Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 27
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The critical values of the Fourier number for the three geometries are given in Table 3.8. The first eigenvalue can be computedaccurately by means of the correlation [151]81,oo8~ = [1 + (8~/~~1~/~tvl,O) Jwhich is valid for all values of Bi. The asymptotic values of the first eigenvalues corresponding to very small and very large values of Bi are given in Table 3.8.
The correlation parametern is also given in Table 3.8. The correlation equation predicts values of 81 that differ from theexact values by less than 0.4 percent.For Fo < Foc, additional terms in the series solutions must be included. It is therefore necessary to use numerical methods to compute the higher-order eigenvalues 8n that lie in theintervals nrc < 8n < (n + 1/2)rt for the plate and (n - 1)r~< 8n < nrt for the cylinder and the sphere.Computer algebra systems are very effective in computing the eigenvalues.Chen and Kuo [12] have presented approximate solutions of O/Oiand Q/Qi for the platesand long cylinders.
The accuracy of these solutions is acceptable for engineering calculations.Multidimensional SystemsThe basic solutions for the infinite plates and infinitely long cylinders can be used to obtainsolutions for multidimensional systems such as long rectangular plates, cuboids, and finitecircular cylinders with end cooling. The texts on conduction heat transfer [4, 11, 23, 29, 38,49, 56, 87] should be consulted for the proofs of the method and other examples.Langston [51] showed how to obtain the heat loss from multidimensional systems using theone-dimensional solutions given above. Two-dimensional systems such as long rectangularplates and finite circular cylinders are characterized by two Biot numbers, two Fourier numbers, and two dimensionless position parameters.
Threedimensional systems such as cuboids are characterized bythree sets of values of Bi, Fo, and ~ corresponding to thethree cartesian coordinates. When the two or three sets ofFourier numbers are greater than the critical values given inTable 3.8, then the first-term approximate solutions dis0cussed above can be used to develop composite solutions.Yovanovich [151] has discussed the application of the basicsolutions to long rectangular plates, cuboids, and finitelength circular cylinders."~2XIDimensionless Temperature and Heat Loss Fraction forFinite-Length Cylinders. The finite-length circular cylin-FIGURE 3.9 Circularcylinder of finite length.der of radius R and length 2X, shown in Fig.
3.9, has constantproperties and is cooled through the sides and the two endsby uniform film coefficients hr and hx, respectively. The sys-3.26CHAPTER THREEtem is characterized by four physical parameters: Bix = hxX/k, Bic = hrR/k, Fox = ou/X 2, and(zt/R 2. The dimensionless temperature within the cylinder is obtained from the productsolution:F o r --(~xr:(~x(~rwhere ~x and ~)r a r e the solutions corresponding to the x and r coordinates, respectively.According to Langston [51], the heat loss fraction can be obtained from(Q)xr=(Q)x+(Q)r--(Q)x'(Q)rwhere Qi = pCp2nXR20i. The subscripts x and r denote solutions corresponding to the x and rcoordinates, respectively.Transient One-Dimensional Conduction in Half-SpacesThe analytical solutions for transient one-dimensional conduction in half-spaces x > 0 are wellknown and appear in most heat transfer texts.
The solutions are given here for completenessand to review important characteristics of the solutions.Equation and Initial and Boundary Conditions.The diffusion equation and the initial andboundary conditions are presented first, followed by the solutions with some important relationships.3203x 21 30tx 3 t 't > 0,x>0(3.74)where 0 = T(x, t ) - Ti is the instantaneous temperature rise within the half-space. The initialcondition is0=0,t=0,x>0(3.75)and the boundary condition at remote points in the half-space is0 --->0,t>0,x --->oo(3.76)There are three options for the boundary condition at x = 0.Dirichlet ConditionO= To- Ti,t>0(3.77)t>0(3.78)where To is the fixed temperature on the surface.Neumann Condition30~=-~3xq0k'where q0 is the constant heat flux imposed on the surface.Robin Condition30~x -hk ( % - 0),t>0(3.79)CONDUCTION AND THERMAL CONTACT RESISTANCES (CONDUCTANCES)3.27where h is the constant film or contact conductance that connects the surface to the heatsource and 0i = Ti - Ti is the constant temperature difference between the heat source temperature and the initial temperature.
The solutions have been obtained by several analyticalmethods. Introducing the dimensionless parameters ~ = ( T(x, t) - Ti)/(To- Ti), 11 = x/(2V'at),and Bi = (h/k)V~t, the three solutions are given below.Dirichlet Solution= erfc (11),11 > 0(3.80)which gives the instantaneous and time-average surface fluxesqo(t) = ~1k(To-Ti)V~2 g(To-Ti)q0(t) = V ~V~and(3.81)(3.82)The time average value of any function f(t) is defined as f(t) = (l/t) ~o f(t) dt.Neumann Solutionk[ T(x, t ) - T,]2qoV~1= V ~ exp(-rl2) - rl erfc (11)(3.83)which gives the following relationships for the instantaneous and time-average surface temperatureskiT(0, t) - Ti]12q0V~: V~andk [ T ( 0 ) - Ti]22q0V~= 3V~(3.84)(3.85)Robin SolutionT(x, t ) - T,.r~-r/= erfc (11)- exp(2rl Bi + Bi 2) erfc (11 + Bi)(3.86)which yields the following two relationships for the instantaneous surface temperature andthe surface heat flux:T(0, 0 - T,Ti-Tiand= 1 - e x p ( B i 2) erfc (Bi)q°(t)Vr~t~t = Bi exp(Bi 2) erfc (Bi)k(Tf- Ti)(3.87)(3.88)For large values of the parameter Bi > 100, the Robin solution approaches the Dirichlet solution.The three one-dimensional solutions presented above give important short-time resultsthat appear in other solutions such as the external transient three-dimensional conductionfrom isothermal bodies of arbitrary shape into large regions.
These solutions are presented inthe next section.3.28CHAPTER THREEExternal Transient Conduction From Long CylindersIntroduction. Transient one-dimensional conduction external to long circular cylinders isconsidered in this section. The conduction equation, the boundary and initial conditions, andthe solutions for the Dirichlet and Neumann conditions are presented. The conduction equation for the instantaneous temperature rise O(r, t) - Ti in the region external to a long circularcylinder of radius a is320 1 301 303-7 + r 3 r - aat't>O,r>a(3.89)The initial condition is0=0,t =0and the boundary condition at remote points in the full space is0 --+ 0,r--+ ooTwo types of boundary conditions at the cylinder boundary r = a will be considered:(1) Dirichlet and (2) Neumann.Dirichlet Condition0 = T0- Ti,t>0where To is the fixed surface temperature.Neumann Condition203r--~qok't>0where q0 is the constant heat flux on the cylinder surface.The solutions for the two boundary conditions are reported in Carslaw and Jaeger [11].The solutions were obtained by means of the Laplace transform method.
The solutions aregiven as infinite integrals and the integrand consists of Bessel functions of the first and secondkinds of order zero.Dirichlet Solution0 _ 1 + -2 F e-F° ~2 Jo(fJr/a)Yo(fS)- Yo(fJr/a)Jo(fS) dr3~ JoJ2(13) + Y2(13)13(3.90)0iwith Fo = (xt/a 2 > 0. The integral can be evaluated accurately and easily for all dimensionlesstimes using computer algebra systems.Instantaneous Surface Heat Flux.at r = a, is given by the integralThe instantaneous surface flux, defined as q(t) =-kaO/3raq(t)4 f : e_FO~2dr3kOi - 71;2[j02(~) + y2(~)]~(3.91)Carslaw and Jaeger [1 1] presented short-time expressions for the instantaneous temperaturerise and the surface heat flux.Short-Time Temperature Rise01[p-1 ] (p-1)~/~oierfc [ p - 10 i - ~ erfc 2x/-F-oo +4133/22x/Go]+(9-29-792 )[P -1 ]32pS/2i2erfc 2N/-~oC O N D U C T I O N AND T H E R M A L CONTACT RESISTANCES (CONDUCTANCES)3.29with p =- r/a > 1.
The special functions ierfc (x) and i2erfc (x) are integrals of the complementary error function and are defined in Carslaw and Jaeger [11].Short-Time Surface Heat Flux.The instantaneous surface heat flux is given byaq(t)kOiV ~ / - ~ o + ~ - - ~-+~-FoThe first term corresponds to the half-space solution when the dimensionless time is verysmall, i.e., Fo < 10-3.N e u m a n n Solution. The instantaneous temperature rise for arbitrary dimensionless timeFo > 0 at arbitrary radius r/a > 1 is given by the integral solution:kO _ - 2 If (1- e-F°~) Jo(fSr/a)Y,([5)- Yo(~F/a)Jl(~) dr3qorrj2 (~) + y2 (~)~2Carslaw and Jaeger [11] presented an approximate short-time solution for arbitrary radius:p-lJk02 ~/~o{ierfc[q0 - V~2V~owith p = r/a > 1 and Fois given by=-(3p+1)i2erfc [ p - 1 ]}4p2~ooff/a 2.
The instantaneous surface temperature rise 00 for short timesk00_ 2 ~ / - ~ o _ 1q0~2Transient External Conduction From SpheresIntroduction. Solutions of transient conduction from a sphere of radius a into an isotropicspace whose properties are constant and whose initial temperature Ti is constant are considered here.
The dimensionless equation isa2(~ap 2 +2 a~p apa~-aFo'Fo > 0,p>1(3.92)where ¢~is the dimensionless temperature, p = r/a, and Fo = m/a 2 is the dimensionless timedefined with respect to the sphere radius. Solutions are available in Carslaw and Jaeger [11]for three boundary conditions: (1) the Dirichlet condition where T(a, t)= To, (2) the Neumann condition where OT(a, t)/br =-qo/k, and (3) the Robin condition where OT(a, t)/br =- ( h / k ) [ T r - T(a, t)]. The thermophysical parameters To, TI, q0, and h are constants. Theseboundary conditions in dimensionless form are ~ = 1, O¢/0p =-1, and 0~/0p =-Bi(1 -¢~) withBi = ha/k for the three boundary conditions, respectively.
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