Nothing Special   »   [go: up one dir, main page]
Scribd Logo
Upload

Welcome to Scribd!

  • 20 views

    Fazal - CFD - Assignment - Example 7.2

    Uploaded by

    Fazal Chawla
    This document contains MATLAB code to solve a 2D steady state heat diffusion equation using the Gauss-Seidel and TDMA methods. It includes code to discretize the PDE using finite volumes on a 12 node grid, set up the coefficient matrix and source vector, and implement the Gauss-Seidel and TDMA line-by-line solvers. The code provides the numerical solution and plots the temperature distribution across the domain.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    Fazal - CFD - Assignment - Example 7.2

    Uploaded by

    Fazal Chawla
    20 views9 pages
    This document contains MATLAB code to solve a 2D steady state heat diffusion equation using the Gauss-Seidel and TDMA methods. It includes code to discretize the PDE using finite volumes on a 12 node grid, set up the coefficient matrix and source vector, and implement the Gauss-Seidel and TDMA line-by-line solvers. The code provides the numerical solution and plots the temperature distribution across the domain.

    Original Description:

    Original Title

    Fazal_CFD_Assignment_ example 7.2

    Copyright

    © © All Rights Reserved

    Available Formats

    PDF, TXT or read online from Scribd

    Did you find this document useful?

    Is this content inappropriate?

    This document contains MATLAB code to solve a 2D steady state heat diffusion equation using the Gauss-Seidel and TDMA methods. It includes code to discretize the PDE using finite volumes on a 12 node grid, set up the coefficient matrix and source vector, and implement the Gauss-Seidel and TDMA line-by-line solvers. The code provides the numerical solution and plots the temperature distribution across the domain.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    20 views9 pages

    Fazal - CFD - Assignment - Example 7.2

    Uploaded by

    Fazal Chawla
    This document contains MATLAB code to solve a 2D steady state heat diffusion equation using the Gauss-Seidel and TDMA methods. It includes code to discretize the PDE using finite volumes on a 12 node grid, set up the coefficient matrix and source vector, and implement the Gauss-Seidel and TDMA line-by-line solvers. The code provides the numerical solution and plots the temperature distribution across the domain.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 9

    Department of Chemical Engineering.

    Assignment:
    “MATLAB Code for Example 7.2:
    2-D Steady State Diffusion using Gauss
    Seidel and TDMA (Line by Line).
    Subject: Computational Fluid Dynamics.

    Submitted by:
    Fazal e Rabbi (MS- Process Engineering).
    10-24-2022
    1
    Gauss Seidel and TDMA (Line by Line).

    INTRODUCTION:
    The Gauss-Seidel method and TDMA-2D method was applied to solve the Two-Dimensional
    Heat Diffusion equation with sources and insulation. Firstly, the differential equation was
    discretized using Finite Volume Method and algebraic equations for 12 nodes were obtained,
    according to the method for 2-D diffusion discussed in class by using (East-West, North-South
    scheme). In order to solve the following matrix equation:
    𝐴12𝑥12 𝑋12𝑥1 = 𝐵12𝑋1

    Fig.1: Example 7.2 Data.

    Fig.2: Solution provided for comparison..


    2
    Gauss Seidel and TDMA (Line by Line).

    MATLAB CODE FOR GAUSS SEIDEL SCHEME:


    %Example 7.2 (2-D Steady State Diffusion) Gauss-Seidel.
    clear all;
    clc;
    %Data:
    L=0.4;
    W=0.3;
    dz=0.01;
    k=1000;
    TN=100;
    qW=500*1000;
    qS=0; qE=0;

    %Discretization:
    Nx=3;
    Ny=4;
    dx=W/Nx;
    dy=L/Ny;
    Ae=dy*dz; Aw=Ae;
    An=dx*dz; As=An;

    Nt=Nx*Ny;
    A=zeros(Nt,Nt);
    B=zeros(Nt,1);

    %Coefficients (East, West, North, South) and source terms):


    aE=k*Ae/dx;
    aW=k*Aw/dx;
    aN=k*An/dy;
    aS=k*As/dy;

    suW=qW*Aw;
    suN=TN*k*An/(dy/2);
    sp=-k*An/(dy/2);

    %For interior points (Nodes 6 and 7) No source:

    for i=6:1:7;
    A(i,i)=aE+aW+aN+aS;
    A(i,i+4)=-aE;
    A(i,i-4)=-aW;
    A(i,i+1)=-aN;
    A(i,i-1)=-aS;
    end

    %For node 1 (q(W) source from west and inulated south):

    A(1,1)=aE+aN;
    A(1,5)=-aE;
    A(1,2)=-aN;
    B(1)=suW;
    3
    Gauss Seidel and TDMA (Line by Line).
    %For nodes 2 and 3 (q(W) source from west):

    for i=2:1:3
    A(i,i)=aE+aN+aS;
    A(i,i+4)=-aE;
    A(i,i+1)=-aN;
    A(i,i-1)=-aS;
    B(i)=suW;
    end

    %For node 4 (q(W) source from west and T(N) from north):

    A(4,4)=aE+aS-sp;
    A(4,8)=-aE;
    A(4,3)=-aS;
    B(4)=suW+suN;

    %For node 5 (south insulated):

    A(5,5)=aE+aW+aN;
    A(5,9)=-aE;
    A(5,1)=-aW;
    A(5,6)=-aN;

    %For node 8 (T(N) source from north):

    A(8,8)=aE+aW+aS-sp;
    A(8,12)=-aE;
    A(8,4)=-aW;
    A(8,7)=-aS;
    B(8)=suN;

    %For node 9 (east and south insulated):

    A(9,9)=aW+aN;
    A(9,5)=-aW;
    A(9,10)=-aN;

    %For node 10 and 11 (east insulated):

    for i=10:1:11
    A(i,i)=aW+aN+aS;
    A(i,i-4)=-aW;
    A(i,i+1)=-aN;
    A(i,i-1)=-aS;
    end

    %For node 12 (T(N) source from north and east insulated):

    A(12,12)=aW+aS-sp;
    A(12,8)=-aW;
    A(12,11)=-aS;
    B(12)=suN;
    4
    Gauss Seidel and TDMA (Line by Line).
    A
    B

    U=Gauss_Seidel(A,B);

    T=zeros(Ny,Nx);

    T(1,1)=U(4);
    T(1,2)=U(8);
    T(1,3)=U(12);
    T(2,1)=U(3);
    T(2,2)=U(7);
    T(2,3)=U(11);
    T(3,1)=U(2);
    T(3,2)=U(6);
    T(3,3)=U(10);
    T(4,1)=U(1);
    T(4,2)=U(5);
    T(4,3)=U(9);

    X=(dx/2):dx:W;
    Y=(dy/2):dy:L;

    surf(X,Y,T);

    Results:

    Fig.3: Gauss-Seidel solution (Convergence in 140 iterations).


    5
    Gauss Seidel and TDMA (Line by Line).

    MATLAB CODE FOR TDMA (LINE BY LINE) SCHEME:


    %Example 7.2 (2-D Steady State Diffusion) TDMA Line-by-Line.
    clear all;
    clc;
    %Data:
    L=0.4;
    W=0.3;
    dz=0.01;
    k=1000;
    TN=100;
    qW=500*1000;
    qS=0; qE=0;

    %Discretization:
    Nx=3;
    Ny=4;
    dx=W/Nx;
    dy=L/Ny;
    Ae=dy*dz; Aw=Ae;
    An=dx*dz; As=An;

    Nt=Nx*Ny;
    A=zeros(Nt,Nt);
    B=zeros(Nt,1);

    %Coefficients (East, West, North, South) and source terms):


    aE=k*Ae/dx;
    aW=k*Aw/dx;
    aN=k*An/dy;
    aS=k*As/dy;

    suW=qW*Aw;
    suN=TN*k*An/(dy/2);
    sp=-k*An/(dy/2);

    %For interior points (Nodes 6 and 7) No source:

    for i=6:1:7;
    A(i,i)=aE+aW+aN+aS;
    A(i,i+4)=-aE;
    A(i,i-4)=-aW;
    A(i,i+1)=-aN;
    A(i,i-1)=-aS;
    end

    %For node 1 (q(W) source from west and inulated south):

    A(1,1)=aE+aN;
    A(1,5)=-aE;
    A(1,2)=-aN;
    B(1)=suW;
    6
    Gauss Seidel and TDMA (Line by Line).
    %For nodes 2 and 3 (q(W) source from west):

    for i=2:1:3
    A(i,i)=aE+aN+aS;
    A(i,i+4)=-aE;
    A(i,i+1)=-aN;
    A(i,i-1)=-aS;
    B(i)=suW;
    end

    %For node 4 (q(W) source from west and T(N) from north):

    A(4,4)=aE+aS-sp;
    A(4,8)=-aE;
    A(4,3)=-aS;
    B(4)=suW+suN;

    %For node 5 (south insulated):

    A(5,5)=aE+aW+aN;
    A(5,9)=-aE;
    A(5,1)=-aW;
    A(5,6)=-aN;

    %For node 8 (T(N) source from north):

    A(8,8)=aE+aW+aS-sp;
    A(8,12)=-aE;
    A(8,4)=-aW;
    A(8,7)=-aS;
    B(8)=suN;

    %For node 9 (east and south insulated):

    A(9,9)=aW+aN;
    A(9,5)=-aW;
    A(9,10)=-aN;

    %For node 10 and 11 (east insulated):

    for i=10:1:11
    A(i,i)=aW+aN+aS;
    A(i,i-4)=-aW;
    A(i,i+1)=-aN;
    A(i,i-1)=-aS;
    end

    %For node 12 (T(N) source from north and east insulated):

    A(12,12)=aW+aS-sp;
    A(12,8)=-aW;
    A(12,11)=-aS;
    B(12)=suN;
    7
    Gauss Seidel and TDMA (Line by Line).
    A
    B

    %U=Gauss_Seidel(A,B);

    %TDMA Line by line:

    alpha = zeros(Nt,1);
    beta = zeros(Nt,1);
    Aj = zeros(Nt, 1) ;
    C=zeros(Nt,1);
    C_prime = zeros(Nt, 1) ;
    % getting the elements
    % d - diagonal elements $ b - below diagonal $ a - above diagonal
    D = diag(A);
    beta(2:Nt) = -diag(A,-1);
    alpha(1:Nt-1) = -diag(A,+1);

    T=zeros(Ny,Nx);
    k=0;
    tol=0.01;
    eval=ones(Ny,Nx);

    while eval>tol
    T_old=T;
    %For line 1 (Nodes 1,2,3,4) aw=0, and East temperatures initial guess.

    %For point 1 only:


    C(1)=B(1)+aE*T(5)
    Aj(1) = alpha(1)/D(1);
    C_prime0=1; Aj0=1; %Any arbitrary values as these. do not exist
    C_prime(1) = (C(1)+beta(1)*C_prime0)/(D(i)-beta(i)*Aj0) ;

    for i=2:1:4
    C(i)=B(i)+aE*T(i+4);
    Aj(i)=alpha(i)/(D(i)-beta(i)*Aj(i-1));
    C_prime(i) = (C(i)+beta(i)*C_prime(i-1))/(D(i)-beta(i)*Aj(i-1)) ;
    end
    for i=4:-1:1
    T(i,1)=Aj(i)*T(i+1)+C_prime(i);
    end

    %For line 2 (Nodes 5,6,7,8) East temperature guess, west temperature


    %calculated (1,2,3,4):

    for i=5:1:8
    C(i)=B(i)+aW*T(i-4)+aE*T(i+4);
    Aj(i)=alpha(i)/(D(i)-beta(i)*Aj(i-1));
    C_prime(i) = (C(i)+beta(i)*C_prime(i-1))/(D(i)-beta(i)*Aj(i-1)) ;
    end
    for i=8:-1:5
    T(i-4,2)=Aj(i)*T(i+1)+C_prime(i);
    end
    8
    Gauss Seidel and TDMA (Line by Line).
    %For line 3 (Nodes 9,10,11,12)aE=0, West temperatures calculated (5,6,7,8)

    for i=9:1:12
    C(i)=B(i)+aW*T(i-4);
    Aj(i)=alpha(i)/(D(i)-beta(i)*Aj(i-1));
    C_prime(i) = (C(i)+beta(i)*C_prime(i-1))/(D(i)-beta(i)*Aj(i-1)) ;
    end
    T13=1; %Does not exist, any arbitrary value.
    T(4,3)=Aj(12)*T13+C_prime(12);
    for i=11:-1:9
    T(i-8,3)=Aj(i)*T(i+1)+C_prime(i);
    end
    eval=norm(T-T_old);
    k=k+1;
    end

    fprintf('Solution converged in %d iterations by TDMA 2D method',k);

    T_order= flipud(T)

    X=(dx/2):dx:W;
    Y=(dy/2):dy:L;

    surf(X,Y,T_order);

    Results:

    Fig.4: TDMA-2D Solution (Convergence in 45 iterations).

    You might also like