Conjugate Heat Transfer Simulation

AIM : To setup the simulation for a flow through a pipe (outer layer is a metal-Aluminium and inner is a fluid region) of thickness 5mm where heat is supplied constantly.

SIMULATION-SETUP :

• Case setup : CONVERGE CFD STUDIO
• Commanad editor : Cygwin
• Post-processor : Paraview

THEORY ( CONJUGATE HEAT TRANSFER )

The word conjugate heat transfer refers to the variation in temperature of both solids and fluids which occurs due to thermal interaction between solids and fluids. In-other words it’s the combination of heat transfer both in solid and fluid. For example, heat sinks are deigned to combine heat transfer by conduction in the heat sink and with the convection in the surrounding fluid.

As we all know that the heat transfer in solid is mainly due to – CONDUCTON, where the heat is transferred by the tight bound molecules inside the solid.

Whereas fluids transfers the heat through CONVECTION, where the heat is transferred by freely movable gaseous or fluid molecules through air or any other medium

Heat Transfer by Solids and Fluids

Heat Transfer in a Solid

In most cases, heat transfer in solids, if only due to conduction, is described by Fourier’s law defining the conductive heat flux, q, proportional to the temperature gradient: .

For a time-dependent problem, the temperature field in an immobile solid verifies the following form of the heat equation:

Heat Transfer in a Fluid

Due to the fluid motion, three contributions to the heat equation are included:

1. The transport of fluid implies that there is energy transport too, which appears in the heat equation as the convective contribution. Depending on the thermal properties on the fluid and on the flow regime, either the convective or the conductive heat transfer can dominate.
2. The viscous effects of the fluid flow produce fluid heating. This term is often neglected, nevertheless, its contribution is noticeable for fast flow in viscous fluids.
3. As soon as a fluid density is temperature-dependent, a pressure work term contributes to the heat equation. This accounts for the well-known effect, for example, compressing air produces heat.

Accounting for these contributions, in addition to conduction, results in the following transient heat equation for the temperature field in a fluid:

THE CASE SETUP

The geometry was designed in converge studio itself and no other designing software was used to build the geometry

The pipe consists of an inner radius of 15mm and outer radius of 20mm and hence the thickness is clearly 5mm.

The outer boundary of the pipe is made of metal – Aluminum and the inner layer is the region where the fluid flow.

Heat is constantly added to the aluminum at the rate of -10000 W/m*2. As explained in the theory, the mode of heat transfer in solids is conduction and therefore it takes magnitudely a large time to heat up a solid when compared to the liquid.

BOUNDARY FLAGGING

The given geometry was divided into 5 different boundaries. The outer boundary was a solid, made of aluminum and the inner was a fluid region.

It is necessary in heat transfer simulation to defined a boundary or region that is common to both solid and fluid regions or in case of hotter and colder body a surface.

The region or space between the solid and the fluids is called as interface. Hence an interface boundary was assigned.

The inlet and the outlet boundaries were assigned.

MATERIALS FOR SIMULATION

Air ( fluid ) is made to pass through the inlet and outlet, which is being heated constantly at the rate of -10000 W/m^2 at a temperature of 300K.

As consists of O2 and N2, were added as a species for the simulation.

The outer boundary was a solid made of aluminum, and the inner was a fluid region.

SIMULATION TIME PARAMETERS

The type of simulation we are doing here is a transient based full-hydrodynamic mode simulation, where the parameters vary with respect to time and the final result also are solved in terms of the time dependent derivatives.

Pressure-based transient solver is chosen here.

The simulation was ran from an initial time step of 0 sec to an end time of 0.5 sec. During this time of simulation the simulation was made to run for an initial and minimum time of 1e-7 sec ie,,, for every interval of 1e-7 sec the simulation was running. The maximum time step was 1 sec, although the simulation won’t reach the maximum time step.

BOUNDARY CONDITIONS

The outer boundary ie,, the solid – aluminum was assigned to a wall-type boundary condition, with the constant heat flux added at the rate of -10000 W/m^2.

In case of inlet, was an inflow boundary and the pressure boundary condition was assigned to a Neumann boundary condition as the inlet reynold’s number of 7000 was given and hence the velocity boundary condition was a Dirichlet boundary condition with a velocity along it’s length (z-axis) as 7 m/s.

The outlet was a out-flow boundary condition, with pressure boundary condition being a Dirichlet boundary condition at an atmospheric pressure ie,,, 101.325 kPa.

The interface boundary was assigned to a interface boundary condition. Incase of interface boundary condition it is necessary to define the forward and the backward direction for the boundary.

This is because when the converge runs the simulation it is going to create a copy of interface. So now we have 2 interface boundaries where one of it belongs to the solid side and the other one belongs to the fluid side. So it is necessary to assign the boundary condition for both the original and replicated interface.

In our case as shown in the above figure, the forward side will be pointing towards the fluid flow domain and the reverse side is clearly pointing towards the remaining domain ie,,, the solid domain.

Since the forward side is pointing towards fluid domain, the velocity and temperature boundary condition will be law of wall. On the other hand as the reverse direction is pointing towards the solid domain the velocity boundary condition will be treated as slip and the temperature will be a Dirichlet boundary condition, where we will be defining the value of temperature.

TURBULENCE-MODELLING :

The turbulence model taken here is RNG K-Ɛ.

MESHING

The main intention of meshing ie,,, discretizing a given model is to visualize the variation of parameters like velocity, pressure etc.. through each cell or element.

The model designed was discretized into finite number of elements in all 3 directions. The simulation was done for three different base mesh size

• dx = dy = dz = 0.004mm
• dx = dy = dz = 0.003mm
• dx = dy = dz = 0.002mm

In the first case converge predicted around 5369 cells, however analytical or graphical solution converges or shows a total cell count of 6000.

In the second case, the mesh refinement here is much more finer and comparatively a large number of cells are created here than in first case. Converge predicted around 12726 cells, however analytical or graphical solution converges or shows a total cell count of 10496.

Similarly in the third case a better and much more finer mesh is refined here to get a better results and convergence, when compared to the previous two cases. Converge predicted around 42951 cells in the simulation.

POST-PROCESSING VARIABLES AND OUTPUT FILES

In addition to post-processing variables like pressure, temperature, viscosity, velocity etc… a dimensionless term called was added.

In a flow simulations or a flow which is bounded by a solid regions (wall), the behavior of flow near the walls is a complicated phenomenon and hence to distinguish different regions of flow near the wall and what is the effect of meshing elements on final results concept of was introduced.

• : It is a dimensionless number where distance from the wall is measured in terms of viscous lengths.

In a flow bounded by a wall, different scales and physical processes are dominant in the inner portion near the wall, and the outer portion approaching the free stream. These layers are typically known as the inner and outer layers.

The flow near the wall has been analyzed in terms of three layers:

• The inner layer, or sublayer, where viscous shear dominates
• The outer layer, or defect layer, where large scale turbulent eddy shear dominates
• The overlap layer, or log layer, where velocity profiles exhibit a logarithmic variation

From the above image we need to be careful to ensure that our values are not so large that the first node falls outside the boundary layer region. If this happens, then the Wall Functions used by our turbulence model may incorrectly calculate the flow properties at this first calculation point which will introduce errors into our pressure drop and velocity results.

Thus it is imporatant that the wall function approach and value required is determined by the flow behavior and the turbulence model being used in the simulation. If we have an attached flow, then generally we can use a Wall Function approach, which mean a larger initial value, smaller overall mesh count and faster run times. If one expects to have flow separation and knows that the accurate prediction of the separation point will be having an impact on the result, then he would be advised to resolve the boundary layer all the way to the wall with a finer mesh. Also unfortunately, as the value is dependent on the local fluid velocity which varies across the wall significantly, it is not possible to know the exact prior to running an initial simulation. Hence, it is a important process for one to get into the habit of checking the values as part of his normal post-processing so that one can make sure to fall in the valid range for the flow physics and turbulence model selection.

SIMULATION RESULTS

The simulation was ran for three different base meshes ie,,,

MESHING

• Base grid of 4mm

• Base grid of 3mm

• Base grid of 2mm

for all this base mesh the super-cycling stage interval was set to 0.03s and the result that were obtained was observed for the convergence criteria ie,,, how the fluid temperature converges to a particular value, when the base mesh is varied.

The second part of the simulation was varying in the super-cycling stage interval. The simulation was done for three different super-cycling stage intervals.

• 01 s
• 02 s
• 03 s

Why do we need Super-Cycling modeling ??

Well, in order to solve simulation like heat transfers and gas-turbines, the simulation time needs to go hand on hand, hence to reduce the computational time involved and make the simulation run in faster time and accurate one with evenly distribution of temperature, Super-Cycling modeling is used.

How does this work ??

When the Super-Cycling modeling is enabled, converge first solves both the fluid and solid regions by using a transient solver for a particular period of time. This time period may be the steady-state for the flow through the pipe or time taken for it to flow throughout it. During this time period all the instantaneous heat-transfer coefficients and temperatures are solved and stored in the interface assigned. Once this process is done converge averages the heat-transfer coefficients and temperatures near the wall by freezing the fluid flow solver.

Now the obtained heat-transfer coefficients and the temperatures are used as boundary conditions for the solid region to solve the equation of heat-transfer pertaining to solid region, by using a steady-state solver. Once the solid side temperatures are solved, these values are stored and updated. Once this process is done the entire process repeats again till the solution converges.

This is what is called as “ Super-Cycling ”

EFFECT OF BASE-MESH ON OUTLET- FLUID AND SOLID TEMPERATURE

How does the fluid temperature converges to a particular value for diffrent mesh size ??

Above figure shows the variation of fluid temperature for diffrent mesh size of 4mm, 3mm, and 2mm for a sub-grid scale stage interval of 0.03s.

Initially as per the boundary conditions the boundary conditions the temperature of the fluid ie,, air is at 300k. Once the heat flux is added at a constant rate of

-10000 W/m^2 ie,,, the heat is given to the system, the temperature of the air increases and reaches a constant value around 325K, after it reaches the steady state. The

temperature can't increase beyond this value, as this is the maximum value of the temperature that a fluid can carry through convection process. For diffrent base although the temperature

couldn't converge to exactly a particular value, but it's around 323 - 325K. If the simulation was ran for a mesh size of 1mm, the solution would have converged, since for 2mm mesh size the

solution took almost 1.5 Hrs, there was no meaning of running it again, to get a value almost around the values obatined.

Above figure shows the variation of Solid temperature for diffrent mesh size of 4mm, 3mm, and 2mm for a sub-grid scale stage interval of 0.03s.

The solid temperature for base mesh size of 2mm was unable to converge with the mesh size of 2 & 3mm, this is because in case of solids the heat-transfer is mainly due to the process of

conduction. Where the solid takes a huge time-interval for getting heated up, and as the mesh size is reduced the elements or cells created will be a smaller one's and it takes larger time for heating up each and every element and hence the convergence is slower in case of 2mm base grid and the reason for slight solution deviation.

The solid temperature hence, is around 600K.

TOTAL CELL-COUNT

The total-cell count is a constant value throughout the simulation. The simulation was ran for 3 diffrent base mesh of 4mm, 3mm and 2mm. Diffrent total number of cells were created for diffrent base mesh sizes.

For 4mm

For 3mm

For 2mm

As shown in the above figures, there are four different ranks shown. Each rank corresponds to a number of processors used during running the simulation. Since the simulation is ran for 4 processors (to reduce the computational time involved and to lower the loads on only 1 processor), hence there are 4 rank indicated. The main intention of creatin the cells or elements is to visualize the flow through each cells and what is causing the flow to vary in that particular region of the cells created.

Effect of Super-Cycling modelling

In order to solve simulation like heat transfers and gas-turbines, the simulation time needs to go hand on hand, hence to reduce the computational time involved and make the simulation run in faster time and accurate one with evenly distribution of temperature, Super-Cycling modeling is used.

The simulation was ran for three diffrent super-cycling stage intervals.

1) 0.03s 2) 0.02s 3) 0.01s

Super-cycling temperature

Above figure shows the effect of super-cycling stage interval on 3 diffrent super-cycling stage interval of 0.01, 0.02, and 0.03s. The solution converges to a temperatureof 605K.

The time taken by the solution to converges for the super-cycling model is of the order 0.01s < 0.02s < 0.03s ie,,, the simulation converges more fastly in case of lower super-cycling stage interval and as the stage interval is increase the time required for convergence also increases. But in all three cases the simulation conerges to a particular value.

Effect of Super-Cycling modelling on fluid outlet temperature

For diffrent Super-Cycling stage interval the temprature of the outlet fluid converges to a particular value of 325K.

Effect of Super-Cycling modelling on Solid temperature

For diffrent Super-Cycling stage interval the temprature of the solid converges to a particular value of 600K, which is same as observed in the above results of silid region.

Y+ results

In addition to post-processing variables like pressure, temperature, viscosity, velocity etc… a dimensionless term called was added.

In a flow simulations or a flow which is bounded by a solid regions (wall), the behavior of flow near the walls is a complicated phenomenon and hence to distinguish different regions of flow near the wall and what is the effect of meshing elements on final results concept of was introduced.

• : It is a dimensionless number where distance from the wall is measured in terms of viscous lengths.

The value of for a mesh size of 4mm and supercycling stage of 0.03s is in the range of 0 to 41

Similarly, The value of for a mesh size of 3mm and supercycling stage of 0.03s is in the range of 0 to 39

And for mesh of much finer size of 2mm, the value of is around 0 to 18

Thus, we need to be observing values of , such that the values are not so large that the first node falls outside the boundary layer region. If this happens, then the Wall Functions used by our turbulence model (RNG K-Ɛ) may incorrectly calculate the flow properties at this first calculation point which will introduce errors into our pressure drop, temperature and velocity results.

Animation Links :

1) For 2mm mesh size.

Temperature profile : https://youtu.be/mZF7dIyliiY

Velocity profile : https://youtu.be/5DXHFUgBceU

2) For 3mm mesh size.

Temperature profile : https://youtu.be/HN9MbmZ52ZQ

Velocity profile : https://youtu.be/hPCgj266FxQ

3) For 3mm mesh size.

Temperature profile : https://youtu.be/kHBn7ChWEg4

Velocity profile : https://youtu.be/Sc_fAIdfxbU