1 INTRODUCTION

Film condensation is a classical and complicated physical phenomenon, which widely exists in nuclear engineering, chemical engineering, aerospace engineering and many other fields. Many researchers have made efforts to get the accurate heat transfer coefficient correlations. The complexity in film condensation involves the following aspects: the non-equilibrium characteristics, the different modes of condensation, vapor and liquid in different flow regimes, various wall geometries and different kinds of vapors and liquids[1].

Originally, Nusselt developed a heat transfer coefficient correlation for laminar film condensation [2]. Then, Nusselt’s theory was extensively improved and extended by considering the film subcooling, the inertia and drag, and the shear stress [3-6]. As an important factor in the film condensation, suction effect was found to influence the heat transfer coefficient, as high as 20% at the maximum [7]. Recently, the film condensation heat transfer in the presence of different species of non-condensable gas has been extensively studied, where the boundary layer method and the diffusion layer model were adopted [8-11]. Furthermore, Wu et al. [12] improved the diffusion layer model based on real gas state equation, and expanded its application to the gas-steam pressurizer. In terms of the analytical model, Sin [13] developed the analytical model of film condensation in Passive Containment Cooling System(PCCS), and Mosayebidorcheh et al.[14] used the new hybrid method to seek the analytical solution of the steady state condensation film on the inclined rotating disk.

Many researchers verified Nusselt’s theory of film condensation with the experimental results. However, large deviation occurred in the laminar and wavy flow regime and the turbulent flow regime. For this reason, many modified heat transfer coefficient correlations in film condensation were proposed [15-18]. Some semi-theoretical correlations for film condensation based on the experimental method were developed [19,20]. One of the classic cases was developed by Butterworth [21,22]. In the Butterworth’s correlation, the whole flow regime is divided into three according to the value of Reynolds number: the laminar and wave-free flow regime, the laminar and wavy flow regime, and the turbulent flow regime. The corresponding correlations of the heat transfer coefficient in different flow regimes are given based on massive experimental databases. Many experiments have also been conducted to investigate the film condensation on the horizontal and vertical plane, inside and outside the tube, in different flow regimes, on declined condensation plane with different inclination angles, in presence of non-condensable gas, and in different applications [23-27].

Most heat transfer coefficient correlations for the film condensation given by previous researchers are the functions of Reynolds number, or adding the modified term itself as a function of Reynolds number. Thus, the accurate calculation of Reynolds number is considered vital to obtain an accurate heat transfer coefficient. According to the basic governing equations of the film condensation, the expressions of the mass flow rate of the condensate based on the mass relation and the energy relation can be obtained respectively. They provided two approaches to calculate the heat transfer coefficient of the film condensation. However, few researches have distinguished the differences between them, and improper applications are usually found in public researches.

In this paper, the effect of Reynolds correlation in the natural convection film condensation on outer tube is studied. Based on the general assumption of the modified term of film condensation heat transfer coefficient, the general forms of heat transfer coefficient are derived in different flow regimes. We also discover the difference between the two approaches to calculate the heat transfer coefficient of the film condensation corresponding to the different Reynolds correlations by using the analytical method and the iterative solution method. The general expressions for Re_mass and Re_energy and the relation between the corresponding heat transfer coefficients are obtained. Re_mass and Re_energy are graphically shown to see the difference. They are also used to correlate the experimental data from the open literature. It is found that the approach based on Re_energy is more accurate.

2 FILM CONDENSATION THEORY

2.1 Fundamentals of Condensation Heat Transfer

Generally, for the film condensation outside the tube, Reynolds number of the liquid film outside tube in the experiment is originally defined as

$\text{Re}=\frac{4\text{Γ}}{{\mu }_{\text{l}}}.$ (1)

The mass rate of condensation Г is measurable in the experiment while calculated in the following theoretical analysis. Based on the Nusselt’s theory, the expression of the condensate mass flow rate can be [28]

$\text{Γ}=\frac{{\rho }_{\text{l}}({\rho }_{\text{l}}-{\rho }_{\text{g}})\text{g}{\delta }^3}{3{\mu }_{\text{l}}}.$ (2)

Substituting Eq.(2) into Eq.(1), and it gives

${\text{Re}}_{\text{mass}}=\frac{4{\rho }_{\text{l}}({\rho }_{\text{l}}-{\rho }_{\text{g}})\text{g}{\delta }^3}{3{\mu }_{\text{l}}^{\text{2}}}.$ (3)

The mass flow rate of the film condensate has another expression based on the energy relation [29], namely

$\text{Γ}=\frac{{\overline{h}}_{\text{film}}(T_{\text{sat}}-T_{\text{w}})L}{h_{{}_{\text{fg}}}^{\text{"}}}.$ (4)

By substituting Eq.(4) into Eq.(1), we obtain Reynolds correlation based on energy relation as

${\text{Re}}_{\text{energy}}=\frac{4{\overline{h}}_{\text{film}}(T_{\text{sat}}-T_{\text{w}})L}{{\mu }_{\text{f}}{h}_{\text{fg}}^{\text{"}}}.$ (5)

According to Nusselt’s model, the correlations of heat transfer coefficient in the laminar and wavy flow regime and the turbulent flow regime are developed by adding the modified term. In this paper, we take full account of the difference of variable coefficients in the modified term and the uncertainty of the demarcation points of the different flow regimes, and obtain the general expressions of the heat transfer coefficient in three flow-regimes film condensation model. The condensation film is divided into three regimes: the laminar and wave-free flow regime (0<Re≤Re₁), the laminar and wavy flow regime (Re₁<Re≤Re₂), and the turbulent flow regime (Re>Re₂), as shown in Fig.1. The demarcation points of the film flow regimes (Re₁, Re₂) are variable.

Fig. 1Full-screen View

Flow regimes (color online)

In the laminar and wavy flow regime the modified term (T_m) is a function of Reynolds number (Re). In the turbulent flow regime, T_m is a function of Reynolds number and Prandtl number (Re, Pr). The coefficients in the modified terms are semi-empirical parameters, and usually can be treated as variable.

In the laminar and wavy-free flow regime (0<Re≤Re₁), Nusselt’s equation is

$h_{\text{film}}={\left[\frac{{\rho }_{\text{l}}({\rho }_{\text{l}}-{\rho }_{\text{g}})\text{g}{h}_{{}_{\text{fg}}}^{\text{"}}{k}_{{}_{\text{l}}}^{\text{3}}}{4{\mu }_{\text{l}}L(T_{\text{sat}}-T_{\text{w}})}\right]}^{1/4}.$ (6)

In terms of the original Reynolds number, it can also be expressed as

$\frac{h_{\text{film}}}{k_{\text{l}}}{\left[\frac{{\mu }_{\text{l}}^{\text{2}}}{{\rho }_{\text{l}}({\rho }_{\text{l}}-{\rho }_{\text{g}})\text{g}}\right]}^{1/3}=1.10{\text{Re}}^{-1/3}.$ (7)

In the laminar and wavy flow regime (Re₁<Re≤Re₂) and in the turbulent flow regime (Re>Re₂), the modified terms are added to the basic equation of local coefficient of heat transfer in the laminar and wavy-free flow regime. The local heat transfer coefficients in all flow regimes are

$\frac{h_{\text{film}}}{k_{\text{l}}}{\left[\frac{{\mu }_{\text{l}}^{\text{2}}}{{\rho }_{\text{l}}({\rho }_{\text{l}}-{\rho }_{\text{g}})\text{g}}\right]}^{1/3}=1.10{\text{Re}}^{-1/3}\cdot T_{\text{m}},$ (8)

where the general expressions of T_m are

$T_{\text{m}}=\{\begin{array}{c}\begin{array}{l}10<\text{Re}\le {\text{Re}}_1\\ m_1{\text{Re}}^{n_{\text{1}}}\begin{array}{cc}& {\text{Re}}_1<\text{Re}\le {\text{Re}}_2\end{array}\end{array}\\ m_2{\text{Re}}^{n_{\text{2}}}{\mathrm{Pr}}^s\begin{array}{ccc}& \text{Re}>{\text{Re}}_2& \end{array}\end{array}.$ (9)

The average film condensation heat transfer coefficient is obtained by the integral average method.

$\frac{1}{{\overline{h}}_{\text{film}}}=\frac{1}{\text{Re}}{\displaystyle \underset{0}{\overset{\text{Re}}{\int }}\frac{1}{h_{\text{film}}}}d\text{Re}.$ (10)

The average heat transfer coefficients in different flow regimes are

(11)

where

${\text{Re}}_{\text{m}}=\frac{1}{1.1m_1(\frac{4}{3}-n_1)}({\text{Re}}_2^{(\frac{4}{3}-n_1)}-{\text{Re}}_1^{(\frac{4}{3}-n_1)})+0.68{\text{Re}}_1^{\frac{4}{3}}.$ (12)

The subcooling of the condensate and temperature jump across the film [28] should be taken into account when the latent heat, h_fg^’, is calculated, as given by

$h_{\text{fg}}^{\text{"}}=h_{\text{fg}}\left[1+0.68\frac{C_{\text{pl}}(T_{\text{sat}}-T_{\text{w}})}{h_{\text{fg}}}\right].$ (13)

The temperature jump across the condensate is considered when calculating the temperature of liquid film [28].

$T_{\text{film}}=T_{\text{w}}+0.25(T_{\text{sat}}-T_{\text{w}}).$ (14)

As seen from Eqs.(3) and (5), Re_energy contains the average heat transfer coefficient${\overline{h}}_{\text{film}}$, whereas, Re_mass is a function of the thickness of the film condensate, which corresponds to the local heat transfer coefficient h_film. In consideration of Eqs.(7) and (11), it is indicated that it provides two approaches to calculate the heat transfer coefficient of the film condensation and Reynolds number, such as pair parameters of (${\overline{h}}_{\text{film}}$, Re_energy) and (h_film, Re_mass). However，the fundamental difference between these two approaches still remains unclear when applied to calculate the film condensation heat transfer coefficient.

2.3 Relation of Re_mass and Re_energy in Butterworth’s case

For further analysis, the coefficients of the Butterworth’s correlation are adopted as an example for Eqs.(18) and (21) (Table 1).

TABLE 1.Full-screen View

Variable coefficients in Butterworth’s correlation

Parameter	m1	n1	m2	n2	s	Re1	Re2
Value	0.6868	0.11	0.2091	7/12	1/2	30	1600

Show more

By substituting all these parameters into Eqs.(18) and (21), the relations between Re_mass and Re_energy in different flow regimes can be obtained.

In the laminar and wave-free flow regime

${\text{Re}}_{\text{mass}}={\text{Re}}_{\text{energy}}.$ (22)

In the laminar and wavy flow regime

${\text{Re}}_{\text{mass}}={(7.12{\text{Re}}_{\text{energy}}^{1.22}-34.29)}^{0.564}.$ (23)

In the turbulent flow regime

${\text{Re}}_{\text{mass}}=228.40{({\text{Re}}_{\text{energy}}^{\text{0}\text{.75}}{\text{Pr}}^{-2.5}+150.86{\text{Pr}}^{-2.0}-253{\text{Pr}}^{-2.5})}^{0.2727}.$ (24)

Under the same thermal hydraulic parameter condition, the relation of Re_mass and Re_energy is significantly different in each flow regime. Obviously, Re_mass and Re_energy are equivalent in the laminar and wavy-free regime. However, Fig.2 shows the relation between Re_massand Re_energy in the laminar and wavy flow regime and the turbulent flow regime, described by Eqs.(23) and (24).

Fig. 2Full-screen View

(Color online) Relation between Re_massand Re_energy

From Fig.2 (a), we can see there is a significant difference between Re_mass and Re_energy in the laminar and wavy regime. Under the same thermal hydraulic parameter condition, Re_energy keeps higher than Re_mass (maximally 4 times). Prandtl number does not play a role in this flow regime.

In the turbulent flow regime, Prandtl number significantly influences the relation between Re_mass and Re_energy (see the line of Re_mass~Re_energy in Fig.2 (b)). Considering the corosspoint of Re_mass=Re_energy and Re_mass~Re_energy, there is an obvious demarcation point at Prandtl number, 0.35. When Prandtl number is less than 0.35, the lines of Re_mass=Re_energy and Re_mass~Re_energy intersect at one point, and Re_mass<Re_energyand Re_mass>Re_energy are presented on either side of the coresspoint. When Prandtl number is larger than 0.35, the line of Re_mass~ Re_energy deviates heavenly from that of Re_mass=Re_energy as Prandtl number increasing, and Re_mass<Re_energy. Especially, when Prandtl number increases to about 0.8 or more, 30<Re_mass<1600, corresponding to the laminar and wavy regime; whereas, Re_energy>1600, the corresponding flow regime is the turbulent flow regime. In conclusion, Prandtl number greatly influences the relation between Re_mass and Re_energy, and their large deviation occurs in the turbulent flow regime.

We can also obtain the heat transfer coefficient and Reynolds number, such as (h_film, Re_mass) and (${\overline{h}}_{\text{film}}$, Re_energy), in different flow regimes by using the iterative solution method. The iterative calculation flow chart is shown in Fig.3.

Fig. 3Full-screen View

Iterative calculation flow chart

According to the result of h_filmand Re_mass, we can obtain${\overline{h}}_{\text{film}}$ by using the average method of Eq.(10). Define the relative deviation of the average heat transfer coefficient as:

$\epsilon =\frac{{\overline{h}}_{\text{mass}}\text{ -}{\overline{h}}_{\text{energy}}}{{\overline{h}}_{\text{energy}}}\times \text{100\%}.$ (25)

This definition of relative deviation not only presents its relative magnitude, but also directly shows whether it is positive or negative.

According to Eqs.(22) and (25), ε = 0 in the laminar and wave-free flow regime, and it means that ${\overline{h}}_{\text{film}}$ and ${\overline{h}}_{energy}$ are equivalent.

In the laminar and wavy flow regime, the value of ε is positive, as shown in Fig.4 (a). The maximum deviation is as high as 30%, corresponding to Re_energy of 1600. It means that ${\overline{h}}_{mass}$>${\overline{h}}_{energy}$exists in the whole laminar and wavy flow regime.

Fig. 4Full-screen View

(Color online) Relative deviation of average heat transfer coefficient

Fig.4 (b) shows a complicated situation in the turbulent flow regime, and ε is a highly non-linear function of Reynolds number and Prandtl number. When Prandtl number is less than 0.35, there are two intersections of the curve of ε and the horizontal coordinate. The value of ε is alternately negative, positive and negative, with the increase of Reynolds number; ${\overline{h}}_{mass}$<${\overline{h}}_{energy}$ and ${\overline{h}}_{mass}$>${\overline{h}}_{energy}$ alternately present in low, moderate and high Reynolds number regime. The minimum value of ε is larger than -60%, corresponding to Pr=0.1 and Re_energy=1600. When Prandtl number is larger than 0.35, ε monotonously decreases from positive value to negative value as Reynolds number increasing, and there is only one crosspoint of ε=0. When Prandtl number increases from 0.35 to 4.0, the value of ε increases from 0 to 35% in the low Reynolds number region (1600<Re<4000), the maximum value of ε is less than 60%, and the Re_energy value corresponding to the crosspoint of ε=0 decreases from 12000 to 4000. In conclusion, when 0.1<Pr< 4.0, ε is generally within the range of -60% ~ +60%, and it is a highly non-linear function of Reynolds number and Prandtl number in the turbulent flow regime.

3 EXPERIMENT VALIDATION

Experimental results from Bum-Jin Chung [15] and G.M.Hebbard and W.L.Badger [18] are used to verify the accuracy of the two Reynolds correlations.

The experiment set by B.J. Chung [15] focuses on the film wise and drop wise condensation of steam on short inclined plates. Table 2 shows the experiment conditions.

The result is shown in Figs.5 and 6. The heat flux and Reynolds number vary according to the temperature difference between wall and bulk. The result of the Nusselt’s theory case is obtained from the basic Nusselt's theory in the laminar and wave-free flow regime (Eqs.(6) or (7)), and the modified term (T_m) is not taken into account in all flow regimes. It is just a reference case.

Fig. 5Full-screen View

(Color online) Heat flux with temperature difference between wall and bulk

Fig. 6Full-screen View

(Color online) Reynolds number with temperature difference between wall and bulk

It shows that the heat flux calculated with Re_energy has a good agreement with the experimental data, especially in high temperature difference region. The result obtained with Re_massover-predicted the test results at an average deviation of 11.0%. The heat flux calculated from Nusselt’s theory is lower than the experimental results, and the deviation increases as ∆T_wb increasing. However, when ∆T_wb is small (10K<∆T_wb<16K) in this experiment, Nusselt’s theory can effectively predict the heat flux, while the method using Re_energy slightly over-predicted the results. Major reason is that Reynolds number corresponding to this condition is less than 80 (Fig.6), and Nusselt’s theory can be effective in the wavy-weak flow regime. Re_mass and Re_energy at the same ∆T_wb are quite different, as shown in Fig.5. The average deviation Reynolds number between Re_energy and Nusselt’s theory is within 10%. It is consistent with the theoretical analysis in the part II-C.

In conclusion, the calculating approach based on Re_energy promises a better performance in the low Reynolds number regime.

Another pure steam condensation experiment performed by G.M. Hebbard and W.L. Badger [15] is selected to assess the Reynolds correlation in the laminar and wavy flow regime and the turbulent flow regime. The experimental conditions are listed in Table 3.

When the bulk temperature varies from 353.15 K to 393.15 K, Reynolds number is generally located in the laminar and wavy flow regime and the turbulent flow regime. According to the value of Reynolds number, different correlations are used for various flow regimes to determine the film condensation heat transfer coefficient. The result is shown in Fig. 7.

Fig. 7Full-screen View

(Color online) Heat transfer coefficient with different steam temperature

Form Fig.7, in the case of different steam temperature, the value curve of the heat transfer coefficient with Re_energy is always located between the value curve of the heat transfer coefficient with Re_mass and that of the heat transfer coefficient with Nusselt’s theory. Furthermore, the heat transfer coefficient with Re_mass is maximum. The experimental data is also sandwiched between the value curves of the heat transfer coefficient with Re_mass and Nusselt’s theory, and accords well with the heat transfer coefficient with Re_energy. It is indicated that when the steam temperature changes, the calculations with Re_energy always show a great accuracy. However, the heat coefficients are over-predicted by Re_mass and underestimated by Nusselt’s theory.

It is necessary to point out that when ∆T_wb is small (3K<∆T_wb<7K), the predictions based on Re_massfit the experimental data better, whereas when ∆T_wb is higher (7K<∆T_wb<25K), namely in the turbulent flow regime, the results based on Re_energy are more accurate. In this experiment, the pure steam in the bulk is not solely under natural convection, but slowly driven by external force. This is the main reason why the predictions based on Re_mass fit the experimental data better when ∆T_wb is small. When ∆T_wb is small, the condensation film is thin, and the weak forced convection is still influential and enhances the heat transfer coefficient. When ∆T_wb is larger and the condensation film becomes thicker, this effect becomes insignificant as the forced convection is no longer dominant. Thus, when ∆T_wb is small, the experiment results are slightly higher than the heat transfer coefficient of natural convection, which is similar to the prediction of Re_energy. But they are closer to the lager-predicted result calculated from Re_mass. In the natural convection of film condensation, the prediction of Re_energy is more accurate.

In addition, there may be a gradual transition between Re_mass and Re_energy cases when ∆T_wb is small, and the experimental value locates in the transition regime. The validation of this gradual transition will be our further study in the future.

According to the previous analysis, Reynolds correlation influences not only the heat transfer coefficient, but also the calculation of the value itself, as they are defined based on different principles, namely the mass relation and the energy relation. The approach based on Re_energy is a better form of formula when calculating the condensation heat transfer coefficient.

4 CONCLUSION

This paper mainly discusses the effect of the Reynolds correlation on the prediction of heat transfer coefficients in the natural-convection film condensation in different flow regimes.

For the natural convection film condensation on the outer wall of tube, the general forms of the heat transfer coefficient correlation are developed in different flow regimes, which can be applied in a wide range of conditions. The difference between the two approaches to calculating the heat transfer coefficient of the film condensation corresponding to Re_mass and Re_energy is distinguished by using the analytical method and the iterative solution method. The general expressions for Re_mass and Re_energy and the relation of the corresponding average heat transfer coefficients are also obtained for three different flow regimes. In the laminar and wave-free flow regime, Re_mass and Re_energy are equivalent. In the laminar and wavy flow regime, Re_mass is much smaller than Re_energy, and the maximal deviation of the average heat transfer coefficients from the two Reynolds correlations is as high as about 30%. In the turbulent flow regime, Prandtl number greatly influences the relation between Re_mass and Re_energy. Compared with the experimental results, the approach based on Re_energy turns out to be a good choice to calculate the heat transfer coefficient.