[ Article ]
Journal of Korea Technical Association of the Pulp and Paper Industry - Vol. 53, No. 4, pp.64-77
ISSN: 0253-3200 (Print)
Print publication date 30 Aug 2021
Received 04 May 2021 Revised 05 Aug 2021 Accepted 06 Aug 2021

# Characterization of the Pressure Drop during Condensation in Channels of Multi-Channel Cylinder Dryer Using Homogeneous and Separated Flow Models

Lijie Qiao1, ; Jixian Dong2, ; Huan Liu3 ; Bo Wang4 ; Sha Wang3 ; Yan Dong3 ; Haozeng Guo3
1College of Mechanical and Electrical Engineering, Shaanxi University of Science & Technology, Xi’an, Shaanxi Province, 710021, Lecturer, People’s Republic of China
2College of Mechanical and Electrical Engineering, Shaanxi University of Science & Technology, Xi’an, Shaanxi Province, 710021, Professor, People’s Republic of China
3College of Mechanical and Electrical Engineering, Shaanxi University of Science & Technology, Xi’an, Shaanxi Province, 710021, Student, People’s Republic of China
4College of Mechanical and Electrical Engineering, Shaanxi University of Science & Technology, Xi’an, Shaanxi Province, 710021, Senior experimentalist, People’s Republic of China

Correspondence to: E-mail: qiaolijie@sust.edu.cn (Adddress: College of Mechanical and Electrical Engineering, Shaanxi University of Science & Technology, Xi’an, Shaanxi Province, 710021, People’s Republic of China) Contributed by footnote: E-mail: djx@sust.edu.cn (Adddress: College of Mechanical and Electrical Engineering, Shaanxi University of Science & Technology, Xi’an, Shaanxi Province, 710021, People’s Republic of China)

## Abstract

The condensation resistance characteristics of steam flow in horizontal rectangular channels of multi-channel cylinder dryer (MCD) is a very important factor that affects the heat transfer, while it is not very clear due to the limited studies. In this paper, the frictional resistance characteristics of the two-phase flow (steam and water) in the rectangular small channels of MCD were investigated with deionized water as the working fluid, and the calculation models, homogeneous and separated flow models, of the two-phase flow pressure drop in the channel were verified and evaluated. It showed that the main part of the total pressure drop was the frictional pressure drop, which means that the calculation of frictional pressure drop was the key to the calculation of the pressure drop of the two-phase flow in the channel. Meanwhile, it was found that the homogeneous model was not suitable for the calculation of the frictional pressure drop and the separated flow model was completely in line with the experimental data. In this study, the converted coefficient of complete steam/liquid phase were concerned to calculate the frictional resistance using five separated flow models. It was found that the Chisholm B model showed better accuracy than prior correlations, which was greatly consistent with experimental data, and the average absolute percentage deviation (MAPE) was 24.9%.

## Keywords:

Two-phase flow, pressure drop, resistance experiment, multi-channel cylinder dryer

## 1. Introduction

With the rapid development of high-speed paper machine, the improvement of efficiency of the cylinder dryer have attracted widespread attention from both academia and industries. Condensate drainage is one of the most important problems hindering the development of dryers. Multi-channel cylinder dryer (MCD), based on the concept of a compact small-channel heat exchanger with high heat transfer specific surface area, was proposed in 2001.1) After that, MCD has attracted great attention due to its efficient forced convection heat transfer mechanism.2,3) Its basic principle is shown in Fig. 1. Driven by the steam pressure, the steam passes through the channels and pushes the condensate out of it, which solves the problem of the accumulation of condensate and improves the drying efficiency.

Schematic diagram of working principle of multi-channel cylinder dryer.

The heat exchange performance of the two phases (steam and water) in the channel is closely related to its flow pressure drop, while a higher flow pressure drop will cause more pressure loss of the two-phase flow resulting in the reduction of the drying efficiency. For more than 70 years, the research on the pressure drop of condensation two-phase flow in small channels has been carried out, and there are extensive theoretical and experimental research results. Many scholars proposed some experimental correlations involving pressure drop calculation and applied them to the industry, which can be divided into homogeneous and separated flow models. For a horizontal conventional channel with equal cross-section, the most focus point is to calculate the frictional pressure drop. For the calculation of the frictional pressure drop of the steam-liquid two-phase flow in the steady-state conventional channel, all the prediction correlations adopt the “equivalent” or “conversion” method. Table 1 showed six models (Eq. 1-6) based on equivalent viscosity of homogeneous model.

Equivalent viscosity μTP correlations of homogeneous model

Table 2 showed six models (Eq. 7-12) based on converted coefficients of complete phase.

Two-phase multipliers of separated flow model

Although there are many correlations for calculating the flow pressure drop, there are large limitations of correlations’ range of application, even large deviations from each other or to specific conditions. Therefore, it is necessary to find a suitable pressure drop calculation model for condensation two-phase flow in channels of MCD.

In this paper, deionized water is used as the working medium to explore the resistance characteristics of steam condensation two-phase flow in the small rectangular channels of MCD. First, the applicability of homogeneous model and separated flow model were discussed, and then the correlation for calculating the frictional pressure drop in this experiment was determined through the verification and evaluation of 12 models. Not only this research can provide data support for the steam condensation flow in the channels of MCD, but it also provides a reference for device design from the perspective of enhancing heat transfer.

## 2. Materials and Methods

### 2.1 Experimental apparatus

The experimental apparatus was shown in Fig. 2. The experimental section was made of an aluminum rectangular plate with a total length of 800 mm and a cross-section size of 4.5 mm (H)×13.5 mm (W), as shown in Fig. 3. On each side of the plate one rectangular cross-section channel was engraved, namely steam condensation channel and cooling water heat-exchange channel. The steam channel was covered with transparent polycarbonate sheet for observation. In this experiment, deionized water was used in steam channel with filtered tap water used as coolant. When the steam generator was turned on, the generated steam directly entered the rectangular cross-section channel, contacted with the subcooled wall surface and underwent condensation and released heat. The steam that had not been cooled in time in the channel would further condense into condensate in the heat exchanger. The collected condensate returned to the steam generator through the loop, thus the entire cycle was completed.

Schematic Diagram of Experiment Apparatus.

Size of steam channel.

The experimental section was placed horizontally, and pressure points are set at both ends of steam channel. A PX409-2.5GI differential pressure transmitter was used to measure the pressure drop on the steam side, the measurement range was 0-17.2 kPa. The six temperature measurement points were evenly distributed in the channel with T-type thermocouples (-200~350℃). The steam flow rate is measured by a turbine flowmeter (FTB1411, 2.3-11.3 LPM). In addition, all the sensors and flowmeter were connected to a data acquisition instrument (Japan HIOKI, LR8400), which is used to record the temperature, pressure and flow rate under steady-state conditions. A high-speed digital camera (PCO.dimax S4), with a shutter speed of 1.5 μs-40 ms, was used for observing the flow patterns. It should be noted that, all outer surfaces of the experimental section were closely attached with thermal-insulation cotton to minimize the heat loss, except for the PC sheet.

### 2.2 Data collection and processing

2.2.1 Pressure drop

The total pressure drop between the inlet and outlet of horizontal channel included the pressure drop of two-phase flow (ΔpTP), the local loss at the inlet sudden contraction and the local loss at the outlet sudden expansion, as shown in the following formula [13]:

 $∆{p}_{\mathrm{T}\mathrm{P}}=∆p-∆{p}_{\text{in}}-∆{p}_{\text{out}}$ [13]

According to formula [13], after measuring the total pressure drop between the inlet and the outlet using a differential pressure transmitter, the local pressure drop at the inlet and the pressure drop at the outlet must be calculated to obtain the pressure drop of the two-phase flow. As mentioned earlier, the pressure drop calculation model is divided into a homogeneous model and a separated model, thus the pressure drop at the sudden contraction of the inlet and the sudden expansion of the outlet will be different.

For homogeneous flow, the local pressure drop at the sudden contraction of the inlet can be calculated by the following equation [14]:

 $\mathrm{\Delta }{p}_{\text{in,H}}=\frac{{G}_{1}^{2}}{2{\rho }_{1}}{\left(\frac{1}{{a}_{c}}-1\right)}^{2}\left[1+\left(\frac{{\rho }_{\text{1}}}{{\rho }_{\text{v}}}-1\right)x\right]$ [14]

ac is a parameter related to a1.16) a1>1, a1 is the ratio of the cross-sectional area of the connecting tube to the cross-sectional area of the channel.

Local pressure drop at the sudden expansion of the outlet17):

 $\mathrm{\Delta }{p}_{\text{out,H}}=\frac{{G}_{2}^{2}}{2{\rho }_{1}}{\left(1-{a}_{2}\right)}^{2}\left[1+\left(\frac{{\rho }_{\text{1}}}{{\rho }_{\text{v}}}-1\right)x\right]$ [15]

a1<1, a2 is the ratio of the cross-sectional area of the connecting tube to the cross-sectional area of the channel.

For separated flow, the local pressure loss of the sudden contraction can be calculated by the following formula:

 $\begin{array}{c}\mathrm{\Delta }{p}_{\text{in,S}}=\frac{{G}_{1}^{2}}{2{\rho }_{1}}\left\{\left(\frac{1}{{a}_{c}}+1\right)\frac{{\rho }_{\text{v}}}{{\rho }_{\text{1}}}\right\\left[\frac{{x}^{2}}{{\alpha }^{2}}+{\left(\frac{{\rho }_{\text{1}}}{{\rho }_{\text{v}}}\right)}^{2}+\frac{\left(1-{x}^{3}\right)}{{\left(1-\alpha \right)}^{2}}\right]\\ -2\left[\frac{{x}^{3}}{\alpha }\left(\frac{{\rho }_{\text{1}}}{{\rho }_{\text{v}}}\right)+\frac{{\left(1-{x}^{2}\right)}^{2}}{1-\alpha }\right]}\end{array}$ [16]

Local pressure loss of sudden expansion:

 $\begin{array}{c}\mathrm{\Delta }{p}_{\text{out,S}}=\frac{{G}_{2}^{2}}{{\rho }_{1}}\left(1-{a}_{2}\right)\left\{\frac{{\rho }_{\text{v}}}{{\rho }_{\text{1}}}\right\\frac{1+{a}_{1}}{2}\left[\frac{\left(1-{x}^{3}\right)}{{\left(1-\alpha \right)}^{2}}+\left(\frac{{\rho }_{\text{1}}}{{\rho }_{\text{v}}}\cdot \frac{{x}^{3}}{{a}^{2}}\right)\right]\\ -{a}_{2}\left[\frac{{\left(1-x\right)}^{2}}{1-\alpha }+\frac{{\rho }_{\text{1}}}{{\rho }_{\text{v}}}\cdot \frac{{x}^{2}}{a}\right]}\end{array}$ [17]
 $\alpha =\frac{1}{1+\frac{1-x}{x}{\left(\frac{{\rho }_{v}}{{\rho }_{1}}\right)}^{2/3}}$ [18]

In a horizontal conventional channel with equal cross-section, the pressure drop of steam condensed two-phase flow can be divided into three parts: two-phase frictional pressure loss, acceleration pressure loss and gravity pressure loss. Because the inclination angle θ between the channel and the horizontal direction was 0, the gravity pressure drop was no longer considered. Due to the steady-state conditions and small steam mass flux, the acceleration pressure loss can be ignored. Finally, for a small horizontal channel with equal cross-section, it can be concluded that the pressure drop of the two-phase flow was equal to the frictional pressure drop. Then, the key of the problem would be the calculation of the frictional pressure loss.

2.2.2 Mean absolute percentage error (MAPE)

The evaluation parameter, of the comparison between the experimental value and the predicted value of the two-phase flow pressure drop (ΔPTP), was the mean absolute percentage error (MAPE). MAPE is mainly used to describe the degree of dispersion of a set of data. The definition of MAPE was shown in equation [19].

 $MAPE=\frac{1}{N}\sum _{N}^{}\frac{\left|\mathrm{\Delta }{P}_{\text{TP,E}}-\mathrm{\Delta }{P}_{\text{TP,P}}\right|}{\mathrm{\Delta }{P}_{\text{TP,E}}}×100%$ [19]

Where MAPE was the mean absolute percentage error, %. N, ΔPTP,E and ΔPTP,P represented the number of data, experimental value of the two-phase flow pressure drops and predicted value of the two-phase flow pressure drops, respectively.

## 3. Results and Discussion

### 3.1 Calculation models of pressure drop in small channels

According to the classic two-phase flow theory, there are two main types of theoretical models for the pressure drop of steam condensation two-phase flow, homogeneous flow model and separated flow model. For two-phase flow, the homogeneous flow model uses the equivalent viscosity to calculate the pressure drop, which is actually a simple extension of the single-phase flow calculation method. However, the ideal calculation result cannot be achieved by adjusting the equivalent viscosity relationship only. Homogeneous model is a convenient concept for simulating pressure drop of two-phase flow. The main idea is to treat a two-phase flow system (generally a gas-liquid or vapor-liquid two-phase flow system) as a same fluid, such as pure gas phase, pure vapor phase and pure liquid phase. When calculating physical properties of the gas (vapor)-liquid two-phase, it needs to be determined according to the average value of its physical properties. In the process of calculating the frictional pressure drop of the homogeneous model, the frictional pressure drop of the two-phase flow can be characterized by the friction coefficient. The two-phase average viscosity μTP used in the calculation of the friction coefficient reflects the concept of “equivalent single-phase”. The difference between the different homogeneous models is the calculation methods of two-phase average viscosity μTP (equivalent viscosity). For this reason, many scholars have proposed an empirical correlation for the two-phase average viscosity. Theoretically, the homogeneous model cannot reflect the influence from the two-phase parameters, such as two-phase flow pattern and the shear force between two phases, on the pressure drop. There would be a certain error between the predicted value of the model and the experimental value.

The separated flow model considers that the gas phase and the liquid phase are two completely separated fluids. It assumes that the gas phase and the liquid phase are continuous media, and the gas phase and the liquid phase are completely separated. There are different speeds and characteristics between the two phases, and the two-phase flow is in the pipeline. They flow at their respective speeds and are coupled by the interaction force between the two phases. At the same time and position, the gas phase and the liquid phase have their own different speeds, temperatures, and densities. Its basic calculation feature is to convert the frictional resistance of two-phase flow into single-phase frictional resistance. Among the many correlations of separated flow proposed by the predecessors, the converted coefficients of the two-phase frictional resistance were involved. The converted coefficients of the two-phase friction resistance are the key to the classification of separated flow models. The first kind of models was based on the “separated phase” concept. The converted coefficients included: the converted coefficients of separated liquid-phase (фl2) and the converted coefficients of separated gas-phase фg2. The second kind of models was based on the “complete phase” concept, and the converted coefficients included: converted coefficient of complete liquid-phase фlo2 and converted coefficient of complete gas-phase фgo2.

The difference between two flow models is that the separated flow model takes into account the different physical parameters and flow speeds of each phase in the two-phase flow, and establishes related model equations for each phase. Theoretically, the separated flow model is closer to the actual two-phase fluid situation.

The following two types of models were analyzed and deduced to obtain the corresponding theoretical models to guide this research.

3.1.1 The selection of homogeneous models

The empirical correlations given in this paper are suitable for gas (steam)-liquid two-phase flow and are cited more frequently, as shown in Table 1. The McAdams et al. (1942) correlation is widely applicated and recognized by many scholars for pressure drop calculation of steam condensation, 8,9,17-19) which was the reason for its selection. Owing to the working medium is the same as this research as mentioned before, the Cicchitti et al. (1960) correlation was also selected as the calculation model. Condensation is non-adiabatic process existing heat transfer between phases. Since Lin et al. mainly considered the influence of mass vapor rate during calculation which was more suitable for non-adiabatic, the correlation was selected. Both the Dukler et al. (1964) and the Beattie-Whalley (1982) correlations are more suitable for adiabatic process, so they cannot be selected for calculation. The Awad -Myuztchka (2008) correlations was not selected for this research, due to the experimental objects and the round mini/micro channel shape. As a result, three correlations including the McAdams et al. (1942), the Cicchitti et al. (1960) and the Lin et al. (1991) were selected to compare with the experimental values, and an appropriate empirical correlation should be decided to predict the friction pressure drop.

3.1.2 The selection of separated models

Table 2 showed a classic model (Eq. 7)10) that was commonly used based on converted coefficients of separated phase, and five models (Eq. 8-12) based on converted coefficients of complete phase. On the one hand, in terms of the form of correlation, the separated flow model (Eq. 7) proposed based on the concept of “gas/liquid phase” was more suitable for two phase flow where the working conditions are basically normal temperature and pressure, and the composition remained unchanged, e. g. an oil-water two-phase flow system. In other words, this model is more suitable for a two-component two-phase flow system without phase change, because such models ignore the interaction force between the two phases. On the other hand, with total mass flux of the two-phase flow unchanged, analysis is still based on the form of correlation. Due to the separated flow models based on the concept of “complete phase” fully consider the interaction force between two phases, it is more suitable for steam-water two-phase flow system. Obviously, for the single-component two-phase flow with phase change (condensation) in this study, it was more reasonable and accurate to use converted coefficient of complete liquid-phase фlo2 and converted coefficient of gas-phase фgo2.

The Chisholm B correlation is suitable for most two-phase flow conditions including this experiment not only due to the experimental object (steam-water), which is consistent with this research, but due to the more comprehensive range of mass flux of the correlation. Cavallini et al. (2002) correlation is suitable for steam condensation, which is the preferred consideration. In addition, the prediction results using Cavallini et al. (2001) correlation show excellent agreement with a large amount of experimental data, and the scale of the experimental tube is similar to this research. Similar to the Cavallini et al. (2002) correlation, Wilson et al. (2003) correlation was also proposed for condensation, and the scale of the experimental tube was similar to this study. At last Wilson et al. (2003) correlation should be selected for calculation of friction pressure drop. As mentioned in introduction the Lockhart-Martinelli correlation is not suitable to the non-adiabatic steam-liquid two-phase flow for single-component, but to gas-liquid two-phase flow. Because the Friedel (1979) correlation takes gravity into account, it is mainly for the vertical pipes. Definitely is it not appropriate for the pressure drop study of horizontal pipes. In addition, the consideration of surface tension indicates it more suitable for micro-channels and mini-channels instead of conventional channel. The Zhang and Webb correlation is not suitable for this study, since it is suitable for refrigerants and for the calculation of frictional pressure drop under adiabatic conditions.

Eventually, three correlations including Chisholm B (1973), Cavallini et al. (2002) and Wilson et al. (2003) were selected.

### 3.2 The results with homogeneous model

The comparison between the calculation results of the homogeneous model and the experimental data was shown in Fig. 4. The error between the predicted value and the experimental value adopted the mean absolute percentage error (Mean Absolute Percentage Error, MAPE) to compare. It can be found that the errors of the correlations of the homogeneous model were different according to the selected two-phase average viscosity correlations. When the steam mass flux was 20~40 kg/(m2·s) (Fig. 4(a)), the results obtained from the Lin correlation and the McAdams correlation had the smaller average error, which were 25.6% and 25.8%, respectively. While MAPE of the results obtained by the Cicchitti correlation was estimated to reach 46.3%. The McAdams correlation was based on the two-phase viscosity correlation in the same form as the two-phase density correlation. In this way, the two-phase viscosity also conformed to the general law of various physical properties changing with working conditions. In addition, its wide applicability had been fully proven, and the experiment object of this type also includes water. Therefore its predicted value error was small enough. The Lin correlation is aimed at the non-adiabatic process of gasification. Similar to the McAdams correlation, the influence of steam quality fraction is also included, also was the error small. Although the Cicchitti correlation is aimed at the experimental object of wet steam, its working condition is the high-pressure working condition of thermal power plant. which obviously does not match the pressure range (0.095~0.5 MPa) of this study, so the error was the largest. After comparison, it was shown that the two-phase average viscosity should be calculated by the Lin (1991) correlation when the pressure drop calculated with the homogeneous model, as the steam mass flux is 20~40 kg/(m2·s) for the conventional channel of this research scale.

Comparison of experimental data of Δpf data with prediction values of homogeneous models.

However, compared with the experimental data, the McAdams (1942) correlation and the Lin (1991) correlation were only 6.9% and 11.7% error, when the steam mass flux was 10 kg/(m2·s). Still large the error of the Cicchitti correlation was, reaching 57.4%, as shown in Fig. 4(b). As a result, the McAdams (1942) and Lin (1991) correlations can predict better the frictional pressure drop under the condition of lower steam mass flux. The Cicchitti correlation was not suitable for the prediction of this study because of its narrow applicability.

Fig. 4(c) presented that the three correlations had very large errors, and the smallest one was more than 151%, when compared with the experimental data with a steam mass flux of 50 kg/(m2·s). The reason was that the homogeneous model assumes that the two-phase fluid mixed fairly uniformly and its gas phase is dispersed in the liquid phase. It appears as a mist flow or bubble flow as for the flow pattern. However, when the steam mass flux is 50 kg/(m2·s) in this study, the flow pattern had been transformed into an annular flow, annular wavy flow, wavy flow and slug flow with a distinct steam-liquid interface. Therefore, the results of this experiment were different from the theory of the homogeneous model. The homogeneous model cannot be used to calculate the frictional pressure drop.

The homogeneous model uses the correlation of the two-phase average viscosity to confirm the frictional pressure drop. However, the results were not satisfactory for the steam mass flux range of this study. Table 3 showed MAPE of three correlations of homogeneous model when the steam mass flux was 10~50 kg/(m2·s).

Evaluation of different homogeneous models

By reviewing the original concept of the homogeneous model, it can be seen that the model treats the two-phase medium as a uniformly mixed medium, and its physical parameters are the average values of the corresponding two phases’ parameters, so the two-phase medium can be regarded as a single-phase fluid. Its flow pattern mainly presented a bubble flow (Fig. 5). The bubble flow closer to the state of single-phase water, as the bubbles were small and dispersed. With condensing, the flow state in the channel will quickly change to a fully condensed state with single-phase water filling the channel. Therefore, this flow state is classified as a mode dominated by single-phase heat transfer. The existence of a large amount of water in the fully condensed state will greatly interfere with the heat transfer effect. In this way, if there is a bubble flow in the MCD, its heat transfer efficiency will be relatively poor.

B ubble flow (typical flow patterns of homogeneous flow).

However, most of the flow patterns in this study were shown as annular flow and wave flow after observation (Fig. 6). The flow rates of the gas and liquid phases of these flow patterns were very different, resulting in non-uniform flow. For annular flow and wavy flow, the liquid film was thin and covered the upper and lower walls of the channel. It can be seen from Fig. 5 and Fig. 6 that although the thickness of the liquid film cannot be ignored, it was still much thinner than the bubble flow. Therefore, their heat transfer effect will be better than the effect of bubble flow. If the steam mass flux in the MCD is appropriate, the annular flow and wavy flow can maintain higher heat transfer efficiency. If the homogeneous model was chosen, there would be bias.

Flow patterns under this experimental condition.

### 3.3 The Results with separated flow model

Fig. 7 showed the comparison between the results of the separated flow model and the experimental data when the steam mass flux was G=20~40 kg/(m2·s). It can be found that the MAPE of the Chisholm B correlation was the smallest by 24.9%. First, the experimental object of Chisholm was a steam-water system, which was consistent with the experimental object of this study. In addition, this correlation was divided according to the ratio of the total vapor and liquid frictional pressure drop and the two-phase total mass flux, which was relatively so comprehensive that can include the mass flux range of this study (G=20~40 kg/(m2·s)). therefore, satisfactory results can be obtained with the Chisholm B correlation. However, both the Cavallini correlation and the Wilson correlation had larger errors, which were 37.3% and 44.2%, respectively. The reason why the Cavallini correlation imprecise was that the influence of Weber number (We) was included in the correlation. Weber number represents the ratio of inertial force to surface tension. The smaller Weber number is, the more important the surface tension is. If there is an interface between different fluids and the interface curvature is large, the Weber number is generally used to analyze the motion of two-phase fluids. It can be considered that the Cavallini correlation is more suitable for the calculation of frictional pressure drop of microchannels, e. g. small-scale problems just like capillary phenomena.

C omparison of experimental data of Δpf data with prediction values of separated flow models.

Within the range of working conditions involved in this study, the lower steam mass flux obtained the relatively smaller Weber number, which exaggerated the role of surface tension in this study. Moreover, due to the different qualitative properties of working mediums, the corresponding surface tensions of the Cavallini correlation and this experiment were never the same. Therefore, the error of the Cavallini correlation was relatively large. While for the Wilson correlation, the prediction of the two-phase frictional pressure drop is only related to the physical parameters of each working medium. Similar to the Cavallini correlation, the Wilson correlation is also a study on refrigerants. The physical properties of refrigerants are obviously different from steam, that caused the predicted values to deviate from the real situation in this study.

When comparing with the experimental data with a steam mass flux of 10 kg/(m2·s), it was found that the correlation errors of the Wilson et al. (2003) and the Cavallini et al. (2002) were so large that the two correlations could not be used to predict, as shown in Table 4. Although the error of the Chisholm B (1973) correlation was relatively large, it was obviously much smaller than the previous two correlations. Under lower steam mass flux, the Wilson et al. (2003) correlation was not suitable for the prediction in this study, because its prediction of the two-phase friction pressure drop was only related to the physical parameters of each working fluid. Furthermore, both the two-phase flow parameters and the inter-phase force were not considered. Besides, the object of this correlation was the refrigerant. For the Cavallini correlation, it was still not possible to predict the pressure drop in this study because it is suitable for the micro-channel.

Evaluation of different separated models

When compared with the experimental data under a steam mass flux of 50 kg/(m2·s), after increasing the steam mass flux, three separated flow models had larger errors in calculating the frictional pressure drop, also shown in Table 4. The Cavallini correlation and the Chisholm B correlation showed slightly smaller errors, but none of these three correlations could predict the frictional pressure drop under this steam mass flux.

The separated flow model uses the correlations of the converted coefficient of complete vapor/liquid phase for calculation of the frictional pressure drop. Result in its wider range of the steam mass flow range, this model is very suitable for the lower steam mass flux range of this study. Table 4 showed the MAPE of the three separated flow models with the steam mass flux being 10~50 kg/(m2·s). The results found that, within the steam mass flux range of this experiment, all the errors from the Chisholm B correlation were small. Especially when the steam mass flux was 20~40 kg/(m2·s), the error was only 24.9%, which showed that the Chisholm B correlation was very suitable for the prediction of frictional pressure drop within this steam mass flux range. In all steam mass flux ranges, the errors of the other two correlations were larger than those of the Chisholm B correlation.

## 4. Conclusions

The current study presented the characteristics of a two-phase flow pressure drop for steam condensation in small rectangular channel of MCD. Main conclusions are as follows:

• 1. For the steam-liquid two-phase condensate flow in the small rectangular channel of MCD, the main part of the total pressure drop was the friction pressure drop, followed by the local pressure drop. Both the gravity pressure drop and the acceleration pressure drop can be ignored.
• 2. Three homogeneous models and three separated flow models for the rectangular small channel proposed by previous studies were evaluated. The results showed that the homogeneous models were not appropriate enough, and the separated flow model was completely consistent with the experimental results. Among the separated flow models, the Chisholm B correlation was in best agreement with the experimental results, and its MAPE was 24.9% with steam mass flux of 20~40 kg/(m2·s).
• 3. Regardless of which model was used, the local pressure drops at the inlet and outlet could not be ignored compared to the total pressure drop. The local pressure drop at the inlet and outlet should be reduced when designing the MCD, due to its none contribution to the enhancement of heat transfer.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 51375286), Shaanxi Provincial Education Department Scientific Research Program of China (Grant No. 19JC004), Shaanxi Provincial Key Research and Development Project of China (Grant No. 2020GY-105).

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### Fig. 1.

Schematic diagram of working principle of multi-channel cylinder dryer.

### Fig. 2.

Schematic Diagram of Experiment Apparatus.

### Fig. 3.

Size of steam channel.

### Fig. 4.

Comparison of experimental data of Δpf data with prediction values of homogeneous models.

### Fig. 5.

B ubble flow (typical flow patterns of homogeneous flow).

### Fig. 6.

Flow patterns under this experimental condition.

### Fig. 7.

C omparison of experimental data of Δpf data with prediction values of separated flow models.

### Table 1.

Equivalent viscosity μTP correlations of homogeneous model

Models Equivalent viscosity correlations
Where μ represents viscocity, and x represents quality fraction of gas, and ρ represents density. Subscript TP means two–phase flow, and v represents gas phase, and subscript l represents liquid phase.
McAdams et al. (1942),4) Eq. [1] $\frac{1}{{\mu }_{\text{TP}}}=\frac{x}{{\mu }_{\text{v}}}+\frac{1-x}{{\mu }_{\text{1}}}$
Cicchitti et al. (1960),5) Eq. [2] ${\mu }_{\text{TP}}=x{\mu }_{\text{v}}+\left(1-x\right){\mu }_{\text{1}}$
Dukler et al. (1964),6) Eq. [3] ${\mu }_{\text{TP}}={\rho }_{\text{TP}}\cdot \left[x\cdot \frac{{\mu }_{\text{v}}}{{\rho }_{\text{v}}}+\left(1-x\right)\cdot \frac{{\mu }_{\text{1}}}{{\rho }_{\text{1}}}\right]$
Beattie-Whalley (1982),7) Eq. [4] ${\mu }_{\text{TP}}={\mu }_{\text{1}}-2.5{\mu }_{\text{1}}\cdot {\left[\frac{{x\rho }_{\text{1}}}{{x\rho }_{\text{1}}+\left(1-x\right){\rho }_{\text{v}}}\right]}^{2}+\frac{{x\rho }_{\text{1}}+\left(1.5{\mu }_{\text{1}}+{\mu }_{\text{v}}\right)}{{x\rho }_{\text{1}}+\left(1-x\right){\rho }_{\text{v}}}$
Lin et al. (1991),8) Eq. [5] ${\mu }_{\text{TP}}=\frac{{\mu }_{\text{1}}{\mu }_{\text{v}}}{{\mu }_{\text{v}}+{x}^{1.4}\left({\mu }_{\text{1}}-{\mu }_{\text{v}}\right)}$
Awad-Myuztchka (2008),9) Eq. [6] ${\mu }_{\text{TP}}={\mu }_{\text{g}}\frac{2{\mu }_{\text{g}}+{\mu }_{\text{1}}-2\left({\mu }_{\text{g}}+{\mu }_{\text{1}}\right)\left(1-x\right)}{{\mu }_{\text{g}}+{\mu }_{\text{1}}+\left({\mu }_{\text{g}}+1\right)\left(1-x\right)}$

### Table 2.

Two-phase multipliers of separated flow model

Models Correlations
Where subscript vo represents complete gas phase, and subscript lo represents complete liquid phase.
Lockhart-Martinelli (1949),10) Eq. [7] ${\varphi }_{1}^{2}=1+\frac{C}{X}+\frac{1}{{X}^{2}}$
C is an empirical constant set as 5, 10, 12, 20 according to the laminar flow and turbulent flow combination within different gas-liquid components, and X is the Martinelli parameter.
Chisholm B (1973),11) Eq. [8]
With the different range of Y, there are different calculation equations about B.
Friedel (1979),12) Eq. [9]
H is a parameter related to physical properties,
Zhang & Webb (2001),13) Eq. [10] ${\varphi }_{1\text{o}}^{2}={\left(1-x\right)}^{2}+2.87{x}^{2}{\left(\frac{p}{{p}_{c}}\right)}^{-1}+{1.68x}^{0.8}{\left(1-x\right)}^{0.25}{\left(\frac{p}{{p}_{c}}\right)}^{-1.64}$
Cavallini et al. (2002),14) Eq. [11] ${\varphi }_{1\text{o}}^{2}=E+\frac{1.262FH}{W{e}^{0.1458}}$
Wilson et al. (2003),15) Eq. [12] ${\varphi }_{1\text{o}}^{2}=12.82{\left(1-x\right)}^{1.8}{X}_{\text{tt}}^{-1.47}$
Xtt is a parameter related to quality and physical properties of two-phase flow,
${X}_{\text{tt}}={\left[\left(1-x\right)/x\right]}^{0.9}{\left({\rho }_{g}/{\rho }_{1}\right)}^{0.5}{\left({\mu }_{1}/{\mu }_{g}\right)}^{0.1}.$

### Table 3

Evaluation of different homogeneous models

Models MAPE/%
G=10 kg/(m2·s) G=20~40 kg/(m2·s) G=50 kg/(m2·s)
McAdams et al.4) 6.9 25.8 169
Cicchitti et al.5) 57.4 46.3 296
Lin et al.8) 11.7 25.6 181

### Table 4.

Evaluation of different separated models

Models MAPE/%
G=10 kg/(m2·s) G=20~40 kg/(m2·s) G=50 kg/(m2·s)
Chisholm B11) 55.7 24.9 68.9
Wilson et al.15) 204 44.2 129
Cavallini et al.14) 245 37.3 66.4