STUDY OF THE COOLING PROCESS OF A HORIZONTAL INFINITELY LONG CYLINDER BY UNSTEADY NATURAL CONVECTION IN AIR

G. I. FUKS, V. V. BANYUKHA

The problem of unsteady natural convection around a cylindrical body was formulated by us in [1] and has the form:

\[ \frac{\partial w_x}{\partial x}+\frac{\partial w_y}{\partial y}=0 \]

\[ \begin{gathered} \frac{\partial w_x}{\partial \tau} +w_x\frac{\partial w_x}{\partial x} +w_y\frac{\partial w_x}{\partial y} = \nu\frac{\partial^2 w_x}{\partial y^2} +g\,\frac{t-t_\infty}{t_\infty}\cdot \sin\beta \\[6pt] \frac{\partial t}{\partial \tau} +w_x\frac{\partial t}{\partial x} +w_y\frac{\partial t}{\partial y} = a\frac{\partial^2 t}{\partial y^2} \\[6pt] -c_{p.m}\rho\,\frac{D}{4}\,\frac{dt_w}{d\tau} = \alpha\,(t_w-t_\infty). \end{gathered} \tag{1} \]

The purpose of the present work is to show the validity of certain assumptions in solving the system of equations (1).

1. In the region of thermal influence \(\Delta\), \(w_y=0\).

2. The process of motion around the body occurs sufficiently slowly in time, on the basis of which we assume:

\[ \frac{\partial w_x}{\partial \tau}=0. \]

Then the expression for the thickness of the region of thermal influence \(\Delta\), obtained in solving the system of equations (1), if it is assumed that the law of temperature variation within the region of thermal influence can be approximated by a quadratic parabola,

\[ \Theta(y_1,\beta)=\Theta_w(1-2\eta+\eta^2), \tag{2} \]

where \(\Theta_w\) is the wall temperature;

\[ \eta=\frac{y}{\Delta}, \qquad y_1=\frac{y}{D}, \]

has the form:

\[ \Delta = D\sqrt[4]{\frac{80}{\mathrm{Gr}\cdot\mathrm{Pr}}} \cdot \sqrt[4]{\frac{\beta}{\sin\beta}}, \tag{3} \]

where \(D\) is the cylinder diameter;
\(\beta\) is the angle measured from the forward critical point of branching (the stagnation point).

When \(\beta = 0\), the expression for the thickness of the region of thermal influence takes the form:

\[ \Delta_0 = D \sqrt{\frac{80}{\mathrm{Gr}\cdot \mathrm{Pr}}}. \tag{4} \]

The pattern of variation of the region of thermal influence \(\Delta\) during the unsteady cooling of a cylinder by natural convection was obtained with the aid of a shadow instrument designed for the qualitative and quantitative study of the distribution and change in the density of gas layers around the models under investigation.

The layout of the experimental setup is shown in Fig. 1. A thin-walled copper cylinder \(1\) was chosen as the experimental specimen; its surface was coated with nickel and carefully polished. At the ends the specimen was closed with tight plugs made of textolite \(2\). The ratio of the cylinder length to its diameter was

\[ \frac{l}{D} = 15. \]

Fig. 1. Layout of the experimental setup

Heating to a temperature of \(180^\circ\mathrm{C}\) was carried out by means of a heater consisting of a single nichrome wire \(3\) of diameter \(\varnothing = 1\ \mathrm{mm}\), whose small heat capacity made it possible to leave it inside the specimen during the cooling process. To prevent sagging of the nichrome wire caused by its residual elongation, the heater design included a spring \(5\), whose contactless attachment to the nichrome wire was accomplished by means of a ceramic bead \(4\). The end of the spring was fastened to the bottom of a thick-walled textolite cup \(6\), pressed into plug \(2\).

To avoid distortion of the shadow pattern and the superposition of additional convective currents on it, the supply wires \(12\) were made of copper of diameter \(\varnothing = 3\ \mathrm{mm}\) and insulated with asbestos cord.

The temperature of the cylinder wall was measured by twelve copper–constantan thermocouples, the junctions of which were caulked around the circumference in the central cross section of the specimen. The EMF of the thermocouples was recorded on the chart of an EPP-09 self-recording potentiometer. The specimen was placed horizontally in the working section of the shadow instrument between the observation

tube 10 and collimator 11. The cylinder was aligned using 8 and 9, fastened to strip 7.

As quantitative methods widely used in practical measurements, we chose two methods: the knife-edge method and the filament-in-focus method. However, the knife-edge method gave poorer results: because of the finite width of the illuminating slit, the shadow boundary was not sharp. There was a certain penumbra zone, which made it difficult to determine the true position of the boundary. The filament method is also not free of these shortcomings. The shadow of the filament likewise constitutes a blurred structure, affected both by the finite width of the light source and by diffraction phenomena; however, it is symmetric, and therefore measurements made with a filament are more reliable.

As the illuminator we used an ultra-high-pressure mercury lamp SVDSh-250 (DRSh). The slit was set equal to 0.2 mm, and the visualizing filament to 0.16 mm.

Fig. 2. Patterns of convective currents around a heated cylinder:

a) $\Theta_w = 180^\circ\mathrm{C}, \quad \tau = 0,$

b) $\Theta_w = 40^\circ\mathrm{C}, \quad \tau = 30$ min

Photography of the shadow pattern was carried out with a “Zenit-3M” single-lens reflex camera with an exposure of $1/30$ sec. Filming was performed during the cooling of the cylinder from $\Theta_w = 180^\circ$ to $\Theta_w = 30^\circ\mathrm{C}$ over 46 min, with an interval of 2 min.

Figure 2 shows the patterns of convective currents around the heated cylinder at the initial instant of time ($\Theta_w = 180^\circ\mathrm{C}$, $\tau = 0$) and 30 min after the start of cooling.

As a result of the work carried out, the following results were obtained:

Fig. 3. Dependence of the variation of the thickness of the region of thermal influence on time \(\tau\):
—— according to equation (3) (solid line),
× × × — experimental values for \(\beta = 90^\circ\),
⊙ ⊙ ⊙ — experimental values for \(\beta = 0^\circ\).

Fig. 4. Dependence of the variation of the thickness of the region of thermal influence on wall temperature:
—— according to equation (3) (solid line),
× × × — experimental values for \(\beta = 90^\circ\).
⊙ ⊙ ⊙ — experimental values for \(\beta = 0^\circ\).

1) within the region of thermal influence \(\Delta\), motion of the air layers in the direction normal to the surface of the specimen is absent \((w_y = 0)\);

2) the process of motion around the body takes place extremely slowly; therefore the assumption of a quasi-stationary process,

\[ \frac{\partial w_x}{\partial \tau}=0, \]

is fully justified (Fig. 3);

3) the nature of the variation in the thickness of the region of thermal influence obeys the laws (3) and (4).

Figure 3 shows the law governing the variation, with time, of the ratio of the region of thermal influence to the diameter of the cylinder.

The deviations of the experimental values from the calculated ones do not exceed 10%.

REFERENCES

1. V. V. Banukha, The influence of thermal unsteadiness on the heat-transfer coefficient under conditions of natural convection. Izvestiya TPI, Vol. 184 (present collection).