Consider a cylindrically shaped zone of conduction (physically realized by the p

Question

Consider a cylindrically shaped zone of conduction (physically realized by the presence of a conducting medium) of radium R and length L where the thermal conductivity is k1(known and constant). Within the zone of conducton an homogenous chemical reacton takes placewith an unkown , yet constant (uniform), volumetric heat generation rate q. The above zone of conduction is enveloped by a layer of a specific type of material of thickness d for which the thermal conductivity is k2 (known and constant ). If the outer temperature value of the afforementioned material layer is known and constant Ts, as is the tempertature value at the interface between the above two ocnduction zones Tint, calculate both: i.) the volumetric heat generation rate q dot, and the rate of heat transfer insdie the two zones of conduction as a function of r (radial distance). Assume also steady state conditions for the overall thermal system.

Explanation / Answer

solution:

1)here heat conduction equation for cylindrical coordiante is poissons equation is written as

(1/r)*(d/dr)(r*dT/dr)+q'/k=0

hence on solving we get equation as

T=(-q'r^2/4k)+c1*lnr+c2

where boundary condition are

at r=Ri,T=Tint

at r=Ro,T=Ts

hence above equation is writtena s

Tint=(-q'Ri^2/4k)+c1*lnRi+c2

Ts=(-q'Ro^2/4k)+c1*lnRo+c2

on solving as (Tint-Ts),we get constnat c1 and C2 as follows

c1=[(Ts-Tint)+(q'/4k)(Ro^2-Ri^2)]/ln(Ro/Ri)

and in putting Ts we get c2 as

c2=Ts+(q'/4k)*Ro^2-[(Ts-Tint)+(q'/4k)(Ro^2-Ri^2)]*[ln(Ro)/ln(Ro/Ri)]

on putting c1 and c2 we get equation as

T=(-q'r^2/4k)+{[(Ts-Tint)+(q'/4k)(Ro^2-Ri^2)]/ln(Ro/Ri)}*lnr+{Ts+(q'/4k)*Ro^2-[(Ts-Tint)+(q'/4k)(Ro^2-Ri^2)]*[ln(Ro)/ln(Ro/Ri)]}

where heat generation rate at any radius r and temperature T is given by

q'=(4k/r^2)*{{[(Ts-Tint)+(q'/4k)(Ro^2-Ri^2)]/ln(Ro/Ri)}*lnr+{Ts+(q'/4k)*Ro^2-[(Ts-Tint)+(q'/4k)(Ro^2-Ri^2)]*[ln(Ro)/ln(Ro/Ri)]}-T}

3)here temperature profile inside cylinder is given by above equation and heat conduction is given by

Q=-k1*Ai*(dT/dr)=-k2*Ao*(dT/dr)

Ai=2*pi*Ri*L

where dT/dr=(-q'*r/2k)+c1/r

hence on putting c1 we get

dT/dr=(-q'*r/2k)+{[(Ts-Tint)+(q'/4k)(Ro^2-Ri^2)]/ln(Ro/Ri)}/r

heat transfer rate is givenby

Qi=-k1*2*pi*Ri*L*(-q'*r/2k)+{[(Ts-Tint)+(q'/4k)(Ro^2-Ri^2)]/ln(Ro/Ri)}/r

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