COMPARISON OF THE THERMAL PERFORMANCE OF FINNED AND SMOOTH PLANE WALLS

V. V. Ivanov

(Presented by Professor, Doctor G. I. Fuks)

1

Let us compare the temperatures of finned and smooth plane walls through which a constant heat flux \(q\) passes. Heat from both walls is transferred to a layer, identical in thermal conductivity, of length \(L\). At the boundary the layer has the same temperature \(t_2\) (Figs. 1 and 2). It is assumed that the fins have high thermal conductivity and a small height \(h\); therefore it may be assumed that their temperature is equal to the temperature of the wall itself, \(t_1\).

Fig. 1.

Fig. 2.

Fig. 3.

We shall solve the stated problem by means of conformal mappings. The function mapping the exterior of the finned plane wall (Fig. 1) onto the exterior of the smooth wall (Fig. 3) has the form [1].

\[ W=\frac{a}{\pi}\operatorname{arc\,cos}\left(-\frac{1}{\operatorname{ch}\frac{\pi h}{a}}\cdot \cos \frac{\pi Z}{a}\right). \tag{1} \]

Here \(a\) is the distance between the ribs, \(Z=x+iy\), (2), \(W=u(x,y)+iv(x,y)\). (3)

The transformation (1) maps the isotherm \(t_2(y=L)\) of the \(Z\)-plane onto the straight line \(v=l\) of the \(W\)-plane. Since the heat fluxes passing through the regions between the isotherms \(t_1\) and \(t_2\) in the one and the other planes are equal, we have

\[ q=\frac{\lambda}{l}(t_1-t_2), \tag{4} \]

where \(\lambda\) is the coefficient of thermal conductivity of the material of the layer.
We determine the value \(l\) from the equation

\[ l=v(x,y)_{y=L}, \]

where, for simplicity of calculation, the value of \(x\) may be taken equal to zero. Substituting (2) and (3) into (1), after transformations we find

\[ l=V_{x=0,\;y=L} =\frac{a}{2\pi}\ln \frac{1}{\operatorname{ch}\frac{\pi h}{a}} \left( \operatorname{ch}\frac{\pi}{a}L+ \sqrt{\operatorname{ch}^2\frac{\pi}{a}L-\operatorname{ch}^2\frac{\pi h}{a}} \right)^2 . \tag{5} \]

Putting \(h=0\) and using the formulas

\[ \sqrt{\operatorname{ch}^2\frac{\pi}{a}L-1} =\operatorname{sh}\frac{\pi}{a}L, \]

\[ \left( \operatorname{ch}\frac{\pi}{a}L+ \operatorname{sh}\frac{\pi}{a}L \right)^2 =\exp\left(\frac{2\pi}{a}L\right), \]

as a special case of equation (5), we obtain \(l=L\).

The heat flux for the case shown in Fig. 2 is expressed as

\[ q=\frac{\lambda}{L}(t_1^*-t_2), \tag{6} \]

where \(t_1^*\) is the temperature of the smooth flat wall.
From (4) and (6) we find

\[ \Delta t=t_1^*-t_1=\frac{q}{\lambda}(L-l) \]

or, equivalently,

\[ \Delta t=t_1^*-t_1= \frac{q}{\lambda}\left[ L-\frac{a}{2\pi}\ln \frac{1}{\operatorname{ch}\frac{\pi h}{a}} \left( \operatorname{ch}\frac{\pi}{a}L+ \sqrt{\operatorname{ch}^2\frac{\pi}{a}L-\operatorname{ch}^2\frac{\pi}{a}h} \right)^2 \right]. \tag{7} \]

II

We shall now compare the heat fluxes passing through the ribbed and the smooth wall. In doing so we shall assume that both walls—

are maintained at the constant temperature \(t_1\). The temperature of the boundary of the interlayer is still equal to \(t_2\).

The heat flux passing through the finned wall, on the basis of equation (4), is expressed as

\[ q^*=\frac{\lambda}{l}(t_1-t_2), \tag{8} \]

where \(l\) is determined by relation (5). The heat flux through a smooth wall is equal to

\[ q=\frac{\lambda}{L}(t_1-t_2). \tag{9} \]

Combining (8) and (9) and using (5), we obtain

\[ q^*-q=\lambda(t_1-t_2)\left(\frac{L-l}{L\cdot l}\right)= \]

\[ =\frac{\lambda(t_1-t_2)}{l} \left\{ 1-\frac{a}{2\pi L}\ln \frac{1}{\operatorname{ch}\frac{\pi}{a}h} \left( \operatorname{ch}\frac{\pi}{a}L+ \sqrt{ \operatorname{ch}^2\frac{\pi}{a}L-\operatorname{ch}^2\frac{\pi}{a}h } \right)^2 \right\}. \tag{10} \]

Equations (7) and (10) make it possible to estimate the thermal performance of a finned wall in comparison with a smooth one, depending on the fin height \(h\) and the distance between the fins \(a\).

LITERATURE

1. M. A. Lavrent'ev and B. V. Shabat. Methods of the Theory of Functions of a Complex Variable. GIFML, 1958.