Calculation of the Surface Temperature of a Heat-Generating Element at Various Positions in a Horizontal Cylindrical Cavity

Yu. A. Zagromov, A. S. Lyalikov

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

In [1] a method is set forth and, in the form of graphs, the results are given of an experimental study of free-convective heat transfer in horizontal cylindrical cavities for various positions of the heat-generating element.

As a result of processing the experimental data on the Minsk-1 computer, equations were obtained that relate the dimensionless surface temperature of the heat-generating element \(\Theta\) to its eccentricity \(\eta\):

\[ \Theta = 1 + 0.12\eta - 0.24\eta^2 + 0.30\eta^3 \quad \text{for } 0 \leq \eta \leq 0.92; \tag{1} \]

\[ \Theta = 1 + 0.0774\eta \quad \text{for } 0 \geq \eta \geq -0.92. \tag{2} \]

The maximum deviation of equations (1) and (2) from the experimental data does not exceed \(\pm 2.5\%\).

As noted in the same work, at maximum positive and negative vertical eccentricities \((0.92 < \eta < 1.0\) and \(-0.92 > \eta > -1.0)\), sharp decreases in the surface temperature of the heat-generating element were observed owing to large temperature gradients in the boundary layer. An analogous decrease in temperature also occurred for horizontal eccentricity \(0.92 < \eta < 1.0\). Since the case of the greatest decrease in the surface temperature of the heat-generating element is of considerable practical interest (when the lowest surface temperature is desirable), the experimental data were also processed for vertical and horizontal eccentricities in the range \((0.92) < \eta < (1)\). The following considerations form the basis of this processing.

For eccentricity \(0.92 < \eta < 1\), as is evident from Fig. 1 (segment BD), the function \(\Theta = f(\eta)\) can be expressed by the equation of a straight line. Point \(B\) with \(\eta = 0.92\), on the basis of equation (1), corresponds to \(\Theta_B = 1.154\). Assuming the linearity of segment BD, one may write

\[ \tan \alpha_1 = \frac{\eta - 0.92}{\Theta_B - \Theta_{DB}}, \]

whence

\[ \Theta_{DB} = \Theta_B - \frac{\eta - 0.92}{\tan \alpha_1} = 1.154 - \frac{\eta - 0.92}{\tan \alpha_1}. \]

An analogous approach for the region \(1.0 < \eta < -0.92\) (segment CE) leads to the equation

\[ \Theta_{CE}=\Theta_C+\frac{\eta+0.92}{\tg \alpha_2}=0.929+\frac{\eta+0.92}{\tg \alpha_2}, \]

and for horizontal eccentricities \(0.92<\eta<1.0\), to the equation

\[ \Theta_{NM}=\Theta_A-\frac{\eta-0.92}{\tg \alpha_3}=1-\frac{\eta-0.92}{\tg \alpha_3}. \]

As a result of processing the experimental data, the following values were obtained:

\[ \tg \alpha_1=0.666;\quad \tg \alpha_2=0.435;\quad \tg \alpha_3=0.750. \]

In this case, the equations for calculating \(\Theta\) in the eccentricity ranges \((0.92)<\eta<(1.0)\) take the following form for vertical eccentricities:

\[ \Theta_{DB}=2.53-1.50\eta \quad \text{for } 0.92<\eta<1.0, \tag{3} \]

\[ \Theta_{CE}=3.05+2.30\eta \quad \text{for } -1.0<\eta<-0.92, \tag{4} \]

for horizontal eccentricity:

\[ \Theta_{NM}=2.22-1.35\eta \quad \text{for } 0.92<\eta<1.0. \tag{5} \]

The maximum deviation of equations (3–5) from the experimental data used to obtain them does not exceed 7–8%.

Fig. 1. \(DBACEF\)—\(\Theta=f(\eta)\) for vertical displacement (a);
\(ANM\)—\(\Theta=f(\eta)\) for horizontal displacement (b): I—for a pair of tubes with \(D=160\) mm, \(d=16\) mm at \(l=0.165a\) (1), 0.22 (2), 0.28 (3);
II—the same for \(D=160\) mm, \(d=28\) mm; at \(l=0.16a\) (4), 0.25 (5), 0.3 (6); III—\(D=160\) mm, \(d=40\) mm, \(l=0.19a\) (7), 0.25 (8), 0.3 (9);
IV—\(D=80\) mm, \(d=40\) mm, \(l=0.19a\) (10), 0.25 (11), 0.3 (12)

Using the relations given above (1–5), it becomes possible to calculate more accurately the surface temperature of a heat-release source at its various positions in a closed cylindrical cavity. In this calculation it is necessary to use the value of the surface temperature of the heat-releasing element when it is centrally located; the methods for calculating this are based on investigations of convective heat transfer in symmetric gaps [1], and the features of performing such calculations as applied to certain particular problems are shown in [2].

LITERATURE

1. Yu. A. Zagromov, A. S. Lyalkov. IFZh, 10, No. 5, 1966.
2. G. I. Fuks, A. S. Lyalkov. Izv. TPI, 137, 1965.