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Three-dimensional radiation boundary with non-steady-state temperature field within the rules

Within a non-orthogonal boundary problem with nonlinear boundary conditions of the steady-state heat conduction problem, the problem can be described as a definite solution of equation r = + {j1 + +1 r2sin28Tq '~ 0 (1)' r ∈ n, 0 ≤ ≤ , 0 ≤ ≤ 2 Ji) where r is the temperature; (r,,) boundary conditions for the spherical polar coordinates r = (In s I) (2) 1-Jun Liu,[link widoczny dla zalogowanych], et al: Radiation boundary with the rules within three-dimensional non-steady state temperature field of 85 + rare: o (the s on) (. Hom hi [n. () a (n +1) b (') a] × type of port for the Stefan-Bohzmann constant; to the outer s the distance between the normal direction of the variable type (3), the radiation boundary condition is nonlinear, it borders the fourth power of temperature, whereas the boundary temperature is to be desire. this nonlinear boundary conditions, the usual analysis is to linearize f3, 5J, for example, the temperature of the power index dropped four times by the boundaries of the discrete method in this paper is to deal with such problems, no need to type (3) as a linear process, while maintaining its nonlinear characteristics, and the solution can be obtained in analytical form. with linear boundary conditions are different, the nonlinear boundary condition, it needs to solve nonlinear algebraic equations, instead of line} Health algebraic equations. 3 boundary equations using discrete and separation of variables, set of partial differential equations of the form (r, 0,) = R (r) 0 (0) vertical (vertical) (4) solve for vertical (down) = Acosm ~ + BsinmO ft (r) = C + _Dr a @ (0) = (cos0) where, B, c, _D, are coefficients; (cos @) for the order along with Legend ~ function. Equation (1) the general solution _6Jr (r, 0,) = c (n +6 a) (cos0) × cosm ~ + (c swollen, + r a) P (cosS) sin., rgP (5) If the formal solution of (5) infinite series truncated to finite terms Ⅳ, the formal solution in a total of 2 (Ⅳ +1) 0 个 series to be determined coefficient (. ... 6,[link widoczny dla zalogowanych], c, d) 4 {border temple Discrete Method discrete boundary in accordance with given parameters law, and on s. side of the interface were collected (N +1) discrete points (uniform) set points were P. and P. boundary surface s and s. on a certain discrete points, their coordinates are ( r, 0. vertical) and (a,,),[link widoczny dla zalogowanych], at each discrete point of the boundary will the boundary conditions equation (2) and (3) The form of solution generation (5), is three. [n, delete (r) ~ (r) A] (cos @) × COSrnC +, [c (r) + d (r) A] × P (cos @) sinm ~ '= To (6) [i = l, 2, .... (N +1)] P '~ (cos @) cosmdPi + [w () ~ a (n +1) d () a] P (c. s) sinm + {[n-fan (') +6 () a] P ( c. s) c. s +[()+(') a] P ~ (cos @) sinng /:~}: 0 (7) [J = 1,2, ..., (+1)] associated vertical ( 6) and (7), obtained on the (n, 6, c,) by 2 (N +1) linear algebraic equations consisting of equations. using quasi-Newton method for solving 【J convergent solution obtained after the formal solution of the series asked for items to be determined coefficient, and then substituting back equation (5), the final form of the solution by series of 5 examples shown in Figure l hollow sphere with a radiation boundary analysis of the temperature field . to take the computational domain material are stainless steel = l2w (m ・ oC),[link widoczny dla zalogowanych], aluminum = 204W (m ・ ℃), Silver = 419w (m ・ oC), external radius of the ball = O.5m, within the three hollow ellipsoid an axle, respectively o = b = 0.4m, c = 0.3m, ? = 0.65,: 373K. The results shown in Table l,[link widoczny dla zalogowanych], Table 2 (taken Ⅳ = 2). Table 1-I cross-section radial temperature distribution, Table 2 Ⅱ Ⅱ section of a radial temperature distribution. can be seen, thermal conductivity is large, spherical external surface temperature is high; thermal conductivity of certain materials, the thin outer surface of the ball at high temperature, which with the actual situation is kiss Taiwan. (a) (b) Figure l section of computational domain and boundary discretization Table 1I-I cross-section of radial temperature distribution 040042044o46o48050 stainless steel aluminum silver 3730 ∞ 371070369.38l367.366582 Ⅻ .4193 groan like O 372372767372672m588372514mO00372939mS86372840372799372763 (continued on page 89)
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