Effects of tube shapes on the performance of recuperative and regenerative heat exchangers

Energy 169 (2019) 1e17

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Energy journal homepage: www.elsevier.com/locate/energy

Effects of tube shapes on the performance of recuperative and regenerative heat exchangers Ngoctan Tran*, Chi-Chuan Wang** Department of Mechanical Engineering, National ChiaoTung University, 1001 University Road, Hsinchu 300, Taiwan

a r t i c l e i n f o

a b s t r a c t

Article history: Received 14 July 2018 Received in revised form 31 October 2018 Accepted 27 November 2018 Available online 7 December 2018

In the present work, the effects of tube shapes on the heat transfer and fluid flow characteristics of the shell-and-tube heat exchangers eincluding square, rectangular, circular, elliptical, equilateral- and isosceles-triangular shapes eare numerically investigated in detail. Six parameters eincluding (1) rectangular-tube ratios (⍺ ¼ tube height/width), (2) triangular-tube ratios (b ¼ two-side edge/bottom edge), (3) elliptical-tube ratios (ɣ ¼ b/a; a: inner semi-major-axis lengths and b: inner semi-minor-axis lengths), (4) the tube lengths from 2 mto 20 m, (5) outer tube diameters from 52 mm to 252 mm, and (6) 500  Re  8000 eare examined variables. For all cases in this study, it is found that at the same boundary conditions, the thermal performance of an elliptical tube is superior to that of a circular tube. A locally thermal optimal triangular tube shape is defined with a maximum effectiveness up to 92.46%. A novel correlation is developed to describe the Nusseltnumbers subject to the examined tube shapes. The proposed correlation is not only in line with the existing round tube correlations but also gives good predictions for other tube geometries with maximum deviation being less than 7.1%. © 2018 Elsevier Ltd. All rights reserved.

Keywords: Shell-and-tube heat exchanger Tube shapes Tube lengths Tube perimeters Regenerators Recuperators

1. Introduction The rapid industrializations of global community significantly contributed to the climate change because the utilization of fossil fuels has incurred a huge amount of greenhouse gases to the natural environment. Muller et al. [1]reported that in the years from 1950 to 2000, the average earth-surface temperature had risen by around 0.9  C. The global warming negatively impacted the lives on the earth, in general, and those of the humanity, in particular. One of the solutions for reducing the energy consumption and decreasing the greenhouse-gases release from industrial zones is that: the polluted and high-temperature exhausts from the factories should be carefully managed for effective waste heat recovery before discharging into the natural environment. Indeed, a survey by Hatamiet al. [2]reported that only 12e25% of the fuel energy is converted to the useful work, and up to 30e40% of the fuel energy is wasted. Therefore, the more the waste heat from the exhausts can be recovered, the fewer the amount of the fossil fuels will be utilized and the less the greenhouse gases will be released.

* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (N. Tran), [email protected]. tw (C.-C. Wang). https://doi.org/10.1016/j.energy.2018.11.127 0360-5442/© 2018 Elsevier Ltd. All rights reserved.

Shell-and-tube heat exchangers (STHX) are one of the cost-effective equipment for this solution because the gases of the hot and cold sides can be separated without leaking in the STHX, and they can sustain high pressure/temperature operation. Therefore, investigations relative to improve the heat transfer and fluid flow characteristics of the STHXshad attracted many researchers. Recently, a great number of studies relative to the STHXshad been reported in the published literature. Various computational methods for designing and optimizing the thermal performance of the STHXswere reported in Refs. [3e9], respectively.In which, a multi-objective optimization was carried out by Fettakaet al. [3]. The main objectives of their optimization are to minimize the heat transfer area and pumping power aiming at reducing the manufacturing and operating costs. Calculating methods of operating parameters for designing a new shell-and-tube heat exchanger were reported by Serthand Lestina[4]. Some well-known correlations in the published literature for predicting heat transfer coefficients and pressure drop for both tube and shell sides were adopted in their study. A comparison of heat transfer performance between rectangular and trapezoidal fin shapes was carried out by Kundu[5]. He concluded that at a defined working condition of unbaffledshell-and-tube heat exchangers, the heat transfer rate of the rectangular fin shape was superior to that of the trapezoidal fin shape. Calculations of pressure drop in a shell-and-tube heat

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Nomenclature A Cp Dh Hm Hsh k Lm Lsh Lt Pe Q Q Re T DTlm Wm Wsh

heat transfer area of the tube, (m2) specific heat at constant pressure, (J/kg,K) tube hydraulic diameter, (m) model height, (m) shell height, (m) distance between outer wall of the tube and outer wall of the model, (m) model length, (m) shell length, (m) tubelength, (m) perimeter, (m) heat transfer rate, (W) heat flux, (W/m2) Reynolds number temperature, (ºC) log mean temperature difference, (ºC) model width, (m) shell width, (m)

Greek symbols ⍺ Rectangular ratio between tube heights and tube widths b Triangular ratio between two-side edge lengths and bottom edge lengths

exchangers were simplified by Parikshitet al. [6]. They employed the finite element method in their predicted model, and they revealed that the finite element method had highly effectively in predicting the pressure drop because it required less time for calculation and achieved high accurate predicted results. An interior spray technique for a triangular-pitch shell-and-tube spray evaporator was proposed by Chang and Yu [7], and they revealed that the heat transfer coefficient on the shell side using the interior spray technique was superior to that using conventional flooded type. Geometrical parameters of shell-and-helically-coiled-tube heat exchanger were optimized by Alimoradiand Veysi[8]. An optimal model, which achieved a minimum heat transfer rate per entropy generation, was reported in their study. Total cost and pressure drop were added as optimal objectives by Raja et al. [9]. Their study was carried out for aiming at maximizing the effectiveness and minimizing total cost, pressure drop, and entropy generation. The multi-objective optimization of Raja's group is more useful in a real design of a new heat exchanger; however, it requires much effortsas compared to that presented in Ref. [8]. The effects of baffles on the heat transfer and fluid flow characteristics of the STHXswere reported in Refs. [5,10e16]. In which, influences of the baffle-helix angle on shell-side thermodynamic performance of shell-and-tube heat exchangers were experimentally investigated by Gaoet al. [10]. They revealed that both pressure drop and heat transfer coefficient of the shell side were increased with the decrease in the helix angle. Various studies on helical baffles of shell-and-tube heat exchangers were summarized by Salahuddinet al. [11]. In their review, they pointed out advantages and disadvantages of the helical and segmental baffles, and they concluded that in a shell-and-tube heat exchanger, the helical baffles were more advantageous than segmental baffles. Single shell-pass and two-layer continuous helical baffles were firstly combined in a shell-and-tube heat exchanger and numerically examined by Yang et al. [12]. They revealed that the heat transfer performance of the combined heat exchanger was superior to that

ɣ

l m r

Elliptical ratio between inner semi-major-axis lengths and inner semi-minor-axis lengths thermal conductivity, (W/m,K) dynamic viscosity of the coolant, (kg/m,s) density of the coolant, (kg/m3)

Subscripts actual c ci co f h hi ho max min

actualheat transfer rate cold side cold-side inlet cold-side outlet fluid hot side hot-side inlet hot-side outlet maximum minimum

Acronyms CFD computational fluid dynamics CFD-ACEþ a commercial computational fluid dynamics solver developed by ESIgroup HTC heat transfer coefficient (W/m2,K) STHX shell and tube heat exchanger

of the original segmental baffles or continuous helical baffles. Trefoil holebaffles and helical baffles were numerically studied in shell-and-tube heat exchanger by Zhou et al. [13]and Wen et al. [14], respectively. The Fluent software package was adopted for their simulations. In their studies, the simulated results were verified by experimental results, and good agreements between the simulated and experimental results were achieved with the maximum percentage errors to be less than 20.3%, and ±3% for comparisons by Zhou and Wen, respectively; this indicates that the heat transfer and fluid flow characteristics in shell-and-tube heat exchangers could be highly accurately predicted by a CFDsoftware package. Other numerical studies on geometrical parameters of the STHXswere also reported in Refs. [8,17]. A two-dimensional finite element computational model was adopted by Khan et al. [17]for examining the effects of fin length, fin thickness and materials for a shell-and-tube heat exchanger. The heat exchanger of their study aimed at applying in latent heat storage system. They revealed that the fin length significantly affected the melting rate when compared to that of the fin thickness, and the melting rates of the system with copper and aluminiummaterials were superior to those of AISI4340, cast iron, tin and nickel due to the higher thermal conductivities of the copper and aluminium. Cavazzutiet al. [18] confirmed that the heat exchangers play a very important role in the heat recovery of high temperature industrial systems, and they carried out an optimization of a finned concentric pipe heat exchanger for applying in industrial recuperative burners. The optimal design could enhance the heat transfer capacity of the recuperative heat exchanger by 10.8% and 6.3% for thermal powers of 12.5 kW and 25 kW, respectively. Lygnerudaand Wernera[19] estimated that the total energy saving can be up to 300  109 kWh/ year if the Euro, Iceland, Norway and Switzerland reuse the residual heat in their district heating networks. Salpingidouet al. [20] revealed that the keys of design for recuperative heat exchangers applying in gas turbines and aero engines are that enhancing the heat transfer and minimizing the pressure drop. They reported that

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materials of the heat exchangers played a very important role in optimizing the waste heat exploitation in aero engines. A recuperative heat exchanger was added into an organic ranking cycle to recover the waste heat from vehicles by Zhao et al. [21]. They revealed that the added recuperatorworked more effectively under heavy-duty engine condition. Akbariet al. [22]carried out experimental study on rotary heat exchanger for recovering the waste heat from the exhaust gas of the dryer. They revealed that the 42.2% efficiency improvement when compared to the original. Existed regenerative heat exchanger methods for various cooling technologies were reviewed by Qianet al. [23]. They reported that regenerative heat exchangers with higher heat transfer area showed superior regeneration effectiveness. Based on the reviewed literature, it is found that a great number of prior researches on heat exchangers had focused on studies of fin-and-tube heat exchangers, but studies on smooth tube heat exchangers were comparatively rare. In heat transfer processes of two different working fluids efor example: the heat is transferred from a hightemperature air to a cold refrigerant (evaporators), or from a high-temperature refrigerant to a cold air (condensers), or similar heat-transfer processes of other working fluids ethe fin-and-tube heat exchangers could significantly enhance the thermal performance of the heat exchangers. The heat exchangers accommodate a lot of surface area on the airsideto augment the heat transfer rate. In recent demands of the heat recovery, the heat is transferred from hot air (e.g. a dusty air such as industrial flue gases or other hightemperature exhaust gases) to cold air (e.g. a clean air, in some cases, the air even treated well before receiving the heat such as the applications in food-and-beverage manufacturing factories or chemical factories). In these applications, the heat exchangers must contain characteristics like leakage free amid hot and the cold airs, easiness to clean within both hot and cold sides, low pressure drop, and high effectiveness. The recent demands of the regenerators indicate that the fin-and-tube heat exchangers are not suitable, and the shell-and-smooth-tube heat exchangers may be more appropriate as far as a reliable operation is concerned. However, the studies of shell-and-tube heat exchangers for such applications were rather rare; and more specially, no studies on the effects of tube shapes on heat transfer and fluid flow characteristics for such applications have been published yet. The existing studies of shelland-tube heat exchangers had been applied for high-pressure applications; therefore, the tubes and shells, which have been applied in the shell-and-tube heat exchangers, are almost in a circular shape; however, in the recent demands of the energy recovering applications, the supplied fan power may be limited; therefore, utilization of the tubes or shells having different shapes may be more beneficial as far as energy saving is concerned. This has motivated the present study to carry out an in-depth investigation regarding the influence of tube shapes applicable for shell-and tube heat exchangers and its effects on heat transfer and fluid flow are thoroughly reported. 2. Methodology 2.1. Work description In the present study, the effects of tube shapes, tube lengths, and tube outer perimeter on heat transfer and fluid flow characteristics of shell-and-tube heat exchangers (STHXs) are individually examined in detail. Six tube shapes eincluding circular, elliptical, square, rectangular, isosceles-triangular, and equilateral-triangular ewere considered as examining tube shapes. The tube shapes are listed in Table 1. As shown in the table, when aequals to b, the rectangular, isosceles-triangular, and elliptical shapes will become square, equilateral-triangular and circular shapes, respectively; therefore,

3

the square, equilateral-triangular and circular shapes are not listed in the table. Rectangular tube shapes, with different tube ratios, are examined in detail in section 3.1. The rectangular tube ratio, a, is calculated as follows:

a¼

Ht b ¼ Wt a

(1)

where Ht and Wt stand for tube height and tube width, respectively. Isosceles-triangular tube shapes, with various tube ratios, are investigated in detail in section 3.2. The isosceles-triangular tube ratio, b, is calculated as follows:

b¼

Ltwoside b ¼ a Lbottom

(2)

where Ltwoside and Lbottom stand for the length of two-side edge and the length of bottom edge, respectively. Elliptical tube shapes, with diverse tube ratios, are studied in detail in section 3.3. The elliptical tube ratio, g, is calculated as follows:

g¼

Rminor b ¼ Rmajor a

(3)

where Rmajor and Rminor stand for the length of the semi-major axis and the length of semi-minor axis, respectively. After examinations of sections 3.1, 3.2 and 3.3, three locally optimal tube shapes would be defined, and the locally optimal tube shapes would be comparatively analyzed in section 3.4. Through further comparisons in sections 3.1, 3.2, 3.3 and 3.4, the outside perimeters of the examining tubes, the tube thickness, and the tube length are fixed at 132 mm, 0.8 mm, and 2 m, respectively. By the foregoing fixation, the outer and inner heat-transfer areas, the solid volume, and the weight of the tubes are kept constant in sections 3.1, 3.2, 3.3 and 3.4. The tube lengths ranging from 2 mto 20 m, and the outer perimeter of the tubes spanning from 52 mm to 252 mm would be examined in detail in section 3.5. Some novel correlations for predicting the Nusselt number of the examined tube shapes are developed and proposed in section 3.6. Stainless steel SS-304 is utilized as the tube material. To keep the outer perimeter of the tubes constantly with different tube shapes and different ratios of a, band gthe constraints listed in Table 2are employed, and to keep the cross-section area of the hot-air inlet equally to that of the cold-air inlet in each single tube case, the constraints listed in Table 3are employed. 2.2. Geometric design of shell-and-tube heat exchangers The main objective of this study is to examine the effects of the smooth-tube shapes applicable for recuperators and regenerators; therefore, the effects of inlet and outlet manifolds as well as of the baffles are not included. For high pressure applications (for instance the steam boilers), shell-and-tube heat exchangers normally contain circular shapes. However, for currently industrial recuperative or regenerative applications, the effective working pressure of exhaust fans is around 1.5 kPa [24]; therefore, the shells can be designed with square or rectangular configurations. In this study, the tubes are assumed to be arranged in a rectangular shell as presented in Figs. 1(a) and 2(a). The hot air is assumed flowing inside the tubes, and the cold air flows outside the tubes in an opposite direction of the hot air (counter flow). A cross-section area for a hot-air inlet of a tube is designed equally to that for the cold air inlet. Fig. 1(a)presents a structure of rectangular tubes within a rectangular shell. The yellow parts are the cold-air volume, the blue parts are the cold-air outlets, the green parts are the tubes, and the red parts are the hot-air inlets. For a symmetric structure, the

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Table 1 Three different tube shapes.

Table 2 Constraintsfor the tube shapes. Tube shape names

Constraints

Rectangle

Peoutside ¼ 132 mm;

Isosceles triangle Ellipse

Notes Pe a 2 Pe  a 2d < a < 2b; b ¼ Pe ;b ¼ b=a; 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pe 2a þ a > 2d; Pe ¼ p½3ða þ bÞ  ð3a þ bÞða þ 3bÞ ; g ¼ b=ab ¼  2p 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Pe 5 Pe  a2 þ a ; g ¼ b=a 3p 12p2 9 a > 2d;a ¼ b=a; b ¼

a ¼ 1, the shape of the tube is a square b ¼ 1,the tube shape is an equilateral triangle g ¼ 1, the shape of the tube is a circle.

Table 3 Constraints for the computational models. For tube shapes Rectangle

Constraints sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðWt þ Ht Þ2 ðWt þ Ht Þ ; þ ðWt  2dÞðHt  2dÞ  2 4 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 00 00   8ba h 00 00 ða þ 2bÞ2 þ  ða þ 2bÞ h ak 2bk 2ak 00 ða þ 2bÞd 0 ða bÞ2 ða Þ2 00 h 0 0 þ ; c ¼ a  a ;a ¼ a  ;h ¼ h þ 2k; h ¼ ;k ¼  a ¼ aþ 8b h h h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h 4 a2 Wm ¼ 2a þ k; Hm ¼ 2b þ k; k ¼ ða þ bÞ2 þ pab þ pða  dÞðb  dÞ  ða þ bÞ; Wm ¼ Wt þ k; Hm ¼ Ht þ k;Wtinside ¼ Wt  2d;Htinside ¼ Ht  2d; k ¼

triangle

Ellipse

rectangular or elliptical tubes are arranged in the shell as illustrated in Fig. 1(b). Unit cells, including a tube and its hot-and-cold air volumes, are considered as computational models. The aforementioned parts of the rectangular and elliptical unit cells are presented in detail in Fig. 1(c) and (e), respectively. Dimensions of the rectangular and elliptical unit cells are presented in Fig. 1(d) and (f), respectively. Isosceles-triangular tubes are arranged within a shell as presented in Fig. 2(a). A unit cell eincluding parts of a tube, inlets and outlets of the hot and cold airs, as well as volumes of the hot and cold airs ewas considered as a computational model for examining the triangular tubes as presented in Fig. 2(b). Dimensions of the shell, triangular tubes, and the unit cell are presented in Fig. 2(c) and (d), respectively. With this structure of the shell-and-tube heat exchanger, the tubes can be arrayed within the shell following both row and column directions. The number of tubes can be calculated for a row, Nr,a column, Nc, or a total number of the tubes within the shell, Nt, respectively, as follows: For rectangular and elliptical tubes:

Nr ¼

Wsh Wm

Nc ¼

Hsh Hm

(5)

For triangular tubes:

Nr ¼

2Wsh ðWbt þ Wt Þ

(6)

Nc ¼

Hsh Hm

(7)

Nt ¼ Nr  Nc

(8)

Where Wsh ; Hsh ; Wm ; Hm ; Wbt ; and Wt are the shell width, shell height, computational-model width, and computational-model height, bottom width of the trapezoidal-computational model (for the triangular tubes), and top width of the trapezoidalcomputational model, respectively. 2.3. Mathematic model

(4) For simplifying the analyzing processes, assumptions of the computational models are analogous to some previous studies

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5

Fig. 1. Distributions of rectangular and elliptical tubes in a shell-and-tube heat exchanger.

[25e28]: (1) thermophysical properties of the fluids and the solid are assumed to be temperature independent; (2) the flow is assumed to be three-dimensional, single phase, steady state, incompressible and laminar for Re  2300, and transitional or turbulent for Re > 2300;

(3) the gravitational force is neglected, and there is no internal heat generation within the model; (4) the walls of the tubes have a no-slip condition for velocity and temperature. According to the above assumptions and some literature [25e28], the governing equations for computing heat transfer and fluid flow processes in this study could be written as follows:

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Fig. 2. A distribution of triangular tubes in a shell-and-tube heat exchanger.

Continuity equation



vu vv vw þ þ ¼0 vx vy vz

r u (9)

(10c)

where u,vand ware the velocity components of the coolant in x-, y,and z-directions, respectively.

where m, rand pare the dynamic viscosity, density, and pressure drop of the air, respectively.

Momentum equations

!  vu vu vu vp v2 u v2 u v2 u þ w ¼  þm r u þv þ þ 2 vx vy vz vx vx2 vy2 vz

Energy equation for the working fluids



!  vv vv vv vp v2 v v2 v v2 v þ w ¼ þ m r u þv þ þ vx vy vz vy vx2 vy2 vz2

!  vw vw vw vp v2 w v2 w v2 w þv þ w ¼ þm þ þ vx vy vz vz vx2 vy2 vz2

kf vT vT vT þv þw ¼ vx vy vz rf cp;f

v2 T v2 T v2 T þ 2 þ 2 2 vx vy vz

!

(10a)

u

(10b)

whereT,cp;f ,rf ,and kf are temperature, specific heat at a constant pressure, density, and thermal conductivity of the fluid (hot and cold airs), respectively.



Energy equation for the solid

(11a)

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v2 T v2 T v2 T þ þ vx2 vy2 vz2

ks

! ¼0

(11b)

where ks is the thermal conductivity of the solid. The boundary conditions for the computational models are assumed as follows: Inlet:uh ¼ uh0 ,uc ¼  uc0 , v ¼ 0; w ¼ 0, Th ¼ Th0 , and Tc ¼ Tc0 Outlet: Ph ¼ Pc ¼ P0 Other outer walls and symmetric interfaces: VTs ¼ 0 For no-slip condition, uwall ¼ vwall ¼ wwall ¼ 0. For a steady state vp vT condition and an incompressible flow, vvtr ¼ 0; vV vt ¼ 0; vt ¼ 0 and vt ¼ 0. For evaluating the thermal performance of the tubes, the following equations are employed: Actual heat transfer rate, Qactual , of the computational models is calculated by the following equation:

Qactual ¼

Qh þ Qc 2

(12)

where Q is heat transfer rate. The subscripts, cand h, stand for cold and hot sides, respectively. Maximum possible heat transfer rate, Qmax , is calculated as follows:

  Qmax ¼ MCp min ðThi  Tci Þ

(13)

whereMand Cp are the mass flow rate and specific heat at constant pressure of the air, respectively. Tis temperature. The subscripts, ci, hi, and min, stand for cold-side inlet, hot-side inlet, and minimum value, respectively. Effectiveness of the computational model, ε, is calculated by the following equation:

ε¼

Qactual  100 Qmax

(14)

The convective heat transfer coefficients of the hot-side, hh , and the cold-side, hc , are calculated as follows:

hh ¼

Qh Atube;h DTlm

(15)

hc ¼

Qc Atube;c DTlm

(16)

where Ais heat transfer area. The subscripts, tube; cand tube; h, represent for cold side and hot side of the tube, respectively. The log mean temperature difference, DTlm ,is calculated as follows [3]:

DTlm ¼

DTmax  DTmin lnðDTmax =DTmin Þ

(17)

With DTmax ¼ maxðThi  Tco ; Tho  Tci Þand DTmin ¼ minðThi  Tco ; Tho  Tci Þ

(18)

where Tis temperature. The subscripts, ci, co, hiand ho, stand for cold-side inlet, cold-side outlet, hot-side inlet, and hot-side outlet, respectively. The friction factor of the tube, f,is calculated by the following equation:

f ¼

2DPD rw2 L

7

(19)

where Lis the length of tube, Dis the tube hydraulic diameter, wis mean velocity, and ris density. The system performance coefficient, x, is calculated by the following equation:

x¼

Qh

DPh

(20)

where Qh and DPh are heat transfer rate, and pressure drop of the hot side, respectively. 2.4. Computational model validation In the present study, three-dimensional, fluid-solid, conjugate modules of the CFD-ACEþ [29]software package are employed to analyze the computational models. A high accurate meshing method, which was proposed in Ref. [26], is employed for the meshing process, and one of the elliptical meshed computational models is presented in Fig. 3(a). According to the assumptions of the computational models as presented in section 2.3, all outer walls of the computational models are symmetrical or thermally insulated, and there is no heat generation within the model. Therefore, heat dissipated from the hot air, will be entirely transferred to the cold air, and Eq. (21)is utilized to verify all the simulated results with maximum percentage errors being less than 1.2%.

Qh ¼ Qc better representation Mh Cp;h ðThi  Tho Þ ¼ Mc Cp;c ðTci  Tco Þ

(21)

whereMis the mass flow rate of the air passed through the computational model, Cp is the specific heat of the air, To is the outlet-air temperature, and Ti is the inlet-air temperature. The subscripts, cand h, stand for the cold and hot sides, respectively. In addition, experimental results of a triple concentric pipe heat exchanger, which was reported by Quadir et al. [30], were utilized to verify the present simulated results. Dimensions of the triple concentric pipes and boundary conditions of a computational model for verifying the present simulating method with the experimental results are as follows: outer diameters of the pipes are 0.0508 m, 0.0762 mand 0.1016 m, respectively. The thickness of each pipe is 1.5 mm. Three fluids enamely hot water (H), normal water (N) and cold water (C) with constant inlet temperatures of 52.11  C, 28.23  C, and 10.23  C, respectively ewere used as working fluids of the concentric heat exchanger as illustrated in Fig. 3(b). Fig. 3(c)presents an outer temperature distribution of a ¼part of the verifying model. The temperature distribution in Fig. 3(c)is similar as that reported in Ref. [30]. A comparison between the present simulation results and the experimental results, which reported in Ref. [30], are presented in Fig. 3(d). Good agreements between the two results in the comparison are achieved with maximum percentage errors to be less than 1.8%. The strict control in meshing process and good agreements between the numerical and experimental results indicate that the simulation results obtained in this study can utilize to predict the heat transfer and fluid flow characteristics of the examined tubes confidently. 3. Results and discussion All the simulations are carried out using constraints and boundary conditions depicted in sections 3.1, 3.2, 3.3, and 3.4: the outer perimeter of the tubes is kept constantly at 132 mm by using

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Fig. 3. A verification of the simulation results.

the constraints listed in Table 2(aiming at maintaining a constant heat transfer area of the tubes); the hot-side-inlet cross-sectional area is kept equally to that of the cold side for each single tube shape and tube ratio by utilizing the constraints listed in Table 3(the mass flow rates of the hot and cold sides are equal). As listed in Tables 2 and 3and illustrated in Figs. 1(d) and (f), 2(c) and (d). The tube length is fixed at 2 m, and the thickness of the tube is fixed at 0.8 mm. The hot and cold air inlet temperatures are fixed at 300  C and 25  C, respectively. Hot air flowed inside of the tube with Reynolds number ranges from 500 to 8000, and the cold air flowed outside of the tube is at the same mass flow rates of the hot air with opposite direction (counter flow). Stainless steel SS-304 is employed as a tube material. The thermophysical properties of the air and stainless steel SS-304 are listed in Table 4. 3.1. A rectangular tube examination To examine the effects of rectangular-tube shapes on the heat transfer and fluid flow characteristics of the shell-and-tube heat exchangers, the tube width, a, is altered from 9 mm to 33 mm leading to the tube height, b, varying from 57 mm to 33 mm, the

Table 4 Thermophysical properties of stainless steel and air. Materials

l(W/m$K)

Cp(J/kg$K)

r (kg/m3)

m (kg/m$s)

Stainless steel Air

15.5 0.0263

510 1007

7955 1.1614

e 1.846E-05

ratios, a ¼ b=a, (a: tube width and b: tube height) decreased from 6.3 to 1, and the tube-inner hydraulic diameter increased from 13.05 mm to 31.4 mm. It is noted that when a ¼ b ¼ 33 mm, the tube shape becomes square. A unit cell as presented in Fig. 1(c)is employed as a computational model in this section. Fig. 4(a) and (b) present outer temperature and velocity distributions of rectangular tubes with different ratios, a ¼ b=a, from 1 to 6.3 at Reynolds number of 2200, respectively. The results show that both of the cold air outlet temperature and the air velocities of the hot-and-cold sides increased with the increase in the ratio a. Fig. 4(c)presents the effectiveness of the rectangular tubes versus the Reynolds number subject to the tube ratio. It is found that the behavior of the effectiveness shows distinct behaviors in the laminar and turbulent flowing regions. Specifically, the effectiveness is decreased with the increase of the Reynolds number ranging from 500 to 2200 or from 3000 to 8000. It is interesting to see that the effectiveness increased sharply with the increase in Reynolds number from 2200 to 3000. Fig. 4(d)presents the friction factor versus Reynolds number subject to the a. The results show that the friction factor is decreased when the Reynolds number is increased from 500 to 2200 or from 3000 to 8000; however, it is increased when the Reynolds number is increased from 2200 to 2400. The obtained friction factor in the present study is in line with previous published studies [4,31]. Fig. 4(e)presents the hot-side convective heat transfer coefficient versus the Reynolds number subject to the effect of a. The results show that the hot-side convective heat transfer coefficient increased with the increase in Reynolds number or the increase in the a. Especially, it increased sharply when the Reynolds number

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9

Fig. 4. Results of square and rectangular tubes.

increased from 2200 to 3000. Fig. 4(f)presents the system performance coefficient versus the Reynolds number subject to the influence of a. The results show that the system performance coefficient decreased with the increase in Reynolds number or the increase in the a. As assumed in section 2.3, the flow regime will become laminar for Re  2300 and transitional or turbulant for Re > 2300. If the laminar flow prevails, the temperature of the air at the tube center of the hot side will be much higher than that near to the tube wall due to its poor mixing behavior. Once the transition or turbulence appears, the difference in temperature of the air between the tube center and near wall will be dramatically reduced for its much better mixing characteristics. This leads to that the temperature of the hot air near the wall is increased rapidly when the Reynolds

number is increased from 2200 to 3000. This explains for the sharp increase in the convective heat transfer coefficient and effectiveness when the Reynolds number is increased from 2200 to 3000 as presented in Fig. 4(c) and (d). As aforementioned, when the ais increased from 1 to 6.3, the hydraulic diameter of the rectangular tubes is decreased from 31.4 mm to 13.05 mm. This leads to a rapid increase in velocity with the increase in the aat the same Reynolds number (i). In addition, although the outer perimeter of the tube is constant for all cases in this section, the ratio between the inner heat transfer area of the tube and the inner volume of the tube increases from 127.3 m2/m3to 306 m2/m3(this is easy to see in Fig. 4(a)) when the aincreases from 1 to 6.3 (ii). The greater ratio is the larger heat transfer area in the same volume of the tube, thereby resulting in the better heat transfer of the greater-ratio

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tube. (i) and (ii) explain for the distributions of the temperature, the effectiveness and the hot-side convectiveheat transfer coefficient depending on aas presented in Fig. 4(a), (c) and 4(e). The increase in velocity is presented in Fig. 4(b), which can be combined with Eq. (16)to explain for the decrease in the friction factor with the increase in Reynolds number from 500 to 2200 or from 2400 to 8000 as presented in Fig. 4(d). When the transitionalor turbulantflow appears, the chaotic changes of velocity led to the increase in the friction factor with the increase in Reynolds number from 2200 to 2400 as also presented in Fig. 4(d). Both of the heat transfer rate and pressure drop increased with the increase in the a; however, the rise in the heat transfer rate is much lower than that of the pressure drop; therefore, the system performance coefficient decreased with the increase in aas presented in Fig. 4(f). For all cases in this section, it is found that the thermal performance of the rectangular tube is superior to that of the square tube, and a locally thermal optimal rectangular tube is defined by a rectangular tube with the tube ratio of 6.3.

increase in gas aforementioned. Fig. 5(f)presents the system performance coefficient versus Reynolds number subject to g. The results show that the system performance coefficient is decreased with the increase in the Reynolds number or theg. Specially, it is significantly decreased when the Reynolds number is increased from 500 to 2200 or the gincreased from 3.5 to 14.4. It is due to that the heat transfer rate is slowly increased in the laminar-flowing zone compared to that of the transitional- or turbulent-flowing zone as also estimated from Fig. 5(e); however, the pressure drop in the laminar-flowing zone increased faster when compared to that of the transitional- or turbulent-flowing zone as presented in Fig. 5(d). For all cases in this section, it is found that the thermal performance of an elliptical tube is superior to that of a circle tube; however, the system performance coefficient of the circular tube is superior to that of the elliptical tubes, and a locally thermal optimal elliptical tube is defined with the tube ratio of 14.4.

3.2. An elliptical-tube examination

To examine the effects of different triangular-tube shapes on heat transfer and fluid flow characteristics of the shell-and-tube heat exchangers, the length of the bottom edge, a, altered from 11 mm to 44 mm, leading to the ratios, b ¼ b=a, (a: bottom edge and b: two-side edge) is decreased from 5.5 to 1, and the tube-inner hydraulic diameter is increased from 8.4 mm to 23.8 mm. The unit cell presented in Fig. 2(b)is adopted as a computational model. Fig. 6(a) and (b)present the outer temperature and velocity distributions of the triangular tubes with four different bat a Reynolds number of 3000, respectively. The results show that the cold air outlet temperature of a model with a greater bis higher than that of the smaller one (Fig. 6(a)). The velocity of the triangular models is increased with the increase in the bfrom 1 to 5.5 (Fig. 6(b)), and the velocity of the equilateral triangular model (b¼ 1) is the smallest among all; Fig. 6(c)presents the effectiveness of the triangular tubes versus Reynolds number subject to b. The results show that the effectiveness is decreased with the increase in Reynolds number from 500 to 2200 or from 3000 to 8000; however, it is increased with the increase in Reynolds number from 2200 to 3000. The effectiveness is also increased with the increase in theb. Fig. 6(d)presents the pressure drop of the triangular tubes versus Reynolds number subject to theb. The results show that the pressure drop is increased with the increase in Reynolds number, and it is also increased with the increase in theb. Fig. 6(e)presents the hotside convective heat transfer coefficient versus Reynolds number subject to theb. The results show that the hot-side convective heat transfer coefficient is increased with the increase in the Reynolds number and the increase in b. Specially, it is increased sharply when the Reynolds number increased from 2200 to 3000 or when the bincreased from 2.5 to 5.5. Fig. 6(f)presents the system performance coefficient, x,versus Reynolds number subject to theb. It is found that the equilateral triangular model (b¼ 1) delivers the best system performance coefficient, and the system performance also decreased with the increase in b. With bof 1, 1.5, 2.5 and 5.5, the corresponding ratios between the hot-side heat transfer area of the tube and the volume of the hot-side air within the tube are 168 m2/ m3, 184 m2/m3, 244 m2/m3, and 473.8 m2/m3, respectively. Apparently, equilateral triangular model offers the smallest ratio. This is because the largest of area of an equilateral triangle prevail at the same perimeter. The ratio is increased with the increase in b. Especially, it rises significantly when the ratio increased from 2.5 to 5.5. In this case, the greater ratio means the greater heat transfer area on a unit volume of the tube; this leads to the higher cold-side outlet temperature or higher effectiveness or higher convective heat transfer coefficient of the greater bas compared to those of the smaller ones as presented in Fig. 6(a) and (c)and 6(e), respectively.

To examine the effects of different elliptical-tube shapes on heat transfer and fluid flow characteristics of the shell-and-tube heat exchangers, the lengths of the tube-inner semi-major axis, a, varying from 20.2 mm to 2.2 mm to have the ratios, g ¼ b=a, (a: inner semi-major-axis lengths and b: inner semi-minor-axis lengths) ranging from 1 to 14.4, and the tube-inner hydraulic diameter is decreased from 40.4 mm to 6.8 mm. The unit cell presented in Fig. 1(e)was adopted as a computational model. Fig. 5(a) and (b)present outer temperature and velocity distributions of the computational models with different gat a Reof 2200, respectively. The results show that at the same boundary conditions, the cold-air temperature of a model with a greater grises faster than that with a smallerg. The velocity of a model with a greater ɣwas higher than that with a smallerg. The effectiveness of the computational model versus Reynolds number subject to the gis presented in Fig. 5(c). It is found that the effectiveness is decreased with the decline of gand the increase in Reynolds number from 500 to 2200 and 3000 to 8000. In this case, the heat transfer areas, and the solid volume of the tubes are not changed; however, the total volume of a tube is decreased with the increase in the ɣ; therefore, the ratio between the heat transfer areas of a tube and the total tube volume is increased with the rise in the g. This led to a higher heat transfer performance with a greater gas presented in Fig. 5(a) and (c). Pressure drop versus the Reynolds number subject to gis presented in Fig. 5(d). The results show that the pressure drop is increased with the increase in the gor the Reynolds number. Especially, the pressure drop is significantly increased when the gis increased from 3.5 to 14.4. Although the inner perimeter of the tubes are kept unchanged, the hydraulic diameter is significantly decreased with the increase of gfrom 3.5 to 14.4; therefore, the velocity of the tube with the g ¼ 14.4 is much higher than that with g ¼ 3.5 as presented in Fig. 5(b), and this is also the reason of pressure distributions as presented in Fig. 5(d). The velocity chaos as mentioned in section 3.1also affected to the pressure drop when the Reynolds number increases from 2200 to 3000 as presented in Fig. 5(d). Fig. 5(e)presents the hot-side convective heat transfer coefficient versus Reynolds number subject to theg. The results show that the hot-side convective heat transfer coefficient is increased with the increase in Reynolds number or the increase in theg. Yet, a sharp rise is encountered when the Reynolds number increased from 2200 to 3000 or when the ɣincreased from 3.5 to 14.4. The causes of these phenomena are due to the better mixing in the transitional or turbulent flows as discussed in section 3.1or the increase of heat transfer area with the

3.3. A triangular-tube examination

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11

Fig. 5. Results of elliptical tubes.

The higher pressure drop of the greater bmodel as presented in Fig. 6(d)is caused by the effects of the hydraulic diameter and velocity as discussed in previous sections. When bis increased, the pressure drop is also increased faster than that of the heat transfer rate; therefore, the system performance coefficient is decreased with the increase in bas presented in Fig. 6(f)and discussed in previous sections. For all cases in this section, it is found that the thermal performance of an isosceles triangular tube with b> 1 is superior to that of the equilateral triangular tube; however, the pressure drop acted by opposite way.

3.4. A comparison among the three tube shapes In this section, three locally thermal optimal tubes eincluding a rectangular tube with a ¼ 6.3, a triangular tube with b ¼ 5.5, and an

elliptical tube with g ¼ 14.4 eare selected for comparisons subject to similarly aforesaid constraints and boundary conditions. Fig. 7(a) and (b)present the outer temperature and velocity distributions of the three tube shapes at a Reynolds number of 3000. The triangular tube shows the highest average cold-side outlet temperature, followed by the elliptical tube and lastly the rectangular tube. In addition, the cold side outlet temperature distribution of the triangular tube is also much uniform than those of the rectangular and elliptical tube. At the same Reynolds number and tube outer perimeter, the velocity of the elliptical tube contains the highest velocity, followed by triangular tube and lastly the rectangular tube. Fig. 7(c)presents a comparison of the effectiveness for these tubes. It is found that the effectiveness of the triangular tube is superior to those of the elliptical and rectangular tubes. At the laminar flow regime, the effectiveness of the rectangular tube

12

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Fig. 6. Results of isosceles and equilateral triangular tubes.

shows the sharpest decline against the Reynolds number, followed by the elliptical tube and lastly the triangular tube. However, with the increase Reynolds number from 2200 to 3000, the rectangular tube shows the fastest improvement in effectiveness, followed by the elliptical tube and finally the triangular tube. Fig. 7(d) presents a comparison of pressure drop among the three selected tubes, and it appears the pressure drop of elliptical tube > triangular tube > rectangular tube. Fig. 7(e)presents a comparison of the hotside convective heat transfer coefficient (HTC) among the three selected tubes. The corresponding HTC for the hot-side of the

triangular tube is highest among all and it also shows the largest increasing trend with the rise of the Reynolds number, followed by the elliptical tube, and lastly the rectangular tube. Fig. 7(f)presents a comparison of the system performance coefficient for the three tube shapes with the following sequence: rectangular tube > triangular tube > elliptical tube. In these three tubes, although the outer tube perimeters are the same, the ratio between the heat transfer area and the volume of the air flowing in the rectangular, triangular and elliptical tubes are 306 m2/m3, 474 m2/ m3, and 583 m2/m3, respectively. The triangular tube acts as half of

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13

Fig. 7. A comparison among three tube types.

a rectangular tube having an additional fin in its diagonal, thereby showing a higher the heat transfer area. Meanwhile, the distance between the center of the tube to the wall will be decreased compared to that to the rectangular tube without the added fin. This leads to a superior performance of the triangular tube relative to that of the rectangular tube as presented in Fig. 7(a) and (c)and 7(e). It is interesting to find that although the heat transfer area of the elliptical tube within a unit volume is greater than that of the triangular tube, the thermal performance of the triangular tube outperforms that of the elliptical tube. The cross-section areas at

the conners of the cold side of the elliptical computational mode are greater than those at the two sides as clearly seen in Fig. 1(f). Hence, the velocity of the air at the conners of the cold side of the elliptical tube is higher than those at the two sides as presented in Fig. 7(b). The smaller cross-section area at the two sides and the lower velocities at these locations mean that (1) the distances between the hot air and the two sides are shorter than those and the conners, and (2) mass flow rate at the two sides of the cold air are also much smaller than those at the conners, indicating that the required energy for raising the temperature of the cold air at the

14

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two sides is much smaller than that at the conners; however, the energy of the hot air at near the conners is smaller than those at the two sides. These explain for the non-uniformed temperature distribution of the cold air of the elliptical tube as presented in Fig. 7(a) and it also explains the better thermal performance of the triangular tube as compared to that of the elliptical tube as presented in Fig. 7(c) and (e). For all case in this section, it is found that the thermal performance of the triangular tube is superior to those of the elliptical or rectangular tube, and the system performance coefficient of the triangular tube is also better than that of the elliptical tube. 3.5. Tube length and outer perimeter examinations In this section, the effects of tube length on the heat transfer and fluid flow characteristics of the shell-and-tube heat exchanger ehaving elliptical tubes with a fixed g ¼ 3.5 and a fixed outer perimeter of 132 mm subject to tube lengths from 2 mto 20 m eare examined in detail. Fig. 8(a)presents outer temperature distributions of the computational models with different elliptical tube lengths at a Reynolds number of 8000. The results show that the mean temperature of the cold-side outlet with a longer tube is higher than that with a shorter tube. Pressure drop, effectiveness, and the hot-side convective heat transfer coefficient versus the tube length are presented in Fig. 8(b). The results show that the

pressure drop is linearly increased with the rise of the tube length. It is found that the convective heat transfer coefficient was decreased with the increase in the tube length from 2 mto 8 m, reaching a plateau at a length of 8 mand leveling off slightly after 8 m. The effectiveness shows prominent increase for a tube length from 2 mto 8 m, and the increasing trend is moderate when the tube length exceeds 8 m. In this case, the velocity in both hot-side and cold-side is fixed. In this regard, the time for both streams in contact with the tube walls would be increased with the increase in the tube length. In addition, the heat transfer areas of a longer tube are greater than those of a shorter tube (due to the same tube perimeters). Hence, the temperature difference between an inlet and outlet of the hot or cold side would be increased with the increase in the tube length as presented in Fig. 8(a). In these cases, the heat transferred from the hot side to the cold side. Along the tube length, the hot-side heat energy would be decreased and the cold-side heat energy would be increased. The heat-energy transfer would be decreased when the heat-energy difference between the hot and cold sides is decreased (coming to an equilibrium). This is the reason why the distributions of the effectiveness and convective heat transfer coefficients depends on the tube length as presented in Fig. 8(b). The effects of the tube-outer perimeter on the heat transfer and fluid flow characteristics of the heat exchanger ehaving eliptical tubes with tube thickness of 0.8 mm, ɣ ¼ 3.5, tube length of

Fig. 8. Results of computational models with different eliptical-tube lengths and tube outer perimeters.

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2 msubject to various outer perimeters from 52 mm to 252 mm associated with the variation of the tube-inner hydraulic diameters from 8.81 mm to 46.11 mm eare investigated in the following. The working boundary conditions are the same as previous sections. Fig. 8(c)presents outside temperature distributions of the computationalmodels for the elliptical tube with outer perimeters of 52 mm, 92 mm, 132 mm, 172 mm, 212 mm and 252 mm, respectively. The results show that the average cold-side outlet temperature is increased with the decrease of the eliptical-tube outer perimeter. Fig. 8(d)presents the hot-side convective heat transfer coefficient and effectiveness versus Reynolds number subject to the eliptical-tube outer perimeter. The results show that the hot-side convective heat transfer coefficients is increased with the rise of the Reynolds number and the decrease in the eliptical-tube outer perimeter. In particular, the convective heat transfer coefficient is increased sharply when the Reynolds number ranging from 2200 to 3000 or when the tube outer perimeter is decreased from 92 mm to 52 mm. It is found that the effectiveness is increased with the decrease in tube outer perimeter and it is increased rapidly when the tube outer perimeter is decreased from 92 mm to 52 mm. In addtion, the effectiveness is decreased with the increase in Reynolds number from 500 to 2200 or from 3000 to 8000; however, it is increased when the Reynolds number increased from 2200 to 3000. When the tube outer perameteris increased, the ratio between the heat transfer area of the tube and the volume of air within the tube is decreased. In essence, with the tube outer perimeter of 52 mm, 92 mm, 132 mm, 172 mm, 212 mm and 252 mm, the ratio between the hot-side heat transfer area of the tube and the hot-air volume are 453.53 m2/m3, 245.71 m2/m3, 168.67 m2/m3, 128.11 m2/m3, 103.47 m2/m3and 86.73 m2/m3, respectively. The greater the ratio is, the more the air volume, which flows within the tube, contacts with the tube wall, and the betterthe heat is transferred. This explains why the heat transfer of the smaller diameter tube is superior to those of the larger diameter tubes as presented in Fig. 8(c) and (d). In addition, the great difference of the ratios between the tubes with outer perimeters of 52 mm and 92 mm explains for the rapid increase in hot-side convective heat transfer coefficient when the tube outer perimeter is decreased from 92 mm to 52 mm as presented in Fig. 8(d). The turbulenteffects causes the rapid increase in effectiveness and convective heat transfer coefficient when the Reynolds number is increased from 2200 to 3000 as discussed in previous sections.

correlations are further classified into laminar and turbulent flow regimes in the following: For laminar flow:

   Nur ¼ 0:002892  a0:2429  0:00079 Re  0:668  a0:6947   3:921 (23) Nut ¼ ð0:00318  0:00023  bÞRe þ ð4:057  0:1473  bÞ (24)   Nue ¼ 0:00421  g0:3499  0:00072 Re   þ 3:186  0:00084  g2:638

  Nur ¼ ð0:00165  0:00011  aÞRe  2:134  a1:193  9:114 (26)   Nut ¼ 0:00197  2:28  105  b Re   5:365  0:00023  b  9:597

(27)

  Nue ¼ 0:00184  5:21  105  g Re   þ 13:13  2:982  g0:4197

(28)

where a, b, and gare tube ratios of the rectangular, triangular, and elliptical tubes, respectively. The subscripts, r, t,and e,stand for rectangular, triangular and elliptical tube shapes, respectively. For the effect of tube lengths:

El ¼ 0:5644 

hh D ¼ El  Ed  Nui kf

 0:7989 Lt þ 0:674 Lg

(22)

whereNu, h, D, kand Eare Nusselt number, convective heat transfer coefficient, inner hydraulic diameter of the tube, thermal conductivity of the working fluid, and effective coefficients, respectively. The subscripts, d, f, h, i,and l,stand for tube hydraulic diameter, working fluid, hot side, tube shapes and tube length, respectively. Since the thermal behaviors of the examined tubes are different in the laminar and turbulent flows; therefore, the dimensionless

(29)

whereLt is a predicting tube length (m), and Lg is a given tube length with (Lg ¼ 1 m). For the effect of tube hydraulic diameters:



The results presented in the previous sections show that the Reynolds number, tube ratios, tube hydraulic diameter, tube length and tube shapes significantly impacts appreciably the overall thermal performance. For faster design based on the aforementioned result, a novel correlation efor predicting the hot-side Nusselt number of the square, rectangular, circular, elliptical, isosceles- and equilateral-triangular tube shapes depending on tube ratios, a, b, andg, tube inner hydraulic diameter, tube length and Reynolds number eis developed as follows:

(25)

For turbulent flow:

3.6. A correlation development

Nu ¼

15

Ed ¼ 6:006 

Dh Dg

0:7209 þ 0:5993

(30)

whereDh is the inner hydraulic diameter of the predicting tube (m), Dg is a given reference diameter with (Dg ¼ 1 m). It is noted that the given tube length, Lg , and given diameter,Dg , are always fixed at 1 mfor any predicting case.

3.7. A verification of the presently proposed correlation For ensuring the accuracies of the predictive results of the present developed correlations, some well-known correlations for predicting Nusselt number of circular tube proposed by DittusBoelter or Gnielinski, are adopted in the verification. The details of the adopted correlations are as follows: Dittus-Boelter correlation:

Nu ¼ 0:023Re0:8 Pr 0:3 Gnielinski correlation:

(31)

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  f 8

conclusions for all cases in this study could be drawn as follows:

ðRe  1000ÞPr Nu ¼  0:5  2  1 þ 12:7 8f Pr 3  1

(32)

f ¼ ð0:790lnRe  1:64Þ2

(33)

Fig. 9(a)presents a comparison of Nusselt number among results obtained by circular, square, and equilateral-triangular tubes of the present study and those obtained by the Dittus-Boelter and Gnielinski correlations. The results show that the present obtained results agree nicely with those obtained by the Dittus-Boelter and Gnielinski correlations. In particular, the results of Nusselt numbers obtained by the present circular and equilateral-triangular tubes are in good agreements with those obtained by the Dittus-Boelter correlation at Reynolds number from 5000 to 6000, and it also shows good agreements with those obtained by the Gnielinski correlation at Reynolds number from 7000 to 8000. The results of Nusselt number obtained by the present equilateral-triangular tube agree closely with those obtained by Dittus-Boelter and Gnielinski correlations at Reynolds number of 3000 and 4000, respectively. It is noted that the maximum deviations between the predictions and correlations fall within ±4%. The reason of the some larger errors at other Reynolds numbers may be attributed to the differences in tube shapes. Fig. 9(b)presents a comparison of Nusselt number between the numerical results obtained in this study and those obtained by the present proposed correlation for all the simulated results in this study. Good agreements between the two results are reported with maximum deviation being less than 7.1%. The good agreements in the comparisons, presented in Fig. 9, indicate that the correlation proposed in this study could be utilized to predict accurately the Nusselt number of the examined tube shapes and Reynolds number. The novelty of the present proposed correlation not only can estimate the influences of various tube shapes, different tube lengths, diverse tube diameters for both laminar and turbulent flows on the heat transfer characteristics, but also can be used to predict the Nusselt number in association with different tube ratios which had never been reported in the published literature. 4. Conclusions Based on the obtained results, analyses and comparisons,

1. The thermal performance of a rectangular tube is superior to that of the square, and the thermal performance of a rectangular tube with a greater ratio, ⍺, outperforms that of the smaller one. A hot-side convective heat transfer coefficient of a rectangular tube with ⍺ ¼ 6.3 at Re ¼ 8000 can be enhanced two-fold as compared to that of the square tube. 2. The thermal performance of an isosceles triangular tube is superior to that of an equilateral triangular tube, and the thermal performance of a greater tube ratio, b, performs better than those with a smaller ratio. The pressure drop of an isosceles triangular tube is larger than that of an equilateral tube. The hotside convective heat transfer coefficient of an isosceles triangular tube with b ¼ 5.5 at Re ¼ 8000 can be improved up to 2 times relative to that of the equilateral tube. 3. The thermal performance of an elliptical tube does better than that of a circular tube, and the thermal performance of an elliptical tube with a larger ratio, ɣ, outperforms that with a smaller ratio. The pressure drop of a circular tube is smaller than those of the elliptical tube. The hot-side convective heat transfer coefficient of an elliptical tube with ɣ ¼ 14.4 at Re ¼ 8000 can be augmented over 3 times as compared to that of the circular one. 4. The hot-side convective heat transfer coefficient of a locally thermal optimal triangular tube is superior to those of the elliptical and rectangular tubes, and its system performance coefficient is also higher than that of the elliptical tube. 5. With the same outer perimeter and boundary conditions, the hot-side convective heat transfer coefficients of an equilateral triangular tube at Re ¼ 8000 can be enhanced up to 65.3% and 72.6% when compared to those of the rectangular and circular tubes, respectively. 6. Based on the obtained numerical results, a novel correlation for predicting the Nusselt numbers is proposed. The present proposed correlation could be utilized to predict the Nusselt number of different tube shapes, diverse tube diameters, various tube lengths and tube shapes for both laminar and turbulent flows.  A novel correlation for predicting Nusselt numbers of the tube shapes is proposed.  The thermal performance of a rectangular tube is superior to that of the square tube.

Fig. 9. Comparisons for Nusselt number.

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