Heat Transfer 2011 Alex Grishin MAE 323 Chapter 8: Grishin 1
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, α(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. Conduction 2. Convection 3. Radiation 2011 Alex Grishin MAE 323 Chapter 8: Grishin 2
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): T k T T T x + ky + kz = ρc p λ x x y y z z t (1) where: k i 0 thermal conduction in direction i (Watts/m/ C) ρ = physical mass (kg) C T p = λ = = 3 volumetric heat generation (W/m ) = 0 specific heat capacity (J/kg/ ) 0 temperature ( ) C C *see http://en.wikipedia.org/wiki/orthotropic_material 2011 Alex Grishin MAE 323 Chapter 8: Grishin 3
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. For steady-state problems with no heat generation in onedimension, we have: T k x T k x x 2 0 2 x = = q (2) where q is an applied heat flux (heat flow per unit area. SI units are W/m 2 ) 2011 Alex Grishin MAE 323 Chapter 8: Grishin 4
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, q 2011 Alex Grishin MAE 323 Chapter 8: Grishin 5
Convection 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: ( ) q = h Ts T (3) where q = T T s = 2 heat flow per unit area (W/m ) = 0 surface temperature ( ) 0 fluid temperature far from surface ( C) C T Ts q o q i 2011 Alex Grishin MAE 323 Chapter 8: Grishin 6
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: where ε1 2 F σ 1 2 4 4 ( ) q = ε F σ T T (4) 1 2 1 2 1 2 = emissivity between body 1 and 2 (dimensionless) = view factor (dimensionless) 2 0 4 =Stefan Boltzmann constant (W/m / K ) Equation (4) is generally nonlinear because and special solver utilities are used to solve these problems (beyond the scope of this course) 2011 Alex Grishin MAE 323 Chapter 8: Grishin 7
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: σ xx + b x = 0 x If no body load is present, then: σ xx = 0 x Then we use the isotropic constitutive law (Chapter 4 again) for a unilateral stress: u E = σ x (7) x (5) (6) 2011 Alex Grishin MAE 323 Chapter 8: Grishin 8
Plugging (7) into (6) gets the equation in terms of the primary variable (displacement) u E x 2 = 0 2 Units: Force/length 2 We can do the same thing with the conductivity equation (1). Assuming steady state conduction with no volumetric heat generation in x-direction only, equation (1) becomes: (8) k x T x 2 = 0 2 Units: Energy/time*Temperature/length 3 (9) 2011 Alex Grishin MAE 323 Chapter 8: Grishin 9
We saw in chapter 2 that we can integrate equation (8) twice and apply boundary conditions to solve it. This leads to the canonical beam truss element: EA 1 1 u1 F1 L 1 1 = u F 2 2 (10) 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): dt k q dx = (11) 2011 Alex Grishin MAE 323 Chapter 8: Grishin 10
Solving (11) for T in terms of q yields an equation very similar to (10). This is a thermal link element: ka 1 1 T1 Q1 L 1 1 = T Q 2 2 (12) Similarly, a convection link element can be constructed from (3) as: 1 1 Ts Q1 ha = (13) 1 1 T Q2 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 acts 2011 Alex Grishin MAE 323 Chapter 8: Grishin 11
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: Structural stiffness Displacement Force K x = F R T = Q Thermal resistance Temperature Heat flow Static Structural problem Steady-State thermal problem 2011 Alex Grishin MAE 323 Chapter 8: Grishin 12
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. This combined system may be expressed as: where: ( R + H) T = Q + Qh T R = B κ BdV V T H N N Q h = = S S N h ds T htds (14) κ = conductivity matrix h = convection coefficient N = vector of shape functions N 0 0 x N B = 0 0 y N 0 0 z 2011 Alex Grishin MAE 323 Chapter 8: Grishin 13
Static Structural with Thermal Loads 2011 Alex Grishin MAE 323 Chapter 8: Grishin 14
The governing equations of static structural continua (such as equation (2) of Chapter 5) always contain a body load term. Thermal loads may be considered body loads. Body temperatures are converted to structural body loads via the coefficient of thermal expansion, α(often referred to in industry by the acronym CTE): α CTE (units: Temperature -1 ) α T Thermal strain α E T Thermal stress (15) (16) Thus,(16) would be implemented in equation (2) of Chapter 5 as: T σ δεdv = α E Tw + FwdS V V S 2011 Alex Grishin MAE 323 Chapter 8: Grishin 15
In an element, the discrete form of the thermal becomes: e N α e V E TdV It is thus characterized by a load vector obtained by integrating every element with a temperature other than the reference temperature. This load vector is then added to the global applied load vector e T is thus the difference between the temperature of the body and the reference temperature at which the CTE was measured. It is easy to see that if two bodies with differing CTE s (calculated at the same reference temperature) are raised to the same temperature, they will experience differing thermal-structural loads. If the two bodies are connected, they may experience stresses due to this thermal mismatch * http://www.ami.ac.uk/courses/topics/0162_sctm/index.html 2011 Alex Grishin MAE 323 Chapter 8: Grishin 16
Coupled-Field (Multiphysics) Problems 2011 Alex Grishin MAE 323 Chapter 8: Grishin 17
A static structural analysis which incorporates thermal loads via a temperature distribution obtained from a thermal analysis is one of the earliest types of coupled-field analysis Most commercial codes offer the capability to perform such an analysis in a sequential manner (sometimes referred to as a 2-phase analysis). The primary assumption behind this approach is that the two fields are weakly coupled in a single direction (from thermal-to-structural that is to say that thermal structural loads are obtained from temperature distributions, instead of thermal heat flows being obtained from displacements, stresses, or strains). This makes the thermal-structural sequence linear Phase 1: Thermal Calculate temperature distribution Phase 2: Structural Calculate displacements, stresses, strains 2011 Alex Grishin MAE 323 Chapter 8: Grishin 18