Lecture 2 - Conduction Heat

Transfer
15.0 Release

Heat Transfer Modeling using

ANSYS Fluent
© 2013 ANSYS, Inc. June 3, 2014 1 Release 15
 Agenda

• Introduction

• Energy equation in solids

• Equation solved in FLUENT
• Shell conduction model
• Non-conformal coupled wall
• Anisotropic conductivity
• Moving solids

• Solver parameters

© 2013 ANSYS, Inc. June 3, 2014 2 Release 15

 Agenda

• Introduction

• Energy equation in solids

• Equation solved in FLUENT
• Shell conduction model
• Non-conformal coupled wall
• Anisotropic conductivity
• Moving solids

• Solver parameters

© 2013 ANSYS, Inc. June 3, 2014 3 Release 15

 Conduction Definition

• Heat transfer is energy in transit due to a temperature difference

• Conduction phenomenon:
• Energy is transported by basic carriers
• Fluids – molecules, atoms
• Solids – free electrons

© 2013 ANSYS, Inc. June 3, 2014 4 Release 15

 Fourier’s Law

• Conduction heat transfer is governed by Fourier’s Law.

• Fourier’s law states that the heat transfer rate is directly proportional
to the gradient of temperature.

• Mathematically, qconduction  k T
Thermal conductivity

• The constant of proportionality is the thermal conductivity (k).

• k may be a function of temperature, space, etc.
• For isotropic materials, k is a scalar value.
• In general (for anisotropic materials), k is a matrix.
• Table of k values for various materials can be found in the Appendix

© 2013 ANSYS, Inc. June 3, 2014 5 Release 15

 Agenda

• Introduction

• Energy equation in solids

• Equation solved in FLUENT
• Shell conduction model
• Non-conformal coupled wall
• Anisotropic conductivity
• Moving solids

• Solver parameters

© 2013 ANSYS, Inc. June 3, 2014 6 Release 15

 Energy Equation for Solid Materials

• Equation solved in FLUENT

  h 
   k T   S h
t
Unsteady Conduction Enthalpy
(Fourier’s Law) Source

• The dependent variable h is the enthalpy,


h  C p dT
0

© 2013 ANSYS, Inc. June 3, 2014 7 Release 15

 Shell Conduction

• In FLUENT, by default, planar heat transfer is ignored if the wall

thickness is not meshed.
Exhaust pipe at 800 K emits
radiation in the direction of the shield

Shield, 2 mm thick

• Results from shell (1 layer)

matches with that obtained 3 Prism layers
Shell Conduction ON
using 3 prism layers Shell Conduction OFF

Plate Temperature
Along the Flow Direction
© 2013 ANSYS, Inc. June 3, 2014 8 Release 15
 Shell Conduction

• To activate shell conduction, select it in the wall boundary condition

panel.

Don’t forget to specify the

material name and wall
thickness!

• Text commands
• To activate shell conduction for all walls with nonzero thickness:
grid/modify-zone/create-all-shell
• To deactivate all shell conduction zones:
grid/modify-zone/delete-all-shell

© 2013 ANSYS, Inc. June 3, 2014 9 Release 15

 Multi-layer shell conduction

• Multi-layer shell conduction allows to simulate heat transfer

through layers of different materials

• Multi-layer shell conduction is applicable if the solids are embedded

into the fluid

© 2013 ANSYS, Inc. June 3, 2014 10 Release 15

 Post processing Shells
• Automatic creation of shell surfaces with default naming convention
• External BC

• Embedded wall

© 2013 ANSYS, Inc. June 3, 2014 11 Release 15

 Shell Conduction

• Shell conduction needed regardless of thermal conductivity

k = 0.01 W/m·K (1D)

k = 0.01 W/m·K (Shell)
k = 200 W/m·K (1D)
k = 200 W/m·K (Shell)

© 2013 ANSYS, Inc. June 3, 2014 12 Release 15

 Shell Conduction – Unsteady

• The shell conduction model takes into account thermal inertia

which can not be included via the thin-wall approach.

© 2013 ANSYS, Inc. June 3, 2014 13 Release 15

 Shell Conduction – Connectivity

Boundary condition on
the edge of the shell?

• Specification of boundary condition at the wall end:

• By default, wall shell is adiabatic
• If shell conducting wall connects:
• Another shell conducting wall – The connecting edge has a coupled
boundary condition.
• Another non-conducting external wall – Edge has the same thermal
boundary condition.

• Heat flux on virtual boundaries is not reported in the total heat

flux report.

© 2013 ANSYS, Inc. June 3, 2014 14 Release 15

 Shell Conduction – Limitations

• Limitations of the shell conduction model:

• Shells cannot be created on non-conformal interfaces.
• Shell conduction cannot be used on moving wall zones.
• Shell conduction cannot be used with FMG initialization.
• Shell conduction is not available for 2D.
• Shell conduction is available only when the pressure-based solver is used.
• Shell conducting walls cannot be split or merged. If you need to split or
merge a shell conducting wall, you will need to turn off the Shell Conduction
option for the wall (in the Wall dialog box, perform the split or merge
operation, and then enable Shell Conduction for the new wall zones.
• The shell conduction model cannot be used on a wall zone that has been
adapted. If you want to perform adaption elsewhere in the computational
domain, be sure to use the mask register described in Section 29.11.1 of the
Fluent User Guide. This will ensure that adaption is not performed on the
shell conducting wall.

© 2013 ANSYS, Inc. June 3, 2014 15 Release 15

 Non-Conformal Coupled Wall

• Non-conformal coupled
wall:
• We can use fine mesh on
fluid zone and coarser mesh
on solid zone
• You can also model baffles.
Note:
Use /display/zone-grid ID
to display the shadow walls

© 2013 ANSYS, Inc. June 3, 2014 16 Release 15

 Anisotropic Thermal Conductivity

• Anisotropic thermal conductivity is only available for solid materials.

• By default, the thermal conductivity is considered to be isotropic.

• For anisotropic materials, the thermal conductivity is a matrix.

T
qi  kij
x j
• The thermal conductivity matrix can be defined using one of five
different methods:
• Orthotropic
• Cylindrical orthotropic
• General anisotropic
• Biaxial (shell conduction only)
• Anisotropic thermal conductivity (UDF)
© 2013 ANSYS, Inc. June 3, 2014 17 Release 15
 Anisotropic Thermal Conductivity for Solid Zones

Biaxial
• Defining parameters may depend on (shell conduction only)
temperature.
• UDF or constant/polynomial definition is also
possible.
Orthotropic Cylindrical Orthotropic

Anisotropic

© 2013 ANSYS, Inc. June 3, 2014 18 Release 15

 Agenda

• Introduction

• Energy equation in solids

• Equation solved in FLUENT
• Shell conduction model
• Non-conformal coupled wall
• Anisotropic conductivity
• Moving solids

• Solver parameters

© 2013 ANSYS, Inc. June 3, 2014 19 Release 15

 Conduction in Moving Solids

• Equation solved in FLUENT (for moving solids) :

 h 
 V    h     k T   S h
t
Unsteady Solid Motion Conduction Enthalpy
(Fourier’s Law) Source

• The convective term comes from an Eulerian description of solid

motion.

• If the mesh moves with the solid like for sliding mesh or rigid body
deforming mesh (Lagrangian representation), then the solid motion
term vanishes

© 2013 ANSYS, Inc. June 3, 2014 20 Release 15

 Conduction in Moving Solids

• The velocity field is taken

from the Solid panel
(rotation and translation)
• Note that those velocity
fields satisfy the continuity
equation.

• Convection in conducting
solids is justified for:
• Solid translation of an
extruded geometry (slab,
plate or sheet…)
• Solid rotation of a geometry
of revolution

© 2013 ANSYS, Inc. June 3, 2014 21 Release 15

 Conduction in Moving Solids

• Example of convection in conducting solids

• Metal or glass sheet in translation in a furnace.
A solid meshed sheet is moving.

Inlet: Prescribed temperature

Outlet: Adiabatic (temperature

gradient is 0.)

• Brake disc with source data

The “red wall” corresponds to an uncoupled pair of a wall and its shadow
© 2013 ANSYS, Inc. June 3, 2014 22 Release 15
 Conduction in Moving Solids

• Moving reference frame (MRF) is not appropriate for the entire solid
zone in the following situation:
• Brake disc with holes
Adiabatic

500 K

300 K

© 2013 ANSYS, Inc. June 3, 2014 23 Release 15

 Conduction in Moving Solids

• Can we treat these problems using a

steady approach?
• Just like for the fluid problem the
multiple reference frame approach
may be a useful approximation.

• Brake disc with holes

• Solid region decomposition
• Solid zone in the MRF (body of
revolution)
• Solid zone in the SRF (part with
holes). This part may actually be Solid Region
moving. The effect of rotation on Decomposition
heat transfer will be provided by the
moving material surrounding this
zone.

© 2013 ANSYS, Inc. June 3, 2014 24 Release 15

 Conduction in Moving Solids

• Can we treat this problem using a steady approach?

• Turbomachinery blade

• Solid zone: Stationary

• Wall / Shadow: Thermally coupled
• Wall on solid side: Stationary wall (absolute)
• Wall/Shadow on fluid side: Moving wall (relative to adjacent cell zone)
© 2013 ANSYS, Inc. June 3, 2014 25 Release 15
 Conduction in Moving Solids
• Unsteady state?
• Moving reference frame
can also be used in
unsteady problems with
the same limitations as in
steady state.

• Sliding mesh or rigid body

deforming mesh is a
rigorous way of treating the
unsteady problem.

• Sliding interface should be

located between two fluid
zones

© 2013 ANSYS, Inc. June 3, 2014 26 Release 15

 Conduction in Moving Solids

• Moving reference frame (MRF) approach is only valid for special

cases.
• Rigid-body translation of an extrusion (slab, plate, sheet, …)
• Rigid-body rotation of a solid of revolution

• Multiple reference frame

• Moving solid can be treated as stationary if the surrounding fluid or solid is
in the same frame of reference

• Sliding mesh is often the most accurate approach

© 2013 ANSYS, Inc. June 3, 2014 27 Release 15

 Agenda

• Introduction

• Energy equation in solids

• Equation solved in FLUENT
• Shell conduction model
• Non-conformal coupled wall
• Anisotropic conductivity
• Moving solids

• Solver parameters

© 2013 ANSYS, Inc. June 3, 2014 28 Release 15

 Solver Parameters

• Convergence difficulties

• Solver parameters affecting solution behavior

• Single-precision/double-precision solver
• Explicit relaxation of the energy equation
• Importance of secondary gradients
• MultiGrid methods

© 2013 ANSYS, Inc. June 3, 2014 29 Release 15

 Convergence Difficulties

• Convergence difficulties can be recognized by the following

symptoms.
• Overall imbalance in heat flux at boundaries.
• Slow convergence rate (several thousand iterations)
• Residuals that diverge
• Local (cell) temperatures reaching nonphysical values

• Skewed cells and improperly-posed boundary conditions can also

cause convergence problems.

• These problems can be either mitigated or avoided completely

through simple modifications to the solution setup.

© 2013 ANSYS, Inc. June 3, 2014 30 Release 15

 Double-Precision Solver

• The double-precision solver is designed to minimize truncation error

and thus improve the overall heat balance.

fluent 2ddp or fluent 3ddp

• As a general rule, the double precision solver should be enabled under

the following conditions:
• Cases with large heat fluxes (order of MW)
• Large, possibly solution-dependent heat sources in the energy equation.
• Widely varying solid properties (functions of temperature) such as nonlinear
solids
• Cases where there are large differences in thermal conductivity among
materials.
• Energy equation numerics become stiff.
• Flux matching conditions become more difficult to maintain at solid interfaces.

© 2013 ANSYS, Inc. June 3, 2014 31 Release 15

 MultiGrid Solver Parameters

• MultiGrid Methods
• The default MultiGrid scheme on
energy equation is Flexible
• Using either the W-Cycle or F-Cycle
scheme is preferred when
diffusion is the predominant effect
• W-Cycle is recommended for serial
processing
• V-Cycle or F-Cycle is recommended
for parallel processing

Modified settings Default settings

(14 iterations) (50 iterations)

© 2013 ANSYS, Inc. June 3, 2014 32 Release 15

 Explicit Under-Relaxation

• Scheme command to activate explicit under-relaxation of temperature (enter as

you would any TUI command).
(rpsetvar ‘temperature/explicit-relax? #t)

• Advantages
• Improved convergence for poor quality meshes
• Improved convergence when material properties are strongly dependent on
temperature
• Motivation
• Energy under-relaxation factor of 1 often recommended
• Temperature under-relaxation may be preferred
• Settings:
• Once the Scheme command is activated, the energy under-relaxation is regarded as a
temperature under-relaxation
• Temperature URF typically 0.2–0.5 and energy URF = 1
• The energy URF is for the enthalpy

© 2013 ANSYS, Inc. June 3, 2014 33 Release 15

 Secondary Gradients

• What is a secondary gradient?

• Secondary gradients are used primarily as a corrective measure (the flux
vector normal to the face does not pass through the cell center)

Tc
q  k T  n
h
T T
 k w c  f T 
Tw h

© 2013 ANSYS, Inc. June 3, 2014 34 Release 15

 Secondary Gradients

• Influence of secondary gradients

• The secondary gradient effect increases with mesh skewness. With poor
mesh (skewness greater than 0.9), disabling secondary gradient treatment
will aid in convergence.

Tc
Tc
r h
r h

Tw Tw
Perfect Hexahedral Mesh Skewed Tetrahedral Mesh
Secondary Gradient = 0 Secondary Gradient
depends on skewness

© 2013 ANSYS, Inc. June 3, 2014 35 Release 15

 Secondary Gradients

• Secondary gradient influence

• With poor mesh (skewness greater than 0.9), disabling secondary gradient
treatment will aid in convergence.
• 3 possibilities :
• Disable secondary gradients in all zones
(rpsetvar 'temperature/secondary-gradient? #f)
• Disable secondary gradients only on wall zones
solve/set/expert/
use-alternate-formulation-for-wall-temperature? yes
• Disable secondary gradients only on shell conduction zones
(rpsetvar 'temperature/shell-secondary-gradient? #f)

© 2013 ANSYS, Inc. June 3, 2014 36 Release 15

 Secondary Gradients

• Is accuracy compromised by neglecting secondary gradients?

Default Without Secondary Gradients

© 2013 ANSYS, Inc. June 3, 2014 37 Release 15

 Appendix
15.0 Release

© 2013 ANSYS, Inc. June 3, 2014 38 Release 15

 Thermal Conductivity of Selected
Materials
Thermal
Conductivity
Material
at 20 °C
(W/m·K)

Silver 430
Copper 387
Aluminum 202
Steel 16
Glass 1
Water 0.6
Wood 0.17
Glass wool 0.04
Polystyrene 0.03
Air 0.024

© 2013 ANSYS, Inc. June 3, 2014 39 Release 15

 Conductive Flux Calculation

• Diffusive flux on an interior face

• φ = Temperature for conduction
• k = Thermal conductivity ds
D f  k f   A es 
ds
1  0 A  A  AA 
 kf  k f    A    e s 
Cell C1 ds A  e s  A  es 

Primary flux Secondary

ds Cell or face centroid gradient
A Node
dr
Face f
Cell C0

The flux at a boundary face has a similar expression,

1 is replaced by f and ds replaced by dr

© 2013 ANSYS, Inc. June 3, 2014 40 Release 15