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364 CHAPTER 11  Boiling and condensation

Radiation heat transfer across the vapor film can be calculated using the following expression:

where εw is the emissivity of solid (wall), and αf is the absorptivity of liquid, which is equal to the emissivity of the liquid.

Bromley (1950) proposed the following expression for the combined effects of convection and radiation:

For hr < hc , Bromley (1950) suggested the following approximation:

11.4  Flow boiling

Flow boiling (or forced convection boiling) occurs when a liquid is forced through a tube or over a surface that is maintained at a temperature greater than the saturation temperature of the liquid. The applications of flow boiling include steam generators for thermal and nuclear power plants. The fluid dynamics and heat transfer mechanisms of flow boiling are more complex than in pool boiling, as the forced flow influences the formation of vapor bubbles, their separation from the heating surface, and the two-phase flow patterns along the tube.

11.4.1  Flow boiling regimes

Fig. 11.5 shows the variation of the wall and fluid temperatures, flow patterns, and heat transfer regions for the flow boiling in a vertical uniformly heated tube (constant wall heat flux). Heat transfer to the subcooled liquid takes place by single-phase convection alone until the wall temperature is higher than the saturation temperature, at which point the boiling commences while the liquid may still be subcooled. There is a decrease in the wall temperature due to the increased heat transfer coefficient as a result of boiling. The size of the vapor bubbles formed increases as the bulk liquid attains saturation temperature. At lower qualities, there are small bubbles dispersed in the liquid (bubbly flow), and the heat transfer mechanism is mainly nucleate boiling. The local heat transfer coefficient mainly depends on the heat flux and is not affected by the quality in nucleate boiling. As the quality (or the void fraction) increases along the flow, bubbles coalesce and form larger bubbles separated by liquid slugs, and the flow pattern is termed as a slug flow. With further increase in the vapor quality along the flow, a vapor core is formed, which is irregular, termed as a churn-annular flow.

11.4 Flow boiling 365

FIGURE 11.5

Wall and fluid temperatures, flow patterns, and heat transfer regions for flow boiling in a vertical heated tube.

A continuous vapor core is then formed, surrounded by the liquid, and is termed as an annular flow. Because of the higher velocities caused by the increased vapor fraction, the nucleation from the heating surface is suppressed. This is due to the thinner thermal boundary layer resulting from the increased velocities. The heat transfer is mainly due to the convective evaporation at the liquid-vapor interface. The vapor core contains entrained liquid droplets formed as a result of the interaction between the liquid and vapor phases. The thickness of the liquid film decreases due to evaporation, and the velocity increases due to the increased quality, both of which result in an increase in the heat transfer coefficient (decrease in the wall temperature) along the flow. At some point, the liquid film completely dries out due to evaporation, and the vapor comes in contact with the wall, which results in a sudden decrease in the

366CHAPTER 11  Boiling and condensation

heat transfer coefficient (increase in the wall temperature). This point is called “dryout”, which is very important from the viewpoint of the design and operating conditions. Further downstream, the evaporation of the entrained liquid droplets increases the velocity, and hence there is a slight increase in the heat transfer coefficient. The vapor flow with entrained droplets is called a spray flow or mist flow, with the wall being liquid deficient. Eventually the tube is completely filled with vapor, at which point a single-phase convection starts. The quality (x) shown in the figure indicates thermodynamic quality obtained from energy balance. The actual quality can be different from the thermodynamic quality, mainly in the subcooled boiling and spray flow conditions.

In actual applications, one may not see all the flow patterns described here, as the flow patterns depend on the inlet condition, mass flux, heat flux, tube diameter, and tube length. For the case of the wall temperature controlled heating, the local heat flux depends on the local two-phase heat transfer coefficient. Fig. 11.6 shows the influence of heat flux on the heat transfer coefficient variation along the flow, for

a constant mass flux. It can be seen that with the increase in heat flux (q2), the heat transfer coefficient increases for the same vapor quality, but the dryout occurs early (at a lower quality). With further increase in the heat flux (q3 and q4), there occurs no

FIGURE 11.6

Heat transfer coefficient variation with quality for constant mass flux and heat flux with q1 < q2 < q3 < q4 < q5.

11.4 Flow boiling 367

transition to annular flow, and there is DNB in the saturated nucleate boiling, which causes a sudden reduction in the heat transfer coefficient. The DNB occurs even in the subcooled boiling at higher heat flux (q5), as shown in the figure. Note that the DNB indicates film boiling similar to that in pool boiling and “dryout” indicates the dryout of liquid film. Normally the wall temperature increase caused by DNB is much higher than that by dryout. Note that q (DNB-subcooled) > q (DNB-saturated) > q (dryout).

It must be noted that boiling (both pool boiling and flow boiling) heat transfer is highly sensitive to the heating surface characteristics. Boiling characteristics change with surface ageing, especially for water (Collier and Thome, 1994; Jayaramu et al., 2019), and hence it is very important that the experimental runs be repeated until the repeatable data is obtained, which can be used for the heat transfer analysis, comparison with the literature data, or development of correlations.

11.4.2  The Chen correlation

By far the most successful and widely used two-phase correlation for flow boiling of water and organic fluids in conventional size tubes is the one proposed by Chen (1963). The correlation predicts the data for both the saturated nucleate boiling region and the convective vaporization region. It assumes that both nucleate boiling and convection occur in some proportion over the entire range of correlation, and their contributions are additive.

where hTP , hNcB, and hc are the local two-phase heat transfer coefficient, the nucleate boiling contribution, and the convection contribution, respectively; hc can be evaluated by a Dittus-Boelter type equation that was presented in Chapter 5.

Chen (1963) argued that heat is essentially transferred from wall to a liquid film in both dispersed and annular flows, so it is appropriate to use the liquid thermal conductivity in Eq. (11.31). Furthermore, the magnitudes of the liquid and vapor Prandtl numbers are approximately the same, and the two-phase value should not be very different from this value (Prf or Prg ).

Eq. (11.31) can be written as

11.4 Flow boiling 369

Substituting Eq. (11.38) in Eq. (11.36),

S approaches unity at low flow rates and zero at high flow rates. Based on the experimental data, S can be expressed as

The experimental data used for the development of the Chen correlation considered the two-phase flow in vertical tubes, and the fluids were water (0.55 to 34.8 bar, upward flow and downward flow), benzene, cyclohexane, heptane, methanol, and pentane, all at 1.03 bar with upward flow. The average deviation between the predicted (calculated) and measured data from the experimental cases was ±12%. Though the Chen correlation finds extensive applications for different fluids and operating conditions, it cannot be used for liquid metals or for horizontal tubes due to the asymmetrical nature of flow patterns, such as the stratified flows in horizontal tubes. Many variations of the Chen correlation were proposed later to extend its applicability to a wider range of fluids and operating conditions, and even for different channel geometries (e.g., offset strip fins). In fact, the Chen correlation’s basic philosophy of the nucleate boiling suppression factor S and the liquid-phase convection enhancement factor F has been used even in the development of some of the correlations (called Chen-type correlations) for flow boiling in mini/micro-channels (Jayaramu et al., 2019).

11.4.3  Critical heat flux in flow boiling

The occurrence of the critical heat flux (CHF) and the nature of the CHF (dryout and DNB) in flow boiling were discussed earlier. This section presents the estimation of CHF for flow boiling in vertical uniformly heated round tubes. Flow is in the upward direction.

For uniformly heated tubes, the onset of the CHF occurs first at the tube (or channel) exit, and hence overheating first occurs at the tube exit, in the absence of flow and thermal instabilities.

Boiling cannot occur if the wall temperature is less than the saturation temperature. Therefore, the lower limiting value of the critical heat flux corresponds to the condition of the tube exit wall temperature being equal to the saturation temperature. The fluid temperature at the tube exit can be determined using the inlet condition, the mass flux, and the heat flux. The tube exit wall temperature can be determined using the heat flux, the single phase heat transfer coefficient (hf 0 ), and the fluid

370 CHAPTER 11  Boiling and condensation

temperature at the exit. For a tube of diameter D, length z, mass flux G and the inlet subcooling ( Tsub )i , the lower limiting value of the critical heat flux is

The upper limiting value of the critical heat flux corresponds to the condition of the exit quality being equal to 1(x (z) = 1),

Based on the extensive experimental data on the CHF for water in vertical uniformly heated round tubes, the following correlation was proposed by Bowring (1972):

with the operating pressure p in bar. The pressure functions F1, F2, F3, and F4 are given in Table 11.2.

The empirical correlation was obtained based on the experiments with water in the following parameter ranges:

p : 2 −190 bar

D: 0.002 − 0.045 m z : 0.15 − 3.7 m

G :136 −18600 kg/m2s

The RMS error is 7%, and 95% of the test data lie within ±14%. The empirical correlation does not differentiate between the types of CHF - dryout and DNB (film boiling). Experimental data on the variation of CHF with different parameters is available in Collier and Thome (1994) and Stephan (1992).