Mixing efficiency

1. Project Aim : To calculate the mixing efficiency of a Tee - Joint in a circular pipe.

2. Relevant theory :

What is a Tee - Joint ?

Tee - Joint is basically a flow control device and these can be found in many industrial applications. It is called a Tee - Joint because it splits the flow and the angle between them is 90 degrees.

How to measure the efficiency of mixing?

The standard deviation of the temperature at the outlet would be calculated with the help of statistics

Higher standard deviation implies poor mixing efficiency

Lower standard deviation implies good mixing efficiency

Application

Let us understand one of the applications of the Mixing Tee for getting the air at a desired temperature from the outlet circular pipe having a Tee-Joint. The figure below is self explanatory and shows the Mixing Tee and its application in the air conditioning systems' design.

3. Objectives of the project :

Main Objectives :

• to set up steady-state simulations to compare the mixing effectiveness
• to use the k-epsilon and k-omega SST model for the first case and based on judgment use the more suitable model for further cases.
• to give the reason for choosing a suitable model

Problem Description / Given Parameters:

• hot inlet temperature is 36⁰C & the Cold inlet is at 19⁰C

• Case 1
• Short mixing tee with a hot inlet velocity of 3m/s.
• Momentum ratio of 2, 4.
• Case 2
• Long mixing tee with a hot inlet velocity of 3m/s.
• Momentum ratio of 2, 4.

Momentum ratio = velocity at cold inlet / velocity at hot inlet

Expected Results :

1. Velocity and temperature contour plots on the cut planes along and across the pipe.
2. Velocity and temperature line plots along and across the length of the pipe.
3. Perform Mesh independent study for any one case.
4. Discuss the effect of length and momentum ratio on results.
5. For each simulation, you need to show the convergence plot.
6. In a table, compare the following values from the 2 cases on the basis of

1. i) Cell count ii) Average outlet temperature iii) Number of iterations for convergence

4. Work Flow for the project :

5. Procedure :

5.1 CASE 1 : Analysis of short mixing Tee

5.1.1 Extracting the Fluid Model from the Solid Model of a short Mixing Tee

The already generated CAD solid model was imported from the SOLIDWORKS and then using Space-Claim the fuid volume model was extracted. Length of the short mixing tee is approximately 0.2 m.

Fig1. Fluid Volume Model of Short Mixing Tee

5.1.2 Meshing and Grid independence studies for short tee analysis wiith momentum ratio 2

Different Meshing stategies were tried for the short mixing tee by varing the element size. The table below shows the grid independence study done after carefully examining the residual,std deviation and avg temperature plots. Finally the element size of 0.002 m was selected as a mesh which was fine enough to get the desired results without extending the nodes restrictions of Student Version of ANSYS. Turbulence model used was K-Epsilon (Realizable).

Table1. Grid Independence Study for Short tee

So, selected tetrahedral mesh for the element size of 0.002 m looks like as shown in figure.

Fig2. Actual tetrahedral mesh with the mesh quality

5.1.3 Setting up physics, boundary conditions, outputs to be plotted and Initializing the solution:

Physics:

• Pressure-based steady state method to be adopted
• Energy equation to be solved along with the NS equations
• Turbulence model (both k epislon (realizable) and k omega (SST) were studied and differences studied will be examined below later)

Boundary Conditions:

• Inlet velocity = 3 m/s
• Hot inlet temperature = 36 degree C
• Cold inlet temperature = 19 degree C
• Momentum ratio = 2 and 4

Output reports:

• Avg temperature of mixing at the outlet (To see how the momentum ratio affects the outlet temperature)
• Std deviation of temperature at the outlet (Lower the std deviation at the outlet, effective is the mixing)

Initialization used :

• hybrid initialization

5.1.4 Converged results obtained for the analysis of short mixing tee:

5.1.4.1 case 1 : Momentum ratio=2

Fig3. a) Residual b) Avg. outlet temperature c) Std. deviation of outlet temperature for realizable k epsilon turbulence model

a)

b)

c)

Fig4. a) Residual b) Avg. outlet temperature c) Std. deviation of outlet temperature for k omega SST turbulence model

a)

b)

c)

From above it can be seen that, k omega SST converged faster than the Realizable k epsilon model. But the std. deviation obtained from k omega is 2 whereas k epsilon captures the std deviation as 1.6. Moreover the avg outlet temperature shown by both the models is 30.25 degree C for momentum ratio of 2 and given inlet temperatures. So, it can be said that the mixing tee effectiveness (atleast in simulation results) is better if we choose realizable k epsilon turbulence model. So, realizable k epsilon model will be used for all further simulations.

5.1.4.2 case 2 : Momentum ratio=4

Fig5. a) Residual b) Avg. outlet temperature c) Std. deviation of outlet temperature for realizable k epsilon turbulence model

a)

b)

c)

For this case the simulation was carried out for 200 iterations to get the converged solutions for realizable k epsilon model for momentum ratio of 4. It can be seen from avg. out temperature that as the cold stream velocity is increased by the factor of 2, the avg. outlet temperature also decresed from 32.5 degree C to 27.6 degree C. Also, the std deviation of outlet temperature was aroud 1.4 degree C.

5.1.5 Post processed resultsa obtained for the analysis of short mixing tee :

5.1.5.1 Comparison of the temperature contours for momentum ratio 2 and 4 along the length of the mixing tee

Fig6. Temperature contour for momentum ratio a) 2 and b) 4

a)

b)

5.1.5.2 Comparison of the velocity contours for momentum ratio 2 and 4 along the length of the mixing tee

Fig7. velocity contours for momentum ratio a) 2 and b) 4

a)

b)

Fig7 a) and b) shows the velocity contour along the length. More turbulence can be seen when the momentum ratio is 4 and higher velocities are also encountered at the outlet.

5.1.5.3 Comparison of the temperature line plots for momentum ratio 2 and 4 along the center line of the mixing tee

Fig8. temperature line plots for momentum ratio a) 2 and b) 4

a)

b)

5.1.5.4 Comparison of the velocity line plots for momentum ratio 2 and 4 along the center line of the mixing tee

Fig9. velocity line plots for momentum ratio a) 2 and b) 4

a)

b)

From above figures, it can be seen that the velocity fluctuations are quite high for higher momentum ratio.

5.1.5.5 Comparison of the temperature contours for momentum ratio 2 and 4 across the length of the mixing tee at outlet

Fig10. temperature contour for momentum ratio a) 2 and b) 4

a)

b)

From figure a) and b) it can be seen that the temperature contour looks quite consistent and physically valid. Contour in figure b) has slightly cooler outlet air as compared to a).

5.1.5.6 Comparison of the velocity contours for momentum ratio 2 and 4 across the length of the mixing tee at outlet

Fig11. velocity contour for momentum ratio a) 2 and b) 4

a)

b)

From figure a) and b) the velocity contour at outlets are identical except for the magnitudes.

5.1.5.7 Comparison of the temperature line plots for momentum ratio 2 and 4 across the outlet of the mixing tee

Fig12. temperature line plots for momentum ratio a) 2 and b) 4

a)

b)

From figure a) it can be seen that the lower walls of the outlet show the highest temperature at the outlet which is exactly opposite to the one observed for the figure b). Thus momentum ratio affects the temperature distribution at the outlet cross-section.

5.1.5.8 Comparison of the velocity line plots for momentum ratio 2 and 4 across the outlet of the mixing tee

Fig13. velocity line plots for momentum ratio a) 2 and b) 4

a)

b)

From figure a) and b) it can be seen that momentum ratio does not affect the velocity profile at the outlet. Just the magnitude of velocity is more in b) while the variation as per the y axis remains almost identical in both cases. As per the physics also the distribution seems correct, because the velocity near the walls are close to zero where as it is highest near the centreline of the cross-section.

5.2 CASE 2 : Analysis of long mixing Tee

Fig14. Solid CAD Model of long mixing tee

5.2.1 Extracting the Fluid Model from the Solid Model of a short Mixing Tee

The already generated CAD solid model was imported from the SOLIDWORKS and then using Space-Claim the fuid volume model was extracted. Length of mixing tee is greater than the previous case.

Fig15. Extracted fluid volume model of long mixing tee

Fig16. Extracted fluid volume mesh of long mixing tee with element size of 0.002 m

Fig17. Fluid volume mesh quality of long mixing tee

5.2.2 Converged results obtained for the analysis of long mixing tee:

The other parameters remained same only the geometry is altered. Therefore, directly the converged results can be used. Turbulence model used again is realizable k epsilon.

5.2.2.1 case 1 : Momentum ratio=2

Fig18. a) Residual b) Avg. outlet temperature c) Std. deviation of outlet temperature for realizable k epsilon turbulence model

a)

b)

c)

5.2.2.2 case 2 : Momentum ratio=4

Fig19. a) Residual b) Avg. outlet temperature c) Std. deviation of outlet temperature for realizable k epsilon turbulence model

a)

b)

c)

The results converged in 170 iterations approximately for both the cases as shown in the figure. Also, the difference in avg outlet temperature can be noticed as shown in figure by changing the momentum ratio. Std. deviation is lower for higher momentum ratio, it means the mixing is better for the higher momentum ratio for the long mixing tee.

5.2.3 Post processed results obtained for the analysis of short mixing tee :

5.2.3.1 Comparison of the temperature contours for momentum ratio 2 and 4 along the length of long mixing tee

Fig20. Temperature contour for momentum ratio a) 2 and b) 4

a)

b)

5.2.3.2 Comparison of the velocity contours for momentum ratio 2 and 4 along the length of long mixing tee

Fig21. velocity contours for momentum ratio a) 2 and b) 4

a)

b)

5.2.3.3 Comparison of the temperature line plots for momentum ratio 2 and 4 along the center line of long mixing tee

Fig22. temperature line plots for momentum ratio a) 2 and b) 4

a)

b)

5.3.3.4 Comparison of the velocity line plots for momentum ratio 2 and 4 along the center line of long mixing tee

Fig23. velocity line plots for momentum ratio a) 2 and b) 4

a)

b)

5.3.3.5 Comparison of the temperature contours for momentum ratio 2 and 4 across the length of long mixing tee at outlet

Fig24. temperature contour for momentum ratio a) 2 and b) 4

a)

b)

5.3.3.6 Comparison of the velocity contours for momentum ratio 2 and 4 across the length of long mixing tee at outlet

Fig25. velocity contour for momentum ratio a) 2 and b) 4

a)

b)

5.3.3.7 Comparison of the temperature line plots for momentum ratio 2 and 4 across the outlet of long mixing tee

Fig26. temperature line plots for momentum ratio a) 2 and b) 4

a)

b)

5.3.3.8 Comparison of the velocity line plots for momentum ratio 2 and 4 across the outlet of the mixing tee

Fig27. velocity line plots for momentum ratio a) 2 and b) 4

a)

b)

Effects of increasing the length?

Only advantage of increasing the length of the mixing tee is to get and improved mixing of air, which is represented by std. deviation values as shown in the table above. Lower the std deviation value the better is the mixing.

Effects of increasing momentum ratio?

Increasing the momentum ratio decreases the outlet temperature to a good extent.

When to prefer k epsilon over the k omega models?

A short and basic guideline can be given as :

• k-omega near wall,
• k-epsilon just a little distance after the boundary layer
• LES in fair enough distance from the wall