Numerical Results

 Some of the results shown below were obtained with a pipe length of 6.096 which is slightly different from the current length of 6.045. So your results might be slightly different from those shown below.

Please make sure your project is saved in Workbench. Double click on Results in the Project Schematic window. This will open CFD-Post (the program used to analyze results from FLUENT computation.) Click on z axis in the triad (at the bottom right of the graphics window) to get the view along the z-axis.

Image Removed

Temperature Contour

<iframe width="560" height="315" src="https://www.youtube.com/embed/7_P7arvK-4Q?rel=0" frameborder="0" allowfullscreen></iframe>

Summary of the Above Video:

1. Create Contours > Name a Contour Plot and name it Temperature Contours
2. Under Details Plot TemperatureGeometry > Locations, choose periodic 1 and set the variable of interest as Temperature.
3. Under View > Apply Scale > Enter , enter (01, 30, 01)
4. Default Transform
   1. Mirror > Mirror Apply Reflection about ZX Plane

You can save the image to a file using the camera icon highlighted indicated in the image below or using the Snipping Tool in Windows 7 (you can search for it under Start > Programs).

...

In developing the experiment, it was assumed that by the end of the adiabatic mixing stage, the flow will be well-mixed. Do the results from the numerical solution simulation support this assumption?

Velocity Vectors

Our next challenge is to produce velocity vectors. This is a very similar process to creating the temperature contours above. 

<iframe width="560" height="315" src="https://www.youtube.com/embed/HX27T78mQfg?rel=0" frameborder="0" allowfullscreen></iframe>

...

Does the flow become fully developed at the end of the first section?

Centerline Temperature Plot

Now let's look at the temperature variation along the center-line of the pipe. To do this we need to first create a line corresponding to the center-line:

Insert > Location > Line

Name it "Centerline" and click OK. On the lower left panel, you will see Details of Centerline. Enter the start and end locations of the line and the sampling frequency. Click Apply.

Image Removed

You will see centerline created under User Locations and Plots.

Insert > Chart 
Please name this chart "Centerline Temperature". You will see Details of Centerline Temperature appear on the lower left.

We'll go through the tabs in the menu to specify the plot that we want. Select the General tab and name the chart "Temperature Variation along Pipe Axis".

Image Removed

Select the Data Series tab. Change Name and Location.

Image Removed

We want to see the variation of temperature with the length of the pipe. Therefore, temperature will be on the "y-axis" of the chart and axial position on the "x-axis" of the chart.

Click on X Axis tab. Next to Variable, choose X.
Image Removed

Click on Y Axis tab. Next to Variable, choose Temperature.

Image Removed

Click Apply. You will see Centerline Temperature created under Report in the Outline tab.

Image Removed

...

You need to repeat the FLUENT simulation with inputs from YOUR MEASUREMENTS in the lab. To compare the FLUENT results with experiment, you can export the FLUENT result into Excel. A sample comparison is shown below.

...

You can export the FLUENT data in Excel format by clicking on the Export button in "Details of centerline temperature"

Image Removed

Wall Temperature Plot

<iframe width="560" height="315" src="https://www.youtube.com/embed/-Cvn7HPp9eY?rel=0" frameborder="0" allowfullscreen></iframe>

...

1. Create location > Line
   1. Point 1: (0,.0294,0) Point 2: (6.045,.0294,0)
2. Create Chart
3. Under Data Series  > Location Select line you just created
4. X Tab 
   1. Plot X location
5. Y Tab
   1. Plot Temperature

...

You need to repeat the FLUENT simulation with inputs from YOUR MEASUREMENTS in the lab and compare the FLUENT results for the wall temperature with experiment. A sample comparison is shown below.

Image Removed

The wall temperature values can be exported to a csv file (suitable for Excel) using the Export button highlighted below.

Image Added

...

Importing Experimental Data and Chart Formatting

We can import experimental data from a csv file using the procedure shown below. The csv file should contain two columns, corresponding to the x-axis and y-axis values, respectively 

<iframe width="560" height="315" src="https://www.youtube.com/embed/oy3xJrmWNLo?rel=0" frameborder="0" allowfullscreen></iframe>

...

1. In details of Chart > New Series > Import Data from file
   1. Find file and click okay
2. Change how experimental data looks in CFD Post
   1. In Line Display under Series 2 > No Line > Symbols > Ellipse
3. Chart Details > General
   1. Give TItleTitle
4. Chart Details > Chart Display
   1. Font Sizes
   2. Grid Sizes

Pressure Plot

Create a plot of the pressure variation along the centerline of the pipe. Steps for this are similar to the plot of the centerline temperature that we did earlier.

There is no need to create a new line. We can use the "centerline" created earlier.

Insert > Chart 

Follow steps from the Centerline Temperature plot above, making appropriate modifications. You should see the following plot.

Image Removed

Axial Velocity Profiles

Let's look at the velocity profiles before and after the heated section. To do this, we need to first create lines at x=1.83 m ((start of heated section), x=4.27 m (end of heated section) and x=6.045 m (end of mixing section).

First, create the line at x=1.83 m.

Insert > Location > Line

Name it "x183" and click OK. Enter the following coordinates (0.0294 m is the pipe radius).

Point 1 (1.83, 0, 0)
Point 2 (1.83, 0.0294, 0)

Enter 100 for Samples. Click Apply.

Similarly create lines at x=4.27 m and x=6.045 m.

Insert > Chart 

Name this chart "Axial Velocity Profiles".

Select the General tab and name the chart "Axial Velocity Profiles".

Select Data Series tab. Change the name of the first data series to x=1.83 m. Under Data Source, specify x183 as Location.

Add a new data series by clicking on the "New" icon as shown below and repeat the above steps but for x=4.27 m.

Image Removed

Add a third data series by clicking on the "New" icon and repeating the steps for x=6.045 m. You should then have three items in the Data Series tab.

Image Removed

Specify x-axis variable: Velocity u

Specify y-axis variable: Y

Complete the plot. Here's what we get.

Image Removed

We notice that the flow accelerates due to the heating. As air is heated, its density decreases. So the velocity has to increase to maintain the same mass flow rate.

...

Note that the centerline temperature and pressure variations can be plotted by duplicating this plot as mentioned in the video.

Mixed Mean Temperature at an Axial Location

From energy conservation, we can show that the mixed mean temperature is constant in the flow development and mixing sections and varies linearly in the heated section. This is shown schematically in the following figure from the MAE 4272 lab manual.

...

Using the above equation, calculate the mixed mean temperature Tm at x=2.67 m. In the Verification & Validation section, you'll check that this value matches the Tm value calculated by Remember to add the inlet temperature, otherwise you will just end up with the temperature difference between the mixed mean temperature and the inlet temperature (where we assumed the flow was fully mixed). An alternate procedure to calculate Tm involves integrating the temperature profile. This should be the case if procedure is covered in the Verification & Validation section in the video entitled Check Energy Conservation via Mixed Mean Temperature Variation. If energy is conserved in the FLUENT simulation, the values calculated using the two procedures should match.

Wall Temperature at an Axial Location

For calculating the Nusselt no. at an axial location, we need the wall temperature at that location. The wall temperature at an axial location can be calculated in two ways:

• By interpolating the Tw vs. x values exported to Excel from the wall temperature plot obtained above. 
• By directly extracting the wall temperature at the desired location using the Probe function in the post-processor. 

The following video shows following video show you the procedure for extracting the wall temperature at x=2.67 m using the Probe function; this value can then be used to calculate the Nusselt number. To repeat the calculation at a different axial location, you can right-click on appropriate items in the tree, duplicate and modify as necessary. You need to double-click on an item in the tree to modify it; this is easy to overlook.

...

1. Create a line at x = 2.67
2. Go to Expressions, right click and click New
   1. Name Tw267
3. Right click in Definition box > Functions > CFD-Post > maxVal
4. Right click within the parenthese of maxVal > Variables > Temperature
5. Right click after the @ sign > Locations > x267Click Apply
   

Nusselt No.

To calculate the Nusselt no.:

• Export values from your Tw vs. x plot to an Excel file by clicking on the Export button, as previously discussed in the Wall Temperature Plot section. You'll see that the Excel worksheet contains many values of Tw vs. x.  
• Then calculate Nu directly in the same worksheet from Tw and Tm. 

Cornell MAE 4272 students, Fall 2020: Please use the following procedure for calculating the Nusselt number at two axial locations in the heated section where the flow is thermally fully-developed.  

The convective heat transfer coefficient, h can be determined from Newton's law of cooling:

Image Added

The wall heat flux on the left hand side is known from the boundary condition. We have shown you how to get the wall temperature T_w at an axial location. Calculate the mixed mean temperature at the same axial location by evaluating the following integral using the procedure shown in the video in the Verification and Validation section. The title of the video is Check Energy Conservation via Mixed Mean Temperature Variation. 

Image Added

Once you determine h, you can non-dimensionalize it to get the Nusselt number.

The Nusselt number in the thermally fully-developed region should be the same at different axial locations. Why? Do your Nu values at the two axial locations compare reasonably well? Why or why not?This yields a nice curve of Nu vs x.

<iframe width="640" height="360" src="//www.youtube.com/embed/WiK1uBTdK-M?rel=0" frameborder="0" allowfullscreen></iframe>

<iframe width="640" height="360" src="//www.youtube.com/embed/NbdGrJTesZg?rel=0" frameborder="0" allowfullscreen></iframe>

<iframe width="640" height="360" src="//www.youtube.com/embed/6RNykoM86xA?rel=0" frameborder="0" allowfullscreen></iframe>

<iframe width="640" height="360" src="//www.youtube.com/embed/Jcht7CAPzAc?rel=0" frameborder="0" allowfullscreen></iframe>

<iframe width="560" height="315" src="https://www.youtube.com/embed/x-flaXXU7xg?rel=0" frameborder="0" allowfullscreen></iframe>

Input Summary

You In older versions of FLUENT, you can view the input summary (model, material properties, boundary conditions, etc) by clicking on Report in the menu bar of FLUENT. A small window will pop up and you can print the selected input summary directly in FLUENT.

...