ON SOME CASES OF DISTRIBUTION OF THE COOLING SURFACE DURING STEAM CONDENSATION IN A HORIZONTAL TUBE BUNDLE

B. F. KALUGIN

(Presented by the scientific seminar of the Department of TPP)

This article presents the results of theoretical studies of the influence of the distribution of the cooling surface among passes on the heat-flux density \(q_0\) (in \( \mathrm{kW}/\mathrm{m}^2\)), averaged over the entire horizontal two-pass tube bundle, during condensation of water vapor without admixture of air for two cases: parallel and sequential admission of steam into the bundles of tubes of the passes.

The heat-flux density \(q_0\) for both cases can be determined from the formula

\[ q_0=q_2\cdot q_1\left[(1-\Delta)q_2+q_1\Delta\right]^{-1}, \tag{1} \]

where \(q_1\) and \(q_2\) are the heat-flux densities of the first and second passes, respectively, along the course of the water, \( \mathrm{kW}/\mathrm{m}^2\);

\(\Delta\) is the fraction of the total steam flow into the bundle that condenses on the tubes of the second pass.

The relation between the total cooling surface \(F_0\) and the cooling surface of the second pass \(F_2\) is found from the expression

\[ F_2/F_0=q_1(q_2/\Delta+q_1-q_2)^{-1}. \tag{2} \]

Dependencies (1) and (2) were used in further calculations. For fixed values of the velocity of the steam supplied to the bundle \(w_{\mathrm{p}}\) and its pressure \(p_{\mathrm{k}}\), tube length \(l\), cooling ratio \(m\), outside \(d_{\mathrm{n}}\) and inside \(d_{\mathrm{v}}\) tube diameters, water temperature at the entrance to the bundle \(T_1\), thickness of the fouling layer \(\delta_{\mathrm{z}}\), and known tube material, \(\Delta\) was varied. By calculation, \(q_1\) and \(q_2\) were determined under the condition of complete steam condensation in the bundle.

In the calculations from [1], [2], [3], relations were used to determine the heat-transfer coefficient from steam to the wall, the thermal resistance of the tube wall, and the heat-transfer coefficient from the wall to the cooling water. By transformations, expressions were obtained for \(q_1\) and \(q_2\) for both cases of steam condensation considered in the article. From formula (1), \(q_0\) was determined; from formula (2), \(F_2/F_0\) was determined. From the graph \(q_0=f(F_2/F_0)\), the optimal ratio \((F_2/F_0)_{\mathrm{op}}\), corresponding to the maximum value of \(q_0\), was determined, and then the optimal ratio \((F_1/F_2)_{\mathrm{op}}\), where \(F_1\) is the cooling surface of the first pass. One of the quantities \((m_{\mathrm{k}}, l, w_{\mathrm{p}}, p_{\mathrm{k}}, d_{\mathrm{n}}, d_{\mathrm{v}}, T_1, \delta_{\mathrm{z}})\) was varied, and again [[unclear: text continues on next page]]

analogous calculations were carried out. All calculations were performed for a bundle with an in-line arrangement of tubes; the pitch in the horizontal row was 32.5 mm. The steam entered the bundle dry and saturated.

Table 1 presents part of the calculations for the case of parallel admission of steam into the pass bundles; the tube bundles of the passes are arranged side by side, and the steam enters from above. The notation in the table is as follows: \(\lambda_z\) and \(\lambda_{\mathrm{st}}\) are the thermal-conductivity coefficients of the fouling and of the tube material; \(\Delta q_0\) is the excess of \(q_0\) at \((F_1/F_2)_{\mathrm{opt}}\) over \(q_0\) at \(F_1/F_2=1\), as a percentage of \(q_0\) at \(F_1/F_2=1\). It is seen from the table that, in a number of cases, \((F_1/F_2)_{\mathrm{opt}}=0.870\div0.888\). This is observed at \(p_k=0.1176;\ 0.196\) bar, less so at \(p_k=0.49\) bar, and is practically absent at \(p_k=0.882\) bar (calculations for this pressure are not given). \(\Delta q_0\) may reach 2.88%.

This effect is very close to the effect obtained when changing from an in-line arrangement of tubes to a staggered one. Calculations show that shifting the ratio \(F_1/F_2\) toward values less than 1, and increasing \(l\), causes a more intensive increase in the heat-transfer coefficient of the first pass \(k_1\) in comparison with the rate of growth of the heat-transfer coefficient of the second pass \(k_2\). This, in combination with the higher value of the logarithmic mean temperature difference between the steam and water in the first pass, leads under certain conditions to obtaining a maximum of \(q_0\) at \(F_1/F_2=0.785\div0.88\). This is observed at small \(m_k\), provided that the average temperature of the cooling water for the entire bundle lies in the zone where the physical properties of water (thermal-conductivity coefficient, coefficient of kinematic viscosity, Prandtl number) vary intensively. At large \(p_k\) and \(m_k\), the effect of shifting \((F_1/F_2)_{\mathrm{opt}}\) toward values less than 1 is small.

Small \(m_k\) and large total values of \(l\) may be observed in some types of multipass bundles with parallel admission of steam to the pass surfaces. In this case it is desirable to have the ratio of the cooling surfaces of the first half of the passes along the water flow and the second half within the limits \(0.785\div0.88\). The exergetic efficiency of the bundle \(\eta_{\mathrm{ex}}\) [4] at \((F_1/F_2)_{\mathrm{opt}}=0.785\div0.88\) (at considerable \(l\) and small \(m_k\)) will not be lower than \(\eta_{\mathrm{ex}}\) at \(F_1/F_2=1\), if the reduction of \(F_0\) in the case \((F_1/F_2)_{\mathrm{opt}}=0.785\div0.88\), relative to the case \(F_1/F_2=1\), is effected by reducing \(l\).

The ratios \(F_1/F_2\) corresponding to the maximum values of \(q_0\) were determined by calculation for the case of condensation of pure steam in a two-pass horizontal bundle, when the steam enters the pass bundles sequentially: first into the bundle of the second pass, then into the bundle of the first (the bundle of the second pass is located above the bundle of the first). In obtaining the dependences for \(q_1\) and \(q_2\), the influence of condensate flowing down from the tubes of the second pass on heat exchange in the bundle of the first pass was taken into account according to [5]. The form of the dependences for \(q_1\) and \(q_2\) is not given here. Calculations show that, during condensation of pure steam with its sequential admission into the pass bundles, the maximum values of \(q_0\) are observed at \(F_1/F_2=0.94\div1.02\). The calculations were performed for variants with the initial data \((p_k,\ T_1,\ w_p,\ l,\ d_n,\ d_v,\ m_k,\ \delta_z,\ \lambda_z,\ \lambda_{\mathrm{st}})\) indicated in Table 1. Deterioration of heat exchange in the tube bundle of the first pass shifts the values of \((F_1/F_2)_{\mathrm{opt}}\) toward 1. It is seen from [6] that the presence of air can shift the values of \((F_1/F_2)_{\mathrm{opt}}\) into the interval from 1.2 to 1.4.

CONCLUSIONS

1. During condensation of pure steam in a two-pass horizontal tube bundle with parallel admission of steam to the cooling surfaces

Table 1

passes the maximum heat-flux density $q_0$ is observed (in the case of small $m_k$ and large $l$) when the ratio between the cooling surfaces of the passes is $(F_1/F_2)_{\mathrm{opt}} = 0.785 \div 0.88$.

1. In the case of condensation of pure steam in a two-pass horizontal tube bundle, with steam entering the pass bundles in sequence, the maximum values of $q_0$ are observed at $(F_1/F_2)_{\mathrm{opt}} = 0.94 \div 1$. The presence of air gives $(F_1/F_2)_{\mathrm{opt}} = 1.2 \div 1.4$.

REFERENCES

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

2. I. N. Kirsanov. Condensation Installations. Moscow, “Energiya,” 1965.

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

4. A. M. Aksel’band, Z. P. Bilder, A. S. Yasnitskii. Exergetic efficiency of water–steam heat exchangers taking hydraulic resistances into account. Izvestiya vuzov, “Energetika,” 1970, No. 7.

5. V. P. Isachenko, A. F. Glushkov. Heat transfer during steam condensation on a horizontal tube and condensate flow from above. Teploenergetika, 1969, No. 7.

6. E. N. Shadrin, V. A. Bragin, B. F. Kalugin, Yu. A. Marakulin. On the influence of the distribution of the cooling surface by water passes on the heat-transfer coefficient during steam condensation in a horizontal tube bundle. Scientific Transactions of OMIIT, vol. 70, 1967.