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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
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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
Thermal conductivity 20 W/m°C Coefficient of thermal expansion 12 10-6 /°C 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
The thermal problem Heat moves within the body by conduction
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.
Hp: non isotropic material the direction of the heat flow in general is not parallel to the direction of the temperature increase fx, fy, fzheat flux for unit area (W/m2) matrix of thermal conductivity(W/m°C) Boundary condition on surface with normal n
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=
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)
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
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.
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
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
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
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.
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
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 KTthe vector contribute toQ If the material conductivity is dependent on temperature a nonlinearity is present
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
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.
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
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
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
Iterative solution based on the method of direct substitution: • - assume an initial temperature field T0 • - generateKTandQ based on these temperature • - solve the equation forT1 • - use the newly computed temperature for new values of k, h, and hr • - generate a newKTandQ • - solve for a new T • Iteration is stopped when a convergence test is satisfied. • For example
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
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
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
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
Model Axisymmetric solid of revolution Plane of symmetry Plane of symmetry Gap Radial Axis of revolution
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. Boundary conditions
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 Temperature field
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. Thermal flux
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. Thermomechanical analysis
Results thermomechanical analysis BC is fixed
Maximum and minimum stresses (in MPa) for different axial restraint along BC
