Simulating flow within a Mixing-Tee.

Aim : To obtain the mixing-efficiency of a Mixing-Tee, varying the length of the geometry and the momentum ratios used.

Operating conditions:

The following are the conditions that have been used for two cases involving the short mixing tee and the long mixing tee:

• Inlet temperature : 19 degrees Celsius
• Outlet temperature : 36 degrees Celsius
• Inlet velocity along X- axis for both cases : 3 m/s
• Inlet velocity along Y- axis for cases with momentum ratio of 2 : 6 m/s
• Inlet velocity along Y- axis for cases with momentum ratio of 4 : 12 m/s

The images shown below display the mesh type and the geometry of the mixing tee, long and short length:

Case 1 : Short length Mixing Tee:

Case 2 : Long length Mixing Tee:

Model Usage:

For the first case, two models were used:

1) K-epsilon

2) K-Omega

The other cases utilize the K-epsilon model, mainly due to the fact that the residuals obtained after running the simulation make it clear at what iteration the convergence of the results has been obtained. In the K-Omega model, the residuals did not seem to converge, and gauging the number of iterations required for convergence is unclear. The following images display this difference.

K-epsilon model:

K- omega model:

The K-epsilon model produces clear convergence, and the number of iterations required for convergence is also made clear, by the repetitive oscillations once converged. However, on further inspection, the K-omega model produces similar results when it comes to mass-average of outlet temperature and velocity. But, the K-epsilon model will be used for the reason mentioned above.

(I)Mesh details:

• The mesh element size used was 0.002 m, after performing a mesh independence test, for a short mixing tee. In order to avoid bias during comparsion, the element size for both mixing tees was taken as 0.002 m.
• The following table shows the comparison between cell count, average temperature, and the number of iterations required for convergence of the results.

Fig I(A) : Comparision of different variables for the Case 1 and Case 2.

Explanation:

1) Cell Count:

The cell count shows the number of cell/nodes that are being used during the computation of the simulation. This is important to note, as it shows how precise the solution will be. The precision of the solution increases with the cell count, and after a certain number of cells, the solution does not change with cell count. This is known as mesh independence, and it shall be discussed in a section below.

The cell count is higher for the longer pipe length, at 28526 cells, which was expected. The shorter length pipes only require 21445 and 21524 cells respective to their cases.

2) Average outlet temperature:

A mass-average of the temperature at the outlet plane has been computed, and the temperatures have been displayed in the table shown in Fig I(A). The cases with a momentum ratio of 2 have an overage outlet temperature of around 30 degrees Celsius. For the

cases with a momentum ratio of 4, the average outlet temperature comes around to 27 degrees Celsius.

These points will be elaborated further in the results and conclusions section.

The average outlet temperature has been obtained by selecting the Surface Integral option, under the post- processing section of Fluent. The following image shows this procedure.

3) Number of iterations required for convergence:

Observing the table displayed above, the cases with a momentum ratio 2 converge faster compared to the cases with a momentum ratio of 4.

The following image displays the residual plots obtained after the simulation in Fluent, and consequently displays the number of iterations the case requires in order to converge.

In the next section, the temperature and velocity profiles will be displayed and elaborated on.

PLANE PLACEMENT FOR CONTOURS:

The following image displays the placement of the planes along and across the geometry:

(II.A) VELOCITY AND TEMPERATURE CONTOURS OF CASE 1:

(a) Velocity and temperature contours across the pipe:

Positions of the contours across the pipe geometry. [length along X-axis]

Inlet	-0.0773323 m
Before Inlet Y	-0.05 m
After Inlet Y	-0.033 m
Middle	0.015 m
Near Outlet	0.07 m
Outlet	0.115046 m

OBSERVATIONS:

CASE 1:

A comparison between the velocity and temperature contours of the same pipe with different momentum ratios will be made in this section (they will be referred to as cases, and are not to be confused with Case I and Case II).

Momentum ratio of 2, short pipe - case(a)

Momentum ratio of 4, short pipe - case(b)

• At the inlet, there is virtually no difference between the two cases. The section before Inlet Y experiences a slight change in temperature near the area affected the increased velocity, as we can see a slightly increased temperature drop in the affected area, in case(b).
• The Inlet Y carries the fluid at a higher velocity, and in the area experiencing the temperature drop is much larger in the case(b), when compared to the case(a). This can be seen in the temperature contours that show the temperature at the section after Inlet Y.
• At the middle section, we can see a drastic difference in temperature, between case(a) and case(b). In case(a), the temperature of the middle section is still at around 36 degrees Celsius near the bottom of the pipe. This is not the case with case(b), as the temperature range in that area dips down to the range of 20 to 28 degrees celsius, as shown in the images above.
• The outlet section of the pipe shows the final temperature distribution and it reveals the difference in efficiency between the two cases. At the outlet section of case(a), the temperature is still upwards of 30 degrees Celsius. For case(b), the temperatures have settled in the range of 23.25 degrees Celsius to below 30 degrees Celsius. As shown in the table above (Fig I(A)) , the average outlet temperature for case(a) is 30.277 degrees Celsius, and case(b) at 27.572 degrees Celsius.
• The velocity of the contours containing the velocity of the fluid across the pipe will make more sense along with the velocity contour of the plane section along the middle of the pipe, and will be elaborated on in the next section II(b).

(b)Velocity and temperature contours along the pipe:

OBSERVATIONS:

• Observing the velocity contours of the sections across the pipe, it can be observed that the velocity just after the inlet Y increases beyond the velocity provided at Inlet Y. This will be observed in more detail in the following image.

• In the above image, it can be observed that the velocity is maximum just after the exit bend of the Inlet Y, in both case(a) and case(b).
• However, the fluid flow from inlet Y in case(a) does not carry its initial momentum to the bottom of the pipe, and slows down considerably. This was an expected outcome, and it can be gleaned that the velocity difference is not sufficient for efficient mixing. In case(b), the fluid flows at a higher velocity, and reaches the bottom of the pipe.
• The temperature contours shown above make this point clearer, as the temperature contour for the plane along the middle of the pipe displays the validity of the point mentioned above.
• In case(a), the difference between the velocities of both inlets is not sufficient to create good mixing. The temperature contour displays this, as the inlet temperature of 36 degrees Celsius is carried throughout, to the end of the pipe, as seen at the bottom of the pipe. This adversely affects the average outlet temperature.
• This is not the case with case(b), as it can be observed that the flow of the fluid at 36 degrees Celsius coming from the inlet is completely checked by the flow of the cooler fluid from Inlet Y, and the bottom of the pipe is in the range of 23.25 to 27.50 degrees Celsius.
• A primitive conclusion that can be made, observing the points mentioned above, is the mixing efficiency of the short pipe with a momentum ratio of 4 is superior to the short pipe with a momentum ratio of 2.

In the section below, the velocity and temperature contours of Case 2 will be discussed.

(II.B) VELOCITY AND TEMPERATURE CONTOURS OF CASE 2:

(c) Velocity and temperature contours across and along the pipe:

Positions of the contours across the pipe geometry. [length along X-axis]

Inlet	-0.0773323 m
Before Inlet Y	-0.05 m
After Inlet Y	-0.033 m
Middle	0.055 m
Near Outlet	0.15 m
Outlet	0.190996 m

OBSERVATIONS:

Momentum ratio of 2, Long length - case(c)

Momentum ratio of 4, Long length - case(d)

• The contours shown above will not differ much at the section after Inlet Y, and the trends between both sections are nearly identical, if Case 1 and Case 2 are compared, and the effect of the length of the pipe is seen at the section near the outlet.
• For all intents and purposes, the differences between case(c) and case(d), are the same as case(a) and case(b).
• The effect of length is very subtle, as shown by the average outlet temperature of both case(a) and case(b) are nearly identical to case(c) and case(d) in table shown in Fig.I(A).
• In case(d), we see some heat generation on the top of the pipe, between the section after Inlet Y and the middle section of the pipe. This heat generation is not seen in case(c). This may be an indication of proper mixing, as the hot fluid will rise due to lesser density, in the mixture of the hot and cold fluids.
• The next section will cover the differences between Case 1 and Case 2, using the help of line plots,and effect of the longer length on velocity.

(III) Line plots of velocity and temperature along and across the pipe geometry:

Line Locations:

For the following line plots showing temperature and velocity along, a line is plotted along the axis of the the pipe, and two other lines are plotted 0.01 m above and below the axis.

For the line plots showing the temperature and across, four lines are plotted across the pipe length, along the diameter of the pipe, after the Inlet Y, middle of the pipe, near the outlet, and the outlet of the pipe.

(Note: The following images use CFD post for pictoral representation of the lines only).

• The image below displays the way the line was plotted across the geometry.
• The post-processing suite in fluent was used to create an XY-plot. Lines are created by using the New surface option, and the co-ordinates of the points are entered, in order create the line.
  

• The following figures show the line plot for velocity and temperature along the pipe for both, Case 1 and Case 2.

Along pipe geometry:

Case 1 and case 2 :

Axis of the pipe - Red

Above axis of the pipe - White

Below axis of the pipe - Green

Case 1 (Short length, momentum ratio = 2/4):

After inlet Y - Green

Middle - Red

Near outlet - Blue

Outlet - White

Case 2 (Long length, momentum ratio = 2/4):

After inlet Y - White

Middle - Red

Near outlet - Green

Outlet - Blue

OBSERVATIONS:

(*Note: MR = Momentum Ratio)

1) TEMPERATURE LINE PLOTS ALONG AND ACROSS THE PIPE:

• The temperature line plots along the length of the pipe prove the conclusions obtained from the temperature contours shown above, and give more information regarding the behaviour of the fluid in the domain.
• The behaviour of the cases with the case momentum ratio are similar to each other in many respects. The major differences between the cases with the MR of 2 and MR of 4 is the behaviour of fluid temperature after Inlet Y. The temperature is much higher along the axis of the pipe and above the axis of the pipe in the cases with MR of 4, compared to the two cases with MR of 2. However, the temperature of the line below the axis is significantly higher in the cases with MR of 2. From this, it can be said that the volume of fluid below the axis by 0.01 m to the base of the pipe, has not been affected by the introduction of the cooler fluid entering via inlet Y in the cases with a MR of 2, as shown in the line plot. The line,'below the axis', does not show a drastic temperature drop near the Inlet Y, and this is a sign of improper mixing. The temperature of the fluid on the line below the axis by 0.01 m remains above 30 degrees Celsius, and the other two lines(line above axis and line along axis) experience a drastic temperature drop and proceeds to reduce in temperature along the length. The cases with MR of 4 display much better mixing efficiency, as it can observed that all three lines are affected by the introduction of the cooler fluid, and experience a temperature drop. None of the temperatures exceed 30 degrees Celsius at the end of the pipe, for these lines.
• Looking at the plots that display the plots across the pipe, along the horizontal pipe diameter, the results make it clear that the temperature fluctuation is much smoother in the cases with MR of 4. This can be observed in the temperature plots of the positions near the outlet, and outlet. The temperature range is between 26 to 30 degrees Celsius for the short pipe and 26 to 29 degrees Celsius for the long pipe in the cases with MR of 4. The temperature range trend followed by the cases with momentum ratio of 2 is high temperature near the circumference of the pipe, and relatively lower temperature around the center. The temperature variation is more severe for a short pipe, compared to the longer pipe( in the same case). This does not mean that the longer pipe having a MR of 2 is superior, for the tendency of the top of the pipe to have a higher temperature increases with increase in length, and in the end, nullifies this minute advantage.

2) VELOCITY LINE PLOTS ALONG AND ACROSS THE PIPE:

• The velocity line plots of the velocity along the pipe reiterate the conclusion that obtained by analysing the temperature plots, as it can be clearly seen that the velocity of the line below the axis line, has a higher velocity in the cases with an MR of 4 compared to the cases with an MR of 2.
• This can only mean one thing, and that is that the velocity provided in the cases, taking a momentum ratio of 2, is not enough for efficient mixing of the cooler fluid with the warm fluid.
• The velocity line plots of the velocity across the pipe display the general trend followed by the fluid at those particular sections, and solidify the point mentioned above.

CONCLUSIONS:

Upon analysing the results displayed above, there are two conclusions that can be made.

1. The cases in which the coolant fluid has a higher entrance speed, assist in increasing the efficiency of reducing the temperature of the main fluid flowing in the pipe, despite being at the same temperature upon entry into the domain. Hence, it can be safely said that upon increasing velocity of the fluid to a certain extent, will increase temperature drop of the hot fluid to a certain extent.
2. Observing the average temperature at the outlet, we can see that the longer pipe is better than the short pipe, by an extremely small amount, in the case with a momentum ratio of 4. However, this difference is not significant enough to justify the use of a longer pipe, and will only play a role in the development of the velocity profile. In the cases where the momentum ratio is 2, the longer length is found to be slightly disadvantageous, also by an extremely small amount. But again, the difference is too small between the cases. This being said, the length does not play a vital role in the mixing efficiency of the fluid, and does not have a major effect on the temperature drop of the fluid.

MESH INDEPENDENCE TEST:

• The mesh independence test was performed by keeping the element size of the mesh as the main parameter. The element size was varied between 0.004 m to 0.0009 m. The cell and element count was noted, and the simulations were run by assigning element size to design points. The results required were the average velocity and temperature at the outlet, and the variation of these results would be compared in order to observe any large jumps between each successive result, with respect to the element size. Increasing the mesh density will yield more accurate results, to a certain extent.
• After a certain limit, the results will not change, and at that point, we may take the previous element size, and use that for computation, preserving both, accuracy and speed of computation.
• The following images will display this process, and will be elaborated on slightly.

1. Outline :

The parameters are created by entering the mesh section of the simulation, as shown below.

2. Mesh parameter selection :

After entering Mesh, a display opens, where the element size, node and number count are seen. Clicking the box beside it classifies these factors as parameters, indicated by a blue coloured 'P'.

3. Fluent parameter selection:

Entering the Fluent module, there is a 'User Defined' section, where 'parameters', is observed. Selecting this allow us to create the output parameters for mass-average of temperature and velocity at the outlet.

4. Design Points:

The mesh element size is then assigned different Design Points, which uses the given parameters to calculate the temperature and velocity mass- averages at the outlet, and then observe the trend.

5. Graph plotting:

The table, once completed , is opened in Excel. upon doing this, two charts are plotted. The image below displays the table and the charts.

The mesh node values are plotted against their respective mass-averages of temperature and velocity, and hence, a graph is plotted by selecting the columns containing the values, using the 'Insert Scatter' tool in excel to plot the graph.

Looking at Mesh Nodes vs Velocity, it can be observed that there is not much change after 20000 nodes, as there are no significant jumps in the value of velocity. Observing this, it can be assumed that the mesh element size of 0.002 m is good enough to carry out the simulation, whilst preserving the accuracy of the results along with better computational speed, compared to the Design Points that exceed this mesh node number value.

Looking at the second graph containing temperature, an arguement can be made that the mesh element size of 0.0015 m may prove more accurate with respect to temperature, but to keep computational time within tolerable limits, the Design Point of 0.002 m has been chosen.