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**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

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**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

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**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

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**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

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**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

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**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

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**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

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