Determination of fin pitches for maximum performance index of L-footed spiral fin-and-tube heat exch

Abstract

Keywords: Fin; tube; heat transfer; heat exchanger; optimum; air-water

Introduction

The air-side performances of conventional spiral finned tube heat exchangers have been studied by a number of researchers. The most productive studies are listed below. Genic et al. [1] investigated air-side pressure drops in conventional spiral finned tube heat exchangers with inline and staggered tube arrangements. Hamakawa et al. [2] investigated the vortexshedding characteristics of conventional spiral finned tubes by using a smoke-wire technique. Lee et al. [3, 4] investigated the air-side heat transfer characteristics of conventional spiral finned tube heat exchangers under frosting and non-frosting conditions. They investigated these characteristics by varying the number of tube rows, the fin pitches, and the fin alignments. Empirical correlations for predicting the heat transfer performance at low Reynolds number was proposed. The L-footed spiral fin utilizes a specialized type of fin geometry. The base of the fin has an L-shape, which can provide a large contact area and ensure a good path for heat transfer from the tube surface to the fin. It can also prevent the corrosion of the tube from the long-term operation. Despite its importance in

Data Reduction

An experimental apparatus and test section are shown in Figs. 1-2. The system consists of two loops: an air loop and a water loop. Ambient air and hot water were used as working fluids. The experimental system is composed of air supply loop (open wind tunnel), hot water cycling loop, test section (L-footed spiral fin-and-tube heat exchanger), instrumentation and data acquisition systems. All the tests were performed at steady state condition, temperatures of working fluid at inlet and outlet, as well as the pressure drop of the air flowing across the test section, 251

were measured. The geometric details of the spiral fin-and-tube heat exchangers are shown in Table 1. The air-side heat transfer rate is given by:

which ∆Ta = Ta ,in − Ta ,out . The water-side heat transfer rate is given as:

Figure 1 Schematic Diagram Of The Experimental Apparatus

FIGURE 2 PHOTOS OF THE TESTED L-FOOTED SPIRAL FIN AND TUBE HEAT EXCHANGERS AND SCHEMATIC DIAGRAM OF THE L-FOOTED SPIRAL FIN.

Notes: Staggered layout are used. Remarks: dc = Fin collar outside diameter, df = Outside diameter of fin, di = Tube inside diameter, do = Fin outside diameter, fp = Fin pitch, ft = Fin thickness, nt = Number of tubes, Nrow = Number of tube rows, PL = Longitudinal tube pitch, PT = Transverse tube pitch. The total thermal resistance (or 1/UA ) is the summation of each of resistances (conduction and convection resistances), as follow:

ln(do di ) ln(dc do ) 1 1 1 = + + + . UA hi Ai 2πkt L 2πk f L ηo ho Ao

∗  K  K  2 K / C∗A  + 1 − e , K = 1 − e − NTU A (CA / 2) (6)  2 2  

The details of the water-flow circuit, and a geometric of the L-footed spiral finned tube heat exchanger are illustrated in Fig. 3.

 k  (Redi − 1000)Pr ( f i 2) , hi =  w  2  d i  1 + 12.7 f i 2  Pr 3 − 1  

where C* = Cmin / Cmax is equal to Cc / Ch or Ch / Cc, depending on the values of the hot and cold fluid heat capacity rates.

The relationship of overall surface effectiveness ( ηo ), fin efficiency (η f ), fin surface area ( A f ), and total heat transfer area ( Ao ) can be illustrated as follows:

FIGURE 3 GEOMETRIC DETAILS AND SCHEMATIC DIAGRAM OF THE HEAT EXCHANGERS ALGORITHM FOR MULTIPASS PARALLEL CROSS FLOW, MULTIPASS COUNTER CROSS FLOW AND MULTIPASS PARALLEL-AND-COUNTER CROSS FLOW ( × AND • SIGNS INDICATE THAT WATER FLOWS INTO OR OUT OF THE PAPER, RESPECTIVELY).

The unfinned base surface (Ab) and fin surface area (Af) can be expressed in form of Eqs. (17) and (18), respectively:

 L A f = nt N row   0.5πd 2f − 0.5πd o2 + πd f f t , f   p

and where Ap is the area in the rectangular profile of the fin:

The fin surface area (Af) based on assumption of the circular

I 1 (φR o ) K 1 (φR i ) − I 1 (φR i ) K 1 (φR o ) 2ψ , (19) φ (1 + ψ ) I 0 (φR i ) K 1 (φR o ) + I 1 (φR o ) K 0 (φR i )

The parameters Ro and Ri are given in terms of the radius ratio (ψ ):

Gardner [18] proposed the fin efficiency for a circular fin (Eq. (19)), it will be applied for our calculation.

system performance index (ζ2) at the operating point, which is the point of intersection between the axial fan performance curve and the system curve of the heat exchanger. Finally, the dimensionless system performance index (ζ3) is a modification of the system performance index obtained by dividing the average heat transfer rate by the fan power (WF) at the operating point. In other words, the dimensionless system performance index (ζ3) could be described as the coefficient of performance, since it represents the ratio of the desired heat transfer rate to the provided fan power (Qave / WF)Sys. The fan power is determined by the fanning friction factor (Eq. (30)), as proposed by Kays and London [19].

 A  ρ  2 ΔPρ1  2  ρ1   , f =  min  m  − 1 + σ − 1 2 ρ  2   Ao  ρ1  Gc

area, Ao is the total heat transfer area, and Amin is the minimum free flow area. For Eq. (30), if the entrance and exit effects are negligible (

ρ1 = ρ2 and ρm = [ ρ1 + ρ2 ] / 2 ), then the friction factor can be determined from

Table 2 illustrates the accuracy of the measurements. The root mean sum square method is used to determine the uncertainties, as shown in Table 3. Moreover, this work presents the experimental results according to a primary analysis based on the energy balance. Fig. 4 shows that the energy balances between the air and water based on the ANSI/ASHRAE 33

where the subscript “Sys” means “system.” The dimensionless system performance index is calculated using: ζ3 = (Qave / WF)Sys ,

where the subscript “HX” means “heat exchanger.” The system performance index is calculated using: ζ2 = (Qave /∆P)Sys,

where σ is the ratio of the minimum free flow area to the frontal

where the air-side heat transfer coefficient (ho) is obtained from Eq. (4). The primary purpose of this study is to determine an optimized fin pitch using the performance index. The performance index is defined as the ratio of the desired output to the required input. However, three performance indices are used for analysis the optimum fin pitch: the heat exchanger performance index (ζ1), the system performance index (ζ2), and the dimensionless system performance index (ζ3), each of which is described below. The heat exchanger performance index is calculated using: ζ1 = (Qave /∆P)HX,

Standards [20], indicating that Qa − Qw ×100/ Qave was less than 5%. All of the tested conditions are shown in Table 4.

Table 2 Accuracy Of The Measurements

The actual performance of the heat exchanger relies on the cooperation between heat exchanger and fan. Hence, the heat exchanger performance index (ζ1) quantitatively evaluates the conventional performance of heat exchangers using the ratio of the average heat transfer rate on the air side to the pressure drop across the heat exchanger. In other words, this study also investigated heat transfer performance in relationship to the influence of commercial fans on the operation of heat exchanger systems. The evaluation criteria were proposed in terms of the

Inlet-air dry-bulb temperature, C Pressure drop, Pa Inlet-water temperature, oC Water flow rate, LPM

drop of heat exchanger, respectively. Therefore, the heat exchanger performance index (ζ1) describes the performance of the heat exchanger. In Fig. 5(a), the effects of fin pitches on the heat exchanger performance index (ζ1) are investigated under different inlet water temperatures and water flow rates. For a Vfr of 3–4 m/s, the results showed that fin pitch significantly influenced the heat exchanger performance index (ζ1). As the fin pitch increased (goes from left to right) in the interval from 2.4 to 3.2 mm, the value of ζ1 increased and the curve rose. On the other hand, as the fin pitch increased at the interval from 3.2 to

Table 3 Uncertainties Of The Derived Experimental Values

Air-side heat transfer rate, Qa Water-side heat transfer rate, Qw Pressure drop, ∆P Frontal velocity, Vfr Reynolds number, Redc Colburn factor, j Friction factor, f

Table 4 Experimental Conditions

Inlet-air-dry bulb temperature, oC 31.5±0.5 Inlet-air frontal velocity, m/s 2-8 or Redc (5,000-15,000) Inlet-water temperature, oC 55-70 Water flow rate, LPM 12-14 9

Results And Discussion

The major finding of this study is that fin pitch had a significant effect on the maximum performance of the L-footed spiral fin-and-tube heat exchangers. The experimental results were related to the three performance indexes ζ1, ζ2, and ζ3 which were the significant methodologies developed to optimize fin pitch. The average heat transfer rate (Qave) and the pressure drop across the heat exchanger (∆P) for all of the tested samples were determined from the experimental data. The results were presented in bar charts showing the three performance indices (ζ1, ζ2, and ζ3) obtained from three types of fans for three fin pitches. The optimization of fin pitch was analyzed using these three performance indices, each of which optimized the fin pitch of the L-footed spiral fin-and-tube heat exchangers. The numerator and denominator represent the air-side heat transfer rate and pressure 256

Additionally, the ζ3 can be defined as “Coefficient of performance” because it represents the ratio of desired output (Qave) to required input (WF) of the system. Fig. 8 shows the variation in the fin pitch of an axial commercial fan curve (A to C) on the performance index (ζ3) at the operating point. It can be

clearly seen from the figure, the ζ3 of a lower-input power fan is always higher than the ζ3 οf higher-input power fans. The result shows the same trend for the ζ3 in different types of fan curves.

FIGURE 5 EFFECT OF FIN PITCH ON THE HEAT EXCHANGER PERFORMANCE INDEX ζ1 HAVING DIFFERENT INLET-WATER TEMPERATURES (a) 55 oC AND (b) 70 oC.

FIGURE 6 EFFECT OF FIN PITCH ON THE HEAT EXCHANGER PERFORMANCE INDEX ζ1 HAVING DIFFERENT WATER FLOW RATES (a) 12 LPM AND (b) 14 LPM.

In addition, when the fp decreases from 4.2 mm to 2.4 mm, the ζ3 increases by about 25% (for fp = 3.2 mm) and 35% (for fp= 2.4 mm), respectively, compared with the ζ3 of the fin pitch of 4.2 mm over the range of experimental conditions. Again, considering the performance indices (ζ1 and ζ2), we found that fin pitches of 2.4 and 3.2 mm seemed to have higher values in the performance indexes than that of 4.2 mm. This result indicates the optimized fin pitch for heat exchangers. However, the dimensionless system performance indices (ζ3) for fin pitches of 2.4 and 3.2 mm seem to represent the highest and the intermediate performance of heat exchangers, respectively. As mentioned, these results show the effect of fin pitch on three performance indices, which were analyzed based on the ratio of the desired output to the required input. The heat exchanger performance index (ζ1), the system performance index (ζ2), and the dimensionless system performance index (ζ3) were used to discover the optimum fin pitch in this study. Therefore, as shown in Table 5, we noted the performance indices (ζ1, ζ2, and ζ3) and compared them for fin pitches of 2.4, 3.2, and 4.2 mm to investigate the optimum fin pitch of L-footed spiral fin-and-tube heat exchangers. The fin pitch of 4.2 mm was clearly inferior to the other fin pitches. The three performance indices were analyzed with the intersection-of-sets method for presenting the optimum fin pitch. It is an effective tool for comparing the three performance indices based on different fin pitches. In the same manner, a fin pitch of 4.2 mm is not suitable for the design of heat exchangers. However, fin pitches of 2.4 and 3.2 mm seem to be the optimum fin pitches. Using the VG-1 criteria, a comparison is done between the total heat transfer area of the L-footed spiral fin and the reference fin (i.e., the plain fin) by keeping the heat transfer rate, temperature difference of the fluids, and fan power constant. Considering those constant parameters in the case of VG-1, the following ratio can be expressed

FIGURE 7 EFFECT OF FIN PITCH SUBJECTED TO FAN CURVE ON THE PERFORMANCE INDEX ζ2 AT OPTIMUM FAN OPERATING POINT (Tw,in = 55 oC AND WATER FLOW RATE OF 14 LPM). 100

FIGURE 8 EFFECT OF FIN PITCH SUBJECTED TO FAN POWER ON THE PERFORMANCE INDEX ζ3 AT OPTIMUM FAN OPERATING POINT (Tw,in = 55 oC AND WATER FLOW RATE OF 14 LPM).

TABLE 5 CHARACTERISTICS OF FIN PITCH AND NUMBER OF TUBE ROW ON PERFORMANCES INDEXES

TABLE 6 AIR-SIDE PERFORMANCE CORRELATIONS OF THE PLAIN, CRIMPED

And L-Footed Spiral Finned TUBE HEAT Exchangers

 jN −0.092  N row  = 0.991 2.24 Redo   j4  4   −0.328  PT    j4 = 0.14 Redo  PL 

PT = 20.35 − 50.73 mm PL = 12.7 − 44.09 mm Redo = 800 − 7500

f - friction factor correlation −0.418  f t    f = 1.039 Redo  do 

d o = 10.51 mm f p = 1.77 − 3.21 mm N row = 2 − 6 PT = 25.4 mm PL = 22 mm

−0.4059 j = 0.2150 Redo −0.2156  f p    f = 0.4852 Redo  do 

Notes: Correlations are based on staggered layout. ratios for the Colburn factor and the friction factor; it also measures the possible reduction of the heat transfer area.

where AL-footed and Aplain are areas of the L-footed spiral fin and plain fin geometries, respectively. The area ratio is related to the

mm to 4.2 mm. At the same Reynolds number, the L-footed fins clearly require about 6–13% less heat transfer area than the plain fin for fp = 2.4 mm, the same heat transfer area as the plain fin for fp = 3.2 mm, and 9–16% more heat transfer area than the plain fin for fp = 4.2 mm to yield the same air-side heat transfer performance. Thus, smaller fin pitch presents better performance and requires less area.

L-Footed Spiral FIN With Relation To The Plain

FIGURE 11 RATIO OF AIR-SIDE HEAT TRANSFER AREA OF CRIMPED SPIRAL FIN WITH RELATION TO THE LFOOTED SPIRAL FIN.

In a comparison between the crimped spiral fin and plain fin configurations, the researchers found that the crimped spiral fin needs less heat transfer area than that of the plain fin: about 39– 48% for fp = 2.4 mm, 32–42% for fp=3.2 mm, and 25–36% for fp = 4.2 mm, as shown in Fig. 10. As demonstrated in Fig. 11, the test results also indicate that the effect of fin pitch on the crimped-to-L-footed spiral fins’ heat transfer area ratio is very small. As expected, the crimped fin requires about 50–70% less heat transfer area than that of the L-footed spiral fin for all fin pitches (fp = 2.4, 3.2, and 4.2 mm), as analyzed according to the VG-1 criteria and reported by Webb [24]. This may be because the crimped spiral fin mixes the turbulent flow across the spaces between fins more effectively than the plain fin and L-footed spiral fin. However, due to the specific configuration of the base of the crimped spiral finned tube, it appears that the crimped spiral fins drop pressure more than plain and L-footed spiral fins do at the same Reynolds number.

Crimped Spiral FIN With Relation To The Plain

In addition, the j-Colburn and f-friction factor correlations for the L-footed spiral fin, crimped spiral fin, and plain fin-andtube heat exchangers, which were proposed by Pongsoi et al. [6], Pongsoi et al. [21], Wang and Chang [22], and Wang et al. [23], respectively, are shown in Table 6. These correlations were selected for comparison with the tested results. In Fig. 9, the VG1 criterion covers the research findings on the comprehensive comparison between the L-footed spiral fin and plain fin for three different fin pitches varying from 2.4 mm to 4.2 mm. The effects of the fin pitches and Reynolds numbers on the variations of the area ratios, as compared with the plain fin (reference fin), the Lfooted-to-plain fins heat transfer area ratio increased as the Reynolds number increased. Moreover, the L-footed-to-plain fins heat transfer area ratio increased when fp increased from 2.4

Conclusion

In this study, optimized fin pitches for L-footed spiral finand-tube heat exchangers were investigated experimentally at 2.4, 3.2, and 4.2 mm (i.e., 10, 8, and 6 fpi), respectively. The heat exchanger design usually involves three performance indices (ζ1, ζ2, and ζ3). The findings are as follows: - Variations in fin pitch had a significant effect on the three performance indices. - For the heat exchanger performance index (ζ1), we found that ζ1 reaches its optimum level at a fin pitch of 3.2 mm. 260

However, the effect of fin pitch on ζ1 can be neglected in a frontal velocity range of 5–6 m/s. - The system performance index ζ2 at fin pitches of 2.4 and

Nomenclature

area, m2 A minimum free flow area, m2 Amin total surface area, m2 Ao cross-sectional or profile area of fin, m2 Ap cp specific heat at constant pressure, J/(kg.K) C heat capacity rate, W/K C* capacity rate ratio, dimensionless Cc cold-fuid capacity rate, W/K Ch hot-fluid capacity rate, W/K COP coefficient of performance, dimensionless Cu copper df outside diameter of fin, m di tube inside diameter, m do tube outside diameter, m f Fanning friction factor, dimensionless fp fin pitch, m ft fin thickness, m fpi fin per inch Gc mass flux of the air based on minimum free flow area, kg/m2.s H height, m h heat transfer coefficient, W/(m2.K) I0 modified Bessel function solution of the first kind, order 0 I1 modified Bessel function solution of the first kind, order 1 k thermal conductivity, W/(m.K) K0 modified Bessel function solution of the second kind, order 0 K1 modified Bessel function solution of the second kind, order 1 L length, m mass flow rate, kg/s m number of tubes in row nt Nrow number of tube rows NTU number of transfer units, dimensionless PL longitudinal tube pitch, m Pt transverse tube pitch, m Pr Prandtl number, dimensionless heat transfer rate, W Q

radius function in terms of the radius ratio, dimensionless Reynolds number based on tube inside diameter (di ) , dimensionless Reynolds number based on fin collar outside diameter (dc) , dimensionless temperature, oC water temperature, oC overall heat transfer coefficient, W/(m2.K) velocity based on tube inside diameter (di ), m/s air frontal velocity, m/s power, W

Greek symbols heat exchanger effectiveness, dimensionless ε η fin efficiency, dimensionless

overall surface effectiveness, dimensionless density, kg/m3 contraction ratio of cross-sectional area, dimensionless dynamic viscosity of air, Pa.s combination of terms, dimensionless

radius ratio, dimensionless pressure drop, Pa ∆P ζ1 heat exchanger performance index, W/Pa system performance index, W/Pa ζ2 ζ3 dimensionless system performance index dimensionless

Subscripts 1 air-side inlet 2 air-side outlet a air ave average b unfinned base surface c multipass counter cross flow or cold fluid crimped crimped spiral fin f fin F fan fr frontal (L × H) h hot fluid i tube-side in inlet L-footed L-footed spiral fin m mean value max maximum min minimum o air-side p multipass parallel cross flow plain plain fin pc multipass parallel-and-counter cross flow t tube w water

Acknowledgments

The authors are indebted to the Thailand Research Fund, the National Science and Technology Development Agency and the National Research University Project for the support.

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