Question

Consider a spherical fuel particle with radius R. Within sphere heat is produced which varies with temperature according to the relation: S=S, [1-a(T-T.)] So is the heat produced per unit volume per unit time and "a" is a constant. Surface temperature of the sphere is kept constant at To a. By constructing a shell balance obtain an O.D.E. describing steady state temperature profile. b. By using dimensionless temperature @= T-TO S.R?/k and dimensionless position x= t/R bring the O.D.E to a dimensionless form. C. Obtain the dimensionless temperature profile in a neat form.

Answer 1

Pg no % 3 a ODE eqn :- Consider an element 2 radius 'r' inside spherical particle with thickness or By applying Energy balance for spherical shell ar Qut dr Rate of heat out ) + (Rate of generation) (Rate Accumulation 2 heat) - (Rate of heat in ) + heat Now, Rate & accumulation = Q ut qg Ar Ar y can Taylor's expansion for Quadr, we get accumulation , Qv + dvi use Ex Qvtal Qv + ol d2 (v) Art dr (Qv) (47)2 dur cohere ď dir Car Car)?=0. => Qvt ar = Qr t ole. Car) AY d = y art CQx) ar Qyt q V Ar. Ar dr d [/ (ar) -2e Ar O. dr 4 b. Ara Area of spherical shell 4182

Pg no. of 3 a at re seli - 017-70)] sabatinite all in eqo Co) le (-kid's Ar) dr Ar) - so [n-a (-70)]* 30 & Cake Design Ar) - 50 [1-367-to)]ursa? د؟ ai Com ultraso (i-a dy (T-T.)]=0 dr (rhat 2) dr K (C) y so ) (-a (Foto)] =0, heat franster, Dimension less form :- ODE Governing b ODE to Given data T-to so к ey so P² T To K de R So di a do di dilar 20.

no3와 3 3 do de da 11 dr do dr r Heve N- R, = = xR. d - 의 dr Prem eqo (c), (cy. dT di da dr se (ig3(1-ao-SR] da dx dr IV ㅗ 0 PL dr SOR 누 R S03적 다 न ad. 21 이 R da R < Here d) dî da ) 응 위국 dr du d do da 2Y SoRY da dV C dx k dx dr AT SOR () dx da Substitute dinlar in (doire a sory 0) 26 외 da k K SerR 1) (6) alda (². SOR do (s) (I- [ ] (-2. (3] - (-ey] = alda [~ - - 2] - K K do Řo a so R da R Dimension less eqn.