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1 heat Transfer and MultiphysicsAnalysis 2011 Alex GrishinMAE 323 Lecture 8: heat Transfer and Multiphysics1 heat Transfer AnalysisHeat Transfer and MultiphysicsAnalysis 2011 Alex GrishinMAE 323 Lecture 8: heat Transfer and Multiphysics2 In engineering applications, heat is generally transferred from one location to another and between bodies. This Transfer is driven by differences in temperature (a temperature gradient) and goes from locations of high temperature to those with low temperature. These temperature differences, in turn, cause mechanical stresses and strains in bodies due to their coefficient of thermal expansion, (sometimes abbreviated CTE in engineering literature) The amount of heat Transfer is directly proportional to the size of the temperature gradient and the thermal resistanceof the material(s) involved In engineering applications, there are three basic mechanisms:1.

2 Conduction2. Convection3. RadiationHeat Transfer and MultiphysicsAnalysis 2011 Alex GrishinMAE 323 Lecture 8: heat Transfer and Multiphysics3 conduction For a thermally orthotropicmaterial*, the heat Transfer per unit volume per unit time can be stated (in SI units of Joules per cu. meter per second, or simply Watts per cu. meter):*see ++= where:0300thermal conduction in direction i (Watts/m/)physical mass (kg)volumetric heat generation (W/m )specific heat capacity (J/kg/)temperature ()ipkCCCTC =====(1) heat Transfer and MultiphysicsAnalysis 2011 Alex GrishinMAE 323 Lecture 8: heat Transfer and Multiphysics4 conduction All the terms on the LHS of (1) represent conduction of heat through material (usually solid bodies) The physical mechanism of this conduction is usually molecular (or electronic) vibration.

3 For steady-state problems with no heat generation in one-dimension, we have:220xxTkxTkqx = = where q is an applied heat flux ( heat flow per unit area. SI units are W/m2)(2) heat Transfer and MultiphysicsAnalysis 2011 Alex GrishinMAE 323 Lecture 8: heat Transfer and Multiphysics5 conduction Equation (2) states that the temperature distribution along a length of material conducting heat along that length is linear and proportional to the heat flow, qHeat Transfer and MultiphysicsAnalysis 2011 Alex GrishinMAE 323 Lecture 8: heat Transfer and Multiphysics6 ConvectionqiqoT Ts Convection is a mechanism of heat Transfer that occurs due to the observable (and measurable) motion of fluids As fluid moves, it carries heat with it. In engineering applications, this phenomenon can be characterized by:()sqh TT = where200heat flow per unit area (W/m )surface temperature ()fluid temperature far from surface ()sqTCTC ===(3) heat Transfer and MultiphysicsAnalysis 2011 Alex GrishinMAE 323 Lecture 8: heat Transfer and Multiphysics7 Radiation Thermal radiation is electromagnetic radiation generated by the thermal motion of charged particles in matter Two different bodies at different temperatures separated by some neutral medium (space or air) will exchange heat through this mechanism according to.

4 ()441 2 1 212qFTT = (4)where1 21 2204emissivity between body 1 and 2 (dimensionless)view factor (dimensionless)=Stefan Boltzmann constant (W/m /)FK == Equation (4) is generally nonlinear because and special solver utilities are used to solve these problems (beyond the scope of this course) heat Transfer and MultiphysicsAnalysis 2011 Alex GrishinMAE 323 Lecture 8: heat Transfer and Multiphysics8 In this course, we will only deal with steady-state thermal analyses with heat sources, conduction , and convection. Element formulations for such phenomena are straightforward and have direct analogies with static structural problems. To see this, let s start with the case of bar/truss and a conduction in 1 dimension From Chapter 4, we have static equilibrium in one direction:0xxxbx += If no body load is present, then:0xxx = Then we use the isotropic constitutive law (Chapter 4 again) for a unilateral stress:xuEx = (5)(6)(7) heat Transfer and MultiphysicsAnalysis 2011 Alex GrishinMAE 323 Lecture 8: heat Transfer and Multiphysics9 Plugging (7) into (6) gets the equation in terms of the primary variable (displacement)220uEx = (8)Units: Force/length2 We can do the same thing with the conductivity equation (1).

5 Assuming steady state conduction with no volumetric heat generation in x-direction only, equation (1) becomes:220xTkx = Units: Energy/time*Temperature/length3(9) heat Transfer and MultiphysicsAnalysis 2011 Alex GrishinMAE 323 Lecture 8: heat Transfer and Multiphysics10 We saw in chapter 2 that we can integrate equation (8) twice and apply boundary conditions to solve it. This leads to the canonical truss element:11221111uFEAuFL = Equation (9) has the same form, so we should expect to be able to create an analogous 1D (thermal link) element Integrating (9) once leads to Fourier s Law of conduction in one dimension (the sign comes from the necessary direction of heat flow from hot to cold over an increasing distance):dTkqdx= (10)(11) heat Transfer and MultiphysicsAnalysis 2011 Alex GrishinMAE 323 Lecture 8: heat Transfer and Multiphysics11 Solving (11) for T in terms of q yields an equation very similar to (10).

6 This is a thermal link element:11221111 TQkATQL = (12) Similarly, a convection link element can be constructed from (3) as:121111sTQhATQ = (13) The elements in (13) connect nodes on the surface of a body at Ts to a common ground node at T . Here the area A is area over which the convection elements actsHeat Transfer and MultiphysicsAnalysis 2011 Alex GrishinMAE 323 Lecture 8: heat Transfer and Multiphysics12 Equations (12) and (13) demonstrate that the thermal link elements in a steady-state thermal Analysis are analogous to structural spring elements. Thus the heat flow, Q is the analog of the structural force F and T is the analog of the structural displacement. These analogies lead directly to the notion of thermal resistance, R: = =K x FR T QStatic Structural problemSteady-State thermal problemStructural stiffnessDisplacement ForceThermal resistanceTemperatureHeat flowHeat Transfer and MultiphysicsAnalysis 2011 Alex GrishinMAE 323 Lecture 8: heat Transfer and Multiphysics13 Without going through the details, we will simply mention that the equations (1) and (3) can be combined to yield the governing equations for a system experiencing both conduction and convection.

7 This combined system may be expressed as:()h+ =+R H T Q Qwhere:TVTSThSdVh dShTdS= == RB BHN NQN(14)conductivity matrixconvection coefficientvector of shape functions000000hxyz=== = NNNBNHeat Transfer and MultiphysicsAnalysis 2011 Alex GrishinMAE 323 Lecture 8: heat Transfer and Multiphysics14 Performing a Steady-State Thermal Analysis in ANSYS Workbench Shell and line body assumptions:Shells: no through-thickness temperature gradients. Line bodies: no through thickness variation. Assumes a constant temperature across the variation will still be considered along the line bodySome Assumptions: As with structural analyses, contact regions are automatically created to enable heat Transfer between parts of Transfer and MultiphysicsAnalysis 2011 Alex GrishinMAE 323 Lecture 8: heat Transfer and Multiphysics15 Performing a Steady-State Thermal Analysis in ANSYS Workbench If parts are initially in contact heat Transfer can occur between them.

8 If parts are initially out of contact no heat Transfer takes place (see pinball explanation below). Summary: The pinball region determines when contact occurs and is automatically defined and set to a relatively small value to accommodate small gaps in the modelInitially TouchingInside Pinball Region Outside Pinball RegionBondedYesYesNoNo SeparationYesYesNoRoughYesNoNoFrictionle ssYesNoNoFrictionalYesNoNoContact TypeHeat Transfer Between Parts in Contact Region? heat Transfer and MultiphysicsAnalysis 2011 Alex GrishinMAE 323 Lecture 8: heat Transfer and Multiphysics16 Performing a Steady-State Thermal Analysis in ANSYS WorkbenchBy default,perfect thermal contact conductancebetween parts is assumed, meaning no temperature drop occurs at the conditions can contribute to less than perfect contact conductance:surface flatnesssurface finishoxidesentrapped fluidscontact pressuresurface temperatureuse of conductive grease.

9 Continued .. TTxHeat Transfer and MultiphysicsAnalysis 2011 Alex GrishinMAE 323 Lecture 8: heat Transfer and Multiphysics17 Performing a Steady-State Thermal Analysis in ANSYS WorkbenchThe amount of heat flow across a contact interface is defined by the contact heat flux q:where Tcontactis the temperature of a contact node and Ttargetis the temperature of the corresponding target node . By default, TCC is set to a relatively high value based on the largest material conductivity defined in the model KXXand the diagonal of the overall geometry bounding box essentially provides perfect conductance between Transfer and MultiphysicsAnalysis 2011 Alex GrishinMAE 323 Lecture 8: heat Transfer and Multiphysics18 Performing a Steady-State Thermal Analysis in ANSYS Workbench heat Flow: A heat flow rate can be applied to a vertex, edge, or surface.

10 The load is distributed for multiple selections. heat flow has units of energy/time. Perfectly insulated ( heat flow = 0): Available to remove surfaces from previously applied boundary conditions. heat Flux: heat flux can be applied to surfaces only (edges in 2D). heat flux has units of energy/time/area. Internal heat Generation: An internal heat generation rate can be applied to bodies only. heat generation has units of energy/ positive value for heat load will add energy to the system. heat Transfer and MultiphysicsAnalysis 2011 Alex GrishinMAE 323 Lecture 8: heat Transfer and Multiphysics19 Performing a Steady-State Thermal Analysis in ANSYS WorkbenchTemperature, Convection and Radiation: At least one type of thermal boundary condition must be present to prevent the thermal equivalent of rigid body motion.