Thermal analysis There is a resemblance between the thermal problem and the stress analysis. The same element types, even the same FE mesh, can be used for both analysis Thermal analysis means primarily calculation of temperature within a solid body A by-product of the temperature calculation is information about the magnitude and direction of heat flow in the body. The results of thermal analysis, nodal temperature, can be used to evaluate the distribution of stresses due to the presence of a thermal gradient within the body Advanced Design for Mechanical System Pract10

Pipe with thermal gradient along the thickness Flux= 0 Internal diameter = 40mm Thickness = 10mm Length =600mm a r T =100°C T =25 °C Flux= 0 Thermal conductivity 20 W/m°C Coefficient of thermal expansion 12 10-6 /°C Advanced Design for Mechanical System Pract10

Gradient along the thickness Advanced Design for Mechanical System Pract10

Thermomechanical analysis Advanced Design for Mechanical System Pract10

Heat moves within the body by conduction The thermal problem Heat moves within the body by conduction Advanced Design for Mechanical System Pract10

Fourier equation of heat flow Consider an isotropic material and image that exist a temperature gradient in the x direction fx, heat flux for unit area (W/m2) k thermal conductivity (W/m°C) Negative sign means that heat flows is in a direction opposite to the direction of temperature increase The temperature represents the potential for the heat flow. Advanced Design for Mechanical System Pract10

 matrix of thermal conductivity (W/m°C) Hp: non isotropic material the direction of the heat flow in general is not parallel to the direction of the temperature increase fx, fy, fz heat flux for unit area (W/m2)  matrix of thermal conductivity (W/m°C) Boundary condition on surface with normal n Advanced Design for Mechanical System Pract10

Conductivity The matrix of conductivity is in general a full 3 by 3 matrix If x,y,z are principal axes of the material  is a diagonal matrix In the special case of isotropy 11=22=33=  Advanced Design for Mechanical System Pract10

qv is the rate of internal heat generation for unit volume (W/m3) By considering a differential element of volume dV and writing the balance equation (rate in – rate out)= (rate of increase within), it is obtained: where: qv is the rate of internal heat generation for unit volume (W/m3) c is the specific heat (J/kg °C) is the density (kg/m3) t is the time (s) Advanced Design for Mechanical System Pract10

Where  is the gradient operator. If the problem becomes steady-state. For material isotropic the equation becomes Where  is the gradient operator. If, in addition, k is independent of the coordinates results Solution: T=T(x,y,z) that also meets prescribed boundary conditions Advanced Design for Mechanical System Pract10

Prescribed temperature are analogous to prescribed displacements The heat flow trough the surface of the body is analogous to surface load in the stress analysis. A distributed internal source is analogous to body forces in stress analysis. Prescribed temperature are analogous to prescribed displacements For a steady state problem, the mathematics of all this leads to the global FE equation where: KT depends on the conductivity of the material ([W/°C]) T is the vector of the nodal temperature ([°C]) Q is the vector of the thermal loads ([W]) Convection and radiation boundary conditions, if present, contribute term to both KT and Q. Advanced Design for Mechanical System Pract10

Hypothesis for FE in thermal analysis In solid body the heat transfer occurs by conduction In fluid the transmission by mean of convection must be take into account In many structural problems the thermal problem and the mechanical problem are not coupled This hypothesis is no longer valid when the deformation can generate heat and cause a variation of the mechanical properties Thermal conductivity and other properties of the material must be considered dependent on the temperature The conductivity matrix can be generated by a direct method for very simple elements, otherwise a formal procedure is required Advanced Design for Mechanical System Pract10

Nodal heat flow is positive when direct within the element Thermal bar element The rate of heat flow, q=Af, is constant and direct axially. T1>0, T2=0 T2>0, T1=0 Nodal heat flow is positive when direct within the element An actual bar can be modeled by several of these elements if temperature at several locations between its ends were required Advanced Design for Mechanical System Pract10

In matrix format, these results are: Temperature within the element are obtained by interpolating nodal temperatures The form of the interpolation determines the complexity of the temperature field that an element can represent The shape function can be exactly those used also to interpolate a displacement field Advanced Design for Mechanical System Pract10

In Cartesian coordinates, temperature gradients in a plane element are For solid a third row expressing T/z is added The expression for an element conductivity matrix can be shown to be The array of thermal conductivities  becomes the scalar k for bar elements. Advanced Design for Mechanical System Pract10

Remarks The thermal problem is a scalar field problem A FE temperature field is continuous within elements and across interelement boundaries Temperature gradient, like strains in stress analysis, are typically not interelement continuous If radiation is not present, the rate of heat flow is proportional to temperature differences Upon assembly of elements, nodal rates of heat flow qi from separate elements are combined at shared nodes. Thus the net flow into node i of the structure: except at nodes where Ti is prescribed, nodes on a structure boundary across which heat is transferred, or at internal nodes where Qi arise from qv Advanced Design for Mechanical System Pract10

Convection boundary conditions Where f is the flux normal to the surface and positive inward h is the heat transfer coefficient Tf e T temperature of the fluid and of the surface of the body For the increment, dS, of the element surface subjected to convection formal procedure gives The matrix combine with the element conductivity matrix and hence contribute to KT the vector contribute to Q If the material conductivity is dependent on temperature a nonlinearity is present Advanced Design for Mechanical System Pract10

Radiation boundary conditions Consider two infinite parallel planes. Let the planes have temperature Tr and T , and each is a perfect absorber and perfect radiator (black body). If SB is the Boltzman constant, the net heat flux received by the surface at temperature T is If the emissivity  (ratio of total emissive power to that of a blackbody at the same temperature) is introduced, than Advanced Design for Mechanical System Pract10

The flux f is an average value. Accounted for the fact that the surface are not parallel, often not flat, and certainly not infinite, a shape factor is introduced F is a factor that accounts for the geometry of the radiating surface and their emissivities The calculation of F is sufficiently complicated that it may be done by a separate computer program. The flux f is an average value. Advanced Design for Mechanical System Pract10

This equation can be written as T is the absolute [K] hr= hr(T) is a temperature dependent heat transfer coefficient This makes the problem non linear and requires iterative solution The flux expressions have the same form for convection and radiation boundary conditions Accordingly, a radiation boundary condition leads to matrix and vector expressions having the same form as for convection, with h and Tf replaced by hr and Tr Advanced Design for Mechanical System Pract10

Modeling considerations Element types, and shapes for a thermal analysis may be dictated less by thermal analysis than by an anticipated stress analysis based on the same mesh The mesh demands of stress analysis are usually more severe Also a modest temperature gradient may create forces of constraint that produce large strain gradients, especially near holes, grooves and other stress raisers. Also for thermal analysis the present symmetry must be considered: heat flux trough a plane of symmetry is zero Heat flux must be parallel to an insulate boundary or a plane of symmetry Temperature contours (isotherms) should be parallel to a boundary of constant temperature and normal to plane of symmetry Advanced Design for Mechanical System Pract10

The problem is non linear Non linearity If in the body there are appreciable difference of temperature, it is necessary to regard conductivity as temperature dependent If there is convection with a temperature dependent coefficient of heat transfer If is present radiation The problem is non linear Advanced Design for Mechanical System Pract10

Iterative solution based on the method of direct substitution: - assume an initial temperature field T0 - generate KT and Q based on these temperature - solve the equation for T1 - use the newly computed temperature for new values of k, h, and hr - generate a new KT and Q - solve for a new T Iteration is stopped when a convergence test is satisfied. For example Advanced Design for Mechanical System Pract10

The solution must be integrated with respect to time Thermal transient When steady state conditions do not prevail, temperature change in a unit volume of material is resisted by “thermal mass” that depends on the mass density r and the specific heat c. The solution become The solution must be integrated with respect to time Advanced Design for Mechanical System Pract10

Dependence on temperature of some material properties Advanced Design for Mechanical System Pract10

Flanged Joint Sections of a pipe are connected by a flanged joint. Each flange is connected to the pipe by two circumferential welds Bolts draw the flanges together and compress a gasket between them Evaluate the steady state temperature field in the pipe and in the flange for use in a subsequent stress analysis Advanced Design for Mechanical System Pract10

Data of the problem Boundary conditions: - fluid in the pipe has temperature 0°C - vapor condensing on the outside of the pipe has temperature 100 °C Heat transfer coefficient h: - inside the pipe is 5000 W/m2 - outside the pipe is 20000 W/m2 The material of the pipe and of the flanges is steel with the following thermal properties - Thermal conductivity = 20 W/m°C - Coefficient of thermal expansion = 12 10-6 /°C - Specific heat = 480 J/Kg °C Advanced Design for Mechanical System Pract10

Preliminary Analysis The maximum possible value of flux through the pipe wall is approximated by regarding the wall as plane and using the limiting surface temperature This is a maximum value because: - the inside of the pipe must be warmer than 0°C - the outside of the pipe must be cooler than 100 °C Advanced Design for Mechanical System Pract10

Axisymmetric solid of revolution Model Axisymmetric solid of revolution Plane of symmetry Plane of symmetry Gap Radial Axis of revolution Advanced Design for Mechanical System Pract10

Boundary conditions Between the welds, along IJ, pipe and flanges touch only at random and isolated point, if at all. Conductivity across this cylindrical surface is very low compared with solid metal and can be assumed equal zero This is a pessimistic assumption for eventual stress analysis, because it increase the thermal gradient and therefore increase stresses. Advanced Design for Mechanical System Pract10

Temperature field The lowest temperature is about 3 °C along the right half of the inside boundary AB The highest temperature is 99.99 °C along the outer surface CDEFG Temperature contour are interelements continuous, except along IJ where discontinuity is expected Advanced Design for Mechanical System Pract10

Advanced Design for Mechanical System Pract10

Thermal flux Flux arrows are perpendicular to the temperature contour because the material is isotropic Flux arrows agree with the expectation of preliminary analysis Radial flux near AB is 170.000 W/m2 which, as expected is less that the approximate limiting value. Advanced Design for Mechanical System Pract10

Flux contour Advanced Design for Mechanical System Pract10

Thermal transient Advanced Design for Mechanical System Pract10

Thermomechanical analysis The mesh used for the thermal analysis is used again, now with computed nodal temperatures used to load the model. Two significant question about boundary arise: - should nodes along BC be fixed against axial motion or not? Allowing movement gives no credit to resistance offered by the gasket and the bolts, while fixity gives too much credit. We elect to run the stress analysis twice, first allowing motion along BC and second preventing it. - is it proper to let nodes along IJ move independently? If sides of the interface move apart, as the results shows, the answer is yes. Advanced Design for Mechanical System Pract10

Results thermomechanical analysis BC is fixed Advanced Design for Mechanical System Pract10

Only node B axially restrained All nodes on BC axially restrained Maximum and minimum stresses (in MPa) for different axial restraint along BC Only node B axially restrained All nodes on BC axially restrained sr sq sz Maximum stress 113 216 244 105 229 257 Location I,J L-N M-B I,J,K Minimum stress -68 -142 -199 -107 -225 -279 G Advanced Design for Mechanical System Pract10