%ONE_DQ% Chemical and Fuels Engineering 6453, Heat Transfer% University of Utah % Prof. Geoff Silcox

% 1-D fully developed duct-flow heat transfer in a slit of height d.% I am using a node-centered grid with n control volumes that are % all the same size. The nodes are midway between the boundaries of% the CVs. I have used a dimensional formulation. The flux along% the top and bottom of slit is uniform and given.

% 28 March 2004

%**************************************************************************

close all;clear all;clc;

% Parameters for duct and fluid flow.

dpdz = -1; % Pressure drop in z-direction, kPa.d = 1; % Height of duct, m.mu = 1; % Viscosity of fluid, kg/(m s).wb = 0; % Velocity at bottom, m/s.wt = 0; % Velocity at top, m/s.

% Parameters for heat transfer.

qw = 1; % Specified flux of energy at top and bottom of duct, W/m^2.k = 1; % Thermal conductivity of fluid, W/(m K).tb = 0; % Temperature at bottom of duct, C. This is arbitrary.

% Parameters for solution

n = 20; % Number of control volumes of equal width.

% Construct grid.

dely= d / n; % Distance between CV boundaries.% yb are the positions of the n + 1 boundaries.yb = linspace(0,d,n + 1);% y are positions of the n + 2 nodes.y = yb + 0.5 * dely;y = [0 y];y(n + 2) = d;

% Set up tridiagonal matrix for system of equations, Aw = b.% Velocity in the z-direction is given by w = A\b.% A = is a nxn tridiagonal matrix. This sparse, diagonal matrix is % defined using the MATLAB function "spdiags".% w = column vector of unknown velocities.% b = column vector giving right hand side of equation.% am(i) multiplies w(i-1). Define am, a, ap as column vectors.% a(i) multiplies w(i).% ap(i) multiplies w(i+1).

am = mu / dely .* ones(n,1);am(1) = 0;am(n) = 4 * mu / (3 * dely);ap = am;

ap(1) = 4 * mu / (3 * dely);ap(n) = 0;a = - (am + ap);a(1) = - (4 * mu / dely);a(n) = a(1);b = dpdz * dely .* ones(n,1);

% When using "spdiags", we must truncate and extend am and ap. % We need to go through these acrobatics because of the way "spdiags"% is defined. Note ap, a, and am must be column vectors.

A = spdiags([[am(2:n);0] a [0;ap(1:n-1)]], [-1 0 1], n, n);

% Solve.

w = A\b;

% Calculate hydraulic diameter, dh, mean fluid velocity, wbar, cfRe.

dh = 2 * d;wbar = sum(w) / n;cfRe = -dpdz * dh ^ 2 / (2 * wbar * mu); % Print to screen.

fprintf('Number of CVs = %3.f\n', n)fprintf('cfRe = %g\n', cfRe)fprintf('wbar, m/s = %g\n', wbar)

% The end of velocity calculation. Start heat transfer calculation.

% Set up tridiagonal matrix for system of equations, At = b.% Temperature is given by t = A\b.% A = is a nxn tridiagonal matrix. This sparse, diagonal matrix is % defined using the MATLAB function "spdiags".% t = column vector of unknown temperatures.% b = column vector giving right hand side of equation.% am(i) multiplies t(i-1). Define am, a, ap as column vectors.% a(i) multiplies t(i).% ap(i) multiplies t(i+1).

am = k / dely .* ones(n,1);am(1) = 0;am(n) = 4 * k / (3 * dely);ap = am;ap(1) = 4 * k / (3 * dely);ap(n) = 0;a = - 2 * k / dely .* ones(n,1);a(1) = - 4 * k / dely;a(n) = - 4 * k / dely;b = (2 * qw * dely / (d * wbar)) .* w;b(1) = b(1) - 8 * k * tb / (3 * dely);tt = tb;b(n) = b(n) - 8 * k * tt / (3 * dely);

A = spdiags([[am(2:n);0] a [0;ap(1:n-1)]], [-1 0 1], n, n);

% Solve.

t = A\b;

% Calculate dimensionless temperature, theta, mean fluid temperature, tbar, % heat transfer coefficient, and nu.

tbar = w' * t * dely / (d * wbar);theta = (t - tb) ./ (tbar - tb); % dimensionless temperatureh = qw / (tb - tbar);nu = h * dh / k;

% As a check, flux must equal qw.

flux = k /(3 * dely) * (8 * tb - 9 * t(1) + t(2));

% Plot theta versus y

theta = [0; theta; (tt - tb) / (tbar - tb)];plot(y,theta);title('Dimensionless temperature profile','FontSize',12)xlabel('Distance from bottom, m','FontSize',12)ylabel('Temperature','FontSize',12) % Print temperatures (dimensionless) to screen. fprintf('Height, m Dim Temperature\n')fprintf('%8.5f %g\n',[y; theta'])

fprintf('Number of CVs = %3.f\n', n)fprintf('Nu = %g\n', nu)fprintf('tbar, C = %g\n', tbar)fprintf('flux, W/m^2 = %g\n', flux)

% The End.