*2.1. Computational Model Description*

The computational model was developed to design the HPHEs parameters. As finned HPs are basic heat transfer components, heat transfer through HPs is calculated according to the formula: .

$$
\dot{Q} = \frac{\Delta T\_m}{\sum R} \tag{1}
$$

The overall thermal resistance of finned heat pipe:

$$
\sum R = R\_{ca} + 2R\_{tube} + R\_{hp} + R\_{ha\nu} \tag{2}
$$

where thermal resistance of conduction through tube wall:

$$R\_{tube} = \frac{\ln \frac{d\_o}{dl}}{2 \cdot \pi \cdot k \cdot L} \text{ \,\,\,\,\,\tag{3}$$

thermal resistance of cold air flow over finned surface:

$$R\_{ca} = \frac{1}{h\_{ca}\varepsilon\_o A\_s} \,\tag{4}$$

thermal resistance of hot air flow over finned surface

*Rha* <sup>=</sup> <sup>1</sup> *hha*·*o*·*As* . (5)

The heat transfer coefficient on the airside was calculated according to [20]. The maximal average velocity of air through finned tube bundle:

$$w\_{\text{max}} = \frac{\dot{V}}{A\_o}.\tag{6}$$

Reynolds number:

$$Re\_d = \frac{\mathcal{W}\_{\max} \cdot d\_r}{v} \tag{7}$$

Nusselt number:

$$Nu = \frac{h \cdot d\_r}{k} \tag{8}$$

Figure 1 shows the finned tube arrangement with characteristic geometrical dimensions.

Minimum airflow area:

$$A\_0 = \left[ \left( \frac{L\_3}{X\_l} - 1 \right) z' + (X\_l - d\_r) - \left( d\_f - d\_r \right) t\_f N\_f \right] L\_1 \tag{9}$$

where *Nf* is the number of fins per meter:

$$N\_f = 1/S\_\prime \tag{10}$$

$$z'=\text{2x}'\text{ if }\text{2x}'<\text{2y}',\tag{11}$$

$$z'=2y'\text{ if }2y'<2x',\tag{12}$$

in which *2x'* and *y'* are given by:

$$\mathbf{2x'} = (\mathbf{X}\_t - d\_r) - \left(d\_f - d\_r\right) t\_f \mathbf{N}\_{f'} \tag{13}$$

$$y' = \left[\left(\frac{X\_l}{2}\right)^2 + \left(X\_l\right)^2\right]^{0.5} - d\_r - \left(d\_f - d\_r\right)t\_f N\_f. \tag{14}$$

Dimensions 2*x'* and *y'* are depicted in Figures 1 and 2.

**Figure 1.** Finned heat exchanger tube arrangement: (**a**) airflow through the heat pipes, (**b**) minimum airflow area (according to [20]).

**Figure 2.** Geometrical dimensions of finned HP.

For high-fin tube banks, the correlation based on experimental heat transfer data is [21]:

$$Nu = 0.1387 Re\_d^{0.718} Pr^{1/3} \left(\frac{s}{l\_f}\right)^{0.296} \text{ \AA} \tag{15}$$

with a standard deviation of 5.1%.

For an equilateral triangular pitch with high-finned tubes, the friction coefficient can be calculated by [21]:

$$f = 9.465 \operatorname{Re}\_d^{-0.316} \left(\frac{X\_l}{d\_r}\right)^{-0.927} \tag{16}$$

with a standard deviation of 7.8%. This is applicable for the following parameter definitions:

$$\begin{aligned} Re\_d &= 2000 - 50 \text{\textdegree} 0000\\ P\_\text{l}/d\_\text{r} &= 1.687 - 4.50 \end{aligned}$$

For isosceles triangular layout [22]:

$$f = 9.465 Re\_d^{-0.316} \left(\frac{X\_l}{d\_r}\right)^{-0.927} \left(\frac{X\_l}{X\_l}\right)^{0.515} \tag{17}$$

This is applicable for:

$$\begin{aligned} d\_r &= 18.6 - 40.9 \, mm/s \\ l\_f/d\_r &= 0.35 - 0.56 \, \mu \\ l\_f/s &= 4.5 - 5.3 \, \mu \\ \text{X}\_l/d\_r &= 1.8 - 4.6 \, \mu \\ \text{X}\_l/d\_r &= 1.8 - 4.6 \, \mu \end{aligned}$$

$$\begin{aligned} \text{N}\_l &\geq 6 \, \mu \end{aligned}$$

where *Nt* is the number of heat exchanger tube rows.

The pressure drop of airflow through finned tube bundles;

$$
\Delta p = 2f \mathcal{N}\_l \rho w\_{\text{max}}^2. \tag{18}
$$

The efficiency of the annular fin (rectangular profile with adiabatic tip) is obtained from correlation [23]:

$$\eta\_f = \frac{\frac{2r\_i}{m}}{r\_o^2 - r\_i^2} \frac{K\_1(mr\_i)I\_1(mr\_o) - I\_1(mr\_i)K\_1(mr\_o)}{K\_0(mr\_i)I\_1(mr\_o) - I\_0(mr\_i)K\_1(mr\_o)'} \tag{19}$$

where *ro = df/2*, *ri = dr/2*, *I*0—modified Bessel function of order 0, *I*1—modified Bessel function of order 1, *K*0—modified Bessel function K of order 0, *K*1—modified Bessel function K of order 1, *m* coefficient for annular fin:

$$m = \sqrt{\frac{2 \cdot h}{k \cdot t\_f}}\tag{20}$$

Finned surface area is obtained by the following equation:

$$A\_{\rm s} = A\_{\rm r} + A\_{\rm f} \tag{21}$$

where unfinned area of tube:

$$A\_r = \left(\mathbf{S} - \mathbf{t}\_f\right) \text{ } \pi \, d\_r \, \mathbf{N}\_{f'} \tag{22}$$

finned area of tube:

$$A\_f = \left(2\,\pi\left(r\_o^2 - r\_i^2\right) + \pi\,d\_f\,t\_f\right)\mathcal{N}\_f.\tag{23}$$

Overall finned surface efficiency:

$$
\varepsilon\_o = \frac{A\_r + A\_f \cdot \eta\_f}{A\_s}.\tag{24}
$$

The thermal resistance of the wickless heat pipe is taken from the regression of experimental data from previous authors' work [22], where the thermal performance of various diameters of wickless HPs were investigated for different working fluids filling ratios. The experiment parameters are summarized in Table 1. Various diameters of HPs were tested to obtain thermal resistance characteristics. On the condenser and evaporator sections of HPs, jacket HEXs were installed. The condenser section HEX was fed with cold water, while through the evaporator section HEX hot water flowed. Because of the low volumetric flow of cooling and heating water, the range of heat fluxes obtained for HPs was nearly identical as predicted in the case of the air-to-air HPHE. The lower convective heat transfer coefficients of the air were compensated for by the area enlargement due to the finned surface (the enlargement factor was around 20). The other aim of the study was to choose the best working fluid of the examined substances. Refrigerant R404A was recognized as the best working fluid (giving the highest thermal throughput) from other tested fluids (R134a, R410A, and R407C). A twenty percent volumetric filling ratio was chosen consecutively as the best of four considered in the study (10%, 20%, 30%, and 40%). Thermal resistance versus heat transfer rate for a 32 mm outer diameter heat pipe is shown in Figure 3. The relative uncertainty of the HPs' thermal resistance is nearly identical to the relative uncertainty of the heat transfer rate stated in [22] (typically 10–25%). Thermal resistances are nearly identical for filling ratios from 20–40%. They are considerably lower for 10%, although this filling ratio was rejected because of dry-out limit occurrence. Thermal resistance increases sharply for a 10% filling ratio near to the highest heat throughput (near 150 W) because the falling film of the refrigerant was broken, and the dry patches at the evaporator section impeded the heat flow.

**Figure 3.** Thermal resistance of 32 mm outer diameter heat pipe vs. heat transfer rate for different filling ratios [22].


**Table 1.** Parameters of the experiment [24].

The next lowest filling ratio was considered the best. Even its thermal resistance was comparable with 30 and 40%; lower filling cuts expenses on working fluid and reduces the total weight of HPHE. Experimental data for 20 and 32 mm heat pipes were fitted by a correlation (25) in Figure 4. The correlation takes into account various diameters of HPs. As it can be seen in Figure 4, Equation (25) successfully fits the experimental data [24] for two examined HPs' outer diameters: 20 and 32 mm.

**Figure 4.** Thermal resistance vs. heat transfer rate for 20% filling ratio, for *d* = 20 mm, and *d* = 32 mm HP. Experimental data were fitted with (25) [23].

Correlation (25) was used to predict the overall thermal resistance of HPs in the computational model:

$$R\_{lp}\left(d\_{\prime}\stackrel{\cdot}{\mathcal{Q}}\right) = 0.9204 \cdot \dot{\mathcal{Q}}^{-0.644} \left(\frac{d}{32}\right)^{-0.69} \tag{25}$$

The computational model uses Equations (1)–(25) for the iterative calculation of heat transfer rate, and air streams outlet temperatures. The initial value of the heat transfer rate is guessed and after each iteration, it is updated until the residual becomes lower than 1 W. The iterative technique to solve the non-linear set of equations is the Gauss–Seidel method with an under-relaxation factor of 0.4. This algorithm is used to find a solution (heat transfer rate) for every HPHE row. The HEX is divided into control volumes according to Figure 5. Nodes were at the inlet and outlet planes to each row. Each node represents

an unknown temperature, besides the known values at the inlets (blue nodes for cold air temperatures: *Tc*0*–TcN*, red nodes for hot air temperatures: *Th*0*–ThN*). After all of the temperatures and heat transfer rates . *<sup>Q</sup>*<sup>0</sup> <sup>−</sup> . *QN* were updated, and the maximum residual of all of the temperatures is calculated. If it is not lower than 0.001 ◦C the temperatures and heat transfer rates are updated again. The iterative technique is also Gauss–Seidel but without the under-relaxation.

**Figure 5.** Scheme of the dividing of HPHE into control volumes.
