Problem no - 7 Heat transfer problem in a spherical cylinder
According to Fourier law -
Q r = qr = - k.* A* dt/dr _----- ( 1)
Putting the area of the sphere and integrating between , r= r1 and r= r2 and T1 & T2.
Qr /4π| -- 1/r | * [ r= r1 to r= r2 = -- k |T| * [ T= T1 TO T2 ] ------ ( 2)
Qr [ ( r2 - r1 )/ 4π .r1.r2 ]= - k [ T2 -T1] --------- ( 3)
Qr = qr = 4.π. k *. r2*.r1 [ T2- T1 ] / ( r2-r1 ) -------- ( 4)
Assumption -
1 . system is in steady state.
2 thermal conductivity k , is constant.
3. system follows Fourier law of the heat conduction.
Non -zero heat flux component , since temperature changing in r - direction only , qr is present , the control volume , dr , as shown -
Heat flux entering the control volume at r= r = qr = 4.π.r^2. qr|r
Heat flux leaving at r = r +Δr= 4πr^2. Qr | r + Δr
Any heat source or sink is not present in the control volume and work done on energy balance is reduced to zero.
[ qr.*4π*r^2|r -- qr* 4*π*r^2|r+dr]= 0 ----- (5 )
Diving the equation ( 5 ) by value of control volume - 4.π.r2*Δr and take the
d/dr ( r2.qr )= 0---------(6)
and integrating the equation no 6 -
qr = C1 /r2 --- ( 7)
where C1 = integration constant
by putting the Fourier law of heat conduction and integrating -
T = C1 /K.r + C2
As subjected to boundary conditions -
At r = R , T = T1 and r = R2, T= T2
Using the above boundary conditions and evaluating the constant of integration -
Qr = qr = 1/r2*k * ( T2 - T1) / ( 1 /R1 - 1/ R2)
IN THE SPHERICAL CO-ORDINATES-
1/r2 *∂/∂r * ( r2* ∂T/∂r ) + 1/ r2. sin?* (sin?.∂r/∂?) + 1/r2*sin^2?.∂2T/∂r2 +q/k = 1/?* ∂T/∂t
At boundary conditions-
T ( y = yi ) = Ti
T ( y = yo ) = To
Hence , T- Ti / To -Ti = ( 1/r1 - 1/r)/(1/r - 1/ro )