Heat Transfer Analysis...Conduction •Equation (2) states that the temperature distribution along a...

Heat Transfer and Multiphysics

Analysis

2011 Alex Grishin MAE 323 Lecture 8: Heat Transfer and Multiphysics

1

Heat Transfer

Analysis


Analysis


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•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


Analysis


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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)


Analysis


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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)


Analysis


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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


Analysis


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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)


Analysis


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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)where

1 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)


Analysis


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•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:

0xx xbx

σ∂ + =∂

•If no body load is present, then:

0xxx

σ∂ =∂

•Then we use the isotropic constitutive law (Chapter 4 again) for a unilateral stress:

x

uE

xσ∂ =

∂

(5)

(6)

(7)


Analysis


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•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)


Analysis


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•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)


Analysis


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•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 QhA

T 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


Analysis


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•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


Analysis


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•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


Analysis


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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.


Analysis


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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?


Analysis


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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


Analysis


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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.


Analysis


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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.


Analysis


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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


Analysis


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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


Analysis


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Static Structural

Analysis with

Thermal Loads


Analysis


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•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


Analysis


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•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


Analysis


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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


Analysis


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Coupled-Field

(Multiphysics)

Problems


Analysis


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•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


Analysis


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•However, solving coupled physical fields can be significantly more complicated (and general).

•ANSYS has the following coupled field capability

HeatTransfer

SolidMechanics

MagnetismFluid

Mechanics

Electricity


Analysis


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• 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


Analysis


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• 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


Analysis


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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)


Analysis


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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


Analysis


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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.

Heat Transfer Analysis...Conduction •Equation (2) states that the temperature distribution along a...

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