ISSN 0040-6015, Thermal Engineering, 2006, Vol. 53, No. 2, pp. 128–133. © Pleiades Publishing, Inc., 2006. Original Russian Text © N.D. Agafonova, M.A. Gotovskii, I.L. Paramonova, 2006, published in Teploenergetika.

Comparative Analysis of Correlations for Calculating Subcooled Boiling Heat Transfer N. D. Agafonova*, M. A. Gotovskii**, and I. L. Paramonova** *St. Petersburg State Technical University, ul. Politekhnicheskaya 29, St. Petersburg, 195251 Russia **OAO NPO Central Boiler and Turbine Institute, ul. Politekhnicheskaya 24, St. Petersburg, 194021 Russia

Abstract—Results of calculations of the tube wall temperature under flow boiling of water subcooled relative to the saturation temperature are presented. These calculations are performed using different calculation techniques. The comparison and the analysis of the calculation techniques were conducted with the use of experimental data obtained by several authors. DOI: 10.1134/S0040601506020091

When a liquid flows in a heated channel, besides forced convection, different regimes of boiling—from nucleate to film—can be observed along the channel length, depending on the heat flux density and other regime parameters. At rather high heat flowrates, boiling can appear if a bulk of fluid has a temperature tl that is lower than the saturation temperature ts. Such a type of boiling is called surface boiling (or subcooled boiling). Naturally, the temperature of the heating surface (wall temperature) tw is higher than ts. Figure 1 clearly demonstrates the change in the dependence of heat flowrate q on tw with an increase in the latter. Using this figure, we will try to show what problems arise while developing calculation techniques for determining temperature regime of the wall under subcooled boiling. As long as the wall temperature is lower that the saturation temperature ts, the correlation between the heat flowrate q and tw is determined by line 1, which corresponds to the forced convection regime. With an increase in the heat flowrate, the wall temperature increases and reaches a temperature of the boiling incipience tb.i at which vapor appears at the wall. Vapor generation can encourage somewhat an increase in the heat transfer rate; the q(tw) curve appears to be above the curve 1. With an increase in tw and q, there are two possible variants of further behavior of the surfaceboiling curve, i.e., it can permanently lie above the pool-boiling curve 2 (in the first alternative, curve 3) or it can intersect this curve (the second alternative, curve 4). Finally, a third alternative of transition from curve 1 to boiling curve 2 is possible, which occurs essentially in the point of intersection of these curves. It is apparent that, with an increase in the wall temperature, curves 3 and 4 should coincide with curve 2. Under subcooled-liquid boiling, vapor bubbles formed in the wall layer are condensed while entering a cold core of the flow. Hence, boiling at a heated wall is

combined with a forced convection of a single-phase liquid far from this wall and vapor condensation at a boundary of the boiling wall layer and the cold core. The intensity of the vapor formation at the wall depends on the liquid superheating there, while the condensation process is determined by the liquid subcooling relative to the saturation temperature. The dimensions of the region, which experiences disturbing effect of the evaporation process, depend on the liquid subcooling and the heat flowrate level. The higher the subcooling is, the thinner the layer where the boiling process occurs. With a small amount subcooling, vapor bubbles detach from the heating surface and are condensed in

128

q

2 4

3 qexp 1 t10

ts tb.i

td.b

tw

Fig. 1. Qualitative diagram of the wall-temperature change under transition from forced convection to surface boiling. (1) Forced convection, (2) pool boiling curve, (3) first version of surface boiling curve, (4) second version of surface boiling curve; points correspond to the wall-temperature values for each of the possible alternatives of boiling-curve behavior.

COMPARATIVE ANALYSIS OF CORRELATIONS

the flow. With high subcooling, vapor bubbles do not reach the departure radius and, as the experimental data show, the processes of liquid evaporation into the bubble and vapor condensation from it alternate in the course of bubbles’ oscillations without their departure from the surface. Here, we mean the case when the heating-surface temperature does not essentially depend on the mass flowrate and the temperature of the liquid, i.e., it is determined only by heat flux density, as developed surface boiling. The beginning of developed surface boiling can be approximately determined from the following equation, given in [1]: ∆t d.b = t s – t l = q/α conv – q/α boil ,

(1)

where αconv and αboil are the heat transfer coefficients under forced convection and boiling, respectively. It is apparent, however, that such an approximate definition corresponds to the simplified first alternative of the surface-boiling curve given above. Actually, mutual influence of these two heat transfer regimes takes place, which is accounted for in the correlations proposed in [2, 3], and, strictly speaking, as is seen from Fig. 1, the boundary between these two regimes should correspond to higher heat flowrates. Sometimes, a section of intense vapor generation is assumed as such a boundary. It is determined by means of the following correlations, proposed in [4] and refined later in [5]: x d.b = – 0.0022qdcp /( λ l r ); Pe = ρwc p /λ l < 7 × 10 ;⎫ ⎬(2) 4 ⎭ x d.b = – 145q/ ( ρwr ); Pe ≥ 7 × 10 , 4

where ρw is the mass flowrate; d is the channel equivalent diameter; cp, λl, and r are the specific heat, the heat conductivity, and the heat of evaporation, respectively; and xd.b is the relative enthalpy of the flow at a boundary point. With such a definition, however, the condition of the independence of the wall temperature on ρw and tl is not satisfied. Reliable determination of the wall temperature tw during boiling in channels and tubes is important from the standpoint of efficient operation of power-engineering equipment. It is significant for provision of their safe operation, as well. In our paper, we present the results of a comparison of the wall temperature values, which were calculated by means of the KORSAR code that uses Chen’s model [2] for determining heat transfer coefficient α, with those obtained with the use of other calculation techniques. The correlations for boiling heat transfer based on Chen’s model are widely used in Western calculation codes, in particular, in one of the most popular, the RELAP5 code. This model is applied both for saturated liquid boiling and subcooled-liquid (surface) boiling. THERMAL ENGINEERING

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Chen’s formula for boiling heat transfer is based on theoretical Forster–Zuber correlation for pool boiling obtained in 1955 [6]. In that work, the authors developed a model of bubble growth under the assumption that the main phase that determines heat transfer intensity is the initial phase of the bubble growth rather than the phase of bubble motion at the instant of its departure. Without going into detail as to whether this assumption is true or not, we can emphasize that it gives a value of the exponent n in the exponential function q ≈ ∆tn that is close to 2, although in most of the later exponential correlations the value of n was close to 3. The former value is more correct in the case of combination of high heat flowrates and high subcoolings, when the dependence q(∆t) is weaker than the cubic one. This problem was analyzed in detail in [7, 8]. At the same time, for relatively small heat flowrates, the dependence of the heat transfer coefficient (or the heat flowrate) on ∆t is stronger than it follows from Chen’s correlation. This formula, which was derived from theoretical postulates, is rather complicated, which makes it not very convenient for practical applications. To illustrate this statement, it is expedient to present this formula in a fully expanded form. Modified for use in the case of subcooled boiling [2], it is as follows: t w – t s⎞ - + α mac F. α = α mic S min ⎛ 1, ------------⎝ t w – t l⎠

(3)

Here, the first term describes heat transfer due to microconvection caused by the motion of bubbles, which are formed under boiling in the wall layer, (αmic), with an account of microconvection decrease with an increase in the two-phase mixture velocity (the boiling-suppression function S is responsible for this effect). The second term describes the contribution of macroconvection, (αmac), into two-phase flow heat transfer taking into account macroconvection enhancement in the twophase mixture (convection-enhancement function F). The above parameters can be calculated with the use of the following correlations [9]: λ l c pl ρ l = 0.00122 -------------------------------------0.5 0.29 0.24 0.24 σ µl r ρv 0.79 0.45

α mic

= ( tw – ts )

0.24

( p s.w – p )

0.49

0.75

;

0.4 λ 0.8 c pl µ l⎞ - ----l ; α mac = 0.023 ( Re ) m ⎛ ---------⎝ λl ⎠ d – 4 ρ l w l ( 1 – ϕ )d 1.25 -F ; ( Re ) m = 10 --------------------------------µl

⎧ 1.0; X tt ≤ 0.10 F = ⎨ 0.736 ⎩ 2.35 ( X tt–1 + 0.213 ) ; –1

100 > X tt > 0.10; –1

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AGAFONOVA et al.

Comparison of boiling heat transfer parameters determined by means of Chen’s correlation and with the use of correlations developed on the basis of generalization of experimental data (∆t is the temperature difference) p, MPa 0.1

0.5

3.0 7.0

∆t = tw – ts , αChen , K kW/(m2 K) 10 20 30 10 20 30 10 20 10

– 15.9 22.8 – 27.8 41.3 40.0 83.6 56.8

0.7αp.b , kW/(m2 K)

qChen , kW/m2

qTsKTI/0.7 , kW/m2

qlin , kW/m2

0.7qlin , kW/m2

qexp , kW/m2

– 21.8 53.8 – 46.8 116.0 32.3 162.0 168.0

– 318 684 – 456 1240 400 1672 568

– 436 1614 – 936 3490 323 3640 1680

– 585 1170 – 830 1540 815 2340 1860

– 410 820 – 580 1080 570 1640 1300

– 350 1000 – 800 1600 900 1700 1200

x 0.9 ρ 0.5 µ 0.1 –1 X tt = ⎛ -----------⎞ ⎛ -----l ⎞ ⎛ -----l ⎞ ⎝ 1 – x⎠ ⎝ ρ v⎠ ⎝ µ v⎠ (the Lockhart–Martinelli parameter), 1.14 – 1

⎧ [ 1 + 0.12 ( Re ) m ] ; S = ⎨ –1 ⎩ [ 1 + 0.42 ( Re ) 0.78 m ] ;

( Re ) m < 32.5; 32.5 < ( Re ) m < 70,

where ρl, cpl, λl, and µl are the density, the specific heat, the heat conductivity, and the dynamic viscosity coefficient of the liquid; ρv and µv are the density and dynamic viscosity coefficients of the vapor; ts and tw are the saturation (at a pressure p in the flow) and the wall temperatures; d is the equivalent diameter of a channel; ps.w is the saturation pressure that corresponds to the wall temperature; ϕ is the void fraction of the twophase mixture; and wl is the liquid velocity. The use of such complicated correlations is justified only if they provide a high calculation accuracy, which, as has been shown earlier, is not the case in reality. In proper time, this fact was pointed out by Zuber, who is one of the authors of the correlation that was chosen as the basis of Chen’s formula. Therefore, direct use of this formula for heat-design calculations is hardly advisable at present. Below, we make some principal comments regarding Chen’s correlation. The boiling-suppression factor S in the correlation does not reflect the actual effect that forced convection has on the boiling process. For this reason, factor F, which is connected with S, cannot be considered as the parameter that determines the actual effect of the two-phase flow velocity on heat transfer as well, although its physical meaning is substantiated to a greater extent. The above factors neglect a very important parameter that determines interaction between the boiling and the forced convection, i.e., heat flowrate q. Therefore, it is considerably more convenient to use interpolation formulas, some of which will be given below. If we

assume that the forced convection velocity is rather small and we can neglect the contribution of the forced convection into the total heat transfer, then it is possible to consider only the part of Chen’s correlation that determines boiling heat transfer. In the table, we find the heat transfer coefficients, which were calculated by means of Chen’s formula (αChen) and with the use of Borishanskii’s correlation corrected for the case of boiling in channels (0.7αp.b). We can also find in the table the values of the heat flowrates calculated for these two cases (qChen and qTsKTI/0.7). In the same table, the values of heat flowrates calculated using the linear dependence proposed by Adiutori [10], q lin = A∆t – B

(4)

are presented also. Here, coefficients A and B were adopted for water as a function of pressure [11] and multiplied by a correction factor of 0.7 (by analogy with the formula for the heat transfer coefficient). The use of (4) enables us to considerably simplify calculations of the wall temperature under boiling. The last column gives orienting experimental data taken from [12–16]. We see from the table that, at high pressures, the range of heat flowrates is somewhat more extended than necessary for boiling heat transfer calculations in a VVER-type reactor. However, just such a wide range of q enables us to draw the following conclusions: (i) for relatively low heat flowrates, the part of Chen’s formula that determines boiling heat transfer gives considerably underestimated results; (ii) for high heat flowrates, Chen’s formula gives quite acceptable results, while the exponential correlation appears to be clearly groundless; and (iii) the best results are provided by the corrected linear correlation (4), while calculating heat flowrate in the entire range of its changing. The above results probably show one of the reasons designers prefer to use Chen’s formula. Another reason THERMAL ENGINEERING

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COMPARATIVE ANALYSIS OF CORRELATIONS

is, apparently, as follows. In Chen’s formula, to account for the effect the vapor phase has on heat transfer, we use the same parameters that are usually applied in hydraulic calculations. The error in the calculations of the nucleate boiling term appears to be not very significant at low heat flowrates because, in this range, the convective term is large. Thus, we can conclude that the use of Chen’s formula, which is very inconvenient for calculations due to its awkwardness and many fractional exponents, provides no gain in calculation accuracy. It is justified to replace it by a simpler one to improve the quality of calculations, as well as to speed them up. In our work, for the calculations of the wall temperature under subcooled boiling, besides Chen’s formula, we will consider further the interpolating correlation of TsKTI [17] and the method recommended by Kreit and Black [18]. The TsKTI correlation written for heat flowrates is as follows: 2

q =

2

q conv + q boil ,

(5)

where qconv = αconv(tw – tl) and qboil = αboil(tw – ts). At Reynolds numbers Re = wlρld/µl > 104, αconv is calculated by means of the following formula: λ 0.8 0.4 α conv = ----l 0.023Re Pr ; d µ l c pl - is the Prandtl number. Pr = ---------λl

0.7

0.14

+ 1.37 × 10 p ) = Cq , –2

2

0.7

where p is the pressure in MPa. Thus, the wall temperature sought for is 0.3

q boil -, t w = t l + -------C

if

α boil = α p.b

(6)

(below, it is designated as tTsKTI), or 0.3

q boil - , if αboil = αp.b tw = tl + -------C

(7)

(below, it is designated as tTsKTI/0.7), where qboil =

q – α conv ( t w – t l ) . 2

2

2

Under subcooled liquid boiling, Kreit and Black [18] recommend determining the wall temperature of a channel using the principle of heat flowrate superposiTHERMAL ENGINEERING

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tcalc, °C

300

200 1 2 3 4

100

0

100

200

2006

300 texp, °C

Fig. 2. Comparison of the calculated and experimental values of the wall temperature under subcooled boiling. (1) tTsKTI/0.7, (2) tTsKTI, (3) tKreit , (4) tChen.

tion that was proposed by Rohsenow in [3] (this principle is used by Chen, as well): q = q conv + q boil ,

(8)

where a value of 0.019 is used instead of 0.023 for the coefficient applied while calculating αconv and αboil is calculated using the full heat flowrate q and equal to αp.b. Then, the wall temperature is determined in accordance with the following formula: q + α conv t l + α boil t s -. t w = ------------------------------------------α conv + α boil

Two methods of calculating boiling heat transfer were considered, i.e., αboil = αp.b and αboil = 0.7αp.b. To determine the heat transfer coefficient during pool boiling, the following Borishanskii’s correlation was used: α p.b = 4.34q ( p

131

(9)

The calculations were carried out for the conditions of the experiments in [13–16]. The results of calculations are presented in Figs. 2 and 3. We see from Fig. 2 that, when using all the calculation techniques, the relative error of determination of the wall temperature does not exceed 10% and is comparable with the experimental error. The points in the figure were obtained in the following range of regime parameters: p = 0.6–17 MPa; q = 0.5–6 MW/m2; subin cooling at the inlet ∆ T sub = 5–95 K; and w = 0.5– 3.8 m/s. Calculations using (6) and (9) give wall-temperature values that are largely underestimated relative to the experimental data. On the one hand, the data in Fig. 3 confirm the assumption regarding the possibility of the existence of different alternatives of the surface boiling; on the other hand, they show that, in the range of the heat flowrates considered, the relative error of calculating correlations (6) and (9) increases with a decrease in pressure. The calculations using Chen’s technique meet well the experimental data in the transition range from convection to developed boiling; with an increase in the heat flowrate, however, they give more and more overestimated (as compared to the experimental data) val-

132

AGAFONOVA et al.

q, MW/m2

q, MW/m2

(a)

6

3.0 6 2.5

7

4

6

5

7

4

2.0 1.5

1

1.0

0 150

3

5

5

0.5

(b)

3 1

2 3

2

170

1 190 tw,°C

0 230

2

4 250

270 tw,°C

Fig. 3. Results of calculations of the wall temperature by means of different methods for the conditions of experiin

ments in [13–16]. p, MPa: (a) 0.6; (b) 4.1; ∆ T sub , K: (a) 36.0; (b) 92.5; w, m/s: (a) 2.05; (b) 2.22; d × 103, m: (a) 8.2; (b) 15. (1) qconv , (2) qp.b , (3) tChen, (4) tKreit [(correlation (9)], (5) tTsKTI [correlation (6)], (6) tTsKTI/0.7 [correlation (7)], (7) tlin: experiment: (
) tw; () ts.

ues of tw. The wall temperatures calculated by means of (7) appear to be somewhat lower than the experimental data but still rather close to them. With an increase in pressure (Fig. 3b), the relative error of tw determination decreases and the calculated values of tw obtained with the use of different algorithms become closer. We should note that the technique recommended by Kreit and Black is rather simple and gives good results in the surface boiling region. It does not provide, however, a changeover to pure convection because of use of the reduced value of the numerical coefficient while calculating αconv. Therefore, this technique cannot be considered as an alternative to Chen’s technique. Both algorithms based on TsKTI formula (5) give very close results; when this happens, use of the coefficient 0.7 while calculating αboil provides a slightly higher wall temperature. This coefficient accounts for the difference in the evaporation conditions in tubes and channels from those under pool boiling (in particular, due to the possible effect of walls, flow velocity, and the like). From the standpoint of confidence in the calculation results, there is no special need to introduce this coefficient into the calculating algorithm, because the difference between the wall temperature values obtained in these two cases is comparable with the error of the experiments. It should be pointed out, however, that the experimental data used for the assessment of

the calculation results were obtained in tubes of small diameter (approximately 0.008–0.015 m) and, hence, boiling conditions at the wall actually can differ from those under pool boiling. From the standpoint of safety of equipment to be designed, the introduction of this coefficient into the calculation algorithm is also justified, because it leads to a slightly higher wall temperature. The results can be improved by optimization of this coefficient. The only factor that complicates the calculations by means of these algorithms is the necessity of using of iteration procedures, due to the form of correlations (6) and (7). It is possible to exclude this shortcoming by changing the second term in (5) with the help of (4). The use of the linear dependence q(∆t) gives the quadratic equation for tw: q = α conv ( t w – t l ) + [ A ( t w – t s ) – B ] . 2

2

2

2

(10)

Calculations with the use of (10) were conducted for the regimes presented in Fig. 3. Coefficients A and B were taken from [11]. With ∆t = tw – ts < B/A, qboil is taken to be zero. We see that such an approach makes it possible to obtain rather simply results comparable with calculations by means of the other calculation techniques as to the error of calculation results. With small corrections (for instance, by introducing a constant into the second term of (10)), this approach can be recommended as an alternative to Chen’s correlation while calculating heat transfer under subcooled liquid boiling. REFERENCES 1. Heat Transfer in Two-Phase Flow, Ed. by D. Butterworth and J. Hewitt (Moscow, Energiya, 1980) [in Russian]. 2. J. C. Chen, “A Correlation for Boiling Heat Transfer to Saturated Fluid in Convective Flow,” Process Design and Development 5, 322–327 (1966). 3. A. I. Bergles and B. M. Rohsenow, “Determination of Heat Transfer During Surface Boiling under Forced Convection Conditions,” Trans. ASME, J.of Heat Transfer 86(3), 83–93 (1964) [in Russian]. 4. P. Saha and N. Zuber, “Point of Net Vapor Generation and Void Fraction in Subcooled Boiling,” in Proc. 5th Int. Heat Transfer Conf., Tokyo, 1974 4, 47–53 (1974). 5. G. G. Bartolomei and V. N. Mikhailov, “Enthalpy of Incipience of Intense Evaporation,” Teploenergetika, No. 2, 17–20 (1987). 6. K. Forster and N. Zuber, “Dynamics of Vapor Bubbles and Boiling Heat Transfer,” AIChE Journ. 4(1), 531–535 (1955). 7. M. A. Gotovskii and Yu. A. Zeigarnik, “On Peculiarities of Mechanism of Boiling Crisis in Channels under High Subcooling,” in Proc. 11th Int. Heat Transfer Conference, Kyongju, Korea 2, 255–260 (1998). 8. M. A. Gotovskii, “Some Peculiarities of Heat Transfer and Crisis under High Subcooling Values,” in Proc. of Int. Symp. “The Physics of Heat Transfer in Boiling and Condensation,” Moscow, Russia, 1997. THERMAL ENGINEERING

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COMPARATIVE ANALYSIS OF CORRELATIONS 9. Yu. N. Kuznetsov, Heat Transfer in the Problem of Nuclear Reactor Safety (Energoatomizdat, Moscow, 1989) [in Russian]. 10. E. F. Adiutori, New Methods in Heat Transfer (Mir, Moscow, 1977) [in Russian]. 11. RD 24.035,05-89 Thermal and Hydraulic Design of Heat Transfer Equipment for NPPs (Methodological Guidelines) (NPO TdsKTI, Leningrad, 1981) [in Russian]. 12. G. P. Chelata, “Critical Heat Flux in Subcooled Flow Boiling,” in Proc. 11th Int. Heat Transfer Conference, Kyongju, Korea 1, 261–277 (1998). 13. N. V. Tarasova, A. A. Armand, and A. S. Kon’kov, “Study of Heat Transfer in a Tube under Boiling of Subcooled Water and Steam-Water Mixture,” in Heat Transfer under High Heat Flux and Other Special Conditions (Gosenergoizdat, Moscow, 1959) [in Russian]. 14. I. T. Alad’ev, L. D. Dodonov, and V. S. Udalov, “Experimental Data on Heat Transfer under Nucleate Boiling of Subcooled Water in Tubes,” in Convective and Radiation

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15.

16.

17.

18.

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Heat Transfer (AN USSR Publ., Moscow, 1960) [in Russian]. B. S. Varshnei, “Study of Heat Transfer Rate under Surface Boiling of Water and Normal Propyl Alcohol in Tubes,” Cand. Diss. in Technical Sciences (MIKHM, Moscow, 1964) [in Russian]. I. T. Alad’ev, L. D. Dodonov, and V. S. Udalov, “Heat Transfer under Subcooled Water Boiling in Tubes,” in Study of Heat Transfer to Steam and Water Boiling in Tubes under High Pressures (Atomizdat, Moscow, 1958) [in Russian]. V. M. Borishanskii, A. A. Andreevskii, B. S. Fokin, et al., “Generalized Correlation for Calculation Heat Transfer under Two-Phase Flow Motion in Tubes and Channels,” in Achievements in the Field of Studying Heat Transfer and Hydraulics of Two-Phase Flows in Elements of Power-Engineering Equipment (Nauka, Leningrad, 1973) [in Russian]. F. Kreit and W. Black, Heat Transfer Fundamentals (Mir, Moscow, 1983) [in Russian].