HEAT TRANSFER OF A VERTICAL ROW OF HORIZONTAL TUBES UNDER CONDITIONS OF NATURAL AIR CONVECTION

Yu. A. Zagromov, Yu. A. Korolenko

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

The process of heat transfer of tube bundles under conditions of natural convection is described by the relation [1, 2, 5, 6]

\[ \mathrm{Nu} = f\left(\mathrm{Gr},\ \mathrm{Pr},\ \frac{S}{D}\right). \tag{1} \]

An experimental investigation was carried out of the phenomenon of heat transfer of a vertical row of horizontal tubes in an air medium. In this case the criterion \(\mathrm{Pr}\) remains constant, and dependence (1) takes the form

\[ \mathrm{Nu} = f\left(\mathrm{Gr},\ \frac{S}{D}\right). \tag{1a} \]

The experiments were conducted on a special installation (Fig. 1). Its main element is a model of a bundle, assembled with a specified vertical pitch \(S\) from individual rods. Each rod is a brass tube with outside diameter \(D = 6\ \mathrm{mm}\) and length \(L = 100\ \mathrm{mm}\). A glass tube with a nichrome electric heater mounted in it is inserted inside the brass tube (Fig. 2). The resistance of the heater ranges from 4.00 to 4.04 ohms (mean \(R = 4.02\) ohms).

The space between the brass and glass tubes was filled with layers of aluminum foil of thickness \(\delta = 0.02\ \mathrm{mm}\).

Each rod was provided with a copper-constantan thermocouple made of wires \(d = 0.1\ \mathrm{mm}\), laid between the brass and glass tubes. The sequence of operations for manufacturing the tubes can be understood from Fig. 3. The thermocouple junction was led out into a hole \(d = 0.3\ \mathrm{mm}\) at the middle of the tube length and soldered. When assembling the installation, the junctions of all thermocouples were on the upper generatrix of the tube [3].

The heaters of all rods were connected in series during assembly. Current was supplied to the heaters through an SNE-220-0.75 voltage stabilizer and a LATR-1 RNO-205-2 autotransformer. The power \(N\) consumed by the installation was measured with an EDV wattmeter of accuracy class 0.2.

For measuring the emf of the thermocouples, a circuit with a common cold junction was adopted. The emf of the thermocouples was measured with a laboratory potentiometer PP of class 3. A switch for 10 thermocouples was used for switching the thermocouples.

Fig. 1. Diagram of the experimental setup: 1—stabilized rectifier; 2—autotransformer; 3—wattmeter; 4—model of the bank; 5—hot junction of the thermocouple; 6—cold junction of the thermocouple; 7—cold-junction thermostat; 8—thermometer; 9—potentiometer; 10—null galvanometer.

Fig. 2. 1—brass tube; 2—nichrome heater; 3—glass tube; 4—copper current lead to the heater; 5—heat-resistant putty; 6—thermocouple; 7—annular space filled with aluminum foil.

Fig. 3. Technology for manufacturing the rod: 1—nichrome wire; 2–3—heater coil and current leads; 4–5—soldering and preparation of the heater; 6—glass tube; 7—heater in the tube; 8—heater wrapped with foil; 9—brass tube; 10—assembled rod; 11—rod coated with soot.

The required number of rods was combined into a bundle with pitch \(S\). The vertical dimension of the pitch was adjusted by support screws.

The model was installed at a height of 40 cm above the table level inside a special casing that insulated it from the motion of air in the room.

The minimum distance from the model to the wall of the casing was not less than 20 cm. The ambient temperature was measured with a TNL thermometer with a scale division of \(0.1^\circ\text{C}\). The bulb of the thermometer was located inside the casing.

The experiments were carried out under steady-state thermal conditions. In the experiments the following were measured: the electric power consumed by the model—\(N\)—with a wattmeter; the cold-junction temperature (at the beginning and end of the experiment) \(t_0^{\mathrm{n}}\) and \(t_0^{\mathrm{k}}\); the air temperature inside the casing (at the beginning and end of the experiment) \(t_f^{\mathrm{n}}\) and \(t_f^{\mathrm{k}}\); the surface temperature of the model rods \(t_w\).

The reading of each thermocouple was taken twice, at intervals of 15 min, to check the steadiness of the regime.

A total of 50 experiments was carried out. For each experiment the following were calculated:

1. The total heat consumption by the model

\[ Q_{\mathrm{el}} = N \quad [\text{W}]. \]

1. Heat loss from the ends of the rods, from the current-carrying and connecting wires

\[ Q_n = Q_n' + Q_n'' + Q_n''': \]

a) losses from the ends of the rods were calculated by the formula

\[ Q_n' = 2n\,2\lambda D \Delta t_{\mathrm{av}}; \qquad [4,\ \text{Table } 3\text{--}1] \]

b) losses from the current-carrying wires

\[ Q_n'' = h \left( \frac{2\pi\lambda}{\ln \dfrac{4h}{d'}} \right) \Delta t_{\mathrm{av}}\,2n; \qquad [4,\ \text{Table } 3\text{--}1] \]

c) losses from the connecting wires

\[ Q_n''' = L' \frac{2\pi\lambda \Delta t_{\mathrm{av}}}{\ln \dfrac{4h}{d'}} . \qquad [4,\ \text{Table } 3\text{--}1] \]

In the above formulas: \(\Delta t_{\mathrm{av}} = t_{w\mathrm{av}} - t_{f\mathrm{av}}\); \(t_{w\mathrm{av}}\) is the mean surface temperature of the rods; \(D\) is the rod diameter [m]; \(n\) is the number of rods; \(d'\) is the diameter of the connecting wires (\(d' = 0.0015\) m); \(h\) is the total length of all jumpers between the rods (\(h = 0.008\) m); \(L'\) is the total length of the connecting wires [m]; \(\lambda\) is the thermal-conductivity coefficient of the material \(\left[\dfrac{\text{W}}{\text{m deg}}\right]\). The heat losses in the experiments did not exceed 10%.

1. The heat emitted by radiation from each of the tubes:

\[ Q_l = C_0 \varepsilon_m H_{\mathrm{sf}} \left[ \left( \frac{T_{w\mathrm{av}}}{100} \right)^4 - \left( \frac{T_{f\mathrm{av}}}{100} \right)^4 \right] \ [\text{W}], \]

where \(\varepsilon_m\) is the degree of blackness of the model rods.

In the experiments the surface of the rods was coated with soot \((\varepsilon_m = 0.952,\ [4])\). \(H_{\mathrm{sf}}\) is the mutual surface.

For all rods (except the 2 outer ones) the quantity \(H_{\mathrm{sf}}\) was calculated by the formula

\[ H_{\mathrm{cf}}=2DL\left(a\,\operatorname{arc\,tg}\sqrt{a^{2}-1}-\sqrt{a^{2}-1}\right), \tag{[4]} \]

where

\[ a=\frac{S}{D}. \]

For the end rods

\[ H_{\mathrm{cf}}^{\mathrm{k}}=DL\left(a\,\operatorname{arc\,tg}\sqrt{a^{2}-1}-\sqrt{a^{2}-1}\right)+\frac{\pi DL}{2}. \tag{[4]} \]

1. Heat released by convection,

\[ Q_{\mathrm{k}}=Q_{\mathrm{el}}-Q_{\mathrm{l}}-Q_{n}\ [\mathrm{W}]. \]

1. Heat-transfer coefficient of the model rods

\[ \alpha_{\mathrm{k}}=\frac{Q_{\mathrm{k}}}{F_{\mathrm{st}}\Delta t_{\mathrm{st}}\,n} \left[\frac{\mathrm{W}}{\mathrm{m}^{2}\,\mathrm{deg}}\right], \]

where \(F_{\mathrm{st}}\) is the lateral surface of a rod; \(n\) is the number of rods.

\[ F_{\mathrm{st}}=\pi DL\cdot 10^{-6}\ \mathrm{m}; \qquad \Delta t_{\mathrm{st}}=t_{w}-t_{f}. \]

From the obtained values of \(\alpha_{\mathrm{k}}\) and \(\Delta t_{\mathrm{st}}\), the criteria \(\mathrm{Nu}\) and \(\mathrm{Gr}\) were calculated and a plot was constructed in the coordinates \(\lg \mathrm{Nu}_{f}\)--\(\lg \mathrm{Gr}_{f}\) (Fig. 4). As can be seen, in logarithmic coordinates the dependence is linear and, consequently, can be described by an equation of the form

\[ \lg \mathrm{Nu}_{f}=\lg C+m\lg \mathrm{Gr}_{f}. \tag{2} \]

The value of \(m\) for each series of experiments was determined as the tangent of the angle of inclination of the straight line to the abscissa axis. By setting up and solving a system of equations (2) from the experimental data and the determined value of \(m\), we find \(C\).

Attempts were made to find the dependence of the coefficient \(C\) in the investigated interval on \(\frac{S}{D}\). For this purpose the obtained experimental values of the coefficients \(C\) were plotted in the coordinates \(C\) and \(\frac{S}{D}\) (Fig. 5). From Fig. 5 it is seen that the dependence is linear and, consequently, can be described by an equation of the form

\[ C=k+b\frac{S}{D}. \]

The value of \(b\), just as in the preceding case, is determined from the graph as the tangent of the angle of inclination to the abscissa axis (\(b=0.08\)). Substituting the value of \(b\) into the equation for each series of experiments and solving them, we find \(k=0.072\).

Similar attempts were made to determine the dependence of \(m\) on \(\frac{S}{D}\). In the investigated interval \(\frac{S}{D}=1.5\text{--}4.0\), the value of \(m\) is calculated from the equation

\[ m=0.38-0.02\frac{S}{D}. \tag{3} \]

Thus, the criterion relation for heat transfer in free convection in a vertical row in the interval \(\frac{S}{D}=1.5\text{--}4.0\) is given by the equation

Fig. 4.

Fig. 5.

\[ \mathrm{Nu}_f = C \mathrm{Gr}_f^m, \tag{4} \]

where \(C = 0.072 + 0.08 \dfrac{S}{D}\) (5), and \(m\) is determined from (3).

Since a considerable increase in the pitch leads to the fact that each tube will be cooled as a single rod, it should be expected that equation (4) will take the form of the known formula [5, 6] for single rods cooled under free convection,

\[ \mathrm{Nu}_f = 0.47 \mathrm{Gr}_f^{0.25}. \]

As is evident from formulas (3) and (5), the values of \(m\) and \(C\) tend to 0.25 and 0.47, respectively.

A comparison was made of the calculated values \(\alpha_k^p\) according to formula (4) with the experimental values \(\alpha_k^{\mathrm{op}}\). The comparison results showed that the maximum deviation of the calculated values from the experimental ones does not exceed \(\pm 5\%\).

REFERENCES

1. A. A. Gukhman. Physical Foundations of Heat Transfer. ONTI, 1934.
2. Yu. A. Korolenko. Izvestiya TPI, vol. 110, 1962.
3. G. I. Kruzhnilin, V. A. Shvab. Journal of Technical Physics, vol. 5, nos. 3 and 4, 1935.
4. S. S. Kutateladze, V. M. Borishanskii. Handbook of Heat Transfer. GEI, 1959.
5. I. M. Mikheeva. Teploenergetika, no. 4, 1956.
6. M. A. Mikheev. Fundamentals of Heat Transfer. GEI, 1956.