Week 1- Mixing Tee

problem statement:

we have created two versions of the mixing tee. One of them is longer than the other.

Your job is to set up steady-state simulations to compare the mixing effectiveness when the hot inlet temperature is 36⁰C & the Cold inlet is at 19⁰C.

Use the k-epsilon and k-omega SST model for the first case and based on your judgment use the more suitable model for further cases. Giving the reason for choosing a suitable model is compulsory.

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

Introduction:

T junction is formed when two pipes are joined perpendicular to each other. It has a wide range of applications from household plumbing to air conditioning ducts in industries etc. as the flow considered as turbulent in this study Reynolds avg navior Stokes equation is used. RANS is further divided into several models which can be chosen according to the problems. Among the several models in RANS k-epsilon and k-omega turbulence models are used in this study.

k-epsilon is a two-equation RANS model, in this model we solve for two variables, turbulent kinetic energy k and turbulent dissipation of kinetic energy epsilon. This model uses wall functions to account for the fluid velocity in the viscous sublayer near the wall.

Main limitaitions for k-epsilon models are:

1. No-slip walls
2. Adverse pressure gradients.

But various models in k-epsilon models were developed to improve the performance of the k-epsilon model. They realizable k-epsilon model, RNG k-epsilon model.

k-omega is another turbulence model where the rate of dissipation of kinetic energy epsilon is replaced by omega. The main objective of this model is to account for near-wall interactions more accurately than the k-epsilon model.

Limitations of l-omega

1. The difficulty of convergence compared to k-epsilon
2. Sensitivity to the initial condition

Due to these limitations, the k-omega sst model was developed. It is a hybrid of the k-omega and k-epsilon models. This model is used in mainly aeronautics where free stream turbulence levels have to be addressed and it can also be applied to the viscous affected region without further modification.

Turbulence:

In fluid dynamics turbulent or turbulence flow I s fluid motion characterized by chaotic changes in pressure and flow velocity. It is in constant to a laminar flow, which occurs when a fluid flows in parallel layers with no disruption between those layers.

Turbulence is commonly observed in everyday phenomena such as surface, fast-flowing rivers, billowing storms clouds, or smoke from a chimney, and most fluid flows occurring in nature or created In engineering applications are turbulent. Turbulence is caused by excessive kinetic energy in parts of a fluid flow, which overcomes the damping effect of the viscosity of the fluid. For this reason, turbulence is commonly realized in low viscous fluids. In general terms, in turbulent flow, unsteady vortices appear many sizes which interact with each other, consequently drag due to friction effect increases. This increases energy needed to pump the fluid through a pipe.

In general fluid, turbulence is the norm and laminar flow is an exemption. Turbulence is a very complicated phenomenon, our understanding of which is very limited. Thus when we try to model turbulence in a simulation there are various approaches that exist. It is on us t select an approach that works well with the physics of the problem.

Convergence:

Since most fluid flow natures are nonlinear, dealing with a phenomenon like turbulence. To solve such problems we use an iterative solver to continuously improve our solution till convergence. Is reached. When solving a problem with multiple iterations convergences becomes related to the accuracy of the solution. Typically when assessing convergence for a steady-state CFD solution we need to keep in mind their major things, they are:

1. Residual values,
2. Solution imbalance
3. Observing quantities of interest.

The criteria for convergence is being applied for:

1. Continuity
2. X-velocity
3. Y-velocity
4. Z-velocity
5. Energy
6. Turbulence related variables

Y+:

Y+ is a nondimensional term that is used to define how fine or coarse is the mesh is, this helps us to determine if we need to use a specific wall functions.

Y+ = (Y*u_t)/nu

U_t is friction velocity

Y is the absolute distance from the wall

V is the kinematic viscosity

When we consider a boundary layer, there exist two regions inner boundary layers and outer boundary layers. The inner boundary consists of 3 main regions, the first is the viscous region which is right after the wall, in this region the viscous forces dominate the internal forces and the flow is laminar (Y+ <10). The next is the transition region (10 < Y+ <30) and the third is the turbulent boundary layer, here the internal forces are more dominant (30 <y+ < 300). The y+ values decide us to choose the type of turbulence model we need to use for the particular problem.

Turbulence models:

Renolds average navier strokes equation:

This equation (RANS) are time-averaged equations of motion for fluid flow. The idea behind the equation s is Reynolds decomposition, where an instantaneous quantity is decomposed into its time-averaged solutions to the navier stokes equations.

1 k-epsilon

k-epsilon turbulence model is the most common model used in CFD to simulate the mean for charters for turbulent flow conditions. It is a two-equation model that gives a general description of turbulence by means of two transport equations (PDEs). The original impetus for the k-epsilon model was to improve the mixing length model. In the k-epsilon model, the kinetic eddy et viscosity is expressed as the product of velocity scale and a length scale l=k3/2/epsilon. The rate of dissipation of kinetic energy is not the only possible length scale determining variable realizable k-epsilon model: an immediate benefit of the realizable k-epsilon mode is that it provides improved predictions for the spreading rate of both planar and round jets. It also exhibits superior performance for flows involving rotations, boundary layers under strong pressure gradients, separation, and recirculation. In virtually every measure of comparison, a realizable k-epsilon demonstrates a superior ability to capture the mean flow rate of complex structures.

2 K-omega:

In CFDs the k-omega turbulence model is a common two-equation turbulence model, that is used as an approximation for the Reynolds-averaged Navier-stokes equations. The model attempts to predict the turbulence by two partial differential equations for two variables, and omega, with the first variable being the turbulence kinetic energy k while the second is the rate of dissipation (of the turbulence kinetic energy k into internal thermal energy).

The k-omega model initially attracted attention because integration to the wall does not require, wall damping function in low Reynolds numbers applications. The value of turbulence kinetic energy k at the wall is set to 0. The frequency omega tends to infinity at the wall. Apply a hyperbolic variation omega,p= 6v(beta1*y^p/2) at the near-wall grid point. Practical experience with the model has shown that the results do not depend too much on the precise details of the treatment.

Procedure:

1. To extract volume open space claim, under prepare menu use volume extract command and select the edges of geometry and suppress the uploaded setup
2. Open the mesh window and enter the mesh parameters and generate the mesh, check for the quality of the mesh which should not be less than 5% update the mesh window at the end
3. Open the setup window to select the turbulent models and enter initial, boundary conditions under the physics menu.
4. To plot the graph of weighted avg temp and stood deviation we need to define it in definitions under the solution window.
5. Initialize the model under the user-defined menu to calculate the values of the initial time step and then enter the number of iteration to perform.

Processing:

Imported geometry: in this, we need to import the geometry(tee-shaped tube) in space claim, where the geometry was in the raw material state as created in cad software, so here we have represented the geometry in trimetric view as given below:

Clean up geometry in space claim: since the geometry was in solid-state where the body was having a thickness which needs to be removed so as to make geometry simple and can be converted into volumetric form. The newly updated geometry in volumetric form was as given below.

Mesh the geometry in Ansys mesh: here wall import the volumetric geometry created in space claim after cleaning up the geometry here we applied to mesh on the volumetric region with base mesh size. We also make sure that mesh quality should be of good quality.

After importing 3d cad geometry in Ansys mesh the first step is to name the boundaries by selecting the surfaces and rt clicking and selecting "Create Name Selection" or just by click N on the selected surface. There are two inlets i.e X-inlet which is along the X-axis (as shown in geometry) and another inlet is perpendicular it called Y- Inlet and the other end of the pipe is Outlet. After this, the element size is provided and mesh is created.

The created mesh is using element type as tetrahedral mesh in this study which is an unstructured type of mesh.

the meshed model:

no of cells:12728

no of nodes: 2827

Mesh quality:

Solving / Setup

Ansys Fluent is used as the solver. Here, We have created the setup for simulation where we have provided double precision and displayed mesh after reading so that We can visualize the mesh obtained in the design module.

Single precision means that the floating-point numbers will be represented in the 32-bit system whereas double-precision means that they will be represented in a 64-bit system. So Calculation in double precision will be more accurate. Though the performance of the CPU takes a hit as it takes longer solution time for solving 64-bit floating-point numbers as well the solution requires more RAM to store those nos.

A steady-state computational study has been performed using a pressure-based solver. Air was used as the fluid entering from both the inlets. The default physical properties defined in Fluent were used.

Selecting boundary markers shows that what are the inlets and outlets. Blue Colours arrows show inlet and red color shows outlet.

Turbulence Model

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.

Boundary conditions - Momentum ratios of 2 and 4 have been used in the study. The hot stream enters in the main pipe (larger diameter) while the colder stream enters from the secondary inlet.

Table 1. Boundary conditions

the velocity of the hot air steam	the velocity of the cold air stream	the temperature of the hot air steam	the temperature of the cold air steam
3	6	36	19
3	12	36	19

(i) Inlet velocity for x -axis :-

Here, We have given the value of velocity for x-axis 3 m/s and gauge pressure 0 Pa, and the value for all other parameters has been taken default.

(ii) Inlet velocity for y-axis :-

Here, We have assumed the value of velocity for y-axis 6 m/s and gauge pressure 0 Pa whereas the value for all other parameters has been taken default.

(iii) Wall condition for pipe:-

Wall conditions have been taken in the following way:-

• Stationary wall motion
• No-Slip condition for shear stress
• Standard Roughness Model have been assumed at the surface
• Other parameters have been defaulting value

(iv) Outlet condition for pipe:-

Here, values have been assumed by default for all the conditions.

the results of the k–ε model are much less sensitive to the (arbitrary) assumed values in the free stream, but its near-wall performance is unsatisfactory for boundary layers with adverse pressure gradients. This led to suggest a hybrid model using (i) a transformation of the k–ε model into a k–ω model in the near-wall region and (ii) the standard k–ε model in the fully turbulent region far from the wall, the ε-equation is transformed into an ω-equation by substituting ε = kω. this became the k–ω sst model.

With the above Initial and Boundary, conditions simulation is done for K epsilon and K omega

Below are the plots for K-epsilon and K-omega for momentum ratio 2 for the short Tee model.

Residuals plots K epsilon

Residuals plots K-omega

temperature plot along with the length short -tee:

K-epsilon

K-omega:

standard deviation at outlet:

k-epsilon:

k-omega:

avrrage temparature plots:

k-epsilon:

k-omega:

simulation for momentum ratio 4 short tee:

we have chosen the k-epsilon model for the further cases here:

temperature plot along the length:

average temparature:

standard deviation:

Inference:

From the above plots, we can infer that the converged solution varies in the order of 10−210-2 between K-epsilon and K-omega turbulent models, and also the mixing efficiency is higher in the K-epsilon model which is supported by the Standard deviation plot, Higher the Standard deviation lower is the mixing efficiency and vice versa. From the residuals plots K-epsilon takes less no. of iterations when compared to K-omega model and also high Reynolds number is involved in which K epsilon results in more accurate solution than K-omega and y+ values are above 5 which favors for K-epsilon turbulence models. Hence for this problem, K-epsilon turbulent model is suitable.

For momentum ratio 4, mixing efficiency is more which can be observed from the standard deviation plot (Higher standard deviation means low mixing efficiency and lower standard deviation means higher mixing efficiency) and the results obtained are closer to the analytical solution.

Simulation for long Tee with momentum ratio 2 using K-epsilon turbulent model.

Residual plot:

temperature plot along the length:

line plot temperature distribution along the tee:

velocity:

pressure:

standard deviation for temperature output:

avg temperature:

velocity:

simulation for mixing ration4:

residual plot:

temperature plot along the length:

temperature plot along the length:

velocity:

pressure:

standard deviation at the outlet temperature:

avg velocity at the outlet:

Table Comparing mixing efficiency of the short tee and long tee model for K-epsilon turbulence model.

Model type	Momentum Ratio	Standard Deviation
Short Tee	2	2.25
Short Tee	4	1.0002
Long Tee	2	1.6
Long Tee	4	0.8

Inference:

From the above plots, the variation of temperature and velocity for the given momentum ratio does not vary when compared with the short Tee. However, from standard deviation at the outlet temperature decreases indicating higher mixing efficiency with an increase in length.

Grid Independence test

In order to verify that the results obtained from the simulation do not vary with the size of the mesh, no. of mesh cells are varied by increasing the mesh element size.

From the above table, we can observe the y+ value range increases as the mesh cell size increases because of which k-epsilon turbulent models result in a more accurate solution which can be seen from the average temperature values at the outlet of the tee model.

Conclusion:

The resultant average temperature for both turbulence model Reliazable K -Epsilon & K omega SST looks similar, but for computing K omega SST takes more time and more number of iterations for convergence. Moreover, K epsilon predicts far from the boundary, and K omega capture well near the wall also for the simulation we are dealing with the wall. Because of this reason for the remaining case, I choose K - Epsilon Turbulence model.

For both case momentum ratio of 4, the outlet temperature drops significantly this occurs because of very high cold velocity air. So this clearly shows higher the momentum ratio greater the efficiency of the tee.

We are getting an almost similar result for short tee & long tee for the momentum ratio of 4 from this we can get to know that the length of the pipe doesn't affect the efficiency of the pipe.

When it comes to standard deviation, the long tee gives a very lower value compared to the short tee. Indeed, we can increase high inlet cold air velocity for the application so we can get pretty good mixing in nature for a short pipe instead of using the long pipe.

Overall, the short tee pipe gives a pretty good result, and also it saves material cost. Increasing the pipe length doesn't create any difference in the outlet temperature.