Effect of Thermal Nonstationarity on the Heat-Transfer Coefficient under Conditions of Natural Convection

V. V. Banyukha

(Presented by Professor G. I. Fuks)

In recent years, interest has grown considerably in processes occurring under nonstationary flow of heat-transfer media. The ability to calculate these processes reliably is necessary for calculating various operating regimes of power installations and technological equipment.

The problem of unsteady convection, intensively studied by domestic and foreign researchers, is at present still very far from being solved, which testifies to its exceptional complexity. The literature data concerning this question are highly contradictory and do not permit well-founded conclusions to be drawn even for comparatively simple cases.

Thus, for example, the authors of work [1], Bichutko, Shitnikov, and others, confirming the conclusions of Juge [2] and Kryukova [3], show by the results of their experiments that the heat-transfer coefficient does not depend on thermal nonstationarity.

At the Kharkov Polytechnic Institute, S. F. Kravtsov and K. A. Polnitskii carried out a series of experiments on the study of heat transfer of cylinders under conditions of natural convection. The authors conclude that the heat-transfer coefficients in stationary and nonstationary regimes do not correspond to one another [4].

B. D. Katsnelson and F. A. Timofeeva [5] investigated the influence of velocity nonstationarity on the heat-transfer coefficient and found that this influence is substantial only during the period of formation of the hydrodynamic boundary layer.

The contradictory nature of the conclusions on this question by various investigators prompted us to carry out a number of special experiments with the aim of identifying the laws governing the behavior of the heat-transfer coefficient under conditions of thermal nonstationarity.

Of great practical interest are cases in which the temperature of the medium is constant, while a body introduced into it has, at the initial instant of time, a temperature noticeably different from it. The temperature of the body will then asymptotically approach the temperature of the medium. This case of so-called “simple” cooling of a cylinder, which, along with a sphere and a plate, is an elementary form of the surface of heat-exchange apparatus, was used by us to determine the laws governing the behavior of the heat-transfer coefficient under conditions of nonstationary natural convection.

Let us take an infinitely long thin-walled cylinder. Let the temperature of the liquid at infinity be \(t = \mathrm{const}\). The temperature of the liquid within the region of thermal influence is \(t\). The temperature of the cooled body is \(t_w\).

The problem has the following mathematical formulation:

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

\[ \frac{\partial w_x}{\partial \tau}+w_x\frac{\partial w_x}{\partial x}+w_y\frac{\partial w_x}{\partial y} = \nu\frac{\partial^2 w_x}{\partial y^2} + g\frac{t-t_\infty}{t_\infty}\cdot \sin\beta \tag{b} \]

\[ \frac{\partial t}{\partial \tau}+w_x\frac{\partial t}{\partial x}+w_y\frac{\partial t}{\partial y} = a\frac{\partial^2 t}{\partial y^2} \tag{c} \]

\[ - c_{pM}\rho_M\frac{D^2-d^2}{4D}\cdot \frac{dt_w}{d\tau} = \alpha\left(t_w-t_\infty\right) \tag{d} \]

\[ \tag{1} \]

Let us make the following assumptions:

a) all thermophysical characteristics of the body and the medium will be considered independent of temperature;

b) in the region of thermal influence \(\Delta\), we assume \(w_y=0\);

c) considering the development of the process to be quasi-stationary, we shall have

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

which fully satisfies the physical meaning, since the development of the process of motion around the body in free convection occurs sufficiently slowly in time;

d) the temperature field of the body depends only on time.

Under the above assumptions, the system of equations (1) was solved, and a relation was obtained for determining the heat-transfer coefficient under nonstationary conditions.

Fig. 1

\[ \overline{Nu} = C(Gr\cdot Pr)^{0.25} \left[ 1+\frac{5}{2}\frac{1}{\Theta_w}\cdot \exp\left(-6\bar{\delta}^{\,2}\cdot F_0\right) \right], \tag{2} \]

where

\[ \Theta_w=\frac{t_w-t_\infty}{t_{w\max}-t_\infty}; \]

\[ \bar{\delta}^{\,2}=\left(\frac{D}{\Delta}\right)^2 \]

is a quantity determined by the expression:

\[ \bar{\delta}^{\,2} = \frac{0.25}{\pi}\sqrt{Gr\cdot Pr} \int_0^\pi \sqrt{\frac{\sin\beta}{\beta}}\, d\beta, \tag{3} \]

\(D\) is the outside diameter of the cylinder;

\(\Delta\) is the thickness of the region of thermal influence.

Equation (2) may be written in another form:

\[ \frac{\overline{Nu_i}}{\overline{Nu_\infty}} = 1+\frac{5}{2\Theta_w}\cdot \exp\left(-6\bar{\delta}^{\,2}\cdot F_0\right). \tag{4} \]

It is interesting to note that for \(F_0=0\), from (4) we obtain the maximum number \(\overline{Nu}\), equal to

\[ \overline{Nu}_{\max}=Nu_{\infty}\left[1+\frac{5}{2}\right]=\frac{7}{2}Nu_{\infty}. \tag{5} \]

In order to verify dependence (4), a special experiment was carried out.

This problem was solved as an inverse heat-conduction problem: from the temperature field of the body known from the experiments, the heat-transfer conditions at the boundary of the body were determined. A simple relation convenient for experimental investigation can be obtained from the equation describing the cooling process of the body,

\[ \frac{d\Theta_w}{d\tau}+\frac{\alpha F}{C_T}\cdot \Theta_w=0, \tag{6} \]

where \(F\) is the area of the heat-emitting surface, \([m^2]\);

\(C_T\) is the heat capacity of the body \(\left[\frac{\mathrm{J}}{\mathrm{deg}}\right]\).

Fig. 2. Schematic of the experimental setup

The schematic of the experimental setup is shown in Fig. 2. The test bodies were copper cylinders coated with nickel and polished; the external and internal diameters of the cylinders were, respectively:

\[ \begin{aligned} \text{specimen 1:}\quad &D=39.9\ \text{mm}, \quad d=33.5\ \text{mm};\\ \text{specimen 2:}\quad &D=34.8\ \text{mm}, \quad d=29.8\ \text{mm};\\ \text{specimen 3:}\quad &D=27.9\ \text{mm}, \quad d=21.8\ \text{mm};\\ \text{specimen 4:}\quad &D=49.9\ \text{mm}, \quad d=46.8\ \text{mm}. \end{aligned} \]

Heating to a temperature of \(\sim 150\text{–}170^\circ\mathrm{C}\) was carried out by means of nichrome heaters inserted inside the cylinders and supplied with alternating current from a 220 V mains supply. The temperature of the cylinder wall was measured by 12 copper–constantan thermocouples, whose junctions were located in the central plane of the cylinder. In order to satisfy the condition of an infinitely long cylinder, the ratio of the length to the external diameter of the specimens was maintained equal to 13–14. To be able to verify the reliability of the experimental data on nonstationary

...ordinary heat transfer, the work was carried out in two stages. The first stage consisted in conducting experiments under steady-state conditions and calculating the heat-transfer coefficients from the relation

\[ q=\alpha(t_w-t), \tag{7} \]

where \(q\) is the heater power, W.

The steady-state regime was maintained for 3–4 hours. Then, from the temperatures known from the experiments, the heat-transfer coefficients were calculated according to the empirical relation of M. A. Mikheev,

\[ Nu_m=0.54(Gr\cdot Pr)^{0.25}. \]

The agreement of the results was good: the deviation of the experimental results did not exceed 5%.

The second stage was the experimental determination of heat-transfer coefficients under nonsteady conditions. The emf of the thermocouples was recorded in time on the tape of an EPP-09 self-recording potentiometer (class 0.5). In order to exclude the influence of the thermal inertia of the heater during the experiment, it was removed from the cylinder. To avoid possible distortion of the boundary layer during cooling of the cylinder, the ends of the thermocouples were applied along the generatrix on the inner surface of the cylinder, since, according to the assumption, it is precisely the peculiarity in the formation of the boundary layer that determines the heat-transfer coefficient under nonsteady conditions.

Fig. 3. Results of comparing the dependence \(\dfrac{\dot{Nu}}{Nu_\infty}=f(Fo)\) according to equation (4)—solid line, and values obtained experimentally—dashed line

The heat-transfer coefficient could have been calculated from equation (6), using the experimental data directly. However, in this case, at small time instants large errors are possible, since the heat-transfer coefficient is determined from the difference of quantities close in value. It was therefore proposed to approximate the experimental temperature cooling curve of the body by a function of the form

\[ \Theta_w=\Theta_{w\max}\cdot e^{-A\tau}\cdot\tau^{B}, \tag{9} \]

where \(A\) and \(B\) are certain coefficients depending on the material and shape of the body; in each individual case they were found by calculation.

in this way. The result proved quite satisfactory. The deviations of the \(\Theta_w\) values calculated from (9) from the experimental data did not exceed 3%. Further, using equation (9), it was not difficult to calculate \(Nu=f(\mathrm{Fo})\). Figure 3 presents the dependence

\[ \frac{\overline{Nu}}{Nu_\infty}=f(\mathrm{Fo}) \]

for specimen No. 1, where the solid line denotes the values of \(\overline{Nu}/Nu_\infty\) calculated from equation (4), and the dashed line denotes the experimental values.

The deviations from the calculated value are maximal only at the very initial moment of cooling, which can apparently be explained by the inertia of the measuring system. In the region \(\mathrm{Fo}>0.5\), the deviations do not exceed 10–15%, which may be considered satisfactory.

Conclusions

1. The introduction of the region of thermal influence made it possible to obtain calculation equations for the convective heat-transfer coefficient during cooling of an infinitely long cylinder by unsteady natural convection.

2. Under natural-convection conditions, the quantity

\[ \frac{\overline{Nu}}{Nu_\infty} \]

significantly exceeds unity only up to the moment when temperature equilibrium is established within the region of thermal influence.

References

1. I. Ya. Bilyutko, V. K. Shitnikov, G. V. Sadovnikov, L. A. Sergeeva. Investigation of Unsteady Heat and Mass Transfer. Minsk, 1966.
2. T. Iuge. RIASM, vol. 5, No. 49, pp. 175–183.
3. M. G. Kryukova. IFZh, No. 4, 1, 1958.
4. S. F. Kravtsov, K. A. Polnitskii. Izvestiya vuzov, Energetika, 1963, No. 7.
5. B. D. Katsnelson, F. A. Timofeeva. Heat Transfer and Aerodynamics. Transactions of TsKTI, issue 3, book 12, Mashgiz, 1949.