Numerical analysis of flow through T-joint pipe and comparison of mixing effectiveness
Objective:
1. To perform numerical analysis of flow through T-joint.
2. To determine the area weighted average and standard deviation of temperature at outlet.
3. To determine the effect of increase in length of pipe and increase in velocity of cold fluid on the mixing effectiveness of the T-joint.
T-joint:
A T-joint is a pipe with 2 inlet(1 inlet perpendicular to flow) and 1 outlet or vise versa.They are widely used to mix fluids with different temperatures(hot and cold) in order to obtain the fluid with desired temperature. The flow and geometric parameters in the T-joint should be carefully adjusted so that the desired temperature is obtained uniformly at the output i.e. fluids should be effectively mixed.
The Geometry of T-joint used in this analysis is shown below,
In current model, cold fluid(air at 10 c) enters the T-joint from inlet_cold and hot fluid(air at 25 c) enters the T-joint through inlet_hot.Since the above pipe has some thickness which is not needed for flow analysis, The computational fluid domian is extracted from above T-joint with volume extraction method in spaceclaim as shown below,
Under the volume extract option, the edges of inlet_cold,inlet_hot and outlet are selected to extract the computational flow domain.
Computational Flow domian:
Case 1:
Case 2:
Case 3: The flow domian of case 3 is same as case 1.
Governing Equations:
Continuity equation:
`frac{delrho}{delt}+ nabla.(rhoV)=0`
Momentum equation:
x component: `frac{del(rhou)}{delt}+grad.(rhouV)=-frac{delp}{delx}+frac{del(tau_(x x ))}{delx}+frac{del(tau_(yx))}{dely}+frac{del(tau_(zx))}{delz}+rhof_x`
Y component: `frac{del(rhov)}{delt}+grad.(rhovV)=-frac{delp}{dely}+frac{del(tau_(x y ))}{delx}+frac{del(tau_(yy))}{dely}+frac{del(tau_(zy))}{delz}+rhof_y`
z component: `frac{del(rhow)}{delt}+grad.(rhowV)=-frac{delp}{delz}+frac{del(tau_(x z ))}{delx}+frac{del(tau_(yz))}{dely}+frac{del(tau_(zz))}{delz}+rhof_z`
Energy equation:
`frac{del}{delt}[rho(e+frac{V^2}{2})]+grad.[rho(e+frac{V^2}{2})V]= rhodotq+frac{del}{delx}(kfrac{delT}{delx})+frac{del}{dely}(kfrac{delT}{dely})+frac{del}{delz}(kfrac{delT}{delz})-frac{del(up)}{delx}-frac{del(vp)}{dely}-frac{del(wp)}{delz}+frac{del(utau_(x x))}{delx}+frac{del(utau_(yx))}{dely}+frac{del(utau_(zx))}{delz}+frac{del(vtau_(xy))}{delx}+frac{del(vtau_(yy))}{dely}+frac{del(vtau_(zy))}{delz}+frac{del(wtau_(xz))}{delx}+frac{del(wtau_(yz))}{dely}+frac{del(wtau_(zz))}{delz}+rhof.V`
For current case, we have considered steady, in-compressible flow.
Turbulence Modeling:
Realizable `k-epsilon` Turbulence Model:
The standard k-ε model is a model based on model transport equations for the turbulence kinetic energy (k) and its dissipation rate (ε). The model transport equation for k is derived from the exact equation, while the model transport equation for ε was obtained using physical reasoning and bears little resemblance to its mathematically exact counterpart.
The realizable k-ε model differs from standard in two important ways:
1.The realizable k-ε model contains an alternative formulation for the turbulent viscosity.
2.A modified transport equation for the dissipation rate, ε has been derived from an exact equation for the transport of the mean-square vorticity fluctuation.
One limitation of the realizable k-ε model is that, it produces non-physical turbulent viscosities in situations when the computational domain contains both rotating and stationary fluid zones. This is due to the fact that the realizable k-ε model includes the effects of mean rotation in the definition of the turbulent viscosity.
Transport Equations for Realizable `k-epsilon` Turbulence Model:
The modelled transport equations for k and `epsilon` in the realizable `k-epsilon` model are
`frac{del(rhok)}{delt}+frac{del(rhoku_j)}{delx_j}=frac{del}{delx_j}[(mu+frac{mu_t}{sigma_k})frac{delk}{delx_j}]+G_k+G_b-rhoepsilon-Y_M+S_k`
Physical Meaning of k equation:
Time derivative of k+ Convective transport = diffusive transport + Rate of production-Rate of destruction
`frac{del(rhoepsilon)}{delt}+frac{del(rhoepsilonu_j)}{delx_j}=frac{del}{delx_j}[(mu+frac{mu_t}{sigma_(epsilon)})frac{del(epsilon)}{delx_j}]+rhoC_1S_epsilon-rhoC_2frac{epsilon^2}{k+sqrt(vepsilon)}+C_(1epsilon)frac{epsilon}{k}C_(3epsilon)G_b+S_epsilon`
Physical Meaning of `epsilon` equation:
Time derivative of ε+ Convective transport = Diffusive transport + Rate of production-Rate of destruction.
where,
k= Turbulent kinetic energy
`epsilon`= Turbulent dissipation rate
`G_k`= generation of turbulence kinetic energy due to the mean velocity gradients
`G_b`= generation of turbulence kinetic energy due to buoyancy
`Y_M`= represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate
`sigma_k`,`sigma_epsilon`= turbulent Prandtl numbers for k and `epsilon` respectively
`C_(1epsilon),C_(2epsilon),C_(3epsilon)`= Constants
Model Constants:
`C_(2epsilon)`= 1.9,
TKE prandtl number= 1
TDR prandtl number = 1.2
Energy prandtl number= 0.85
wall prandtl number= 0.85
Boundary Conditions:
Suitable boundary conditions have been applied according to the physics of the flow.
1. At Inlet_hot:
Velocity inlet of magnitude 3 m/s normal to boundary,
Temperature= 25 c,
Turbulent intensity(%)= 5 and turbulent viscosity ratio= 10.
2. At Inlet_cold:
Velocity Inlet of magnitude 6 m/s (for case 1 and case 2), 12 m/s(for case 3) normal to boundary,
Temperature= 10 c,
Turbulent intensity(%)= 5 and turbulent viscosity ratio= 10.
3. At outlet:
Pressure outlet at zero gauge pressure(atmospheric pressure),
Turbulent intensity(%)= 5 and turbulent viscosity ratio= 10.
4. AT wall:
Stationary wall with no slip condition(velocity of fluid which is in contact with wall surface is zero),
Zeroo heat flux.
Fluid Properties:
Schemes:
1. Pressure velocity coupling: SIMPLE
The SIMPLE algorithm uses a relationship between velocity and pressure corrections to enforce mass conservation and to obtain the pressure field.
2. Spatial discretization:
Gradient: Least squares cell based
Pressure: Second order
Momemtum: Second order upwind
Turbulent kinectic energy: First order upwind
Turbulent dissipation energy : Second order upwind
Energy: Second order upwind
First-order upwind scheme- When first-order accuracy is desired, quantities at cell faces are determined by assuming that the cell-center values of any field variable represent a cell-average value and hold throughout the entire cell; the face quantities are identical to the cell quantities.
Second-order upwind scheme- When second-order accuracy is desired, quantities at cell faces are computed using a multidimensional linear reconstruction approach . In this approach, higher-order accuracy is achieved at cell faces through a Taylor series expansion of the cell-centered solution about the cell centroid.
Initialization:
Hybrid Initialization: Hybrid Initialization is a collection of recipes and boundary interpolation methods. It solves the Laplace equation to produce a velocity field that conforms to complex domain geometries, and a pressure field which smoothly connects high and low pressure values in the computational domain. All other variables (that is temperature, turbulence, VOF, species, and so on) will be patched based on domain averaged values or a predetermined recipe.
Mesh:
Case 1:
Element size: 0.002
Number of elements: 106916
Case 2:
Element size: 0.002
Number of elements: 141714
Case 3:
Element size: 0.002
Number of elements: 106916
For case 1,2 and 3,since the mesh quality index for maximum number of elements is at approx 0.90, therefore The above mesh can be considered of good quality and can be used for further numerical calculations.
Residual Plots:
case 1:
Case 2:
Case 3:
From above plots, we can observe that the residuals for case 1,2 and 3 have decreased and solution have been converged. For case 1 and 2, convergence have been achieved at 600 iterations and for case 3, the convergence have been achieved at nearly 800 iterations.The convergence criteria used for all the three cases is 1e-03.
RESULTS:
Case 1:
1. Velocity Contour at symetric plane:
2. Temperature contour at symetric plane:
3. Area Weighted average temperature plot at output:
4. Standard Deviation of temperature plot at outlet:
5. Variation of temperature at outlet cross-section:
6. Variation of Velocity at outlet cross section:
For case 1, At inlet_hot, air enters at a velocity 3m/s with a temperature of 25 c(298.15 K). At inlet_cold, air enters at a velocity 6m/s with a temperature of 10 c(283.15 K), which can be observed from temperature and velocity contours. No slip condition applied on the walls can be observed from the velocity of fluid near wall(in the velocity contour).
For case 1, from temperature contour we can observe that at outlet fluid at lower bottom is still at 25 c, therefore the mixing of cold fluid fluid is not homogenous.Therefore the area weighted average of temperature at outlet is around 20.15 c and standard deviation of temperature at oulet is around 1.83.
From variation of temperature at outlet cross section, we can observe that at outlet cross section, temperature is maximum at lower bottom and the temperature of fluid is least near the centre of the outlet.
Case 2:
1. Velocity Contour at symetric plane:
2. Temperature contour at symetric plane:
3. Area Weighted average temperature plot at output:
4. Standard Deviation of temperature plot at outlet:
5. Variation of temperature at outlet cross-section:
6. Variation of Velocity at outlet cross section:
For case 2, At inlet_hot, air enters at a velocity 3m/s with a temperature of 25 c(298.15 K). At inlet_cold, air enters at a velocity 6m/s with a temperature of 10 c(283.15 K), which can be observed from temperature and velocity contours. No slip condition applied on the walls can be observed from the velocity of fluid near wall(in the velocity contour).
The area weighted average of temperature at outlet is around 20.05 c and standard deviation of temperature at oulet is around 1.23.
From variation of temperature at outlet cross section, we can observe that at outlet cross section, temperature is maximum at lower bottom and the temperature of fluid is least near the centre of the outlet.
Comparing the temperature at lower bottom of outlet for case 1 and case2, we can observe that there is decrease in temperature(from 297.2 to 295.5), therefore we can conclude that increasing the length of pipe leads to further mixing of the temperatures of fluids, this effect may be due to the diffusion.
Case 3:
1. Velocity Contour at symetric plane:
2. Temperature contour at symetric plane:
3. Area Weighted average temperature plot at output:
4. Standard Deviation of temperature plot at outlet:
5. Variation of temperature at outlet cross-section:
6. Variation of Velocity at outlet cross section:
For case 3, At inlet_hot, air enters at a velocity 3m/s with a temperature of 25 c(298.15 K). At inlet_cold, air enters at a velocity 12m/s with a temperature of 10 c(283.15 K), which can be observed from temperature and velocity contours. No slip condition applied on the walls can be observed from the velocity of fluid near wall(in the velocity contour).
The area weighted average of temperature at outlet is around 17.5 c and standard deviation of temperature at oulet is around 0.96.
From variation of temperature at outlet cross section, we can observe that at outlet cross section, temperature is least at lower bottom and the temperature of fluid is maximum near the centre of the outlet.This behaviour is opposite to that of case 1and case2.
Comparing case 1 and case 3, we can observe that increasing the velocity of cold fluid entering through inlet_cold results in lesser area weighted average temperature at the outlet and also lesser standard distribution of temperature at outlet, hence case 3 results in better mixing effectiveness compared to case 1.
Conclusion:
Comparing the results of case 1, case2 and case3 ,the following can be concluded,
1. Comparing the temperature at lower bottom of outlet for case 1 and case2, we can observe that there is decrease in temperature(from 297.2 to 295.5), therefore we can conclude that increasing the length of pipe leads to further mixing of the temperatures of fluids, this effect may be due to the diffusion.
2. Comparing case 1 and case 3, we can observe that increasing the velocity of cold fluid entering through inlet_cold results in lesser area weighted average temperature at the outlet and also lesser standard distribution of temperature at outlet, hence case 3 results in better mixing effectiveness compared to case 1.
3. Both Increasing the length of the pipe and increasing the velocity of the cold fluid results in better mixing effectiveness, but by comparing the Area weighted average of temperature at outlet and standard deviation of temperature at outlet, we can conclude that increasing the velocity of cold fluid is more effective than increasing the length of the T-joint pipe.