Conjugate Heat Transfer Simulation of Pipe Flow
Conjugate Heat Transfer Simulation of Pipe Flow
I. Overview
In the current project conjugate heat transfer analysis of flow through a circular pipe is performed using CONVERGE CFD, with the following objectives:
i. To study the effect of grid size on the simulation results: The following three grid sizes are used:
(a) dx=dy=dz=0.004m
(b) dx=dy=dz=0.003m
(c) dx=dy=dz=0.002m
The results of this study are discussed in section 1 of Results and discussions.
ii. To study the effect of super-cycle stage interval on the simulation results: The following four intervals are used for the super-cycle stage:
(a) 0.01s
(b) 0.02s
(c) 0.03s
(d) 0.05s
The results of this study are discussed in section 2 of Results and discussions.
II. Geometry
The geometry used for performing simulations consists of a cylindrical pipe having the following dimensions:
Inner diameter=0.03m
Outer diameter=0.04m
Thickness=0.005m
Length=0.2m
The figure given below shows the geometry:
III. Materials
For performing gas simulation, air (predefined mixture) is used as the fluid. Accordingly, the following global properties are utilized:
Turbulent Prandtl number: 0.9
Turbulent Schmidt number: 0.78
For performing solid simulation, Aluminum (predefined solid) is used.
IV. Solver
1. The three dimensional Navier Stokes equations are solved using CONVERGE solver with the following settings:
i. Solver: Transient, Pressure-based
ii. Navier-Stokes solver scheme: PISO
iii. Simulation mode: Full hydrodynamic
iv. Gas flow solver: Compressible
v. Start time: 0
vi. End time: 0.5s
vii. Initial time-step: 1e-7s
viii. Minimum time-step: 1e-7s
ix. Maximum time-step: 1s
2. For modeling the inherent turbulence in the problem, RNG `K-epsilon` model is employed.
3. Super-cycle model is enabled (to speed-up the convergence to steady-state) with the following settings:
i. First time interval after which the super-cycle data is stored: 0.05s
ii. Solid side heat transfer solver: Steady-state solver
iii. Time length for each cycle stage: 0.05s (for grid-independence study)
iv. In order to check the convergence, an output point is created within the solid region.
V. Initial conditions
Before setting up the initial conditions, two regions are defined, each corresponding to either of the two phases (solid and fluid). The region corresponding to the fluid phase is assigned a Stream ID of 0 whereas the region corresponding to the solid phase is assigned a Stream ID of 1. The following initial conditions are applied to the two regions:
Fluid Region:
Stream ID=0
Velocity: X component=0; Y component=0; Z component=0
Temperature: 300K
Pressure: 101325Pa
Turbulence Kinetic Energy: 1m^2/s^2
Turbulent Dissipation: 100m^2/s^3
Species: Mass fractions: Oxygen: 0.23; Nitrogen: 0.77
Solid Region:
Stream ID=1
Temperature: 300K
Species: Mass fractions: Aluminum: 1
VI. Boundary conditions
The figure attached below shows the different boundaries assigned to the geometry.
The following boundary conditions are applied at the assigned boundaries:
1. Solid_Outer_Wall: WALL type boundary condition is applied with the following settings:
Wall motion type: Stationary
Surface movement: FIXED
Velocity: Slip
Temperature: Heat flux: -10000W/m^2
2. Solid_Wall_Thickness: WALL type boundary condition is applied with the following settings:
Wall motion type: Stationary
Surface movement: FIXED
Velocity: Slip
Temperature: Zero normal gradient
3. Inlet: INFLOW type boundary condition is applied with the following settings:
Pressure: Zero normal gradient
Velocity: X component=0; Y component=0; Z component=3.7077m/s
Temperature: 300K
Species: Mass fractions: Oxygen: 0.23; Nitrogen: 0.77
Turbulent Kinetic Energy: Intensity: 0.02 (fraction)
Turbulent Dissipation: Length scale: 0.003m
4. Outlet: OUTFLOW type boundary condition is applied with the following settings:
Pressure: 101325Pa
Velocity: Zero normal gradient
Temperature backflow: 300K
Species backflow: Mass fractions: Oxygen:0.23; Nitrogen: 0.77
Turbulent Kinetic Energy Backflow: Intensity: 0.02 (fraction)
Turbulent Dissipation Backflow: Length scale: 0.003m
5. Interface: INTERFACE type boundary condition is applied with the following settings:
Wall motion type: Stationary
Surface movement: FIXED
Forward side (Fluid region)
Velocity: Law of wall
Temperature: Law of wall (coupled)
Turbulent Kinetic Energy: Zero normal gradient
Turbulent Dissipation: Wall model
Reverse side (Solid region)
Velocity: Slip
Temperature: Dirichlet (coupled)
VII. Grid
For study i (effect of grid size), three different grid sizes are used:
(a) dx=dy=dz=0.004m
(b) dx=dy=dz=0.003m
(c) dx=dy=dz=0.002m
For study ii (effect of super-cycle stage interval), a grid size of dx=dy=dz=0.004m is used.
VIII. Results and discussions
In this section, the simulation results are discussed. Section 1 deals with the simulation results of study i (Effect of grid size) whereas section 2 deals with the results of study ii (Effect of super-cycle stage interval).
1. Effect of grid size
As mentioned previously, for this study the case is simulated using the following three base grid sizes:
i. dx=dy=dz=0.004m: Mesh 1
ii. dx=dy=dz=0.003m: Mesh 2
iii. dx=dy=dz=0.002m: Mesh 3
1.1. Mean temperature in the solid region
The mean temperature in the solid region is plotted over time for all three meshes.
The plot shows that steady-state is achieved for all three meshes. It is also evident that the solution is grid-dependent. The values of mean temperature in the solid region (at 0.5s) for different meshes are as follows:
i. Grid 1 (0.004m): 858.32K
ii. Grid 2 (0.003m): 824.10K
iii. Grid 3 (0.002m): 747.73K
The various super-cycle stages, after 0.05s each, can also be observed on the plot. At each step, the super-cycle data is stored and the steady-state computations are performed for the solid region.
1.2. Super-cycle temperature in the solid region
The super-cycle temperature at the output point, defined in the solid region, is plotted over time for all three meshes. These plots are generated using the steady-state converged values of temperature at the output point after each super-cycle stage interval.
Note: The super-cycle stage intervals begin after the first time interval, at the end of which the storing of super-cycle data begins.
Again, the grid dependence of the solution is quite evident from the plot. The values of super-cycle temperature at the output point for different meshes are as follows:
i. Grid 1 (0.004m): 858.16K
ii. Grid 2 (0.003m): 823.96K
iii. Grid 3 (0.002m): 747.59K
1.3. Average fluid temperature at the outlet
The average temperature of the fluid region at the outlet is plotted over time for all three meshes.
The plot suggests that the average fluid temperature at the outlet converges to nearly equal values for all three meshes.
i. Grid 1 (0.004m): 379.50K
ii. Grid 2 (0.003m): 380.01K
iii. Grid 3 (0.002m): 379.19K
1.4. Temperature contours
For all three meshes the temperature contours are plotted at a plane passing through the centre of the pipe.
i. Grid 1 (0.004m)
ii. Grid 2 (0.003m)
iii. Grid 3 (0.002m)
The temperature contour plots show that maximum temperature is achieved by the solid region. It is also evident that the contour plots become smoother with a decrease in grid size. The temperature gradients near the wall, in the fluid, are best resolved by Grid 3 (0.002m). However the solution is still grid-dependent for Grid 3 and hence is not reliable. Also the value of Y+ needs to be checked before employing the Standard Wall Function in the `K-epsilon` model.
1.5. Y+ values
In turbulence modeling, Y+ values can be used to decide whether the Standard Wall Function needs to be employed.
For 0<Y+<10: The first cell center near the wall lies in the viscous sub-layer region. Within this region, the Standard Wall function is not required and the no-slip condition should be used to inform the solution field near the wall.
For 10<Y+<30: The first cell center near the wall lies in the buffer layer. In this region, the Standard Wall Function yields incorrect values. Modeling transition flows in this region is quite a challenge.
For Y+>30: The first cell center near the wall lies in the logarithmic layer (turbulent region). For this region, the Standard Wall Function can be employed.
Hence if turbulence model (with Standard Wall Function) is employed for a particular problem, Y+ should be greater than 30. This puts a lower limit on the grid size.
For all three meshes the Y+ contours are plotted at a plane passing through the centre of the pipe.
i. Grid 1 (0.004m)
Average Y+ at either wall: 46.24
ii. Grid 2 (0.003m)
Average Y+ at either wall: 48.25
iii. Grid 3 (0.002m)
Average Y+ at either wall: 15.49
The average value of Y+ for Grid 3 (0.002m) indicates that the first cell location lies in the buffer layer and since `K-epsilon` turbulence model (with Standard Wall Function) is employed for the present problem, the results for Grid 3 can\'t be trusted. Hence the solution of the present problem remains grid-dependent in the region Y+>30.
In order to achieve grid independence, the grid needs to be a lot finer such that Y+ becomes smaller than 10. In such cases, the Standard Wall Function is not required. However simulating the current problem with such grid sizes is computationally quite expensive and hence is not attempted in the current project.
1.6. Animation
The animations attached below show the evolution of temperature contours with time.
Note: The solution is not time-accurate
i. Grid 1 (0.004m)
ii. Grid 2 (0.003m)
iii. Grid 3 (0.002m)
2. Effect of super-cycle stage interval
For the present study, the following four super-cycle stage intervals are used:
i. SSI 1: 0.01s
ii. SSI 2: 0.02s
iii. SSI 3: 0.03s
iv. SSI 4: 0.05s
For all cases, a grid size of dx=dy=dz=0.004m is adopted.
2.1. Mean temperature in the solid region
The mean temperature in the solid region is plotted over time for all three intervals.
The values of mean temperature in the solid region (at 0.5s) for different super-cycle stage intervals are given below:
i. SSI 1 (0.01s): 858.41K
ii. SSI 2 (0.02s): 858.39K
iii. SSI 3 (0.03s): 858.37K
iv. SSI 4 (0.05s): 858.32K
It can be seen from the plot that for all super-cycle stage intervals the mean temperature in the solid region converges to nearly equal values. The plots also show that with a decrease in super-cycle stage interval the number of super-cycle stages required to achieve steady-state increases.
2.2. Super-cycle temperature in the solid region
The super-cycle temperature at the output point, defined in the solid region, is plotted over time for all three intervals.
The super-cycle temperature values at the output point for different super-cycle stage intervals are given below:
i. SSI 1 (0.01s): 858.26K
ii. SSI 2 (0.02s): 858.24K
iii. SSI 3 (0.03s): 858.22K
iv. SSI 4 (0.05s): 858.16K
For all super-cycle stage intervals the super-cycle temperature converges to nearly identical values.
2.3. Average fluid temperature at the outlet
The average temperature of the fluid region at the outlet is plotted over time for all three intervals.
The values of average temperature of the fluid region at the outlet for different super-cycle stage intervals are as follows:
i. SSI 1 (0.01s): 379.53K
ii. SSI 2 (0.02s): 379.53K
iii. SSI 3 (0.03s): 379.52K
iv. SSI 4 (0.05s): 379.50K
For all super-cycle stage intervals the average temperature of the fluid region at the outlet converges to nearly identical values.
2.4. Simulation time
The table attached below shows the effect of super-cycle stage interval on the simulation time.
The table shows that with an increase in super-cycle stage interval, the simulation time increases.
ix. Conclusion
The following can be concluded from the simulation results:
i. The solution remains grid-dependent over the three employed grid sizes. For achieving grid independence even smaller grid sizes need to be investigated. However due to limitations on the available computational resources, such a task is not attempted in the current project.
ii. With a decrease in the super-cycle stage interval, the number of super-cycle stages required to achieve steady-state increases.
iii. With an increase in the super-cycle stage interval, the simulation time increases.