Question

2-4 spherical reactor is generating heat according to ġ = ĝol1 – ()*7 per unit of volume. Find the surface temperature in terms of o, sphere radius ro, surrounding convection coefficient h, surrounding temperature T.

Answer 1

Assumption GA dr element 12 Tw (1) One dimerosional radial conduction of heat (1) steady state heat conduction () Uniform heat generation (iv) Constant thermal conductivity Generalized heat conduction equation for sphere is, a®Т aq IT rasgo doa 92 1 + KOT + Dr 2 Sino IT ga Suho do Dr Lo 0 ƏT + 9 g -(1) Reduces to -(1) (a) Hå or k Zg radius a Also, no is the outer radius of sphere k is the thomal Conductivity is the uniform hat generation per unit volume, pes Unit time within the solod Tw is the temperature of surface of sphere and Ta is the Ambient terperatuose. and thickness da, Cornider a Heat conduction radius r is, Qa: -KADT. -K 4599 dT dr da Heat conducted out at radius arda is, Qardn = Qat d Qada [ approximation wing Taylon series . Heat generated qg Ada = -x 494² dT da at -(2) (3 uring] Ta is Qg

여 23 A Substitute å 12 - Cixa) +qq u 2 ( 92 EO 22 DT DA K Ida ga %, (1-12 I a (аз от K da Dr ss di Integrate once -qo 563 K CAROT da - Q. 3 ra DT 22 Shar 2013 -4 ( 1-2) Jantl С. + DT Dr [ i da Integrate again IT 2 24 -90 9 6%3 K

T-9 (notes) 4+ Her, C and C are obtained from Constants of integration, that are Boundary conditions (BC) At center of sphere q=0 so, o=0 ie, al r-ro , T = Tw BC roa rs •]- a to 3070 do q At boundary Tw = -2 e 28 7--14-16 Tw = 9 - 2014 28? C 96 K 15 C. 2 ro TW ro. Twro C 17 286 15 T Substitute C. Ca in 3 + Twro + 1 28 1280 r K 15 T= -4: 1990 85 30% 3 equal to generated # Heat conducted heat state , heat Conducted should be convected from outer surface. # Under steady equal to heat