On the Temperature Regime of a Shell under Free-Convective Heat Transfer through Symmetric and Asymmetric Horizontal Cylindrical Layers

A. S. Lyapikov

(Presented by Prof. Dr. G. I. Fuks)

In free-convective heat transfer through horizontal cylindrical layers, the temperature field of the shell is formed under the influence of the heat-transfer conditions on its inner and outer surfaces. In particular, the temperature of the shell around the perimeter is the same if, on the outer surface of the shell, the condition \(\mathrm{Bi}\to\infty\) is provided artificially or naturally, with an unchanged temperature of the external heat-transfer medium. In other cases the temperature of the shell varies around its perimeter. Therefore, in systems consisting of a heat-generating element and a shell, it is of interest to study the qualitative and quantitative characteristics of heat transfer not only for the body generating heat, but also for the shell.

Despite the considerable number of works on free-convective heat transfer through layers of gas and liquid, no attention has been paid to the temperature regime of the shell, chiefly because experimental studies of heat transfer through layers were carried out at an artificially created uniform temperature over the entire surface of the shell. In works [1, 2, 3] a method is described for studying free-convective heat transfer through gaseous and liquid symmetric and asymmetric horizontal cylindrical layers, as well as the results of generalizing experimental data on the temperature regime of the inner body (heater). These studies were conducted with free-convective heat exchange of the outer surface of the shell with the surrounding air, and the temperature of the shell around the perimeter was variable. As a result of processing the data of these studies on the temperature regime of the shell, certain regularities were established that are of interest primarily from the point of view of a qualitative characterization of the temperature field of the shell.

The processing was carried out in the form \(\Theta_k=f(\omega)\), and as \(\Theta_k\) the following was used:

\[ \Theta_k=\frac{t_\omega-t_{\min}}{t_{\max}-t_{\min}}, \tag{1} \]

where \(\Theta_k\), \(t_\omega\), \(t_{\max}\), \(t_{\min}\) are, respectively, the dimensionless temperature at a point of the shell with angular coordinate \(\omega\), and the dimensional temperatures of the shell: at the point with angular coordinate \(\omega\), the maximum, and the minimum.

1. For the central position of the heater in a housing made of thin copper foil (\(\Delta = 0.25\) mm), filled with air at atmospheric pressure, the dependence of the dimensionless temperature of the housing \(\Theta_k\) on the angular coordinate \(\omega\) of a point of the housing is represented by the curves in Fig. 1.

Fig. 1. Dependence \(\Theta_k = f(\omega)\) for the coaxial arrangement of heater \(H\) and housing \(K\) (\(\eta = 0\)). Housing made of copper foil, convecting medium—air. \(D = 160\) mm: 1—\(d = 40\), 2—\(d = 28\), 3—\(d = 20\), 4—\(d = 16\) mm; \(D = 80\) mm: 5—\(d = 40\), 6—\(d = 28\), 7—\(d = 20\), 8—\(d = 16\) mm; 9—averaged dependence \(\Theta_k = f(\omega)\); 10—\(\Theta_k \cos \omega\).

Each of curves 1–8 at values \(\omega = 0,\ \dfrac{\pi}{6},\ \dfrac{\pi}{3}\), etc., has several points, obtained under different regimes, in terms of power, corresponding to a temperature difference between the surfaces of the heater and the housing of approximately from 40–50 to 150–160°C. The identical character of the curves \(\Theta_k = f(\omega)\) for different ratios \(\dfrac{D}{d} = 2 \div 10\) (curves 2–8 are displaced one relative to another by \(\dfrac{\pi}{6}\)) indicates that the results presented can be expressed by the averaged curve 9, whose points represent averaged values of the function \(\Theta_k\) at the corresponding values of the argument \(\omega\). The equation of this curve was obtained in the form

\[ \left. \begin{aligned} \Theta_k &= \frac{t_\omega - t_\pi}{t_0 - t_\pi} = 1.07 - 0.42\,\omega &&\left(0 \leq \omega \leq \frac{\pi}{2}\right),\\ \Theta_k &= -0.45 + \frac{1.35}{\omega} &&\left(\frac{\pi}{2} \leq \omega \leq \pi\right). \end{aligned} \right\} \tag{2} \]

Curve 10 represents \(\Theta_k = \cos(\omega)\). Approximation of the experimental data considered by such a simple equation would be desirable; however, this would lead to substantial errors.

Proceeding from (2), the mean-integral dimensionless temperature of the housing is

\[ \overline{\Theta}_k = 0.442. \tag{3} \]

The arithmetic-mean dimensionless temperature of the housing, calculated directly from the points of curve 9, is \(\overline{\Theta}_{k.\mathrm{a}.} = 0.441\).

The observed tendency toward similarity of the temperature field of the shell (in particular, the case of low values of the Bi criterion for the inner and outer surfaces of the shell is discussed) indicates the following possibilities:

a) in connection with the existence of the generalized relation \(\Theta_k = f(\omega) = \Psi(t_\omega, t_0, t_\pi)\), from known \(t_0\) and \(t_\pi\) (for example, measured experimentally) it is possible to find the temperature at any point of the shell, as well as its mean-integral value;

b) on the basis of the generalized mean-integral temperature \(\overline{\Theta}_k = f(\omega_{\bar t}) = \mathrm{const}\), one can determine the angular coordinate \(\omega_{\bar t}\) corresponding to the mean-integral temperature of the shell \(\bar t\); in this case, for similar systems, the measurement of \(\bar t\) can be reduced to a measurement at one point with coordinate \(\omega_{\bar t}\) (or at two points symmetric with respect to the vertical plane passing through the center of the shell).

1. A typical dependence \(\Theta_k = f(\omega)\) for various vertical eccentricities of the heater relative to a shell made of copper foil, the air inside which is at atmospheric pressure, is shown in Fig. 2. Similar graphs over the entire range of vertical eccentricities, constructed for \(D = 160\) mm and \(d = 40, 28, 16\) mm, are not presented here.

Fig. 2. Dependence \(\Theta_k = f(\omega)\) for various vertical eccentricities of the heater relative to the shell. Shell made of copper foil, convecting medium—air. \(D = 80\) mm, \(d = 40\) mm; eccentricities \(\eta = \pm \dfrac{l}{\tfrac12(D-d)}\) for the curves:
\(1\)—\((+0.9)\), \(2\)—\((+0.8)\), \(3\)—\((+0.7)\), \(4\)—\((+0.6)\), \(5\)—\((+0.5)\), \(6\)—\((+0.4)\), \(7\)—\((+0.25)\), \(8\)—\((0)\), \(9\)—\((-0.25)\), \(10\)—\((-0.5)\), \(11\)—\((-0.7)\), \(12\)—\((-0.8)\), \(13\)—\((-0.9)\).

The character of the curves in all the graphs is identical. Curve 8, shown in Fig. 2 by a bold dashed line, corresponds to the coaxial arrangement of the heater relative to the shell. It is seen from Fig. 2 that, for various vertical eccentricities of the heater, similarity of the temperature field of the shell is observed, except for the case of negative eccentricities close to the maximum values (curves 11, 12, and 13). Analysis of the tables \(\Theta_k = f(\omega)\), from which the graph curves of Fig. 2 were constructed [2], made it possible to establish the reason for the special character of curves of types 11, 12, and 13. For negative eccentricities \(\eta\), corresponding in the experiments under consideration to a gap between the heater and the shell (along the lower generatrix) \(\delta \ll 6\) mm, heating of the shell occurs

thermal conductivity in the zone of its lower generatrix. In this case, \(t_{\min}\) shifts from the point \(\omega=\pi\) the more strongly, the greater the eccentricity. A “turnover” of the curve is possible (\(t_{\max}\) passes from the point \(\omega=0\) to the point \(\omega=\pi\)—curve 13). But a “turnover” of the curve does not mean an increase in temperature at the point of the casing \(\omega=\pi\). Analysis of the tables has established that the smallest deviation of the current temperature values of the casing from its mean value (i.e., the best approximation to a uniform temperature distribution) occurs for negative eccentricities corresponding to the gap between the heater and the casing \(\delta \approx 2\) mm. It should be noted that Beckman [4] and Kraussold [5] placed the heating rod in the heater casing with negative vertical eccentricity, thereby achieving equalization of the heater-surface temperature around the perimeter and thus ensuring a more accurate experimental setup. Since in Fig. 2, in the region of negative eccentricities (up to \(\delta \approx 6\) mm), the curves are similar to the curve for the case \(\eta=0\), in this region \(\Theta_k=f(\omega)\), regardless of eccentricity, is also described by equation (2). For the region of positive eccentricities, the curves in Fig. 2 are analogous to the curve for the case \(\eta=0\); however, here the quantitative deviations are more appreciable.

1. For horizontal eccentricities of the heater relative to a copper-foil casing filled with air at atmospheric pressure, the typical dependence \(\Theta_k=f(\omega)\) is shown in Fig. 3 (analogous plots for \(D=160\) mm, \(d=40\) and 16 mm are not given here). In this case, the temperature of the casing must be considered in the range \(\omega=0 \div 2\pi\), since there is no symmetry with respect to the vertical plane passing through the center of the casing. Here, in all the plots, the same character is also observed for curves corresponding in eccentricity; however, as is seen from Fig. 3, for different eccentricities the curves are not similar. As the horizontal eccentricity increases, the maximum \(\Theta_k\) lags behind the displacement of the heater. On the other hand, the minimum \(\Theta_k\) tends to retain its position (\(\omega=\pi\)), and only for horizontal eccentricities close to

Fig. 3. Dependence \(\Theta_k=f(\omega)\) for various horizontal eccentricities of the heater relative to the casing. Copper-foil casing, convecting medium—air. \(D=160\) mm, \(d=28\) mm; eccentricities
\[ \eta=\frac{l}{\frac{1}{2}(D-d)} \]
for curves 1—(0), 2—(0.157), 3—(0.303), 4—(0.454), 5—(0.606), 6—(0.758), 7—(0.833), 8—(0.909), 9—(0.939), 10—(0.97).

to limiting values, a certain displacement of the point of the minimum \(\Theta_k\) in the direction of displacement of the heater is noted. Owing to the variability of the interval \(\omega\), in which, for different eccentricities, \(\Theta_k\) changes from 1 to 0, the curves \(\Theta_k=f(\omega)\) have unequal steepness and, thus, are not similar. Analysis of tables of experimental data for layers with a horizontal eccentricity of the heater showed that horizontal eccentricities practically do not lead to a more uniform temperature field of the body.

1. Figure 4 gives the dependences \(\Theta_k=f(\omega)\) for various vertical and horizontal eccentricities of the heater relative to the body. In these experiments the body used was a glass burette with internal diameter \(D=59.4\) mm, wall thickness \(\Delta=3\) mm, while the heater had \(d=10\) mm; the convecting medium was air.

Fig. 4. Dependence \(\Theta_k=f(\omega)\) for various vertical and horizontal eccentricities of heaters relative to the body. Body—a glass burette. \(D=59.4\) mm, \(\Delta=3\) mm, heater \(d=10\) mm, convecting medium—air. Eccentricities

\[ \eta=\frac{l}{\tfrac12(D-d)} \]

for the curves: 1—\((+0.918)\), 2—\((+0.797)\), 3—\((+0.433)\), 4—\((0)\), 5—\((-0.433)\), 6—\((-0.797)\), 7—\((-0.918)\), 8—\((0.433_{\mathrm{pr}})\), 9—\((0.797_{\mathrm{pr}})\), 10—\((0.918_{\mathrm{pr}})\).

Qualitatively, with respect to the temperature field of the body, the same regularities are observed here as on bodies made of thin copper foil. However, as was established by analysis of the table of experimental data, the “overturning” of the curve at maximum negative vertical eccentricities was accompanied by an increase in the body temperature at the point \(\omega=\pi\), which is associated with the lower thermal conductivity of the glass burette; the greatest uniformity of the body temperature here was observed at a negative eccentricity corresponding to a gap \(\delta \approx 5\) mm (curve 6); horizontal eccentricities here also do not exert a favorable influence on the temperature distribution over the body.

1. The results for heat exchange through a layer of water (body—a glass burette) proved to be specific—Fig. 5. First, no “overturning” of the curves at large negative eccentricities is noted; second, two zones of the temperature regime of the body are observed, depending only on the vertical eccentricity (see curves 4, 5, 6, 7, as well as 4 and 8, 9, 10); third, the maximum temperature of the body at large positive vertical eccentricities is close to the temperature of the heater, while the temperature differences over

the housing are considerable in this case (analysis of the table of experimental data); fourth, at the largest negative eccentricities, a temperature distribution over the housing close to uniform is attained (analysis of the table). All these features are associated with the relatively high heat-transfer coefficients of the droplet liquid.

Fig. 5. Dependence \(\Theta_k=f(\omega)\) for various vertical and horizontal eccentricities of the heater relative to the housing. Housing—a glass burette, \(D=59.4\ \mathrm{mm}\), \(\Delta=3\ \mathrm{mm}\), heater \(d=10\ \mathrm{mm}\), convecting medium—water. Eccentricities

\[ \eta=\frac{l}{\tfrac{1}{2}(D-d)} \]

for curves: 1—\((+0.918)\), 2—\((+0.797)\), 3—\((+0.433)\), 4—\((0)\), 5—\((-0.433)\), 6—\((-0.797)\), 7—\((-0.918)\), 8—\((0.433_{\mathrm{pr}})\), 9—\((0.797_{\mathrm{pr}})\), 10—\((0.918_{\mathrm{pr}})\).

In addition, it follows from the table of results of the experiments considered that the temperature difference between the heater and the housing, depending on the vertical eccentricity, changes by a factor of 2.5–3; consequently, the influence of vertical eccentricity on the heat-transfer coefficient through liquid interlayers is quite substantial, and its further well-founded study is of practical interest.

The results presented are purely qualitative, but even so, on the one hand, they make it possible to approach the solution of certain research and engineering problems in a more substantiated manner, and, on the other hand, they raise the question of further investigation of the temperature distribution along the perimeter of the housing, in particular with respect to an analytical solution of the problem of the temperature regime of the housing taking into account its thickness and the values of the Bi criterion for the inner and outer surfaces of the housing.

REFERENCES

1. A. S. Lyallikov, Yu. A. Zagromov. On the question of free-convective heat transfer in a closed volume when the heat-release source is displaced. Izv. TPI, vol. 137, 1965.

2. Yu. A. Zagromov, A. S. Lyallikov. Free-convective heat transfer in a horizontal cylindrical interlayer for various positions of the heat-releasing element. I. F. Zh., X, No. 5, 1966.

3. Yu. A. Zagromov, A. S. Lyallikov. Calculation of the surface temperature of a heat-releasing element for its various positions in a horizontal cylindrical cavity. In the present collection.

4. W. Beckmann. Wärmeübertragung in Zylindrischen Gasschichten. Forsch. auf dem Gebiete des Ingenieurwes., 2, No. 5, 6. 1931.

5. H. Kraussold. Wärmeabgabe von Zylindrischen Flüssigkeitsschichten. Forsch. auf dem Gebiete des Ingenieurwes., 5, No. 4, 1934.