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SECTION 1

HEAT TRANSFER ANALYSIS

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THERMAL ANALYSIS WITH MSC.MARC

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OVERVIEW

Why a Structural Analyst may have to perform Thermal Analysis

Modes of Heat Transfer Available in MSC.Marcand MSC.Patran support Conduction Convection Radiation Transient Analysis

versus Steady State Analysis

Linear versus Nonlinear Minimum Allowable Time

Increment Thermal Analysis How to calculate it Physical Interpretation What happens if you violate the

formula Sequentially Coupled Problems

versus

Fully Coupled Problems

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HEAT TRANSFERMotivation

When the solution for the temperature field in a solid (or fluid) is desired, and the temperature is not influenced by the other unknown fields, a heat transfer analysis is appropriate.

q= -k [dT/dx]

T1 T2

T1>T2

q

Conduction

Modes of Heat Transfer

Advection Radiation

T1 T2

T1>T2

q1

q2

T1

T1>T2

q

Convection

T2

Moving Fluid Removes

Heat From Solid

T1>T2

q

T2T1

Heat Moves IN Moving

Fluid

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Motivation

When the solution for the temperature field in a solid (or fluid) is desired,and is not influenced by the other unknown fields, heat transfer analysis is appropriate.

HEAT TRANSFER MODES

Conduction

Convection

Radiation

Contact Heat Transfer:

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HEAT TRANSFER

Steady State Heat Transfer Definition

Heat transfer analysis is used to determine the temperature field of a given structure. It can also be used as a starting point in a thermal expansion analysis. This is a structural analysis where the applied load is temperature.

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HEAT TRANSFER

Objective To determine the temperature field of a structure. Secondary results of interest include heat flux.

Assumptions The typical heat transfer problem is a conduction problem where

convections and radiation are boundary conditions.

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HEAT TRANSFER Some terms to keep in mind.

Flux: the rate of energy flow across a surface. Denoted by a symbol q’’ = q flux. (單位時間單位面積通過的熱量或能量 )(通量）

Energy: amount of usable power. In this case we are talking about heat as the usable power. So energy would be the flux times the area of the surface times time. (熱傳的能量指熱量 )

Heat: energy in transit due to a temperature difference. This is also called heat transfer. (熱量 )

Modes of Heat Transfer: Conduction, Convection, Radiation

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Conduction: Internal energy transfer due to thermal

gradients interior to a body or across perfect contact of

two bodies.

x

Tkq cond

"

HEAT TRANSFER

q in

x

T

T1

T2

T(x)q out

K is a given property of the material.q = q” x AAnalysis is converged when the heat flow is balanced qin – qout = 0 (steadyState)

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Conduction (cont’d)

where, q is the heat-transfer rate (heat per unit time)

k is the thermal conductivity (power per distance temperature)

A is the area normal to the heat flow

T is the temperature

x is the direction of heat flow

x

Tkq cond

''

x

TkAqcond

(conduction flux)

HEAT TRANSFER

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TS

TB or

q

)(" BSfconv TThq

hf is either a) given from the geometry of the body

and the state of the fluid or b) the result of a thermal

CFD analysis. hf is called the convection coefficient.

Used mainly as a boundary condition so the user would input hf and TB.

T

HEAT TRANSFER Convection: Energy transfer between a solid body and a

surrounding fluid (gas or liquid).

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Convection

where, q” is the heat-transfer rate per unit area hf is the convective film coefficient

TS is the surface temperature

TB is the bulk temperature A is the area that the heat is transferred across

)(" BSfconv TThq )( BSfconv TThq

HEAT TRANSFER

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)(" 44jiijrad TTFq

i

jQ

Electromagnetic

wave

σ = 5.67E-8 W / m² •

= 0.1714E-8 Btu / h • ft² •

Fij is solved for each element pair. This is done before the solution is run and may be either automated or a user input.

4K4R

HEAT TRANSFER Radiation: Energy transfer by electromagnetic waves.

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Radiation

where, q is the heat flow rate from surface i to j. σ is the Stephan-Boltzman constant ε is the emissivity (放射率 ) Ai is the area of surface i

Fij is the form factor between surfaces i & j

Ti, Tj are the absolute temperatures of the surfaces

)( 44jiijirad TTFq

HEAT TRANSFER

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HEAT TRANSFER Equation of State: 1D Example

We will first derive the basic differential equation for the one dimensional conduction only problem. This will allow us an insight to the physical problem that is needed before that FEA formulation is understood.

Begin with the conservation of energy and apply it to a control volume.

E in + E generated = E out + U (change in stored energy)

E inE outQ

Insulated boundaries

Insulated boundaries

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)()()( timeareafluxEnergy

HEAT TRANSFER

Ein is the energy entering the volume in units of J (joules).

qx” is the conduction heat flux into the volume

Ein = q”x • A • dt = qx • dt (flux | power into system)

Eout = q”x+dx • A • dt = qx+dx • dt (flux | power out of system)

Egenerated = Q • A • dx • dt (power per unit volume)

Q is the internal heat source1 J = 1 N • MA is the cross sectional area of the control volume

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HEAT TRANSFER Put all the parts together:

Use Fourier’s Law:

outgeneratedin E

dxx

gyStoredEnerEE

x dtqUdxdtQdtq

""

dxxxxdxx x

TKq

"

qx+dxqx

area A

Q

dx

EQ 1

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HEAT TRANSFER And apply Taylor’s Expansion Theorem

This results in

Stored energy term

dx

dx

dTK

dx

d

dx

dTK"q xxxxdxx

...2

dx

x

fdx

x

fff

2

2

2

xdxx

U = (specific heat) x (mass) x (change in temperature)

dT)dx(c

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HEAT TRANSFER Substituting the preceding into the energy balance

equation (eq 1), we have

Simplifying and dividing by Adxdt we are left with:

t

TcQ

x

TK

x xx

0 for steady state

out

generated

inE

xxxx

storedEEE

x dtdxdx

dTK

dx

d

dx

dTKdxdTcdxdtQdt

dx

dTK

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HEAT TRANSFER For steady state, any differential with respect to time is

zero. Also assume that the conductivity of the material is constant.

The following boundary conditions are possible 1) T = Tb, Tb would be a known temperature DOF constraint

2) = constant. This is a prescribed heat flux.

3) on an insulated boundary. This is the “natural” boundary condition. It exists if nothing is applied.

0Qx

TK

2

2

xx

dx

dTKq xx

"x

0q"x

steady state conduction

Add/subtract energy

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Now convection can be added.

The convection replaces the insulated boundary from before. qh acts on an area therefore let P be the perimeter around area A.

HEAT TRANSFER

qx qx+dxQ

qh

dx

area A

Pdxdtqn)(convectioE hout

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HEAT TRANSFER

Add this term to EQ 1 and simplify as before:

Convection

h

E

dxx

storedEEE

x Pdxdt"qdt"qUdxdtQdt"qoutgeneratedin

Convection

xx )TT(hP

t

TcQ

x

TK

x

)TT(h"q BSfh

EQ 2

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HEAT TRANSFER Again, assume no time dependence and a constant

conduction coefficient:

Now one more boundary condition can be added to the previous three: 4) the heat flow at the solid/surface interface is

balanced.

)orEE(Convection

E

E

2

2

xx

outin

gen

in

)TT(hP

Qx

TK

)TT(hdx

dTKq xx

"

x

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Transient Heat Transfer Definition

Heat transfer analysis where the temperature of the structure is now a function of time.

Don’t forget to add density, , and specific heat, , to the material definition.

t

Tc)TT(

hPQ

x

TK

2

2

xx

c

HEAT TRANSFER

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Special Topics Non-Linear Heat Transfer

Examine the heat equation (no radiation)

if the material properties are functions of temperature, the analysis is non-linear (in temp).

t

Tc)TT(

hPQ

x

TK

2

2

xx

h,c,,K

HEAT TRANSFER

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Special Topics Non-Linear Heat Transfer - Radiation

Add the radiation flux term to EQ 2 (acts over the same surface area as convection)

radiation is naturally non-linear as it depends on the fourth power of temperature. Emissivity may also be a function of temp.

Radiation

rad

Convection

h

E

dxx

storedEEE

x Pdxdt"qPdxdt"qdt"qUdxdtQdt"qoutgeneratedin

gyStoredEnerRadiation

44

Convection

Generation

Conduction

2

2

xx t

Tc)TT(

FP)TT(

hPQ

x

TK

HEAT TRANSFER

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B

q”x

Ta

Tb

q”gap

q”contact

HEAT TRANSFER

Special Topics Thermal Contact Resistance

Where two structures meet, the temperature drop across the interface may be appreciable. The temperature difference results from the fact that on a microscopic (or not so micro) scale, the surfaces are not in perfect contact i.e. gaps exist.

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Special Topics Thermal Contact Resistance

The resistance due to thermal contact is defined as: (R=1/hA)

most values of R” are derived experimentally which is more reliable than the current theories for predicting R”.

x

BAc,t "q

TT"R

HEAT TRANSFER

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HEAT TRANSFER

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HEAT TRANSFER

Summary 1D Steady State Definitions and Examples

Conduction

Convection

Radiation

Derivation of state equation

Transient Heat Transfer

Special Topics

Non-linear Heat Transfer

Thermal Contact Resistance

Units

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HEAT TRANSFER EXAMPLE

Computed TemperaturesTemperature given at bottom left and right end surfaces

Convection given about connector

Example: Steady State Analysis of Radiator (Workshop 4)

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Thermal equilibrium between heat sources, energy flow density and temperature rate is expressed by the Energy Conservation Law, which may be written:

Energy flow density is given by a diffusion and convection part:

where is is the conductivity matrix.

Assume that the continuum is incompressible and that there is no spatial variation of and Cp; then the conservation law becomes:

HEAT TRANSFER MATHEMATICS

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HEAT TRANSFER LOADS & BOUNDARY CONDITIONS

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contact

HEAT TRANSFER LOADS & BOUNDARY CONDITIONS (CONT.)

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contact6) Contact conduction:

h : Transfer coeff.

= Temp.Body 2 = Temp.Body 1

HEAT TRANSFER LOADS & BOUNDARY CONDITIONS (CONT.)

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Only in transient analysis: MSC.Marc uses a backward difference scheme to approximate the time derivative as:

resulting in the finite difference scheme:

where

C: heat capacity matrix

K: conductivity and convection matrix

F: contribution from convective boundary condition

: vector of nodal fluxes

HEAT TRANSFER INITIAL CONDITIONS

T =.

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THERMAL RADIATION

New Radiation LBC

Radiation Viewfactor calculation Uses Monte Carlo Simulation 2D Axisymmetric 3D

New Convective Velocity LBC

Thermal or Coupled analysis

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RADIATION OVERVIEW

Basic equation: q = F A (Ti4 – Tj

4) T is in absolute units

Kelvin or Rankine

is the Stefan Boltzmann Constant 5.6696E-8 Watt/(m2-K4) 1.7140E-9 Btu/(hr-ft2-R4)

F (script-F) is the gray body configuration factor F is a function of

Fij, the geometric view-factor from surface-i to surface-j

ei, the emissivity of surface-i

ej, the emissivity of surface-j

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1-i

iAi

1-j

jAj

1AiFij

RADIOSITY CONCEPT

Radiosity node added for each surface node ( < 1.0) FA is separated into two terms

A “space” resistance which is a function of only Fij

Each “surface” resistance a function of either and only

This technique facilitates Change in surface properties ( without new view factor run Independently variable emissivity values

= 1.0 will eliminate radiosity nodes Facilitates debugging of surface-to-surface connections

Eb1 Eb2J1 J2

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SIMPLE RADIOSITY NETWORK

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HEAT TRANSFER IN MSC.PATRAN MARC PREFERENCE

All three modes of heat transfer may be present in an MSC.Marc analysis. There are two basic types of analyses: Transient analysis: to obtain

the history of the response over time with heat capacity and latent heat effects taken into account

Steady state analysis: when only the long term solution under a given set of loads and boundary conditions is sought (No heat accumulation).

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Time

Temperature

THERMAL NONLINEAR ANALYSES

Either type of thermal analysis can be nonlinear.

Sources of nonlinearity include: Temperature dependence of

material properties. Nonlinear surface conditions:

e.g. radiation, temperature dependent film (surface convection) coefficients.

Loads which vary nonlinearly with temperature. These loads are described using Fields in MSC.Patran.

Latent heat (phase change) effects

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HEAT TRANSFER SHELL ELEMENT

Prior to version 2001, shell elements could have a linear or a quadratic temperature distribution in thickness direction; per node, the number of degrees of freedom is:

2: top and bottom surface temperature (linear)3: top, bottom and mid surface temperature (quadratic)

Especially for certain composite structures, a linear or quadratic distribution might be insufficient to accurately describe the actual temperature profile

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HEAT TRANSFER SHELL ELEMENT (CONT.) New in version 2001: take a linear or quadratic temperature profile

per layer ; per node, the number of degrees of freedom is:

M+1 (M = # of composite layers; linear distribution):

1 = top surface temperature2 = temperature at layer 1-2 interface3 = temperature at layer 2-3 interface…M+1 = bottom surface temperature 2*M+1 (M = # of composite layers; quadratic distribution):

1 = top surface temperature2 = temperature at layer 1-2 interface3 = mid surface temperature of layer 14 = temperature at layer 2-3 interface…2*M+1 = mid surface temperature layer M

The new temperature distribution can also be user for non-composite elements

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FULLY COUPLED PROBLEMS Thermal field affects the

mechanical field as above Thermal loads induce deformation.

Mechanical field affects thermal field Mechanically generate heat-due to

plastic work or friction Deformation changes modes of

conduction, radiation, etc.

Fully coupled problems are supported in MSC.Marc 2001 but not by MSC.Patran 2001

Fully coupled problems are supported in MSC.Patran 2002