| HumanistEngineer |
January 14, 2022 03:37 |
Transient Heat Transfer | Simple One-Layer Wall
Hello All,
I try to build a transient heat transfer model for a simple one-layer wall by use of the Resistance-Capacitance (RC) approach. However, the transient temperature distribution shows weird numbers in time. Would you please check my approach and code?
https://www.linkpicture.com/q/Webp.n...zeimage_28.jpg
Heat balance for each capacitance can be written as:
https://www.linkpicture.com/q/2022-0..._53-Window.png
Forward-Euler Discretization (one-dimensional) gives:
https://www.linkpicture.com/q/2022-0..._31-Window.png
At each time step this calculation procedure is run:
https://www.linkpicture.com/q/2022-0..._59-Window.png
Here is the Python code to calculate the temperature distribution in time:
Code:
import numpy as np
""" Input Parameters """
# Wall
height_w = 3 # [m] height of the wall
wide_w = 5 # [m] wide of the wall
thick_w = 0.3 # [m] thickness of the wall
volume_w = height_w * wide_w * thick_w # [m3] volume of the wall
area_w= height_w * wide_w # [m2] wall area
# Room
deep_r = 4 # [m] deepness of room
volume_r = height_w * wide_w * deep_r # [m3] volume of the room air
# Initial Temperature
T_in = 20 # [°C] indoor temperature
T_out = 2 # [°C] outdoor temperature
# Thermal
h_in = 8 # [W/m2K] heat transfer coefficient (indoor)
h_out = 15 # [W/m2K] heat transfer coefficient (outdoor)
# see: Balaji NC et al. Thermal performance of the building walls
k_w = 0.811 # [W/mK] thermal conductivity (brick)
rho_w = 1820 # [kg/m3] density (brick)
Cp_w = 880 # [J/kgK] specific heat capacity (brick)
# Air
rho_a = 1.204 # [kg/m3] density (air at 20 °C)
Cp_a = 1006 # [J/kgK] specific heat capacity (air at 20 °C)
""" Preliminary Calculations """
delta_t = 1 # [s]
C_a = rho_a * Cp_a * volume_r # [J/K]
C_w = rho_w * Cp_w * volume_w # [J/K]
R_in = 1 / (h_in * area_w) # [K/W]
R_out = 1 / (h_out * area_w) # [K/W]
R_w = thick_w / (k_w * area_w) # [K/W]
# print('R_in = ',R_in)
# print('R_out = ',R_out)
# print('R_w = ',R_w)
""" Numeric Constants """
A = np.zeros( (4, 4) )
A[0][0]=1+delta_t/(C_a*R_in)
A[0][1]=-delta_t/(C_a*R_in)
A[1][0]=4*delta_t/(C_w*R_in)
A[1][1]=1-4*delta_t/C_w*(1/R_in+2/R_w)
A[1][2]=8*delta_t/(C_w*R_w)
A[2][1]=4*delta_t/(C_w*R_w)
A[2][2]=1-2*delta_t/C_w*(2/R_w+2/R_w)
A[2][3]=4*delta_t/(C_w*R_w)
A[3][2]=8*delta_t/(C_w*R_w)
A[3][3]=1-4*delta_t/C_w*(2/R_w+1/R_out)
B = np.zeros((4,1))
B[3][0]=4*delta_t*T_out/(C_w*R_out)
""" Numerical Simulation """
T_init=np.ones((4,1))*15
for i in range(1):
T=np.matmul(A,T_init)+B
T_init=T
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