Engine oil at 100℃ and velocity of 0.1 m/s flows over both surfaces of a 1 m long flat plate maintained at 20 ℃. Using MATLAB plot the boundary layer thickness and local value of the surface shear stress, convection coefficient, and heat flux as a function of x for 0≤x≤1 m

Answer

Given Data

Temperature of engine Oil 100  

Velocity = 0.1 m/s

Length of surface =1 m

Temperature of that surface= 20

Matlab code for boundary layer thickness

function varargout = boundaryLayerGUI(varargin)

gui_Singleton = 1;

gui_State = struct('gui_Name',       mfilename, ...

    'gui_Singleton',  gui_Singleton, ...

    'gui_OpeningFcn', @boundaryLayerGUI_OpeningFcn, ...

    'gui_OutputFcn',  @boundaryLayerGUI_OutputFcn, ...

    'gui_LayoutFcn',  [] , ...

    'gui_Callback',   []);

if nargin && ischar(varargin{1})

    gui_State.gui_Callback = str2func(varargin{1});

end

if nargout

    [varargout{1:nargout}] = gui_mainfcn(gui_State, varargin{:});

else

    gui_mainfcn(gui_State, varargin{:});

end

function boundaryLayerGUI_OpeningFcn(hObject, eventdata, handles, varargin)

 [X, map] = imread(fullfile( matlabroot,'toolbox','matlab','icons','csh_icon.gif'));

icon = ind2rgb(X,map);

uipushtool(handles.uitoolbar1,'CData',icon,...

                 'TooltipString','Help',...

                 'ClickedCallback',@AppHelp);

H = str2double(get(handles.H,'String'));

L = str2double(get(handles.L,'String'));

set(handles.contour,'XLim',[0 L],'YLim',[0 H])

title(handles.contour,'u/U Velocity Contour')

xlabel(handles.contour,'x (m)'),ylabel(handles.contour,'y (m)')

set(handles.vector,'XLim',[0 L],'YLim',[0 H])

title(handles.vector,'Velocity Profile and Boundary Layer Thickness (0.99U)')

xlabel(handles.vector,'x (m)'),ylabel(handles.vector,'y (m)')

set(handles.strline,'XLim',[0 L],'YLim',[0 H])

title(handles.strline,'Streamlines')

xlabel(handles.strline,'x (m)'),ylabel(handles.strline,'y (m)')

annotation(gcf,'textbox',get(handles.Re,'Position'),....

    'String',{'      Re_x = \rhoUx/\mu       \delta_x ~= 5x/{\surd}Re_x'},...

    'LineStyle','none')

handles.output = hObject;

% Update handles structure

guidata(hObject, handles);

function varargout = boundaryLayerGUI_OutputFcn(hObject, eventdata, handles)

varargout{1} = handles.output;

function boundarylayer(handles)

rho = str2double(get(handles.rho,'String')); % kg/m^3

U = str2double(get(handles.U,'String'));   % m/s

mu = str2double(get(handles.mu,'String')); % kg/(m*s)

nu = mu/rho; % m^2/s

H = str2double(get(handles.H,'String')); % m

L = str2double(get(handles.L,'String')); % m

nx = str2double(get(handles.nx,'String'));

ny = str2double(get(handles.ny,'String'));

dx = L/(nx-1);

dy = H/(ny-1);

[X,Y] = meshgrid(0:dx:L,0:dy:H);

Re_L = U*L/nu;

delta_L = 5/sqrt(U/nu/L);

v = zeros(ny,nx);

u = zeros(ny,nx);

u(:,1) = U; % Incoming

v(:,1) = 0; % Incoming

u(1,:) = 0; % Bottom

v(1,:) = 0; % Bottom

u(ny,:) = U; % Top, free stream

% March in x direction

i = 0;

while i < nx-1

    i = i+1;

    for j = 2:ny-1

        if j == 2

            A(j) = 0;

            B(j) = 2*nu/dy^2 + u(j,i)/dx;

            C(j) = - nu/dy^2 + v(j,i)/2/dy;

            D(j) = u(j,i)^2/dx - (- nu/dy^2 - v(j,i)/2/dy)*u(j-1,i+1);

        else if j > 2 && j < ny-1

            A(j) = - nu/dy^2 - v(j,i)/2/dy;

            B(j) = 2*nu/dy^2 + u(j,i)/dx;

            C(j) = - nu/dy^2 + v(j,i)/2/dy;

            D(j) = u(j,i)^2/dx ;

        else if j == ny-1

            A(j) = - nu/dy^2 - v(j,i)/2/dy;

            B(j) = 2*nu/dy^2 + u(j,i)/dx;

            C(j) = 0;

            D(j) = u(j,i)^2/dx - (- nu/dy^2 + v(j,i)/2/dy)*u(j+1,i+1);

        end

    end

    usol = tdma(A(2:end),B(2:end),C(2:end),D(2:end));

    u(2:ny-1,i+1) = usol;

    % solve for v(j,i+1) based on known u

    for j = 2:ny

        v(j,i+1) = v(j-1,i+1) - dy/2/dx*(u(j,i+1)-u(j,i)+u(j-1,i+1)-u(j-1,i));

    end

end

annotation(gcf,'textbox',get(handles.Re,'Position'),...

    'LineStyle','none','BackgroundColor',[0.941 0.941 0.941]);

annotation(gcf,'textbox',get(handles.Re,'Position'),....

    'String',{'      Re_x = \rhoUx/\mu       \delta_x ~= 5x/{\surd}Re_x',...

             ['      Re_L = ',num2str(Re_L),'       \delta_L ~=',num2str(delta_L)]},...

    'LineStyle','none');

% u/U velocity contour

set(handles.contour,'XLim',[0 L],'YLim',[0 H],'NextPlot','replacechildren');

contourf(handles.contour,X,Y,u/U,[0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99]);

colorbar('peer',handles.contour,'SouthOutside');

set(handles.vector,'XLim',[0 L],'YLim',[0 H],'NextPlot','replacechildren');

[~,hdelta] = contour(handles.vector,X,Y,u/U,0.99);

set(hdelta,'Color','r','LineWidth',3,'ShowText','On');hold(handles.vector,'on');

quiver(handles.vector,X,Y,u,v,'b'),hold(handles.vector,'off');

 [strx,stry] = meshgrid(0,0:2*dy:H);

cla(handles.strline);

set(handles.strline,'XLim',[0 L],'YLim',[0 H],'NextPlot','replacechildren');

streamline(handles.strline,X,Y,u,v,strx,stry);

function X = tdma(A,B,C,D)

Cp = C;

Dp = D;

n = length(A);

X = zeros(n,1);

Cp(1) = C(1)/B(1);

Dp(1) = D(1)/B(1);

for i = 2:n

Cp(i) = C(i)/(B(i)-Cp(i-1)*A(i));

Dp(i) = (D(i)-Dp(i-1)*A(i))/(B(i)-Cp(i-1)*A(i));

end

X(n) = Dp(n);

for i = n-1:-1:1

X(i) = Dp(i)-Cp(i)*X(i+1);

end

function AppHelp(varargin)

dlgname = 'About Boundary Layer App';

txt = {'Laminar boundary layer problem solved numerically with TDMA method';

    '';

    'Start - Start the simulation';

    'Open Code - Open a pre-written script to show the solution code';

    '';

    '* Play with flow property and geometry information to study the change';

    '  in velocity profile.';

    '* Use zoom, pan and data curser in the toolbar to interact with the';

    '  plots as usual';

    '* Equation of boundary layer thickness printed on the App is based on';

    '  analytical solution.';

    '* If Reynolds number is too high (>500,000 for flow over flat plate),';

    '  the computation is not valid, since flow becomes turbulent.';

    '';

    'Copyright 2013 The MathWorks, Inc.'};

helpdlg(txt,dlgname);

function start_Callback(hObject, eventdata, handles)

boundarylayer(handles);

function showcode_Callback(hObject, eventdata, handles)

open('boundaryLayerScript.m');

MATLAB CODE for local value of the surface shear stress

function [pointcld_WSS_SurfGrad] = WSS_Surface_Gradient(pointcld_norm_WSSmag_comp, NN)

ptCloud = pointCloud(pointcld_norm_WSSmag_comp(:, 1:3));

for j = 1:1:length(pointcld_norm_WSSmag_comp)

    query_point = pointcld_norm_WSSmag_comp(j,1:3);

    [I] = findNearestNeighbors(ptCloud, query_point, NN);

    for i = 1:1:length(I)

        ind = I(i);

        dx(i) = pointcld_norm_WSSmag_comp(ind, 1) - query_point(1, 1);

        dy(i) = pointcld_norm_WSSmag_comp(ind, 2) - query_point(1, 2);

        dz(i) = pointcld_norm_WSSmag_comp(ind, 3) - query_point(1, 3);

        dMag(i) = pointcld_norm_WSSmag_comp(ind, 7) - pointcld_norm_WSSmag_comp(j, 7);

    end

A = [dx; dy; dz]';

B = [dMag]';

X(j, :) = A\B;

alpha(j,1) = pointcld_norm_WSSmag_comp(j,8)/pointcld_norm_WSSmag_comp(j,7); 

alpha(j,2) = pointcld_norm_WSSmag_comp(j,9)/pointcld_norm_WSSmag_comp(j,7);

alpha(j,3) = pointcld_norm_WSSmag_comp(j,10)/pointcld_norm_WSSmag_comp(j,7);

%Beta vector is the vector normal to the direciton of flow

beta(j,:) = cross(pointcld_norm_WSSmag_comp(j,4:6),alpha(j,:));

dTdA(j,:) = dot(X(j,:),alpha(j,:));

dTdB(j,:) = dot(X(j,:),beta(j,:));

WSS_SurfGrad(j) = sqrt(dTdA(j)^2 + dTdB(j)^2);

end

pointcld_WSS_SurfGrad = [pointcld_norm_WSSmag_comp(:,1) pointcld_norm_WSSmag_comp(:,2)...

    pointcld_norm_WSSmag_comp(:,3) WSS_SurfGrad' dTdA dTdB];

MATLAB CODE for convection coefficient

 function [ out] = a( X,T )

out=lambda(X,T)/((cp(X,T)/Mm(X))*rho(X,T));

end

MATLAB CODE for heat flux

[qx,qy] = evaluateHeatFlux(thermalresults,xq,yq)

[qx,qy,qz] = evaluateHeatFlux(thermalresults,xq,yq,zq)

[___] = evaluateHeatFlux(thermalresults,querypoints)

[___] = evaluateHeatFlux(___,iT)

[qx,qy] = evaluateHeatFlux(thermalresults)

[qx,qy,qz] = evaluateHeatFlux(thermalresults)

 Question

The boundary layer associated with parallel flow over an isothermal plate may be tripped at any x-location by using a fine wire that is stretched across the width of the plate. Determine the value of the critical Reynolds number  that is associated with the optimal location of the trip wire from the leading edge that will result in maximum heat transfer from the warm plate to the cool fluid. Assuming that fluid is atmospheric air at 300K determine the distance .

Answer

According to the Table A from the chapter 7 of the book can be seen that

figure(1);
plot(V,Q); hold on
xlabel('V (m/s)')
ylabel('Q (W/m)')
title('Air')
end
%Saturated water
for V_w = 0.5:0.5:10
Re_w = 13824.757*V_w;
Nu_w = ((0.3)+((1.0024*Re_w^(0.5))*(1+(Re_w/282000).^(0.625)).^(0.8)));
h_w = (62.3*Nu_w);
Q_w = (0.9425*h_w);
figure(2);
plot(V_w,Q_w); hold on
xlabel('V (m/s)')
ylabel('Q (W/m)')
title('Saturated water')
end
%Engine oil
for V_e = 0.5:0.5:10
Re_e = 28.01*V_e;
Nu_e = ((0.3)+((10.015*Re_e^(0.5))*(1+(Re_e/282000).^(0.625)).^(0.8)));
h_e = (14*Nu_e);
Q_e = (0.9425*h_e);
figure(3);
plot(V_e,Q_e); hold on
xlabel('V (m/s)')
ylabel('Q (W/m)')
title('Engine oil')
end

Question

Consider a sphere with a diameter of 20 mm and a surface temperature of 60that is immersed in a fluid at a temperature of 30 and a velocity of 2.5 m/s. calculate the drag force and the heat rate when the fluid is

Water

Air at atmospheric pressure

Answer

From the Table A of the Chapter 7

According to that table the property of water at 30 

Answer

From a given preheater

Condensing steam will be at 100 

Pressure will be maintained at 1atm

And temperature =25

Length of each tube will be equal to 10mm

According to the Table A it can be seen that