On Determining the Heat Flux Density During Vapor Condensation in a Horizontal Tube Bundle

B. F. Kalugin

(Presented by the scientific seminar of the Department of Thermal Power Engineering)

In the usual method for determining the required cooling surface for the condensation of a specified amount of steam, the calculation is carried out according to the following scheme: the cooling surface is prescribed and a checking calculation is performed, in which the thermal resistances are determined for heat transfer from the steam to the tube wall, through the wall itself, and in heat transfer from the wall to the water; then the heat-transfer coefficient is found and the dimensions of the cooling surface are checked. The calculation is carried out by the method of successive approximations: the temperature of the cooling water at the outlet from the bundle is prescribed, and then it is checked. The calculation becomes cumbersome and laborious, especially when the number of tubes in the bundle changes. The formula recommended in [6] for the heat-transfer coefficient can be applied at a cooling-water temperature not exceeding 35°C.

In the present work, a relation is given for the heat flux density over a wider temperature range. A calculation using it is simpler and requires little labor. The heat flux density is expressed through quantities that are used in analyzing the operation of condensers. In deriving the formula for the heat flux density, use was made of the relation from [1] for determining the heat-transfer coefficient from the wall to the water, and of the relations from [2] for the cooling surface and the heat-transfer coefficient, averaged over the bundle surface, from the steam to the tube wall. In deriving the formula, a number of fractional-power dependences of the physical parameters of water on temperature and of the friction coefficient in a round tube on the temperature and velocity of water according to [1] were approximated by polynomials of the first and second degree. A two-pass bundle with an in-line arrangement of tubes is considered; steam is supplied to the passes in parallel. The specified quantities are the water temperature at the inlet to the bundle \(T_1\), the cooling ratio \(m_k\), the steam flow rate \(G_k\), the saturation temperature \(T_s\), the pressure at which condensation occurs, the specific volume of the steam entering the bundle \(v_p\), the steam velocity \(\omega_p\) (chosen by us or determined from certain dimensional conditions), the pitch in the horizontal row \(t\), the outside \(d_n\) and inside \(d_v\) diameters of the tubes, their length \(l\), the thermal-conductivity coefficient of the tube material \(\lambda\), the amount of heat given up to the water by each kilogram of steam \(r_k\), and the fraction of the total steam flow condensing in the second pass \(\Delta\).

It is assumed that in each pass the steam condenses completely, and that all tubes without exception take part in operation. On the steam side, here—

the same assumptions are made as in [2]. For the heat-flux density of the first pass, the following formula was obtained:

\[ q_1 = \frac{\rho_{\mathrm{w}1} S_{19}\left[(R_4\Delta t_1 + R_3)S_{21} - 0.238 R_4 Z'_1(1-\Delta)d_{\mathrm{v}}^2 r_{\mathrm{k}}\right]} {\left(0.238 R_4 S_{19} Z'_2 + 45000\right)4m_{\mathrm{k}} d_{\mathrm{n}} l}; \tag{1} \]

for the second pass:

\[ q_2 = \frac{\rho_{\mathrm{w}2} S_{19}\left[(R_4\Delta t_2 + R_3)S_{21} f(\Delta t_2) - 0.238 R_4 Z''_1 \Delta d_{\mathrm{v}}^2 r_{\mathrm{k}}\right]} {\left[0.238 R_4 S_{19} Z''_2 + 45000 f(\Delta t_2)\right]4m_{\mathrm{k}} d_{\mathrm{n}} l}; \tag{2} \]

for the entire bundle:

\[ q_0 = \frac{q_2 \cdot q_1}{(1-\Delta)q_2 + \Delta q_1}. \tag{3} \]

When the number of tubes is the same, and likewise \(d_{\mathrm{n}}\), \(d_{\mathrm{v}}\), \(w_{\mathrm{p}}\), \(l\), and \(t\) are the same for the passes, the calculation is carried out by (2); in this case \(\Delta = 1\), \(l \cdot 2\), and \(\Delta t_2\) is taken for the entire bundle.

In these formulas, \(\rho_{\mathrm{w}1}\) and \(\rho_{\mathrm{w}2}\) are the density of water in the corresponding pass; \(\Delta t_1\) and \(\Delta t_2\) are the logarithmic mean temperature differences for the corresponding pass, determined from the usual relationships, which, for known \(\Delta\), is not difficult to do.

\[ R_3 = 8.5 + 0.052(T_s - 273). \tag{4} \]

\[ R_4 = 5.601 + 0.0345(T_s - 273). \tag{5} \]

\[ S_{21} = 0.05 m_{\mathrm{k}} d_{\mathrm{n}} l. \tag{6} \]

\[ S_{19} = 121\left[ \frac{w_{\mathrm{p}}^{0.09} r_{\mathrm{k}}^{0.138}} {d_{\mathrm{n}}^{0.718}\sigma_{\mathrm{p}}^{0.01}(t-d_{\mathrm{n}})^{0.07}} \right]^{0.791}. \tag{7} \]

\[ Z'_1 = M_1 d_{\mathrm{v}}^{0.125}\left[0.757C_5 f(\mathrm{Pr}_1) + 0.3175C_7 d_{\mathrm{v}}^{0.125} M_1\right]. \tag{8} \]

\[ Z'_2 = M_1 d_{\mathrm{v}}^{0.125}\left[0.757C_6 f(\mathrm{Pr}_1) + 0.3175C_8 d_{\mathrm{v}}^{0.125} M_1\right] + 52.5\rho_{\mathrm{w}1}\sum R_{\mathrm{st}}. \tag{9} \]

\[ Z''_1 = M_2 d_{\mathrm{v}}^{0.125}\left[0.757C_5 f(\mathrm{Pr}_s) + 0.3175C_7 f(\Delta t_2) M_2 d_{\mathrm{v}}^{0.125}\right]. \tag{10} \]

\[ Z''_2 = M_2 d_{\mathrm{v}}^{0.125}\left[0.757C_6 f(\mathrm{Pr}_s) + 0.3175C_8 f(\Delta t_2) M_2 d_{\mathrm{v}}^{0.125}\right] + \]

\[ + 52.5\rho_{\mathrm{w}2} f(\Delta t_2)\sum R_{\mathrm{st}}. \tag{11} \]

\[ M_1 = 5.35 + 0.013(T_s - \Delta t_1 - 273). \tag{12} \]

\[ M_2 = 5.35 + 0.013(T_s - \Delta t_2 - 273). \tag{13} \]

\[ f(\mathrm{Pr}_1) = \mathrm{Pr}_1^{0.71} - 1. \tag{14} \]

\[ f(\Delta t_2) = \left(1 - \frac{\Delta t_2}{T_s - 273}\right)^{0.5}. \tag{15} \]

\[ f(\mathrm{Pr}_s) = \mathrm{Pr}_s^{0.71} - f(\Delta t_2). \tag{16} \]

\[ \mathrm{Pr}_1^{0.71} = 15.8(T_s - \Delta t_1 - 273)^{-0.5}; \tag{17} \]

\[ \mathrm{Pr}_s^{0.71} = 15.8(T_s - 273)^{-0.5}. \tag{18} \]

Dimensions of the quantities:

\[ [q_2] = [q_1] = [q_0] = \mathrm{kW}/\mathrm{m}^2; \]

\[ [t] = [l] = [d_{\mathrm{v}}] = [d_{\mathrm{n}}] = \mathrm{m}; \qquad [G_{\mathrm{k}}] = \mathrm{kg}/\mathrm{s}; \]

\[ [\rho_{\mathrm{w}1}] = [\rho_{\mathrm{w}2}] = \mathrm{kg}/\mathrm{m}^3; \qquad [r_{\mathrm{k}}] = \mathrm{kJ}/\mathrm{kg}; \]

\[ [w_{\mathrm{p}}] = \mathrm{m}/\mathrm{s}; \qquad [\sigma_{\mathrm{p}}] = \mathrm{m}^3/\mathrm{kg}; \qquad [T_s] = [T_1] = {}^\circ K. \]

When calculating the thermal resistance of a wall with deposits, the wall and deposit thicknesses in \(\sum R_{\rm st}\) should be substituted in m, and the thermal-conductivity coefficients in \({\rm W}/({\rm m}\cdot{\rm deg})\). For the water-velocity intervals \(C_{\rm v}=0.1\div0.5;\ 0.4\div1;\ 1\div4\ {\rm m/sec}\), the values of the constants are, respectively,

\[ C_5=0.72;\ 0.82;\ 0.93; \]

\[ C_7=0.5;\ 0.65;\ 0.85; \]

\[ C_6=0.387;\ 0.177;\ 0.07, \]

\[ C_8=0.683;\ 0.346;\ 0.152. \]

The correctness of the choice of constants is checked by determining the velocities of the water motion.

For the first pass,

\[ C_{\rm v1}=\frac{4\cdot l m_{\rm k} d_{\rm n} q_1}{d_{\rm v}^{2}\rho_{\rm v1}(1-\Delta)r_{\rm k}}. \tag{19} \]

For the second pass,

\[ C_{\rm v2}=\frac{4\cdot l m_{\rm k} d_{\rm n} q_2}{d_{\rm v}^{2}\rho_{\rm v2}\Delta r_{\rm k}}. \tag{20} \]

When the heat-flux densities and steam flow rate are known, the cooling surface is easily determined both as a whole and by passes.

The number of tubes in the first pass:

\[ n_1=\frac{(1-\Delta)G_{\rm k}r_{\rm k}}{q_1\pi d_{\rm n}l}. \tag{21} \]

The number of tubes in the horizontal row of the first pass:

\[ n_{1{\rm r}}=\frac{(1-\Delta)G_{\rm k}v_{\rm p}}{w_{\rm p}l(t-d_{\rm n})}. \tag{22} \]

For the second pass they are determined analogously; instead of \((1-\Delta)\), \(q_1\), one should substitute only \(\Delta\), \(q_2\).

With a staggered arrangement of tubes, \(S_{19}\), determined from (7), must additionally be multiplied by 1.035; in this case, in (7) and in (22), by \(t-d_{\rm n}\) one should understand the distance between tubes in the first row.

Relations (1), (2), and (3) were checked by calculation: the heat-flux density determined from (1) and (2) was compared with the heat-flux density obtained in the usual way—by separately determining the heat-transfer coefficients from the steam to the wall according to [2], from the wall to the water according to [3] (the relation in [3] is also recommended in [5]), and the thermal resistance of the wall, after which the overall heat-transfer coefficient was determined. The discrepancy did not exceed \(\pm 2.8\%\). The calculations were carried out as applied to condenser tubes [4] and to the operating conditions of condenser tube bundles when transferred to operation with a degraded vacuum, down to a pressure of about 0.9 bar. The strict limits of the article do not allow the inclusion of material illustrating the considerable reduction in the work required when using relations (1), (2), (3).

Conclusions

1. Formulas have been obtained for determining the heat-flux densities averaged over the passes and over the entire bundle during steam condensation in a two-pass horizontal tube bundle with parallel steam supply by passes.

1. A new method is proposed for determining the required cooling surface as applied to a two-pass bundle with parallel supply of the condensing steam.

2. Formulas (1), (2), and (3) make it possible to analyze the operation of the entire bundle as a function of various factors.

REFERENCES

1. G. P. Petersikh. Friction and heat transfer in turbulent flow. Khimicheskaya promyshlennost’, No. 8, 1954.

2. L. D. Berman. An approximate method for calculating heat transfer during steam condensation in a horizontal tube bundle. Teploenergetika, No. 3, 1964.

3. B. S. Petukhov, V. V. Kirillov. On heat transfer in turbulent flow of liquid in tubes. Teploenergetika, No. 4, 1958.

4. I. N. Kirsanov. Condensing Installations. Energiya, 1965.

5. V. P. Isachenko, V. A. Osipov, A. S. Sukomel. Heat Transfer. Energiya, 1965.

6. L. D. Berman. Heat-transfer coefficient of a surface condenser. Izvestiya VTI, No. 2, 1952.