- #1

kartini99

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- TL;DR Summary
- Problem finding the correct temperature value for unexposed heat by using separation variables.

Hello everyone,

I'm trying to solve the transient heat transfer problem within the ID wall.

The material is steel, and it is isotropic.

The properties are given below :

L = 5 mm

qin = 0

Tinf = 100 deg C

Tini = 20 deg C

rho = 7850 kg/m3

cp = 460 W/Kg.K

k = 45.8 W/m.K

h = 20 W/m^2.K

alpha = k / rho*cp

t = 360 second

Fo = alpha * t / (L**2)

By using the attached formulation, I got the root of the function lambda is 0.03303492.

However, I increased the time, and the temperature in the unexposed wall was reduced. and when I change the time to 1 (one), the value shows almost like the T imposed, which is 100. Kind of the opposite way, Kindly any body help me with that. I also attached the script below

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I'm trying to solve the transient heat transfer problem within the ID wall.

The material is steel, and it is isotropic.

The properties are given below :

L = 5 mm

qin = 0

Tinf = 100 deg C

Tini = 20 deg C

rho = 7850 kg/m3

cp = 460 W/Kg.K

k = 45.8 W/m.K

h = 20 W/m^2.K

alpha = k / rho*cp

t = 360 second

Fo = alpha * t / (L**2)

By using the attached formulation, I got the root of the function lambda is 0.03303492.

However, I increased the time, and the temperature in the unexposed wall was reduced. and when I change the time to 1 (one), the value shows almost like the T imposed, which is 100. Kind of the opposite way, Kindly any body help me with that. I also attached the script below

Python:

```
from scipy.optimize import fsolve
import numpy as np
def xtanxmbi(x, Bi):
# tan = np.tan(x*np.pi/180)
# return np.arctan(tan)*Bi-x
# return (1/np.tan(x))*Bi-x
return (np.cos(x)/np.sin(x))*Bi-x
# W = 1
# H = 0.1
L = 2.5 / 1000
qin = 0
Tinf = 100
Tini = 20
rhocp = 7850 * 460
k = 45.8
h = 20
alpha = k / rhocp
t = 360
Fo = alpha * t / (L**2)
print(f"The Fo is = {Fo}")
N = 6
n = 6
Bi = h * L / k
theta = np.zeros((n, N))
x0 = np.arange(n) * np.pi + 0.3
ans = fsolve(xtanxmbi, x0=x0, args=(Bi), full_output=True)
if ans[2] != 1:
print("error solving roots of transcendental equation")
roots = ans[0]
Bi5 = Bi
print(f"Bi = {Bi5} , {roots}")
for i in np.arange(n):
An = 2* np.sin(roots[I]) / (roots[I]+np.sin(roots[I])*np.cos(roots[I]))
print(f"An = {An}")
x = 1
theta[I] = An * np.exp((-roots[I] ** 2 )* Fo) * np.cos(roots[I] * x)
print(f"value of theta = {theta}")
print(type(theta))
# Cn_0 = 4.0 * np.sin(roots[0]) / (2.0 * roots[0] + np.sin(2.0 * roots[0]))
# print(f"value of Cn is ={Cn_0}")
# theta_0 = Cn_0 * np.exp(-roots[0] ** 2 * Fo) * np.cos(roots[0] * x)
# print(f"value of theta is ={theta_0}")
sum_theta = theta[0, 0]
# print(sum_theta)
T = Tini + (Tinf - Tini) * sum_theta
print(f"value T={T}")
```

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