Heat Transfer: 2D, Steady State, qdot = 0

*This program is still a working progress, I would like to add a way of increasing the refinement at specific points. This will likely be done by making the algorithm run, check areas for high temperature gradients and then run again at those location.

# 2D Heat Conduction - Finite Difference Method

# Steady State, Uniform Properties, No internal heat generation

from matplotlib import pyplot as plt

import numpy as np

# ================================================ #

# case 1 - 4 nodes, del(x) = 0.25 [m], q_dot = 0

# - T_top = 100

# - T_bottom = 300

# - T_right = 200

# - T_left = 50

# ------------------------------------------------ #

# [T1] [0 1 1 0] [T1] [100 + 50.]

# |T2| = 1/4 |1 0 0 1| |T2| + 1/4 |100 + 200|

# |T3| |1 0 0 1| |T3| |300 + 50.|

# [T4] [0 1 1 0] [T4] [300 + 200]

# ------------------------------------------------ #

def create_A_matrix(m):

C = np.zeros((m, m))

C[0, 0:m - 1] = 1

C[m - 1, 1:m - 1] = 1

C[0:m - 1, 0] = 1

C[1:m - 1, m - 1] = 1

for i in range(m):

C[i, i] = -m

C = 1/m * C

return C

def final_temp_distribution_matrix(n, T_t, T_b, T_r, T_l):

Temp_final = np.zeros((n, n))

# Wall Temps

Temp_final[0, 1:n - 1] = T_t

Temp_final[n - 1, 1:n - 1] = T_b

Temp_final[1:n - 1, 0] = T_l

Temp_final[1:n - 1, n - 1] = T_r

Temp_final[0, 0] = (T_t + T_l) / 2

Temp_final[0, n-1] = (T_t + T_r) / 2

Temp_final[n-1, 0] = (T_b + T_l) / 2

Temp_final[n-1, n-1] = (T_b + T_r) / 2

# Calculated Node Temps

counter = 1

for i in range(1, n - 1):

for j in range(1, n - 1):

Temp_final[i][j] = T[counter - 1][0]

counter += 1

return Temp_final

def display_temp_distribution(Temp_final):

plt.matshow(Temp_final, cmap='magma', interpolation='gaussian')

for (i, j), z in np.ndenumerate(Temp_final):

plt.text(j, i, '{:0.1f}'.format(z), ha='center', va='center',

bbox=dict(boxstyle='round', facecolor='white', edgecolor='0.3'))

plt.title("Temperature Distribution in 2D Flat Plate")

plt.show()

refinement = 1

n = 4 * refinement

A = create_A_matrix(n)

T_t, T_b, T_r, T_l = 100, 300, 200, 50

B = 1/n * np.array([-(T_t+T_l), -(T_t+T_r), -(T_b+T_l), -(T_b+T_r)])

B.shape = (n, 1)

A_inv = np.linalg.inv(A)

T = np.matmul(A_inv, B)

deg = '\N{DEGREE SIGN}'

for i in range(n):

print("Temperature [", deg, "C] at node ", i, " = ", round(T[i][0], 2), sep='')

Temp_final = final_temp_distribution_matrix(n, T_t, T_b, T_r, T_l)

display_temp_distribution(Temp_final)