# Chap5 - CHAP 5 FINITE ELEMENTS FOR HEAT TRANSFER PROBLEMS...

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1 CHAP 5 FINITE ELEMENTS FOR HEAT TRANSFER PROBLEMS FINITE ELEMENT ANALYSIS AND DESIGN Nam-Ho Kim

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2 HEAT CONDUCTION ANALYSIS • Analogy between Stress and Heat Conduction Analysis – In finite element viewpoint, two problems are identical if a proper interpretation is given. • More Complex Problems – Coupled structural-thermal problems (thermal strain). – Radiation problem Structural problem Heat transfer problem Displacement Temperature (scalar) Stress/strain Heat flux (vector) Displacement B.C. Fixed temperature B.C. Surface traction force Heat flux B.C. Body force Internal heat generation Young’s modulus Thermal conductivity
3 THERMAL PROBLEM •G o a l s : – Solve for temperature distribution for a given thermal load. • Boundary Conditions – Essential BC: Specified temperature – Natural BC: Specified heat flux [] { } { } T = KT Q Thermal load Nodal temperature Conductivity matrix

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4 • Fourier Heat Conduction Equation: – Heat flow from high temperature to low temperature • Examples of 1D heat conduction problems STEADY-STATE HEAT TRANSFER PROBLEM Thermal conductivity (W/m/ ° C ) Heat flux (Watts) x dT qk A dx =− T high q x T low q x T high T low
5 GOVERNING DIFFERENTIAL EQUATION • Conservation of Energy – Energy In + Energy Generated = Energy Out + Energy Increase • Two modes of heat transfer through the boundary – Prescribed surface heat flow Qs per unit area – Convective heat transfer – h: convection coefficient (W/m 2 / ° C ) Q g Q s T A dx x x dq qx dx x q + =+ Δ in generated out EE E U ( ) h Qh T T =−

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6 GOVERNING DIFFERENTIAL EQUATION cont. • Conservation of Energy at Steady State – No change in internal energy ( Δ U = 0) – P: perimeter of the cross-section • Apply Fourier Law – Rate of change of heat flux is equal to the sum of heat generated and heat transferred ( ) gen in out x xs g x E E E dq qQ P x h T T P x Q A x q x dx ⎛⎞ + Δ =+Δ ⎝⎠ ±²²³²²´ ±²²²²²²²²²²²²²²³²²²²²²²²²²²²²²´ ±²²²²²²³²²²²²²´ ( ) ,0 x gs dq QA hPT T QP x L =+ + () 0, 0 dd T kA Q A hP T T Q P x L dx dx ++ +=
7 GOVERNING DIFFERENTIAL EQUATION cont. • Boundary conditions – Temperature at the boundary is prescribed (essential BC) – Heat flux is prescribed (natural BC) – Example: essential BC at x = 0, and natural BC at x = L: 0 (0) L x L TT dT kA q dx = = =

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8 DIRECT METHOD • Follow the same procedure with 1D bar element – No need to use differential equation • Element conduction equation – Heat can enter the system only through the nodes Q i : heat enters at node i (Watts) – Divide the solid into a number of elements – Each element has two nodes and two DOFs ( T i and T j ) – For each element, heat entering the element is positive T i e () e i q T j j i e j q 1 2 N Q 1 Q 2 Q N Q 3 L ( e ) x i x j
9 ELEMENT EQUATION • Fourier law of heat conduction • From the conservation of energy for the element • Combine the two equation

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## This note was uploaded on 08/22/2011 for the course EML 4500 taught by Professor Staff during the Fall '08 term at University of Florida.

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Chap5 - CHAP 5 FINITE ELEMENTS FOR HEAT TRANSFER PROBLEMS...

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