2.3.3. Velocity, Temperature, and Pressure Profiles

The longitudinal velocity, the temperature, and the static pressure contours are shown in Figures 8 and 9, respectively, for the simulation: *Xt* <sup>=</sup> 9.00 mm, *Fp* <sup>=</sup> 5.00 mm, *Uf r* <sup>=</sup> 4.40 m·s<sup>−</sup>1. Figure <sup>8</sup> shows the contours at different locations of the heat exchanger, i.e., the two first tubes (entrance region), the 17th and 18th tubes (center) and the last two tubes (exit region), respectively. Figure 9 shows the contours at different minimum cross sections normal to the airflow, i.e., the first tube (entrance region), the 17th tube (center), and the last tube (exit region), respectively.

Velocity (longitudinal direction) ܷm·sƺ1)

**Figure 8.** Longitudinal velocity, temperature, and static pressure contours at the entrance (**left**), middle (**center**), and exit (**right**); (side view); *Xt* <sup>=</sup> 9.00 mm, *Fp* <sup>=</sup> 5.00 mm, *Uf r* <sup>=</sup> 4.40 m·s<sup>−</sup>1.

Velocity (longitudinal direction) ܷ (m·sƺ1)

**Figure 9.** Longitudinal velocity, temperature, and static pressure contours at the entrance (**left**), middle (**center**), and exit (**right**); (frontal view); *Xt* <sup>=</sup> 9.00 mm, *Fp* <sup>=</sup> 5.00 mm, *Uf r* <sup>=</sup> 4.40 m·s<sup>−</sup>1.

The velocity contours on Figures 8 and 9 indicate that the flow develops and reaches almost fully developed velocity contour at the center region compared with the exit region. The temperature contours indicate similarity at the center and the exit region, which also confirm that the flow becomes fully developed. Additionally, the pressure change during the inlet contraction and outlet expansion are easily observable in Figure 8. Figure 10 indicates recirculation in the wake of the channels with locally low heat transfer coefficient.

**Figure 10.** Velocity streamlines (**left**) and velocity vectors (**right**); *Xt* = 9.00 mm, *Fp* = 5.00 mm, *Uf r* = 4.40 m·s<sup>−</sup>1.

#### *2.4. Data Reduction*

The data reduction followed simple equations to calculate the involved surface areas and flow areas etc., and matched the CFD implementation including rounding effects within 2% deviation (including the maximum core velocity, *Uc*). These equations, considering the computational domain, are given as follows:

$$A\_{fr} = X\_t \cdot F\_{p,r} \tag{1}$$

$$A\_{\mathbf{c}} = (\mathbf{X}\_{\mathbf{l}} - \mathbf{t}\_{\mathbf{l}}) \cdot \mathbf{F}\_{\mathbf{p}} - \frac{\mathbf{P}\_f \cdot \mathbf{F}\_{\mathbf{l}}}{2},\tag{2}$$

$$P\_f = 2 \cdot \left( \left[ \left( \mathbf{X}\_t - t\_h \right)^2 + F\_p \right. \right]^{1/2} - F\_t \right) \tag{3}$$

$$A\_f = P\_f \cdot \mathbf{L}\_{l\prime} \tag{4}$$

$$A\_{\rm tube} = \left[ 2 \cdot (t\_{\rm w} + t\_{\rm h}) \cdot F\_p - 2 \cdot F\_{\rm t} \cdot t\_{\rm w} \right] N\_{\rm l} \tag{5}$$

$$A\_{\rm tot} = A\_{\rm tube} + A\_{f'} \tag{6}$$

where *Af r* is the frontal area, *Ac* the minimum free flow area, *Pf* the fin perimeter, *Af* the fin area, *Atube* the bare tube area, *Atot* the total heat transfer area. To calculate the Colburn j-factor, the effectiveness-NTU method was used with the assumption of constant wall temperature,

$$NTU\_{air} = -ln\left(1 - \frac{T\_o - T\_i}{T\_{w} - T\_i}\right) \tag{7}$$

$$h = NTU\_{\rm air} \cdot \frac{C\_{\rm min}}{A\_{\rm tot}},\tag{8}$$

$$j = h \cdot \frac{Pr^{\frac{2}{3}}}{\rho \cdot \mathcal{U}\_c \cdot \mathbf{c}\_p} \,, \tag{9}$$

where *To*, *Ti*, and *Tw* are the outlet, inlet, and wall temperature, respectively, *h* the heat transfer coefficient, *Cmin* the minimum heat capacitance rate, *Pr* the Prandlt number, ρ the density, and *cp* the specific heat capacity at constant pressure.

These equations were used to calculate the tube local heat transfer coefficient and global heat transfer coefficients, respectively. The tube local heat transfer coefficients were based on the mass-flow averaged inlet and outlet air temperatures of each tube row and local surface area. On the other hand, the global heat transfer coefficient was based on the mass flow averaged inlet temperature of the heat exchanger and the mass flow averaged outlet temperature of each tube row and cumulated local area.

Figure 11 illustrates the tube local and global heat transfer coefficients, and the extended global heat transfer coefficient, calculated by further integrating the fully developed tube local heat transfer coefficient:

$$h\_{\rm ext} = \frac{1}{L\_{l,\rm sim}} \cdot \int\_0^{L\_{l,\rm im}} h\_{\rm loc} \cdot dL\_l + \frac{1}{L\_l - L\_{l,\rm sim}} \cdot \int\_{L\_{l,\rm im}}^{L\_l} h\_{fd,l\rm loc} \cdot dL\_{l,\rm} \tag{10}$$

where *Ll*,*sim* is the longitudinal length of the simulated geometry, *hloc* and *hf d*,*loc* are the local and fully developed local heat transfer coefficient, respectively. The extended global heat transfer coefficient was integrated to provide global heat transfer coefficients for 90 tube rows in total for each of the 42 CFD simulations.

**Figure 11.** Extension of the global heat transfer coefficient (*Xt* <sup>=</sup> 13.00 mm, *Fp* <sup>=</sup> 7.50 mm, *Uf r* <sup>=</sup> 4.40 m·s<sup>−</sup>1).

The standard deviation of the five rearmost tube rows of the simulated geometry in terms of tube local heat transfer coefficients range from 0.07 to 1.23 W·m−2·K−<sup>1</sup> for all the considered simulations. These values were considered reasonable for assuming thermally developed flow.

Similarly, the total pressure drop was reconstructed and extended by calculating the contraction and expansion pressure drop at the inlet (*i*) and the outlet (*o*), as well as the local core pressure drop (*core*),

$$
\Delta p\_{\text{tot}} = \Delta p\_{\text{i}} + \Delta p\_{\text{core}} + \Delta p\_{\text{o}} \tag{11}
$$

$$
\Delta p\_i = \frac{G\_c^{\prime 2}}{2 \cdot p\_i} \cdot \left(1 - \sigma^2 + K\_c\right) \tag{12}
$$

$$
\Delta p\_o = \frac{G\_c}{2\cdot\rho\_i} \mathbf{(1} - \sigma^2 - K\_e) \cdot \frac{\rho\_i}{\rho\_o} \,\tag{13}
$$

$$
\Delta p\_{\text{core}} = \frac{G\_c^{\cdot 2}}{2 \cdot \rho\_i} \cdot \left[ f \cdot \frac{A\_{\text{tot}}}{A\_c} \cdot \frac{\rho\_i}{\rho\_m} + 2 \cdot \left( \frac{\rho\_i}{\rho\_o} - 1 \right) \right] \tag{14}
$$

where Δ*p* is the pressure drop, *Gc* the maximum mass velocity, σ the contraction ratio, *Kc* and *Ke* the contraction and expansion coefficient, respectively, and ρ*<sup>m</sup>* the mean density. Here it was assumed that the contraction and expansion pressure drops were independent on the number of tube rows and could be directly added to the averaged local core pressure. Furthermore, the acceleration pressure drop in Equation (14) was assumed negligible. Figure 12 illustrates the simulated pressure drop, the reconstruction, and extension of the reconstruction.

**Figure 12.** Reconstruction and extension of the pressure drop (*Xt* <sup>=</sup> 13.00 mm, *Fp* <sup>=</sup> 7.50 mm, *Uf r* <sup>=</sup> 4.40 m·s<sup>−</sup>1).

Finally, the extended pressure drop was converted into a total friction factor, which incorporates the contraction and the expansion pressure drops, respectively, consistent with usual practice regarding compact heat exchanger pressure drop correlations. This was done by solving Equations (11)–(14) for *f* with *Kc* = *Ke* = 0 and negligible core acceleration pressure drop (term 2 in Equation (14)).

Appendix A.2 demonstrates that the extension of the global heat transfer coefficient, as well as the reconstruction and extension of the pressure drop, are indeed valid by simulating geometrical designs with 18, 35, 53, and 70 tube rows. Moreover, the results were almost identical and independent of the number of tube rows.

#### **3. Results**

#### *3.1. Heat Transfer and Pressure Drop Regression*

The reduced CFD results in terms Colburn j-factor and Fanning friction f-factor were regressed using multiple linear and nonlinear regression techniques. Moreover, the asymptotic model was used to model the transition between the entrance region (*ent*) and the fully developed (*f d*) region, respectively,

$$y^n = y\_{ent}^n + y\_{fd}^n.\tag{15}$$

where *y* denote the Colburn j-factor or Fanning f-factor, respectively. Four nondimensional parameters based on the hydraulic diameter were used to model the entrance and fully developed regions,

$$y\_{ent} = b\_1 \cdot Re\_{d\_h} l^{2} \cdot \left(\frac{L\_l}{d\_h}\right)^{b3} \cdot \left(\frac{X\_t}{d\_h}\right)^{b4} \cdot \left(\frac{F\_p}{d\_h}\right)^{b5},\tag{16}$$

$$\log y\_{fd} = b\_6 \cdot \text{Re}\_{d\_h} \, ^{b7} \cdot \left(\frac{X\_t}{d\_h}\right)^{b8} \cdot \left(\frac{F\_p}{d\_h}\right)^{b9} \, \tag{17}$$

where *Re* is the Reynolds number, *b*1,2... regression coefficients, and *dh* the hydraulic diameter given by,

$$d\_h = \frac{4 \cdot A\_{\text{c}} \cdot L\_l}{A\_{\text{tot}}}.\tag{18}$$

Notice that the fully developed equation was independent longitudinally, in contrast to the entrance equation. The regression procedure followed the four steps:


Figure 13 illustrates the regression methodology for the j- and f-factor, respectively.

**Figure 13.** Regression methodology for j-factor (**a**) and friction factor (**b**). (*Xt* = 13.00 mm, *Fp* = 7.50 mm, *Uf r* <sup>=</sup> 4.40 m·s<sup>−</sup>1).

Equations (15)–(17) resulted in very accurate correlations compared with the CFD simulation results. Table 2 indicates the coefficients to be used for the j- and f-factor correlations and Figure 14 shows the resulting parity plots. A total number of 42 × 90 = 3780 simulations points were used to derive the heat transfer and pressure drop correlations.


**Table 2.** Coefficients for the heat transfer and friction correlations.

*Fluids* **2019**, *4*, 205

**Figure 14.** Parity plots for j-factor (**a**) and f-factor (**b**), dashed lines indicate 10% error.

Furthermore, the parity plots indicate the mean average deviation (*MAD*), mean relative deviation (*MRD*), and coefficient of determination *R*2 . These values were computed by using the following equations:

$$MAD = \frac{1}{n} \cdot \sum\_{i=1}^{n} \left| \frac{y\_{i,pred} - y\_{i,sim}}{y\_{i,sim}} \right|, \tag{19}$$

$$MRD = \frac{1}{n} \cdot \sum\_{i=1}^{n} \left( \frac{y\_{i,prod} - y\_{i,sim}}{y\_{i,sim}} \right),\tag{20}$$

$$\mathcal{R}^2 = 1 - \frac{\sum\_{i=1}^n \left( y\_{i,sim} - y\_{i,pred} \right)^2}{\sum\_{i=1}^n \left( y\_{i,sim} - \overline{y}\_{sim} \right)^2},\tag{21}$$

where *n* is the number of samples, and *pred* and *sim* denote prediction and simulation, respectively. The accuracy of the correlations cannot be guaranteed when the correlations are applied beyond the ranges of the simulation points. The ranges of the simulation points were as follows:


#### *3.2. Analysis of Entrance Region*

The results of this work indicated that the heat transfer effects of the entrance region are significant and necessary to include in the heat transfer correlation. The thermally developed region is typically claimed when the heat transfer coefficient is within 98% of the fully developed value. Figure 15 shows the number of tube rows for which this criterion is reached at different frontal velocities as a function of the hydraulic diameter, the Reynolds number, and the fin angle.

The results show that the thermally developed flow criterion is reached at different tube rows depending on mainly the air velocity and hydraulic diameter. The highest entrance regions are found at low air velocity and high hydraulic diameter and vice versa. No particular tendencies are found with respect to fin angle.

**Figure 15.** Number of tube rows to reach thermally developed flow at different frontal velocities vs. hydraulic diameter (**a**), Reynolds number (**b**), and fin angle (**c**).

#### *3.3. Volume Goodness Factor*

The volume goodness factor, defined for extended surfaces by Shah and Sekulic [32], is used to compare the microchannel geometries with the baseline finned-tube evaporator for industrial refrigeration (see Section 2.2 for comparisons of hydraulic diameter and compactness). The volume goodness factor compares the heat transfer rate per unit temperature difference and unit core volume versus the friction power expenditure per unit core volume, both defined by:

$$
\hbar \eta\_o \cdot \hbar \cdot \beta = \frac{c\_p \cdot \mu}{Pr^{2/3}} \cdot \eta\_o \cdot \frac{4 \cdot \sigma}{d\_h^{-2}} \cdot j \cdot Re,\tag{22}
$$

$$E \cdot \beta = \frac{\mu^3}{2 \cdot \rho^2} \frac{4 \cdot \sigma}{d\_{\text{fl}} 4} \cdot f \cdot \text{Re}^3,\tag{23}$$

where η*<sup>o</sup>* is the overall surface efficiency calculated using the fin efficiency for rectangular fins (see Appendix A.1), μ is the viscosity, and *E* is the friction power per unit surface area.

Most correlations for finned-tube evaporators in the literature are developed for staggered tube layouts as pointed out by Webb and Kim [33]. The correlations are typically developed for designs with lower fin pitch and lower number of tube rows compared with the baseline finned-tube evaporator for industrial refrigeration. This complicates the choice of correlations to compare with our results. In the following comparison, the plain finned-tube correlations by Kaminski and Groß [34] are used to calculate the j- and f- factors and the overall surface efficiency, as outlined by Fraß et al. [35]. Figure 16 shows the comparisons of the microchannel evaporator having 35 tube rows and the baseline finned-tube evaporator having eight tube rows.

**Figure 16.** Volume goodness factors for the microchannel evaporator geometries (Table 1) and the baseline finned-tube industrial refrigeration evaporator (*Uf r* <sup>=</sup> 2.93 m·s<sup>−</sup>1, *Nl* <sup>=</sup> 35 and 8, respectively).

The volume goodness factors reveal that the microchannel evaporator is indeed more attractive than the baseline finned-tube evaporator, transferring more heat per unit volume at the same fluid flow power, and vice versa. In other words, the microchannel performs the best from the viewpoint of heat exchanger volume. There is however a single point (*Xt* = 21 mm, *Fp* = 2.5 mm) where the pressure drop of the microchannel evaporator increases more than the heat transfer, and results in similar performance as the baseline finned tube evaporator. This is mainly due to the low fin angle effects for this geometry. Furthermore, the variation of the number of tube rows had an insignificant effect on the volume goodness factor.

#### **4. Discussion**

The correlations obtained herein are based on (or fixed by) the microchannel profile design. For providing general correlations, the tube width, the tube height, and the longitudinal tube pitch must be parametrized too. This work did not attempt to reach beyond the actual dimensions of the extruded microchannel profile. The work must rather be viewed as a first attempt to deliver correlations for the design of such evaporators, and to be used for future research and development, especially devoted to the refrigerant charge minimization in industrial refrigeration systems. The developed correlations can be used to design the new microchannel evaporator for this purpose in dry conditions.

Frosting, defrosting, and water condensate drainage are furthermore dependent on the total size of the evaporator, especially the height as the water condensate need to travel downwards through the triangular fins. These considerations are considered for future work. A prototype evaporator is already outlined at this moment and it will be tested experimentally at the Danish Technological Institute laboratory in the near future. These tests will be used to compare the correlations accuracy. Furthermore, tests are planned to study the cooling capacity during frost build-up and defrost performances.

Additionally, the CFD simulations should be viewed as idealized flows compared with the total evaporator flow in a real installation. There are many peculiarities in real evaporators such as airside and tube-side temperature nonuniformity, fluid flow maldistribution, nonidealized fin conduction, transitional fluid flow regimes, imperfect contact between tubes and fins, fin geometry manufacturing uncertainties, etc. These factors must be incorporated in the anticipated uncertainty during the design of the microchannel evaporator.

The correlated heat transfer coefficient is surface averaged. To be used in heat exchanger simulation codes, it should be used to calculate the fin efficiency as well. In Appendix A.1, it is demonstrated that the fin efficiency for rectangular fins can be used with good accuracy, even though a heat flux concentration (2D effect) occurs near the base of the fin at the microchannel walls.

The entrance region was found to be significant in the current analysis. Disregarding the effect of the entrance region might lead to significant underestimations of the global heat transfer coefficient, especially at lower frontal velocities where the highest entrance regions were found. It should be stressed that the current investigation considers plain triangular fins with large fin pitches. The developing region might be insignificant for other types of fins and fin pitches, e.g., because of larger secondary flows in louvered fins. No clear entrance length trends were found in terms of Reynolds number or fin angle. However, Shah and London [36] found that the entrance region reached a minimum for triangular duct flow with angles around 2ϕ ≈ 55◦.

Additionally, in Appendix A.2, the extension of the global heat transfer coefficient longitudinally as well as the reconstruction and extension of the pressure drop longitudinally are assessed and discussed. Indeed, the methodology can be applied to minimize CFD simulation points and simplify the computational domain.

#### **5. Conclusions**

This paper presented heat transfer and pressure drop correlations for a new microchannel evaporator design, based on a newly developed microchannel profile with condensate drainage slits and use of triangular shaped plain fins with large fin pitch. The chosen evaporator geometry corresponds to evaporators for industrial refrigeration systems with long frosting periods. Heat transfer and pressure drop correlations were developed using computational fluid dynamics (CFD) and defined in terms of Colburn j-factor and Fanning f-factor. The computational domain covered the complete thermal entrance and developed regions, which made it possible to extract virtually infinite longitudinal heat transfer and pressure drop characteristics. Indeed, the entrance region was found to be significant compared to the typical longitudinal evaporator length. Therefore, the asymptotic model was used to correlate the entrance and developed regions, respectively. The developed Colburn j-factor and Fanning f-factor correlations were able to predict the numerical results with 3.41% and 3.95% deviation, respectively.

**Author Contributions:** Writing—original draft, B.R. and M.R.K; Writing—review and editing, B.R., W.B.M., J.H.W. and M.R.K.

**Funding:** The research was funded by the Danish Energy Agency (DEA) through the Energy Technology Development and Demonstration Programme (EUDP), project number: 64017-05128—FARSevap.

**Acknowledgments:** The funding by the DEA is greatly acknowledged. Furthermore, the authors acknowledge the collaboration regarding the new microchannel evaporator design and fin design with our industrial partners: The Danish Technological Institute, Aluventa and Hydro Precision Tubing.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Nomenclature**



#### *Fluids* **2019**, *4*, 205


#### **Appendix A**

#### *Appendix A.1. Fin E*ffi*ciency*

The computation of the heat transfer coefficient and fin efficiency are equally important for the design optimization of the new microchannel heat exchanger. In order to examine the fin efficiency, two heat conduction Finite Element (FEM) simulations were carried out, one with a smaller fin and another with a larger fin. Symmetry plans were used again to minimize the computational domain. The temperature of the channel internal walls was specified to 6 ◦C. A constant heat transfer coefficient was applied to the fin and channel external surfaces, corresponding to the thermally developed local heat transfer coefficient extracted from the CFD simulations. The contact between the channels and the fins is assumed perfect. The fin efficiency was calculated based on the results of the heat conduction simulations as follows.

$$\eta\_f = \frac{\dot{Q}\_{\text{actual}}}{\dot{Q}\_{\text{ideal}}} = \frac{h \cdot \int\_f (T\_a - T\_f) \cdot dA}{h \cdot A\_f \cdot (T\_a - T\_b)},\tag{A1}$$

where *Ta* is the mean fluid temperature, *Tb* is the fin base (or contact) temperature, and *Tf* is the fin temperature. The fin efficiency was compared with the analytical fin efficiency evaluated for rectangular fins

$$\eta\_f = \frac{\tan \text{h}(m \cdot l\_c)}{m \cdot l\_c},\tag{A2}$$

$$m = \left(\frac{2\cdot h}{k\_f \cdot F\_t}\right)^{1/2},\tag{A3}$$

with *lc* = *Pf* /4. The comparison is shown in Figure A1 and the temperature contours of the heat conduction simulations are shown in Figure A2.

**Figure A1.** Fin efficiency vs. thermal conductivity for two geometries. Symbols indicate the analytical fin efficiency evaluated for rectangular fins.

**Figure A2.** Temperature contours of the channel and fins. (*Fp* × *Xt* = 5.00 × 9.00 mm (left) and *Fp* <sup>×</sup> *Xt* <sup>=</sup> 10.00 <sup>×</sup> 17.00 mm (right), *Uf r* <sup>=</sup> 4.40 m·s<sup>−</sup>1).

The results demonstrate that the analytical fin efficiency for rectangular fins can be used with good accuracy to model the fin efficiency of the current fin design. This holds true even though heat flux concentration (2D effects) occurs near the base of the fin at the microchannel walls.

#### *Appendix A.2. Longitudinal Extrapolation Analysis*

In this section, the validity of the extended global heat transfer coefficient is assessed. Moreover, three additional simulations were performed at different number of tube rows. The geometrically centered dimensions (*Fp* = 7.50 mm, *Xt* = 13.00 mm) and highest air velocity (*Uf r* = 4.40 m·s−1) were used in these simulations. The results in terms of Colburn j-factor and the Fanning f-factor are represented in Figure A3 including the prediction of our correlation (Equation (15)).

**Figure A3.** Colburn j-factor and Fanning f-factor vs. longitudinal length (or number of tube rows) (*Fp* <sup>=</sup> 7.50 mm, *Xt* <sup>=</sup> 13.00 mm, *Uf r* <sup>=</sup> 4.40 m·s<sup>−</sup>1).

The results showed very good agreement between the results of simulations that were close to identical, and well predicted using the developed correlations. The MAE of the four simulated Colburn j-factors compared with Equation (15) were 1.9%, 1.9%, 2.4%, and 3.2% for the 18, 35, 53, and 70 tube rows, respectively. The MAE of the four simulated Fanning f-factors compared with Equation (15) were 8.2%, 3.7%, 2.2%, and 2.6% for the 18, 35, 53, and 70 tube rows, respectively. This indicated that 35 tube rows were sufficient for developing the correlations in the paper.
