation obtaine for self-similar free stream velocity istributions can be use ith high accuracy for an arbitrary pressure graient, but in this case instea of epression (3.37) for β, a similar formula for m shoul be use: m= U/ U( ξ) ξ (3.) by Taylor & Francis Group, LLC

32 86 Conjugate Problems in Convective Heat Transfer g, g 3, g 4 g , 4 g 3 g , , 4 g g n Figure 3. Depenence of coefficient g on the Prantl number an eponents n an m for non-netonian flui s = n, m=,pr =, m=,pr =, m = 3 /, Pr >, 3 m =, Pr >, 4 m=, Pr >. For the general case of arbitrary eponents n an s in las (3.94), only approimate solutions for arbitrary nonisothermal surface have been obtaine [3 5] The Effect of Mechanical Energy Dissipation [] The effect of issipation is minor for incompressible fluis because this effect is proportional to the square of velocity, hich in this case is typically relatively small. Therefore, the effect of issipation in the case of an incompressible flui can be significant only for large Prantl numbers. The effect of issipation is etermine for both Netonian an non-netonian fluis by the secon term of Equation (3.). The coefficient g in this equation can be calculate by integration of orinary ifferential equation (3.9), similar to computing coefficients g. Some results are given in Figure 3.3. Computing a heat of issipation is important for recovery factor an for etermining aiabatic all temperature. To obtain these quantities, the case of non heat flu istribution shoul be consiere. This problem can be solve using the same approach as in Section 3.4, here this problem is consiere ignoring issipation. Again, using relation by Taylor & Francis Group, LLC

33 Heat Transfer from Arbitrary Nonisothermal Surfaces in a Laminar Flo 87 g 6 m = m = n Figure 3.3 Depenence of coefficient g in etermining the issipative term on eponents n an m for non-netonian an Netonian ( n = ) fluis. (Equation 3.58) as an orinary ifferential equation efining the temperature hea as a sum, θ q = + h g U cp, (3.3) the same ifferential (Equation 3.59) an integral (Equation 3.64) relations are obtaine. Assuming in sums (Equations 3.59 an 3.64) for the case of aiabatic alls q = yiels the epression for recovery factor: T r = U T U = g + h U (3.4) a. / c p = The corresponing integral form follos from Equation (3.64): gc r = ξ Γ( C C U ) Γ( ) C C ( ) C C ξ U ( ξ) ξ ξ (3.5) Coefficients h in Equation (3.4) for Netonian fluis are given in Section 3.4. They can be similarly calculate for non-netonian fluis using Equation (3.6) an non g (Figure 3.) Aisymmetric Streamline an Rotating Boies [] Stepanov [6] an Mangler [7] suggeste variables that transform a problem for an aisymmetric streamline boy to an equivalent to-imensional by Taylor & Francis Group, LLC