ON THE PROBLEM OF HEAT TRANSFER THROUGH STRAIGHT FINS UNDER CONDITIONS OF INTENSIVE HEAT EXCHANGE

A. A. GURCHENOK, A. F. FEDOROV

(Presented by Prof. G. I. Fuks)

The intensification of thermal processes makes it necessary to investigate the features of heat transfer through straight fins at large heat fluxes.

The calculation of heat transfer through straight fins (Fig. 1) is usually carried out under the assumption that the temperature inside the fin is constant across the width and varies only with height [1]. In this case the temperature head and the amount of heat transferred are determined from the equations:

\[ \vartheta_2=\vartheta_1 \frac{1}{\operatorname{ch} mh+\dfrac{\alpha_2}{m\lambda}\operatorname{sh} mh} \tag{1} \]

\[ Q=\lambda m f \vartheta_1 \frac{\dfrac{\alpha_2}{m\lambda}+\operatorname{th} mh} {1+\dfrac{\alpha_2}{m\lambda}\operatorname{th} mh}. \tag{2} \]

If heat transfer from the end face is neglected, then (1) and (2) take the following form:

\[ \vartheta_2=\vartheta_1\frac{1}{\operatorname{ch} mh}; \tag{1a} \]

\[ Q=\lambda m f \vartheta_1 \operatorname{th} mh. \tag{2a} \]

Here:

\[ m=\sqrt{\frac{\alpha_1}{\lambda a}}; \]

\(\alpha_1, \alpha_2\) are the heat-transfer coefficients, respectively, of the lateral (1) and end (2) surfaces;

\(\lambda\) is the coefficient of thermal conductivity of the material;

\(2a, h\) are the thickness and height of the fin;

Fig. 1.

\(\vartheta_1, \vartheta_2, \vartheta\) are the temperature differences at the base (1), at the end of the fin (2), and the current value.

The value of \(\vartheta_1\) is usually taken equal to the temperature difference on a smooth surface.

In the present work, the results obtained from (1a) are compared with data from calculating heat transfer through straight fins as a two-dimensional problem. An equation is obtained for calculating the temperature variation along the smooth surface, which makes it possible to determine the temperature at the base of the fin with allowance for the outflow of heat to the fin. In conclusion, a comparison is given of the calculation results with data obtained on an electrothermal model.

1.

The temperature field inside the fin is two-dimensional [2]. The differential equation of the two-dimensional field has the form [3]

\[ \frac{\partial^2 \vartheta}{\partial x^2} + \frac{\partial^2 \vartheta}{\partial y^2} = 0, \tag{3} \]

where \(x, y\) are the current values of the fin width and length.

Boundary conditions:

\[ \frac{\partial \vartheta}{\partial x} = \pm n \vartheta \quad (4) \quad \text{for} \quad x = \pm a, \]

where

\[ n = \frac{\alpha}{\lambda}. \]

Solving equation (3), we obtain

\[ \vartheta_2 = \vartheta_1 \sum_{\kappa=1}^{\infty} A_{\kappa}\exp\left(-\frac{\delta_{\kappa}}{a}y\right)\cdot \cos \frac{\delta_{\kappa}x}{a}. \tag{5} \]

Here \(\delta_{\kappa}\) is the root of the \(\kappa\)-th solution, determined from the equation

\[ \operatorname{ctg}\delta_{\kappa} = \frac{\delta_{\kappa}}{n\cdot a}. \tag{6} \]

\(A_{\kappa}\) are the coefficients of the series, determined from the expression

\[ A_{\kappa} = \frac{2\sin\delta_{\kappa}}{\delta_{\kappa}+\sin\delta_{\kappa}\cos\delta_{\kappa}}. \tag{7} \]

The index “\(\kappa\)” runs through the entire set of natural numbers. In Table 1, for the example given below, the results obtained from (1) and from (5) are presented. Calculation practice using (5) showed that the value is determined mainly by the first term \((k=1)\). If the values of the remaining terms of series (5) are neglected, then for the case \(x=0\) solution (5) can be written in the form

\[ \vartheta_2 \simeq \vartheta_1 A_1 \exp\left(-\frac{\delta_1}{a}y\right). \tag{5a} \]

It is of interest to determine the conditions under which the solutions according to (1a) and (5a) are identical.

Equating (1a) and (5a) and expanding the expressions

\[ \exp\left(-\frac{\delta_1}{a}y\right) \quad \text{and} \quad \operatorname{ch} my = \frac{\exp(my)+\exp(-my)}{2} \]

in a series up to the third term, we obtain

\[ 2\left(1+\frac{\delta_1 y}{a}+\frac{\delta_1^2 y^2}{2a^2}\right) = A_1\left(1+my+\frac{m^2y^2}{2}+1-my+\frac{m^2y^2}{2}\right). \]

After transformations, the conditions for equality of (1a) and (5a) take the form

\[ a_{\mathrm{pr}}=\frac{2\lambda a}{y^2 A_1} \left(1-A_1+\frac{\delta_1}{a}y+\frac{\delta_1^2}{2a^2}y^2\right). \tag{8} \]

The conditions found may be regarded as limiting ones; for the application of relations (1) and (2) the condition \(a \leqslant a_{\mathrm{pr}}\) is necessary. For

Table 1

the example given below, condition (8) has the following meaning: for a fin of height \(h=10.5\ \mathrm{cm}\), \(a \leqslant 400\ \mathrm{W}/\mathrm{m}^2\cdot{}^\circ\mathrm{C}\); \(h=4.8\ \mathrm{cm}\)—\(a \leqslant 150\ \mathrm{W}/\mathrm{m}^2\cdot{}^\circ\mathrm{C}\); \(h=1.5\ \mathrm{cm}\)—\(a \leqslant 0\).

2.

In [4], [5] it is shown that in a number of cases the temperature variation along a smooth surface is a large quantity, comparable with the difference \(\vartheta_1-\vartheta_2\). Let us find relations for determining it [6].

Let us isolate, in the smooth surface adjacent to the region of the fin, an element with cross section \(dx\) (Fig. 2). We shall make the assumptions that: a) the neighboring fin does not affect the heat transfer of the fin under investigation; b) the temperature variation through the thickness of the smooth surface is small and may be neglected. Then, according to Fig. 2, the heat balance in the isolated element will be equal to

\[ Q_1+Q_2=Q_3+Q_4. \tag{9} \]

The values of the quantities entering into (9) are as follows:

\[ Q_1=q e dx. \tag{10} \]

Fig. 2.

Here \(q\) is the heat flux through the smooth surface;
\(e\) is the length of the fin, surface.

\[ Q_2=-\lambda\delta\,\frac{d\vartheta}{dx}, \tag{11} \]

\[ Q_3=-\lambda\delta\,\frac{d}{dx}\left[\vartheta+\frac{d\vartheta}{dx}(-dx)\right], \tag{12} \]

\[ Q_4=\alpha\vartheta e dx. \tag{13} \]

After substitution into (9) and transformations, one can obtain the differential equation

\[ \frac{d^2\vartheta}{dx^2}+\frac{\alpha\vartheta}{\lambda\delta}=\frac{q}{\lambda\delta}. \tag{14} \]

The solution of equation (14) has the form

\[ \vartheta=\left(\vartheta_1-\frac{q}{\alpha}\right) \cos\left[\sqrt{\frac{\alpha}{\lambda\delta}}(x-a)\right] +\frac{q}{\alpha}. \tag{15} \]

In equation (15), the values \(\vartheta\) and \(\vartheta_1\) are quantities to be determined. Therefore, in order to solve by equation (15), it is first necessary to determine one of them. The following procedure is possible. For

smooth surface, without taking into account the influence of heat transfer to the base of the fin, the expression for the heat flux has the form

\[ q=\frac{\vartheta_0}{\dfrac{\delta}{\lambda}+\dfrac{1}{\alpha}}, \tag{16} \]

where \(\vartheta_0\) is the total temperature head. For the steady-state regime, from (16) we obtain

\[ \vartheta=q\cdot 1/\alpha. \tag{17} \]

For the region occupied by the fin, under the conditions specified for (16), the heat-flux equation will have the form

\[ q_p=\frac{\vartheta_0}{\dfrac{1}{\lambda/\delta}+\dfrac{1}{\lambda m\,\operatorname{th} mh}}, \tag{18} \]

whence

\[ \vartheta_1=\frac{q_p}{\dfrac{1}{\lambda m\,\operatorname{th} mh}}. \tag{19} \]

It is obvious that the magnitude of the temperature head on the smooth surface will turn out to be overestimated, while for points at the base of the fin it will be underestimated relative to the true value, since in reality the heat flux is partly directed along the surface toward the base of the fin.

Let us try to take this part of the heat flux into account. From (15), determine the value of \((x-a)\) for which the influence of the fin is absent, that is, condition (17) is satisfied:

\[ \cos\left[\sqrt{\frac{\alpha}{\lambda\delta}}\,(x-a)\right]=0. \tag{20} \]

The right-hand side in (20) may be represented as a particular value of the same cosine function.

Then we obtain

\[ \sqrt{\frac{\alpha}{\lambda\delta}}\,(x-a)=\frac{\pi}{2}. \]

Consequently, for

\[ x<a+\frac{\pi/2}{\sqrt{\dfrac{\alpha}{\lambda\delta}}}, \]

the temperature on the smooth surface will be lower than the temperature on the surface beyond the boundary of the zone of influence of heat outflow to the fin. If the boundary of the fin’s zone of influence is located at a distance \(b\), then the temperature on the surface of this boundary will be equal to

\[ \vartheta_b=\left(\vartheta_1-\frac{q}{\alpha}\right)\cos\sqrt{\frac{\alpha}{\lambda\delta}}\cdot b+\frac{q}{\alpha}. \tag{15a} \]

Using the found value \(\vartheta_b\), one can determine a refined value of the heat flux on the smooth surface

\[ q'=\vartheta_b\cdot \alpha_1. \tag{21} \]

The difference \(q-q'\) will be the heat flux flowing through the base of the fin. Then the total flux through the fin will be equal to

\[ q'_p = q_p + (q - q'). \tag{22} \]

From the value found for \(q'_p\), a new value of \(\vartheta'_k\) should be determined. Then, from the new value of \(\vartheta'_1\), we find a new value of \(\vartheta'_b\) and again carry out the calculation according to (21), (22), (19) in order to determine the value of \(\vartheta'_1\). We end the calculation when the values \(\vartheta^{\,n-1}_1 = \vartheta^n_1\) coincide.

3.

To compare the calculation methods described, electrical modeling of the heat-transfer process in a straight fin was carried out (Fig. 3).

Fig. 3. Diagram of the electrical model:
1 — computational region; 2 — fictitious resistances; 3 — current-collecting buses; 4, 5 — resistance boxes; 6 — galvanometer; 7 — source of constant regulated voltage; 8 — measuring probe; 9 — region corresponding to the pipe cross-section.

The electrical model of the fin was made from electrically conductive paper with specific resistance \(\rho = 15000\ \dfrac{\mathrm{ohm}}{\mathrm{cm}}\). The outline of the fin section was drawn on the paper. Since the problem is symmetric, only one half of the section was constructed.

The boundary conditions on the model were specified by means of special sections (2), located between the current-receiving lines (3) and the boundaries of the rib cross-section contour (cooling surface). The width of sections (2) was determined from the condition [7]

\[ l=\frac{\lambda}{\alpha}. \tag{23} \]

The source of constant regulated voltage was a rectifier of the VS-12 type. The wires from the rectifier are connected to the corresponding busbars.

The electric potential at various points of the computational domain was measured according to a balanced-bridge circuit. Between the busbars two resistance boxes are connected, forming two arms of the bridge. The two other arms are formed by conducting paper from the point of potential measurement \(M\) to the corresponding busbars. A null galvanometer is connected into the diagonal of the bridge. Contact at the point \(M\) is made by means of a measuring needle fixed at the end of a wire, which can be moved over the surface of the model.

The computational domain of the model was divided into a grid. At each grid point the electric potential was determined (when the needle is placed at these points the galvanometer must read zero), which can be calculated from the condition

\[ \frac{\vartheta-\vartheta_2}{\vartheta_1-\vartheta_2} = \frac{R_1}{R_1-R_2}, \tag{24} \]

where \(R_1\) and \(R_2\) are known. The following 9 examples were solved and modeled.

A section of a finned surface (Fig. 1) is given with the following data: the fin is rectangular, has width \(2a=2.0\ \mathrm{cm}\), height \(h\), respectively: a) \(10.5\ \mathrm{cm}\), b) \(4.8\ \mathrm{cm}\), c) \(1.5\ \mathrm{cm}\); spacing between fins \(2.0\ \mathrm{cm}\), base thickness \(1.4\ \mathrm{cm}\); thermal conductivity of the material \(\lambda=58.2\ \mathrm{W}/\mathrm{m}\,^\circ\mathrm{C}\), heat transfer by convection: d) \(\alpha=582\ \mathrm{W}/\mathrm{m}^2\,^\circ\mathrm{C}\), e) \(\alpha=1163\ \mathrm{W}/\mathrm{m}^2\,^\circ\mathrm{C}\), f) \(\alpha=2326\ \mathrm{W}/\mathrm{m}^2\,^\circ\mathrm{C}\); temperature of the heated surface \(100^\circ\mathrm{C}\), of the surrounding medium \(20^\circ\mathrm{C}\).

Table 2

It is required to determine the temperature value inside the fin along the \(y\)-axis at heights \(0\), \(h/4\), \(h/2\), \(h\). It is required to determine the temperature gradient on the smooth surface at the base of the fin.

Table 1 presents the results of calculations of the one-dimensional (I) and two-dimensional (II) temperature fields, according to the data of the electrical model (III). It is evident from the table that, for all 9 examples, the greatest deviation of the calculated results from the data of the electrical model was obtained for the case of calculation by the formulas of the one-dimensional temperature field.

Table 2 gives the temperature values on the smooth surface and at the base of the fin, calculated by the formulas of the one-dimensional case (I), by the method of successive approximation to the two-dimensional flux (II), and according to the data of the electrical model (III). It is evident from the table that calculation by the method of successive approximations to the two-dimensional flux makes it possible to determine the temperature variation on the smooth surface with sufficient accuracy.

Conclusion

In investigating heat transfer through straight fins under conditions of intensive heat exchange, it is necessary to take into account the change in temperature on the smooth surface and to carry out the solution using the formulas of the two-dimensional problem. Reduction of the solution to a one-dimensional problem is possible only under the condition

\[ \alpha \ll \alpha_{\mathrm{pr}}. \]

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