CHAPTER 8: Thermal Analysis

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1 CHAPER 8: hermal Analysis hermal Analysis: calculation of temperatures in a solid body. Magnitude and direction of heat flow can also be calculated from temperature gradients in the body. Modes of heat transfer: by conduction within a body by convection and radiation to/from a body here may be internal heat generation due to electric current, chemical reaction, dielectric heating, etc. emperatures may be prescribed on a boundary or in the interior.

2 Modes of Heat ransfer and Boundary Conditions

3 Finite Element Form for Steady State Problem A domain can be discretized for thermal analysis, i.e., a FE mesh can be created. he steady-state heat-transfer discretized euation is written: K = Q where : vector of nodal temperatures Q : vector of thermal loads due to internal heat generation, convection and radiation to/from the body K : matrix depending on conductivity of the material and convection and radiation

4 Nonlinearity Material properties such as conductivity depend on temperature. K = K () in general Nonlinear problem!! Radiative heat transfer is inherently nonlinear. Note the similarity of K = Q to the euilibrium euation KD=R in stress analysis. Same element types, same FE mesh can be used for both stress and thermal analysis. Genesis uses same names. Internal heat source: analogous to body force in stress analysis. Prescribed temperatures: analogous to prescribed displacements 4

5 Heat Flow Euations: Isotropic Material Fourier law of heat conduction for an isotropic material: 5

6 Heat Flow Euations: Anisotropic Material Heat conduction for an anisotropic material: where κ is a matrix of thermal conductivities. (Scalar matrix for an isotropic material: κ=ki) 6

7 Energy Balance Energy balance euation for a solid body under thermal loading: first term on the left : net energy that moves into a unit volume of the body in one second due to heat exchange via conduction with the surroundings second term on the left : internal heat generation per unit volume. he right hand side represents the rate of energy storage as a result of temperature change Convection and radiation to be added when present. 7

8 8 Energy Balance: Isotropic Material & Steady-State For steady-state problems, For an isotropic material, conduction euation in D is Under these conditions, the energy balance euation is where the derivatives are expressed as a gradient vector: = 0 t = z y x k f f f z y x k j i ) ( ) ( ) ( z y x + + =

9 Uniform Conductivity Uniform conductivity throughout the solid body: Care must be exercised in choosing units. Some units: energy : joule, J (N.m) (heat) power : watt, W (J/s) heat flux : W/m conductivity k : W/m/K or W/m/ C ; K= C + 7 heat gener. v : W/m (rate of internal heat generation) 9

10 Finite Elements in hermal Analysis Element conductivity matrices can be derived with a direct method or a formal procedure as in stress analysis. Direct Method, Bar Element: rate of heat flow through an area = = area * flux = Af = - Ak(/x) If temperature is a linear function of x = Ak he direct method works by applying a temperature to node of a bar element while keeping node at zero (or, reference) temperature and then repeating for the other node. 0

11 Direct Method Applied Heat flow into a node is taken positive. he flow is uniform within the element since A is uniform. Figures (a) and (b) establish the first and second columns of the element conductivity matrix, respectively.

12 Element Conductivity Matrix with the Direct Method Gathering the results in matrix form or k = e with k being the element conductivity matrix, which is similar to the stiffness matrix of a bar element. A tapered bar can be modeled with multiple elements with each element treated as uniform.

13 Formal Procedure: Shape Functions A formal procedure is needed for the conduction matrix of general elements. Hence, where N and e are the vectors of shape functions and element nodal temperatures, respectively. =(x) is the temperature at an arbitrary point along the bar. he shape (interpolating) functions can be identical to those used for displacement fields for a given type of finite element.

14 Element Conductivity Matrix with the Formal Procedure From the previous expression, the temperature gradients are he general expression for an element conductivity matrix: k = B κbdv For a bar element, N is the same as for stress analysis and κ becomes k. 4

15 Remarks hermal finite elements are assembled in the same manner as structural ones. emperature is scalar unlike stress. A FE model has only one dof per node, the nodal temperature. Either temperature or heat flow can be prescribed at a node, but not both. On an insulated boundary, nodal temperatures are unknown and nodal heat flows are zero. Same mesh can be used for a thermal and a subseuent stress analysis. Mesh layout is usually governed by stress analysis because it reuires a more detailed FE model. 5

16 6 Example: Prob. 8. Constant heat flow imposed at node. Node kept at constant temperature. Find expressions for and. Do and give the expected value at node? Individual element matrices and euations: = =, Ak Ak

17 7 Example: Assembly Assembly of the two euations give = Ak is given. herefore, Ak Ak = cannot be prescribed and the last euation would be used to compute. Plays role of reaction force in stress analysis.

18 8 Example: Solution Eliminating the last euation, 0 0 Ak Ak = he solution: + + = Ak Ak value at node : =( - )Ak/ or =( - )Ak/ From above, these are both eual to - as expected..

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