*2.3. Parametric Optimization of HPHE*

The initial arrangement of finned heat pipes is shown in Figure 7 (three first rows). It is a start positioning for the further thermal and flow analysis carried out with the use of the computational model. This arrangement is considered for the first set of plots: Figures 8–10. Volumetric flow rates for hot and cold airstreams are held constant (300 m3/h). Inlet cold air temperature is kept at 10 ◦C and hot air inlet temperature at 30 ◦C. The counterflow arrangement was chosen as it produces a higher mean temperature difference between streams than parallel flow. A staggered HPs arrangement was also selected over the inline as it ensures better mixing and temperature uniformity of air streams. Initially, the isosceles layout was assumed due to a minimal pressure drop [20]. Figure 8 shows the heat transfer

rate for inline and staggered finned HPs arrangement as a function of the number of HPHE rows. It is a validation of the correctness of the computational model calculations. As expected in the whole, considered range of the number of rows, the staggered arrangement produces a higher heat transfer rate than inline. As the number of rows increases, thermal effectiveness grows non-linearly (Figure 9)—the shape of the function is similar as in the case of heat transfer rate. The previously stated goal is 60% effectiveness (red horizontal line), and it corresponds to a 17-rows HPHE. Because of thermal calculations uncertainty, 20 rows were chosen (three extra rows) to obtain the assumed effectiveness. Pressure drop increases linearly with the number of rows and exceeds 40 Pa at approximately nine rows (Figure 10). For 20 rows HPHE, Δ*P* of 100 Pa is not exceeded, which makes that acceptable value for small air conditioning systems (it can be handled by a typical air duct fan). Further increase in effectiveness by adding more rows (more than 20) is not viable, because the addition of two rows produces approximately 10 Pa extra pressure drop (which is 10% relative increase) while only 2–3% raise of effectiveness. It supports 20 rows as the final choice, taking into account that a pressure drop increases linearly, while the effectiveness slope becomes less steep for increasing row number (adding more rows becomes even less viable in terms of effectiveness vs. Δ*P* change as row number increases).

**Figure 7.** The initial arrangement of the HPs in HPHE (staggered).

**Figure 8.** Heat transfer rate for inline vs. staggered heat pipe arrangement, for inlet cold air temperature 10 ◦C and hot air inlet temperature 30 ◦C.

**Figure 9.** Thermal effectiveness of HPHE as a function of number of rows, for inlet cold air temperature 10 ◦C and hot air inlet temperature 30 ◦C.

**Figure 10.** Pressure drop for HPHE vs. number of rows, for inlet cold air temperature 10 ◦C and hot air inlet temperature 30 ◦C.

The dependence of parameters on hot air inlet temperature is shown in Figures 11–13. All computations for the plots were made with the assumption of 20 HPs rows and 0 ◦C cold air stream inlet temperature. Figure 11 shows the dependence of the Reynolds number, which is decreasing for increasing hot air inlet temperature. It is an effect of an increase in air kinematic viscosity with temperature. There is approximately a 15% Reynolds number decrease relative to the value for the lowest considered temperature: 8 ◦C. Figure 12 presents the pressure drop as a function of the hot air inlet temperature. The pressure drop decreases linearly by 10% for the assumed temperature range—it is caused mainly by a decrease in air density with temperature. The combined effect of air thermophysical properties changes with temperature (variation with pressure is negligible) and mean temperature rise impacts HPHE effectiveness, yet very weakly for hot air temperatures greater than 30 ◦C (Figure 13). The relative change from 30 to 42 ◦C is just a fraction of a percent. The above analysis shows that the assumption of approximately 60% effectiveness is valid for a broad range of temperatures for 20 rows of HPHE. There is a potential benefit in reducing a pressure drop for higher temperatures, although it is calculated for a constant volumetric flow of air, which could decrease in a real situation, where a fan is inducing the flow. The working point of a fan can change for lower air density and flow conditions could vary.

**Figure 11.** Reynolds number vs. hot air inlet temperature, for cold stream inlet temperature 0 ◦C and 20 HPs rows.

**Figure 12.** Pressure drop vs. hot air inlet temperature, for cold stream inlet temperature 0 ◦C and 20 HPs rows.

**Figure 13.** Effectiveness vs. hot air inlet temperature, for cold stream inlet temperature 0 ◦C and 20 HPs rows.

Figure 14 shows hot air heat transfer coefficient dependence on the number of rows (inlet hot air temperature: 30 ◦C, inlet cold air temperature: 10 ◦C). The computational model allows very small dependency of the heat transfer coefficient as equation (15) is valid for the number of rows greater than six; therefore, the calculated heat transfer coefficients for a smaller number of rows are uncertain. This inaccuracy, however, is not very important, as the practical number of rows which produces reasonable HPHE efficiency (ε > 50%) is greater than 10. The dependence of the hot air heat transfer coefficient on the hot air inlet temperature, for cold air inlet temperature 0 ◦C, is shown in Figure 15. For the broad range of temperatures, heat transfer coefficient changes only slightly—it drops by 5%. This proves the stability of the working point of HPHE under various temperature conditions. Three-dimensional graphs are introduced for defining the optimal spacing between the center of the tubes—longitudinal and traversal in the respect to the airflow direction. The subsequent analysis was carried out with an assumption of constant hot and cold air inlet temperatures (10 and 30 ◦C). Figure 16 presents the exchanged heat transfer rate as a function of traversal and longitudinal spacings. Heat transfer rate changes stepwise at *Xt* = 0.062 m because below that spacing it is possible to add one extra HP in the row. Further decrease in traversal distance results in weak growth. At minimal traversal spacing *Xt* = 0.05 m (contact of finned HPs), heat transfer rate attains the maximal value. Longitudinal spacing does not affect the heat transfer rate. HPHE effectiveness, shown in Figure 17, exhibits similar behavior to heat transfer rate, with the peak value of 64% for minimal traversal spacing. The pressure drop rises more sharply with the decrease in *Xt* than with the reduction in *Xl*. As longitudinal spacing does not affect heat transfer significantly, moving away HPs in the longitudinal direction is advised, yet excessive spacing could cause an unacceptable increase in the total length of HPHE. The peak of a pressure drop is noted for minimal spacings (approx. 160 Pa).

**Figure 14.** Hot air heat transfer coefficient vs. the number of rows, for hot air inlet temperature 30 ◦C and cold air inlet temperature 10 ◦C.

**Figure 15.** Hot air heat transfer coefficient vs. hot air inlet temperature, for cold air inlet temperature 0 ◦C.

**Figure 16.** Heat transfer rate vs. traversal and longitudinal spacing of the tubes.

**Figure 17.** Thermal efficiency vs. traversal and longitudinal spacing of the tubes.

The hot air heat transfer coefficient grows by 20% with traversal spacing reduction (Figure 18). Enhancement by altering the *Xl* is negligible. Figure 19 presents the dependence of HPHE effectiveness on *Xt* and hot air inlet temperature. A maximal value of 66% is attained for minimal *Xt* and the maximal temperature difference between hot and cold airstreams. As expected for higher temperature differences, HPHE performs better.

**Figure 18.** Heat transfer coefficient vs. traversal and longitudinal spacing of the pipes.

**Figure 19.** Thermal efficiency vs. traversal spacing of the pipes and hot air inlet temperature.
