HEAT TRANSFER FROM A HORIZONTAL TUBE BUNDLE TO AIR UNDER FREE-CONVECTION CONDITIONS

Yu. A. KORCHENKO

(Presented by Prof. G. I. Fuks, Doctor of Technical Sciences)

The problem of heat transfer from tube bundles under conditions of free convection is one of the least studied in the field of free convection.

Treatment of the differential equations describing the phenomenon by methods of similarity theory leads to the relation

\[ Nu=f\left(Gr;\ Pr;\ \frac{S_1}{D};\ \frac{S_2}{D}\right), \tag{1} \]

where \(D\) is the diameter of the rods, and \(S_1\) and \(S_2\) are the horizontal and vertical pitch of the rods in the bundle.

In order to find the specific form of relation (1) for a bundle of tubes with identical heat release, transferring heat to air, an experimental apparatus was built. Its main part consisted of brass calorimetric tubes with internal electric heaters. To reduce end losses, the tube length \(L\) was selected so that

\[ \frac{L}{D} \geq 15. \]

The heaters of all tubes of one diameter had resistances differing within \(\pm 0.5\%\). The surface of the tubes was coated with candle soot (degree of blackness \(\varepsilon = 0.952\) [4]). From horizontally arranged calorimetric tubes of the required diameter, a bundle of the required type with the prescribed horizontal pitch \(S_1\) and vertical pitch \(S_2\) was assembled in a special holder. At least 3/5 of the calorimetric tubes in each row of the bundle were provided with copper-constantan thermocouples made of wires 0.1 mm in diameter, laid along the inner wall of the brass tubes. The thermocouple junctions were led out to the surface of the tube at the middle of its length and soldered there.

Some of the tubes (not fewer than two in each experiment) also had thermocouples at the ends. When the bundle was assembled, the junctions of all thermocouples were directed upward [2]. The thermocouples were preliminarily calibrated to an accuracy of \(0.1^\circ\)C. In the experiments, the thermocouple emf was measured with a PPTR-1 potentiometer.

The heaters of the calorimetric tubes were connected in series. Their power supply was provided by alternating current through a sta-

Figure labels

Vertical axis: $Nu_f$

Horizontal axis: $Gr_f$

Legend:

• ● — tubes $\varnothing 6\ \mathrm{mm}$
• ▼ — tubes $\varnothing 10\ \mathrm{mm}$
• ■ — tubes $\varnothing 14\ \mathrm{mm}$
• ▲ — tubes $\varnothing 22\ \mathrm{mm}$
• ♦ — tubes $\varnothing 28\ \mathrm{mm}$

Curve labels:

• $\dfrac{s}{D}=1.82$
• $\dfrac{s}{D}=1.36$
• $\dfrac{s}{D}=1.28$
• $\dfrac{s}{D}=1.22$
• $\dfrac{s}{D}=1.169$
• $\dfrac{s}{D}=1.15$
• $\dfrac{s}{D}=1.082$

Fig. 1.

voltage stabilizer and an LATR-1 autotransformer. The power dissipated in the heaters was measured with a class 0.2 electrodynamic wattmeter.

The ends of the tubes were protected against heat loss by a 25 mm layer of glass wool. The difference between the temperature at the center and at the ends of the rods in the experiments did not exceed \(1^\circ\mathrm{C}\).

The model was placed at a height of 0.5 above the table, inside a casing measuring \(1000 \times 700 \times 600\), which protected it from the influence of air motion in the room.

In the experiments, the diameter and pitch of the tubes were measured, as well as the temperature of their surface \(t_w\), the temperature inside the casing \(t_f\), the temperature of the cold junction of the thermocouples \(t_0\), the room temperature \(t_{\mathrm{п}}\), and the power dissipated in the heaters.

All temperature and power measurements were carried out only after a steady-state thermal condition of the model had been rigorously established.

The heat-transfer coefficient from the tubes of the model was calculated from the relation

\[ \alpha = \frac{Q_k}{F(t_w - t_f)} \frac{\mathrm{kcal}}{\mathrm{m}^2\,\mathrm{h}\,^\circ\mathrm{C}}}, \]

where

\(F\)—the total lateral surface area of all tubes of the model,

\(t_w\)—the surface temperature of the hottest tube in the bundle. (For bundles of tubes with identical heat release, for example, rod-type shunts, this temperature is usually limiting.)

\(Q_k\)—the heat dissipated by convection,

\[ Q_k = Q_{\mathrm{el}} - Q_l - Q_p; \]

\(Q_{\mathrm{el}}\)—the heat released by the current in the heaters;

\(Q_l\)—the heat dissipated by radiation from the tubes of the bundle;

\(Q_p\)—the heat lost through the tube ends and through the current leads of the heaters,

\[ Q_p < 0.12 Q_{\mathrm{el}}. \]

On the basis of the experimental data obtained, the values of the Nusselt criterion \(Nu = \dfrac{\alpha l}{\lambda}\) and the Grashof criterion \(Gr = \dfrac{g l^3}{\nu^2}\,\beta\,\Delta t\) were calculated.

The tube diameter \(D\) was taken as the characteristic dimension, and the temperature of the air far from the bundle, \(t_f\), as the characteristic temperature ([1] [3] [5]).

The error in determining the heat-transfer coefficient was composed of the errors of the instruments, the error in measuring the dimensions of the model, and the error in calibrating the thermocouples. The maximum error possible in determining \(\alpha\) does not exceed 6% at tube temperatures above \(50^\circ\mathrm{C}\).

Three series of experiments were carried out on the apparatus:

1. Experiments with a single row of horizontal tubes.
2. Experiments with a multirow in-line tube bundle.
3. Experiments with a multirow staggered tube bundle.

The experiments with a single row of horizontal tubes were intended to establish the specific form of dependence (1) for this case and the minimum distance between tubes at which their influence on one another disappears.

In conducting the experiments, the following were varied:

1. Tube diameter \(D\), from 6 mm to 28 mm.
2. Relative pitch of the tubes in a row, \(\dfrac{S_1}{D}\), from 1.082 to 4.33.

1. The surface temperature of the most heated tube \(t_w\)—from \(25^\circ\mathrm{C}\) to \(240^\circ\mathrm{C}\).

The values of the Grashof criterion \(Gr_f\) in the experiments varied from 800 to 520000. About 250 experiments were carried out on the model, the results of which are presented in Fig. 1.

It is evident from Fig. 1 that, at small values of the relative pitch

\[ \frac{S_1}{D}<1.36 \]

the influence of neighboring tubes on convective heat transfer is felt at all values of the Grashof numbers, which substantially reduces the heat-transfer coefficient (the Nusselt criterion) in comparison with a single tube at the same values of \(Gr_f\). It is also obvious that at \(Gr_f \simeq 3200\) a change occurs in the regularity of the heat-transfer process. At \(Gr_f<3200\) the slope of the lines \(Nu_f=f(Gr_f)\) is different and depends on the relative pitch \(\frac{S_1}{D}\). At values \(Gr_f>3200\) the slope of the lines \(Nu_f=f(Gr_f)\) is the same for all \(\frac{S_1}{D}<1.36\).

At

\[ 1.36<\frac{S_1}{D}<1.82 \]

the influence of neighboring tubes on one another is felt only at \(Gr_f>3200\). In this case the influence of the relative pitch on the heat-transfer coefficient disappears. At \(\frac{S_1}{D}>1.82\), the influence of neighboring rods on one another ceases, and the heat-transfer process develops according to the laws of free convection of individual rods in an unbounded volume.

It is evident from Fig. 1 that \(\lg Nu_f=f(\lg Gr_f)\) is rectilinear for all \(\frac{S_1}{D}\). This makes it possible to seek the specific form of the function in the form

\[ Nu_f=CGr_f^n . \tag{2} \]

Fig. 2. Graph for determining \(C\) at \(Gr_f<3200\).

The values of \(C\) and \(n\) are different for different regimes. At \(Gr_f<3200\)

\[ n=\frac{1.14}{\left(\dfrac{S_1}{D}\right)^{1.45}}, \tag{3} \]

\(C\)—from the graph (Fig. 2).

At \(Gr_f>3200\), \(n=0.17\).

For

\[ \frac{S_1}{D}<1.36 \quad C=2.93\frac{S_1}{D}-3.16. \tag{4} \]

For

\[ 1.36<\frac{S_1}{D}<1.82 \quad C=0.82. \tag{5} \]

For all \(Gr_f\) at

\[ \frac{S_1}{D}>1.82 \]

\[ n=0.25,\quad C=0.47, \]

which agrees with the known dependence obtained by P. M. Mikheev ([1]; [3]).

The values of the heat-transfer coefficients calculated from formulas (2)—(5) differ from the experimental data by no more than 3%.

In carrying out the second and third series of experiments, the aim was to determine the dependence of the heat-transfer coefficient (the Nusselt criterion \(Nu_f\)) on the Grashof criterion \((Gr_f)\), the geometrical characteristics of the tube bundle—the relative horizontal pitch \(\dfrac{S_1}{D}\) and vertical pitch \(\dfrac{S_2}{D}\)—and the number of rows of tubes in the bundle \(n_2\).

In studying the in-line bundle, the following were varied:

1. The tube diameter \(D\)—from 6 to 19 mm.

2. The relative horizontal pitch of the tubes in a row \(\dfrac{S_1}{D}\)—from 1.5 to 3.5.

3. The relative vertical pitch of the rows of tubes \(\dfrac{S_2}{D}\)—from 1.5 to 3.5.

4. The surface temperature of the most strongly heated tube of the bundle \(t_c\), from 60 to 220.

5. The number of horizontal rows of tubes in the bundle \(n_2\)—from two to seven.

6. The number of tubes in a horizontal row \(n_1 = 5\).

The value of the Grashof criterion in the experiments varied from 1225 to 228000. More than 200 experiments were carried out on the model. It was found that, at pitches \(\dfrac{S_1}{D} < 2.0\) and \(\dfrac{S_2}{D} < 2.0\) (a compact bundle), the inner tubes of the bundle become sharply overheated in comparison with the tubes located on the surface of the bundle. Heat transfer from the inner tubes of a compact bundle is practically absent. Bundles with a relative pitch \(\dfrac{S_1}{D} \ge 2.0\) and \(\dfrac{S_2}{D} \ge 2.0\) were studied in detail. The results of the experiments are given in Fig. 3. It is seen from it that, as in the first series of experiments, at \(Gr_f \approx 3200\) there occurs a change in the regularity of the relation \(Nu_f = f(Gr_f)\). The experiments showed that the heat-transfer coefficient \((Nu_f)\) depends to the same degree on the values of \(\dfrac{S_1}{D}\) and \(\dfrac{S_2}{D}\), which makes it possible to take their influence on heat transfer into account jointly, i.e., in the form of a dependence on the product \(\dfrac{S_1}{D} \cdot \dfrac{S_2}{D}\).

The influence of the number of horizontal rows of tubes \(n_2\) on the heat transfer of an in-line bundle was studied on models with \(\dfrac{S_2}{D} = \dfrac{S_1}{D} = 2.5\). The results of these experiments are shown in Fig. 4. From the experiments it was found that, with an increase in the number of horizontal rows of tubes \(n_2\), the heat-transfer coefficient decreases up to \(n_2 = 6\). Beginning with the number of horizontal rows \(n_2 = 6\), the heat-transfer coefficient ceases to depend on \(n_2\).

As is seen from Figs. 3 and 4, the dependence \(\lg Nu_f = f(\lg Gr_f)\) for an in-line bundle is rectilinear. Therefore (1) can be written in the form

\[ Nu_f = C_z \left(\frac{S_1}{D}\,\frac{S_2}{D}\right)^k Gr_f^m . \tag{6} \]

for

\[ \frac{S_1}{D} \ge 2.0,\qquad \frac{S_2}{D} \ge 2.0 \quad \text{and} \quad Gr_f > 3200 \]

\[ m = 0.25,\qquad k = 0.34. \]

\[ C_n = 0.182 - 0.012\,(n_2 - 2) \quad \text{for } 2 \le n_2 \le 6 \]

for

\[ n_2 > 6,\qquad C_n = 0.134. \]

Fig. 3. In-line bundle

\[ 1 — \frac{S_1}{D} \times \frac{S_2}{D} = 2 \times 2,\qquad 2 — \frac{S_1}{D} \times \frac{S_2}{D} = 2 \times 2.5 = 2.5 \times 2, \]

\[ 3 — \frac{S_1}{D} \times \frac{S_2}{D} = 2 \times 3 = 3 \times 2,\qquad 4 — \frac{S_1}{D} \times \frac{S_2}{D} = 2 \times 3.5 = 3.5 \times 2, \]

\[ 5 — \frac{S_1}{D} \times \frac{S_2}{D} = 2.5 \times 3.5 = 3.5 \times 2.5, \]

\[ 6 — \frac{S_1}{D} \times \frac{S_2}{D} = 3 \times 3.5 = 3.5 \times 3, \]

\[ 7 — \frac{S_1}{D} \times \frac{S_2}{D} = 3.5 \times 3.5,\qquad 8 — \frac{S_1}{D} \times \frac{S_2}{D} = 2.5 \times 1.5. \]

Calculations by relation (6) give values of the heat-transfer coefficients differing from the experimental data by no more than 3.5%. The investigation of a staggered tube bundle was carried out by the same method as the investigation of the in-line bundle. In the experiments the following were varied:

1. Tube diameter \(D\)—from 6 to 19 mm.

2. Relative tube pitch: \(\dfrac{S_1}{D}\)—from 2.0 to 4.5, \(\dfrac{S_2}{D}\)—from 1.5 to 4.5.

3. Number of horizontal rows of tubes in the bundle \(n_2\)—from 2 to 7; the number of tubes in an even horizontal row was 4, and in an odd row—5.

4. The value of the Grashof criterion varied in the experiments from 1119 to 224,000.

As in the preceding experiments, in the investigations of the staggered tube bundle, at \(Gr_f \approx 3200\) a change was observed in the form of the dependence \(Nu_f = f(Gr_f)\).

“Close” in the experiments were tube bundles with $\dfrac{S_1}{D} < 2.5$ and $\dfrac{S_2}{D} < 2$.

Staggered tube bundles with $\dfrac{S_1}{D} \geq 2.5$ and $\dfrac{S_2}{D} > 2$ at $Gr_f > 3200$ were investigated in detail. The experiments showed that, for staggered tube bundles with the indicated parameters, the convective heat-transfer coefficient

Fig. 4. Influence of the number of horizontal rows of tubes $n_2$ on the heat transfer of an in-line tube bundle to air

$1—n_2=2,\quad 2—n_2=3,\quad 3—n_2=4,$

$4—n_2=5,\quad 4—n_2>5.$

depends only very weakly on the vertical tube pitch, $\dfrac{S_2}{D}$, and in calculations the influence of $\dfrac{S_2}{D}$ may be neglected.

The influence of the number of horizontal rows of tubes $n_2$ on the value of the heat-transfer coefficient (Nusselt criterion $Nu_f$) is felt up to $n_2=5$. Bundles with a number of rows greater than 5 have the same $Nu_f$ under otherwise equal conditions.

The heat-transfer coefficient of a staggered tube bundle at $\dfrac{S_1}{D} \geq 2.5$, $\dfrac{S_2}{D} > 2.0$, and $Gr_f > 3200$ can be calculated from the formula

\[ Nu_f = C_n \left( \frac{S_1}{D} \right)^{0.37} Gr_f^{0.25}. \tag{7} \]

Here,

\[ C_n = 0.241 - 0.012(n_2 - 2) \quad \text{for } 2 \leq n_2 \leq 5, \]

\[ C_n = 0.205 \quad \text{for } n_2 > 5. \]

The given relationship describes the experimental data quite well (Fig. 5). The deviations of the experimental data from the values found from formula (7) do not exceed 4%.

Legend:
● — \(D = 6\) mm
■ — \(D = 10\) mm
▼ — \(D = 19\) mm

Vertical axis: \(\dfrac{\alpha d}{\lambda}\big/ C\left(\dfrac{s}{d}\right)^{0.37}\)
Horizontal axis: \(Gr_f Pr_f\)

Fig. 5. Heat transfer from a staggered tube bundle to air under free-convection conditions.

Conclusion

The results are presented of an experimental study of the little-studied phenomenon of heat transfer from various bundles of horizontal tubes to air under conditions of free convection. A change in the form of the dependence \(Nu_f=f(Gr_f)\) at \(Gr_f \simeq 3200\) has been established. Calculation formulas are given for computing the coefficient of heat transfer by free convection to air from a horizontal row, staggered bundles, and in-line bundles of tubes. The formulas presented agree well with the experimental data.

References

1. Mikheev M. A. Fundamentals of Heat Transfer. GEI, 1956.
2. Kruzhilin G. I., Shvab V. A. Journal of Technical Physics, vol. 5, issues 3 and 4, 1935.
3. Mikheeva I. M. Thermal Power Engineering, No. 4, 1956.
4. Kutateladze S. S. and Borishanskii V. M. Handbook of Heat Transfer. GEI, 1959.
5. Tukhman A. A. Physical Foundations of Heat Transfer. ONTI, 1934.