Arab J Sci Eng DOI 10.1007/s13369-016-2226-z
RESEARCH ARTICLE - MECHANICAL ENGINEERING
Numerical Study of Flow Over Annular Elliptical Finned Tube Heat Exchangers H. Nemati1 · S. Samivand2
Received: 14 November 2015 / Accepted: 18 May 2016 © King Fahd University of Petroleum & Minerals 2016
Abstract In the present study, performance of a tube row with annular elliptical fin was compared to the circular type. It was shown that heat transfer coefficient and pressure drop are functions of ratio of horizontal diameter to vertical diameter. It means that not only the diameter ratio, but also ellipse orientation affects heat transfer and pressure drop. Interestingly, it was found out that the pressure drop may be as low as one half of a circular fin tube. Moreover, with the same incoming air velocity, heat transfer coefficient on vertical fin is higher than circular type. Because of the lower pressure drop, higher incoming velocity may be applied, and therefore, higher heat transfer can be achieved. Altogether, tube with annular elliptical fin may be a good candidate for circular type when there is space restriction or severe limitation on pressure drop. Keywords drop
Annular fin · Elliptical fin · Fin tube · Pressure
Nomenclature A = Af + At Af At Eu H˙ Nu P
B
Total fin tube area per unit length of fin tube Fin surface area per unit length of fin tube Tube surface area per unit length of fin tube Euler number Air flow rate enthalpy Nusselt number Pressure
H. Nemati
[email protected]
1
Department of Mechanics, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran
2
Department of Mechanical Engineering, Aligoodarz Branch, Islamic Azad University, Aligoodarz, Iran
Q˙ Tin Tout Tw Uin d h k kf L1 L2 n p q r1 r2 rh rv s tf u max P η θ μ ρ
Heat exchange between fin tube and air Air inlet temperature Air outlet temperature Tube wall temperature Air inlet velocity Tube outside diameter Air side heat transfer coefficient Air thermal conductivity Fin thermal conductivity Larger fin length Smaller fin length Number of tube rows, n = 1 Tube pitch, 63.5 mm Heat exchange between fin tube and air per unit air mass flow rate Larger fin radius Smaller fin radius Horizontal fin radius Vertical fin radius Fin spacing Fin thickness Maximum air velocity in tube bundle Air pressure drop in bundle Fin efficiency Log mean temperature difference Air viscosity Air density
1 Introduction Annular fins are used widely in heat recovery systems and ventilation industries [1]. Circular type of annular fin is a common type which has been studied widely experimentally
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Arab J Sci Eng r2 rh rv s tf umax P µ
Smaller fin radius Horizontal fin radius Vertical fin radius Fin spacing Fin thickness
Maximum air velocity in tube bundle Air pressure drop in bundle Fin efficiency Log mean temperature difference Air viscosity Air density
Fig. 1 A schematic view of an annular elliptical finned tube Fig. 2 Computational domains: a horizontal orientation, b vertical orientation and c top view
and numerically [2–6]. Numerical study is relatively a new method in analysis of annular finned tube behavior. The first numerical work on annular finned tube was performed by Jang et al. [7]. They studied numerically and experimentally fluid flow and heat transfer performance in four-row annular finned tube heat exchangers, assuming laminar flow regime. Their work was followed later by Mon and Gross considering turbulent flow regime [8] and then by other researchers [4–6]. In spite of usefulness of annular circular fin, there are many situations in which it may not perform well, i.e., where the space restriction, pressure drop restriction or weight limitation exists. Under such circumstance, annular elliptical fin is recommended. Pressure drop across elliptical fin is lesser than circular fin. Moreover, since local heat transfer coefficient on the frontal side of an annular fin is higher than lateral sides, elliptical fin is more economical. A schematic view of an annular elliptical finned tube is shown in Fig. 1. Nagarani and Mayilsamy [9,10] performed some limited experiments to study the natural convection on elliptical fin with a specified geometry. Kundu and Das [11] proposed a semi-analytical method similar to sector method to approximate elliptical fin efficiency. However, this method ends to a tedious series of Bessel function families. Later, Nemati and samivand [12,13] proposed a correlation to predict elliptical fin performance based on circular fin performance. However, nothing may be found in prediction of heat transfer or pressure drop of annular elliptical finned tube. As a first attempted, in this study, behavior of air passing through one row of annular elliptical finned tube is studied numerically and results are compared with those of annular circular fin.
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Since, it is the first time that annular fin with elliptical shape has been considered, its weakness and strength in comparison with annular circular fin have been studied.
2 Numerical Simulation 2.1 Computational Domain and Boundary Conditions A schematic view of computational domain is shown in Fig. 2. Two orientations may be assumed for elliptical fin, horizontal orientation in which the ratio of radius parallel to fluid stream (rh ) to radius perpendicular to fluid stream (rv ) is more than unity, and the vertical orientation in which the ratio rh /rv is less than unity. Symmetry lines are shown by dotted lines. The upstream boundary is located at 8 times, and the downstream boundary is set as 22 times fin longest diameter from the center of the tube. 2.2 Governing Equations Momentum, continuity and energy equation must be solved numerically in this study by assuming three-dimensional, incompressible and steady flow. The Reynolds-averaged equations are as follows: ∂ (ρu i ) = 0 ∂ xi
(1)
Arab J Sci Eng Fig. 3 Generated mesh for annular elliptical fin (horizontal orientation)
∂P ∂ ui u j = − ∂x j ∂x j ∂u j ∂u i ∂ 2 ∂u l μ + + − δi j ∂x j ∂x j ∂ xi 3 ∂ xl ∂ + −u i u j (2) ∂x j ∂ ∂T ∂ (in fluid zone) E + P)) = + k (u i (ρ (k t) ∂ xi ∂ xi ∂ xi (3) ∂ ∂T ks = 0 (in solid zone) (4) ∂ xi ∂ xi
ρ
where:
− u i u j = μt
∂u j ∂u i + ∂x j ∂ xi
−
2 ∂u l ρk + μt 3 ∂ xl
(a)
Fig. 4 Elliptical fins are inscribed inside the circular fins a horizontal, b vertical Table 1 Numerical results #fin/m
Uin
N u [17]
Nu
Eu [18]
Eu
433
1.5
23.1
25.6
0.88
0.90
2
27.3
30.0
0.77
0.81
2.5
31.0
33.8
0.69
0.74
3
34.4
36.8
0.63
0.70
1.5
23.6
25.9
0.89
0.84
2
27.9
30.1
0.77
0.84
2.5
31.8
33.6
0.70
0.80
3
35.2
36.6
0.64
0.67
δi j
(5)
The transition SST model is used for turbulence simulation. The transition SST model is based on the coupling of the SST K-ω transport equations with two other transport equations, one for the intermittency and one for the transition onset criteria, in terms of momentum-thickness Reynolds number. So, it is a useful model for cases in which both laminar and turbulent flows exist. This condition can be observed in a finned tube bundle [14]. Detail of this method may be found in the literature [15].
2.3 Boundary Conditions The fluid is assumed dry air which enters normal to the bundle with inlet velocity Uin and inlet temperature Tin (300 K) and turbulent intensity I (2 %) [3]. The aluminum fin and tube has constant temperature Tw (500 K). Heat flux and velocity normal to symmetry plane are zero. Structured mesh was employed to discretize domain. Grids near the fin tip are finer than the other parts of domain. An example of generated mesh is shown in Fig. 3.
(b)
394
3 Case Study and Data Reduction For annular circular finned tube, tube diameter is selected equal to 25.4 mm and fin diameter equal to 57.15 mm with fin thickness equal 0.4 mm. Two different fin densities (433 and 394 fins/m) are also selected. These geometries are common in finned tube air-cooled heat exchangers [16]. The elliptical fins are assumed to be inscribed inside the circular fins (Fig. 4). So, the maximum diameters of all ellipses are constant and equal to 57.15 mm. The ratio rh /rv varies among 0.7 to 1/0.7 = 1.43. Accordingly, circular fin is a kind of elliptical fin with rh /rv = 1. For those aforementioned geometry, the inlet velocity Uin varies among 1.5–3 m/s which are again common in industrial air-cooled heat exchangers. Defining H˙ as flow rate enthalpy, the amount of heat trans˙ is calculated as: fer from air to the fin, Q,
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Arab J Sci Eng
Q˙ = H˙ out − H˙ in
(6)
m=
2h k f tf
(11)
Accordingly, heat transfer coefficient is defined as: Q˙ h= (A − (1 − η)Af )θ
(7)
in which A is total heat transfer area, Af is fin surface area and θ is log mean temperature difference, θ=
Tin − Tout ln(Tin − Tw ) − ln(Tout − Tw )
in which kf and tf are fin thermal conductivity and fin thickness, respectively, and d is tube outside diameter and also: L1 + L2
2 r1 r2 R¯ f = (d/2)2
L=
(12) (13)
(8)
4 Results where Tin is air inlet temperature, Tout is air outlet temperature and Tw is tube wall temperature. The fin efficiency η can be calculated iteratively by Eqs. 7–13 [12].
η=
tanh(ψm L)
ψ
ψm L
ψ = 1 + 0.179 ln( R¯ f )
Fig. 5 Stream lines on fins with a rh /rv = 1.43, b rh /rv = 1 and c rh /rv = 0.7
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(9) (10)
4.1 Data Comparison The first step is comparing calculated heat transfer and pressure drop in circular fins with experimental results. So, mass-weighted average of outlet air temperature and pressure drop were calculated. Accordingly, the following dimensionless parameters were calculated:
Arab J Sci Eng Fig. 6 Stream lines on tubes with a rh /rv = 1.43, b rh /rv = 1 and c rh /rv = 0.7
hd kf P Eu = ρu 2max
Nu =
(14) (15)
Eqs. 16 and 17 were used for comparison. N u = 0.134Re0.681 Pr 1/3 ( p/L 1 )0.2 ( p/tf )0.1134 0.134 1 + u max /n 2 , (Ref. [17])
(16)
Eu = 138.3Re−0.478 ( p/d)−1.454 u 2max n, (Ref. [18])
(17)
The results are presented in Table 1. A relatively good agreement can be observed. It is worldwide to emphasize that all experimental correlations are presented for at least four tube rows and they should be adjusted for one tube row by correc0.134 which reduce their tion factors such as 1 + u max /n 2 accuracy significantly. 4.2 Local Flow Behavior Local behavior of flow on a fin tube with three different diameter ratios (0.7, 1 and 1.43) and fin density equal to 433 #/m
and the inlet velocity equal to 3 m/s are presented here. Figure 5 shows stream lines on those three geometries. Also stream lines on tubes are presented in Fig. 6. It is clear that stream lines on both tubes and fins are affected by fin geometry. Velocity vectors are also shown in Fig. 7 for both finned space (right side) and unfinned space (left side). Separation points are indicated by small circles. Interestingly, the separation from tube wall occurs near to 90◦ which is similar to laminar flow, while the separation from fine edge occurs on obtuse angle which is similar to turbulent flow. Moreover, separation from circular fin edge occurs farther in comparison with elliptical one. Separation from horizontal ellipse occurs also farther in comparison with vertical ellipse. However, fine geometry does not affect the separation point from tube wall considerably. Fluid temperature contours passing from fins and tubes are shown in Figs. 8 and 9, respectively. Temperature distribution is more uniform in elliptical fins. And finally, variation in temperature on fins are shown in Fig. 10. As it is clear, the cooled areas are much less in elliptical fins, which promises more average heat transfer per unit area.
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Arab J Sci Eng
Fig. 7 Velocity vectors on tube (right side) and on fins (left side) with a rh /rv = 1.43, b rh /rv = 1 and c rh /rv = 0.7
4.3 Global Flow Behavior Heat transfer coefficient and pressure drop are results of global flow behavior. Total heat transfer of each fin is normalized by dividing it by circular fin heat transfer. The results are shown in Fin 11. As it is clear, the total amount of heat transfer from circular fin is more than the others. However, it is not a good base for judgment, since the area of circular fin is also more. Interestingly, elliptical fins with the same area have different heat transfer. It shows that fin orientation is also a matter of consideration (Fig. 11). For this reason, normalized heat transfer from unit area (q/ ˙ q˙circular ) is shown in Fig. 12. Interestingly, heat transfer intensity from vertical ellipse is 30 % higher than circular fin, while the horizontal ellipse is lower only by 20 %.
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In this regard, normalized Nusselt numbers are shown in Fig. 13. It is evident that this ratio is not a strong function of air velocity. Nusselt number on vertical ellipse is 30 % higher than circular one. The restricting parameter in fin tube bundles is pressure drop. Since the inlet velocity is limited only by allowable pressure drop, the normalized pressure drops are also shown in Fig. 14. Pressure drop across tube banks with vertical elliptical fins are less than circular fins. So, vertical elliptical fins are superior in both heat transfer and pressure drop. Moreover, although Nusselt number is lower than circular fin by about 30 %, pressure drop is about one half of that fin. So, heat transfer can be accommodated by increasing inlet velocity. Finally, to have a balance between heat transfer and pressure
Arab J Sci Eng Fig. 8 Fluid temperature contours passing fin with a rh /rv = 1.43, b rh /rv = 1 and c rh /rv = 0.7
drop, normalized generated entropy is shown in Fig. 15. The generated entropy is calculated as: s −
q = sgen Tw
in which s is defined as: P Tout s = Cp ln − R ln 1 − Tin Pin
(18)
(19)
and Cp = 1006.43 J/KgK and R = 287.16 J/KgK. As it is expected, the most generated entropy belongs to circular tube.
5 Conclusions In this paper, the annular elliptical fin tube is studied deeply. It is found that the vertical annular elliptical fin shows superior performance in comparison with circular type. The pressure drop is lower, while the Nusselt number is higher. Moreover, it was shown that although the Nusselt number is lower in a horizontal annular elliptical fin tube, pressure drop can be as low as one half of a circular type. It was shown also that the Nusselt number ratio as well as pressure drop ratio is independent of inlet velocity or fin density. All in all, an annular elliptical fin tube can be a good candidate whenever there is space or pressure drop restriction.
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Arab J Sci Eng Fig. 9 Fluid temperature contours passing tube a rh /rv = 1.43, b rh /rv = 1 and c rh /rv = 0.7
Fig. 10 Variation in temperature on fins a rh /rv = 1.43, b rh /rv = 1 and c rh /rv = 0.7
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Arab J Sci Eng 1.1
1.1
v=1.5 m/s
1
v=2 m/s
0.9
v=3 m/s
0.8
v=2 m/s
0.9
v=2.5 m/s
P/ Pcircular
Q /Q circular
v=1.5 m/s
0.7 0.6
v=2.5 m/s v=3 m/s
0.7
0.5
0.5 0.4
0.5
0.7
0.9
1.1
1.3
0.3 0.5
1.5
0.7
0.9
1.1
1.3
Diameter ratio
Fig. 14 Normalized pressure drop
Fig. 11 Normalized total heat transfer
1.1
v=1.5 m/s
1.4
v=2.5 m/s
sgen /sgen(Circular)
v=2 m/s
1.2
v=2.5 m/s v=3 m/s
1
v=2 m/s
1
v=1.5 m/s
q /qcircular
1.5
Diameter rao
0.9
v=3 m/s
0.8 0.7 0.6
0.8
0.5
0.6 0.5
0.5
0.7
0.9
1.1
1.3
0.9
1.1
1.3
1.5
Diameter ratio
1.5
Diameter rao
0.7
Fig. 15 Normalized entropy generation
Fig. 12 Normalized total heat transfer from unit area
References 1.3
v=1.5 m/s v=2 m/s
1.1
v=2.5 m/s
Nu
v=3 m/s 0.9
0.7
0.5
0.5
0.7
0.9
1.1
Diameter rao
Fig. 13 Normalized Nusselt number
1.3
1.5
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