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Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics1
Heat Transfer
Analysis
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics2
•In engineering applications, heat is generally transferred from one
location to another and between bodies. This transfer is driven by
differences in temperature (a temperature gradient) and goes from
locations of high temperature to those with low temperature.
•These temperature differences, in turn, cause mechanical stresses
and strains in bodies due to their coefficient of thermal expansion,
α (sometimes abbreviated CTE in engineering literature)
•The amount of heat transfer is directly proportional to the size of
the temperature gradient and the thermal resistance of the
material(s) involved
•In engineering applications, there are three basic mechanisms:
1. Conduction
2. Convection
3. Radiation
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics3
Conduction
•For a thermally orthotropic material*, the heat transfer per unit
volume per unit time can be stated (in SI units of Joules per cu.
meter per second, or simply Watts per cu. meter):
*see http://en.wikipedia.org/wiki/Orthotropic_material
x y z p
T T T Tk k k C
x x y y z z tρ λ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + = − ∂ ∂ ∂ ∂ ∂ ∂ ∂
where:0
3
0
0
thermal conduction in direction i (Watts/m/ )
physical mass (kg)
volumetric heat generation (W/m )
specific heat capacity (J/kg/ )
temperature ( )
i
p
k C
C C
T C
ρλ
===
=
=
(1)
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics4
Conduction
•All the terms on the LHS of (1) represent conduction of heat
through material (usually solid bodies)
•The physical mechanism of this conduction is usually
molecular (or electronic) vibration.
•For steady-state problems with no heat generation in one-
dimension, we have:
2
20x
x
Tk
xT
k qx
∂ =∂∂ = −∂
where q is an applied heat flux (heat flow per
unit area. SI units are W/m2)
(2)
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics5
Conduction
•Equation (2) states that the temperature distribution along a
length of material conducting heat along that length is linear
and proportional to the heat flow, q
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics6
Convection
qi
qo
T∞∞∞∞Ts
•Convection is a mechanism of heat transfer that occurs due to the
observable (and measurable) motion of fluids
•As fluid moves, it carries heat with it. In engineering applications, this
phenomenon can be characterized by:
( )sq h T T∞= − where2
0
0
heat flow per unit area (W/m )
surface temperature ( )
fluid temperature far from surface ( )
s
q
T C
T C∞
==
=
(3)
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics7
Radiation
•Thermal radiation is electromagnetic radiation generated by the
thermal motion of charged particles in matter
•Two different bodies at different temperatures separated by some
neutral medium (space or air) will exchange heat through this
mechanism according to:
( )4 41 2 1 2 1 2q F T Tε σ− −= − (4)
where1 2
1 2
2 0 4
emissivity between body 1 and 2 (dimensionless)
view factor (dimensionless)
=Stefan Boltzmann constant (W/m / )
F
K
ε
σ
−
−
==
•Equation (4) is generally nonlinear because and special solver utilities
are used to solve these problems (beyond the scope of this course)
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics8
•In this course, we will only deal with steady-state thermal analyses
with heat sources, conduction, and convection. Element formulations
for such phenomena are straightforward and have direct analogies
with static structural problems. To see this, let’s start with the case of
bar/truss and a conduction in 1 dimension
•From Chapter 4, we have static equilibrium in one direction:
0xxxb
x
σ∂ + =∂
•If no body load is present, then:
0xx
x
σ∂ =∂
•Then we use the isotropic constitutive law (Chapter 4 again)
for a unilateral stress:
x
uE
xσ∂ =
∂
(5)
(6)
(7)
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics9
•Plugging (7) into (6) gets the equation in terms of the
primary variable (displacement)
2
20
uE
x
∂ =∂
(8)Units: Force/length2
•We can do the same thing with the conductivity equation (1).
Assuming steady state conduction with no volumetric heat
generation in x-direction only, equation (1) becomes:
2
20x
Tk
x
∂ =∂
Units: Energy/time*Temperature/length3 (9)
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics10
•We saw in chapter 2 that we can integrate equation (8) twice and
apply boundary conditions to solve it.
•This leads to the canonical truss element:
1 1
2 2
1 1
1 1
u FEA
u FL
− = −
•Equation (9) has the same form, so we should expect to be able to
create an analogous 1D (thermal link) element
•Integrating (9) once leads to Fourier’s Law of Conduction in one
dimension (the sign comes from the necessary direction of heat flow
from hot to cold over an increasing distance):
dTk q
dx= −
(10)
(11)
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics11
•Solving (11) for T in terms of q yields an equation very
similar to (10). This is a thermal link element:
1 1
2 2
1 1
1 1
T QkA
T QL
− = −
(12)
•Similarly, a convection link element can be constructed from
(3) as:
1
2
1 1
1 1sT Q
hAT Q∞
− = −
(13)
•The elements in (13) connect nodes on the surface of a
body at Ts to a common ground node at T∞. Here the area A
is area over which the convection elements acts
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics12
•Equations (12) and (13) demonstrate that the thermal link elements in
a steady-state thermal analysis are analogous to structural spring
elements. Thus the heat flow, Q is the analog of the structural force F
and T is the analog of the structural displacement. These analogies
lead directly to the notion of thermal resistance, R:
⋅ =⋅ =
K x F
R T Q
Static Structural problem
Steady-State thermal problem
Structural
stiffnessDisplacement Force
Thermal
resistance Temperature Heat flow
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics13
•Without going through the details, we will simply mention that the
equations (1) and (3) can be combined to yield the governing
equations for a system experiencing both conduction and convection.
This combined system may be expressed as:
( )h+ ⋅ = +R H T Q Q
where:
T
V
T
S
Th
S
dV
h dS
hTdS
= ⋅ ⋅
=
=
∫
∫
∫
R B κ B
H N N
Q N
(14)
conductivity matrix
convection coefficient
vector of shape functions
0 0
0 0
0 0
h
x
y
z
===
∂ ∂
∂ = ∂
∂ ∂
κ
N
N
NB
N
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics14
Performing a Steady-State Thermal Analysis in ANSYS
Workbench
•Shell and line body assumptions:
Shells: no through-thickness temperature gradients.
Line bodies: no through thickness variation. Assumes a
constant temperature across the cross-section.
Temperature variation will still be considered along the
line body
Some Assumptions:
•As with structural analyses, contact regions are automatically
created to enable heat transfer between parts of assemblies.
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics15
Performing a Steady-State Thermal Analysis in ANSYS
Workbench
•If parts are initially in contact heat transfer can occur between them.
•If parts are initially out of contact no heat transfer takes place (see
pinball explanation below).
•Summary:
•The pinball region determines when contact occurs and is automatically
defined and set to a relatively small value to accommodate small gaps in
the model
Initially Touching Inside Pinball Region Outside Pinball RegionBonded Yes Yes NoNo Separation Yes Yes NoRough Yes No NoFrictionless Yes No NoFrictional Yes No No
Contact TypeHeat Transfer Between Parts in Contact Region?
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics16
Performing a Steady-State Thermal Analysis in ANSYS
Workbench
By default, perfect thermal contact
conductance between parts is assumed,
meaning no temperature drop occurs at the
interface.
Numerous conditions can contribute to less
than perfect contact conductance:
surface flatness
surface finish
oxides
entrapped fluids
contact pressure
surface temperature
use of conductive grease
. . . .
Continued . . .
∆T
T
x
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics17
Performing a Steady-State Thermal Analysis in ANSYS
Workbench
The amount of heat flow across a contact interface is defined by the
contact heat flux q:
where Tcontact is the temperature of a contact “node” and Ttarget is the
temperature of the corresponding target “node”.
By default, TCC is set to a relatively ‘high’ value based on the largest
material conductivity defined in the model KXX and the diagonal of the
overall geometry bounding box ASMDIAG.
This essentially provides ‘perfect’ conductance between parts.
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics18
Performing a Steady-State Thermal Analysis in ANSYS
Workbench
• Heat Flow:
– A heat flow rate can be applied to a vertex, edge, or surface. The load is distributed for
multiple selections.
– Heat flow has units of energy/time.
• Perfectly insulated (heat flow = 0):
– Available to remove surfaces from previously applied boundary conditions.
• Heat Flux:
– Heat flux can be applied to surfaces only (edges in 2D).
– Heat flux has units of energy/time/area.
• Internal Heat Generation:
– An internal heat generation rate can be applied to bodies only.
– Heat generation has units of energy/time/volume.
A positive value for heat load will add energy to the system.
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics19
Performing a Steady-State Thermal Analysis in ANSYS
Workbench
Temperature, Convection and Radiation:• At least one type of thermal boundary condition must be present to
prevent the thermal equivalent of rigid body motion.
• Given Temperature or Convection load should not be applied on surfaces that already have another heat load or thermal boundary condition applied to it.
• Perfect insulation will override thermal boundary conditions.
• Given Temperature:– Imposes a temperature on vertices, edges, surfaces or bodies
– Temperature is the degree of freedom solved for
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics20
Performing a Steady-State Thermal Analysis in ANSYS
Workbench
• Convection:– Applied to surfaces only (edges in 2D analyses).
– Convection q is defined by a film coefficient h, the surface area A, and the difference in the surface temperature Tsurface & ambient temperature Tambient
– “h” and “Tambient” are user input values.
– The film coefficient h can be constant or temperature dependent
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics21
Static Structural
Analysis with
Thermal Loads
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics22
•The governing equations of static structural continua (such as
equation (2) of Chapter 5) always contain a body load term. Thermal
loads may be considered body loads. Body temperatures are
converted to structural body loads via the coefficient of thermal
expansion, α (often referred to in industry by the acronym CTE):
α CTE (units: Temperature-1)
Tα∆ Thermal strain
E Tα ∆ Thermal stress
•Thus, (16) would be implemented in equation (2) of Chapter
5 as:
(15)
(16)
T
V V S
dV E Tw wdSδ α= ∆ +∫ ∫ ∫σ ε F
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics23
•In an element, the discrete form of the thermal becomes:
e
e e
V
E TdVα ∆∫ N
•It is thus characterized by a load vector obtained by integrating every
element with a temperature other than the reference temperature.
This load vector is then added to the global applied load vector
•∆T is thus the difference between the temperature of the body and
the reference temperature at which the CTE was measured.
•It is easy to see that if two bodies with differing CTE’s (calculated at
the same reference temperature) are raised to the same temperature,
they will experience differing thermal-structural loads. If the two
bodies are connected, they may experience stresses due to this
“thermal mismatch”*
http://www.ami.ac.uk/courses/topics/0162_sctm/index.html
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics24
Performing a Single-Phase Structural analysis with thermal
loads in ANSYS Workbench
•Workbench has the capability of adding a constant thermal body
load to bodies (parts) in Mechanical interface. One can add different
uniform temperatures to different bodies. This is done in the “Static
Structural” branch in the tree outline by selecting “thermal
Condition”
•if a temperature distribution is to be applied, this can only be done
via an imported load object (either through the “External Data” tool
in the toolbox of the project page, or via a linked thermal analysis)
Note that a global reference temperature (for all defined
CTE’s) can be set in the Details view of the “Static Structural”
branch
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics25
Coupled-Field
(Multiphysics)
Problems
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics26
•A static structural analysis which incorporates thermal loads via a
temperature distribution obtained from a thermal analysis is one of the
earliest types of coupled-field analysis
•Most commercial codes offer the capability to perform such an analysis
in a sequential manner (sometimes referred to as a 2-phase analysis). The
primary assumption behind this approach is that the two fields are weakly
coupled in a single direction (from thermal-to-structural– that is to say
that thermal structural loads are obtained from temperature
distributions, instead of thermal heat flows being obtained from
displacements, stresses, or strains). This makes the thermal-structural
sequence linear
Phase 1: Thermal
Calculate temperature distribution
Phase 2: Structural
Calculate displacements, stresses, strains
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics27
•However, solving coupled physical fields can be significantly more
complicated (and general).
•ANSYS has the following coupled field capability
HeatTransfer
SolidMechanics
MagnetismFluid
Mechanics
Electricity
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics28
• Two basic types of multiphysics coupling
– direct
– sequential
• Each method has several common names
– Direct versus Sequential
– Matrix versus Load Vector
– Direct versus Indirect
– Strongly versus Weakly
– Tightly versus Loosely
– Fully versus Partly
] most common
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics29
• Direct Method is
– used to simultaneously calculate the DOFs from multiple fields
– only necessary when the individual field responses of the model are dependent
upon each other
• Directly coupled analyses are usually
– nonlinear since equilibrium must satisfied based on multiple criteria
– more costly than comparably sized single-field models, because more DOFs are
active per node
[K11] [K12][K21] [K22]
[X1] [X2]
[F1] [F2]=[ {] } { }
Direct Method:
• Subscript 1 represents one physics• Subscript 2 represents the other physics• Coupled effects are accounted for by the off-diagonal coefficient terms K12 and K21• Provides for coupled response in solution after one iteration
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics30
Sequential Method:
[K11] [ 0 ][ 0 ] [K22]
[X1] [X2]
[F1] [F2]=[ {] } { }
• Subscript 1 represents one physics• Subscript 2 represents the other physics• Coupled effects are accounted for by the load terms F1 and F2• At least two iterations, one for each physics, in sequence, are needed
to achieve a converged coupled response• Separate results files for each physics
– jobname.rst (structural)
– jobname.rth (thermal, electrostatics)
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics31
Performing a Two-Phase Coupled Thermal-Structural
Analysis in Workbench
•In this course, we will only ever deal with sequential weakly
coupled analyses. For thermal/structural analyses, this can be
achieved by:
• Inserting the “Steady-State Thermal” from the Workbench toolbox will set up a SS Thermal system in the project schematic.
• In Mechanical the “Analysis Settings” can be used to set solution options for the thermal analysis.
Step 1:
Solve the
Thermal
Analysis
Heat Transfer and Multiphysics
Analysis
2011 Alex GrishinMAE 323 Lecture 8: Heat Transfer and
Multiphysics32
Performing a Two-Phase Coupled Thermal-Structural
Analysis in Workbench
Step 2:
Solve the
structural
model
•link a structural analysis to the thermal model at the Solution level.
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