ON THE CALCULATION OF THE COOLING SURFACE DURING STEAM CONDENSATION IN A SINGLE-PASS HORIZONTAL TUBE BUNDLE

B. F. KALUGIN

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

This article presents the derivation of a formula for determining the heat-flux density \(q\) averaged over a bundle \((\mathrm{kW}/\mathrm{m}^2)\) during steam condensation in a single-pass horizontal tube bundle, and the procedure for calculating the cooling surface \(F\) \((\mathrm{m}^2)\) using the obtained dependence for \(q\). Using the same assumptions as in [1], as well as the features of the calculations in the processing of the experimental data in [1], one can obtain

\[ q=\vartheta^{0.891} M^{0.791} N^{0.791} C_*^{0.791} d_{\mathrm{n}}^{\,\,} \frac{\lambda_k^{0.891}}{\mu_k^{0.1642}} [f(\Delta)]^{0.791}\cdot \Delta^{-0.0552}, \tag{1} \]

\(\vartheta\) — difference between the temperatures of the steam and the tube wall, deg;
\(G_k\) — total amount of steam entering the bundle, kg/sec;
\(\Delta\) — fraction of \(G_k\) condensed on the tubes of the bundle.

\[ f(\Delta)=\Delta\cdot[1-(1-\Delta)^{0.84}]^{-1}, \tag{2} \]

\[ M=19.5\,(w_{\mathrm{n}}^{2}\rho_{\mathrm{p}})^{0.08}\cdot(g\cdot d_{\mathrm{n}}\rho_k)^{-0.08}, \tag{3} \]

\(\rho_{\mathrm{p}}\) and \(\rho_k\) — densities of steam and condensate, respectively, kg/m\(^3\);
\(g\) — acceleration due to gravity, m/sec\(^2\);
\(d_{\mathrm{n}}\) — outside diameter of the tube, m;
\(w_{\mathrm{n}}\) — velocity of the steam flow, m/sec.

\[ N=a_1 G_k^{0.416}\cdot r^{0.208}\cdot d_{\mathrm{n}}^{-0.638}, \tag{4} \]

\(r\) — amount of heat given up to the cooling water during condensation of 1 kg of steam, kJ/kg;
\(\lambda_k\) — coefficient of thermal conductivity for the condensate forming on the tubes, W/(m·deg);
\(\mu_k\) — dynamic viscosity of the condensate, N·sec·m\(^{-2}\).
\(a_1\) and \(a_2\) — coefficients arising from conversion of the formulas [1] to the SI system.

\[ C_*=0.84\pi^{0.07}[r\rho_{\mathrm{p}}w_{\mathrm{n}}(t-d_{\mathrm{n}})]^{-0.07}, \tag{5} \]

\(t\) — pitch in the horizontal row of tubes, m.

Initially, a bundle with an in-line arrangement of tubes is considered. The number of horizontal rows in the bundle in deriving dependence (1) was determined by the formula

\[ n_{\mathrm{r}}=r\Delta\rho_{\mathrm{p}}w_{\mathrm{n}}(t-d_{\mathrm{n}})\cdot(q\pi d_{\mathrm{n}})^{-1}. \tag{6} \]

At a condensate temperature \(T_{\mathrm{k}}=278 \div 368^\circ\mathrm{K}\), the expression \(a_{2}^{\lambda_{\mathrm{k}}^{0.891}}\times \psi_{\mathrm{k}}^{-0.1642}\) can be written in the following form:

\[ a_{2}^{\lambda_{\mathrm{k}}^{0.891}}\cdot \psi_{\mathrm{k}}^{-0.1642} = b_{1}+b_{2}(T_{\mathrm{k}}-273), \tag{7} \]

\(b_{1}\) and \(b_{2}\) are constants. According to [1], \(T_{\mathrm{k}}=T_{\mathrm{p}}-0.75\vartheta\); \(T_{\mathrm{p}}\) is the steam temperature in \({}^{\circ}\mathrm{K}\). The heat-transfer coefficient from the tube wall to the water, \(\alpha_{\mathrm{v}}\), can be determined by the formula from [2]

\[ \alpha_{\mathrm{v}}= f'/2\cdot c_{p}C_{\mathrm{v}}\rho_{\mathrm{v}} \left[12\sqrt{f'/2}\left(Pr^{0.71}-1\right)+1\right]^{-1}, \tag{8} \]

\(c_{p}\) is the specific heat capacity of water, \(\mathrm{kJ}/(\mathrm{kg}\cdot\mathrm{deg})\);

\(C_{\mathrm{v}}\) is the velocity of water in the tube, \(\mathrm{m}/\mathrm{s}\);

\(Pr\) is the Prandtl number for water;

\(f\) is the friction coefficient;

\([\alpha_{\mathrm{v}}]\) is \(\mathrm{kW}/(\mathrm{m}^{2}\cdot\mathrm{deg})\);

\(\rho_{\mathrm{v}}\) is the density of water, \(\mathrm{kg}/\mathrm{m}^{3}\).

For turbulent flow (up to Reynolds number \(Re=400000\)), \(f/2\) is approximated as follows:

\[ \frac{f}{2}=b_{3}\left(b_{4}+\frac{100}{Re}\right)^{0.250}, \tag{9} \]

\[ b_{3}=0.0125; \]

\[ b_{4}=0.00006. \]

When \(\vartheta\) varies from 1.7 to 41 degrees, the following equality holds:

\[ \vartheta^{0.891}=1+0.663\vartheta. \tag{10} \]

For \(T_{\mathrm{v}}=278 \div 373^\circ\mathrm{K}\), \(Pr^{0.71}\) can be written in the following form

\[ Pr^{0.71}=15.8\cdot (T_{\mathrm{v}}-273)^{-0.5}. \tag{11} \]

\(T_{\mathrm{v}}\) should be taken as averaged over the entire bundle, \(T_{\mathrm{v}}=T_{\mathrm{p}}-\Delta t\). The sum of the thermal resistance of the tube wall with fouling, \(\sum R_{\mathrm{st}}\) \(\mathrm{m}^{2}\cdot\mathrm{deg}\cdot\mathrm{kW}^{-1}\), and \(1/\alpha_{\mathrm{v}}\) has the form:

\[ \frac{1}{\alpha_{\mathrm{v}}}+\sum R_{\mathrm{st}}= \]

\[ = \frac{ 12\sqrt{f'/2}\,Pr_{t\mathrm{p}}^{0.71} +\left(1-\frac{\Delta t}{T_{\mathrm{p}}-273}\right)^{0.5} \cdot \left(1-12\sqrt{f'/2}+\frac{f}{2}c_{p}C_{\mathrm{v}}\rho_{\mathrm{v}}\sum R_{\mathrm{st}}\right) }{ f'/2\left(1-\frac{\Delta t}{T_{\mathrm{p}}-273}\right)^{0.5} \cdot c_{p}C_{\mathrm{v}}\rho_{\mathrm{v}} }, \tag{12} \]

\(\Delta t\) is the mean logarithmic temperature difference between the steam and the cooling water, determined in the usual way;

\(Pr_{t\mathrm{p}}\) is the Prandtl number for water at \(T_{\mathrm{p}}\). Relation (12) can be given the following form:

\[ \frac{1}{\alpha_{\mathrm{v}}}+\sum R_{\mathrm{st}} = \left[ b_{3}c_{p}C_{\mathrm{v}}\rho_{\mathrm{v}} \left(1-\frac{\Delta t}{T_{\mathrm{p}}-273}\right)^{0.5} \right]^{-1} \cdot \left\{ 12b_{3}^{0.5} \left[ \left(\frac{100}{Re}\right)^{-0.125} \right. \right. \]

\[ \left. \left. -0.125b_{4}\left(\frac{Re}{100}\right)^{1.125} \right] Pr_{t\mathrm{p}}^{0.71} + (b_{3}c_{p}C_{\mathrm{v}})^{-1} \left[ \left(\frac{Re}{100}\right)^{0.250} -0.250b_{4}\left(\frac{Re}{100}\right)^{1.250} \right. \right. \]

\[ \left. \left. -12b_{3}^{0.5} \left[ \left(\frac{Re}{100}\right)^{0.125} -0.125b_{4}\left(\frac{Re}{100}\right)^{1.125} \right] +b_{3}c_{p}C_{\mathrm{v}}\rho_{\mathrm{v}}\sum R_{\mathrm{st}} \right\}. \tag{13} \]

An estimate shows that, in the first approximation, the terms with \(b_4\) may be neglected. We may write

\[ C_{\mathrm{v}}^{0,125}=b_5+b_6 C_{\mathrm{v}};\qquad C_{\mathrm{v}}^{0,250}=b_7+b_8 C_{\mathrm{v}}. \tag{14} \]

The constants for the intervals of variation \(C_{\mathrm{v}}=1\div 4\); \(C_{\mathrm{v}}=0,4\div 1\); \(C_{\mathrm{v}}=0,1\div 0,5\ \mathrm{m/sec}\), respectively, are:
\(b_5=0,93;\ 0,82;\ 0,72;\)
\(b_6=0,07;\ 0,177;\ 0,387;\)
\(b_7=0,85;\ 0,65;\ 0,5;\)
\(b_8=0,152;\ 0,346;\ 0,683.\)

After this, expression (13) can be reduced to the form

\[ \frac{1}{\gamma_{\mathrm{v}}}+\sum R_{\mathrm{st}} = \frac{Z_*+Z_{**}C_{\mathrm{v}}} {b_3 c_p \rho_{\mathrm{v}} C_{\mathrm{v}} \left(1-\frac{\Delta t}{T_{\mathrm{p}}-273}\right)^{0,5}}, \tag{15} \]

where \(Z_*\) and \(Z_{**}\) are functions of a number of quantities:

\[ Z_* = f_1(b_5,b_3,b_7,M_2,d_{\mathrm{v}},Pr_{\mathrm{tp}},\Delta t,T_{\mathrm{p}}); \]

\[ Z_{**}=f_2(b_6,b_3,b_8,M_2,d_{\mathrm{v}},Pr_{\mathrm{tp}},\Delta t,T_{\mathrm{p}},\sum R_{\mathrm{st}},\rho_{\mathrm{v}}). \]

For \(T_{\mathrm{v}}=275\div 373^\circ K\)

\[ M_2=\left(\frac{1}{\nu}\right)^{0,125} =5,35+0,013(T_{\mathrm{v}}-273); \tag{16} \]

\(\nu_{\mathrm{v}}\) is the coefficient of kinematic viscosity for water, \(\mathrm{m^2/sec}\);
\(d_{\mathrm{v}}\) is the internal diameter of the tubes, \(\mathrm{m}\). \(C_{\mathrm{v}}\) is determined from the formula

\[ C_{\mathrm{v}}=4l\cdot m'_k q d_{\mathrm{n}}\cdot (d_{\mathrm{v}}^2 \rho_{\mathrm{v}} r \Delta)^{-1}; \tag{17} \]

\(l\) is the tube length, \(\mathrm{m}\); \(m'_k=W:G_k\); \(W\) is the flow rate of cooling water, \(\mathrm{kg/sec}\). \(m'_k\) is the conventional cooling ratio: not all the steam \(G_k\) is condensed in the bundle. The actual cooling ratio \(m_k\) is understood as the ratio of \(W\) to the flow rate of steam \(\Delta\cdot G_k\) condensed in the bundle. At \(\Delta=1\), \(m'_k=m_k\).

\(\vartheta\) is found from the following equation:

\[ \vartheta=\Delta t-q(1/\alpha_{\mathrm{v}}+\sum R_{\mathrm{st}}). \tag{18} \]

The dependence for \(\vartheta\), obtained after the simultaneous solution of equations (1), (7), (10), (15), (17), (18), is substituted into the equation

\[ q=(\Delta t-\vartheta)(1/\alpha_{\mathrm{v}}+\sum R_{\mathrm{st}})^{-1}. \tag{19} \]

Instead of the second bracket in equation (19), we substitute expression (15). Finally, from equation (19) we obtain

\[ q= \frac{ \rho_{\mathrm{v}}S_{11}\varphi(\Delta)\cdot \left[(R_4\Delta t+R_3)f(\Delta t)\cdot S_{21} -0,238R_4Z_*\Delta d_{\mathrm{v}}^2 r\right] }{ \left[0,238R_1S_{11}Z_{**}\varphi(\Delta)+45000f(\Delta t)\right] 4m_k d_{\mathrm{n}}l }; \tag{20} \]

\[ \varphi(\Delta)=[f(\Delta)]^{0,791}\cdot \Delta^{-0,052}; \tag{21} \]

\[ S_{21}=0,05m'_k d_{\mathrm{n}}l; \tag{22} \]

\[ S_{11}=121\left[ \frac{ w_{\mathrm{p}}^{0,09}\cdot r^{0,138} }{ d_{\mathrm{n}}^{0,718}\nu_{\mathrm{p}}^{0,01}(t-d_{\mathrm{n}})^{0,07} } \right]^{0,791}; \tag{23} \]

\[ f(\Delta t)=\left[1-\Delta t(T_{\mathrm{p}}-273)^{-1}\right]^{0,5}; \tag{24} \]

\[ R_3=8,5+0,052(T_{\mathrm{p}}-273); \tag{25} \]

\[ R_4=5,601+0,0345(T_{\mathrm{p}}-273); \tag{26} \]

\[ Z_*= M_2 d_{\mathrm{v}}^{0,125} \left\{0,757b_5[Pr_{\mathrm{tp}}^{0,71}-f(\Delta t)] +0,3175b_7 f(\Delta t)M_2 d_{\mathrm{v}}^{0,125}\right\}; \tag{27} \]

\[ Z_{**}= M_2 d_{\mathrm{v}}^{0,125} \left\{0,757b_6[Pr_{\mathrm{tp}}^{0,71}-f(\Delta t)] +0,3175b_8 f(\Delta t)M_2 d_{\mathrm{v}}^{0,125}\right\} +52,5\rho_{\mathrm{v}}f(\Delta t)\sum R_{\mathrm{st}}; \tag{28} \]

\(v_{\text{p}}\) is the specific volume of steam, \(m^3/kg\); in equation (28) \([\sum R_{\text{st}}]\) is \(m^2\cdot deg/W\); in equation (20) \([q]\) is \(kW/m^2\). The dimensions of the other quantities in expressions (20)–(28) were indicated earlier in the text. Dependence (20) was checked by calculation. The value of \(q\) from equation (20) was compared with \(q\) obtained in the usual way: the heat-transfer coefficient from the steam to the tube wall was determined separately according to [1], \(\alpha_{\text{w}}\) was determined according to [3], and \(\sum R_{\text{st}}\) was determined, after which \(q\) was found. The discrepancy between the values of \(q\) determined by the different methods was \(2\)–\(2.8\%\). The check was carried out at different \(G_k\), with \(d_{\text{v}} = 14\)–\(28\ mm\) and a tube-wall thickness of \(1\ mm\); the steam pressure did not exceed \(0.9\ bar\), and the flow regime of the cooling water was turbulent. In the calculations it was assumed that practically pure steam enters the bundle [1]. The entire cooling surface participates in the operation.

To determine the required cooling surface, the following quantities must be specified (some of them may be selected): \(G_k\), steam pressure, \(d_{\text{v}}\), \(d_{\text{n}}\), \(t\), \(l\), the bundle width or \(w_{\text{p}}\), the wall material and data on tube fouling, \(W\) or \(m'_k\), \((m'_k)\), \(\Delta\), \(T_1\) (the temperature of the cooling water at the inlet to the bundle), and the required temperature difference \(T_{\text{p}} - T_2\), where \(T_2\) is the temperature of the cooling water at the outlet from the bundle. The dryness fraction of the steam at the inlet to the bundle must be specified. First, \(\Delta t\), \(\varphi(\Delta)\), \(S_{19}\), \(S_{21}\), \(f(\Delta t)\), \(R_3\), \(R_4\), \(T_{\text{v}} = T_{\text{p}} - \Delta t\), \(Pr_{t_{\text{p}}}\), \(Z_*\), \(Z_{**}\) are determined; then \(q\) is found from equation (20). After this, \(C_{\text{v}}\) is calculated from equation (17), and, if required, the constants \(b_5\) and \(b_7\) in expression (27), \(b_6\) and \(b_8\) in expression (28), \(Z_*\), \(Z_{**}\), and \(q\) are refined. The repeated calculation is very simple, since it consists of only three operations. The cooling surface is determined by the formula

\[ F=\frac{\Delta G_k r}{q}, \tag{29} \]

and the total number of tubes is determined from the expression

\[ n_{\text{tr}}=\frac{F}{\pi d_{\text{n}} l}. \tag{30} \]

The number of horizontal rows is calculated from expression (6). With a staggered arrangement of tubes in the bundle, in expression (23) \(125.2\) should be substituted instead of \(121\); by \(t-d_{\text{n}}\) one should understand the distance between the tubes in the first horizontal row. Calculations show that determining \(F\) by the proposed method requires little labor compared with existing methods, since the latter consist of trial calculations of \(F\), which is assigned preliminarily and then refined, with complicated calculations repeated at each refinement step.

Conclusions

1. A formula has been obtained for determining the heat-flux density during steam condensation on the tubes of a horizontal single-pass bundle, making it possible to analyze the operation of the bundle as a function of various factors.

2. A method is given for calculating the required cooling surface of a single-pass horizontal tube bundle, characterized by comparatively low labor intensity and a sufficient degree of accuracy.

References

1. L. D. Berman. Approximate method for calculating heat transfer during steam condensation on a bundle of horizontal tubes. Teploenergetika, 1964, No. 3.

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

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