Steady-State Heat Conduction Model using FDM Solution to 2D Laplace Equation

Topic: Finite Difference Method of Laplace Equation in 2D;

Subject: Heat Transfer and Numerical Methods;

Tool: Scilab;

By: Gani Comia, Dec. 2024;

The steady-state heat conduction model can be represented mathematically by Laplace equation. Shown is the Laplace equation in 2D:

$$\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2}\;=\;0$$

The solution, \(T(x,y)\), represents the temperature distribution. The Laplace equation can be discretized using the Finite Difference Method (FDM) to approximate the solution numerically. This article will present the Scilab script of the FDM solution to the Laplace equation.

The FDM solution for a uniform grid, \((\Delta x = \Delta y)\), with the boundary temperatures specified based on Dirichlet conditions, is shown:

$$T_{i,j}\;=\;\frac{T_{i+1,j}+T_{i-1,j}+T_{i,j+1}+T_{i,j-1}}{4}$$

The final value of  \(T(x,y)\) is the result of iteration based on the convergence criterion:

$$max\left\vert T_{i,j}^{new}\;-\;T_{i,j}^{old}\right\vert\;<\;\epsilon$$

Consider a sample 2D domain of dimensions 30x30 units with the following constant temperature conditions. 

Fig. 1. Sample 2D Domain with Constant Temperature.

Shown below is the temperature profile as represented by \(T(x,y)\) using the FDM solution to the Laplace equation.

Fig. 2. Temperature Profile \(T(x,y)\).

The Scilab script as a toolbox can be used to implement the FDM numerical solution for the steady-state heat conduction model represented mathematically using the Laplace equation.

Scilab Script

// Copyright (C) 2024 - Gani Comia
// Date of creation: 8 Dec 2024
// FDM Solution to 2D Laplace Eq, d2T/dx2 + d2T/dy2 = 0
clear;clc;
// domain and grid parameters
Lx = 30;                    // units, domain length at x-axis
Ly = 30;                    // units, domain length at y-axis
dx = 1;                     // grid spacing at x-axis
dy = 1;                     // grid spacing at y-axis
nx = Lx./dx;                // no. of grid points in x-direction
ny = Ly./dy;                // no. of grid points in y-direction
max_iter = 1000;            // max no. of iterations
tolerance = 1e-6;           // convergence tol.

// temperature field initial value
T = zeros(nx, ny);
// boundary value conditions
    T(:,1) = 0;             // T = 0 C @ y = 1
T(1,:) = 0;                 // T = 0 C @ x = 1
T(:,ny) = 0;                // T = 0 deg C @ y = 30
T(nx,:) = 100;              // T = 100 deg C @ x = 30

// finite difference method (Jacobi iteration)
for iter = 1:max_iter
    Tnew = T;
    for i = 2:nx-1
        for j = 2:ny-1
            Tnew(i,j)=0.25*(T(i+1,j)+T(i-1,j)+T(i,j+1)+T(i,j-1));
        end
    end
    // convergence check
    if norm(Tnew - T) < tolerance then
        break;
    end
    T = Tnew;
end
// calculation status 
mprintf("\n\tCalculation completed!\n")

// plot properties
clf;
f = gcf()
f.figure_size = [700,650]
ax = gca()
ax.tight_limits = ["on" "on" "off"];

// plotting results
surf(T);
title(["$Temperature\;Profile$","$T(x,y)$"],"fontsize",4);
xlabel("x-axis");
ylabel("y-axis");
zlabel("$T$");
colormap(jet);
colorbar(0,100);

Feel free to comment for inquiry, clarification, or suggestion for improvement. Drop your email to request the soft copy of the file.

Curve Fitting with Polynomials using Python

Topic: Method of Curve Fitting with Polynomial Functions;

Subject: Numerical Methods;

Tool: Python;

By: Gani Comia, Dec. 2024;

Curve fitting with polynomials finds a polynomial equation of a given degree that approximates the set of data points. Curve fitting is used in the following applications:

• Data Smoothing
• Predictive Modelling
• Scientific Analysis

The general form of a polynomial function of degree n is:

$$P(x)\;=\;{a_n\,x^n}+{a_{n-1}\,x^{n-1}}+\ldots+{a_1\,x}+{a_0}$$

This article shows a Python script as toolbox to implement the method of curve fitting with polynomials. The script requires numerical data for both dependent and independent variables. For a given sample of data points in Fig.1., an approximate polynomial curve of 3rd degree shown in Fig.2. can be calculated and plotted using Python script.

$$xdata = [ 0 \quad 1 \quad 2 \quad 3 \quad 4 \quad 5 \quad 6 \quad 7 ]$$

$$ydata = [ 5 \quad 10 \quad 60 \quad 50 \quad 130 \quad 220 \quad 300 \quad 280 ]$$

Fig.1. Sample Data Points for Approximating Polynomial Curve.

Below is the approximate polynomial curve that closely fit the given data points.

Fig.2. Polynomial Curve of 3rd Degree from Data Points.

Python Script

# -*- coding: utf-8 -*-
"""
Created on Mon Nov  7 21:08:48 2024
@author: Gani Comia
"""
'''
Curve Fitting with Polynomials for a given xdata and ydata
'''
import numpy as np
import matplotlib.pyplot as plt

# given: xdata and ydata
xdata = np.array([0. , 1. , 2. , 3. , 4. , 5. , 6. , 7.])
ydata = np.array([5. , 10. , 60. , 50. , 130. , 220. , 300. , 280.])

# solution:
# solving for coefficients of polynomial fit for cubic (degree=3)
# z is an array of coefficients , highest first , i.e.
# z = np.array ([c_3, c_2 , c_1 , c_0])
z = np.polyfit(xdata, ydata, 3)
print("\n",z,"\n") 

# ‘poly1d‘ returns the polynomial function from coefficients    
p = np.poly1d(z)
print(p)

# polynomial's curve data points
xs = [0.1 * i for i in range (70)]  # from 0 to 7, step=0.1
ys = [p(x) for x in xs]   # evaluate p(x) for all x in list xs

# creating plot of data points and polynomial curve for visualization
plt.plot(xdata, ydata, 'o', label='data points') # data points
plt.plot(xs, ys, label='fitted curve')    # polynomial fit
plt.title('Curve Fitting with Polynomials')
plt.ylabel('y-value')
plt.xlabel('x-value')
plt.legend()
plt.grid()
plt.show()

On the above script, Python Numpy’s utilities, np.polyfit(), returns the polynomial coefficients of the given degree that approximately fits on the given data points.  Numpy’s np.poly1d() creates a polynomial functions for plotting and calculation.

Feel free to comment for inquiry, clarification, or suggestion for improvement. Drop your email to request the softcopy of the file.

Temperature Profile of Heating Device using Data Logger and Scilab

Topic: Heating Device Temperature Profiling and Analysis;

Subject: Heat Transfer;

Tool: Temperature Data Logger and Scilab;

By: Gani Comia, Dec. 2024;

Data logger nowadays has a built-in software and computer interface to visualize in real-time as in this case the temperature profile of a device being monitored. However, for data analysis and reporting, a software like Scilab comes into picture.

Presented in this article is an example of visual report of temperature data over time. Data points from the logger can be extracted in the form of .CSV file. Transforming this data may sometimes be needed before an analysis be taken. The Scilab scripting facilitates the analysis and presentation of data for reporting.

Fig.1. Temperature Profile using Scilab

Shown below is the script to generate the graph of the data points.

Scilab Script:

// Copyright (C) 2024 - Gani Comia
// Date of creation: 6 Dec 2024
// Heating Oven Temperature Profile using Data Logger & Scilab
clear;clc;
// data extraction
clear importdata;

function [data]=importdata(filename)
    data = csvRead(filename, ",", ".", "double")
endfunction

[A] = importdata("temp_profile_after_repair_xform.csv");

// data transformation
time = A(2:$,13);
ovenWallTemp = A(2:$,2);
partTemp = A(2:$,9);
// average temperature
avgWallTemp = mean(ovenWallTemp)
avgPartTemp = mean(partTemp)

// data visualization
clf;
f=gcf();
f.figure_size=[900,600];
plot(time,ovenWallTemp,"-b","linewidth",1.75)
plot(time,partTemp,"--r","linewidth",1)
legend(["Oven Wall","Workpiece"],"in_upper_right")
title("Heating Oven Temp Profile from Data Logger")
xlabel("Time [ min ]")
ylabel("Temperature [ deg C ]")

a=gca(); 
// x & y-axis range
a.data_bounds = [0 230;780 250]; 

// additional plot properties
t = 0:60:840;
temp = [230 232 250];

// hourly vertical lines
for i = 2:length(t)
    plot([t(i),t(i)],[temp(1),temp(3)],"--y","linewidth",1)
end

// target temperature horizontal lines
plot([t(1),t($)],[temp(2),temp(2)],"--k","linewidth",1)
xstring(5,232,"Std = 232 C")
xstring(625,232,"https://gani-mech-toolbox.blogspot.com")

// hourly label 
minute = 60;
hour = 1;
for i = 2:length(t)
    note = [string(hour),"hour(s)"]
    xstring(minute,235,note,-90)
    minute = minute + 60;
    hour = hour + 1;
end

// average oven temp lines
plot([t(1),t($)],[avgWallTemp,avgWallTemp],"--b","linewidth",1)
noteWall = ["Avg Wall Temp:",string(round(avgWallTemp)),"C"]
xstring(5,avgWallTemp+0.8,noteWall)

// average part temp lines
plot([t(1),t($)],[avgPartTemp,avgPartTemp],"--r","linewidth",1)
notePart = ["Avg Part Temp:",string(round(avgPartTemp)),"C"]
xstring(5,avgPartTemp+0.5,notePart)

This toolbox of Scilab script can be handy reference to analyze data points and present it in the form of graphs for better understanding of numerical data.

Feel free to comment for inquiry, clarification, or suggestion for improvement. Drop your email to request the softcopy of the file.