1. HEAT TRANSFER ANALYSIS
SECTION 1
HEAT TRANSFER ANALYSIS

2. THERMAL ANALYSIS WITH MSC.MARC

3. OVERVIEW Why a Structural Analyst may have to perform Thermal Analysis
Modes of Heat Transfer Available in MSC.Marc and 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

4. HEAT TRANSFER Motivation Modes of Heat Transfer
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. T1
T1>T2 q
Convection T2
Moving Fluid Removes Heat From Solid T1 T2
T1>T2 q
Conduction
T1>T2 q T2 T1
Heat Moves IN Moving Fluid T1 T2
T1>T2 q1 q2
q= -k [dT/dx]
Advection
Radiation
Modes of Heat Transfer

5. HEAT TRANSFER MODES Contact Heat Transfer: Conduction Convection
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.
Conduction
Convection
Radiation
Contact Heat Transfer:

6. 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.

7. HEAT TRANSFER Objective Assumptions
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.

8. 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

9. HEAT TRANSFER Conduction: Internal energy transfer due to thermal
gradients interior to a body or across perfect contact of
two bodies.
q in x T T1 T2
T(x)
q out
K is a given property of the material.
q = q” x A
Analysis is converged when the heat flow is balanced qin – qout = 0 (steady
State)

10. HEAT TRANSFER 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
(conduction flux)

11. HEAT TRANSFER Convection: Energy transfer between a solid body and a
surrounding fluid (gas or liquid).
TB or q
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. TS

12. HEAT TRANSFER 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

13. HEAT TRANSFER Radiation: Energy transfer by electromagnetic waves. i jQ
Electromagnetic
wave
σ = 5.67E-8 W / m² •
= E-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.

14. HEAT TRANSFER 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

15. 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 in
E out Q
Insulated boundaries

16. 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 source
1 J = 1 N • M
A is the cross sectional area of the control volume

17. HEAT TRANSFER Put all the parts together: Use Fourier’s Law: EQ 1 dx
qx+dx qx
area A Q dx

18. HEAT TRANSFER And apply Taylor’s Expansion Theorem This results in
Stored energy term
U = (specific heat) x (mass) x (change in temperature)

19. HEAT TRANSFER
Substituting the preceding into the energy balance equation (eq 1), we have
Simplifying and dividing by Adxdt we are left with:
0 for steady state

20. 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.
steady state conduction
Add/subtract energy

21. HEAT TRANSFER 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. qx
qx+dx Q qh dx
area A

22. HEAT TRANSFER
Add this term to EQ 1 and simplify as before:
EQ 2

23. 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.

24. HEAT TRANSFER 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.

25. HEAT TRANSFER 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).

26. HEAT TRANSFER 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.

27. HEAT TRANSFER Special Topics Ta B Tb Thermal Contact Resistance q”x
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. B
q”x Ta Tb
q”gap
q”contact

28. HEAT TRANSFER 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”.

29. HEAT TRANSFER
1Btu= KJ,1Btu/h=2.9307e-4KW

30. HEAT TRANSFER

31. 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

32. Convection given about connector Computed Temperatures
HEAT TRANSFER EXAMPLE
Example: Steady State Analysis of Radiator (Workshop 4)
Convection given about connector
Temperature given at bottom left and right end surfaces
Computed Temperatures

33. HEAT TRANSFER MATHEMATICS
Energy flow density is given by a diffusion and convection part:
Thermal equilibrium between heat sources, energy flow density and temperature rate is expressed by the Energy Conservation Law, which may be written:
where is L is the conductivity matrix.
Assume that the continuum is incompressible and that there is no spatial variation of r and Cp; then the conservation law becomes:

34. HEAT TRANSFER LOADS & BOUNDARY CONDITIONS

35. HEAT TRANSFER LOADS & BOUNDARY CONDITIONS (CONT.)
contact

36. HEAT TRANSFER LOADS & BOUNDARY CONDITIONS (CONT.)
6) Contact conduction:
contact
h : Transfer coeff.
= Temp.Body = Temp.Body 1

37. HEAT TRANSFER INITIAL CONDITIONS
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
T = .

38. THERMAL RADIATION New Radiation LBC Radiation Viewfactor calculation
Uses Monte Carlo Simulation 2D
Axisymmetric 3D
New Convective Velocity LBC
Thermal or Coupled analysis
Two new LBCs have been added: Radiation and Convective Velocity
If radiation conditions are present, you can run MonteCarlo simulation to create a view factor file for use in thermal radiation analysis. 2D elements (shell faces - top & bottom, and element edges) and 3D elements (solid faces) can have radiation boundary conditions for use in 2D, 3D or axisymmetric analyses.

39. RADIATION OVERVIEW Basic equation: q = s F A (Ti4 – Tj4)
T is in absolute units
Kelvin or Rankine
s 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

40. RADIOSITY CONCEPT
Radiosity node added for each surface node (e < 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 ei and ej only
This technique facilitates
Change in surface properties (e) without new view factor run
Independently variable emissivity values
e = 1.0 will eliminate radiosity nodes
Facilitates debugging of surface-to-surface connections
1-ei
eiAi
1-ej
ejAj 1
AiFij
Eb1 J1 J2
Eb2
輻射網路法

41. SIMPLE RADIOSITY NETWORK

42. 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).

43. THERMAL NONLINEAR ANALYSES
Time
Temperature
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

44. 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

45. 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 temperature
2 = temperature at layer 1-2 interface
3 = temperature at layer 2-3 interface …
M+1 = bottom surface temperature
2*M+1 (M = # of composite layers; quadratic distribution):
3 = mid surface temperature of layer 1
4 = 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

46. 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