Week 6: Conjugate Heat Transfer Simulation

                                                  SIMULATION OF CONJUGATE HEAT TRANSFER ANALYSIS OF AIR-FLOW THROUGH PIPE USING SUPER-CYCLING MODEL

 

l. OBJECTIVES

1. Simulate the flow of air through a pipe using conjugate heat transfer analysis and super-cycling model.

2. Perform a grid dependence test.

3. Analyse the effects of variation of super-cycle stage interval on the convergence and total simulation time.

 

ll. INTRODUCTION

1. Super-cycling Model

Super-cycling is a method used by CONVERGE Studio in the case of conjugate heat transfer problems with solid and liquid regions.

The integral problem here is that both solvers cannot run simultaneously since solving in the fluid domain is much faster due to the time scale difference for heat transfer in liquids compared to solids. This causes problems during the solution given that the solid side solver would not have reached convergence in the time the liquid solver does. This is where the concept of super-cycling can be used.

The basic idea of super-cycling is that the solver for the fluid domain is paused until the solver for the solid domain converges. This pausing is done in intervals that can be set by the user.

2. Wall Functions and Y+

Gradients of velocity, temperature, etc. close to the wall are large and hence a fine grid resolution is required to accurately resolve the gradients.

Wall Functions are empirical functions that are fitted to the observed behavior close to the wall.

In the turbulent region, we have three layers: the viscous sub-layer, the buffer layer, and the log-law layer. We can use the no-slip condition to resolve the flow in the viscous sub-layer and hence no wall function is required. The wall function can help to resolve the flow in the log-law layer.

We need a method to determine the layer of the flow to s=resolve the flow. This is done by the non-dimensional term Y+, which can be used to understand how coarse or fine our grid is. This can be used to determine if wall functions need to be used or not.

Note that the wall functions cannot resolve the flow accurately at the buffer region, hence CFD codes don't recommend placing cells in the buffer region. This leads to resolved meshes (y+ < 5) and wall-function based meshes (y+ > 30). Also, note the wall functions (y+ > 30) are likely to be inaccurate if the models are subjected to adverse pressure gradients, separation of flow, to the presence of curvatures.

lll. SIMULATION OF FLOW OF AIR THROUGH A PIPE - BASELINE CONFIGURATION

A. Model

The model considered in this project is as shown below - 

 

B. Dimensions

The pipe has the following dimensions - 

• Outer Cylinder Diameter = 0.04 m
• Inner Cylinder Diameter = 0.03 m
• Length of Pipe = 0.2 m

C. Geometry Setup And Boundary Flagging

The geometry is set up and the boundaries are flagged in converge studio as shown in the figure below-

 

 

D. Diagnosis Check

A diagnosis check is performed to check for errors - 

Diagnosis Check 1 - After Geometry Setup

 

The Nonmanifold Problems occur due to the presence of three surfaces of a triangle element in contact with each other which are not recommended in ordinary simulations but are unavoidable in the CHT analysis. This error will automatically disappear after the case setup procedure as CONVERGE identifies this simulation to be a CHT analysis -

Diagnosis Check 2 - After Case Setup

 

E. CASE SETUP

1. Application Type: Time-Based

2. Materials:

• Predefined Mixtures: Air
• Gas Simulation: Default
• Solid Simulation:
  • Predefined Solids: Aluminium
• Species:
  • Solid: Aluminium
  • Gas: `O_2 and N_2`

3. Solver: Transient

4. Simulation Time Parameters:

• Start Time: 0 s
• End Time: 0.5 s
• Initial time-step: 1e-7 s
• Minimum time-step: 1e-7 s
• Maximum time-step: 1 s

5. Regions and Initialization:

• Fluid Region:
  • Stream ID: 0
  • Velocity: 3.66 m/s (z-direction)
  • Total Pressure: 101325 Pa
  • Temperature: 300 K
  • Species: Air
    • Mass Fraction of Nitrogen: 0.77
    • Mass Fraction of Oxygen: 0.23
• Fluid Region:
  • Stream ID: 1
  • Total Pressure: 101325 Pa
  • Temperature: 300 K
  • Species: Aluminium
    • Mass Fraction of Aluminium: 1

6. Boundary Conditions:

We will be defining the inlet Reynold's Number as 7000.

• `Re= 7000`

Diameter of the Fluid Region,

• `D = 0.030 m`

At an inlet temperature of 300 K, the density and viscosity of air is as follows -

• Density of Air at 300 K,

        `rho = 1.177 (kg) / m^3`

• Viscosity of Air at 300 K,

         `mu=1.846 *10^(-5) (kg)/(ms)`

The Reynold's Number is defined as -

• `Re=(rho VD)/(mu)`

From the above equation, the inlet velocity can be calculated as follows

• `V=(Re.mu)/(rho.D)`
• `=>V = 3.66m/s`

 

Boundary	Region	Boundary Type	Dirichlet Boundary Condition	Neumann Boundary Condition
Outer Wall	Solid	WALL	Velocity: Stationary and Slip     Temperature: Heat Flux =              -10000 W/m2	-
Solid Side	Solid	WALL	Velocity: Stationary and Slip	Temperature
Inlet	Fluid	INFLOW	Velocity = 3.66 m/s Temperature = 300 K Species = Air	Pressure
Outlet	Fluid	OUTFLOW	Pressure = 101325 Pa Temperature = 300 K (Backflow) Species = Air (Backflow)	Velocity
Interface	Forward Boundary: Fluid Region	INTERFACE	Velocity: Law of Wall Temperature: Law of Wall	-
	Reverse Boundary: Solid Region		Velocity: Slip Temperature: Specified Value

  7. Physical Models:

• Turbulance Model: RNG `kepsi`
• Super-cycle Model:
  • Begin storing super-cycle data: 0.05 s
  • Solid side heat transfer solver: Steady-state solver
  • Time Length of each super-cycle stage: 0.05 s
  • Total number of stages: 1
  • Conduction CFL number: 100
  • SIE tolerance: 1e-7
  • SIE relaxation Factor: 1.4
  • Super-cycle Monitor Point: (0.012933, 0.01179, 0.18)

8. Grid Control:

• Base Grid:
  • dx = dy = dz = 0.004 m

9. Output Files:

• Time interval for writing 3-D output files: 0.012 s
• Time interval for writing text output: 1e-6 s
• Time interval for writing restarting output: 0.1 s

 

F. OUTPUTS

1. Mesh

Figure 1 - Mesh

2. Flow Properties Contour Animation

Animation 2.1 - Temperature Contour

 

Animation 2.2 -  Y+ Contour

 

3. Plots

Figure 3.1 - Super-cycle At The Monitor Point

Figure 3.2 - Temperature Profile In the Solid Region

Figure 3.2 - Temperature Profile In the Fluid Region

 

G. RESULTS AND INFERENCES

1. Temperature Contour

The temperature contour animation shows that the solid gradually heats up which is followed by the heating of the fluid. The contour is not well defined due to the coarseness of the mesh.

2. Y+ Contour

The y+ contour shows that most of the y+ values at the wall are in the range of 10-30. This region is the buffer region where the laminar flow is transitioning to the turbulent flow and is the least desirable range as the CFD codes are not able to accurately predict the boundary effects in this region. Hence, it is recommended to either increase the mesh size to obtain a y+ value of above 30 or decrease the mesh size to obtain a y+ value of below 10.

Increasing the size of the mesh will result in a coarse mesh which further results in inaccurate results. Decreasing the mesh size will increase the accuracy and increase the computational resources required to simulate the flow. An alternative and recommended approach is to only refine the mesh at the boundary to capture the effects accurately.

3. Super-cycle At The Monitor Point

At monitor, a point is made in the Solid Thickness boundary to observe the effects of super-cycling. The plot shows that the steady-state solver is employed in the said region at fixed intervals of time (i.e. the Super-cycle Stage Interval) and the results are computed while the transient solver computing the fluid flow is temporarily paused.

Super-cycling is done as the heat flow in the solids when depicted by a transient solver, is time-consuming. Hence, it is better to apply steady-state analysis for the heat transfer through the solid while studying the behavior of the fluid flow through the pipe.

4. Temperature profile In The Solid Region

As mentioned above, the steady-state solver is employed in the solid region resulting in the sudden increase in temperature at various time intervals. This does not depict the actual temperature flow in the solid and only the steady-state temperature. Note that the increase in the temperature in the solid regions is due to the heat flux added to the solid outer wall.

5. Temperature Profile In The Fluid Region

The transient solver is used to simulate the flow in the fluid region and hence the actual increase in the temperature is observed in this case. The increase in temperature is due to the transfer of heat from the solid to the fluid.

 

lV. SIMULATION OF FLOW OF AIR THROUGH A PIPE - REFINED CONFIGURATION

The dimensions and geometry setup for the pipe remain the same as that for the baseline configuration.

A. CASE SETUP

In the case setup, the grid size is further refined and one embedded layer is also added to the boundary of the fluid region to properly capture the fluid behavior at the boundaries.

Also, The super-cycling stage interval is reduced to ensure that the solid region reaches equilibrium conditions within the simulation time.

All the other conditions and values remain the same as that of the baseline configuration.

1. Physical Models:

• Turbulence Model: RNG `Kepsi`
• Super-cycle Model:
  • Time Length for each super-cycle stage: 0.01 s
• Fixed Embedding:
  
  • Entity Type: Boundary 
  • Boundary ID: Interface
  • Region: Fluid Region
  • Model: Permanent
  • Scale: 1
  • Embed layers: 1

B. OUTPUTS

1. Mesh

Figure 1 -Mesh

2. Flow Properties Contour Animation

Animation 2.1 - Temperature Contour

Animation 2.2 - Y+ Contour

3. Plots

Figure 3.1 - Temperature Profile In the Solid Region

Figure 3.2 - Temperature Profile In The Fluid Region

C. RESULTS AND INFERENCES

1. Temperature Contour

The temperature contour animation shows that the solid gradually heats up which is followed by the heating of the fluid. The contour is much more well-defined as compared to the contour generated by the base mesh.

2. Y+ Contour

The y+ contour shows that most of the Y+ values at the wall are in the range of 0-10. This region is the laminar region where the flow is laminar and the no-slip condition is applied. Although the generated y+ values are not the most recommended value(i.e. Y+ `le` 1), it is still within the acceptable range and the results are much more accurate than those obtained in the baseline configuration.

3. Temperature Profile In The Solid Region

As mentioned before, the steady-state temperature results do not depict the actual temperature flow in the solid and only the equilibrium temperature. Since we have reduced the super-cycle stage interval, the number of cycles executed to achieve steady-state temperature has increased. This is done to ensure that steady-state temperature is achieved within the defined simulation time.

4. Temperature Profile In The Fluid Region

The transient solver is used to simulate the flow in the region and hence the actual increase in the temperature is observed in this case. Note that the mean temperature of the fluid is higher than that obtained in the base mesh as the refined mesh can capture the fluid flow more accurately.

V. GRID DEPENDENCE TEST

The grid dependence test is carried out by refining the mesh in the baseline configuration while keeping the remaining parameters unchanged. We shall be considering the following three cases - 

• Case 1: Grid Size = 0.003 m
• Case 2: Grid Size = 0.0025 m
• Case 3: Grid Size = 0.002 m

A. OUTPUTS

1. Mesh

2. Flow properties Contour Animation

Animation 2.1 - Temperature Contour

Animation 2.2 - Y+ Contour

3. Plots

Figure 3.1 - Temperature Profile In The Solid Region

Figure 3.2 - Temperature Profile In The Fluid Region

B. RESULTS AND INFERENCES

1. Temperature Contour

From the temperature contour animation, it can be observed that as the mesh size decreases, the contour becomes more well defined and the number of contour bands in the fluid region also increases. Decreasing the size of the mesh helps in increasing the accuracy of the results at each cell, which further results in an overall increase in the accuracy of the solution. 

2. Y+ Contour

The values of Y+ decrease as the size of the mesh decreases. The average value of Y+ for the most refined mesh is in the range of 10-20, which is still in the buffer region and the results at the boundary can still not be considered to be accurate.

3. Temperature Profile In The Solid And Fluid Region

The temperature profile shows that as the grid decreases, the mean temperature of the solid and the fluid decreases. This is because, as the mesh size decreases, the temperature distribution is captured more accurately at each cell, which results in an overall decrease in the mean temperature in the solid and fluid region.

Vl. VARIATION OF SUPER-CYCLE STAGE INTERVAL

In this part, we shall be varying the time length for each cycle stage i.e. the super-cycle stage interval, and observe the effects on the mean temperature in the solid region for the entire simulation interval. We shall be considering the following three cases -

• Case 1: Super-cycle Stage Interval = 0.01 s
• Case 2: Super-cycle Stage Interval = 0.02 s
• Case 3: Super-cycle Stage Interval = 0.03 s

 

A. OUTPUT

 

B. RESULTS AND INFERENCES

1. Effect of SSI on the Convergence Time

Decreasing the super-cycle stage interval results in faster convergence of the solution concerning the simulation time.

It is better to have a small duration and need the simulation to converge within the given simulation time.

In the given cases, the solution converges in about 0.39s for an SSI of 0.01s.

2. Effect of SSI on the Total Simulation Time

Decreasing the super-cycle stage interval results in an increase in the number of cycles executed by the solver. This results in more pauses in the fluid solver, which will further increase the computation time of the solution.

It is better to have a larger SSI when we have a larger simulation time for the steady-state solver to converge. However, the condition is that the solution should completely converge within the stipulated simulation time.

 

Vll. CONCLUSIONS

The flow of air through a pipe using conjugate heat transfer analysis and super-cycling model has been simulated successfully. The grid dependence test is performed and the effects of variation of super-cycle stage interval on the convergence and total simulation time are analyzed.