A Review on Condensation Process of Refrigerants in Horizontal Microfin Tubes: A Typical Example

Abstract

Heat transfer performance of refrigerant on the condensation process is very important in the designing of condensation equipment, especially in air conditioning and refrigeration systems. The outstanding advantages of microfin tubes are reducing the weight and size of condensers and also reducing the amount of refrigerant on the system. Reviewing the general concept of condensation and detailing the formation of flow pattern map that is also the procedure to determine heat transfer coefficient and pressure drop during condensation process of refrigerant inside the horizontal microfin tubes would be considered. Also, a typical example will be presented to illustrate a detailed procedure to calculate the value of heat transfer coefficient and pressure drop during the condensation process in horizontal microfin tubes. The data results show that microfin tube J60 with 60 number of fins inside 8.96 mm inner diameter, 0.2 mm height of fin, 40o of apex angle and 18o of helix angle, the condensation procedure of R1234ze at 35°C, heat flux of 8.62 kW/m2, and mass flux of 222 kg/m2s that could be estimated with heat transfer coefficient would be reduced from 8160.4 [W/m2 K] to 1413.8 [W/m2 K] follow with quality x changes from 0.99 to 0.01, and the maximum pressure drop of this process is 3173.8 [Pa/m].

1. Introduction

The condensation procedure of refrigerant is an important issue in the refrigeration industry. It is strongly supported to the optimization of equipment by using the right refrigerant with the very good thermodynamics properties and friendly environment and also by applying the microfin tube to conceive, design, implement, and operate in order to improve the heat transfer performance of system and to reduce the size, weight of the system, and the amount of refrigerant used in the system. The best performances of microfin tubes are consequences of the increase of the heat transfer and the increase of the pressure drop. To understand this phenomenon, many researchers these days are demonstrating to analyze and create the new models for flow pattern map, heat transfer, and pressure drop.

The group research of Thome and Cavallini with their colleagues [1, 2, 3, 4, 5, 6, 7, 8] is concentrated in condensation phenomena from the past to now. They presented so much information from experiments to empirical correlation for these fields, condensation on tubes (smooth, microfin tubes, micro channel, etc.) to consider about the formation of flow regime and also the penalization terms for heat transfer and pressure drop. With the group of Muzzio, Colombo and Lucchini [9, 10, 11, 12], they have been considering about an experimental facility to determine the flow patterns map and heat transfer performance of condensation on the difference geometries of microfin tubes with vary of refrigerant from R22 to R134a, and now they keep working with new refrigerant R1234ze and R1234yf. Besides, the group research of Sunden and Wu [13, 14, 15] is also focused on two-phase phenomena on microfin tube with some difference geometries for the current refrigerants by using their experimental data as well as collected data results from the others group to make an overview for their purposes.

More specific, about the flow pattern maps, until now is no new update of the maps for microfin tubes. Those are still presented the flow pattern map of condensation procedure of refrigerant on smooth horizontal tube, with the maps for condensation in microfin tubes are come from experimental map, it is not represented to build for other conditions. So that, apply the map on smooth tubes to estimate and get the stepping stone of the behavior and the changing of two-phase flow in the tube in circumstances. The map of Breber [16] was introduced in 1980 after 2 years the map of Tandon [17] was published, and in 2003 the map of Hajal [1], the group research of Thome and Cavallini, was presented.

About the performance of heat transfer, it cannot be denied that the outstanding performance of heat transfer in microfin tubes for condensation is normative to compare with smooth tubes. A huge number of experimental researches have been done to contribute to a lot of databases for establishing the predictive and empirical models to determine the heat transfer coefficients and pressure drop. General correlations for heat transfer of condensation processes on microfin tubes are published and used to estimate data results with an acceptable value to design condensers in refrigeration systems. Those are Yu and Koyama [18] Kedzieski and Goncalves [19], Han and Lee [20], Cavallini et al. [4]. Also, for the pressure drop of two-phase flow, some correlations were built for both evaporation and condensation procedure of Choi et al. [21], Goto et al. [22], and the others used for separated purposes. Some particular pressure drop correlations could be mentioned for condensation procedure of Kedzieski and Goncalves [19], Haraguchi et al. [23], and Cavallini et al. [24].

This chapter is focused on the condensation procedure of refrigerants in horizontal microfin tubes. Summary, the phenomenon, the effectiveness of fluid dynamics on the formation of flow pattern map, heat transfer coefficient, and pressure drop during condensation process under the empirical correlations are the examples to calculate condensation parameters.

2. Flow pattern map

With the aim of understanding the trend of increasing and decreasing heat transfer coefficient and pressure drop during the changing phase of refrigerants during condensation processes, the formation and transient of flow regime during the changing phase of condensation should be considered.

In order to draw a map of flow mechanism, there needs to be a classification of different regimes of flow in microfin tube, which relies on flow patterns such as slug, fully stratified, stratified wavy, helix, annular, and some regimes which are the combination of two or three flow pattern simultaneously happened. Many parameters have to define as an input parameter such as geometries of tube, refrigerants (consider about thermophysical and thermodynamic of refrigerant) heat flux, and mass flux that could take microfin tube from Colombo et al. as an example as shown in Figure 1, and the geometrical of microfin tube consists of number of fins, fin height, apex angle, and helix angle besides the inner and outer diameters of tubes.

Figure 1.

A geometrical parameter of microfin tube J60 [9].

It is could be considered about an experimental result of flow pattern map in microfin tube from the group of Colombo et al. [10], and they classified the pattern into five regimes: slug flow, stratified wavy flow, intermittent (slug, stratified wavy, and helix), helix flow, and annular flow. They showed their map as an indifferent dot of data points and compared it with the map of Hajal et al. as presented in Figure 2.

Figure 2.

Flow pattern results observed from experiment of Colombo et al. [10] draw on the map of Hajal [9].

2.1 The flow pattern map of Breber et al.

Breber et al. [16] published their map in 1980, and the map could be used for smooth tube that divided into four quadrant zones based on two parameters: dimensionless gas velocity J_G– and turbulent-turbulent Martinelli parameter X_tt (annular and mist-annular flow, wavy and stratified flow, slug and plug flow, and bubble flow) and some transition zones between each other.

Dimensionless gas velocity J_G is given by:

$$J_G=\frac{G.x}{\sqrt{g.d.{\rho }_G\left({\rho }_L-{\rho }_G\right)}}$$E1

Turbulent-turbulent Martinelli parameter X_tt is given by:

$$X_{\mathit{tt}}={\left(\frac{1-x}{x}\right)}^{0.9}{\left(\frac{{\rho }_v}{{\rho }_l}\right)}^{0.5}{\left(\frac{{\mu }_l}{{\mu }_v}\right)}^{0.1}$$E2

Transition criteria for flow regimes on the map:

Zone 1	Zone 2	Zone 3	Zone 4
Annular flow	Wavy and stratified	Slug flow	Bubble (at high reduced pressure)
JG > 1.5; X < 1.0	JG < 0.5; X < 1.0	JG < 1.5; X > 1.5	JG > 1.5; X > 1.5

2.2 The flow pattern map of Tandon et al.

The map of Tandon et al. [17] is presented in five zones of patterns: spray, annular and semi-annular, wavy, slug, and plug. They used two dimensionless numbers to draw their map: gas velocity J_G and (1 − α)/α with α is the void fraction.

Dimensionless gas velocity J_G is given by:

$$J_G=\frac{G.x}{\sqrt{g.d.{\rho }_G\left({\rho }_L-{\rho }_G\right)}}$$E3

Void fraction is defined from Smith’s equation (1970) is given as follows:

$$\alpha ={\left\{1+\left(\frac{{\rho }_G}{{\rho }_L}\right)\left(\frac{1-x}{x}\right)\left[0.4+0.6\sqrt{\frac{\frac{{\rho }_L}{{\rho }_G}+0.4\left(\frac{1-x}{x}\right)}{1+0.4\left(\frac{1-x}{x}\right)}}\right]\right\}}^{-1}$$E4

Transition criteria for flow regimes on the map.

Spray	Annular and semi-annular	Wavy	Slug	Plug
JG ≥ 6; (1-α)/α ≤ 0.5	1 ≤ JG ≤ 6; (1-α)/α ≤ 0.5	1 ≤ JG; (1-α)/α ≤ 0.5	0.01 ≤ JG ≤ 0.5; (1-α)/α ≥ 0.5	JG ≤ 0.01; (1-α)/α ≥ 0.5

2.3 The flow pattern map of Hajal et al.

The map of Hajal et al. [1] introduced the logarithmic mean void fraction instead of homogeneous void fraction to define the stratification of liquid and vapor and is based on the transient line between mass velocity and vapor quality to classify the regimes of patterns. The procedure to calculate step by step is shown as follows:

• Step 1: Calculate logarithmic mean void fractions ε based on homogeneous and nonhomogeneous void fractions:

  Homogeneous: ε_h

  $${\epsilon }_h={\left[1+\left(\frac{1-x}{x}\right)\left(\frac{{\rho }_V}{{\rho }_L}\right)\right]}^{-1}$$E5

  Nonhomogeneous: ε_ra

  $${\epsilon }_{\mathit{ra}}=\frac{x}{{\rho }_V}{\left(\left[1+0.12\left(1-x\right)\right]\left[\left(\frac{x}{{\rho }_V}\right)+\left(\frac{1-x}{{\rho }_L}\right)\right]+\frac{1.18\left(1-x\right){\left[\mathit{g\sigma }\left({\rho }_L-{\rho }_V\right)\right]}^{0.25}}{G{{\rho }_L}^{0.5}}\right)}^{-1}$$E6

  Logarithmic mean void fractions: ε

  $$\epsilon =\frac{{\epsilon }_h-{\epsilon }_{\mathit{ra}}}{\mathit{ln}\left(\frac{{\epsilon }_h}{{\epsilon }_{\mathit{ra}}}\right)}$$E7

• Step 2: Determine the stratified angle θ_strat and geometrical parameters for two-phase flow in a circular tube:

  + The stratified angle θ_strat:

  $${\theta }_{\mathit{\text{strat}}}=\begin{array}{l}2\pi -2\{\pi \left(1-\epsilon \right)+{\left(\frac{3\pi }{2}\right)}^{1/3}\left[1-2\left(1-\epsilon \right)+{\left(1-\epsilon \right)}^{\frac{1}{3}}-{\epsilon }^{\frac{1}{3}}\right]\\ -\frac{1}{200}\left(1-\epsilon \right)\epsilon \left[1-2\left(1-\epsilon \right)\right]\left[1+4\left({\left(1-\epsilon \right)}^2+{\epsilon }^2\right)\right]\}\end{array}$$E8

  + Geometrical parameters for two-phase flow in a circular tube:

  $$A_{\mathit{LD}}=\frac{A_L}{D^2}=\frac{A\left(1-\epsilon \right)}{D^2}$$E9
  $$A_{\mathit{VD}}=\frac{A_V}{D^2}=\frac{\mathit{A\epsilon }}{D^2}$$E10
  $$h_{\mathit{LD}}=0.5\left(1-\mathit{cos}\left(\frac{2\pi -{\theta }_{\mathit{\text{strat}}}}{2}\right)\right)$$E11
  $$P_{\mathit{iD}}=\mathit{sin}\left(\frac{2\pi -{\theta }_{\mathit{\text{strat}}}}{2}\right)$$E12

• Step 3: Use the value from step 1 and 2 to determine the transient lines:

  + Transient line of stratifine wavy: G_wavy:

  $$G_{\mathit{\text{wavy}}}=\begin{array}{l}{\left\{\frac{16A_{\mathit{VD}}^3\mathit{gD}{\rho }_L{\rho }_V}{x^2{\pi }^2{\left(1-{\left(2h_{\mathit{LD}}-1\right)}^2\right)}^{0.5}}\left[\frac{{\pi }^2}{25h_{\mathit{LD}}^2}{\left(\frac{\mathit{\text{We}}}{\mathit{Fr}}\right)}_L^{-1.023}+1\right]\right\}}^{0.5}+50\\ -75e^{-{\left(x^2-0.97\right)}^2/x\left(1-x\right)}\end{array}$$E13

G_wavy_min occur at x_min.

then, for x > x_min → G_wavy = G_wavy_min.

+ Transient line of fully stratified: G_strat:

$$G_{\mathit{\text{strat}}}={\left\{\frac{{\left(226.3\right)}^3{A}_{\mathit{LD}}{A}_{\mathit{VD}}^2{\rho }_V\left({\rho }_L-{\rho }_V\right){\mu }_{L}g}{x^2\left(1-x\right){\pi }^3}\right\}}^{1/3}+20x$$E14

+ Marttineli parameter: X_IA:

$$X_{\mathit{IA}}={\left\{\left[0.2914{\left(\frac{{\rho }_V}{{\rho }_L}\right)}^{-1/1.75}{\left(\frac{{\mu }_L}{{\mu }_V}\right)}^{-1/7}\right]+1\right\}}^{-1}$$E15

+ Transient line of mist flow: G_mist:

$$G_{\mathit{\text{mist}}}={\left\{\frac{7680A_{\mathit{VD}}^2\mathit{gD}{\rho }_L{\rho }_V}{x^2{\pi }^2\xi }{\left(\frac{\mathit{Fr}}{\mathit{\text{We}}}\right)}_L\right\}}^{0.5}$$E16

$$\xi ={\left[1.138+2\mathit{log}\left(\frac{\pi }{1.5A_{\mathit{LD}}}\right)\right]}^{-2}$$E17

G_mist_min occur at x_min.

then, for x > x_min → G_mist = G_mist_min.

+ Transient line of bubbly flow: G_bubbly:

$$G_{\mathit{\text{bubbly}}}={\left\{\frac{256A_{\mathit{VD}}{A}_{\mathit{LD}}^2{D}^{1.25}{\rho }_L\left({\rho }_L-{\rho }_V\right)g}{0.3164{\left(1-x\right)}^{1.75}{\pi }^2{P}_{\mathit{iD}}{\mu }_L^{0.25}}\right\}}^{1/1.75}$$E18

• Step 4: Draw the flow pattern map of condensation in 2D with the transient lines of mass velocity and vapor quality. The result of Hajal flow pattern map of condensation R134a in a smooth tube of 8 mm diameter at 40°C temperature with initial value of mass velocity 300 kg/m²s is shown in Figure 3.

Figure 3.

Flow pattern map of Hajal et al. [1].

3. Two-phase heat transfer coefficient in condensation

3.1 The correlation of Cavallini et al.

The correlation of Cavallini et al. [4] was published in 2009 which derived 558 data points, experimental results, and validated databases from other researches. The working range of microfin tubes: h/D < 0.04 (fin height/fin tip diameter) and helix angle is from 0° to 30°.

D: fin tip diameter; n_g: fins number; h: fin height; β: spiral angle; γ: apex angle.

Heat transfer coefficient of two phases could be determined from two zones as follows:

$$\alpha ={\left[{\alpha }_A^3+{\alpha }_D^3\right]}^{0.333}$$E19

Heat transfer coefficient for the ΔT independent zone α_A is given as follows:

$${\alpha }_A={\alpha }_{\mathit{AS}}.A.C$$E20

$${\alpha }_{\mathit{AS}}={\alpha }_{\mathit{LO}}\left[1+1.128x^{0.817}{\left(\frac{{\rho }_L}{{\rho }_G}\right)}^{0.3685}{\left(\frac{{\mu }_L}{{\mu }_G}\right)}^{0.2363}{\left(1-\frac{{\mu }_G}{{\mu }_L}\right)}^{2.144}{\mathit{Pr}}_L^{-0.1}\right]$$E21

$${\alpha }_{\mathit{LO}}=0.023\frac{{\lambda }_L}{D}{\mathit{Re}}_{\mathit{LO}}^{0.8}{\mathit{Pr}}_L^{0.4}=0.023\frac{{\lambda }_L}{D}{\left(\frac{\mathit{GD}}{{\mu }_L}\right)}^{0.8}{\mathit{Pr}}_L^{0.4}$$E22

$$A=1+1.119{\mathit{Fr}}^{-0.3821}{\left(\mathit{Rx}-1\right)}^{0.3586}$$E23

$$\mathit{Fr}=\frac{G^2}{\mathit{gD}{\left({\rho }_L-{\rho }_G\right)}^2}$$E24

$$\mathit{Rx}=\left\{\frac{2h.n_g\left[1-\mathit{sin}\left(\gamma /2\right)\right]}{\mathit{\text{πDcos}}\left(\gamma /2\right)}+1\right\}\frac{1}{\mathit{\text{cosβ}}}$$E25

$$C=1\mathit{\text{if}}\left(n_{\mathit{opt}}/n_g\right)\ge 0.8$$E26

$$\begin{array}{ccc}C={\left(n_{\mathit{opt}}/n_g\right)}^{1.904}& \mathit{\text{if}}& \left(n_{\mathit{opt}}/n_g\right)\end{array}<0.8$$E27

$$n_{\mathit{opt}}=4064.4D+23.257$$E28

Heat transfer coefficient for the ΔT dependent zone α_D is given as follows:

$${\alpha }_D=C\left[2.4x^{0.1206}{\left(\mathit{Rx}-1\right)}^{1.466}{C}_1^{0.6875}+1\right]{\alpha }_{D,S}+C\left(1-x^{0.087}\right)\mathit{Rx}.{\alpha }_{\mathit{LO}}$$E29

$${\alpha }_{D,S}=\frac{0.725}{1+0.741{\left(\frac{1-x}{x}\right)}^{0.3321}}{\left[\frac{{\lambda }_L^3.{\rho }_L\left({\rho }_L-{\rho }_G\right)g.h_{\mathit{LG}}}{{\mu }_L.D.∆T}\right]}^{0.25}$$E30

$$\begin{array}{ccc}{C}_1=1& \mathit{\text{if}}& J_G\end{array}\ge J_G^{\ast }$$E31

$$\begin{array}{ccc}{C}_1=J_G/J_G^{\ast }& \mathit{\text{if}}& J_G\end{array}<J_G^{\ast }$$E32

$$J_G=\frac{\mathit{xG}}{{\left[\mathit{gD}{\rho }_G\left({\rho }_L-{\rho }_G\right)\right]}^{0.5}}$$E33

$$J_G^{\ast }=0.6{\left\{{\left[\frac{7.5}{4.3X_{\mathit{tt}}^{1.111}+1}\right]}^{-3}+{2.5}^{-3}\right\}}^{-0.3333}$$E34

$$∆T=T_s-T_w$$E35

3.2 The correlation of Han et al.

Heat transfer coefficient correlation of Han et al. [20] was proposed in 2005, which was formed from a database of three various refrigerant R134A, R22, and R410A and four different inner diameter of microfin tubes di = 8.92, 6.46, 5.1, 4 mm. Some hypotheses were used to build this correlation as the uniform of liquid film thickness and turbulent flow in vapor core, neglect the radial temperature gradients and temperature in the vapor, and assume that the saturation temperature is constant at temperatures in the vapor core and at liquid-vapor interface.

Heat transfer coefficient of two-phase flow is given as follows:

$$h=\frac{\rho c_p{u}_t}{0.904{e^{+}}^{0.592}{\mathit{Pr}}^{0.729}+2.1335\mathit{ln}\left(\delta /e\right)}$$E36

Liquid film thickness is given as follows:

$$\delta =\frac{\left(1-\lambda \right)D_i}{4}$$E37

Void fraction from Yashar λ is given as follows:

$$\lambda ={\left[1+\left(\frac{1-x}{x}\right)\left(\frac{{\rho }_g}{{\rho }_l}\right)\left\{1+1.3{\mathit{Re}}_{\mathit{lo}}^{-0.19}{\left(\frac{{\rho }_l}{{\rho }_g}\right)}^{0.72}{\left(\frac{x}{1-x}\right)}^{0.5}\right\}\right]}^{-1}$$E38

With Re_lo is Reynolds of liquid only, e⁺ is roughness Reynolds number and u_t is turbulent friction (or shear velocity) and is given as follows

$$e^{+}=\frac{e{u}_t}{v}$$E39

$$u_t={\left(\frac{\tau }{\rho }\right)}^{0.5}={\left\{\frac{D_i}{4\rho }{\left(\frac{\mathit{dP}}{\mathit{dz}}\right)}_{\mathit{fr}}\right\}}^{0.5}$$E40

where e is the fin height, p = πD_i/ntanβ is the axial fin pitch, Di is the inside tube diameter, n is the number of fins, and β is the spiral angle.

If liquid film thickness is less than fin height, the term of logarithm ln(δ/e) is neglected.

3.3 The correlation of Kedzieski and Goncalves

Kedzieski and Goncalves in 1999 [19] collected 1704 data points from their experimental database to regress correlation. Their experimental data recorded in various working conditions as from four different refrigerants R134a, R410A, R125, and R32 in one microfin tube that has 8.91 mm root diameter, 0.2 mm fin height, and 50°apex angle, and the model was derived as follows:

$$\mathit{Nu}=\frac{h_{2\varnothing }{D}_h}{K_l}=2.256{\mathit{Re}}^{\beta 1}{\mathit{Ja}}^{\beta 2}{\mathit{Pr}}^{\beta 3}{\left(\frac{P_r}{P_c}\right)}^{\beta 4}{\left(-{\mathit{log}}_{10}\frac{P_r}{P_c}\right)}^{\beta 5}{\mathit{Sv}}^{\beta 6}$$E41

$\beta 1=0.303$; $\beta 2=0.232x_q$; $\beta 3=0.393$; $\beta 4=-0.578x_q^2$; $\beta 5=-0.474x_q^2$; $\beta 4=2.531x_q^{2.}$

The authors also supplied the simpler form with larger uncertainty:

$$\mathit{Nu}=\frac{h_{2\varnothing }{D}_h}{K_l}=4.94{\mathit{Re}}^{\beta 1}{\mathit{Pr}}^{\beta 3}{\left(\frac{P_r}{P_c}\right)}^{\beta 4}{\left(-{\mathit{log}}_{10}\frac{P_r}{P_c}\right)}^{\beta 5}{\mathit{Sv}}^{\beta 6}$$E42

$\beta 1=0.235$; $\beta 3=0.308$; $\beta 4=-1.16x_q^2$; $\beta 5=-0.887x_q^2$; $\beta 4=2.708x_q^2$

$$\mathit{Ja}=\frac{i_{\mathit{fg}}}{C_{\mathit{pr},l}∆T_s}$$E43

$$∆T_s=T_s-T_w$$E44

$$\mathit{Sv}=\frac{v_v-v_l}{v}$$E45

$$v=x_q{v}_v+\left(1-x_q\right)v_l$$E46

$$\mathit{Re}={\mathit{GD}}_h/{\mu }_{r,l}$$E47

where Dh is the hydraulic diameter.

3.4 The correlation of Yu and Koyama 1998

The correlation of Yu and Koyama [18] for microfin tubes was updated from the correlation of Haraguchi et al. for smooth tubes and was introduced in 1998. The procedure to determine heat transfer coefficient by Nusselt number is presented as follows:

The Nusselt number of two-phase flow is given by:

$$\mathit{Nu}={\left({\mathit{Nu}}_F^2+{\mathit{Nu}}_B^2\right)}^{1/2}$$E48

$${\mathit{Nu}}_F=0.152\left[0.3+0.1{\mathit{Pr}}_L^{1.1}\right]\left({\mathrm{\Phi }}_v/X_{\mathit{tt}}\right){\mathit{Re}}_L^{0.68}$$E49

$${\mathrm{\Phi }}_v=1.1+1.3{\left[\frac{G{X}_{\mathit{tt}}}{\sqrt{g{d}_i{\rho }_v\left({\rho }_l-{\rho }_v\right)}}\right]}^{0.35}$$E50

$${\mathit{Nu}}_B=\frac{0.725}{{\mathrm{\eta }}_A^{\frac{1}{4}}}.H\left(\xi \right){\left(\frac{\mathit{Ga}.{\mathit{Pr}}_L}{{\mathit{Ph}}_L}\right)}^{1/4}$$E51

$$H\left(\xi \right)=\xi +A\sqrt{\xi }\left(1-\sqrt{\xi }\right)$$E52

$$A=10{\left(1-\xi \right)}^{0.1}-8.0$$E53

where di is the mean inner diameter of tube, η_A is the enlargement ration of heat transfer area, and Ph_L is the phase change number.

$$\xi ={\left[1+\frac{{\rho }_v}{{\rho }_l}\left(\frac{1-x}{x}\right)\left(0.4+0.6\sqrt{\frac{\frac{{\rho }_v}{{\rho }_l}+0.4\frac{1-x}{x}}{1+0.4\frac{1-x}{x}}}\right)\right]}^{-1}$$E54

$${\mathit{Re}}_L=\frac{G\left(1-x\right)d_i}{{\mu }_L}$$E55

$$\mathit{Ga}=\frac{g{\rho }_L^3{d_i}^3}{{{\mu }_L}^2}$$E56

$${\mathit{Ph}}_L=\frac{{\mathit{Cp}}_L\left(T_{\mathit{sat}}-T_{\mathit{wi}}\right)}{∆h_{\mathit{vl}}}$$E57

4. Pressure drop

4.1 The correlation of Han et al.

Han et al. [20] not only presented the correlation for heat transfer coefficient as aforementioned but also introduced a new correlation to define pressure drop. It is could be considered as follows:

Friction pressure gradient of two phases is given as follows:

$${\left(\frac{\mathit{dP}}{\mathit{dz}}\right)}_{\mathit{fr}}={\mathrm{\Phi }}_l^2\frac{f_l{\left[G\left(1-x\right)\right]}^2}{2D_i{\rho }_l}$$E58

$$f_l=0.193{\left[\frac{G\left(1-x\right)D_i}{{\mu }_l}\right]}^{-0.024}{\left(\frac{p}{e}\right)}^{-0.539}$$E59

$${\mathrm{\Phi }}_l^2=2.684X_{\mathit{tt}}^{-1.946}$$E60

$$X_{\mathit{tt}}={\left(\frac{1-x}{x}\right)}^{0.9}{\left(\frac{{\rho }_v}{{\rho }_l}\right)}^{0.5}{\left(\frac{{\mu }_l}{{\mu }_v}\right)}^{0.1}$$E61

4.2 The correlation of Kedzieski and Goncalves

In 1999, Kedzieski and Goncalves [19] modified the correlation of Pierre 1964 with confirmation by their own experimental data from four refrigerants such as R134a, R410A, R125, and R32 in their microfin tube that has 8.91 mm root diameter, 0.2 mm fin height, and 50°apex angle, and presented the correlation as follows:

$$\frac{∆p}{L}=\frac{∆p_f}{L}+\frac{∆p_m}{L}=G^2\left[f\frac{\left(v_{\mathit{tp},\mathit{\text{out}}}+v_{\mathit{tp},\mathit{\text{in}}}\right)}{d_h}+\frac{\left(v_{\mathit{tp},\mathit{\text{out}}}-v_{\mathit{tp},\mathit{\text{in}}}\right)}{L}\right]$$E62

$$d_h=4A_c\mathit{\text{cosβ}}/\left(n.S_p\right)$$E63

$$f=0.002275+0.00933{\mathit{exp}}^{\left[e/\left(-0.003d_{\mathit{fr}}\right)\right]}{\mathit{Re}}_{h,\mathit{LO}}^{-1/\left(4.16+532e/d_{\mathit{fr}}\right)}{K}_f^{0.211}$$E64

$${\mathit{Re}}_{h,\mathit{LO}}=G{d}_h/{\mu }_l$$E65

$$K_f=\frac{∆x.h_{\mathit{lv}}}{\mathit{gL}}$$E66

where d_h is the hydraulic diameter, d_fr is the fin root diameter, and e is the fin height.

4.3 The correlation of Harraguchi

Harraguchi et al. [23] presented their correlation for two-phase friction pressure drop from their experimental results with R134a, R123, and R22 in a horizontal smooth tube and a horizontal microfin tube in 1993.

Two-phase friction pressure drop is given as follows:

$${\left(\frac{\mathit{dP}}{\mathit{dz}}\right)}_f={\mathrm{\Phi }}_v^2{\left(\frac{\mathit{dP}}{\mathit{dz}}\right)}_v={\mathrm{\Phi }}_v^{2}2f_{e,v}{\left(\mathit{Gx}\right)}^2/\left({\rho }_v{d}_e\right)$$E67

$$d_e=\sqrt{\frac{4A_c}{\pi }}$$E68

$$f_{e,v}=0.046{\mathit{Re}}_{e,v}^{-0.2}$$E69

$${\mathrm{\Phi }}_v=1.1+1.3{\left\{X_{\mathit{tt}}G/{\left[g{d}_e{\rho }_v\left({\rho }_l-{\rho }_v\right)\right]}^{0.5}\right\}}^{0.35}$$E70

$${\mathit{Re}}_{e,v}=\mathit{Gx}{d}_e/{\mu }_v$$E71

$$X_{\mathit{tt}}={\left(\frac{1-x}{x}\right)}^{0.9}{\left(\frac{{\rho }_v}{{\rho }_l}\right)}^{0.5}{\left(\frac{{\mu }_l}{{\mu }_v}\right)}^{0.1}$$E72

5. Example

The condensation of refrigerant R1234ze inside horizontal microfin tube at 35°C, heat flux of 8.62 kW/m², and mass flux of 222 kg/m²s using the microfin tube J60 with the geometrical parameters: 60 number of fins, 8.96 mm inner surface diameter, 0.2 mm fin height, 18^o of helix angle, and 40^o of apex angle is considered. Heat transfer coefficient and pressure drop during condensation process are determined. Saturation to wall temperature difference is kept constant (5°C).

Solve:

Thermal properties of refrigerant R1234ze at saturation temperature t_sat = 35°C is listed in Table 1.

Refrigerant: R1234ze, molar mass: M = 114 g/mol, p_critical = 3.64 MPa

Reduce pressure: p_r = p_sat/p_critical = 0.66756/3.64 = 0.1834

Heat flux: q = 15.3 kW/m²

Mass flux: G = 220 kg/m² s

Microfin tube: N = 60 number of fins, dr = 8.96 mm, h = 0.2 mm, β = 18^o; γ = 40^o

5.1 Heat transfer coefficients

Quality x = 0.5 could be taken as an example to present the procedure to calculate heat transfer coefficient and pressure drop of boiling refrigerant inside horizontal microfin tube, with the data point of quality x from 0 to 1 that can be determined with the same method.

Applied the correlation of Cavallini et al. to calculate heat transfer coefficient as below:

• Heat transfer coefficient of two phases could be determined from two zones as follows:

  $$\alpha ={\left[{\alpha }_A^3+{\alpha }_D^3\right]}^{0.333}$$

• Heat transfer coefficient for the ΔT independent zone α_A is give as follows:

  $${\alpha }_A={\alpha }_{\mathit{AS}}.A.C$$

+ Calculate α_AS:

$${\alpha }_{\mathit{LO}}=0.023\frac{{\lambda }_L}{D}{\left(\frac{\mathit{GD}}{{\mu }_L}\right)}^{0.8}{\mathit{Pr}}_L^{0.4}=0.023\frac{7.098.{10}^{-2}}{8.96.{10}^{-3}}{\left(\frac{220.\left(8.96.{10}^{-3}\right)}{1.772.{10}^{-4}}\right)}^{0.8}{3.548}^{0.4}$$

$${\alpha }_{\mathit{LO}}=521.87$$

$${\alpha }_{\mathit{AS}}={\alpha }_{\mathit{LO}}\left[1+1.128x^{0.817}{\left(\frac{{\rho }_L}{{\rho }_G}\right)}^{0.3685}{\left(\frac{{\mu }_L}{{\mu }_G}\right)}^{0.2363}{\left(1-\frac{{\mu }_G}{{\mu }_L}\right)}^{2.144}{\mathit{Pr}}_L^{-0.1}\right]$$

$${\alpha }_{\mathit{AS}}=\begin{array}{l}521.87[1+1.128\left({0.5}^{0.817}\right){\left(\frac{1129}{35.32}\right)}^{0.3685}{\left(\frac{1.772.{10}^{-4}}{1.269.{10}^{-5}}\right)}^{0.2363}(1\\ {-\frac{1.269.{10}^{-5}}{1.772.{10}^{-4}})}^{2.144}{3.548}^{-0.1}]\end{array}$$

$${\alpha }_{\mathit{AS}}=2199.82$$

+ Calculate A:

$$\mathit{Fr}=\frac{G^2}{\mathit{gD}{\left({\rho }_L-{\rho }_G\right)}^2}=\frac{{220}^2}{9.81.\left(8.96.{10}^{-3}\right){\left(1129-35.32\right)}^2}=0.46035$$

$$\mathit{Rx}=\left\{\frac{2h.n_g\left[1-\mathit{sin}\left(\gamma /2\right)\right]}{\mathit{\text{πDcos}}\left(\gamma /2\right)}+1\right\}\frac{1}{\mathit{\text{cosβ}}}$$

$$\mathit{Rx}=\left\{\frac{2.\left(2.{10}^4\right).60.\left[1-\mathit{sin}\left(40/2\right)\right]}{\pi .\left(8.96.{10}^{-3}\right).\mathit{cos}\left(40/2\right)}+1\right\}\frac{1}{cos18}=1.79$$

$$A=1+1.119{\mathit{Fr}}^{-0.3821}{\left(\mathit{Rx}-1\right)}^{0.3586}$$

$$A=1+1.119\left({0.46035}^{-0.3821}\right).{\left(1.79-1\right)}^{0.3586}=2.383$$

+ Calculate C:

$$n_{\mathit{opt}}=4064.4D+23.257=4064.4\left(8.96.{10}^{-3}\right)+23.257=59.7$$

$$\begin{array}{ccc}C=1& \mathit{\text{if}}& \left(n_{\mathit{opt}}/n_g\right)\end{array}\ge 0.8$$

$$\begin{array}{ccc}C={\left(n_{\mathit{opt}}/n_g\right)}^{1.904}& \mathit{\text{if}}& \left(n_{\mathit{opt}}/n_g\right)\end{array}<0.8$$

In this case:

$$\frac{n_{\mathit{opt}}}{n_g}=\frac{59.7}{60}=0.995\ge 0.8$$

• C = 1

  Heat transfer coefficient for the ΔT independent zone α_A is given as follows:

  $${\alpha }_A={\alpha }_{\mathit{AS}}.A.C=\left(2199.82\right).\left(2.383\right).\left(1\right)=5242.213$$

• Heat transfer coefficient for the ΔT dependent zone α_D is given as follows:

+ Calculate C1:

$$J_G=\frac{\mathit{xG}}{{\left[\mathit{gD}{\rho }_G\left({\rho }_L-{\rho }_G\right)\right]}^{0.5}}=\frac{\left(0.5\right).\left(220\right)}{{\left[\left(9.81\right).\left(8.96.{10}^{-3}\right).\left(35.32\right).\left(1129-35.32\right)\right]}^{0.5}}$$

$$J_G=1.8878$$

Martinelli number is given as follows:

$${\mathrm{X}}_{\mathrm{tt}}={\left(\frac{1-\mathrm{x}}{\mathrm{x}}\right)}^{0.9}{\left(\frac{{\mathrm{\rho }}_{\mathrm{G}}}{{\mathrm{\rho }}_{\mathrm{L}}}\right)}^{0.5}{\left(\frac{{\mathrm{\mu }}_{\mathrm{L}}}{{\mathrm{\mu }}_{\mathrm{G}}}\right)}^{0.1}={\left(\frac{1-0.5}{0.5}\right)}^{0.9}{\left(\frac{35.32}{1129}\right)}^{0.5}{\left(\frac{1.772.{10}^{-4}}{1.269.{10}^{-5}}\right)}^{0.1}=0.2302$$

$$J_G^{\ast }=0.6{\left\{{\left[\frac{7.5}{4.3X_{\mathit{tt}}^{1.111}+1}\right]}^{-3}+{2.5}^{-3}\right\}}^{-0.3333}$$

$$J_G^{\ast }=0.6{\left\{{\left[\frac{7.5}{4.3\left({0.2302}^{1.111}\right)+1}\right]}^{-3}+{2.5}^{-3}\right\}}^{-0.3333}=1.3994$$

$$\begin{array}{ccc}{C}_1=1& \mathit{\text{if}}& J_G\end{array}\ge J_G^{\ast }$$

$$\begin{array}{ccc}{C}_1=J_G/J_G^{\ast }& \mathit{\text{if}}& J_G\end{array}<J_G^{\ast }$$

In this case: J_G = 1.8878 > J*_G = 1.3994 = > C1 = 1

+ Calculate α_D,S:

$$∆T=T_s-T_w=5$$

$${\alpha }_{D,S}=\frac{0.725}{1+0.741{\left(\frac{1-x}{x}\right)}^{0.3321}}{\left[\frac{{\lambda }_L^3.{\rho }_L\left({\rho }_L-{\rho }_G\right)g.h_{\mathit{LG}}}{{\mu }_L.D.∆T}\right]}^{0.25}$$

$${\alpha }_{D,S}=\frac{0.725}{1+0.741{\left(\frac{1-0.5}{0.5}\right)}^{0.3321}}{\left[\frac{{\left(7.098.{10}^{-2}\right)}^3.\left(1129\right).\left(1129-35.32\right).\left(9.81\right).\left(159.86\right)}{\left(1.772.{10}^{-4}\right).\left(8.96.{10}^{-3}\right).5}\right]}^{0.25}$$

$${\alpha }_{D,S}=226.31$$

= > Heat transfer coefficient for the ΔT dependent zone α_D is given as follows:

$${\alpha }_D=C\left[2.4x^{0.1206}{\left(\mathit{Rx}-1\right)}^{1.466}{C}_1^{0.6875}+1\right]{\alpha }_{D,S}+C\left(1-x^{0.087}\right)\mathit{Rx}.{\alpha }_{\mathit{LO}}$$

$$\begin{array}{l}{\alpha }_D=1.\left[2.4\left({0.5}^{0.1206}\right).{\left(1.79-1\right)}^{1.466}\left(1^{0.6875}\right)+1\right].\left(226.31\right)+1.\left(1-{0.5}^{0.087}\right).\left(1.79\right).\\ \left(521.87\right)\end{array}$$

$${\alpha }_D=634.51$$

Heat transfer coefficient of condensation at x = 0.5 is given as follows:

$$\alpha ={\left[{\alpha }_A^3+{\alpha }_D^3\right]}^{0.333}$$

$$\alpha ={\left[{5242.213}^3+{634.51}^3\right]}^{0.333}=5200.58W/m^{2}K$$

5.2 Pressure drop

5.2.1 The correlation of Harraguchi

Harraguchi et al. [23] presented their correlation for two-phase friction pressure drop from their experimental results with R134a, R123, and R22 in a horizontal smooth tube and a horizontal microfin tube in 1993.

Two-phase friction pressure drop is given as follows:

$${\mathrm{X}}_{\mathrm{tt}}={\left(\frac{1-\mathrm{x}}{\mathrm{x}}\right)}^{0.9}{\left(\frac{{\mathrm{\rho }}_{\mathrm{G}}}{{\mathrm{\rho }}_{\mathrm{L}}}\right)}^{0.5}{\left(\frac{{\mathrm{\mu }}_{\mathrm{L}}}{{\mathrm{\mu }}_{\mathrm{G}}}\right)}^{0.1}={\left(\frac{1-0.5}{0.5}\right)}^{0.9}{\left(\frac{35.32}{1129}\right)}^{0.5}{\left(\frac{1.772.{10}^{-4}}{1.269.{10}^{-5}}\right)}^{0.1}=0.2302$$

$$d_e=\sqrt{\frac{4A_c}{\pi }}\cong D=8.96.{10}^{-3}\mathit{mm}$$

$${\mathit{Re}}_{e,v}=\frac{G.x.d_e}{{\mu }_v}=\frac{220x0.5x0.00896}{1.269.{10}^{-5}}=77667.45$$

$$f_{e,v}=0.046{\mathit{Re}}_{e,v}^{-0.2}=\left(0.046\right).\left({77667.45}^{-0.2}\right)=0.004838$$

$${\mathrm{\Phi }}_v=1.1+1.3{\left\{X_{\mathit{tt}}G/{\left[g{d}_e{\rho }_v\left({\rho }_l-{\rho }_v\right)\right]}^{0.5}\right\}}^{0.35}$$

$${\mathrm{\Phi }}_v=1.1+1.3{\left\{\left(0.2302\right).220/{\left[\left(9.81\right).\left(0.00896\right).\left(35.32\right).\left(1129-35.32\right)\right]}^{0.5}\right\}}^{0.35}$$

$${\mathrm{\Phi }}_v=2.3378$$

$${\left(\frac{\mathit{dP}}{\mathit{dz}}\right)}_f={\mathrm{\Phi }}_v^2{\left(\frac{\mathit{dP}}{\mathit{dz}}\right)}_v=\frac{{\mathrm{\Phi }}_v^{2}2f_{e,v}{\left(\mathit{Gx}\right)}^2}{\left({\rho }_v{d}_e\right)}=$$

$${\left(\frac{\mathrm{dP}}{\mathrm{dz}}\right)}_{\mathrm{f}}={\mathrm{\Phi }}_{\mathrm{v}}^2{\left(\frac{\mathrm{dP}}{\mathrm{dz}}\right)}_{\mathrm{v}}=\frac{{\mathrm{\Phi }}_{\mathrm{v}}^{2}2{\mathrm{f}}_{\mathrm{e},\mathrm{v},\mathrm{Go}}{\left(\mathrm{Gx}\right)}^2}{\left({\mathrm{\rho }}_{\mathrm{v}}{\mathrm{d}}_{\mathrm{e}}\right)}=\frac{{\left(2.3378\right)}^2.2.\left(0.004838\right){\left(220.\left(0.5\right)\right)}^2}{\left(35.32).(0.00896\right)}=2022.105\left(\frac{\mathit{Pa}}{m}\right)$$

Condensation process of R1234ze at 35°C inside horizontal microfin tube J60 with quality change from x = 0.99 to x = 0.01 could be estimated with heat transfer performance which are performed in heat transfer and pressure drop in Table 2.

Property	Unit	Value	Property	Unit	Value
Tsat	[°C]	35	Psat	[MPa]	0.66756
ρL	[kg/m^3]	1129	λL	[W/m-K]	7.098E−02
ρV	[kg/m^3]	35.32	λV	[W/m-K]	1.452E−02
νL	[m^3/kg]	8.857E−04	μL	[Pa-s]	1.772E−04
νv	[m^3/kg]	2.831E−02	μV	[Pa-s]	1.269E−05
hLV	[kJ/kg]	159.86	PrL	[−]	3.548
CpL	[kJ/kg-K]	1.421	PrV	[−]	0.904
CpV	[kJ/kg-K]	1.034	σ	[N/m]	7.578E−03

Table 1.

Properties of R1234ze.

x	α	dP/dz
[−]	[W/m^2K]	[Pa/m]
0.99	8160.4	2448.5
0.95	7930.4	2968.4
0.90	7641.1	3151.4
0.85	7349.0	3173.8
0.80	7053.8	3112.0
0.75	6755.3	2996.4
0.70	6453.0	2843.2
0.65	6146.9	2662.9
0.60	5836.4	2462.4
0.55	5521.1	2247.4
0.50	5200.6	2022.1
0.45	4874.1	1790.4
0.40	4541.0	1555.5
0.35	4200.2	1320.5
0.30	3850.1	1088.6
0.25	3489.1	862.8
0.20	3114.7	646.6
0.15	2722.8	444.1
0.10	2306.4	260.7
0.05	1849.2	104.9
0.01	1413.8	13.1

Table 2.

Heat transfer performance of condensation.

6. Conclusions

An overview of the condensation procedure of refrigerant inside microfin tubes has been done with the flow pattern maps and heat transfer performance. The list of some correlations to determine heat transfer and pressure drop during the condensation process has been presented. Besides, an example has been introduced step by step to define heat transfer coefficient and pressure drop of condensation process with refrigerant R1234ze inside horizontal microfin tube J60. Until now, this field is still not getting too much data to get the general correlation for all types of microfin tubes and all kinds of refrigerants. Therefore, these fields are strongly attracted by scientists around the world.

Acknowledgments

We acknowledge the support of time and facilities from Ho Chi Minh City University of Technology (HCMUT), VNU-HCM for this study.