Week 1- Mixing Tee

Aim: Simulation of flow-through mixing tee using Ansys fluent.

Objective:

• To determine the changes in flow properties for different lengths of geometry, and momentum ratio.
• Perform mesh independent study for any geometry.
• Use different turbulence models, and determine which is suitable for accurate results.

Introduction:

Momentum ratio: It is a ratio of velocity at the cold inlet to the velocity at the hot inlet(Dimensionless quantity).

Mr = velocity at cold inlet/velocity at hot inlet

 

Mixing tee:

The mixing tee consists of two pipe sections joined at a right angle to each other, one stream passes straight through the tee while the other enters perpendicular, this arrangement is generally called a side-tee.

Mixing tee is the case of internal flow through the pipe, during the flow the fluid faces two types of forces generally inertial force responsible for the fluid flow and viscous force which resists its flow.

The effect of the above two forces defines whether the flow will be streamlined or turbulent flow, In mixing tee we are generally interested in turbulent flows since mixing of two flow stream perpendicular to each other with Reynolds number greater than 4000 yields dissipation of heat, momentum, energy transfer at a faster rate.

For turbulence modelling we generally have many models i.e. k-epsilon, k-omega, k-omega SST, etc.

In k-epsilon we have categories such as RNG, standard k-epsilon, and realizable.

Application of mixing tee:

• Low viscosity mixing such as wastewater treatment.
• Blending some oils(injection of additives).
• Petrochemical products.
• Air conditioner.

 

History and theory behind k-ϵ, k-ω, and k-ω SST model.

The k-ϵ model was developed in 1973, and it was the only accurate model to predict eddy viscosity until the next 20 years.

There was some drawback in the k-ϵ model which lead to the development of the k-ω model in 1990 and it was changed many times until 2006, this model had drawback it depends on free stream turbulence conditions.

The k-ϵ models were not able to predict accurate results near the wall in viscous sublayer, and the k-ω model had demerit for far wall turbulence prediction, So what scientists and engineers tried is to combine both the model k-ω in viscous sublayer adjacent to wall and k-ϵ far from the wall, this was base for the development of new model named k- ω SST.

Nowadays k- ω SST is commonly used for simulations where there are adverse pressure effects such as in aerofoils, and turbomachinery.

Many other models are being developed based on the same base which can also be used for particular applications.

 

k-ϵ model:

When we convert the momentum equation into RANS(Reynolds averaged Navier stokes equation), we get an extra term: Reynolds stress.

To find this quantity the most common approach is Boussinesq equations.

The quantities in RHS are known, except the eddy viscosity μ­_t, so μ­_t  needs to be calculated to close the RANS equation, for calculation of eddy viscosity various methods are being proposed.

In the k-epsilon method the formula used is:

Where ‘k’ stands for turbulent kinetic energy, and ‘ϵ’ stands for turbulent dissipation.

The above two quantities are calculated using transport equations.

 

The transport equation for turbulent kinetic energy is the same for RNG, realizable, and standard k- ϵ, but the co-efficient in turbulent dissipation(C1, C2) decides the method used.

Once the value of k. and ϵ is calculated we can calculate eddy viscosity, and close the RANS equation.

 

 

k- ω model:

The only change in the k-ω model is in place of ‘ϵ’  we calculate ‘ω’.

Both ‘ϵ’ and ‘ω’ are dissipation rate, and ‘ω’ depends on ‘ϵ’ as given in the formula below.

The transport equation for turbulent kinetic energy is the same, therein change in transport equation for ‘ω’.

Once both values are calculated we can use them to calculate eddy viscosity and close the RANS equation.

 

Cases to be simulated:

 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.

Case 3:

• Short mixing tee with momentum ratio 4 and mesh sizes 0.005,0.003, and 0.002.

Case 4:

• Short mixing tee with momentum ratio 4 mesh size 0.003, models used k- ω, and k-ϵ.

 

 Tabular column for initial and boundary condition:

Properties	Short mixing tee		Long mixing tee
	Vinlet x  = 6m/s	Vinlet y  =12m/s	Vinlet x  = 6m/s	Vinlet y  =12m/s
Velocity of hot inlet	3m/s	3m/s	3m/s	3m/s
Turbulence model	k- ϵ	k- ϵ	k- ϵ	k- ϵ
Hot inlet temperature	36 ˚C	36 ˚C	36 ˚C	36 ˚C
Cold inlet temperature	19 ˚C	19 ˚C	19 ˚C	19 ˚C

 

Geometry:

 

Short tee:

 

 

Long tee:

 

Extracted volume:

Since we are not calculating wall heat transfer or conduction through the thickness of the wall we extract the internal volume of the geometry for our simulations. 

 

Case setup:

Quality check of mesh:

• After generating mesh we should check its quality whether the 80% of cells in our geometry are under our specified mesh size.
• Once there are no errors we can move forward with setup.
• The next step is to define the physics, In this section, we assign fluid properties, boundary conditions, initial conditions, and turbulence models used. all these values are mentioned in the tabular column.
• The final step is to generate results, in the solution section we can assign convergence criteria, initialization of solution, definition(the plots we need), and no of iterations. Once completed setting all values we can click calculate.

 

RESULTS:

Case1:

 

Short tee momentum ratio 2:

Residuals:

 

Standard deviation of temperature at outlet:

 

Temperature at outlet:

 

Velocity at outlet:

 

Enthalpy at outlet:

 

Pressure throughout the geometry at central axes:

 

Temperature throughout the geometry at central axes:

 

Velocity throughout the geometry at central axes:

 

 

Short tee with momentum ratio 4:

Residuals:

 

Standard deviation of temperature at outlet:

 

Temperature at outlet:

 

Velocity at outlet:

 

Enthalpy at outlet:

 

Pressure throughout the geometry at central axes:

 

Temperature throughout the geometry at central axes:

 

Velocity throughout the geometry at central axes:

 

Contour plots for both cases.

Short tee with momentum ratio 2:

 

Short tee with momentum ratio 4:

When the momentum of cold air is increased the more amount of cold air is dissipated into hot air decreasing its overall temperature.

What decrease in temperature generally mean?

The hot particles have greater vibrations and lots of collisions are occurring in them causing the release of energy hence that particle has a greater temperature when such particle mix with particles of less energy the overall temperature of it decreases.

Also, the overall heat content of air decreases due to a decrease in temperature, and velocity attends the average velocity of hot and cold air.

From the pressure and velocity graph plotted across the geometry, we see that pressure decreases and velocity increases at the point where both the cold air and hot air are getting mixed, the velocity increases because higher velocity cold fluid is mixed with lower velocity hot fluid, and pressure generally decreases which we can presume from the behavior of Bernoulli's equation.

 

 

Case 2:

Long tee momentum ratio 2:

Residuals:

 

Standard deviation of temperature at outlet:

 

Temperature at outlet:

 

Velocity at outlet:

 

Enthalpy at outlet:

 

Pressure throughout the geometry at central axes:

 

Temperature throughout the geometry at central axes:

 

Velocity throughout the geometry at central axes:

 

 

Long tee momentum ratio 4:

Residuals:

 

Standard deviation of temperature at outlet:

 

Temperature at outlet:

 

Velocity at outlet:

 

Enthalpy at outlet:

 

Pressure throughout the geometry at central axes:

 

Temperature throughout the geometry at central axes:

 

Velocity throughout the geometry at central axes:

There is minimal effect on outlet velocity, temperature, and other parameters with an increase in the length of the geometry. but there is a difference in the standard deviation of temperature which can't be neglected.

The standard deviation defines how dispersed our value is, if the values are more dispersed we generally say that our solution is not stable.

A standard deviation of < 1 is expected for any solution.

 

 

Case 3: 

Different mesh sizes:

Short tee with momentum ratio of 4	Mesh size	Nodes	Elements
	0.005	3,152	14.330
	0.003	8,343	39,803
	0.002	21,347	1,05,865

 

Mesh size 0.005:

 

Mesh size 0.003:

 

 

Mesh size 0.002:

 

Residuals:

0.005:

 

0.003:

 

0.002:

 

Temperature sd:

0.005:

 

0.003:

 

0.002:

 

Velocity:

0.005:

 

0.003:

 

0.002:

 

Pressure across the geometry:

0.005:

 

0.003:

 

0.002

 

Tabular column for the output value comparisons:

Outlet properties	Short mixing tee		Long mixing tee		Short mixing tee, with velocity ratio of 4, and diff mesh sizes.
	Vin y = 6m/s	Vin y = 12m/s	Vin y = 6m/s	Vin y = 12m/s	0.005	0.003	0.002
Outlet temperature	30.25 ˚C	27.5 ˚C	30.25 ˚C	27.5 ˚C	27.5 ˚C	27.5 ˚C	27.5 ˚C
Outlet velocity	4.505 m/s	6.03 m/s	4.505 m/s	6.03 m/s	6.03 m/s	6.03 m/s	6.03 m/s
Number of iteration	300	170	300	220	170	145	300
The standard deviation of temperature at the outlet	1.125 ˚C	1.2 ˚C	0.9 ˚C	0.6 ˚C	1.2 ˚C	0.98 ˚C	1 ˚C

Generally with an increase in the number of cells the accuracy of the solution increases since the equations are solved for more cells, but in the simulation of mixing tee it is grid-independent since flow parameters do not change in value.

Sometimes using more number of cells may lead to instability of solution which we can see for 0.002 mesh size, It doesn't get converged even for 300 iterations.

 

 

Case 4: 

 

Velocity:

k-epsilon model:

K-omega SST model:

 

 

Static temperature at the centerline:

k-epsilon model:

 

K-omega SST model:

 

Contour plots:

k-epsilon model:

K-omega SST model:

 

 

Pressure and velocity comparison for models:

Short mixing tee, with velocity ratio of 4, and mesh size 0.003, The line probe was placed at the center of geometry in Y-axis to extract the pressure and velocity values at the wall. Since we know from experiments that the k-ϵ model has errors in viscous sublayer, and k- ω SST is not prone to it so we get different values for both the cases.
Parameters	k-ϵ	k- ω SST
Pressure	5.25 pa	5.95 pa
Velocity	9.5 m/s	10 m/s

Pressure:

k-epsilon model:

K-omega SST model:

 

 

Velocity:

k-epsilon model:

K-omega SST model:

 

Overall outlet velocity and pressure have negligible differences so, for our flow problem, we can use the k-ϵ model.

The k-ϵ model is used in the high Reynolds number problem, and k- ω SST works for any simulations.

 

Inference:

• With the increase in momentum, more molecular diffusion takes place which will result in greater velocity of air at the hot inlet, and also a good mixture temperature and composition.
• With the increase in length, there is no major effect in any flow parameters.
• The flow simulation in mixing tee is grid-independent.
• The k-epsilon model is suitable for the prediction of solution in the case of mixing tee.