Prediction of Friction Factor and Heat Transfer Coefficient for Single-Phase Forced Convection Inside Microfin Tubes

Abstract

Microfin tubes are widely used to enhance heat transfer in heat exchangers in order to reduce volumes, costs and refrigerant charge. Much experimental work has been published for the flow of liquids, while some experimental work is available for the flow of gases for the chemical, refrigeration and air conditioning industry. This work reviews the literature and presents new experimental friction factors for the flow of the superheated vapor of R1234ze(E) in a 5 mm outside diameter microfin tube. The authors have also collected an extensive data bank of heat transfer coefficients (around 648 points from different research laboratories) and friction factors (around 536 points), covering 45 different geometries of inner finned tubes. After comparing the predictions from available correlations with experimental data, the present paper suggests the best performing equations for the calculation of the friction factor and of the Nusselt number during forced convection flow of liquids and gases. The suggested model for friction factor estimates the experimental values with a relative and absolute deviation of −0.3% and 7.9%, respectively, whereas the suggested model for the heat transfer coefficient predicts the experimental data bank with a relative and absolute deviation of −3.3% and 13.9%, respectively. The validity range of the two correlations is extremely wide, covering microfin tubes with diameters from 2.6 mm to 24.4 mm, and Reynolds number from about approximately 1000 to 300,000 for the friction factor, and from 3000 to 1,000,000 for the heat transfer coefficient.

1. Introduction

Heat transfer enhancement is an important factor in achieving higher energy efficiencies and in reducing heat exchangers volumes and costs and refrigerants charge. Microfin tubes are widely used to obtain enhancement on the inner fluid side heat transfer coefficients, while maintaining sufficiently small pressure drops. They are used in heat exchangers with liquids or gases or in condensers and evaporators of refrigeration and air-conditioning systems and heat pumps. Single-phase flow is recurring in almost every vapor compression system, on the brine side or on the refrigerant side. Refrigerant exits as superheated vapor from the evaporator, and it enters as superheated vapor into the condenser from which it often exits in subcooled conditions. Therefore, the proper design of evaporators and condensers considers both the single-phase and the two-phase regions.

Many experimental and theoretical studies cover the condensation and evaporation of different fluids in microfin tubes; many empirical equations are available to calculate the heat transfer coefficient and the frictional pressure drop in these tubes. Relatively less studies are available for liquid or gas flow in these tubes, and few equations to calculate the friction factor and heat transfer coefficient during single-phase forced convection are published in the open literature. There is a lack of tools to efficiently design heat exchangers with microfin tubes in which water, brines, compressed air and superheated vapors of refrigerants flow. This work reviews the available literature, presents some new never-before-published experimental data and finally suggests the most promising equations to calculate the heat transfer coefficient and friction factor.

The geometrical parameters that characterize a microfin tube are (Figure 1): inner diameter at the fin root d_i, fin height e, mean fin thickness s, helix angle of the fins β, the number of fin starts N, the fin pitch p_f and the apex angle of the fin γ. Tubes with inner fins are generally considered microfin tubes if they present a ratio H = 2 e/d_i ≤ 0.06 [1]. Cross-grooved tubes are helical finned tubes with the addition of notches cut into the fins.

Several authors presented experimental single-phase data in turbulent flow and empirical correlations able to estimate the heat transfer coefficient and the friction factor, among whom are: Jensen and Vlakancic [1], Carnavos [2], Ravigururajan and Bergles [3], Webb et al. [4], Zdaniuk et al. [5]. Some researchers focused their experimental work on the inlet effect on the transition from laminar to turbulent flow: Meyer and Oliver [6,7] and Tam et al. [8]. Other researchers analyzed the effect of the Prandtl number: Brognaux et al. [9], Li et al. [10] and Raj et al. [11]. Meyer and Olivier [6], Tam et al. [8], Siddique and Alhazmy [12], Li et al. [10], Brognaux et al. [9], Jensen and Vlakancic [1] and Wang et al. [13] found that a secondary transition existed in the transition flow. All of the research presented in [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] were relative to water, brines (water-ethylene glycol), air or oil. Li et al. [17] tested the subcooled liquid of R22; Eckels et al. [18] tested superheated vapors of R410A, R134a and R407C. Diani et al. measured heat transfer coefficients during the liquid flow of R1234yf inside a 3.0 mm [19] and inside a 4.0 mm OD microfin tube [20]. Diani et al. [21] tested liquid R1234ze(E) inside a 7.0 mm OD microfin tube. Mori et al. [22] tested superheated vapor of R134a, while superheated vapor of R11 was considered by Nozu et al. [23]. Lee et al. [24] and Kuwahara et al. [25] tested supercritical carbon dioxide cooling in microfin tubes.

Some analytical work is available too. Wu et al. [26] compared predictions from a modified version of the model by Ravigururajan and Bergles [2] with experimental heat transfer coefficients by Brognaux et al. [9], Li et al. [10] and Wang et al. [14]. Wang and Rose [27] compared six friction factor correlations with a database from six experimental investigations, covering a wide range of microfin geometrical characteristics.

In order to test the ability of the available correlations to predict friction factors and Nusselt numbers for the flow of liquids and gases in inner finned tubes, the present authors have collected, from the open literature, a wide data bank of heat transfer coefficients (around 648 points from different research laboratories) and friction factors (around 536 points), covering 45 different geometries of inner finned tubes.

The present paper displays new friction factor data too: the authors have performed pressure drop measurements in a new microfin tube with an outside diameter of 5 mm and small fin pitch p_f.

In order to suggest the most reliable equations to the designer of heat exchangers, this paper compares the predictions from equations from the literature against experimental data, presents the results of the analysis and suggests a correction of the friction factor model for microfin geometries with small values of fin pitch.

The main purpose of this paper is the extension of the validity of easy-to-use correlations in terms of geometrical parameters of the microfin tube as well as the Reynolds number range. This is conducted because, until now, none of the empirical equations satisfactorily predicts the friction factor and heat transfer coefficient in a wide spectrum, as they have been targeted for a specific range of validity.

2. Experimental Data Bank

A data base of measured friction factors and Nusselt numbers has been compiled by the present authors. Fluids, tubes and geometrical characteristics of the microfin and high fin tubes are summarized in

Table 1. Experimental data sets and geometrical characteristics of microfin and high fin tubes.

Reference	Fluid	N (-)	β (°)	e (mm)	di (mm)	s (mm)	γ (°)	H (-)	DP, HT
Jensen and Vlakancic [1], Vlakancic [28]	Water/ ethyl. glycol	54	45	0.44	22.08	0.54	40 *	0.0399	DP, HT
		54	45	0.33	24.13	0.9	40 *	0.0274	DP, HT
		54	45	0.22	22.1	0.58	40 *	0.0199	DP, HT
		30	30	0.36	24.18	0.64	40 *	0.0298	HT
		54	30	0.18	24.03	0.4	40 *	0.015	HT
		54	30	0.36	24.41	0.7	40 *	0.0295	HT
		54	15	0.36	24.05	0.62	40 *	0.03	HT
		36	25	0.68	23.00	1.14	40 *	0.0591	HT
		8	30	1.16	23.64	1.0	40 *	0.0981	DP, HT
		14	30	1.2	23.78	1.02	40 *	0.1	DP, HT
		30	30	1.3	23.70	0.82	40 *	0.11	DP, HT
		14	30	2.06	24.28	0.64	40 *	0.17	HT
		14	15	1.18	23.72	0.92	40 *	0.0995	HT
		14	0	1.18	23.64	0.70	40 *	0.0998	HT
		22	25	0.62	21.86	1.84	40 *	0.0567	HT
Brognaux et al. [9]	Water	78	27	0.35	14.86	0.2	30	0.0471	DP, HT
		78	17.5	0.35	14.86	0.2	30	0.0471	DP, HT
		78	20	0.35	14.86	0.2	30	0.0471	DP, HT
Nozu et al. [23]	R11	47	20	0.24	8.44	0.2	63	0.0569	DP
		65	24	0.16	8.82	0.19	87.1	0.0363	DP
Li et al. [17]	R22	35	18	0.12	4.54	0.036 ***	25	0.0529	DP, HT
		40	18	0.15	4.6	0.073 ***	40	0.0652	DP, HT
Webb et al. [4]	Water	45	45	0.327	15.54	0.1345	41	0.0421	DP, HT
		30	45	0.398	15.54	0.136	41	0.0512	DP, HT
		10	45	0.43	15.54	0.1405	41	0.0553	DP, HT
		40	35	0.466	15.54	0.1355	41	0.0600	DP, HT
		25	35	0.493	15.54	0.136	41	0.0635	DP, HT
		25	25	0.532	15.54	0.136	41	0.0685	DP, HT
		18	25	0.554	15.54	0.136	41	0.0713	DP, HT
Mori et al. [22]	R134a	50	18	0.21	6.5	0.09 ***	35 **	0.0646	DP
Meyer and Olivier [6,7]	Water	25	18	0.399	14.648	0.231 ***	46.97	0.0545	DP, HT
		35	27	0.395	14.56	0.212 ***	43.93	0.0543	DP, HT
		25	18	0.48	17.66	0.223 ***	38.49	0.0544	DP, HT
		35	27	0.467	17.82	0.239 ***	41.92	0.0524	DP, HT
Eckels et al. [18]	R134a, R410A, R407C	60	18	0.2	9.12	0.095	51	0.0440	HT
		50	18	0.2	7.54	0.11	57	0.0531	HT
		72	0	0.2	8.88	0.093	50 ****	0.0451	HT
		60	0	0.19	7.21	0.089	50 ****	0.0527	HT
Diani et al. [19,20,21]	R1234ze(E)	50	18	0.18	6.5	0.092 ***	42	0.0554	HT
	R1234yf	40	18	0.12	3.64	0.063 ***	43	0.066	HT
	R1234yf	40	7	0.12	2.64	0.063 ***	43	0.091	HT
Present work	R1234ze(E)	54	30	0.15	4.58	0.021		0.0655	DP

* Estimated, as suggested in [27]. ** Estimated from the area enhancement ratio; *** Calculated supposing triangular fin, s = 4/3 e tan(γ/2), as suggested in [7]. **** Estimated.

.

Predictions from the correlations for friction factors have also been compared with the data for cross-grooved tubes, whose geometry is reported in

Table 2. Experimental data sets and geometrical characteristics of cross-grooved tubes.

Reference	Fluid	N (-)	β (°)	e (mm)	di (mm)	s (mm)	γ (°)	e (mm) Second Groove
Brognaux et al. [9]	Water	78	17.5	0.35	14.86	0.2	30	0.14
								0.21
								0.28

.

Table 3. Experimental uncertainties.

Reference	Friction Factor Uncertainty	HTC Uncertainty
Vlakancic [28]	±3.3%	±19.5%
Brognaux et al. [9]	±4.9%	±9.65%
Nozu et al. [23]	±17% for (−Δp/L)f	-
Li et al. [15]	-	±9.6%
Webb et al. [4]	±5% *	±8% **
Mori et al. [22]	±7% *	-
Meyer and Olivier [6]	0.58–55.7%	1.04–1.22%
Eckels et al. [18]	2–12% (Re > 100,000)	4.86–18.6% (Re > 100,000)
Diani et al. [19,20,21]	-	1.9–7.2%
Present work	±5.0%	-

* Compared with the Blausius equation for the smooth tube. ** Compared to the Gnielinski equation for the smooth tube (Equation (21)).

reports the declared associated uncertainties.

3. New Experimental Friction Factor Data Points

The present authors present new friction factor data, measured during the adiabatic flow of R1234ze(E) superheated vapor, in a microfin tube with a diameter at the fin root d_i = 4.58 mm, number of fin starts N = 54 and fin height e = 0.15 mm (Table 1). As it appears from the geometrical parameters of the data reported in the table, this microfin tube, being relatively recent, presents an extreme high ratio between the number of fins and inner diameter, whose characteristic was uncommon for more classical microfin tubes. This will help to extend the validity of the correlations that will be proposed in the following. The test rig and the experimental procedure are widely described in [19,20,21].

The facility is located in the Heat Transfer in MicroGeometries Laboratory of the Department of Industrial Engineering at the University of Padua. The test rig is a liquid pumped circuit which permits a wide range of working conditions to be set. A schematic of the test rig is shown in Figure 2. A magnetically coupled pump circulates the refrigerant in the Coriolis-effect flow meter and then into an evaporator, at the exit of which the superheated vapor flows into a pre-condenser. A small heat flow rate is exchanged in the pre-condenser for these tests, since they are in superheated conditions. The evaporator is a brazed plate heat exchanger, where the refrigerant receives the heat flow rate given by hot water, supplied in the hot water loop. A water temperature of 60 °C at the inlet of the evaporator is set by means of a PID controller, which guarantees superheated conditions since the saturation temperature in the refrigerant loop is equal to approximately 30 °C. Thus, the pre-condenser is used to adjust the superheating degree at the inlet of the test section. A post-condenser is used to condense and subcool the refrigerant after the test section, prior to returning to the pump.

The test section is made by the microfin tube with two pressure ports: the upstream pressure port is connected to an absolute pressure transducer (accuracy of ±1950 Pa), whereas both the upstream pressure port and the downstream one are connected to a differential pressure transducer (accuracy of ±25 Pa), for the measurement of the pressure drop. Each datum comes from the average value considering a 100 reading in steady-state conditions taken at a frequency of 1 Hz.

The isothermal Fanning friction factor is defined by the following equation:

$$∆p=2f\frac{L}{d_i}\frac{G^2}{\rho }$$

(1)

where d_i is the diameter at the fin root and the mass velocity G is based on the nominal flow area $A_n$ of the microfin tube:

$$A_n=\frac{\pi d_i^2}{4}$$

(2)

The experimental uncertainty of the friction factor is calculated following the procedure suggested by Kline and McClintock [29], according to which the uncertainty (i_y) on a general physical quantity y, which is a function of n independent parameters x_i, can be calculated as:

$$i_i=\sqrt[2]{{\sum }_{i=1}^n{\left(\frac{\partial y}{\partial x_i}{i}_i\right)}^2}$$

(3)

where i_i is the uncertainty on the i-th parameter. Following this procedure, the mean, maximum and minimum uncertainties on the experimental friction factor are 5.0%, 9.5% and 4.5%.

Figure 3 reports the experimental friction factors against the Reynolds number and the interpolating equation. Being experimental tests with super-heated vapor, all the data are in a turbulent regime. The classical behavior of the friction factor with respect to the Reynolds number is confirmed. As the Reynolds number increases, the friction factor decreases with a dependence from the Reynolds number to a power of −0.3.

4. Friction Factor Modeling and Comparison of Predictions with Experimental Data

The isothermal friction factor is defined by Equation (1). For oil, whose viscosity at the wall differs from the viscosity at the bulk temperature, during diabatic tests, Li et al. [10] suggest the Petukhov [28] correction, developed for smooth tubes:

$$\frac{f}{f_{cp}}=\left\{\begin{array}{ll}\frac{7-\frac{{\mu }_b}{{\mu }_W}}{6}& \mathrm{healing}\\ {(\frac{{\mu }_W}{{\mu }_b})}^{0.24}& \mathrm{cooling}\end{array}\right.$$

(4)

In a fully developed laminar flow, Meyer and Olivier [6] and Shome and Jensen [30] found a small increase of the friction factor compared to the smooth tube, due to the roughness height of the fin. Meyer and Olivier [6] suggested the following equation for the friction factor in laminar flow:

$$f_{lam}=\frac{16}{\mathrm{R}\mathrm{e}}\left[1+88{\left(\frac{e}{d_i}\right)}^{2.2}{\mathrm{R}\mathrm{e}}^{0.2}\right]$$

(5)

The above equation is valid up to the transition to turbulent flow which starts at the critical Reynolds number. Meyer and Olivier [6] suggest the following equation for the critical Reynolds number:

$${\mathrm{R}\mathrm{e}}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}=2200{\left[1+9.13·{10}^9{\left(\frac{e}{d_i}\right)}^{5.8}\right]}^{-1/10}$$

(6)

based on data with 0.022 ≤ e/d_i ≤ 0.057.

Wang and Rose [27] compared six friction factor correlations (Jensen and Vlakancic [1], Carnavos [2], Ravigururajan and Bergles [3], Webb et al. [4], Wang et al. [14] and the Cavallini et al. [31] correlation developed for condensation) for a fully developed turbulent flow with a database from six experimental investigations, covering a wide range of microfin geometrical characteristics. The Jensen and Vlakancic [1] correlation was found to be the best, representing the database within ±21%.

For microfin tubes with 0.015 ≤ H ≤ 0.06, 3000 < Re < 70,000, 30 ≤ N ≤ 54 and 15 ≤ β ≤ 45 degrees, Jensen and Vlakancic [1] suggest:

$$\frac{f_{JV}}{f_{st}}={\left(\frac{l_{csw}}{d_i}\right)}^{-1.25}{\left(\frac{A_n}{A_{xs}}\right)}^{1.75}-\frac{0.0151}{f_{st}}\left\{{\left(\frac{l_{csw}}{d_i}\right)}^{-1.25}{\left(\frac{A_n}{A_{xs}}\right)}^{1.75}-1\right\}\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{-\mathrm{R}\mathrm{e}}{6780}\right)$$

(7)

$$\left(\frac{l_{csw}}{d_i}\right)=1-a{\left(\frac{N\sin \beta }{\pi }\right)}^b{\left(\frac{2e}{d_i}\right)}^c{\left[\left(\frac{\pi }{N}-\frac{s}{d_i}\right)\cos \beta \right]}^d$$

(8)

a = 1.577, b = 0.64, c = 0.53, d = 0.28 for e/d_i ≤ 0.02

a = 0.994, b = 0.89, c = 0.44, d = 0.41 for 0.02 < e/d_i ≤ 0.03

$$A_{xs}=A_n-Nes$$

(9)

The friction factor for smooth tubes, $f_{st}$, from [32]:

$$f_{st}={\left[1.58\ln \left(\mathrm{R}\mathrm{e}\right)-3.28\right]}^{-2}$$

(10)

Jensen and Vlakancic [1] tested high fin tubes. For these tubes, in a fully developed turbulent flow, in the ranges 5000 < Re < 70,000, 0.057 ≤ H ≤ 0.17, 8 ≤ N ≤30 and 0 ≤ β ≤ 30°, they suggest:

$$\frac{f_{JV}}{f_{st}}={\left(\frac{l_{csw}}{d_i}\right)}^{-1.25}{\left(\frac{A_n}{A_{xs}}\right)}^{1.75}$$

(11)

$$\left(\frac{l_{csw}}{d_i}\right)=\frac{l_c}{d_i}\left[1-0.203{\left(\frac{N\sin \beta }{\pi }\right)}^{0.65}{\left(\frac{2e}{d_i}\right)}^{0.2}\right]$$

(12)

$$\left(\frac{l_c}{d_i}\right)=\frac{A_n{\left(1-H\right)}^2}{A_{xs}}\left(1-H\right)+\frac{A_{fin}}{A_{xs}}\left[\frac{\pi }{N}\left(1-\frac{e}{d_i}\right)-\frac{s}{d_i}\right]$$

(13)

$$A_{fin}=A_{xs}-A_n{\left(1-H\right)}^2$$

(14)

The friction factor for smooth tubes, $f_{st}$, from [32] is reported in Equation (9). For tubes with β = 0°, Jensen and Vlakancic [1] suggest applying Equations (10)–(14).

Meyer and Olivier [6] present an equation for the friction factor in the transition region, lying between the laminar and turbulent flow regime, as

$$f_{trans}={\left(\frac{16}{{\mathrm{R}\mathrm{e}}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}}\right)}^{0.94}\exp \left(0.57\frac{\mathrm{R}\mathrm{e}}{{\mathrm{R}\mathrm{e}}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}}\right){\left(\frac{\beta }{90}\right)}^{0.37}{\left(\frac{e^2}{p_a{d}_i}\right)}^{0.028}{\left(\frac{p_a}{d_i}\right)}^{-0.009}{\left(\frac{e}{d_i}\right)}^{0.06}$$

(15)

The range of validity of the above correlation (15) is [6]: 18° ≤ β ≤ 79°, 6.14 × 10⁻⁴ ≤ e²/p_f d_i ≤ 0.004, 6.48 × 10⁻⁴ ≤ p_f/d_i ≤ 1.23 and 0.022 ≤ e/d_i ≤ 0.057 and for the Reynolds numbers higher than the critical Reynolds number, up to the Reynolds number at the intersect of equation (15) with the Jensen and Vlakancic equations [1].

The predictions from the above model (Equations (1)–(15)) have been compared with the data bank of experimental friction factors. Figure 4 reports the ratio of the predicted friction factor to the measured one against the Reynolds number. With reference to Table 1, Table 2 and Table 3, Jensen and Vlakancic friction factors [1] are relative to the first three microfin tubes in Table 1 and to the first three high fin tubes with H = 0.1 in Table 1. The equations for high fin tubes have been applied to predict data relative to the Jensen and Vlakancic tubes [1] of Table 1 with H = 0.1 and to the Webb et al. data [4] (Table 1) with the number of starts N equal to 10, 25 and 18.

Meyer and Olivier’s model [6] for the laminar transition zone and Jensen and Vlakancic’s model [1] for the turbulent flow zone up to Re = 70,000, in their validity range, work quite well. At Re > 70,000, the experimental data points are overestimated.

It is worth highlighting that the experimental database used in this analysis covers a huge range both of the parameters of microfin tubes and of fluids. The new tested microfin tube presents a very high value of the ratio between the number of fins (54) and the fin root diameter (4.58 mm), whereas less recent microfin tubes analyzed in the comparison present a much lower number of fins with a diameter larger than 20 mm. Furthermore, the Reynolds numbers span from approximately 1000 up to more than 200,000. This so wide range explains the scattering of the data reported in Figure 4. Moreover, as it appears in Table 3, the experimental uncertainty on friction factors is not negligible for some data.

In order to evaluate the accuracy of the implemented correlations, the relative deviation (dev_rel) and the absolute deviation (dev_abs) are used and defined in the following:

$${dev}_{rel}=\frac{1}{Np}{\sum }_{i=1}^{Np}\left(\frac{y_{calc}-y_{exp}}{y_{exp}}\right)·100$$

(16)

$${dev}_{abs}=\frac{1}{Np}{\sum }_{i=1}^{Np}\left|\frac{y_{calc}-y_{exp}}{y_{exp}}\right|·100$$

(17)

y_calc and y_exp are calculated and experimental generic quantities (friction factor or Nusselt number), respectively. The model (Equations (1)–(15)) is able to estimate 536 experimental values of the friction factors with a relative deviation ${dev}_{rel}=+5.2\%$ and an absolute deviation ${dev}_{abs}=10.5\%$ (

Table 4. Mean and absolute deviations between calculated friction factors and experimental ones.

Model	Relative Deviation (%)	Absolute Deviation (%)	Np
Equations (1)–(15) model	5.24	10.5	536
Equation (18) model	−0.317	7.9	536

).

Brognaux et al. [9], Nozu et al. [23], Li et al. [10], Mori et al. [22], and present work data sets are overestimated by the model from 20 to 70%. Referring to Figure 4, they show the highest deviations with the Jensen and Vlakancic model [1] for Re > 20,000. These data were taken in tubes with a number of fins per unit perimeter (N/π d_i) greater than 1.2 fins/mm. The other data sets show values lower than 1 fin/mm. The present work tube has the highest value (3.75 fins/mm). In order to reduce the deviations, the model (Equations (1)–(15)) is corrected as follows for the data taken in tubes with p_f ≤ 0.001 m:

$$f=f_{model(1-15)}{\left(\frac{p_f}{0.001}\right)}^{0.25}\mathrm{for}\hspace{1em}p\le 0.001\hspace{0.17em}\mathrm{m}$$

(18)

Figure 5 gives the ratio of the friction factor predicted by Equation (18) to the measured one against the Reynolds number. Relative and absolute deviations are ${dev}_{rel}=-0.317\%$ and ${dev}_{abs}=7.9\%$, respectively, with Np = 536 (Table 4). The experimental data at the Reynolds numbers higher than 70,000 are now much better estimated. However, further experimental investigation is needed in the high Reynolds number region.

5. Heat Transfer Coefficient Modeling and Comparison of Predictions with Experimental Data

In the following, the heat transfer coefficient is defined as:

$$HTC=\frac{q}{\pi {·d}_i·L_{HT}·\left|∆T\right|}$$

(19)

where ΔT is the mean temperature difference between wall and fluid bulk.

Several correlations have been proposed to calculate the heat transfer coefficient.

Diani et al. [19] suggest calculating the heat transfer coefficient as:

$${\mathrm{N}\mathrm{u}}_D=\frac{HTC{·d}_i}{\lambda }={\mathrm{N}\mathrm{u}}_{\mathrm{st},\mathrm{Gnielinski}}\hspace{0.17em}Rx$$

(20)

Equation (20) is here applied with Nu_{st,Gnielinski} from [33]:

$$\mathrm{N}{\mathrm{u}}_{\mathrm{st},\mathrm{Gnielinski}}=\frac{\left(f/2\right)\left({\mathrm{Re}}_D-1000\right)\hspace{1em}\mathrm{Pr}}{1+12.7{\left(f/2\right)}^{0.5}\left({\mathrm{Pr}}^{2/3}-1\right)}{\left(\frac{\mathrm{P}\mathrm{r}}{{\mathrm{P}\mathrm{r}}_w}\right)}^{0.11}$$

(21)

$$\mathrm{R}\mathrm{e}=\frac{4\dot{m}}{\pi \left(d_i\right)\mu }$$

(22)

with f from Equation (10) and the enhancement area factor Rx, which in [19] is referred to as the area of the smooth tube with fin tip diameter, is here calculated as:

$$Rx=\left\{\frac{2hN\left[1-\sin \left(\gamma /2\right)\right]}{\pi \left(d_i-2e\right)\cos \left(\gamma /2\right)}+1\right\}\frac{1}{\cos \left(\beta \right)}\frac{\left(d_i-2e\right)}{d_i}$$

(23)

Figure 6 gives the ratio of the Nusselt number predicted by Equations (20)–(23) to the measured one against the Reynolds number for the data sets in Table 1.

Jensen and Vlakancic [1] suggest for Re > 10,000:

$$\mathrm{N}\mathrm{u}=0.012{\hspace{0.17em}\mathrm{Pr}}^{0.4}\left[{\mathrm{R}\mathrm{e}}^{0.87}-280\right]{\left(\frac{{\mu }_b}{{\mu }_W}\right)}^n{\left(\frac{l_{csw}}{d_i}\right)}^{-1.2}{\left(\frac{A_n}{A_{xs}}\right)}^{0.8}\left[f\left(\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}\right)\right]$$

(24)

$$n=0.11\hspace{1em}\mathrm{for}\hspace{1em}\mathrm{heating},\hspace{1em}n=0.25\hspace{1em}\mathrm{for}\hspace{1em}\mathrm{cooling}$$

For microfin tubes, in the ranges 0.015 ≤ H ≤ 0.06, 12,000 < Re < 70,000, 30 ≤ N ≤ 54 and 15 ≤ β ≤ 45 degrees, ($l_{csw}/d_i$) is calculated from Equation (8) and

$$\left[f\left(\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}\right)\right]=\frac{{SA}_{act}}{{SA}_n}\left[1-0.059{\left(\frac{N\sin \gamma }{\pi }\right)}^{-0.31}{\left[\left(\frac{\pi }{N}-\frac{s}{d_i}\right)\cos \beta \right]}^{-0.66}\right]$$

(25)

For high fin tubes, in the ranges 12,000 < Re < 70,000, 0.057 ≤ H ≤ 0.17, 8 ≤ N ≤ 30 and 0 ≤ β ≤ 30°, ($l_{csw}/d_i$) is calculated from Equation (12) and

$$\left[f\left(\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}\right)\right]={\left(\frac{{SA}_{act}}{{SA}_n}\right)}^{0.29}\left[1-1.792{\left(\frac{N\sin \gamma }{\pi }\right)}^{0.64}{\left(H\right)}^{2.76}{\left(\mathrm{R}\mathrm{e}\right)}^{0.27}\right]$$

(26)

The authors [1] gave no equation to calculate the ratio actual inside the heat transfer area divided by the nominal area. Here, it is calculated with Equation (23).

Carnavos [2] instead gives for 10,000 < Re < 100,000, 0.7 < Pr < 30 and 0 < β < 30°:

$${\mathrm{N}\mathrm{u}}_C=0.023{\mathrm{P}\mathrm{r}}^{0.4}{\mathrm{R}\mathrm{e}}_C^{0.8}{\left(\frac{A_{xs}}{A_n}\right)}^{0.1}{\left(\frac{{SA}_n}{{SA}_{act}}\right)}^{0.5}{\left(\mathrm{c}\mathrm{o}\mathrm{s}\beta \right)}^{-3}$$

(27)

In the above equation, the Nusselt number Nu_C and the Reynolds number Re_C are based on the actual heat transfer area and the actual flow area and on the hydraulic diameter. The author [2] gave no equation to calculate the ratio actual inside the heat transfer area divided by the nominal area. Again, here it is calculated with Equation (23).

Ravigururajan and Bergles [3] presented the following equation (tested in the ranges 5000 < Re < 250,000, 0.66 < Pr < 37.6, 0.01 < e/d_i < 0.2, 0.1 < p_a/d_i < 7 and 0.3 < β/90 < 1), referred to the equation for a smooth tube by Petukhov and Popov [34,35]:

$$\frac{\mathrm{N}\mathrm{u}}{{\mathrm{N}\mathrm{u}}_{st}}={\left\{1+{\left[2.64{\mathrm{R}\mathrm{e}}^{0.036}{\left(\frac{e}{d_i}\right)}^{0.212}{\left(\frac{p_a}{d_i}\right)}^{-0.21}{\left(\frac{\beta }{90}\right)}^{0.29}{\mathrm{P}\mathrm{r}}^{0.024}\right]}^7\right\}}^{1/7}$$

(28)

$${\mathrm{N}\mathrm{u}}_{st}=\frac{\left(f_{st}/2\right)\mathrm{R}\mathrm{e}\mathrm{P}\mathrm{r}}{1+12.7{\left(f_{st}/2\right)}^{0.5}\left({\mathrm{Pr}}^{2/3}-1\right)}$$

(29)

The friction factor f_st should be calculated with Equation (10).

Webb et al. [4] gave:

$$\mathrm{N}\mathrm{u}=0.00933{\mathrm{P}\mathrm{r}}^{1/3}{\mathrm{R}\mathrm{e}}^{0.819}{N}^{0.285}{\left(\frac{e}{d_i}\right)}^{0.323}{\beta }^{0.505}$$

(30)

Based on the experimental data for water with 20,000 < Re < 70,000, 5.08 < Pr < 6.29, 0.021 < e/d_i < 0.035, 18 < N < 45 and 25° < β < 45°.

Meyer and Olivier [7] published a correlation for the laminar flow up to Re = 2198 and a correlation for 3500 < Re< 8000 that compared well with their own experimental data. In the transition zone 1900 < Re < 4000, they suggested a combination of the two equations.

The predictions from the above models have been compared with the data bank of experimental heat transfer coefficients measured during liquid or vapor flow. Table 1 and Table 2 report the data sets considered, the geometrical characteristics of the tubes and the associated declared uncertainties. The data of Eckels et al. [18] have been considered only for Re > 100,000, since large drops of uncertainty occur at high Re.

Table 5. Mean and absolute deviations between calculated Nusselt numbers and experimental ones.

Model	Relative Deviation (%) Equation (16)	Absolute Deviation (%) Equation (17)	Np
Present model Equations (20)–(23)	3.3	13.9	648
Jensen and Vlakancic [1] Equations (24)–(26)	−18.9	21.4	648
Carnavos [2] Equation (2)	23.0	31.4	648
Ravigururajan and Bergles [3] Equations (28) and (29)	22.1	24.9	590
Webb et al. [4] Equation (30)	9.84	17.3	590

summarizes the relative and absolute deviations between the calculated values from the above models and the experimental data. Experimental data taken in tubes with helix angle β = 0° are not considered in the comparisons with the Ravigururajan and Bergles model [3] or with the Webb et al. model [4]. The present model (Equations (20)–(23)) better predicts the experimental values of the heat transfer coefficients.

Figure 6 gives the ratio of the Nusselt number predicted by Equations (20)–(23) to the measured one against the Reynolds number for the data sets in Table 1. The proposed model can predict the experimental values of the heat transfer coefficient with a mean relative and absolute deviation of 3.3% and 13.9%. The data whose prediction falls outside the ±20% bars are characterized by having water as a working fluid. Almost all data with refrigerants are within the ±20% bars.

6. Conclusions

New friction factor data, measured during the adiabatic flow of R1234ze(E) superheated vapor, in a tube with a diameter at the fin root d_i = 4.58 mm, number of fin starts N = 54, fin height e = 0.15 mm and fin pitch p_f = 0.266 mm, are presented in the paper.

A data bank of heat transfer coefficients (around 648 points from different research laboratories) and friction factors (around 536 points), covering 45 different geometries of inner finned tubes, has been collected. The database covers different fluids, such as water and refrigerants including R134a, R410A, R1234ze(E) and so on, tube diameters ranging between 2.6 mm to 24.2 mm and a Reynolds number approximately from 1000 to 1,000,000.

The friction factor can be estimated with the Olivier and Meyer model [6] for laminar and transition flow and with the Jensen and Vlakancic correlations [1] for Re > 3000 corrected with Equation (18) for tubes with small fin pitches. The relative and absolute deviation are ${dev}_{rel}=-0.3\%$ and ${dev}_{abs}=7.9\%$, respectively, with Np = 536. The proposed correlation is now able to satisfactorily predict the friction factor also in the high Reynolds number region.

The model that estimates the heat transfer coefficient with better accuracy is the Diani et al. model [19] (Equations (20)–(23)). The model suggests evaluating the heat transfer coefficient by multiplying the one for a smooth tube (calculated with Gnielinski’s equation [33]) for the area enhancement Rx to take into account the augmented surface area of the microfin tube. This easy-to-use correlation is able to estimate the experimental values with a relative and absolute deviation of ${dev}_{rel}=3.3\%$ and ${dev}_{abs}=13.9\%$, respectively, with Np = 648.