08 Laminar Forced Convection Over a Heated Flat Plate

Laminar Forced Convection Over a Heated Flat Plate ME433 COMSOL INSTRUCTIONS

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LAMINAR FORCED CONVECTION OVER A HEATED FLAT PLATE Problem Statement Ambient room temperature air at standard atmospheric pressure flows over semi infinite flat plate that is being heated by surface heat flux sq . Air starts to flow at x = 0 with a uniformly distributed velocity profile V. The plate has an insulated section extending from x = 0 to x = x0 and experiences applied heat flux ,0sq x from x = x0 to x = L (the considered plate length L is not shown). Of general interest is to learn how to use COMSOL in obtaining the flow and temperature distribution fields and compare them with the Blasius and Pohlhausen solutions (or more general curve fits of them). It is desired to obtain qualitative, as well as quantitative perspectives about boundary layer flow concept from COMSOL solutions.

nown quK antities:

luid: Air F

= 0.1 m/s V

T = 20 C sq = 1000 W/m2

b

n external flow, forced convection problem. Both fluid and temperature

elt

e of the

ng the thickness of the plate, the flow and heat transfer processes can be

O

servations

This is afields are essential parts of the problem. COMSOL model must include steady state analyses for both heat transfer and Navier Stokes application modes.

Subject to all 16 assumptions given in section 7.2.1, Blasius solution applies.

Although Pohlhausens solution does not apply directly due to a lack of plate temperature knowledge, it still can be used to develop equations for local Nussnumber and plate surface temperature distribution. Reference equations for these quantities will be presented in Postprocessing and Visualization section.

Although one of the assumptions for analytic solution is that of constant

properties, COMSOL can easily handle material property variations. Somkey properties of air strongly depend on temperature variations. We will discuss which properties of air should be varied in Options and Settings, along with equations that achieve this. Property variation will be included in our COMSOL model.

Neglecti

modeled with a simple rectangular geometry. However, plate boundary must thenbe split into two separate but connected boundaries in order to allow the correct boundary condition setup.

Flow over an Isothermal Flat Plate with an Insulated Leading Section

L = 10 cm x0 = 2 cm


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Assignment

1. State and calculate the conditions under which the flow field in this problem can .

2. Use COMSOL to solve for and save 2D color distributions of velocity and

3. Use COMSOL to solve for and save 2D color distributions of key air properties.

4. Use COMSOL to plot and save T(0.08,y).

5. Use COMSOL to plot and extract surface temperature data

be considered laminar and that the concept of boundary layer flow can be applied

temperature fields.

Use your textbooks Appendix C to examine whether or not these properties wereaccurately determined by COMSOL.

,0T x . Use this data

lid? [Note:

6. Use COMSOL data for

to compare it with surface temperature reference equations given in Postprocessing and Visualization section. Are COMSOL results vaIn this instruction set, part of this assignment question will be done with MATLAB, but you are free to use any software of your choice]

,0T x on 0x x L and Newtons law of cooling to

determine COMSOL h(x) for 0x x L . Compute and plot analytically determined local h(x) given by a reference equation and COMSOL h(x) osame graph. [Note: In this instruction set, part of this assignment question will done with MATLAB, but you are free to use any software of your choice]

7. Calculate and plot the percent error between COMSOL h(x) and theoretical

n the be

h(x). .

MATLAB to graph on the same plot

Base your error analysis on assumption that COMSOL h(x) is the correct solutionCan you conclude that COMSOL results are valid? [Note: In this instruction set, part of this assignment question will be done with MATLAB, but you are free to use any software of your choice]

8. [Extra Credit]: Use COMSOL and

theoretical and COMSOL determined boundary layer . Comment on differences in the solutions you notice. Which results would you trust? Thinstructions for COMSOL boundary layer data extraction and sample MATLscripts that will plot

e AB

are given separately in the appendix.

9. [Extra Credit]: Determine wall shear o induced by the flow on the plate and friction coefficient Cf.


Modeling with COMSOL Multiphysics MODEL NAVIGATOR To start working on this problem, we first need to enable two application modes in the model navigator to create a Multiphysics model. The correct application modes are located under COMSOL Multiphysics Fluid Dynamics and Heat Transfer sections. These modes will be responsible for setting up and calculating temperature and velocity distribution fields, respectively. For this setup:

1. Start COMSOL Multiphysics.

2. From the list of application modes, select COMSOL Multyphysics Fluid Dynamics Incompressible Navier Stokes Steady state analysis.

3. Click the Multiphysics button.

4. Click the Add button.

5. From the list of application modes, select COMSOL Multyphysics Heat Transfer Convection and Conduction Steady state analysis.

6. Click the Add button.

7. Click OK.

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OPTIONS AND SETTINGS: DEFINING CONSTANTS In this section, we will define material properties of air (Applying them to geometry is done in Subdomain Settings section). Some of the properties strongly depend on temperature while others do not. Since we are working with a rather large heat flux and would like to include property variation in the model, we first need to determine which of the properties exhibit strong temperature dependence. This is done by examining Appendix C Properties of dry air at atmospheric pressure. Since we do not know the high temperature extreme in this problem, we will take the largest temperature available in Appendix C. Notice that with increasing temperature, properties of air either increase or decrease in the temperature range of 20C to 350C. Notice further that no property reaches a maximum or a minimum in this temperature range. This enables us to concentrate our attention on the extremes of the temperature range in evaluating temperature dependence of the properties. The following table lists numerical values for properties of air at these temperature extremes and shows the percent difference in those properties based on these extremes.

pC k Pr EVALUATED AT T 1006.1 1.2042 18.17x10-6 0.02564 0.713 EVALUATED AT 350C 1056.8 0.5665 31.07x10-6 0.04692 ~ 0.7

% DIFFERENCE (based on 20C) 5.04 53 71 83 1.86

Based on these calculations, it is now clear that for air in this temperature range, , , and strongly depend on temperature while p and r are weakly dependent propertieswith respect to temperature. Therefore, and will be set as constants while

k CPr

P

pC , , and k will be modeled as varying properties. The following equations will be used to calculate air properties that vary strongly with temperature:

10

30

3.723 0.865log

6 8

, [kg/m ]

10 , [W/m K]6 10 4 10 , [Pa s]

w

T

P MRT

kT

ng[Ref.: J.M. Coulson and J.F. Richardson, Chemical E ineering, Vol. 1, Pergamon Press, 1990, appendix]

Where,

0 (atmospheric pressure) 101.3 kPa,(molecular weight of air) 0.0288 kg mol,

(universal gas constant) 8.314 J/mol Kw

PMR


Armed with these equations, let us now define temperature dependent air properties in COMSOL.

1. From the Options menu select Expressions Scalar Expressions

2. Define the following names and expressions:

NAME EXPRESSION UNIT DESCRIPTION

k_air 10^(-3.723+0.865*log10(abs(T[1/K])))[W/(m*K)] W/(mK) Air Conductivity

rho_air 1.013e5[Pa]*28.8[g/mol]/(8.314[J/(mol*K)]*T) kg/m3 Air Density

mu_air 6e-6[Pa*s]+4e-8[Pa*s/K]*T Kg/(ms) Air Viscosity

3. Click OK.

COMSOL automatically determines correct property unit under the Unit column. If it does not, you are most likely entering wrong expressions. Carefully check the expression you typed and make corrections, if necessary. The description column is optional and can be left blank. Although Prandtls number is essential, it is a composite property that is defined by , pC , and k , most of which have now been defined. The only constant property that needs to be defined as well is . We will define and apply it to geometry in Subdomain Settings section.

pC

GEOMETRY MODELING In this model we will create a 2D rectangular geometry by drawing it. This is particularly useful since we need to create a boundary for the insulated part as a separate entity.

1. Start by clicking on the Line button located on the draw toolbar.

2. Position your cursor at the origin (0,0) in the main drawing area and start making a line by pressing on the left mouse button (LMB) once and moving the mouse to the right. You should be getting a line that looks like this one .

3. Move your cursor to the (0.2,0) coordinate and press the left mouse button (LMB)

once to create the first line. As you do this, the line segment from (0,0) to (0.2,0) should turn red, as shown here .

4. Continue to make the line segments outlined in the previous step for the following

coordinates; from: (a) (0.2,0) to (1,0); (b) (1,0) to (1,0.4); (c) (1,0.4) to (0,0.4); and (d) (0,0.4) to (0,0). The geometry you are creating should look rectangular.

5. Once back at the origin (0,0), press on the right mouse button (RMB) to finish the

rectangle.

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We now must scale the geometry down to centimeters. (Recall that COMSOLs default system of units is the MKS. Therefore, we just made a 1 meter long rectangle).

6. To scale the geometry, go under Draw Modify Scale menu and type 0.1 as a scale factor for both x and y fields as shown below:

7. Click OK.

8. Click on Zoom Extents button in the main toolbar to zoom into the

geometry. Your geometry should now be complete and highlighted in red, as shown below.

PHYSICS SETTINGS Physics settings in COMSOL consist of two parts: (1) Subdomain settings and (2) boundary conditions. The subdomain settings let us specify material types, initial conditions, modes of heat transfer (i.e. conduction and/or convection). The boundary conditions settings are used to specify what is happening at the boundaries of the geometry. In this model, we will have to specify and couple physics settings for the flow of air and heat transfer. Let us begin with the air flow physics settings.

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Incompressible Navier Stokes (ns) Subdomain Settings:

1. From Mulptiphysics menu, select 1 Incompressible Navier Stokes (ns).

2. From the Physics menu select Subdomain Settings (equivalently, press F8).

3. Select subdomain 1 in the Subdomain selection window.

4. Enter rho_air and mu_air in the fields for density and dynamic viscosity .

5. Click OK.

Incompressible Navier Stokes (ns) Boundary Conditions:

1. From the Physics menu open the Boundary Settings (F7) dialog box.

2. Apply the following boundary conditions:

BOUNDARY BOUNDARY TYPE BOUNDARY CONDITION COMMENTS

1 Inlet Velocity Enter 0.1 in U0 field (Normal Inflow velocity) 2, 4 Wall No Slip

3, 5 Open boundary Normal stress Verify that field f0 is set to 0

3. Click OK to close the boundary settings window.

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Convection and Conduction (cc) Subdomain Settings:

1. From Mulptiphysics menu, select 2 Convection and Conduction (cc) mode.

2. From the Physics menu, select Subdomain Settings (F8).

3. Select Subdomain 1 in the subdomain selection field.

4. Enter k_air, rho_air and 1006 in the k(isotropic), , and Cp fields, respectively.

5. Enter u and v in the u and v fields, respectively.

6. Click OK to close the Subdomain Settings window.

Convection and Conduction (cc) Boundary Conditions:

1. From the Physics menu open the Boundary Settings (F7) dialog box.

2. Apply the following boundary conditions:

BOUNDARY BOUNDARY CONDITION COMMENTS

1 Temperature Enter 273.15+20 in T0 field 2, 3 Thermal Insulation

4 Heat Flux Enter 1000 in q0 field 5 Convective flux

3. Click OK to close Boundary Settings window.

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MESH GENERATION To minimize the computational time without compromising much accuracy of the solution, we must change the default meshing parameters. To do this,

1. Go to the Mesh menu and select Free Mesh Parameters option.

2. Change Predefined mesh sizes from Normal to Finer.

3. Switch to Boundary tab.

4. Select boundaries 1 and 5 in the Boundary selection field while holding the Control (ctrl) key on your keyboard.

5. Switch to Distribution tab.

6. Enable Constrained edge element distribution option.

7. Enter 20 in the Number of edge elements field.

8. Select boundary 2. (Do not hold the Control (ctrl) key on your keyboard)

9. Switch to Distribution tab and enable Constrained edge element distribution.


11. Select boundary 4. (Do not hold the Control (ctrl) key on your keyboard)

12. Switch to Distribution tab and enable Constrained edge element distribution.


14. Switch to Point tab.

15. Select points 1 and 3 in the Point selection field while holding the Control (ctrl) key on your keyboard.

16. Enter 0.0001 in the Maximum element size field.

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17. Click the Remesh button.

18. Click OK to close the Free Mesh Parameters window.

As a result of these steps, you should get the following triangular mesh:

We are now ready to compute our solution.

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COMPUTING AND SAVING THE SOLUTION In this step we define the type of analysis to be performed. We are interested in steady state analysis here, which we previously selected in the Model Navigator. Therefore, no modifications need to be made. To enable the solver, proceed with the following steps:

1. From the Solve menu select Solve Problem. (Allow few seconds for solution)

2. Save your work on desktop by choosing File Save. Name the file according to the naming convention given in the Introduction to COMSOL Multiphysics document.

The result that you obtain should resemble the following boundary color map:

By default, your immediate result will be given in Kelvin instead of degrees Celsius. (In fact, the first result you will see is the velocity field, not temperature). Furthermore, it will be colored using a jet colormap and the velocity field (represented by arrows in the above) will not be shown. We will use distinct colormap options to represent the air velocity and temperature fields. The next section (Postprocessing and Visualization) will help you in obtaining the above and other diagrams, such as 2D color distributions of key air properties, a plot of T(y) at x = 8 cm, a plot of local ,0xq x for 0x x L . We will also show how to use COMSOL to compute the total heat transfer rate per unit length, Tq and use MATLAB to determine h(x) from COMSOL ,0xq x data and analytically.

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POSTPROCESSING AND VISUALIZATION After solving the problem, we would like to be able to look at the solution. COMSOL offers us a number of different ways to look at our temperature (and other) fields. In this problem we will deal with 2D color maps, velocity (and other) vector fields, extraction of plate surface temperature , as well as computation of local heat transfer coefficient and 1D temperature distribution plot. You will then address the questions of COMSOL solution validity and compare the results to theoretical predictions mainly by using MATLAB.

,0T x

Displaying T(x, y) and Vector Field V(x, y) Let us first change the unit of temperature to degrees Celsius:

1. From the Postprocessing menu, open Plot Parameters dialog box (F12).

2. Under the Surface tab, change the unit of temperature to degrees Celsius from the drop down menu in the Unit field.

3. Change the Colormap type from jet to hot.

4. Click Apply to refresh main view and keep the Plot Parameters window open.


The 2D temperature distribution will be displayed using the hot colormap type with degrees Celsius as the unit of temperature. Lets now add the velocity vector field V(x,y).

5. Switch to the Arrow tab and enable the Arrow plot check box.

6. Choose Velocity field from Predefined quantities.

7. Enter 20 in the Number of points for both x and y fields.

8. Press the Color button and select a color you want the arrows to be displayed in. (Note: choose a color that produces good contrast. Green and white are good choices here).

9. Click Apply to refresh main view and keep the Plot Parameters window open.

At this point, you will see a similar plot as shown on page 11. It is a good idea to save this colormap for future use. Before you do save it, however, experiment with the Number of points field in Plot Parameters window and adjust the velocity vector field to what seems the best view to you. Put 30 for the y field and update your view by pressing Apply button. Notice the difference in velocity vector field representation. Try other values.

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You may also want to see other quantities as vector fields. Available quantities are: (1) Temperature gradient, (2) Conductive heat flux, (3) Convective heat flux, and (4) Total heat flux. To see these quantities represented by a vector field:

10. Change the color of the arrow (see step 8).

11. Choose the quantity you wish to plot from Predefined quantities.

12. Click Apply.

13. Click OK when you are done displaying these quantities to close the Plot Parameters window.

Saving Color Maps: After you have selected a view that shows the results clearly, you may want to save it as an image for future discussion. This may be done as follows:

1. Go to the File menu and select Export Image. This will bring up an Export Image window.

For a 4 by 6 image, acceptable image quality settings are given in the figure below. If you need higher image quality, increase the DPI value.

2. Change your Export Image value settings to the ones in the above figure. 3. Click the Export button. 4. Name and save the image.

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Displaying V(x, y) as a Colormap:

1. From the Postprocessing menu, open Plot Parameters dialog box (F12). 2. Under the Arrow tab, disable the Arrow plot checkbox

3. Switch to Surface tab.

4. From Predefined quantities, select Velocity field.

5. Change the Colormap type from hot to jet.

6. Click Apply to refresh main view and keep the Plot Parameters window open. The 2D Velocity distribution will be displayed using the jet colormap. Displaying Variations of Key Air Properties as Colormaps: With the Plot Parameters window open, ensure that you are under the Surface tab,

7. Type k_air in Expression field (without quotation marks).

8. Click Apply. (Note: The unit will change automatically) These steps produce a colormap that displays variations in airs thermal conductivity k. Note the values on the color scale and compare them with Appendix C of your textbook.

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To produce colormaps for density and viscosity variations, repeat steps 6 and 7 while typing rho_air and mu_air, respectively in the Expression field in step 6. When done, click OK to close the Plot Parameters window. Note: You may also view composite properties, such as kinematic viscosity and Prandtls number simply by entering their definitions in the Expression field. Thus, to view kinematic viscosity variation, enter mu_air/rho_air. For Prandtl number, enter 1006*mu_air/k_air. It is even possible to enter expressions for other desired quantities, such as local Reynolds number. For Reynolds number evaluated at every x and y using x as the computational value in its definition, enter 0.1[m/s]*rho_air*x/mu_air. For Reynolds number evaluated at every x and y using y as the computational value in its definition, enter 0.1[m/s]*rho_air*y/mu_air. Plotting T(0.08, y) (or T(y) at x = 8 cm):

1. From Postprocessing menu select Cross Section Plot Parameters option.

2. Switch to the Line/Extrusion tab.

3. Change the Unit of temperature to degrees Celsius.

4. Change the x axis data from arc length to y.

5. Enter the following coordinates in the Cross section line data: x0 = x1 = 0.08; y0 = 0 and y1 = 0.04.

6. Click OK.

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These steps produce a plot of T(y) at x = 8 cm, from y = 0 cm (plate surface) to y = 4 cm (ambient environment conditions). Temperature T is plotted on the y axis and y coordinates are plotted on the x axis. To save this plot,

7. Click the save button in your figure with results. This will bring up an Export Image window.

8. Follow steps 2 4 as instructed on page 14 to finish with exporting the image.

Plotting Plate Surface Temperature 0T x, For 0 x x L To plot for ,0T x 0x x L using COMSOL,

1. Select Cross Section Plot Parameters option from Postprocessing menu. 2. Switch to the Line/Extrusion tab.

3. From Predefined quantities, select Temperature.

4. Change the Unit of temperature to degrees Celsius.

5. Change the x axis data from y to x.

6. Enter the following

coordinates in the Cross section line data: x0 = 0.02, x1 = 0.1; y0 = y1 = 0.

7. Click OK to close Cross

Section Plot Parameters window.

As a result of these steps, a new plot will be shown that graphs ,0T x for

0x x L . Do not close this plot just yet. We are going to extract this data to a text file for comparative analysis with MATLAB.


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Exporting COMSOL Data to a Data File:

1. Click on Export Current Plot button in the graph created in the previous s

tep.

2. Click Browse and navigate to your saving folder (say Desktop).

3. Name the file comsol_temperature.txt. (Note: do not forget to type the .txt

4. Click OK to save the file.

his completes COMSOL modeling procedures for this problem.

extension in the name of the file).

T


Modeling with MATLAB

This part of modeling procedures describes how to create comparative graphs of local heat transfer coefficient h(x) (along the heated portion of the plate) using MATLAB. Obtain MATLAB script file named heated_plate.m from Blackboard prior to following these procedures. Save this file in the same directory as the data file(s) (comsol_temperature.txt) from COMSOL. (Note: heated_plate.m file is attached to the electronic version of this document as well. To access the file directly from this document, select View Navigation Panels Attachements and then save heated_plate.m in a proper directory) Comparing COMSOL solution with Approximated Pohlhausen Solution: The reference analytic equations for heated plates with an insulation section are:

13

1 13 2

13

1 13 2

0

0

0.417 1 Pr Re

2.396 1Pr Re

xx

ss

h x xNuk x

q x xT x Tk x

MATLAB script (heated_plate.m) is programmed to use exported COMSOL data for surface temperature and Newtons Law of cooling to determine the local heat transfer coefficient h(x) along the heated portion of the plate. The script is also programmed to calculate analytical local heat transfer coefficient h(x) and surface temperature according to analytic reference equations given above. These equations represent a more general approximation to Pohlhausen solution that is suitable for plates with insulated section and applied heat flux. The script will ultimately produce comparative graphs that will plot both solutions. Follow the steps below to complete this problem:

,0T x

1. Open MATLAB by double clicking its icon on the Desktop. 2. Load heated_plate.m file by selecting File Open Desktop

heated_plate.m. The script responsible for COMSOL data import and data comparison will appear in a new window.

3. Press F5 key to run the script. MATLAB editor will display a warning message.

Click Change Directory to run the script. Approximated Pohlhausens and COMSOL solutions for h(x) and sT x will be plotted in Figures 1 and 3. Figures 2 and 4 plot the percent error between quantities considered according to the equations printed on the figures. These results are shown on the next page.

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Results plotted with MATLAB:

The results shown above were based on varying properties of air determined by the equations given in Options and Settings: Defining Constants section. By default, the script is programmed to use constant air properties determined at film temperature. This, however, introduces greater error. If you wish to use varying properties, you must export them to the same folder where the MATLAB script is. You must export varying Prandtls number, conductivity k, and kinematic viscosity along heated portion of the surface of the plate. Refer to steps 1 7 on page 17 and 1 4 on page 18 to properly extract these quantities. Type the following expressions in the Expressions field of Cross Section Plot Parameters window to extract these properties and give them the following file names:

PROPERTY EXPRESSION FILE NAME

Pr 1018*mu_air/k_air Pr_comsol.txt

k k_air k_comsol.txt mu_air/rho_air eta_comsol.txt

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Further, you will need to un suppress a section in MATLAB script. This is explained in the script itself under Varying Quantities Import From COMSOL section. While in MATLAB, you may zoom into plots to notice departures in results based on the solution methods. Armed with these results, you are in a position to answer most of the assigned questions. (Approaches that show how to answer extra credit questions are given in appendix). This completes MATLAB modeling procedures for this problem.

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APPENDIX MATLAB script

If you could not obtain this script from the Blackboard or the PDF file, you may copy it here, then paste it into notepad and save it in the same directory where you saved COMSOL data file(s). You will most likely get hard to spot syntax errors if you copy the script this way. It is therefore highly advised that you use the other 2 methods on obtaining this script instead of the copying method. % ######################################################################### % ME 433 - Heat Transfer % Sample MATLAB Script For: % (X) Laminar Thermal Boundary Layer [Specified Surface Heat Flux] % IMPORTANT: Save this file in the same directory with % "comsol_temperature.txt" file. % ######################################################################### % clc; % Clears the UI prompt clear; % Clears variables from memory %% Constant Quantities Tinf = 20; % Ambient temperature, [degC] Vinf = 0.1; % Velocity at the inlet, [m/s] k_air = 0.03525; % Air conductivity at Tfilm, [W/m-K] Pr_air = 0.701; % Air Prandtl number at Tfilm, [1] eta_air = 29.75e-6; % Air viscosity at Tfilm, [m^2/s] x0 = 0.02; % A speical plate coordinate!, [m] qs = 1000; % Applied Surface Heat Flux, [W/m^2] %% Varying Quantities Import From COMSOL % ######################################################################### % Un-suppress the quantities below to perform verification of results % using varying air properties determined by COMSOL. Make sure to export % the following data files from COMSOL and save them in the same % directory as this script: Pr_comsol.txt, k_comsol.txt, eta_comsol.txt. % Otherwise, leave this section suppressed. When suppressed, the results % you get are determined at T film and introduce larges errors. % ######################################################################### % load Pr_comsol.txt % Pr_air = Pr_comsol(:,2)'; % load k_comsol.txt % k_air = k_comsol(:,2)'; % load eta_comsol.txt % eta_air = eta_comsol(:,2)'; % clear Pr_comsol k_comsol eta_comsol; %% COMSOL Data Import and h(x) Computation load comsol_temperature.txt x = comsol_temperature(:,1)'; Ts_comsol = comsol_temperature(:,2)'; hx_comsol = qs./(Ts_comsol - Tinf); clear comsol_temperature; %% Finding Ts(x) and h(x) Analytically (Correlation Equation) Rex = Vinf*x./eta_air; qs = 1000; % Applied Uniform Surface Heat Flux, [W/m^2] Ts_analyt = Tinf + 2.396*qs./k_air.*(1-x0./x).^(1/3).*x./... (Pr_air.^(1/3).*Rex.^(1/2)); % Correlation Eq. for T(x) Nux = 0.417*(1-x0./x).^(-1/3).*... Pr_air.^(1/3).*Rex.^(1/2); % Local Nusselt number, [1] hx_analyt = Nux.*k_air./x; % Local Heat transfer coefficient, [W/(m^2-C)] %% Error analysis in h(x) and Ts(x) deltah = abs(hx_comsol - hx_analyt); %| -> Simple % Error errorh = deltah./hx_comsol*100; %| -> calculation for h

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deltaT = abs(Ts_comsol - Ts_analyt); %| -> Simple % Error errorT = deltaT./Ts_comsol*100; %| -> calculation for T %% h(x) Plot Begins Here: figure1 = figure('InvertHardcopy','off',... %\ 'Colormap',[1 1 1 ],... % | -> Setting up the figure 'Color',[1 1 1]); %/ plot(x,hx_comsol,'b',x,hx_analyt,'r--') % Plots COMSOL vs. Theory h %% Plot cosmetics for figure 1 begin here: annotation(figure1,'textbox',... 'String',{'Flow Over a Heated Plate','with Insulated Edge'},... 'HorizontalAlignment','center',... 'FontSize',14,... 'FontName','Times New Roman',... 'FitBoxToText','off',... 'LineStyle','none',... 'BackgroundColor',[1 1 1],... 'Position',[0.5324 0.6079 0.3669 0.1669]); annotation(figure1,'textbox',... 'String',{'q_s" = 1000W/m^2','T_\infty = 20\circC','V_\infty = 0.1 m/s','L = 10 cm','x_0 = 2 cm'},... 'FontSize',14,... 'FontName','Times New Roman',... 'FontAngle','italic',... 'FitBoxToText','off',... 'EdgeColor',[1 1 1],... 'BackgroundColor',[1 1 1],... 'Position',[0.599 0.3462 0.2552 0.3127]); legend('COMSOL Solution','Analytic Equation') box off grid on title('\fontname{Times New Roman} \fontsize{16} \bf Local Heat Transfer Coefficient') xlabel('x, [m]') ylabel('h(x), [W/m^2-\circC]') set(get(gca,'YLabel'),... 'fontsize', 14,... 'FontName','Times New Roman',... 'FontAngle','italic') set(get(gca,'XLabel'),... 'fontsize', 14,... 'FontName','Times New Roman',... 'FontAngle','italic') %% Error Plot in h(x) begins here: figure2 = figure('InvertHardcopy','off',... %\ 'Colormap',[1 1 1 ],... % | -> Setting up the figure 'Color',[1 1 1]); %/ plot(x,errorh) % Plots % Error in h %% Plot cosmetics for figure 2 begin here: box off grid on title('\fontname{Times New Roman} \fontsize{16} \bf Error Analysis in h(x)') xlabel('x, [m]') ylabel('Error in h(x), [%]') set(get(gca,'YLabel'),... 'fontsize', 14,... 'FontName','Times New Roman',... 'FontAngle','italic') set(get(gca,'XLabel'),... 'fontsize', 14,... 'FontName','Times New Roman',... 'FontAngle','italic') str1(1) = {... '$${\%err={h_{x_{analyt}}-h_{x_{comsol}}\over h_{x_{comsol}}}\times 100} $$'}; text('units','normalized', 'position',[.35 .2], ... 'fontsize',14,...

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'FontName', 'Times New Roman',... 'FontAngle', 'italic', ... 'BackgroundColor',[1 1 1],... 'interpreter','latex',... 'string', str1); %% Ts(x) Plot Begins Here: figure3 = figure('InvertHardcopy','off',... %\ 'Colormap',[1 1 1 ],... % | -> Setting up the figure 'Color',[1 1 1]); %/ plot(x,Ts_comsol,'b',x,Ts_analyt,'r--') % Plots COMSOL vs. Theory h %% Plot cosmetics for figure 3 begin here: annotation(figure3,'textbox',... 'String',{'Flow Over a Heated Plate with Insulated Edge'},... 'FontSize',14,... 'FontName','Times New Roman',... 'FitBoxToText','off',... 'LineStyle','none',... 'BackgroundColor',[1 1 1],... 'Position',[0.1493 0.8014 0.6401 0.08788]); annotation(figure3,'textbox',... 'String',{'q_s" = 1000W/m^2','T_\infty = 20\circC','V_\infty = 0.1 m/s','L = 10 cm','x_0 = 2 cm'},... 'FontSize',14,... 'FontName','Times New Roman',... 'FontAngle','italic',... 'FitBoxToText','off',... 'EdgeColor',[1 1 1],... 'BackgroundColor',[1 1 1],... 'Position',[0.6266 0.1593 0.2552 0.3127]); legend('COMSOL Solution','Analytic Equation','location', 'southwest') box off grid on title('\fontname{Times New Roman} \fontsize{16} \bf Plate Surface Temperature') xlabel('x, [m]') ylabel('T_s(x), [\circC]') set(get(gca,'YLabel'),... 'fontsize', 14,... 'FontName','Times New Roman',... 'FontAngle','italic') set(get(gca,'XLabel'),... 'fontsize', 14,... 'FontName','Times New Roman',... 'FontAngle','italic') %% Error Plot in Ts(x) begins here: figure4 = figure('InvertHardcopy','off',... %\ 'Colormap',[1 1 1 ],... % | -> Setting up the figure 'Color',[1 1 1]); %/ plot(x,errorT) % Plots % Error in h %% Plot cosmetics for figure 4 begin here: box off grid on title('\fontname{Times New Roman} \fontsize{16} \bf Error Analysis in T_s(x)') xlabel('x, [m]') ylabel('Error in T_s(x), [%]') set(get(gca,'YLabel'),... 'fontsize', 14,... 'FontName','Times New Roman',... 'FontAngle','italic') set(get(gca,'XLabel'),... 'fontsize', 14,... 'FontName','Times New Roman',... 'FontAngle','italic') str1(1) = {... '$${\%err={T_{s_{analyt}}-T_{s_{comsol}}\over T_{s_{comsol}}}\times 100} $$'};

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text('units','normalized', 'position',[.35 .2], ... 'fontsize',14,... 'FontName', 'Times New Roman',... 'FontAngle', 'italic', ... 'BackgroundColor',[1 1 1],... 'interpreter','latex',... 'string', str1); COMSOL Hints and Sample MATLAB Scripts For Extra Credit Question The goal of this question is to obtain boundary layer from COMSOL and compare it directly with analytical boundary layer solution obtained by Blasius. This is particularly tricky, since there is no clear definition as to where the viscous boundary layer thickness

/occurs. Notice that in our textbook, the definition is given based on the condition that

be 0.994, from which, with the use of table 7.1, equation 7.11 is derived. We could have taken as close to unity as we wish and equation 7.11 would therefore change.

layer, since it implies t V . The number 0.994, however, is special because it corresponds to a Prandtls r of 1.0 on Pohlhausens solution given in figure 7.2From figure 7.2 and equation 7.19, it follows that for air (Pr = 0.7), / 1t

u V/u V

Physically, the closer /u Vhat u

nu

is to unity, the better the distinction in the viscous boundary

.

mbe , since

6.5t .

e therefore need to program MATLAB with the following analytical equation foW r :

5.2Rex

x

To compute Rex , properties at Tfilm must be found. This is easily done since both temperature extremes are now known. Variable x ranges between 0 0.1x meters. In COMSOL, we have to use Contour plot type to single out velocity iso curve that orrespond to condition. To extract boundary layerc / 0.994u V from COMSOL,

ckbox on the top left portion of the window.

1. From the Postprocessing menu, open Plot Parameters dialog box (F12).

2. Under the Arrow tab, disable the Arrow plot checkbox

3. Switch to Contour tab.

4. Enable Contour plot che

5. Type u in the Expression field. (without quotation marks)


6. Enable Vector with isolevels radio button option. (The entry field right below it enables us to enter the single out the velocity for which we want the iso curve to be mapped out).

7. Enter 0.0944 in the entry field below Vector with isolevels. (This is the x

component velocity that satisfies / 0.994u V condition).

8. Switch to General tab.

9. Disable all other plot types except Contour and Geometry edges.

10. Use the Plot in drop down menu (located on the bottom left of the window) to switch from Main axes to New figure.

11. Click OK. Viscous COMSOL boundary layer satisfying will be

shown in a new plot figure. / 0.994u V

12. Click on Export Current Plot button .

13. Name the file fluid_blayer.txt. (Note: do not forget to type the .txt

extension in the name of the file).

14. Click OK to save the file. The file is saved in the same directory where you first saved COMSOL model file with extension .mph.

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The following MATLAB script sample shows how the above analytical equations can be programmed. It also imports COMSOL boundary layer data saved in fluid_blayer.txt text file and uses it to plot comparative graphs. %% Preliminaries clc; % Clears the UI prompt clear; % Clears variables from memory %% Constant Quantities Vinf = 0.1; % Velocity at the inlet, [m/s] k_air = 0.03525; % Conductivity at Tf, [W/m-K] rho_air = 0.8150; % Density at Tf, [kg/m^3] Pr_air = 0.701; % Prandtl number at Tfilm, [1] mu_air = 24.24e-6; % Viscosity at Tf, [m^2/s] eta_air = mu_air/rho_air; % Specific viscosity at Tf, [m^2/s] %% COMSOL Data Import load fluid_blayer.txt; xcomsol = fluid_blayer(:,1); ycomsol = fluid_blayer(:,2); %

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