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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


HumanistEngineer January 15, 2022 05:54

I found the error:

The first heat balance shall be replaced with:

C_in*(dT_in/dt)=- (T_in - T_w1) / R_in

agustinvo July 29, 2022 20:03

If you are using Python you could use Scipy to solve your state-space equations.


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