Question

Select the correct answer to the following multiple-choice questions. Briefly explain your choice.

10. In heat conduction analysis, the error of the heat flux is estimated using a formula in which an unknown constant, a measure of the element shape, and the element size appear. This formula tells us
    A) where the locations with the largest error are,
    B) how to control the error,
    C) both of the above,
    D) none of the above.

Answer 1

Solution : C

Example:

The heat fluxes at the wall faces can be evaluated from Fourier’s law, x '' x dx dT  k)x(q Using the temperature distribution T(x) to evaluate the gradient, find dx d k)x(q '' x  [a+bx2 ]= -2kbx. The flux at the face, is then x=0 2 x '' '' 20 x x '' m/W000,10)L(q m050.0)m/C2000(K.m/W502kbL2)l(q,LatX 0)0(q      Comments: from an overall energy balance on the wall, it follows that g 0EEE . out . in .  0Lq)L(q)0(q . x '''' x  35 " 2 x " x . m/W100.2 050.0 m 0m/w00,10 L )0(q)L(q q

Conduction Heat Transfer

We will start by examining conduction heat transfer. We must first determine how to relate the heat transfer to other properties (either mechanical, thermal, or geometrical). The answer to this is rooted in experiment, but it can be motivated by considering heat flow along a "bar" between two heat reservoirs at TA, TB as shown in Figure 2.1. It is plausible that the heat transfer rate Q& , is a HT-6 function of the temperature of the two reservoirs, the bar geometry and the bar properties. (Are there other factors that should be considered? If so, what?). This can be expressed as

Q& = f1 (TA , TB , bar geometry, bar properties)

It also seems reasonable to postulate that Q& should depend on the temperature difference TA - TB. If TA – TB is zero, then the heat transfer should also be zero. The temperature dependence can therefore be expressed as

Q& = f2 [ (TA - TB), TA, bar geometry, bar properties]

L TA TB Q& Figure 2.1: Heat transfer along a bar An argument for the general form of f2 can be made from physical considerations. One requirement, as said, is f2 = 0 if TA = TB. Using a MacLaurin series expansion, as follows:

f( T) f(0) f ( T) T 0 ∆ ∆ = + ∆ ∂ ∂ +L (2.3)

If we define ∆T = TA – TB and f = f2, we find that (for small TA – TB),

f (T T ) Q f (0) f (T T ) TT . 2A B 2 2 A B TA TB

0 − == + A B ∂ ∂ − ( ) − + ⋅ − = L (2.4)

We know that f2(0) = 0 .

The derivative evaluated at TA = TB (thermal equilibrium) is a measurable property of the bar. In addition, we know that Q TT f T T A B 2 A B ⋅ > > ∂ ∂ − ( ) 0 0 if or > . It also seems reasonable that if we had two bars of the same area, we would have twice the heat transfer, so that we can postulate that Q& is proportional to the area. Finally, although the argument is by no means rigorous, experience leads us to believe that as L increases Q& should get smaller. All of these lead to the generalization (made by Fourier in 1807) that, for the bar, the derivative in equation (2.4) has the form HT-7 ∂ ∂ − ( ) = − = f T T kA L 2 A B TA TB 0 . (2.5) In equation (2.5), k is a proportionality factor that is a function of the material and the temperature, A is the cross-sectional area and L is the length of the bar. In the limit for any temperature difference ∆T across a length ∆x as both L, TA - TB → 0, we can say ( ) ( ) dx dT kA L T T kA L T T Q kA A B B A = − − = − − = & . (2.6) A more useful quantity to work with is the heat transfer per unit area, defined as q A Q & & = . (2.7) The quantity q& is called the heat flux and its units are Watts/m2 . The expression in (2.6) can be written in terms of heat flux as dx dT q& = −k . (2.8) Equation 2.8 is the one-dimensional form of Fourier's law of heat conduction. The proportionality constant k is called the thermal conductivity. Its units are W / m-K. Thermal conductivity is a well-tabulated property for a large number of materials. Some values for familiar materials are given in Table 1; others can be found in the references. The thermal conductivity is a function of temperature and the values shown in Table 1 are for room temperature.

Table :

Thermal conductivity at room temperature for some metals and non-metals Metals Ag Cu Al Fe Steel k [W/m-K] 420 390 200 70 50 Non-metals H20 Air Engine oil H2 Brick Wood Cork k [W/m-K] 0.6 0.026 0.15 0.18 0.4 -0 .5 0.2 0.04