Mixing efficiency
OBJECTIVE:
To analyze the mixing effectiveness of the mixing tee (Like pipe who has two Inlet and one Outlet) using different momentum ratios and different lengths.
a) Find out Velocity and temperature contour plots along and across the pipe.
b) Find out Velocity and temperature line plots along and across the length of the pipe.
c) Perform Mesh independent study.
d) Discuss the effect of length and momentum ratio on results.
Given Inputs:
Inlet-x temperature = 36 degree celcius
Inlet-y temperature = 19 degree celcius
Momentum ratio= Inlet-y velocity(cold Inlet)/ Inlet-x velocity(hot Inlet)
Inlet-x Velocity = 3 m/s
Analytical Solution
The reference formulas will be used from the challenge and can be seen below:
Here only the formula for temperature will be used because
`h=mcunderset(p)( )dt`
here the mass flow rate and the value of Specific Heat will be same because the initial conditions and the working fluid is same.
Thus with the help of teh reference formula
T mixture=30.32C or 303.47K for the momentum ratio 2
T mixture=27.4915C or 300.6415K for the momentum ratio 4
where,
Radius Hot pipe=16.94
Radius Cold pipe=8.48
Turbulence Models
RANS based two turbulence models are used to analyze the mixing effectiveness of the mixing tee.
1) K-epsilon realizable Turbulence model with standard wall function
An immediate benefit of the realizable k-ɛ model is that it provides improved predictions for the spreading rate of both planar and round jets and captured turbulence in the bulk flow. While wall function added here which capture the information near the wall.
Pros: Robust, Easy to implement. Valid for fully turbulent flows only. Suitable for bulk flow, initial screening of alternative designs, and parametric studies.
Cons: Performs poorly for complex flows involving severe pressure gradient, separation, strong streamline curvature. And wall shear stress overestimated by wall function so that delay in separation occurs. Another shortcoming is numerical stiffness when equations are integrated through the viscous sublayer which is treated with damping functions that have stability issues.
2) SST K-Omega Turbulence Model
Shear Stress Transport (SST) is a variant of the standard k–ω model and used when there are wall effects present within the case.
Pros: The SST model accounts for the transport of turbulent shear stress and gives highly accurate predictions of the onset and the amount of flow separation under adverse pressure gradients. SST is recommended for high accuracy boundary layer simulations.
Cons: Dependency on wall distance makes this less suitable for free shear flows compared to standard k-w. It requires mesh resolution near the wall.
Short Mixing T Geometry:
For solving this problem for short tee geometry we will start with the grid dependency test first and find the best grid size to solve the problem such that the final results become grid independent. For preforming the grid dependency test we will start with the coarse mesh size i.e. 5e-3. then we will refine the mesh and check the results further.
Before proceeding to the grid dependence test we will extract the fluid volume in which our liquid will flow. The volume is extracted with the help of space claim module in Ansys Workbench. The extracted volume can be seen below:
The mesh of sizing 5e-3 can be seen below:
The quality of this mesh can be seen below:
The number of elements are 12789 and number of nodes are 2761.
After giving the proper name to the boundaries the geometry was imported in the Ansys fluent domain where the working fluid was given as air. The properties of the air can be seen below:
The grid dependency test was done for the first case .i.e for realizable k-epsilon model with the momentum ratio of 2. Further the inlet velocity was kept as 3m/s with the temperature of 36C for the hot fluid wheras for the cold fluid the velocity was kept as 6m/s and temperature as 19C. The energy equation was turned on for the method such that the temperature could be solved under this method. Fyrther the walls were kept at noslip BC and the outlet at constant prssure of 0 guage.
The results from this mesh size can be seen below:
1.) Residual Plot
2.) Area average of temperature at outlet
3.) Standard deviation of temperature at outlet
4.) Temperature Contour at mid plane
4.) Velocity Contour at mid plane
5.) Temperature Steamline
6.) Velocity Streamline
Further two lines were made in fluid volume as shown and velocity and temperature plots were made over these line.
1.) Temperature change along outlet
2.) Velocity at Outlet
3.) Velocity at mid line
4.) Temperature at Mid Line
Further with the similar Boundary and initial conditions the problem was solved for mesh size of 3e-3. The results can be seen below:
The mesh of sizing 3e-3 can be seen below:
The quality of this mesh can be seen below:
The number of elements are 58890 and number of nodes are 11494.
The results from this mesh size can be seen below:
1.) Residual Plot
2.) Area average of temperature at outlet
3.) Standard deviation of temperature at outlet
4.) Temperature Contour at mid plane
5.) Velocity Contour at mid plane
6.) Temperature Steamline
7.) Velocity Streamline
8.) Temperature change along outlet
9.) Velocity at Outlet
10.) Velocity at mid line
11.) Temperature at Mid Line
Further with the similar Boundary and initial conditions the problem was solved for mesh size of 1e-3. The results can be seen below:
The mesh of sizing 1e-3 can be seen below:
The quality of this mesh can be seen below:
The number of elements are 1553949 and number of nodes are 273797.
The results from this mesh size can be seen below:
1.) Residual Plot
2.) Area average of temperature at outlet
3.) Standard deviation of temperature at outlet
4.) Temperature Contour at mid plane
5.) Velocity Contour at mid plane
6.) Temperature Steamline
7.) Velocity Streamline
8.) Temperature change along outlet
9.) Velocity at Outlet
10.) Velocity at mid line
11.) Temperature at Mid Line
Further to get more concrete data to figure out with which mesh size to proceed the values at different points have been tabulated below:
The values have been tabulated at the two planes that were made in between the pipe and the outlet plane. The two planes can be seen in the figure below:
The tabulated data can be seen below:
| Distance of the Plane from the origin | |||
| Mesh Size | 0 | 57.5 | Outlet |
| 5e-3 | 302.59 | 303.424 | 303.557 |
| 3e-3 | 302.458 | 303.347 |
303.478 |
| 1e-3 | 302.59 | 303.704 | 303.771 |
Here it can be seen that first the values of velocity drop upto the mesh size of 3e-3 and then increase slightly for the mesh size of 1e-3. For the size og 5e-3 and 3e-3 it can be noticed that the values are almost equal. It can also be observed that with increse in the mesh size the mesh quality degrades thus a more finer mesh cant be used as it may give errors ehile solving the problem. It can also be obsereved that with increase in the mesh size the number of iterations required to solve the model also increase. Further the diffusion of the hot and cold liquid can be seen more in the with increase in mesh size thus the final mesh size will be taken as 3e-3 and the further problems will be solved with that mesh size.
Further proceeding in the problem we will solve this problem with the mesh size of 3e-3 and for the momentum ratio of 4. For this momentum ratio the inlet velocity for cold fluid will change from 6m/s to 12m/s.
The mesh size and the quality will be same as shown above the further plots and residuals can be seen below:
1.) Residual
2.) Area Average of temperature at Outlet
3.) Standard Deviation of Temperature at Outlet
4.) Temperature Contour at Mid Plane
5.) Velocity Contour at Mid Plane
6.)Temperature Streamline
7.) Velocity Streamline
8.) Temperature at outlet
9.) Velocity at Outlet
10). Temperature at Mid Line
11). Velocity at Mid Line
Further the problem will be solved at the momentum ratio of 2 for the k omega sst model:
For this problem again the inlet velocity at the hot inlet is 3m/s and temperature is 36C and at cold inlet the velocity is 6m/s and the temperature is 19C. The residual plots and various velocities can be seen below:
1.) Residual
2.) Area Average of temperature at Outlet
3.) Standard Deviation of Temperature at Outlet
4.) Temperature Contour at Mid Plane
5.) Velocity Contour at Mid Plane
6.)Temperature Streamline
7.) Velocity Streamline
8.) Temperature at outlet
9.) Velocity at Outlet
10). Temperature at Mid Line
11). Velocity at Mid Line
Further the problem will be solved at the momentum ratio of 4 for the k omega sst model:
For this problem again the inlet velocity at the hot inlet is 3m/s and temperature is 36C and at cold inlet the velocity is 12m/s and the temperature is 19C. The residual plots and various velocities can be seen below:
1.) Residual
2.) Area Average of temperature at Outlet
3.) Standard Deviation of Temperature at Outlet
4.) Temperature Contour at Mid Plane
5.) Velocity Contour at Mid Plane
6.)Temperature Streamline
7.) Velocity Streamline
8.) Temperature at outlet
9.) Velocity at Outlet
10). Temperature at Mid Line
11). Velocity at Mid Line
Discussion For Shot T Method
For the different methods the average temperature at outlet is computed below along the number of iterations needed to solve the equation:
| Method | Average Output Temperature | Number of iterations |
| K-epsilon momentum ratio 2 | 303.478 | 350 |
| K-epsilon momentum ratio 4 | 300.753 | 400 |
| K-omega momentum ratio 2 | 303.554 | 375 |
| K-omega momentum ratio 4 | 300.793 | 350 |
Here it can be seen that for both the methods the number of iterations are almost eqaual or similar for solving both the cases. In the average output temperature it can be seen that the Kepsilon method has more closer values to that of the k omega sst model as compared to the analystical solution given in the reference formula. As we know that bothe the methods are correct but have different abilities to capture the different phenomenon, but here we are interested in the output results and they are closer in k-epsilon method thus we will go with the k-epsilon realisable method.
Long Tee Geometry
The long tee geometry can be seen. This can be seen below:
The extracted geometry can be seen below:
We will be solving this for the same grid size of 3e-3 as the radius of volume remains the same only the change in length is observed thus the same grid size will give the similar results in the problem. The mesh for the long tee can be seen below:
The mesh quality can be seen below:
The number of elemwnts in mesh are 80600 and the number of nodes in the mesh are 15619.
Further for solving the equation for the momentum ratio of 2 the inputs are given as shown above for short tee .i.e. the inlet velocity of 3m/s and temperature of 36C for hot fluid and for that of cold fluid 6m/s and 19C respectively.
The residuals and various other plots can be seen below:
1.) Residual
2.) Area Average of temperature at Outlet
3.) Standard Deviation of Temperature at Outlet
4.) Temperature Contour at Mid Plane
5.) Velocity Contour at Mid Plane
6.)Temperature Streamline
7.) Velocity Streamline
8.) Temperature at outlet
9.) Velocity at Outlet
10). Temperature at Mid Line
11). Velocity at Mid Line
Further for solving the equation for the momentum ratio of 4 the inputs are given as shown above for short tee .i.e. the inlet velocity of 3m/s and temperature of 36C for hot fluid and for that of cold fluid 12m/s and 19C respectively.
The residuals and various other plots can be seen below:
1.) Residual
2.) Area Average of temperature at Outlet
3.) Standard Deviation of Temperature at Outlet
4.) Temperature Contour at Mid Plane
5.) Velocity Contour at Mid Plane
6.)Temperature Streamline
7.) Velocity Streamline
8.) Temperature at outlet
9.) Velocity at Outlet
10). Temperature at Mid Line
11). Velocity at Mid Line
Further the average temperature of the outlet have been tabulated in the table below:
| Method | Number of Iterations | Average Outlet Temperature |
| K omega momentum ratio 2 | 400 | 303.53 |
| K omega momentum ratio 4 | 450 | 300.688 |
Disscussion:
- It can be seen that the length of the Tee doesnt impact much on the outlet temperature.
- It can also be noticed that the increse in momentumm ration decrese the outlet temperature of the mixing Tee.
- For all the geometry (long and short) the flow can be seen in developing region and it has not fully developed yet.
- Further it is observed that the temperature at outlet for momentum ratio two is minimum in the center region but it is exact opposite in case of momentum ration 4 where the minimum temperature stream lines are concentrated at the walls.
- For the velocity and temperature plots in the mid line it can be seen that in all the cases these paremeters first increse and as it touches the cold inlet these values suddenly drop and then start to increse steadily.
- The velocity at the walls can be seen zero this is due to the no slip boundary condition.
- A low pressure area can be seen developed near the entry of cold fluis in the main tube this is due to which the velocity falls and the temperatures increases in that area.