**2.2 The flow pattern map of Zhuang et al.**

The flow pattern map of Zhuang et al. [4] in **Figure 4** was form from the model of Kim et al. combine with their experimental data for R170 with working range of saturation pressures from 1.5 MPa to 2.5 MPa on mass flux from 100 kg/m<sup>2</sup> s to 250 kg/m<sup>2</sup> s. The map was built in the terms of dimensionless weber number We and Mattinelli parameter Xtt.

**Figure 4.** *Flow pattern map of Zhuang et al. [4].*

*A Review on Convective Boiling Heat Transfer of Refrigerants in Horizontal Microfin-Tubes… DOI: http://dx.doi.org/10.5772/intechopen.100024*

Four transition curves were presented to draw the transition line to separate five different zones, each zone is named as smooth annular, wavy annular, transition, slug, plug.

Marttinelli number: Xtt

$$\mathbf{X\_{tt}} = \left(\frac{\mathbf{1} - \mathbf{x}}{\mathbf{x}}\right)^{0.9} \left(\frac{\rho\_{\rm G}}{\rho\_{\rm L}}\right)^{0.5} \left(\frac{\mu\_{\rm l}}{\mu\_{\rm G}}\right)^{0.1} \tag{12}$$

Determine Weber number (We) based on the Reynold range of liquid flow:

$$\text{Re}\_{\text{L}} = \frac{\mathbf{G}(\mathbf{1} - \mathbf{x})\mathbf{D}}{\mu\_{\text{L}}} \tag{13}$$

With Re <sup>L</sup> ≤ 1250

$$\mathrm{We}^\* = 2.45 \frac{\mathrm{Re}\_G^{0.64}}{\mathrm{Su}\_G^{0.3} \left(1 + 1.09 \mathrm{X}\_{\mathrm{tt}}^{0.039} \right)^{0.4}} \tag{14}$$

With Re <sup>L</sup> > 1250

$$\text{We}^\* = 0.85 \frac{\text{Re}\_{\text{G}}^{0.79} \text{X}\_{\text{tt}}^{0.157}}{\text{Su}\_{\text{G}}^{0.3} \left(1 + 1.09 \text{X}\_{\text{tt}}^{0.039}\right)^{0.4}} \left[ \left(\frac{\mu\_{\text{G}}}{\mu\_{\text{L}}}\right)^2 \left(\frac{\rho\_{\text{L}}}{\rho\_{\text{G}}}\right) \right]^{0.084} \tag{15}$$

$$\text{Su}\_{\text{G}} = \frac{\rho\_{\text{G}} \sigma \text{D}}{\mu\_{\text{G}}^2} \tag{16}$$

The transition line between patterns: Smooth-annular to wavy-annular flow:

$$\text{We}^\* = \text{29.25X}^{0.27}\_{\text{tt}} \tag{17}$$

Wavy-annular to transition flow:

$$\mathbf{We}^\* = \mathbf{18.91X}^{0.33}\_{\text{tt}} \tag{18}$$

Transition to slug flow:

$$\text{We}^\* = \text{9.62X}^{0.35}\_{\text{tt}} \tag{19}$$

Slug to plug flow:

$$\text{We}^\* = 4.38 \text{X}^{0.45}\_{\text{tt}} \tag{20}$$

#### **2.3 The flow pattern map of Yang et al.**

The map of Yang et al. [5] introduced two maps in **Figure 5**, one map is presented for plug flow, slug flow, other is separate between slug flow and annular flow. Two transitions line on two difference maps as a function of Martinelli parameter Xtt, the transition lines based on three dimensionless numbers K1, K2, K3 which depended on the inertia force, surface tension force, shear force, gravity force and evaporation momentum force.

**Figure 5.** *Flow pattern map of Yang et al. [5].*

Dimensionless number K1 is a ratio of evaporation momentum force with inertia force:

$$\mathbf{K}\_{\mathbf{l}} = \frac{\text{evaporation momentum force}}{\text{inertia force}} = \frac{\left(\frac{\mathbf{q}}{\mathbf{h}\_{\text{lv}}}\right)^{2} \frac{1}{\rho\_{\text{v}}}}{\frac{\mathbf{G}^{2}}{\rho\_{\text{l}}}} = \left(\frac{\mathbf{q}}{\mathbf{h}\_{\text{lv}}}\right)^{2} \frac{\rho\_{\text{l}}}{\rho\_{\text{v}}} \tag{21}$$

Dimensionless number K2 is a ratio of evaporation momentum force with surface tension force:

$$\mathbf{K}\_2 = \frac{\text{evaporation momentum force}}{\text{surface tension force}} = \frac{\left(\frac{\mathbf{q}}{\mathbf{h}\_\mathbf{v}}\right)^2 \frac{1}{\rho\_\mathbf{v}}}{\frac{\sigma}{\mathbf{D}}} = \left(\frac{\mathbf{q}}{\mathbf{h}\_\mathbf{v}}\right)^2 \frac{\mathbf{D}}{\sigma \rho\_\mathbf{v}}\tag{22}$$

Dimensionless number K3 is a ratio of shear force with gravity force:

$$\mathbf{K}\_3 = \frac{\text{shear force}}{\text{gravity force}} = \frac{\frac{\mu\_\text{l}}{\rho\_\text{l}} \frac{\text{G}}{\text{D}}}{(\rho\_\text{l} - \rho\_\text{v})\text{gD}} = \frac{\mu\_\text{l}\text{G}}{(\rho\_\text{l} - \rho\_\text{v})\rho\_\text{l}\text{gD}^2} \tag{23}$$

The transition line from plug to slug and slug to annular based on function of Xtt:

Plug to slug:

$$\mathbf{K} = \mathbf{K}\_{\mathbf{P}-\mathbf{S}} = \mathbf{K}\_1^{-0.8385} \mathbf{K}\_2^{1.1388} \mathbf{K}\_3^{-0.3993} = \mathbf{14.87} \mathbf{X}\_{\mathbf{tt}}^{1.269} \tag{24}$$

Slug to annular:

$$\mathbf{K} = \mathbf{K}\_{\text{S}-\text{A}} = \mathbf{K}\_1^{-0.2963} \mathbf{K}\_2^{0.3620} \mathbf{K}\_3^{0.1941} = \mathbf{0}.3044 \mathbf{X}\_{\text{tt}}^{0.5671} \tag{25}$$

#### **3. Two phases heat transfer coefficient in boiling**

Due to the changing phase during the convective boiling, the proportion of liquid and vapor is also changed, which affected the mechanism of boiling. In order to indicate the boiling heat transfer coefficient, the nucleate boiling and convective boiling are considered. More detailed about the methodology, the boiling number and some other factors are presented in the method of each group authors as below: *A Review on Convective Boiling Heat Transfer of Refrigerants in Horizontal Microfin-Tubes… DOI: http://dx.doi.org/10.5772/intechopen.100024*

#### **3.1 The correlation of Han et al.**

Base on the basic form of boiling heat transfer coefficient, Han et al. [20] was established their correlation from the updating their experimental result. The experimental data results were done on the working range of mass flux G = [100; 250] kg/m<sup>2</sup> s, heat flux q = [11.76; 52.94] kW/m2 , temperature T = [�5, 8]°C with fluid R161 for microfin tube with 6.34 mm average inside diameter, 15° helix angle, 30° fin angle, 0.1 mm fin height, 65 number of fins.

Heat transfer coefficient of two-phase flow:

$$\mathbf{h}\_{\mathbf{r},\mathbf{tp}} = \mathbf{F}\mathbf{h}\_{\mathbf{r},\mathbf{l}} + \mathbf{S}\mathbf{h}\_{\mathbf{r},\mathbf{nb}} \tag{26}$$

Convective heat transfer with convective fin factor:

$$\mathbf{h}\_{\rm r,l} = \mathbf{E}\_{\rm RB} \mathbf{h}\_{\rm l} \tag{27}$$

$$\mathbf{h}\_{\rm l} = \mathbf{0}.023 \,\mathrm{Re}^{0.8}\_{\rm l} \mathrm{Fr}^{0.4}\_{\rm l} \left(\frac{\mathbf{k}\_{\rm l}}{\mathbf{d}\_{\rm l}}\right) \tag{28}$$

$$\mathbf{E\_{RB}} = \left\{ \mathbf{1} + \left[ 2.64 (\,\mathrm{Re}\,\mathrm{l})^{0.036} \left( \frac{\mathrm{e}}{\mathrm{d}\_{\mathrm{i}}} \right)^{0.212} \left( \frac{\mathrm{p}}{\mathrm{d}\_{\mathrm{i}}} \right)^{-0.21} \left( \frac{\mathrm{\beta}}{90^{\mathrm{o}}} \right)^{0.29} (\mathrm{Pr\_{L}})^{-0.024} \right]^{7} \right\}^{1/7} \tag{29}$$

$$\mathrm{Re}\_{\mathrm{rl}} = \frac{\mathbf{G}(\mathbf{1} - \mathbf{x})\mathbf{d}\_{\mathrm{i}}}{\mu\_{\mathrm{l}}} \tag{30}$$

Heat transfer of nuclear boiling

$$\mathbf{h\_{r,nb}} = \mathbf{55P\_r^{0.12}} \left(-\log\_{10} \mathbf{P\_r}\right)^{-0.55} \mathbf{M}^{-0.5} \mathbf{q}^{0.67} \tag{31}$$

The new function of F and S

$$\mathbf{F} = \mathbf{1} + \mathbf{a} \mathbf{B} \mathbf{o}^{1.16} + \mathbf{b} \left(\frac{\mathbf{1}}{\mathbf{X}\_{\text{tt}}}\right)^{0.86} \tag{32}$$

$$\mathbf{S} = \frac{1}{\mathbf{1} + \mathbf{c} \mathbf{F}^{\mathrm{d}} \operatorname{Re}\_{\mathrm{l}}^{1.17}} \tag{33}$$

In which a = 7196.741; b = 1.5135; c = 2.703; d = 1.94,

where e: microfin height; p: axial pitch from fin to fin; β: helix angle; N: number of fin; Pr: reduced pressure.

#### **3.2 The correlation of Rollmann and Spindler**

The model of Rollmann and Spindler [13] for heat transfer coefficient was derived with total 1614 data points experiment for refrigerant R407C on microfin tube at 8.95 mm fin root diameter, 0.24 mm fin height, 15° Helix angle, 25° Apex angle and 55 fins. Heat flux q = [1000; 20000] W/m<sup>2</sup> , mass flux G = [25; 300] kg/m<sup>2</sup> s, saturation temperature Tsat = [�30; 10]°C,

The model defined by Nuselt number:

$$\text{Nu}(\mathbf{x}, \mathbf{B}\mathbf{o}, \mathbf{R}\mathbf{e}, \mathbf{P}\mathbf{r}) = \mathbf{C}\_{4} \left(\frac{\mathbf{C}\_{1}}{\mathbf{P}\mathbf{r}^{2}} + \mathbf{C}\_{2}\right) \text{Re}^{2/3} [\ln \left(\mathbf{B}\mathbf{o}\right) + \mathbf{C}\_{3}] \mathbf{x}^{\left(\frac{\mathbf{C}\_{1}}{\mathbf{P}\mathbf{r}^{2}} + \mathbf{C}\_{2}\right)}\tag{34}$$

where C1 ¼ �3*:*7; C2 ¼ 0*:*71; C3 ¼ 12*:*17; C4 ¼ 1*:*2 Bo <sup>¼</sup> <sup>q</sup> GΔhv ; Re <sup>¼</sup> GDFR ηL ; Pr <sup>¼</sup> <sup>η</sup>LCp,L <sup>λ</sup><sup>L</sup> ; Nu <sup>¼</sup> hDFR λL DFR: diameter at fin root Working range: Bo >5.1837.10�6; Pr > 2.2828

#### **3.3 Correlation for boiling model of Chamra and Mago**

The semi empirical model of Chamra and Mago [11] was derived based on 380 collected data points from available literature.


Developed on the basic of Cavallini model

$$\mathbf{h\_{tp}} = \mathbf{h\_{pb}}\mathbf{15160}(\mathbf{X\_{tt}})^{11610}\mathbf{F\_1}^{-17640} + \mathbf{h\_l}\mathbf{@Rx}^{2.6220}(\mathbf{Bon}^{\rm w}\mathbf{F}\mathbf{v\_l})^{-0.2158}\mathbf{F\_2^{0.992}}\mathbf{F\_3^{0.0582}}\tag{35}$$

$$\mathbf{h\_{pb}} = \mathbf{55P\_R^{0.12}} \left(-\log\_{10} \mathbf{P\_R}\right)^{-0.55} \mathbf{M}^{-0.5} \mathbf{q^{0.67}} \tag{36}$$

$$\mathbf{q} = \mathbf{G}.\mathbf{i}\_{\mathbf{f}\mathbf{g}}.\Delta\mathbf{x} \tag{37}$$

$$\mathbf{X\_{tt}} = \left(\frac{\mathbf{1} - \mathbf{x}}{\mathbf{x}}\right)^{0.9} \left(\frac{\rho\_{\mathbf{v}}}{\rho\_{\mathbf{l}}}\right)^{0.5} \left(\frac{\mu\_{\mathbf{l}}}{\mu\_{\mathbf{v}}}\right)^{0.1} \text{ If } \mathbf{X\_{tt}} > \mathbf{1 then } \mathbf{X\_{tt}} = \mathbf{1} \tag{38}$$

$$\mathbf{F\_1 = 0.01d\_i^{-1}}\tag{39}$$

$$\mathbf{h}\_{\rm l} = \mathbf{0}.023 \frac{\mathbf{k}\_{\rm l}}{\mathbf{d}\_{\rm i}} \,\mathrm{Re}\_{\rm l}^{0.8} \mathrm{Pr}\_{\rm l}^{0.4} \tag{40}$$

$$\Phi = \left[ (\mathbf{1} - \mathbf{x}) + 2.63 \mathbf{x} \left( \frac{\rho\_{\rm l}}{\rho\_{\rm v}} \right)^{\frac{1}{2}} \right]^{0.8} \tag{41}$$

$$\mathbf{Rx} = \left\{ \frac{2\mathbf{en}\_{\mathbf{f}} (\mathbf{1} - \sin \left( \mathbf{\beta}/2 \right))}{\pi \mathbf{d}\_{\mathbf{i}} \cos \left( \mathbf{\beta}/2 \right)} + \mathbf{1} \right\} \frac{\mathbf{1}}{\cos \left( \mathbf{\gamma} \right)} \tag{42}$$

$$\mathbf{Bon}^{\rm w} = \frac{\mathbf{g}\rho\_{\rm L}\mathbf{e}\pi\mathbf{d}\_{\rm i}}{8\sigma\mathbf{n}\_{\rm f}}\tag{43}$$

$$\mathbf{Fr\_V} = \frac{\mathbf{G^2}}{\rho\_{\text{vgd\_i}}^2} \tag{44}$$

$$\mathbf{F\_2 = 0.01d\_i^{-1}}\tag{45}$$

$$\mathbf{F}\_3 = \mathbf{100G}^{-1} \tag{46}$$

#### **3.4 The correlation of Yun et al.**

Using the database with 749 data points of five different refrigerants to create a generalized correlation for boiling heat transfer in horizontal microfin tubes.

*A Review on Convective Boiling Heat Transfer of Refrigerants in Horizontal Microfin-Tubes… DOI: http://dx.doi.org/10.5772/intechopen.100024*

Heat transfer coefficient of two-phase flow:

$$\mathbf{h}\_{\rm tp}/\mathbf{h}\_{\rm l} = \left[ \mathbf{C}\_{\rm l} \mathbf{B} \mathbf{o}^{\rm C\_2} \left( \frac{\mathbf{P}\_{\rm sat} \mathbf{d}\_{\rm l}}{\sigma} \right)^{\rm C\_3} + \mathbf{C}\_{\rm t} \left( \frac{\mathbf{1}}{\mathbf{X}\_{\rm tt}} \right)^{\rm C\_5} \left( \frac{\mathbf{G} \mathbf{f}}{\mu\_{\rm l}} \right)^{\rm C\_6} \right] \mathbf{R} \mathbf{e}\_1^{\rm C\_7} \mathbf{Pr}\_{\rm l}^{\rm C\_8} \left( \frac{\mathbf{8}}{\mathbf{f}} \right)^{\rm C\_9} \tag{47}$$

$$\mathrm{Bo} = \frac{\mathrm{q}}{\mathrm{G} \mathrm{h}\_{\mathrm{lv}}} \tag{48}$$

$$\mathbf{h}\_{\rm l} = \mathbf{0}.023 \,\mathrm{Re}\_{\rm l}^{0.8} \mathrm{Fr}\_{\rm l}^{0.4} \left(\frac{\mathbf{k}\_{\rm l}}{\mathbf{d}\_{\rm i}}\right) \tag{49}$$

$$\text{Re}\_{\text{l}} = \frac{\mathbf{G}(\mathbf{1} - \mathbf{x})\mathbf{d}\_{\text{i}}}{\mu\_{\text{l}}} \tag{50}$$

$$\delta = \frac{\mathbf{d}\_{\mathbf{r}}(\mathbf{1} - \mathbf{e})}{\mathbf{4}} \tag{51}$$

$$\mathbf{e} = \frac{\mathbf{x}}{\rho\_{\rm G}} \left[ (\mathbf{1} + \mathbf{0}.\mathbf{12}(\mathbf{1} - \mathbf{x})) \left( \frac{\mathbf{x}}{\rho\_{\rm G}} + \frac{\mathbf{1} - \mathbf{x}}{\rho\_{\rm L}} \right) + \frac{\mathbf{1}.\mathbf{18}(\mathbf{1} - \mathbf{x}) [\mathbf{g}\sigma(\rho\_{\rm L} - \rho\_{\rm G})]^{0.25}}{\mathbf{G}\rho\_{\rm L}^{0.5}} \right]^{-1} \tag{52}$$

$$\mathbf{X}\_{\rm tt} = \left(\frac{\mathbf{1} - \mathbf{x}}{\mathbf{x}}\right)^{0.9} \left(\frac{\rho\_{\rm v}}{\rho\_{\rm l}}\right)^{0.5} \left(\frac{\mu\_{\rm l}}{\mu\_{\rm v}}\right)^{0.1} \tag{53}$$


#### **3.5 The correlation of Cavallini et al.**

The correlation of Cavallini et al. [8] produced for not only microfin tube but also for cross groves tubes with 643 data points collected from available literatures.

Heat transfer coefficient of two phase flow:

$$
\alpha = \mathfrak{a}\_{\text{nb}} + \mathfrak{a}\_{\text{cv}} \tag{54}
$$

Nucleate boiling component:

$$\mathbf{a\_{nb}} = \mathbf{a\_{cooper}} \mathbf{S.F\_1(d\_i)} = \left[ \mathbf{55P\_R^{0.12}} (-\log\_{10} \mathbf{P\_R})^{-0.95} \mathbf{M}^{-0.5} \mathbf{q^{0.67}} \right] \mathbf{S.F\_1(d\_i)} \tag{55}$$

$$\mathbf{S} = \mathbf{A} . \mathbf{X}\_{\rm tt}^{\rm B} = \mathbf{A} . \left[ \left( \frac{\mathbf{1} - \mathbf{x}}{\mathbf{x}} \right)^{0.9} \left( \frac{\rho\_{\rm G}}{\rho\_{\rm L}} \right)^{0.5} \left( \frac{\mu\_{\rm L}}{\mu\_{\rm G}} \right)^{0.1} \right]^{\rm B} \\ \text{If } \mathbf{X}\_{\rm tt} > \mathbf{1} \text{ then } \mathbf{X}\_{\rm tt} = \mathbf{1} \tag{56}$$

F1(di): function of fin tip tube diameter

*Heat Exchangers*

$$\mathbf{F\_1(d\_i)} = \left(\frac{\mathbf{d\_o}}{\mathbf{d\_i}}\right)^{\mathbf{C}} \tag{57}$$

Convective term:

$$\mathbf{a}\_{\text{cv}} = \frac{\lambda\_{\text{L}}}{\text{d}\_{\text{i}}}.\text{Nu}\_{\text{cv,smooth\,tube}}.\text{Rx}^{\text{S}}.(\text{Bo.Fr})^{\text{T}}.\text{F}\_{2}(\text{d}\_{\text{i}}).\text{F}\_{3}(\text{G})\tag{58}$$

$$\mathbf{Nu}\_{\text{cv,smooth tube}} = \mathbf{Nu}\_{\text{LO}} \Phi = \left[ 0.023 \left( \frac{\text{Gd}\_{\text{l}}}{\mu\_{\text{L}}} \right)^{0.8} \text{Pr}\_{\text{L}}^{1/3} \right] \left[ (1 - \mathbf{x}) + 2.63 \mathbf{x} \left( \frac{\rho\_{\text{L}}}{\rho\_{\text{G}}} \right)^{1/2} \right]^{0.8} \tag{59}$$

$$\mathbf{Rx} = \left\{ \frac{2\mathbf{h}\mathbf{n}\_{\mathbf{g}} (\mathbf{1} - \sin\left(\mathbf{\gamma}/2\right))}{\pi \mathbf{d}\_{\mathbf{i}} \cos\left(\mathbf{\gamma}/2\right)} + \mathbf{1} \right\} \frac{\mathbf{1}}{\cos\left(\boldsymbol{\upbeta}\right)}\tag{60}$$

$$\text{Bo} = \frac{\text{g}\rho\_{\text{L}}\text{h}\pi\text{d}\_{\text{i}}}{8\sigma\text{n}\_{\text{g}}} \tag{61}$$

$$\text{Fr} = \frac{\text{u}\_{\text{GO}}^2}{\text{gd}\_{\text{i}}} \tag{62}$$

$$\mathbf{F\_2(d\_i)} = \left(\frac{\mathbf{d\_o}}{\mathbf{d\_i}}\right)^{\mathbf{V}} \tag{63}$$

$$\mathbf{F}\_{\mathbf{3}}(\mathbf{G}) = \left(\frac{\mathbf{G}\_{\mathbf{o}}}{\mathbf{G}}\right)^{Z} \tag{64}$$

uGO: velocity of gas phase with total flow rate


Working range:


#### **3.6 The correlation of Thome et al.**

The model was derived to predict the microfin with the test data for R134a, R123, mass flux G = [100; 500] kg/m<sup>2</sup> s, quality x = [0.15; 0.85] and heat flux q = [2; 47] kW/m2

$$\mathbf{h} = \mathbf{E\_{mf}} \left[ \left( \mathbf{h\_{nb}} \right)^{3} + \left( \mathbf{E\_{RB}} \mathbf{h\_{cv}} \right)^{3} \right]^{1/3} \tag{65}$$

$$\mathbf{h\_{nb}} = \mathbf{55P\_R^{0.12}} \left(-\log\_{10}\mathbf{P\_R}\right)^{-0.95} \mathbf{M}^{-0.5} \mathbf{q}^{0.67} \tag{66}$$

*A Review on Convective Boiling Heat Transfer of Refrigerants in Horizontal Microfin-Tubes… DOI: http://dx.doi.org/10.5772/intechopen.100024*

$$\mathbf{h}\_{\rm cv} = \mathbf{0}.0133 \,\mathrm{Re}\_{\rm l}^{0.69} \mathrm{Pr}\_{\rm l}^{0.4} \lambda\_{\rm l} / \delta \tag{67}$$

$$(\text{Re}\_{\text{l}})\_{\text{film}} = \frac{4\mathbf{G}(\mathbf{1} - \mathbf{x})\mathbf{6}}{(\mathbf{1} - \mathbf{e})\mu\_{\text{l}}} \tag{68}$$

$$\boldsymbol{\mathfrak{G}} = \mathbf{d}\_{\mathbf{r}}(\mathbf{1} - \boldsymbol{\mathfrak{e}})/\mathbf{4} \tag{69}$$

$$e = \frac{\mathbf{x}}{\rho\_{\rm G}} \left[ (\mathbf{1} + \mathbf{0}.\mathbf{1}\mathbf{2}(\mathbf{1} - \mathbf{x})) \left( \frac{\mathbf{x}}{\rho\_{\rm G}} + \frac{\mathbf{1} - \mathbf{x}}{\rho\_{\rm L}} \right) + \frac{\mathbf{1}.\mathbf{1}\mathbf{8}(\mathbf{1} - \mathbf{x}) \left[ \mathbf{g}\sigma(\rho\_{\rm L} - \rho\_{\rm G}) \right]^{0.25}}{\mathbf{G}\rho\_{\rm L}^{0.5}} \right]^{-1} \tag{70}$$

$$\mathbf{E\_{mf}} = \mathbf{1.89} \left( \frac{\mathbf{G}}{\mathbf{G\_{ref}}} \right)^2 - \mathbf{3.7} \left( \frac{\mathbf{G}}{\mathbf{G\_{ref}}} \right) + \mathbf{3.02} \tag{71}$$

$$\mathbf{E\_{RB}} = \left\{ \mathbf{1} + \left[ 2.64 \left( \mathbf{Re\_D} \right)^{0.036} \left( \frac{\mathbf{e}}{\mathbf{d\_r}} \right)^{0.212} \left( \frac{\mathbf{p}}{\mathbf{d\_r}} \right)^{-0.21} \left( \frac{\emptyset}{90^{\circ}} \right)^{0.29} \left( \mathbf{Pr\_L} \right)^{-0.024} \right]^{7} \right\}^{1/7} \tag{72}$$

$$\mathrm{Re}\_{\mathrm{D}} = \frac{\mathbf{G}(\mathbf{1} - \mathbf{x})\mathbf{d}\_{\mathrm{r}}}{\mu\_{\mathrm{l}}} \tag{73}$$

$$\mathbf{p} = \frac{\pi \mathbf{d}\_{\mathbf{r}} / \mathbf{N}}{\tan \beta} \tag{74}$$

$$\mathbf{G}\_{\text{ref}} = \mathbf{500kg/m^2s} \tag{75}$$

where e: microfin height; p: axial pitch from fin to fin; β: helix angle; N: number of fins; dr: root diameter

#### **4. Pressure drop**

#### **4.1 The correlation of Choi et al.**

The model form of Choi et al. [14] was carryout from Pierre 1964 model with 831 data pointed collected from NIST database with 626 data point for boiling and 205 data points for condensation for some different fluids R134a, R22, R125, R32, R407C, R410A and R32/R134a. Those data points derived with test section has 8.92 mm root diameter, 9.52 mm outside diameter, 18° helix angle.

Total pressure drop:

$$\frac{\Delta \mathbf{p}}{\mathbf{L}} = \frac{\Delta \mathbf{p}\_{\text{f}}}{\mathbf{L}} + \frac{\Delta \mathbf{p}\_{\text{m}}}{\mathbf{L}} = \mathbf{G}^{2} \left[ \mathbf{f} \frac{(\mathbf{v}\_{\text{tp,out}} + \mathbf{v}\_{\text{tp,in}})}{\mathbf{d}\_{\text{h}}} + \frac{(\mathbf{v}\_{\text{tp,out}} - \mathbf{v}\_{\text{tp,in}})}{\mathbf{L}} \right] \tag{76}$$

Two phase friction factor:

$$\mathbf{f} = \mathbf{0}.00506 \,\mathrm{Re}\_{\mathrm{h}, \mathrm{LO}}^{-0.0951} \mathrm{K}\_{\mathrm{f}}^{0.1554} \tag{77}$$

$$\mathbf{Re}\_{\mathrm{h,LO}} = \mathbf{Gd}\_{\mathrm{h}} / \mu\_{\mathrm{l}} \tag{78}$$

Hydraulic diameter:

$$\mathbf{d}\_{\mathrm{h}} = 4 \mathbf{A}\_{\mathrm{c}} \cos \mathfrak{P} / \mathrm{(n.S}\_{\mathrm{p}}) \tag{79}$$

Two phase number:

$$\mathbf{K}\_{\mathbf{f}} = \frac{\Delta \mathbf{x}. \mathbf{h}\_{\text{lv}}}{\mathbf{g} \mathbf{L}} \tag{80}$$

"Specific volumes of the two-phase fluid, vtp,out and vtp,in, are quality-weighted sums of the vapor and liquid specific volumes at either the outlet or inlet of the tube"

#### **4.2 The correlation of Goto et al.**

Model of Goto et al. [15] derived from experiment of R41A and R22 for both boiling and condensation at mass flux G = [200; 340] kg/m<sup>2</sup> s inside spiral groove tube 7.3 mm mean inside diameter and herring-born groove tube 7.24 mm mean inside diameter.

Frictional pressure gradient:

$$
\left(\frac{\mathrm{dP}}{\mathrm{d}\mathbf{z}}\right)\_{\mathrm{f}} = \Phi\_{\mathrm{v}}^{2} \left(\frac{\mathrm{dP}}{\mathrm{d}\mathbf{z}}\right)\_{\mathrm{v}} = \Phi\_{\mathrm{v}}^{2} 2\mathbf{f}\_{\mathrm{e,v,Go}} (\mathrm{G}\mathbf{x})^{2} / (\rho\_{\mathrm{v}}\mathrm{d}\_{\mathrm{e}}) \tag{81}
$$

$$\Phi\_{\rm v} = \mathbf{1} + \mathbf{1}.64 \mathbf{X}\_{\rm tt}^{0.79} \tag{82}$$

Or could be compute based on the liquid phase form

$$
\left(\frac{\mathrm{dP}}{\mathrm{d}\mathbf{z}}\right)\_{\mathrm{f}} = \Phi\_{\mathrm{l}}^{2} \left(\frac{\mathrm{dP}}{\mathrm{d}\mathbf{z}}\right)\_{\mathrm{l}} = \Phi\_{\mathrm{l}}^{2} 2\mathbf{f}\_{\mathrm{e},\mathrm{l},\mathrm{Lo}} \left(\mathbf{G}(\mathbf{1}-\mathbf{x})\right)^{2} / \left(\mathbf{p}\_{\mathrm{l}}\mathbf{d}\_{\mathrm{e}}\right) \tag{83}
$$

$$\Phi\_{\rm l} = \mathbf{1} + \text{7.61X}\_{\rm tt}^{-1.7} \tag{84}$$

$$Re\_{\varepsilon,v} = G\varepsilon d\_{\varepsilon}/\mu\_v \text{ Diameter of inner tube de} \tag{85}$$

$$\mathbf{X}\_{\rm tt} = \left(\frac{\mathbf{1} - \mathbf{x}}{\mathbf{x}}\right)^{0.9} \left(\frac{\rho\_{\rm v}}{\rho\_{\rm l}}\right)^{0.5} \left(\frac{\mu\_{\rm l}}{\mu\_{\rm v}}\right)^{0.1} \tag{86}$$

$$\text{Re}\_{\text{e,v}} \le 2000; \ = > \mathbf{f}\_{\text{e,v,Go}} = \mathbf{16/Re\_{e,v}} \tag{87}$$

$$2000 \le \text{Re}\_{\text{e,v}} \le 2\text{600}; = > \mathbf{f}\_{\text{e,v,Go}} = \mathbf{0}.000147 \,\text{Re}\_{\text{e,v}}^{0.53} \tag{88}$$

$$2\text{\(600} \le \text{Re}\_{\text{e,v}} \le \text{\(6500; } = \text{\(>} \mathbf{f}\_{\text{e,v,Go}} = \mathbf{0}.\text{\(046\)} \text{Re}\_{\text{e,v}}^{-0.2}\text{\)}\tag{89}$$

$$\text{16500} \le \text{Re}\_{\text{e,v}} \le \text{12700}; \quad = \text{>f}\_{\text{e,v,Go}} = \text{0.00123 } \text{Re}\_{\text{e,v}}^{0.21} \tag{90}$$

$$12700 \le \text{Re}\_{\text{e,v}}; = > \text{f}\_{\text{e,v,Go}} = 0.0092\tag{91}$$

#### **4.3 The correlation of Wongsa et al.**

The correlation for boiling with high mass flux was proposed by Wongsa et al. [16]. Derived from their experimental results for R134a on microfin tube with 8.92 mm inner diameter, saturation temperature Tsat = [10; 20]°C, mass flux G = [400; 800] kg/m2 s.

Two phase friction pressure drop

$$
\left(\frac{\text{dP}}{\text{dz}}\right)\_{\text{f}} = \Phi\_{\text{VO}}^2 \left(\frac{\text{dP}}{\text{dz}}\right)\_{\text{h,VO}} = \Phi\_{\text{VO}}^2 2\mathbf{f}\_{\text{h,VO}} \mathbf{G}^2 / (\rho\_{\text{v}} \mathbf{d}\_{\text{h}}) \tag{92}
$$

$$4\text{f}\_{\text{h,VO}} = \frac{1.325}{\left[\ln\left(\frac{\text{Rx}\_{\text{h}}}{\text{3.7}} + \frac{5.74}{\text{Re}\_{\text{h},VO}^{0.7}}\right)\right]^2} \tag{93}$$

*A Review on Convective Boiling Heat Transfer of Refrigerants in Horizontal Microfin-Tubes… DOI: http://dx.doi.org/10.5772/intechopen.100024*

$$\text{Re}\_{\text{h,VO}} = \frac{\text{Gd}\_{\text{h}}}{\mu\_{\text{v}}} \tag{94}$$

$$\mathbf{Rx\_h} = \frac{\mathbf{0.3(e/d\_h)}}{(\mathbf{0.1} + \cos\beta)}\tag{95}$$

$$\Phi\_{\rm VO}^{2} = 2.3263 - 1.8043 \left\{ \frac{\rm X\_{tt}G}{\left[ \rm gd\_h\rho\_v (\rho\_l - \rho\_v) \right]^{0.5}} \right\}^{0.0802} \tag{96}$$

#### **4.4 The correlation of Kou and Wang**

The correlation of Kou and Wang 1996 [17] was analyzed from their own experimental data results for boiling of R22 and R407C in a 9.52 mm diameter micro-fin tube and a smooth tube at two different evaporation temperatures (6°C and 10°C). The mass flux was between 100 and 300 kg/m<sup>2</sup> s and the heat fluxes 6 and 14 kW/m<sup>2</sup> .

$$\left(\frac{\text{dP}}{\text{dz}}\right)\_{\text{f}} = \frac{0.0254 \text{G}^2 [\text{x} \mathbf{v}\_{\text{v}} + (\mathbf{1} - \mathbf{x}) \mathbf{v}\_{\text{l}}]}{\text{d}\_{\text{fr}}} \tag{97}$$
