Performance of Air-Cooled Heat Exchanger with Laminar, Transitional, and Turbulent Tube Flow

. Some air-cooled heat exchangers, especially in air conditioning and heating installations, heat pumps, as well as car radiators, work in a wide range of loads when the liquid flow in the tubes can be laminar, transitional or turbulent. In this paper, a semi-empirical and empirical relationship for the Nusselt number on the liquid-side in the transitional and turbulent range was derived. The friction factor in the transition flow range   was calculated by linear interpolation between the values of the friction w,tre Re 3,000  . Based on experimental data for a car radiator, empirical heat transfer relationships for the air and water-side were found by using the least squares method. The water temperature at the outlet of the heat exchanger was calculated using P-NTU (effectiveness-number of transfer units) method. The heat flow rate from water to air was calculated as a function of the water flow rate to compare it with the experimental results. The theoretical and empirical correlation for the water-side Nusselt number developed in the paper were used when determining the heat flow rate. The calculation results agree very well with the


Introduction
Heat transfer correlations for calculating heat transfer coefficients on both air and water side are needed in the thermal calculations of heat exchangers using engineering methods such as LMTD, -NTU, and P-NTU [1][2][3][4]. Turbulent flow regime of the fluid in tubes is usually assumed in the design and performance calculations of the plate fin and tube heat exchangers (PFTHEs). Usually, the power-type heat transfer correlations are applied for calculating the water and airside heat transfer coefficients [1][2][3][4][5][6]. The most popular heat transfer correlations on the water -side are the wellknown relationships of Dittus-Boelter and Sieder and Tate. The Gnielinski [7] relationship is also frequently used. These correlations are appropriate only for high Reynolds numbers when the Rew is larger than 4000. Recently, Taler [8][9] proposed a new correlation to calculate the Nusselt number for the transitional and turbulent fluid flow inside the tubes at a given tube surface temperature or constant heat flux. In this paper, theoretical and empirical relationships were found. The tube-side heat transfer correlations proposed in [8,9] were generalized to account that transitional flow begins at Rew,trb and ends at Rew,tre. It was assumed in [8,9] that Rew,trb = 2300 and Rew,tre = 3000. However, the experimental results obtained by Ghajar et al. [10] and Meyer and co-workers [11] showed that the values of Rew,trb and Rew,tre can differ from the values 2300 and 3000, respectively. Air and water-side heat transfer correlations for a PFTHE made of circular tubes were determined by the least squares method using 70 experimental data sets obtained in transitional and turbulent flow regime. To calculate the air and water temperature at the outlet of the heat exchanger, the P-NTU method was used. Empirical water and air side correlations were determined based on 70 measurement series including transitional and turbulent flow regimes. Experimental tests were carried out on the new-design car radiator made of circular tubes. The water-side friction factor was linearly interpolated between w = 64/Rew for laminar flow at Rew = Rew,trb and the friction factor for the turbulent flow w(Rew,trb) both in a theoretical and empirical relationship for the Nusselt number. The water-side friction factor in the turbulent flow regime, when the Reynolds number is higher than Rew,trb was determined using explicit formula recently proposed by Taler [12]. The Gnielinski formulas [7] was used to calculate the heat transfer coefficient for the laminar flow of the liquid in heat exchanger tubes. The heat flow rate transferred in the studied heat exchanger from the hot water to the air was calculated by the P-NTU method in which new heat transfer correlations were used. It was shown that computed and experimentally determined heat flow rates are in good agreement.

Performance calculations of the investigated PFTHE
Performance calculations of the investigated PFTHE, i.e., determining the temperature of the water and air at the outlets of the investigated heat exchanger was conducted using the P-NTU method. a) b) Fig. 1. Flow arrangement of the two-row car radiator with two passes; a) front and top view, b) flow system; 1the first tube row in the first (upper) pass, 2the second tube row in the first (upper) pass; 3the first tube row in the second (lower) pass, 4the second tube row in the the second (lower) pass.
Thermo-hydraulic tests of air-cooled plate fin-andtube heat exchanger were carried out in a laminar, transition and turbulent flow regime of water in the tubes. The tested PFTHE was a car radiator for a spark engine with a displacement volume of 1,600 cm 3 . Two-pass car radiator has two rows of round tubes. There are twenty tubes in the first pass, ten in each row, i.e. nu = 10. The number of tubes in the second pass is smaller and is eighteen, nine in each row, i.e. nl = 9. The tube arrangement is in-line. Water flows in parallel through the first and second row of tubes in both first and second pass. Between the first and second pass, there is a chamber in which water from the first and second row of tubes is mixed and then flows into the second pass. The flow arrangement of an automobile radiator is shown in Fig. 1. The internal diameter of the tube with the wall thickness t = 0.5 mm is din = 6.2 mm. The transverse pitch of the tube arrangement is equal to p1 = 18.5 mm, and the longitudinal pitch is p2 = 10 mm. The plate fins with a thickness of 0.08 mm are set on tubes with the pitch of 1.5 mm. Dimensions of the heat exchanger are as follows: length Lch = 520 mm, height HLch = 359 mm, and thickness Wch = 2p2 = 24 mm. The mean water Tcm and air temperature um T  after the first pass and after the second pass w T  and lm T  (Fig. 1) were calculated using the P-NTU (Effectiveness -Number of Transfer Units) method.
The mean water Tcm and air temperature um T  after the first pass and after the second pass w T  and lm T  (Fig. 1) were calculated using the P-NTU (Effectiveness -Number of Transfer Units) method.
Formulas for calculating the water-side effectiveness Pw and air-side effectiveness Pa were derived based on the mathematical model of the heat exchanger developed by Taler [13].
The numbers of heat transfer units for water NTUw and air NTUa are defined in relation to one tube located in the first (upper) or second (lower) pass.
The mass flow rate (1) and (2) The following formulas were obtained using the mathematical model of the exchanger presented in the paper [13] The water temperature Tcm at the outlet from the first pass as well as the average air temperature um T  after the first pass can be determined using only the formula (5) for water-side effectiveness Pw,u. The temperature Tcm can be obtained from the definition (3) where Pw,u is calculated using the relationship (5). Solving the heat balance equation for the upper pass gives the air temperature The air mass flow rate u m through the upper heat exchanger pass is With the known temperature Tcm, the liquid temperature w T  at the heat exchanger outlet and the air mean temperature lm T  behind the second pass can be determined similarly. The air-side and water sideeffectiveness of the second pass is defined as , lm am al cm am The relationships for Pa,l and Pw,l are as follows The water temperature w T  at the outlet from the second pass and the average air temperature lm T  after the second pass can be determined using the relationships (11) for the air-side effectiveness Pa,l and water-side effectiveness Pw,l. The water temperature w T  at the outlet from the second pass is where Pw,l is given by Eq. (13).
The presented procedure for calculating the water and air outlet temperature based on the P-NTU method can also be applied to other types of heat exchangers with different flow systems. Many relationships for calculating the effectiveness P as a function of the number of transfer units NTU can be found in the books on the heat exchangers [1][2][3][4]. Overall heat transfer coefficients ho,l and ho,u referred to the outer surface area of the bare tube for the lower and upper pass are given by ,, ,, The effective heat transfer coefficient ho taking into account the presence of fins, based on the surface area of the bare tube was determined using the following relationship The continuous fin was divided into 38 rectangular fins due to the symmetry of the temperature field in the continuous fin. The fin efficiency f was determined assuming that the air-side heat transfer coefficient ha is constant. With this assumption, the fin efficiency f can be determined using the formula where fin T is the mean temperature of the fin surface. The surface temperature of the fin Tfin and the average surface temperature of the fin T were calculated by the finite element method (FEM) using the software ANSYS v. 16. The thermal conductivity of the aluminum fin was kfin = 207 W/(m 2. K). The calculations were carried out for the air-side heat transfer coefficient ha varying from 5 W/(m 2. K) to 300 W/(m 2. K) with a 5 W/(m 2. K) step. The temperature of the fin base Tb was assumed to be 100ºC. Temperature distributions on the fin surface for heat transfer coefficients of 5, 50, 100 and 300 W/(m 2. K) are depicted in Fig. 2. The analysis of the results presented in Figure 3 shows that the temperature of the fin decreases with the increase in the air-side heat transfer coefficient ha. The calculated fin efficiencies f for different heat transfer coefficients ha using the ANSYS 16 were approximated using the least squares method. When operating the heat exchanger in a wide range of loads, the flow regime inside the tubes can be laminar, transitional or turbulent.
The heat transfer coefficient for the laminar flow on the waterside depends on the assumed boundary condition on the tube surface. It is possible to set a constant tube wall temperature or a constant heat flux. In the case of tubular cross-flow heat exchangers, better compatibility with experimental results was obtained assuming a constant heat flux at the inner surface of the tubes. Fluid temperature variations in the tubular crossflow heat exchanger are similar to changes in fluid temperatures that occur in counter-flow heat exchangers. The difference between the temperatures of both fluids is almost constant over the length of the entire heat exchanger. It can, therefore, be assumed that the heat flux at the inner surface of the tubes is constant.

Air-side heat transfer correlation
The Nusselt number on the air side was assumed in the following form 2 1/3 1

Nu
Re Pr cross-section between two neighboring tubes located in the same row. All physical air properties in Eq. (28) were evaluated at the average temperature

Liquid-side heat transfer correlation
In this section, heat transfer correlations for laminar flow as well as for transition and turbulent flow in tubes used in the mathematical model of the exchanger are presented. The average Nusselt number Nu m,q for hydraulically and thermally developing laminar fluid flow in a tube with the uniform wall heat flux, which usually occurs in heat exchangers was used. The following formula recommended by the VDI Heat Atlas [7] The limit value of the Reynolds number Rew,trb, in which the laminar flow ends, and the transitional starts, is usually taken as 2300 [7] or 2100 [14][15]. However, recent experimental studies [10][11] show that the limit value of the Reynolds number Rew,trb is influenced by the type and shape of the inlet to the tube. Eq. (31) was derived by Lévêque [16] using the local Nusselt number for small values of the parameter (x/din)/(RewPrw), where x is the distance from the tube inlet.
The Nusselt number Num,q,3 appearing in the correlation developed for a tube with a constant wall heat flux is given by The relationship (32) was obtained by an approximation of Nusselt numbers obtained from a numerical solution of the momentum and energy conservation equations assuming a uniform velocity and temperature profiles at the inlet of the tube.
Taler [8] determined the Nusselt number for the turbulent flow in the tube as a function of Prandtl and Reynolds numbers by integrating the energy conservation equation assuming a radial velocity distribution determined experimentally. Using the Nusselt number values obtained by Taler [8], the correlation for transitional and turbulent flow on the water-side was found by the method of least squares. The friction factor w was evaluated using the explicit relationship proposed by Taler [12] for turbulent flow regime. In this paper, the relationship proposed by Taler [8] was generalized to the following form The formula (35) for the friction factor results from linear interpolation between w = 64/Rew for laminar flow at Rew = 2100 and turbulent friction factor given by Eq.(34) at Rew = 3000. The water-side Reynolds number Rew = wwdin/w is based on the inner diameter of the circular tube. The inner diameter of the tubes was din = 6.2 mm. The physical properties of the water were determined at the mean temperature In the relationship (35) there is a Reynolds number Rew,trb, at which the transitional flow begins. It was assumed that Rew,trb = 2100. The heat transfer correlation (33) can also be used when the transitional flow regime starts at other values of the Reynolds number [10][11]. In heat exchangers, the influence of free convection inside tubes is negligibly small. Natural convection in horizontal pipes occurs due to the temperature difference of the liquid inside the pipe. The temperature of the fluid in the central part of the tube cross-section is different than near its inner surface. The formation of natural convection is favored by the large diameter and length of the pipe and the high heat flux at the inner surface of the pipe. In heat exchangers, there are usually several passes. Between two neighboring passes, there are reversible chambers or bends, in which the fluid is intensively stirred. This, in turn, equalizes the temperature distribution inside the cross-sections of the tubes. At the inlet to each pass, the fluid temperature is almost uniform. The ratio of the length of pipes to their diameters is not very high in heat exchangers. Therefore, there are no conditions to create a large difference in fluid temperature inside the cross-section of the heat exchanger tubes, contributing to the formation of natural convection. For this reason, in the heat transfer correlations (33) and (36) intended for use in heat exchangers, natural convection is omitted.
The water-side correlation Nuw = f(Rew, Prw) was assumed for the investigated heat exchanger in the form similar to the correlation (36) valid for flow in straight tubes  