1. FEA Course Lecture V – Outline
UCSD - 10/30/03
Review of Last Lecture (IV) on Plate and Shell Elements.
Thermal Analysis
Summary of Concepts of Heat Transfer
Thermal Loads and Boundary Conditions
Nonlinear Effects
Thermal transients and Modeling Considerations
Example FE
Finite Elements Principles and Practices - Fall 03

2. Finite Elements Principles and Practices - Fall 03
Thermal Analysis - Introduction:
Thermal Analysis involves calculating:
Temperature distributions
Amount of Heat lost or gained
Thermal gradients
Thermal fluxes.
There are two types of thermal analysis:
Steady-state analysis
Transient thermal analysis
Finite Elements Principles and Practices - Fall 03

3. Finite Elements Principles and Practices - Fall 03
Two types of Thermal Analysis:
Steady-state Thermal Analysis. It involves determining the temperature distribution and other thermal quantities under steady-state loading conditions. A steady-state loading condition is a situation where heat storage effects varying over a period of time can be ignored.
Some examples of thermal loads are:
Convections
Radiation
Heat Flow Rates
Heat Fluxes (Heat Flow/unit area)
Heat Generation Rates (heat flow/unit volume)
Constant Temperature Boundaries
Steady State thermal analysis may be linear or nonlinear (due to material properties not geometry). Radiation is a nonlinear problem.
Transient Thermal Analysis. It involves determining the temperature distribution and other thermal quantities under conditions that vary over a period of time.
Finite Elements Principles and Practices - Fall 03

4. Theoretical Basis for Thermal Analysis
[KT] {T} = {Q} where [KT] = f (conductivity of material).
T = vector of nodal temperatures
Q = vector of thermal loads.
[KT] is nonlinear when radiation heat transfer is present. Note that convection and radiation BCs contribute terms to both [KT] and {Q}.
Heat is transferred to or from a body by convection and radiation.
Heat Flow across boundary due to radiation (in-out)
Prescribed rate of heat flow across boundary (in or out)
Heat generated internally (eg., Joule heating)
Prescribed temperature (BC) – insulated for example.
Heat Flow across boundary due to radiation (in-out)
Finite Elements Principles and Practices - Fall 03

5. Equation of Heat Flow (1D Systems)
fx = -k dT/dx [Fourier Heat Conduction Equation]. Heat flows from high
temperature region to low temperature region.
Q = -kA dT/dx Q – heat flow
fx = Q/A where fx = heat flux/unit area, k = thermal conductivity, A = area of cross-section, dT/dx = temperature gradient
In general, {fx, fy, fz} = -k{dT/dX, dT/dY, dT/dZ} T
For an elemental area of length dx, the balance of energy is given as:
-KA dT/dx +qAdx = rca dT/dt dx – [KA dT/dx + d/dx(KA dT/dx)dx]
d/dx(KA dT/dx) + Aq = rca dT/dt
rate in – rate out = rate of increase within
For generally anisotropic material
-[d/dx d/dy d/dz] {fx fy fz}T +qv = cr dT/dt
where c is specific heat, t is time, r – mass density and qv – rate of internal heat generation / unit volume.
Above equation can be re-written as:
Steady state if dT/dt = 0
dT/dx fx fy
Finite Elements Principles and Practices - Fall 03

6. Finite Elements Principles and Practices - Fall 03
Some Notes
If the body is plane and there is convection and or radiation heat transfer across its flat lateral surfaces, additional equations for flux terms are needed:
Convection BC
f = h(Tf – T) (Newton’s Law of cooling)
[K] += f(h)
{Q} = f(h,Tf)
where f = flux normal to the surface; Tf – temperature of surrounding fluid; h – heat transfer coefficients (which may depend on many factors like velocity of fluid, roughness/geometry of surface, etc) and T- temperature of surface.
Radiation BC
f = hr(Tr – T)
[K] += f(hr)
{Q} = f(hr,T)
where, Tr – temperature of the surface; hr – temperature dependent heat transfer coefficients.
hr = Fs(Tr2+T2)(Tr+T).
Where F is a factor that accounts for geometries of radiating surfaces.s is Stefan-Boltzmann constant.
Finite Elements Principles and Practices - Fall 03

7. FEs in Thermal Analysis
1D Bar Element
Uniform bar whose lateral surface is insulated
2D Elements
PLANE35, PLANE55, PLANE77 etc….
3D Elements
SOLID70, SOLID90 etc….
Transient Thermal Analysis
[KT]{T} + [C]{Ť} = Q where Q = Q(t)
Ť = dT/dt,
C = Summation of c
Solution: Use Crank Nicholson Method, etc.
Finite Elements Principles and Practices - Fall 03