# Which boundary condition is coorect or what will be the correct alternative?

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Joydeep Bagchi . 2023년 9월 26일
댓글: Torsten . 2023년 9월 27일
% Constants
Ly = 0.001; % Thickness of the sheet (meters)
T = 15; % Total simulation time (seconds)
Ny = 10; % Number of spatial grid points
Nt = 300; % Number of time steps
k1 = 0.15; % Thermal conductivity of PVC W/m-K
rho = 1250; % density in kg/m3
cp = 1350 ; % specific heat in J/kg-K
alpha = k1/(rho*cp); % Thermal diffusivity (m^2/s)
T_ambient = 25; % Ambient temperature (°C)
velocity_x = 0.1667; % Velocity in x-direction (m/s)
%contact with hot roller
k2 = 50; %Thermal conductivity of steel roller W/m-K
h_roller = 500; %convective heat transfer coefficient of roller W/m2-K
d_roller = 0.4; %diameter of roller in m
contact_angle = 180; %in °
contact_length = pi*d_roller*contact_angle/360; %contact length with roller
T_roller = 100; % Roller temperature
%Contact in air
distance_x = 0.5; % Distance traveled in x-direction (meters)
h_air = 12; % Convective heat transfer coefficient of air W/m2-K
%Contact in IR field
distance_ir = 0.3 ; % in m
absorption = 50; % in %
performance = 80; % in %
%Contact with air again
distance_v = 1 ; % in m
%Naming the distance variables
y1 = contact_length;
y2 = contact_length + distance_x;
y3 = contact_length + distance_x+ distance_ir;
y4 = contact_length + distance_x+ distance_ir+distance_v;
% Discretization
dy = Ly / (Ny - 1);
dt = T / Nt;
y = linspace(0, Ly, Ny);
t = linspace(0, T, Nt);
% Initial temperature distribution
initial_temperature = 25;
distance_travelled = zeros(1,Nt); %initial distance travelled
% Initialize temperature matrix
u = zeros(Ny, Nt);
u(:, 1) = initial_temperature;
% Forward euler method
r = alpha * dt / (dy^2); % as 0 < r < 0.5
% Time-stepping loop (explicit method)
for n = 1:Nt - 1
% Update position in x-direction
distance_travelled(n+1) = distance_travelled(n) + velocity_x * dt;
% Check if the total distance is within the contact length
if distance_travelled(n) <= y1
for i = 2:Ny - 1
u(i, n + 1) = u(i, n) + r * (u(i + 1, n) - 2 * u(i, n) + u(i - 1, n));
end
% Apply boundary conditions for contact with the roller
u(1, n + 1) = ((h_roller * T_roller * dy) + (k1 * u(2, n + 1))) / (k1 + h_roller * dy);
u(Ny, n + 1) = ((h_air * T_ambient * dy) + (k1 * u(Ny - 1, n + 1))) / (k1 + h_air * dy);
% For the next part in contact with air
elseif distance_travelled(n) <= y2
for i = 2:Ny - 1
u(i, n + 1) = u(i, n) + r * (u(i + 1, n) - 2 * u(i, n) + u(i - 1, n));
end
% Apply boundary conditions for contact with air
u(1, n + 1) = ((h_air * T_ambient * dy) + (k1 * u(2, n + 1))) / (k1 + h_air * dy);
u(Ny, n + 1) = ((h_air * T_ambient * dy) + (k1 * u(Ny - 1, n + 1))) / (k1 + h_air * dy);
%In contact with IR field radiation
elseif distance_travelled(n) <= y3
for i = 2:Ny - 1
u(i, n + 1) = u(i, n) + r * (u(i + 1, n) - 2 * u(i, n) + u(i - 1, n));
end
% Apply boundary conditions for contact with IR field
u(1, n + 1) = ((h_air * T_ambient * dy) + (k1 * u(2, n + 1))) / (k1 + h_air * dy);
u(Ny, n + 1) = u(Ny - 1, n + 1) + ((Net_radiative_intensity * dy) / k1);
% Distance in contact with air again
else
for i = 2:Ny - 1
u(i, n + 1) = u(i, n) + r * (u(i + 1, n) - 2 * u(i, n) + u(i - 1, n));
end
% Apply boundary conditions for contact with air
u(1, n + 1) = ((h_air * T_ambient * dy) + (k1 * u(2, n + 1))) / (k1 + h_air * dy);
u(Ny, n + 1) = ((h_air * T_ambient * dy) + (k1 * u(Ny - 1, n + 1))) / (k1 + h_air * dy);
end
% Break the loop when the total distance is reached
if distance_travelled(n + 1) >= y4
break;
end
end
% Plot the temperature distribution over time
figure;
for n = 1:min(Nt,numel(distance_travelled))
plot(y, u(:, n));
xlabel('Thickness (m)');
ylabel('Temperature (°C)');
title(['Time = ', num2str(t(n)), ' seconds']);
axis([0 Ly 20 150]); % Adjust the axis limits if needed
pause(0.1); % Pause to create animation effect
% Break the plotting loop when the loop breaks
if distance_travelled(n) >= y4
break;
end
end
which boundary condition is correct? The PVC material's 1st element is in contact with the roller whose temperature is maintained at 100°C so ideally the mode of heat transfer should be conduction. In that case, either dirichlet boundary condition or neumann boundary condition should be implemented
If i apply dirichlet boundary condition u(1,n+1) = T_roller so it is fixing the temperature to 100°C until it is contact with roller which will not be the case . In reality, the temperature of the first element will rise more and try to reach 100°C as compared to other elements but won't be 100°C
For Neumann boundary condition, I need the information of temperature gradient or flux but unfortunately, I don't have neither of that.
I am not sure the boundary condition which I have applied here is correct or not? Can someone please clarify this or help me in this issue?

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### 채택된 답변

Torsten 2023년 9월 26일
편집: Torsten 님. 2023년 9월 26일
If you can estimate the heat transfer coefficient k between PVC material and roller, you could use
-lambda_PVC * dT_PVC/dn = k * (T_PVD - T_roller)
where d/dn is the spatial derivative in the direction of the outer normal. Look up "Robin boundary condition".
If k is large, this means that the PVC temperature follows the roller temperature very fast. In this case, it is justified to use the Dirichlet boundary condition for T_PVC.
But if the two materials are in direct contact, they should share the same temperature at this location, shouldn't they ?
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Joydeep Bagchi 2023년 9월 27일
Once agian thanks Torsten for the reply.
There is no transition zone used in the last code I posted.
Ok, but I need a continuous transition not like the temperature distribution at each time step rather a continuous temperature distribution through the whole time span.
Which question ? You mean the boundary condition you have to use on both ends of the web ?
The link of the pdf which I have posted has the answer in it so I have implemented that condition which is right.
Your boundary condition terms use values for u at time t(n+1) on the right-hand sides. Thus you have a mixture of explicit and implicit Euler for which I'm not able to predict whether it will work out correctly.
I need to give the term for gradient thatswhy I have written like that which is working fine for me.
Thanks a lot for your effort and time.
Torsten 2023년 9월 27일
Ok, but I need a continuous transition not like the temperature distribution at each time step rather a continuous temperature distribution through the whole time span.
The temperature distribution over the web will be continuous if you choose more curves to plot.
The link of the pdf which I have posted has the answer in it so I have implemented that condition which is right.
I cannot find a boundary condition for the roller on page 31 of the document.
I need to give the term for gradient thatswhy I have written like that which is working fine for me.

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