**1. Introduction**

Cross-flow finned heat exchangers are used in many industrial fields. In the case of heat exchangers operating at high parameters, individual round fins are welded to the outer surfaces of the tubes. On the other hand, heat exchangers in which the temperature of the working medium is just above the ambient temperature consist

of aluminum tubes fitted with aluminum fins. These are compact heat exchangers known as plate-fin and tube heat exchangers (PFTHEs).

CFD simulations were used in [15–17] to determine the uniform heat transfer

*The CFD Based Method for Determining Heat Transfer Correlations on Individual Rows…*

Sun et al. presented the results of CFD simulations and experimental studies of the optimized PFTHE in their paper [15]. The optimization of the finned heat exchanger consisted of directing the airflow to the back of the tubes using appropriately prepared channels. This procedure was intended to stop the formation of dead zones, thus improving heat transfer and reducing the pressure drop on the air-side.

Numerical and experimental studies of different types of fins are presented in the paper [16]. Wavy and plain fins with radially spaced winglets around the tubes in the plate fin tube heat exchanger were discussed. Numerical calculations were carried out in the form of CFD simulations in which laminar flow was assumed for air velocities ranging from 1.5 m/s to 3.5 m/s. For air velocities above 3.5 m/s, the turbulent flow was assumed. On the other hand, experimental studies allowed the

Nagaosa in paper [17] used direct numerical simulation (DNS) to determine the air-side heat transfer correlation, which was then verified experimentally. Good agreement of the obtained results was achieved. Unfortunately, computer-based

In the available literature, few works are describing mathematical models of heat exchangers considering different heat transfer coefficients on different rows of heat exchanger tubes. The works [19–20] used mathematical models of PFTHEs with different heat transfer coefficients on the first and second row of tubes to simulate steady-state operation. In work [21], calculations of PFTHEs operating under

The paper [22] describes an analytical method for determining the required air mass flow rate of a heat exchanger used in the propane pre-cooling cycle of an LNG plant. The method can be used to control the air mass flow rate through the fin tubes of an air-cooled heat exchanger to control the outlet temperature of the

The paper [23] deals with the numerical modeling of a heat exchanger with finned tubes with large fin pitch. Computer simulations were carried out to propose the optimal perforation size and design for maximum heat transfer coefficient. Finand-tube compact heat exchangers (FTCHEs) were investigated in [24]. The results of a three-dimensional CFD simulation for an FTCHE heat exchanger constructed from oval tubes with smooth and corrugated fins are shown. Thermal–hydraulic studies were carried out for Reynolds number in the range of 200–900. The computational results show that the average Nusselt number for FTCHE with corrugated fins can be increased up to 20% compared to the case with plain fins. The average performance value for the single- and three-wave finned oval tube compact heat exchangers increased by 5% and 15%, respectively, relative to the plain tube

When turbulent flow occurs on the air-side (high Reynolds numbers), the Nusselt number in the first pipe row of the PFTHE takes on lower values than the Nusselt numbers in the subsequent pipe rows. This phenomenon occurs both in heat exchangers made of smooth tubes [1–3] or single finned tubes [18] and in heat exchangers with continuous fins [11]. Kearney and Jacobi [18] determined by experimental study the Nusselt number for each row of tubes in cross-flow finned tube heat exchangers with two-rows of tubes. The results showed that for a heat exchanger with an in-line arrangement of tubes, the Nusselt number in the first row was 34% lower than the Nusselt number in the second row for a Reynolds number of about 5000 and 45% lower for a Reynolds number of about 28000. In a heat exchanger with a staggered arrangement of tubes, a 45% lower Nusselt number over the entire range of Reynolds numbers was obtained for the first row of tubes

determination of correlations for the air-side Nusselt number.

DNS calculations require a long time to perform.

compared to the second row.

working media.

**31**

transient conditions were performed.

coefficient for the entire PFTHE.

*DOI: http://dx.doi.org/10.5772/intechopen.97402*

In finned heat exchangers with both inline and staggered tube arrangements with continuous fins, the heat transfer coefficient reaches its highest values on the first tube row (laminar flow). In subsequent rows, the HTC decreases. This is due to the large HTC values in the inlet part of channels between the fins compared to the mean coefficient on the entire fin surface.

Different methods are used to calculate heat exchangers. In the ε-NTU (Effectiveness - Number of Transfer Units) method, one effectiveness value is determined for the entire heat exchanger [1]. In the P-NTU (Effectiveness - Number of Transfer Units) method, P efficiency is determined for each medium [2]. Both the above methods and the LMTD (Log Mean Temperature Difference) method assume a constant HTC value on the gas side.

Kuppan collected in his book [3] relations and diagrams to determine the efficiencies of typical heat exchangers. The temperature of the working media in large heat exchangers built from several rows of tubes can be determined by numerical models [4] based on the finite difference method or the finite volume method.

The determination of the average air-side HTC of PFTHEs built with one, two, three, and four rows of tubes have received much attention in the literature. For finned tube heat exchangers with more than four tube rows, the air-side average HTC for the entire heat exchanger is determined as for a heat exchanger built with four tube rows. Kim et al. in [5] proposed formulas to calculate wavy PFTHEs. The formulas are valid for both in-line and staggered tube arrangements and allow the determination of the air-side Nusselt number and the friction coefficient. On the other hand, the paper [6] describes correlations based on experimental measurements and applicable only to PFTHEs with a linear tube system. The paper [7] describes a study showing that increasing the number of tube rows in a continuous plain fin and tube heat exchanger under dehumidifying leads to a decrease in heat transfer. Similar studies were presented by Halici et al. in the paper [8].

The authors studied PFTHEs with a staggered tube arrangement, made of copper tubes and aluminum fins. The results of the study allowed the determination of friction and Colburn coefficients and HTC for air. The determined friction factor and heat transfer coefficient for wet surfaces were greater than for dry surfaces and decreased with the increasing number of pipe rows. Correlations for the Colburn factor presented in [5–8] were determined for whole finned heat exchangers. No correlations for Colburn factor were determined for individual tube rows in PFTHEs. Also in the works [9–10], the correlations determined were for the entire car radiator and not for individual rows.

Rich [11] performed a study of PFTHEs with a staggered tube arrangement with a higher number of tube rows (5 and 6 tube rows). The results also showed that the HTC on the air side decreases with the increasing number of rows. The tests were conducted for air velocity of less than 3.5 m/s and Reynolds number of 12000.

Marković et al. [12] based on experimental data proposed a formula to determine the air pressure drop in a finned heat exchanger with a staggered tube arrangement. In the paper [12], the Darcy-Weisbach friction factor on the air-side depends on the Reynolds number and the ratio of the total external surface area of the finned tube to the surface area of the unfinned tube.

The works [1, 3, 13, 14] contain information describing the effect of pipe dimensions, shape and wall thickness, fin size, the longitudinal and transverse pitch of tube spacing on heat transfer in PFTHE.

In addition, in [14], it is discussed how the heat output of the heat exchanger changes depending on different methods of heat transfer intensification on the inner surfaces of the tubes.

*The CFD Based Method for Determining Heat Transfer Correlations on Individual Rows… DOI: http://dx.doi.org/10.5772/intechopen.97402*

CFD simulations were used in [15–17] to determine the uniform heat transfer coefficient for the entire PFTHE.

Sun et al. presented the results of CFD simulations and experimental studies of the optimized PFTHE in their paper [15]. The optimization of the finned heat exchanger consisted of directing the airflow to the back of the tubes using appropriately prepared channels. This procedure was intended to stop the formation of dead zones, thus improving heat transfer and reducing the pressure drop on the air-side.

Numerical and experimental studies of different types of fins are presented in the paper [16]. Wavy and plain fins with radially spaced winglets around the tubes in the plate fin tube heat exchanger were discussed. Numerical calculations were carried out in the form of CFD simulations in which laminar flow was assumed for air velocities ranging from 1.5 m/s to 3.5 m/s. For air velocities above 3.5 m/s, the turbulent flow was assumed. On the other hand, experimental studies allowed the determination of correlations for the air-side Nusselt number.

Nagaosa in paper [17] used direct numerical simulation (DNS) to determine the air-side heat transfer correlation, which was then verified experimentally. Good agreement of the obtained results was achieved. Unfortunately, computer-based DNS calculations require a long time to perform.

When turbulent flow occurs on the air-side (high Reynolds numbers), the Nusselt number in the first pipe row of the PFTHE takes on lower values than the Nusselt numbers in the subsequent pipe rows. This phenomenon occurs both in heat exchangers made of smooth tubes [1–3] or single finned tubes [18] and in heat exchangers with continuous fins [11]. Kearney and Jacobi [18] determined by experimental study the Nusselt number for each row of tubes in cross-flow finned tube heat exchangers with two-rows of tubes. The results showed that for a heat exchanger with an in-line arrangement of tubes, the Nusselt number in the first row was 34% lower than the Nusselt number in the second row for a Reynolds number of about 5000 and 45% lower for a Reynolds number of about 28000. In a heat exchanger with a staggered arrangement of tubes, a 45% lower Nusselt number over the entire range of Reynolds numbers was obtained for the first row of tubes compared to the second row.

In the available literature, few works are describing mathematical models of heat exchangers considering different heat transfer coefficients on different rows of heat exchanger tubes. The works [19–20] used mathematical models of PFTHEs with different heat transfer coefficients on the first and second row of tubes to simulate steady-state operation. In work [21], calculations of PFTHEs operating under transient conditions were performed.

The paper [22] describes an analytical method for determining the required air mass flow rate of a heat exchanger used in the propane pre-cooling cycle of an LNG plant. The method can be used to control the air mass flow rate through the fin tubes of an air-cooled heat exchanger to control the outlet temperature of the working media.

The paper [23] deals with the numerical modeling of a heat exchanger with finned tubes with large fin pitch. Computer simulations were carried out to propose the optimal perforation size and design for maximum heat transfer coefficient. Finand-tube compact heat exchangers (FTCHEs) were investigated in [24]. The results of a three-dimensional CFD simulation for an FTCHE heat exchanger constructed from oval tubes with smooth and corrugated fins are shown. Thermal–hydraulic studies were carried out for Reynolds number in the range of 200–900. The computational results show that the average Nusselt number for FTCHE with corrugated fins can be increased up to 20% compared to the case with plain fins. The average performance value for the single- and three-wave finned oval tube compact heat exchangers increased by 5% and 15%, respectively, relative to the plain tube

case. CFD modeling has also been used by the authors of a paper [25] to evaluate the thermo-hydraulic conditions of heat exchangers with finned tubes of circular, oval and flat cross-sections, with in-line or staggered tube arrangements. The study was carried out for small Reynolds numbers (400–900). For the selected performance criteria, it has been shown that the oval tube with the highest axial ratio is the optimum configuration that provides a heat transfer coefficient increase of 14% at Reynolds number Re = 400 and 5% at Reynolds number Re = 900. In turn, the effect of different combinations of circular and elliptical tubes on the air-side flow and heat transfer characteristics of the heat exchanger was investigated in [26]. Inlet air velocities were in the range of 0.5–2.5 m/s. The CFD modeling results obtained show that at low inlet velocities, the system with the elliptical tube in front of the circular tube is better than the system with the circular tube alone. At higher air inlet velocities, the system composed of elliptical and circular tubes performs better than the heat exchanger with elliptical tubes alone. Also, the heat exchanger performance was found to be better when elliptical tubes were grouped in the upper region and round tubes in the lower region compared to the alternative arrangement of elliptical and round tubes.

**2.1 Analytical model for the first row of pipes**

*DOI: http://dx.doi.org/10.5772/intechopen.97402*

*d x*<sup>þ</sup> ¼ �*N<sup>I</sup>*

temperature of the air in the first row of pipes.

� � (**Figure 2**). The average air temperature *T<sup>I</sup>*

*TI*

*<sup>a</sup> Tw*,1 *<sup>x</sup>*<sup>þ</sup> ð Þ� *<sup>T</sup><sup>I</sup>*

the first tube row are

*<sup>a</sup> x*þ, *y*<sup>þ</sup> 1 � � *∂ y*<sup>þ</sup> 1

defined as follows

where: *A<sup>I</sup>*

where: *AI*

**Figure 2.**

**33**

*AI*

*∂T<sup>I</sup>*

*x*þ, *y*<sup>þ</sup> 2

*dTw*,1 *x*<sup>þ</sup> ð Þ

<sup>¼</sup> *<sup>N</sup><sup>I</sup>*

*p*<sup>2</sup> is determined by the expression

*NI*

*bout* <sup>¼</sup> *nI*

*bin* <sup>¼</sup> *<sup>n</sup><sup>I</sup>*

pass, *Pin* inner circumference of the tube, *hI*

*Diagram of a single-pass two-row heat exchanger.*

*<sup>w</sup>* <sup>¼</sup> <sup>2</sup>*U<sup>I</sup> <sup>A</sup><sup>I</sup>*

*uPoutLc*, *nI*

where: *x*<sup>þ</sup> ¼ *x=Lc*, *y*<sup>þ</sup>

The governing energy conservation equation equations for the water and air in

*m a <sup>x</sup>*<sup>þ</sup> ð Þ � � <sup>0</sup>≤*x*<sup>þ</sup> <sup>≤</sup>1 (1)

<sup>1</sup> ¼ *y*1*=p*<sup>2</sup> dimensionless coordinates; *x*, *y*<sup>1</sup> Cartesian coordi-

*<sup>a</sup>* <sup>¼</sup> *<sup>U</sup><sup>I</sup> <sup>A</sup><sup>I</sup>*

*δw kw* þ 1 *hI oe*

*uPinLc* inner surface area of the pipes in the first row of the first

*<sup>u</sup>* total number of pipes in the first row of the first pass,

<sup>1</sup> ≤ 1 (2)

*<sup>a</sup>* are

*<sup>w</sup>* to *T*<sup>0</sup>

*<sup>w</sup>*, and

(5)

*am* (**Figure 2**).

1 � � and

*m a x*<sup>þ</sup> ð Þ over the first-row width

<sup>1</sup> (3)

*bout<sup>=</sup> <sup>m</sup>*\_ *<sup>a</sup> cp a* � � (4)

*am* to *T*<sup>00</sup>

*in* HTC on the tube inner surface in the

*<sup>w</sup>* and air-side *N<sup>I</sup>*

*<sup>a</sup>* the

*<sup>w</sup> Tw*,1 *<sup>x</sup>*<sup>þ</sup> ð Þ� *<sup>T</sup><sup>I</sup>*

*The CFD Based Method for Determining Heat Transfer Correlations on Individual Rows…*

*<sup>a</sup> x*þ, *y*<sup>þ</sup> 1 � � � � 0 ≤*x*<sup>þ</sup> ≤ 1 0≤ *y*<sup>þ</sup>

nates (**Figure 2**) *Tw*,1 the temperature of the water in the first row of pipes, *T<sup>I</sup>*

1

*TI <sup>a</sup> x*þ, *y*<sup>þ</sup> 1 � �*dy*<sup>þ</sup>

*y*þ <sup>1</sup> ¼0

*bout<sup>=</sup> <sup>m</sup>*\_ *<sup>w</sup> cp w* � � *<sup>N</sup><sup>I</sup>*

*bout* outer surface area of the bare pipes in the first row of the first pass, *Pout* outer circumference of the plain tube, *Lc* length of a tube in the PFTHE, The symbol*cpw* means the average specific heat of water in the temperature range from *T*<sup>00</sup>

Two local dimensionless coordinate systems were introduced *x*þ, *y*<sup>þ</sup>

*m a <sup>x</sup>*<sup>þ</sup> ð Þ¼ <sup>ð</sup>

The number of heat transfer units on the water-side *N<sup>I</sup>*

the average specific heat of air*cpa* in the temperature range from *T*<sup>0</sup>

*bout AI bin*

1 *hI in* þ *AI bout AI wm*

The overall HTC *U<sup>I</sup>* is calculated using the expression

1 *<sup>U</sup><sup>I</sup>* <sup>¼</sup> *<sup>A</sup><sup>I</sup>*

An analytical mathematical model of a two-pass car radiator taking into account the different air-side HTCs on the first and second row of pipes was presented in this chapter. The results of modeling car radiators made of round and oval pipes were compared with the results of experimental research.
