46CHAPTER 2  One-dimensional, steady state heat conduction

Example 2.8: Consider steady state conduction in a composite slab with heat generation in the middle portion of the slab. The problem description is shown in Fig. 2.24.

Given T2 = 270 °C, T3 = 220 °C, determine the thermal conductivity of the middle portion of the slab and the volumetric heat generation in the middle portion of the slab, for steady state one dimensional conduction with constant properties of all the materials.

qright = 135714 W/m2

qtotal = 83478 + 135714 = 219192 W/m2

FIGURE 2.24

Schematic representation of a composite plane wall with heat generation in the middle and convection boundary conditions at the two ends.

48 CHAPTER 2  One-dimensional, steady state heat conduction

is one in which the effects are known (typically temperature measurements in heat transfer problems), and the causes (like thermal conductivity) have to be estimated.

2.9  Fin heat transfer

For maintaining the temperatures of equipment as low as possible, it is necessary to maximize the heat transfer from them, subject to various constraints such as cost, pressure drop, and space. One practical way of increasing heat transfer from surfaces is by providing more surface area, and these extended surfaces are called fins.

The convection heat transfer from a surface is given by

Here the goal is to increase “Q.”

Methods to increase Q:

1.Increase heat transfer coefficient (h): when velocity increases, h will increase. However, the pressure drop also increases.

2.Increase surface area (A): the passive method.

3.Increase (Tw −T∞ ): restrictions may be there on the capabilities of the material or legislation or safety in respect of the maximum temperatures allowed.

Maximizing the heat transfer with an increase in surface area is an effective method. We accomplish this by using extended surfaces or fins on the base surface (Kraus et al., 2002). Addition of fins may marginally decrease the heat transfer coefficient. However, (hA) with fins is usually much greater than (hA) without fins, if they are engineered based on sound scientific principles to be discussed here.

Examples: Electrical transformers, condensers, economizers, motor cycle, IC engines, etc.

Desirable properties of fin material:

1.High thermal conductivity to maximize heat transfer.

2.Density of fin material is as low as possible; otherwise, the weight of the system increases significantly, outweighing the benefits.

3.Reasonable cost.

2.10  Analysis of fin heat transfer

Fig. 2.25 shows a schematic representation of a typical variable area fin with a rectangular cross-section.

The following assumptions are in order:

1.One-dimensional, steady state heat transfer, i.e., T = f(x) alone

2.Constant properties

50 CHAPTER 2  One-dimensional, steady state heat conduction

In Eq. (2.169), m is the fin parameter and is known for any fin a priori.

For a rectangular fin of constant cross sectional area shown in Fig. 2.26, the perimeter, p and the cross sectional area Ac of the fin are given by,

FIGURE 2.26

Schematic representation of a rectangular fin with constant cross section area.

52 CHAPTER 2  One-dimensional, steady state heat conduction

Eq. (2.182) indicates that the temperature distribution across the fin is exponential.

The heat transfer from the fin can now be calculated as

Fig. 2.27A represents the temperature variation along the length of the fin for the ideal case. In this case, the fin temperature is equal to the base temperature everywhere.

However for a typical fin, the temperature varies along the length, exponentially, and is shown in Fig. 2.27B.

As seen in Fig. 2.27C, the unhatched portion indicates the penalty we pay for the fin not being isothermal. If the fin is not maintained isothermally throughout the length, its efficiency to transfer heat (we will formally introduce the term efficiency in the next sub-section) will get reduced and is shown with the unhatched portion in Fig. 2.27C. Therefore, we try to make the exponential variation as close to the horizontal line as possible in order to get maximum efficiency (close to ideal efficiency).

cosh(m(L − x))

θ = θb cosh (2.187)

(mL)

The fin tip temperature can be obtained by evaluating θ at x = L.

FIGURE 2.27

Graphical representation of fin performance. (A) ideal case, (B) typical fin case, and (C) reduction in θ for a fin not being isothermal.

54 CHAPTER 2  One-dimensional, steady state heat conduction

FIGURE 2.28

Schematic representation of n number of fins over a base plate.

Eq. (2.197) confirms that it is more advantageous to use high thermal conductivity fins in situations where the heat transfer coefficient “h” is low. In so far as the geometry of the fin is concerned, lower the fin cross section and higher the perimeter, higher is the effectiveness. We can now work out the effectiveness relation for a rectangular fin.

Rectangular fin

For a rectangular fin, the effectiveness which turns out to be the key engineering metric, as already discussed is

We would like to have an effectiveness as high as possible. Hence, we choose a material with a high thermal conductivity and we try to make a fin thin, as already discussed. A surprising and counterintuitive result is that as h is increasing while Qfin keeps increasing, ε comes down. So, we are confronted with a situation where a fin appears to be good but not effective. In simple English, effficiency tells us if a fin is thermally the best of many competing fin designs, while effectiveness adresses the fundamental question as to whether we need to use a fin, however efficient it may be in an engineering problem.