Please solve both questions, thank you. The 1- D and steady conduction equation
ID: 2993404 • Letter: P
Question
Please solve both questions, thank you.
The 1- D and steady conduction equation in cylindrical coordinates (with the radial direction r being the heat flow direction) is k1/r d/dr (rdT/dr) = -q"' Solve this equation for the following conditions: T(ri) = Ti, T{r2) = T2, q" = 0, where T1 and T2 are specified constants and rj and r2 are the inner and outer surfaces of the cylinder; this is a pipe-type configuration. What happens to the solution in part when rx = 0, i.e., a solid pipe? Does this problem even have a solution? What happens to the solution in part a) when ri is finite and nonzero and r2 ? Does this solution exist? A solid cylinder with T(r2) = Ts and q'" = constant. What is the boundary condition at the center of the cylinder? The 1 D and steady conduction equation in spherical coordinates (with the radial direction r again being the heat flow direction) is Solve this equation for the following conditions: T(rj) = 7, T(r infinity) - T2, q'" = 0. Note that the 'domain' (the region to which the DE is applied) is rjExplanation / Answer
2 and 3) d^2(T)/dx^2 + q'''/k = 0
Integrate with x
dT/dx = -(q'''/k)*x +C1
Integrate with x
T(x) = -(q'''/k)*x^2/2 +C1*x + C2
a) T(0) = To T(L) =TL q'''' = 0
T(x) = C1*x+C2
at x = 0
C2 = To
and at x = L
C1 = (TL-To)/L
T(x) = (TL-To)/L*x + To
Heat Flux
q''(x) = -K*dT/dx = -K*C1
q''(0) = -K*To
q''(L) = -K*To
Net Heat Flux = 0
b)T(x) = C1*x+C2
dT(x)/dX = C1
at x = 0
C2 = To
at x = L
C1 = q(s)''/K
T(x) = q(s)''/K*x + To
Heat Flux
q(x) = -K*dT/dx = -q(s)
q(0) = -q(s)
q(L) = -q(s)
Net heat Flux = 0
c)
T(x) = C1*x+C2
C2 = To
C1 = -h/k(TL-Tamb)
T(x) = -h/k(TL-Tamb)*x + To
Heat Flux
q(x) = h*(TL-Tamb)
q(0) = h*(TL-Tamb)
q(L) = h*(TL-Tamb)
Net Heat Flux = 0
d)
T(x) = -(qo'''/k)*x^2/2 +C1*x + C2
At x = 0
C2 = To
At x= L
TL = -(qo'''/k)*L^2/2 +C1*L +To
C1 = (TL-To)/L + (qo'''/k)*L/2
T(x) = -(qo'''/k)*x/2*(x-L) + (TL-To)/L*x + To
Heat Flux
dT/dx = -(qo'''/k)*1/2(2x-L) + (TL-To)/L
q''(x) = -k*{-(qo'''/k)*1/2(2x-L) + (TL-To)/L}
q''(0) = -k*{(qo'''/k)*L/2 + (TL-To)/L}
q''(L) = -k*{-(qo'''/k)*L/2 + (TL-To)/L}
Net heat transfer
q''net = (qo''')*L
e)
T(x) = -(q'''/k)*x^2/2 +C1*x + C2
T(x) = -(qo'''*e(-kx)/k)*x^2/2 +C1*x + C2
dT/dx = -(qo'''*e(-kx)/k)*x + qo'''*k*e(-kx)*1/k*x^2/2 +C1
dT/dx = -(qo'''*e(-kx)/k)*x + qo'''*e(-kx)*x^2/2 +C1
At x = 0
dT/dx = C1 = h*(T(0)-Tamb)
dT/dx = -(qo'''*e(-kx)/k)*x + qo'''*e(-kx)*x^2/2 + h*(T(0)-Tamb)
Heat flux
q''(x) = -k*{-(qo'''*e(-kx)/k)*x + qo'''*e(-kx)*x^2/2 + h*(T(0)-Tamb)}
q''(0) = -k*h*(T(0)-Tamb)
q''(L) = -k*{-(qo'''*e(-kL)/k)*L + qo'''*e(-kL)*L^2/2 + h*(T(0)-Tamb)}
Net Heat Flux
q''(net) = -k*{-(qo'''*e(-kL)/k)*L + qo'''*e(-kL)*L^2/2}