1. 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
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2. 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 /°C
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3. Gradient along the thickness
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4. Thermomechanical analysis
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5. Heat moves within the body by conduction
The thermal problem
Heat moves within the body by conduction
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6. 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.
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7.  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
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8. 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= 
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9. 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)
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10. 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
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11. 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.
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12. 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
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13. 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
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14. 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
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15. 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.
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16. 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
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17. 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
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18. 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
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19. 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.
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20. 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
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21. 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
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22. 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
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23. 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
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24. 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
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25. Dependence on temperature of some material properties
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26. 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
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27. 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 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 = /°C
- Specific heat = 480 J/Kg °C
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28. 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
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29. Axisymmetric solid of revolution
Model
Axisymmetric solid of revolution
Plane of symmetry
Plane of symmetry
Gap
Radial
Axis of revolution
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30. 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.
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31. Temperature field
The lowest temperature is about 3 °C along the right half of the inside boundary AB
The highest temperature is °C along the outer surface CDEFG
Temperature contour are interelements continuous, except along IJ where discontinuity is expected
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33. 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 W/m2 which, as expected is less that the approximate limiting value.
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34. Flux contour
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35. Thermal transient
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36. 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.
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37. Results thermomechanical analysis
BC is fixed
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38. 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
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