Week 1- Mixing Tee

Aim: To set-up steady state simulations for computing mixing efficacy of a T-joint. 

 

Objectives:

1. To set-up a steady state simulation to compute mixing effectiveness of the hot and cold fluid.

2. To use RANS based turbulence models and compare them to obtain best suitable model for the given case.

3. To perform mesh independent study.

4. To discuss the effect of length of t-joint and momentum ratio on results. 

 

Software used for simulation: Ansys Fluent

 

Introduction:

In this project, we are simulating and analyzing thermal mixing in T-junction / T-joint. So let us approach required theory in baby steps.

What is T-Joint? (Mixing Tee)

• T-joint or T-junction is a very common component in pipe networks, mainly used to distribute (diverge) flow from main pipe to several branching pipes and to accumulate (converge) flow from many pipes to a main pipe of the system. In our case, we will be dealing with the T-joints used in Air Conditioning systems. 
• In this scenario, the function of T-joint will be used to supply chilled air from the AC unit to the air conditioned space. The air coming out from the AC system will have much lower temperatures. However to have user defined air temperatures at the outlet, the concept of mixing-tee comes into existence. 
• Mixing Tee is same as T-joint but it is used to mix the chilled air with hot air in order to get the mixing temperatures of both of these at the outlet. It uses the concept of air thermal mixing which allows the user to obtain varied temperatures in the air conditioned space as per the human comfort conditions.
• By adjusting th flow rate of the chilled air, user can have desired temperature of air at the outlet, which would be connected to the air conditioned space. 
• The mixing tee has various applications in the field Refrigeration and Air Conditioning (RAC), Nuclear reactor cooling (with different fluid medium, other than air), Power plant engineering and many more.

• The mixing temperature can be calculated as follows,

  `T_(mixture) = (dotm_(hot)xxT_(hot) + dotm_(cold)xxT_(cold))/(dotm_(hot) + dotm_(cold))`

 

                                                              

                                                            Fig (a): T-joint / Mixing tee used in Air Conditioning System

As we can see in the above picture, there is turbulent flow after mixing of both the outside and chilled air. This is because of the orientation of the fluid flow is orthogonal at the mixing point. Due to this there is highly transient temperature fluctuations at the pipe wall, which gives rise to thermal striping phenomenon.

Note:

What is thermal striping?

Thermal striping is a random temperature fluctuation produced by the incomplete mixing of fluid streams at different temperatures. It causes thermal fatigue damage of the structures exposed to such highly transient temperature fluctuations. 

Moreover, we also have heat transfer from the fluid medium (air) to the solid pipe walls, which is nothing but "Conjugate Heat transfer". However, our pipe walls will be insulated so there won't be any heat transfer from pipe wall to surrounding. This leads to transfer of heat again in the reverse direction (solid walls) to fluid medium (air). In this case we are not focusing on wall effects and so it can be added to future scope of the project.  

As we know after two fluids interaction at the mixing point, the flow conditions will become turbulent. This turbulence will ensure better mixing of hot air and cold air in the system. So it is important to resolve turbulence in more approximate manner to get better results. So in order to resolve for turbulence, we will be using Reynold's Average Navier-Stokes Equations (RANS/URANS). The solver will solve RANS/URANS equations in order to calculate flow field variables. However, to capture the turbulence variables such as (turbulence kinetic energy, turbulence eddy dissipation, specific dissipation rate etc) we need to solve additional equations to get turbulence quantities.

For the same purpose, we will use turbulence model. In this case, we will be using two turbulence models

a) K-epsilon (Realizable)

b) K-omega (SST) 

 

A] K-epsilon (K- `epsi`) (Realizable):

• K-epsilon turbulence model is the most common model used in CFD to simulate mean flow conditions.
• It is a two equation that gives a general description of turbulence by means of two transport equations ((PDE's).
• The first transport variable is turbulence kinetic energy (k).
• The second transport variable is the rate of dissipation of turbulent kinetic energy (`epsi`).
• Generally k-epsilon turbulence model is used for compressible/incompressible, external flow interactions with complex geometries.
• Limitations: It is not accurate for no-slip walls, adverse pressure gradients, strong curvature into flow and jet flows. 

B] K-omega (k- `omega`) (SST):

• K-omega turbulence model is a common two equation turbulence model, that is used as an approximation for the RANS equation.
• It attempts to predict turbulence by two partial differential equation for two variables, K and `omega` .
• 'K' represents 'Turbulence kinetic energy' and '`omega`' represents ' Specific turbulence dissipation rate'.
• It is similar to k-epsilon model but with improved accuracy for internal flows, curvatures, strong adverse pressure gradients, separated flows and jet flows. 
• The k-omega SST model has uses a combination of K-epsilon (in the outer region and outside of the boundary layer) and K-omega (in the inner boundary layer)

Prominently, the turbulent kinetic energy dissipation is carried out by convection (which is a combination of advection and diffusion).

The above theory explained is sufficient to understand the physics occuring in Mixing-Tee joint analysis.

So now we will move forward to set-up the CFD case for the thermal analysis of Mixing-Tee Joint.

Case studies:

In the case studies for the same momentum ratio, two different mixing-tee lengths (short and long) will be evaluated. Then only the effect of mixing-tee length on the mixing efficiency can be sensibly resulted. 

Cases	Momentum ratio	Mixing Tee length	Turbulence model
Case 1	2	Short	K-epsilon
Case 2	2	Short	K-epsilon
Case 3	2	Short	K-omega
Case 4	2	Long	*
Case 5	4	Short	*
Case 6	4	Long	*

 * = It depends upon the best optimum turbulence model after comparing K-epsilon and K-omega

 

CFD Case set-up:

General case set-up workflow:

 

                                

                                                 Fig (b): Workflow in Ansys Fluent

The workflow is followed sequentially as a part of CFD case set-up. In this workflow, there are intermediatary tools like "Space-Claim" (Geometry creation and clean-up / Pre-processing) , "Fluent" (Processing) and "CFD post" (Post processing).

Now let's get started with following workflow. 

1. Geometry

Initially, we will import the geometry to the Space-claim. We can also create the geometry here itself in Space-claim.                                                                                                                          

                                  

                                               Fig (c): Mixing-Tee geometry (short)

 

However, in this analysis, the only matter of concern with the geometry is the "Wet volume" (It is the volume occupied by the fluid).

                                      

                                                   Fig (d): Solid and Wet volume

1 (a). Geometry clean-up

The wet volume has been extracted from the solid volume in order to reduce extensive use of computation resources. 

                           

                                                             Fig (e): Wet volume

2. Mesh

In this step, initially name the patches according to the boundary conditions. 

2(a). Named selections:

I] Inlet_y       

It is the inlet for the chilled air coming out from the AC system.                                                                                            

       

II] Inlet_x

It is the inlet for the hot air from which it will enter the T-joint and mix with the chilled air.

III] Walls

The surface colored in red are the T-joint walls.                                                                                                          

             

IV] Outlet

The patch colored in red is defined as the outlet of the T-joint.

Note: This first two steps are common for each and every case set-up, remaining workflow has been maintained accordingly for different case studies.

 Case 1:

This case will be have less number of cells (coarse mesh) in order to execute mesh independent study. It will act as a baseline case, and results from this case would not be taken into consideration for the conclusion results.

Parameters:

T-joint length	Short
Momentum ratio	2
Hot fluid velocity and temperature	3m/s ; `36^0c`
Cold fluid velocity and temperature	6m/s ; `19^0c`
Element size	1e-02
Total number of cells	12,760
Turbulence model	K-epsilon (Realizable)

 

Mesh:

 

Mesh quality:

The mesh quality chart gives the insight about the number of elements following the quality criteria. 

 

Residuals plot:

Here all residuals of the variables in transport and governing equations are resolved as a function of time advancement. At 175 to 200 iterations, almost every variable achieves convergence at a pre-defined tolerance. Because there is no negative slope (decrease in the residuals) and the plot is showing straight line, we can assert convergence at such instances, instead running it for more iterations.

 

Area-Weighted Average of Temperature:

 

Standard Deviation of Temperature:

 

Report:

 

 

Post-Processing in CFD-post:

Contours:

1. Temperature                                                                                                              2. Velocity:

 

 

3. Turbulence kinetic energy:                                                                                   4. Turbulence eddy dissipation:

 

 

Discussion:

The above contour plot shows temperature, velocity, TKE (Turbulence kinetic energy) and TED (Turbulence eddy dissipation). From this generated contours, we can have a general idea about the distribution of the scalar and vector fields inside the mixing-tee. Moreover, the resolution is not so good to capture clear physics. This because of the coarse mesh used in the mixing-tee. But at the macroscopic level, we have an insight about the turbulence driven mixing. The two fluids at different velocities and mass flow rates are mixed at the critical point in space and heat transfer is taking place at the fluid contact interface.

Note: Detailed post-processing will be discussed at the end of the all case studies. For now, we will set-up different cases based on the turbulence models, momentum ratio and mixing-tee lengths.

 __________________________________________________________________________________________ 

Case 2:

Parameters:

T-joint length	Short
Momentum ratio	2
Hot fluid velocity and temperature	3m/s ; `36^0c`
Cold fluid velocity and temperature	6m/s ; `19^0c`
Element size	1.5e-03
Total number of cells	2,11,968
Turbulence model	K-epsilon (Realizable)

 

Mesh:

 

Mesh quality:

 

Residuals plot:

 

 Area-Weighted Average of Temperature:

 

 

Standard Deviation of Temperature:

 

Report:

 

Post-Processing:

Contours:

1. Temperature                                                                                                              2. Velocity:

 

 

3. Turbulence kinetic energy:                                                                                   4. Turbulence eddy dissipation:

 

 

Case 3:

In this case K-omega turbulence model is selected and results are derived to develop a comparison for best optimum turbulence model for further case studies.

Parameters:

T-joint length	Short
Momentum ratio	2
Hot fluid velocity and temperature	3m/s ; `36^0c`
Cold fluid velocity and temperature	6m/s ; `19^0c`
Element size	1.5e-03
Total number of cells	2,11,968
Turbulence model	K-omega (SST)

 

Mesh:

 

Mesh quality:

 

Residuals plot:

 

Area-Weighted Average of Temperature:

 

Standard Deviation of Temperature:

 

Report:

 

Post-Processing:

Contours:

1. Temperature                                                                                                              2. Velocity:

 

 

 

3. Turbulence kinetic energy:                                                                                   4. Turbulence eddy dissipation:

 

 

 

 

Intermeditary Results and turbulence model selection:

Analytical solution and Observation Table:

The temperature of the air at outlet can be given as,

`T_(mixture) = (dotm_(hot)xxT_(hot) + dotm_(cold)xxT_(cold))/(dotm_(hot) + dotm_(cold))`

Given data:

`T_(hot) = 36^0c` ; `T_(cold) = 19^0c`

Momentum ratio is 2, 

`V_(hot) = 3`m/s ; `V_(cold) = 6`m/s

`dotm_(hot) = rhoA_(hot)V_(hot) = 1.225xx225.9131xx3 = 3313.0959 (kg)/(s)`

`dotm_(cold) = rhoA_(cold)V_(cold) = 1.225xx225.9131xx6 = 1660.4612 (kg)/(s)`

`T_(mixture) = ((3313.0959xx36)+(1660.4612xx19))/(3313.0959+1660.4612)`

`T_(mixture) = 30.32^oc`

 

Cases	Turbulence model	Momentum ratio	Mixing-Tee length	Total number of cells	Area weighted average of outlet temperature	Standard deviation of temperature at the outlet	Number of iterations
Case 1 (Baseline case)	K-epsilon	2	Short	12,760	30.21	1.62	175-200
Case 2	K-epsilon	2	Short	2,11,968	30.34	2.0903	250-300
Case 3	K-omega	2	Short	2,11,968	30.41	2.1986	350-400

Discussion: 

The results for first three case studies has been obtained. The analytical solution for the outlet temperature (mixed air temperature) is used for case validation. 

All the three cases shows similar pattern in area-weigted average outlet temperature, just with minute differences. However, as we know that case 1 is having coarse mesh (less number of cells) and case 2 and 3 are having refined mesh/fine mesh (more number of cells). Still the differences between the both groups (coarse and fine mesh) are not that significant. There is little accuracy advancement in the results with such drastic mesh refinement. This can validate the mesh independent study by comparing the results. 

It is natural to have more number of iterations for refined mesh, as the solver needs to solve governing and transport equations for more number of cells, consumes time. Similarly in case of standard deviation we see good amount difference, this is because at the outlet in case 1 there are less number of cells. And as a consequence, the temperature variation across the mesh is not properly captured instead it is averaged (Less standard deviation). However, in refined mesh set-up, comparatively more of temperature variation is captured at the outlet (More standard deviation). 

The above explained comparison is sufficiently strong enough to assert "Mesh Independent Study" for this case.

Secondly, we need to select best optimum turbulence model for further case studies. In this mixing-tee thermal analysis, there is lot of physics to explore. However, we are interested in just deriving the mixing-tee efficiency in terms of outlet temperature. The analytical mixing temperature at outlet is `30.32^0c`. In this case studies, case 2 study (k-epsilon and refined mesh) approximates the analytical solution more accurately. However, case 3 uses k-omega as a turbulence model which tends to be less accurate in calculating mixing temperature at the outlet. As k-omega (SST) is generally used for internal flows though it tends to be less efficient as compared K-epsilon. Why such paradox? According to me, K-omega (SST) tends to capture wall effects more accurately than that of K-epsilon, however in this case we are not focusing on the wall effects (Heat flux diffusion). By taking wall effects in account, K-omega will be undoubtedly the best turbulence model to resolve this problem. 

In contrast to it, K-epsilon (Realizable) shows better results with less iterations for convergence. The standard deviation is also less than that for K-omega. As per our requirement for mixing efficiency, K-epsilon tends to be the best optimum choice as a turbulence model. In further case studies, we will use K-epsilon as turbulence model to resolve mixing temperature at the outlet.

 

Case 4:

Parameters:

T-joint length	Long
Momentum ratio	2
Hot fluid velocity and temperature	3m/s ; `36^0c`
Cold fluid velocity and temperature	6m/s ; `19^0c`
Element size	1.7e-03
Total number of cells	2,11,094
Turbulence model	K-epsilon (Realizable)

Mesh:

 

 

Mesh quality:

 

 

Residuals plot:

 

Area-Weighted Average of Temperature:

 

 

Standard Deviation of Temperature: 

 

 

Report:

 

 

Post-Processing:

Contours:

1. Temperature                                                                                                              2. Velocity:

 

 

3. Turbulence kinetic energy:                                                                                   4. Turbulence eddy dissipation:                                                             

 

 

Analytically derived outlet temperature of the mixed air:

The temperature of the air at outlet can be given as,

`T_(mixture) = (dotm_(hot)xxT_(hot) + dotm_(cold)xxT_(cold))/(dotm_(hot) + dotm_(cold))`

Given data:

`T_(hot) = 36^0c` ; `T_(cold) = 19^0c`

Momentum ratio is 4, 

`V_(hot) = 3`m/s ; `V_(cold) = 12`m/s

`dotm_(hot) = rhoA_(hot)V_(hot) = 1.225xx225.9131xx3 = 3313.0959 (kg)/(s)`

`dotm_(cold) = rhoA_(cold)V_(cold) = 1.225xx225.9131xx12 = 3320.9225 (kg)/(s)`

`T_(mixture) = ((3313.0959xx36)+(3320.9225xx19))/(3313.0959+3320.9225)`

`T_(mixture) = 27.48^oc`

 

Case 5: 

Parameters:

T-joint length	Short
Momentum ratio	4
Hot fluid velocity and temperature	3m/s ; `36^0c`
Cold fluid velocity and temperature	12m/s ; `19^0c`
Element size	1.5e-03
Total number of cells	2,11,968
Turbulence model	K-epsilon (Realizable)

 

Mesh:

 

Mesh quality:

 

 

Residuals plot:

 

 

Area-Weighted Average of Temperature:

 

 

Standard Deviation of Temperature:

 

 

Report:

 

 

Post-Processing:

Contours: 

1. Temperature                                                                                                              2. Velocity:

        

 

3. Turbulence kinetic energy:                                                                                   4. Turbulence eddy dissipation:

 

 

 

 

Case 6: 

Parameters:

T-joint length	Long
Momentum ratio	4
Hot fluid velocity and temperature	3m/s ; `36^0c`
Cold fluid velocity and temperature	12m/s ; `19^0c`
Element size	1.7e-03
Total number of cells	2,11,094
Turbulence model	K-Epsilon (Realizable)

 

Mesh:

 

 

Mesh quality:

 

Residuals plot:

 

 

Area-Weighted Average of Temperature:

 

Standard Deviation of Temperature:

 

 

Report:

 

 

Post-Processing:

Contours:

1. Temperature                                                                                                              2. Velocity:

 

 

3. Turbulence kinetic energy:                                                                                   4. Turbulence eddy dissipation:

 

Technical Discussion:

When the short and long t-joints are compared, we observe that mixing is driven by the turbulence for the respective momentum ratio's. More momentum ratio signifies generation of more turbulence. Turbulence drives the two fluids in random and chaotic manner which assists the mixing of the fluids.

At the fluid contact interface (critical point), advection assists heat tranfer whereas at the right side of the cold fluid layer there is generation of turbulence. This is because of the adverse pressure gradient and recirculation of the fluid. The turbulence generated consists of eddies, which are responsible for momentum and energy exchange (conserved quantities) forming the energy cascade in the flow directions. In turbulence kinetic energy plot we can see the magnitude of kinetic energy associated with the eddies (or more precisely turbulence). However, this turbulence (momentum and energy exchange between fluid layers) converts kinetic energy of the turbulent fluid into thermal energy (i.e. turbulence dissipation). The turbulence dissipation also contributes to the generation of heat inside the system. However, which is not so dominant in this case as we can observe it from turbulence eddy dissipation plot. 

The mixing to be sufficiently efficient length of the mixing-tee also contributes equivalent assistance. However, mixing efficiency as a function of t-joint length helps or sometimes overindulge (more advective heat transfer) in mixing of the two fluids. So length should be optimum to get expected outcomes.

 

Validation of the Numerical Results:

Analytical solution for momentum ratio as 2,

`T_(mix) = 30.32^0c`

Analytical solution for momentum ratio as 4,

`T_(mix) = 27.48^0c`

Observation Table:

Cases	Total number of cells	Turbulence Model	Momentum Ratio	Mixing-Tee Length	Standard Deviation	Area-Weighted Average of Temperature	No.of.iterations
Case 2	2,11,968	K-epsilon	2	Short	2.0903	30.3403	250-300
Case 4	2,11,094	K-epsilon	2	Long	1.6204	30.3994	300-350
Case 5	2,11,968	K-epsilon	4	Short	1.2027	27.5688	250-300
Case 6	2,11,094	K-epsilon	4	Long	0.5613	27.5036	300-350

 

Discussion: 

By comparing the case studies for the same momentum ratio and different lengths resulted in interesting outcomes. Initially for the momentum ratio of two (2), case study with short tee length produced more approximate results. However, the standard deviation of outlet temperature is more than that of long tee. The standard deviation of long tee is less because mixing of the fluids linearly varies with flow length (can be called as characteristic length). More the length of tee joint, more will be the mixing of the two fluids in a constraint domain. However, as stated above, length can sometimes increase the temperature of fluid (because mass flow rate of hot fluid is more in this case), which is undesirable. In this case, short tee is nearly approximating the analytical solution more accurately than that of long tee for the momentum ratio of two (2). 

Reason for more standard deviation in case 2 though giving accurate results!!

The chilled air coming out from the small inlet acts as a jet stream which is diverged by the hot fluid coming from the comparatively big inlet. Due to sudden encounter (orthogonal) of the fluid streams, there is momentum and energy exchange (faster to slow moving layers), due to this there is viscous dissipation with the advection. Though viscous heating is not that dominant, but still plays a crucial role. This jet stream (chilled air stream) is not normalized by the hot air (in short tee case), which asserts incomplete advection (the temperature at the layer of fluids interaction is not normalized by the hot air). Because of this we see more temperature at the bottom of the outlet (in vertical orientation). However, in long tee the jet stream is thermally normalized by the hot air therefore we have distributed air temperature of the fluid interaction layer. This is the cause for variation in standard deviation between two cases.

Secondly, for the momentum ratio of four (4), the chilled air possess higher velocity which is 4 times the hot fluid velocity. Undoubtedly, more turbulence will be generated in the mixing flow. This will increase the rate of heat transfer between the two fluids. However, here in this case length of the t-joint should be sufficiently enough to hold the mixing phenomenon inside the domain before exit or else improper mixing would result. The turbulence kinetic energy is also more skewed to the outlet because of fluid momentum. However as in case 5 (short tee length) the domain is not sufficiently long enough to hold mixing phenomenon, that's why resulting in more standard deviation and less approximation of the outlet temperature. In contrast to it, case 6 is resulting in less standard deviation with more approximate outlet temperature. This is only by the effect of mixing length which confirms sufficient mixing of air without additional undesirable temperature variations.

 

Conclusion: 

A simple thermal analysis of a mixing tee produced interesting results with many dependent variables responsible to achieve better mixing. As per the case study and results extracted from CFD analysis, we can see much dependence of mixing lengths and momentum ratios on the mixing efficiency of a mixing-tee. In terms of standard deviation of the outlet temperature, the t-joint with long length tends to be more effective. However for less momentum ratio and long t-joint length, we observed undesirable heating of the air (because of hot air over-indulgence in the flow) which gave less approximate results. So short tee length is the best fit choice for less momentum ratio in order to achieve efficient mixing.

In contrast to it, short t-joint with higher momentum ratio produced less approximate results (more standard deviation and more outlet temperature) because of insufficient length for proper mixing. The long t-joint produced more approximate temperature at the outlet with less standard deviation. This is only possible because of the flow span provided to the mixing fluids was sufficiently enough for efficient mixing.  

 

References:

1. https://www.sciencedirect.com/science/article/pii/S0029549309005676

2. https://skill-lync.com/knowledgebase/how-to-check-for-convergence

3. https://www.comsol.com/blogs/which-turbulence-model-should-choose-cfd-application/

4. https://www.computationalfluiddynamics.com.au/tips-tricks-turbulence-wall-functions-and-y-requirements/