Accepted Manuscript Title: Numerical investigation of heat transfer intensification in shell and helically coiled finned tube heat exchangers and design optimization Authors: Ashkan Alimoradi, Meysam Maghareh PII: DOI: Reference:
S0255-2701(17)30410-5 http://dx.doi.org/doi:10.1016/j.cep.2017.08.005 CEP 7051
To appear in:
Chemical Engineering and Processing
Received date: Revised date: Accepted date:
24-4-2017 29-7-2017 6-8-2017
Please cite this article as: Ashkan Alimoradi, Meysam Maghareh, Numerical investigation of heat transfer intensification in shell and helically coiled finned tube heat exchangers and design optimization, Chemical Engineering and Processinghttp://dx.doi.org/10.1016/j.cep.2017.08.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Numerical investigation of heat transfer intensification in shell and helically coiled finned tube heat exchangers and design optimization Ashkan Alimoradi1*, Meysam Maghareh2 1
Mechanical Engineering Department, Faculty of Engineering, Razi University, Kermanshah, Iran 2
Mechanical Engineering Department, Faculty of Engineering, Amirkabir University of technology, Tehran, Iran 1*
E-mails:
[email protected],
[email protected] (Tel: +989381875326, Fax: +988346132250) 2 E-mails:
[email protected] Graphical abstract
1
Highlights:
The heat transfer intensification due to installation of the annular fins on the outer surface of the helical coiled tube is studied. Realizable K-ε was selected as the turbulence model. With increase of (ω) and (σ), the heat transfer rate is generally increases. The optimum values for (ω) and (σ) are obtained which maximize the heat transfer and minimize the material consumption.
Abstract In this work, the heat transfer intensification in shell and helically coiled tube heat exchangers via installing annular fins on the outer surface of the helical coil, has been numerically investigated. Thirteen heat exchangers were designed for this purpose. All geometrical parameters of the heat exchangers are same except, fin's number or height. All of the heat exchangers have been studied at three different shell side Reynolds number (Resh=7500, 15000 and 30000). In order to validation of the numerical model two method has been used. In the first method, the calculated heat transfer has been compared with the approximate method which is based on the experimental correlations for both coil and shell side Nusselt numbers and consideration of the E-NTU relations of these types of heat exchangers. In the second method, the heat transfer coefficient of the shell side has been compared with the experimental heat transfer coefficients of the previous works. An acceptable agreement has been observed in these comparisons. Furthermore, the optimum cases and some correlations has been obtained for prediction of heat transfer coefficient of the shell side. Results indicate that, in the range of 7500≤ Resh≤ 30000, the heat transfer rate can increase up to 44.11%.
Nomenclature A
Area (m2)
c
Specific heat capacity (J/kgK)
C
heat capacity (J/K)
2
Cµ
Coefficient of turbulent viscosity as defined in E.q. (11)
d
Diameter (m)
E
Effectiveness
e
Turbulent dissipation (J)
f
The distance between inlet and outlet of the shell
H
Height (m)
h
Heat transfer coefficient (W/m2K)
k
Thermal conductivity (W/mK)
K
Turbulent kinetic energy (J)
l
Length of coiled tube (m)
M
Mass (kg)
m°
Mass flow rate (kg/s)
Nf
Number of fins
NTU
Number of transfer units
Nu
Nusselt number
P
Pitch (m)
p
Pressure (Pa)
Pr
Prandtl number
Q
Heat transfer rate (W)
q
Heat flux (W/m2)
r
Radius (m)
Ra
Rayleigh number
Re
Reynolds number
s
Coefficient of fin efficiency as defined in E.q. (23)
Sij
Strain rate tensor
t
Thickness of fin (m)
T
Temperature (K)
U
Overall heat transfer coefficient (W/m2K)
V
Velocity (m/s)
v
Volume (m3)
xi, xj
Cartesian coordinates 3
Greek symbols α
Bypass area factor
β
Expansion coefficient (K-1)
η
Fin efficiency
ρ
Density (kg/m3)
τij
Viscous stress tensor
θ
Enthalpy (j)
ω
dimensionless height of fins
σ
Number of fins per meter of the coiled tube (fin/m)
µ
Viscosity (Pa.s)
Subscripts c
Coil
dh
Hydraulic diameter
f
Fin
i, j
x, y or z directions
in
Inlet
min
Minimum
o
Overall
opt
Optimum
out
Outlet
r
Relative
sh
Shell
T
Turbulent
t
Tube
tot
Total
v
Shell inlet
Keywords: Heat exchanger, Numerical method, Heat transfer intensification, Helical coil, Annular fin, Optimal design
4
1. Introduction Shell and helically coiled tube heat exchangers, are one of the important devices in many industrial applications like: HVAC (Heating, Ventilation and Air Conditioning), petroleum, power production and waste heat recovery. These types of the heat exchangers are used more than the double pipe heat exchangers, when there are space limitations. Furthermore, the inner side of the coils has a greater heat transfer coefficient than the inner side of straight pipes at same conditions. The heat transfer process in these types of heat exchangers, has been studied by many researchers. Numerous correlations have been proposed for the coil side as well as the shell side Nusselt numbers, by use of numerical or experimental methods. For example, Ashkan et al. [1] studied the effect of the operational and geometrical parameters on the Nusselt numbers of the both (coil and shell) sides by use of both numerical and experimental methods. They found that, the Nusselt number of the coil side is usually more than twice the Nusselt number of the shell side for these heat exchangers. Thus, the shell side (with its high heat transfer resistance) can substantially prevent the heat transfer rate enhancement (even though the heat transfer coefficient of the coil side is too high). A method for solving this problem is, using the finned tubes instead of finless tubes. They are used when the heat transfer coefficient on the outside of the tubes is much lower than the heat transfer coefficient inside the tubes [2]. Thus, installing the fins on the outer side of the helical coils maybe a good choice for the heat transfer intensification. Fins enhance the heat transfer rate via increasing the surface area of the heat exchangers. The second example for the studies on shell and helically coiled finless tube is Ghorbani et al. [3] work. They studied the mixed convection heat transfer in these types of exchangers for various Reynolds numbers, tube to coil diameter ratio and dimensionless coil pitch, experimentally. They proposed a correlation for the shell side Nusselt number as a function of the Reynolds and Rayleigh numbers. Finned tube heat exchangers are widely studied in the past years. Chauvet et al. [4] investigated natural convection heat transfer in a finned helically coiled tube heat exchanger immersed in hot water. They proposed a correlation for the prediction of the Nusselt number of the outer side of the coil as a function of the Rayleigh number. Bahadori et al. [5] suggested simple equations for the prediction of efficiency, heat transfer coefficients of fin side and fin tip temperature for 5
uniform thickness finned tube heat exchangers for df/dt up to 3. Choi et al. [6] studied plate finned tube heat exchangers with large fin pitches. The effect of fin pitch, fin alignment, the number of tube rows, and vertical fin space on the heat transfer was obtained. They found that, the Colburn j-factors of the discrete plate finned tube heat exchangers are 6.0–11.6% higher than those of the continuous plate finned tube heat exchangers. Cui et al. [7] experimentally studied the boiling heat transfer characteristics of a new kind of micro finned helically coiled tube heat exchangers with R134a as the working fluid. They suggested a correlation for calculating the boiling heat transfer coefficient. Gupta et al. [8] obtained an optimized geometrical parameters for the finned coiled tube heat exchangers which are used in small and medium helium refrigerators/liquefiers. Also they found that, the change of clearance can adjust the thermal and pressure drop performance. They also experimentally studied the coiled finned tube heat exchangers used in the cryogenic applications and obtained correlations for the heat transfer coefficient of the shell side and the friction factors for both sides [9-10]. Karmo et al. [11] reported a numerical analysis of the heat transfer and pressure drop in a finned tube heat exchanger and suggested a method for the effective design of these heat exchangers. Kim et al. [12-13] tested flat plate finned tube heat exchangers with large fin pitch. They found that, if the fin pitch decreases and the number of tube row increases the air side heat transfer coefficient will decrease. They also proposed correlations for the heat transfer coefficient and the j-factor as a function of the operational as well as geometrical parameters. Li et al. [14] experimentally tested internally finned and micro finned helical tubes for Reynolds number between 1000 and 8500. They found that, the average heat transfer intensification ratio (compared with the smooth helical tube) for the two finned tubes was 71% and 103%, but associated with a flow resistance increase of 90% and 140%, respectively. Lu et al. [15] numerically investigated the effect of geometrical parameters (i.e. fin pitch (pf), tube pitch (pt), fin thickness (t), and tube diameter (dt)) on the performance of a two row finned tube heat exchanger. It was found that, the heat transfer rate per pressure drop (i.e. Q/∆P) and the heat transfer rate per the assumed power in the pump increases with longitudinal tube pitch or with transverse tube pitch, and it decreases with larger tube diameter or fin thickness. Naphon et al. [16] studied the effect of the operational parameters on the thermal performance and pressure drop of the helical coil heat exchanger with and without helical crimped fins at different coil diameters. Promvonge et al. [17] experimentally studied turbulent flow and heat transfer characteristics in a square duct fitted diagonally with 30° angle 6
finned tapes. They obtained the effect of the fin blockage ratio on the heat transfer and friction factor. Sun et al. [18] numerically studied the fluid flow and the heat transfer of both air side and water side of elliptical finned tube heat exchangers. Response surface methodology was used to obtain the effect of the number of rows, axis ratio, transversal tube pitch, longitudinal tube pitch, fin pitch, air velocity and water volumetric flow rate. Suzuki et al. [19] used the fluent code to study the heat transfer characteristics of an air cooler with finned tubes. The RNG K–e model selected to calculate the thermal hydraulics of the finned heat transfer tube banks. It was found that, the velocity vector patterns, temperature distributions, and loss coefficients are not different largely between the turbulence models. Zdaniuk et al. [20] experimentally studied the heat transfer coefficients and the friction factors for eight internally finned helically coiled tubes and one smooth tube for Reynolds number between 12000 and 60000. They recommended a coil because of its high Colburn j-factors and moderate friction factors at all Reynolds numbers. There are various studies about the application of the annular fins on tubes, for enhancement of heat transfer. For example, Kundu et al. [21] studied radiative heat transfer and heat generated by a nuclear reactor for annular stepped fins through linearization of the radiation terms. They found that, at the optimum case, the heat dissipation rate of the annular stepped fins are higher than this for annular disc fins under constrained volumes. Senapati et al. [22] numerically studied natural convection heat transfer for a vertical cylinder with annular fins. They found that, with the installing of the fins to the outer side of the tube, heat transfer increases for laminar flow and for turbulent flow heat transfer first increases and gets a maximum value then starts to decrease. They also obtained the optimum fin spacing for the maximum heat transfer for the cases of turbulent flow and developed correlations for prediction of Nusselt number for both laminar and turbulent regimes. A. Aziz et al. [23] performed an analytical study to investigate the performance of an annular fins and obtained the design parameters for various types of boundary conditions. B. Kundu et al. [24] suggested an analytical method for a wide range of thermopsychrometric parameters, namely, differential transformation to obtain the temperature field in wet annular fins with triangular and rectangular geometries. They performed an optimization study to determine the optimum design variables. Huang et al. [25] used conjugate gradient method (CGM) to obtain the optimum shapes of partially wet annular fins based on the desired fin efficiency and fin volume. They found that, the optimum annular fin shape always has the highest fin efficiency among all five common annular fins. Arslanturk [26-27] obtained simple 7
correlation equations for optimum design of annular fins with uniform cross section. The fin volume (or mass) was constant to obtain the dimensionless geometrical parameters of the fin with maximum heat transfer rates. He found the optimum radius ratio of an annular fin which maximizes the heat transfer rate as a function of Biot number (hdt/2k) and the fin volume. According to the literature review, there few studies about the helically coiled fined tube heat exchangers. The aim of this study is to investigate the heat transfer intensification in cylindrical shell and helically coiled tube heat exchangers due to installation of the annular fins on the outer surface of the helical coil tube which has not been investigated in the previous works for these special types of heat exchangers. This study is the first numerical simulation of these types of heat exchangers thus, the design details and the results, can be useful for the next researchers. The effect of the number of fins per meter of the coiled tube length and fins dimensionless height on the heat transfer rate will be obtained. Furthermore, a diagram and correlations are obtained for the prediction of the heat transfer coefficient of the fin side which has been not suggested in previous works (however, determination of this parameter is an important part of the design of these types of heat exchangers). At the end of this study, the optimal values of the fin geometry which maximize the heat transfer rate and minimize the material consumption (for construction of the fins), are obtained. This study shows that, how the distribution of the material (for construction of the fins) can be effective on the heat transfer rate.
2. Heat exchanger geometry, mesh and numerical method
The heat exchangers which are designed in this work, include a cylindrical shell and a helical coiled tube. Annular fins with uniform cross section have been installed on the outer surface of the coiled tube to determine their effect on the intensification of the heat transfer. The annular fins are widely used for circular tubes cause of its comfortable installation. The use of the annular fins reduces the difficulties of the transformation of the straight finned tubes to helical coils. The heat exchanger with its finned coiled tube is shown in Fig.1. Furthermore, its dimensions and the geometrical parameters of the fins are shown in and Fig. 2. According to this figure, the following parameters is defined:
8
ω= σ=
2𝐻𝑓
(1)
𝑃𝑐 −𝑑𝑡 𝑁𝑓
(2)
𝑙
A numerical code is used for the simulation of the steady state, single phase forced convection heat transfer on both sides and conductive heat transfer in the solid fins. The governing Equation (i.e. continuity, momentum and energy) are, respectively as follows: 𝜕 𝜕𝑥𝑖
𝜕 𝜕𝑥𝑗
𝜕 𝜕𝑥𝑗
(𝜌𝑉𝑖 ) = 0
(3)
(𝜌𝑉𝑗 𝑉𝑖 ) = −
𝜕𝑝 𝜕𝑥𝑖
+
𝜕𝜏𝑖𝑗
(4)
𝜕𝑥𝑗
1
𝜕
2
𝜕𝑥𝑗
(𝜌𝑉𝑗 (𝜃 + 𝑉𝑖 𝑉𝑖 )) =
(𝑉𝑖 𝜏𝑖𝑗 ) +
𝜕 𝜕𝑥𝑗
𝜕𝑇
(𝑘 𝜕𝑥 )
(5)
𝑗
Where τij is the viscous stress tensor and is defined as follows: 𝜏𝑖𝑗 = 2µ𝑆𝑖𝑗
(6)
Sij is the strain rate tensor which is: 1 𝜕𝑉𝑖 2 𝜕𝑥𝑗
𝑆𝑖𝑗 = (
+
𝜕𝑉𝑗 𝜕𝑥𝑖
)
(7)
The first term on the right-hand side of Equation (5) represents the heat transfer due to convection which is the main form of energy transfer in the fluid regions. However; the second term represents the heat transfer due to conduction. This is the only term which remains in the energy equation at the solid regions (i.e. fins). The turbulent flow is considered for the both sides thus, a model should be selected for the consideration of the turbulence effects in addition to the other governing Equation. Realizable K9
e is selected for the simulation of the turbulence effects. The transfer equations for this model are as follows: 𝜕 𝜕𝑥𝑗
𝜕 𝜕𝑥𝑗
(𝜌𝐾𝑉𝑗 ) =
(𝜌𝑒𝑉𝑗 ) =
𝜕 𝜕𝑥𝑗
𝜕 𝜕𝑥𝑗
[(µ + µ 𝑇 )
[(µ + µ 𝑇 )
𝜕𝐾 𝜕𝑥𝑗
𝜕𝑒 𝜕𝑥𝑗
] + 2µ 𝑇 𝑆𝑖𝑗 𝑆𝑖𝑗 − 𝜌𝑒
(8)
2𝑆𝑖𝑗 𝑆𝑖𝑗 𝐾
] + 𝜌𝑒√2𝑆𝑖𝑗 𝑆𝑖𝑗 max [0.43, 2𝑆𝑖𝑗𝑆𝑒𝑖𝑗𝐾 𝑒
+5
] − 1.9𝜌
𝑒2 µ
𝐾+√𝜌𝑒
(9)
Where µT is the turbulent viscosity and defined as follows [28]:
µ 𝑇 = 𝜌𝐶µ
𝐾2
(10)
𝑒
Where the coefficient Cµ is not constant in the Realizable K-e model (unlike the standard K-e model) and is obtained from the following equation:
𝐶µ =
1
(11)
𝑆𝑖𝑗 𝑆𝑗𝑘 𝑆𝑘𝑖 𝐾 1 4.04+ 𝑒 √6 cos[3 cos−1 (√6 2 )] (𝑆𝑖𝑗 𝑆𝑖𝑗 )3
This model can be used successfully for simulation of the heat transfer phenomena in these types of heat exchangers [1]. For enhancement of the simulation accuracy, the second-order upwind scheme has been used as the discretization pattern in all governing equations. Furthermore, the SIMPLE algorithm is considered for the coupling the pressure and velocity. The following assumptions have been considered during the simulation: a. The fluids are incompressible. b. There is no leakage. c. Contact resistance, tube thickness, radiation, Buoyancy force and heat dissipation from the outer surfaces of the shell are negligible. d. Fin, tube and shell have smooth surfaces. e. b. No-slip conditions at the solid surfaces (i.e. inner and outer surfaces of the tube, fin surface and inner surfaces of the shell). Furthermore, the boundary conditions are as follows:
10
a. Hot water at the temperature of 90 °C flows through the coil side while, the cold water at the temperature of 10 °C flows through the shell side. For all heat exchangers, the inlet velocity for coil side is assumed 1 m/s however; the inlet velocity for the shell side is changed from 0.75 to 3 m/s (i.e. 0.75, 1.5 and 3 m/s) to obtain the effect of the Reynolds number of the shell side on the heat transfer intensification. Thus, thirty nine cases are totally studied. b. The outlet gauge pressure is equal to zero on both sides. c. The outer surfaces of the shell are insulated. Convergence is accepted when the residuals of the continuity, momentum, energy and K-e equations are smaller than 10-4, 10-4, 10-7 and 10-3, respectively. To investigate the effect of the dimensionless height of the fins and the number of the fins per meter of the coiled tube length on the heat transfer intensification, they are changed separately while, the other specifications of the heat exchanger (i.e. operational and other geometrical specifications of the heat exchanger) are kept constant. The range of change of geometrical parameters of the fins is shown in Table 1. According to this table, thirteen heat exchangers should be designed. The fin sizes of the most finned tube heat exchangers are in this range. Furthermore, the fin parameters are dimensionless thus, the results may be used for an arbitrary fin sizes. The working fluid of the both shell and coil side is water. Furthermore, the tube and fins are made from copper. The physical properties of these materials are shown in Table 2. In order to generate an appropriate mesh for this complex geometry, it was divided into four parts: a) the coil, b) the internal cylinder, c) the fins and d) the rest of the shell. By use of Gambit, separate (but connected) mesh have been generated for each parts shown in Fig. 3a and 3b. The interval size between nodes is considered 1.0, 3.0 and 1.0 (mm) for parts (a), (b) and (c), respectively while, this value varies from 1.0 to 3.0 (mm) for part (d). The regular hexahedral elements have been used in the parts (a), (b) and (c) while, irregular tetrahedral elements have been used for part (d). When the solution converged, the of heat transfer rate is obtained as follows: 𝑄𝑐 = 𝑚̇𝑐 𝑐(𝑇𝑖𝑛,𝑐 − 𝑇𝑜𝑢𝑡,𝑐 )
(12)
𝑄𝑠ℎ = 𝑚̇𝑠ℎ 𝑐(𝑇𝑜𝑢𝑡,𝑠ℎ − 𝑇𝑖𝑛,𝑠ℎ )
(13) 11
𝑄𝑡𝑜𝑡 =
𝑄𝑠ℎ +𝑄𝑐
(14)
2
Furthermore, the total mass of the fins and the Reynolds number of the shell side based on the outer diameter of the tube will be obtained as follows: 2
𝑀𝑓 = 𝑁𝑓 𝜌𝑐𝑜𝑝𝑝𝑒𝑟 𝜋𝑡((𝑑𝑡 + 𝐻𝑓 ) − 𝑑𝑡2 )/4
𝑅𝑒𝑠ℎ =
(15)
𝜌𝑉𝑖𝑛 𝑑𝑡
(16)
µ
3. Approximate method for calculating the heat transfer rate
In order to validate the model, a comparison is performed between predicted heat transfer rates which are obtained based on the present numerical model and experimental correlations proposed in previous works. These correlations are as follows: a. The Nusselt number of the coil side is predicted using the following correlation which is proposed by Hardik et al. [29]: 𝑑
𝑁𝑢𝑐 = 0.0456( 𝑐 )−0.16 𝑅𝑒𝑐0.8 𝑃𝑟𝑐0.4
(17)
𝑑𝑡
Where Rec is the coil side Reynolds number based on inside diameter of the tube and prc is the Prandtl number. b. The Nusselt number of the shell side is predicted using the following correlation which is proposed by Ashkan et al. [1]: 𝑑
𝑑
𝑑𝑡
𝑑𝑡
𝑑𝑠ℎ −0.82 𝐻𝑐 0.043 𝐻𝑠ℎ −1.03 𝑓 0.561 𝑃𝑐 0.138 ) ( ) ( ) ( ) ( ) 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑑𝑡
0.723 𝑐 0.378 𝑣 0.556 𝑁𝑢𝑠ℎ = 0.247𝑅𝑒𝑠ℎ ( ) ( ) (
0.717 𝑃𝑟𝑠ℎ (18)
This equation (i.e. Equation (18)) has been obtained for the finless coiled tubes thus, it is expected that, the accurate heat transfer may not be obtained. Where Resh is the shell side Reynolds number based on outside diameter of the tube and prsh is the Prandtl number. dc, dsh, dv
12
are coil, shell, inlet of shell diameter and Hc, Hsh, f and pc are coil height, shell height, the distance between the inlet and outlet of the shell and coil pitch, respectively. When the heat transfer coefficient of both side is obtained, the overall heat transfer coefficient is obtained as follows [30]: 1
=
𝑈𝐴𝑡𝑜𝑡
1 ℎ𝑐 𝐴𝑡
+
1
(19)
ɳ𝑜 ℎ𝑠ℎ 𝐴𝑡𝑜𝑡
Where the overall fin efficiency is obtained as follows:
ɳ𝑜 = 1 −
𝑁𝑓 𝐴𝑓 𝐴𝑡𝑜𝑡
(1 − ɳ𝑓 )
(20)
Where: 𝐴𝑡𝑜𝑡 = 𝑁𝑓 𝐴𝑓 + 𝐴𝑡
2𝑟𝑡
ɳ𝑓 =
(21)
𝐾1 (𝑠𝑟𝑡 )𝐼1 (𝑠(𝑟𝑡 +𝐻𝑓 +𝑡/2))−𝐼1 (𝑠𝑟𝑡 )𝐾1 (𝑠(𝑟𝑡 +𝐻𝑓 +𝑡/2))
𝑠((𝑟𝑡 +𝐻𝑓 +𝑡/2)2 −𝑟𝑡2 ) 𝐾0 (𝑠𝑟𝑡 )𝐼1 (𝑠(𝑟𝑡 +𝐻𝑓 +𝑡/2))+𝐼0 (𝑠𝑟𝑡 )𝐾1 (𝑠(𝑟𝑡 +𝐻𝑓 +𝑡/2))
(22)
Where I0 and K0 are modified, zero-order Bessel functions of the first and second kinds, I1 and K1 are modified, first-order Bessel functions of the first and second kinds, respectively and (s) is defined as follows: 2ℎ𝑠ℎ 0.5 ) 𝑘𝑡
𝑠=(
(23)
Then the heat transfer rate is obtained using E-NTU method. Ashkan [31] suggested to use the ENTU relations as follows:
𝑁𝑇𝑈 =
𝑈𝐴𝑡𝑜𝑡
(24)
𝐶𝑚𝑖𝑛
13
𝐸 = 0.874
1−exp[−𝑁𝑇𝑈(1+𝐶𝑟 )]
(25)
1+𝐶𝑟
𝑄 = 𝐸𝐶𝑚𝑖𝑛 (𝑇𝑖𝑛,𝑐 − 𝑇𝑖𝑛,𝑠ℎ )
(26)
4. Results and discussions
4.1 Mesh and turbulence model independency Mesh independency analysis is studied for the case with ω=0.8 and σ=392 at Resh=15000. Three grids with different number of cells have been considered for this case. The attached cells to the outer wall of the finned tube and the cells of the fins, have been refined from grid #1 to #3. These changes can be seen in Fig. 4a and 4b in a section that is passing through a set of fins and perpendicular to the shell's cross sections. This section has been shown in Fig. 5. Total heat transfer and the temperature of the outlet fluid of the shell side (i.e. Tosh) have been obtained and compared between each grids. Results have been shown in Table 3. As it can be seen from this table, the relative changes of the total heat transfer rate and the temperature of the outlet fluid of the shell side is less than 1% from the grid #2 to #3 thus, the grid #2 is suitable for this geometry and more refinement does not decrease the errors in any appreciable way. To investigate the effect of turbulence models on the results, some cases are re-analyzed with considering the standard K-ω or LES (Large Eddy Simulation) as the turbulence model. In each cases, the attached cells to the outer wall of the finned tube have been refined, until the values of y+ be equal between the turbulence models. This ensures that, this analysis is not a function of y+. As it can be seen from Table 4, the predicted heat transfer rate differences between the turbulence models are negligible (the same results was obtained by Suzuki et al. [19]).
4.2 Model validation In these section, the calculated numerical heat transfer rate for all cases is compared with the heat transfer rate obtained based on the approximate method discussed in the section 3. Results have been shown in Fig. 6. As it can be seen, there is an acceptable agreement between the predicted heat transfer rates obtained via these two methods for some cases (the average and 14
maximum differences are 8.11% and 30.92%, respectively). However; the differences for the other cases are relatively significant. The reason of errors are probably because, in the analytical method Equation (18) has a significant error in the prediction of outer side heat transfer coefficient because it does not consider the effect of the fins geometry. Furthermore, the consideration of the only one heat transfer coefficient for the shell side is probably the second reason of errors. This is clearly visible in Fig 7 which shows the contour of the local heat transfer rate coefficient for the case with σ=192 and ω=0.4 at Resh=30000. The local heat transfer rate coefficient has been obtained from the following formula: ℎ𝑙𝑜𝑐𝑎𝑙 =
𝑞̅ (𝑇̅𝑓,𝑡 −𝑇̅𝑠ℎ )
(27)
Thus, obtaining the effect of fins geometry on the outer side heat transfer coefficients is essential for the prediction of the accurate heat transfer rate. The outer side heat transfer coefficient has been obtained for all cases. The results are shown in Figs. 8. As it can be seen, the heat transfer coefficients of the fin side (i.e. hf), outer side of the tube (i.e. ht) and the total (i.e. htot or hsh) have been obtained as function of Resh, ω and σ. The following equations have been used to obtain these heat transfer coefficients (the bar refers to the average values):
ℎ𝑓 =
ℎ𝑡 =
𝑄𝑓
(28)
𝐴𝑓 (𝑇̅𝑓 −𝑇̅𝑠ℎ )
𝑄𝑡
(29)
𝐴𝑡 (𝑇̅𝑡 −𝑇̅𝑠ℎ )
ℎ𝑡𝑜𝑡 =
𝑄𝑡𝑜𝑡
(30)
𝐴𝑡𝑜𝑡 (𝑇̅𝑓,𝑡 −𝑇̅𝑠ℎ )
It can be seen from Figs. 8 that, there is a significant difference between the heat transfer coefficients of the fin side (hf) and the outer side of the tube (ht). It can also be seen from these figures, when the σ and Resh are held constant, all heat transfer coefficients will generally increase with increase of the ω (or height of the coil). The similar result was obtained by Gupta et al. [8]. However, there are some notable exceptions. For example, for cases with (Resh=7500, σ=192 ω=0.6), (Resh=7500, σ=96 ω=0.6) and (Resh=15000, σ=96 ω=0.6) when the ω increases, ht, hf and htot will increase, respectively (instead of 15
reduction). This figure can be used by the designer of these types of heat exchanger for the prediction of the outer side heat transfer coefficients. The predicted heat transfer coefficients of the shell side (i.e. htot), can be compared with the experimental works of Ghorbani et al. [3] and Gupta et al. [9]. Ghorbani et al. [3] proposed the following correlation for the calculation of the Nusselt number of the shell side in finless coiled tube heat exchangers: 0.3 0.4533 Nudh =0.0041Re0.2 dh Prsh Radh
(31)
Although this correlation is for finless coiled tube heat exchangers, it can also be used for the finned tube heat exchangers because, the Reynolds and Rayleigh numbers are functions of the hydraulic diameter. This parameter involves all geometrical parameters of the shell side and defined as follows: 𝑑ℎ =
4𝑣𝑠ℎ
(32)
𝜋𝑑2 𝑠ℎ
𝐴𝑡𝑜𝑡 + 2 +𝜋𝑑𝑠ℎ 𝐻𝑠ℎ
The Reynolds and Rayleigh numbers (based on hydraulic diameter) are defined as follows: 𝑅𝑒𝑑ℎ =
4𝑚̇𝑠ℎ
(33)
𝜋𝑑ℎ 𝜇
𝑅𝑎𝑑ℎ =
𝑔𝛽|𝑇̅𝑠ℎ − 𝑇̅𝑤 |𝑑ℎ3 𝜌2 𝑐 𝜇𝑘
Where the expansion coefficient β of the water is 207E (-6) K-1. The proposed equation by Gupta et al. [9] for the prediction of the heat transfer coefficient is as follows: 𝛼
𝑅𝑒𝑤𝑜𝑘 0.703 1/3 ) 𝑃𝑟𝑠ℎ 𝛼+1
ℎ𝑡𝑜𝑡 = 0.19( )( 𝑑𝑡
(34)
Where coefficient α is the bypass area factor and it is given by: 𝜋𝑑2
𝛼=
( 4𝑠ℎ −𝜋𝑑𝑐 𝑑𝑓 ) (𝜋𝑑𝑐 (1−𝜎𝑡))
(35)
And the Rewok is defined as follows: 𝑅𝑒𝑤𝑜𝑘 =
𝑚̇𝑠ℎ 𝑑𝑡
(36)
𝜋𝑑2
𝜇 4𝑠ℎ
Then the Nusselt number is obtained as follows: Nudh =
ℎ𝑡𝑜𝑡 𝑑ℎ
(37)
𝑘
16
As it can be seen from Figs. 9 for the most cases, the Nusselt number of the present work has a good agreement with one or both of these two works (i.e. Ghorbani et al. [3] and Gupta et al. [9] works). According to this figures, with increase of the Rhdh, the Nusselt number generally decreases. Furthermore, the data of the present work has a lower dispersion than the others. The following correlation can be proposed for the prediction of the shell side heat transfer coefficient in shell and helically coiled finned tube heat exchangers which are functions of Resh (Equation (16)) and Redh (Equation (36)): a) For Resh=7500 and 2200≤ Redh≤ 5500: −1.863 𝑁𝑢𝑑ℎ = (3𝐸 + 8)𝑅𝑒𝑑ℎ
(38)
b) For Resh=15000 and 2200≤ Redh≤ 5500: −1.572 𝑁𝑢𝑑ℎ = (1𝐸 + 8)𝑅𝑒𝑑ℎ
(39)
c) For Resh=30000 and 2200≤ Redh≤ 5500: −1.584 𝑁𝑢𝑑ℎ = (6𝐸 + 8)𝑅𝑒𝑑ℎ
(40)
4.3 Effect of the geometrical parameters of the fins on the heat transfer and design optimization
The contours of the temperature of the fins and the velocity magnitude of the shell side fluid, have been shown in Figs. 10 and 11 on the section shown in Fig. 5. These contours are obtained for the cases with σ=192 and Resh=30000 and ω=0.2, 0.4, 0.6 and 0.8. In these contours, the ranges for the velocity and temperature are considered 0≤V≤1.5 m/s and 300≤T≤330 K, respectively in order the differences be more sensible. In Fig. 11, there are some discontinuity in the contours. This is because, the velocity in these regions are out of the mentioned range (i.e. 0≤V≤1.5).
17
It can be seen from Figs. 10a-10d, the temperature of the fins reduces with increase of ω (or height of the fins). This is probably because, when the height of the fins increases, the contact area of the cold fluid of the shell side with the fins increases. The effect of σ and ω on the heat transfer rate is shown in Figs. 12a, 12b and 12c at Resh=7500, 15000 and 30000, respectively. It should be noted that, there are two effective factors on the heat transfer rate that change when the fins geometrical parameters are changed. The first factor is the total area (i.e. Atotal) which increases as the σ or ω are increased. The second is the velocity of the shell side fluid near the fins and tube which usually decreases when the σ or ω are increased according to Figs. 11a-11d. Thus, the first factor is in contrast to the second factor. Figs. 12a, 12b and 12c show which of these two factors are more effective on the heat transfer. The Instabilities which exist in these figures are due to the contrast between these two factors. According to these figures, the following results can be obtained: a) For an arbitrary Reynolds number in the range 7500≤ Resh≤ 30000, it can be concluded that, if the σ is held constant, the heat transfer rate generally increases with the increase of ω. Thus, the maximum heat transfer rate is obtained when the ω is equal to 0.8. This is similar to the result that Senapati et al. [22] concluded. The average of maximum increase of the heat transfer rate at Resh=7500, 15000 and 30000 is 17.25%, 35.73% and 44.11% for σ =96, 192 and 392, respectively. b) For an arbitrary Reynolds number in the range 7500≤ Resh≤ 30000, it can be concluded that, if the ω is held constant, the heat transfer rate always increases with the increase of σ. It is obvious that, the maximum heat transfer rate is obtained when the σ is equal to 392. The average of maximum increase of the heat transfer rate at Resh=7500, 15000 and 30000 is 29.35%, 32.24%, 41.74% and 44.11% for ω equals to 0.2, 0.4, 0.6 and 0.8, respectively. c) By holding the ω and σ constant, the heat transfer rate always increases with the increase of Resh. Furthermore, it can be found from these figures that, there are some cases with same heat transfer rate that have the different Mf. Thus, it can be concluded that an optimal cases should be exist, which has the maximum heat transfer rate and minimum material consumption. In order to find the optimal cases, the following parameters have been obtained for each cases:
18
∆𝑄 𝑄𝑓𝑖𝑛𝑙𝑒𝑠𝑠 𝑡𝑢𝑏𝑒
∆𝑄 𝑀𝑓
=
%=
𝑄𝑓𝑖𝑛𝑛𝑒𝑑 𝑡𝑢𝑏𝑒 −𝑄𝑓𝑖𝑛𝑙𝑒𝑠𝑠 𝑡𝑢𝑏𝑒 𝑄𝑓𝑖𝑛𝑙𝑒𝑠𝑠 𝑡𝑢𝑏𝑒
× 100
(38)
𝑄𝑓𝑖𝑛𝑛𝑒𝑑 𝑡𝑢𝑏𝑒 −𝑄𝑓𝑖𝑛𝑙𝑒𝑠𝑠 𝑡𝑢𝑏𝑒
(39)
𝑀𝑓
The designer of the finned tube heat exchangers usually try to design fins which have the maximum heat transfer enhancement and consume the minimum material. These two aims are usually in contrast together (according to Figs. 12a, 12b and 12c). In other words, a cases which has the maximum heat transfer enhancement, usually does not consume the minimum material. Thus, it is reasonable idea to select a case which has the average (or more) heat transfer enhancement and the average (or less) material consumption instead of the cases with the maximum heat transfer enhancement and material consumption or cases with the minimum heat transfer enhancement and material consumption, as the optimum cases. Thus, the following conditions (or criteria) are defined to find the optimum cases: ∆𝑄
a) (
𝑀𝑓
b) (%
̅̅̅̅̅̅ ∆𝑄 )>( )
(40)
𝑀𝑓
∆𝑄 𝑄𝑓𝑖𝑛𝑙𝑒𝑠𝑠 𝑡𝑢𝑏𝑒
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ∆𝑄 ) > (% )
(41)
𝑄𝑓𝑖𝑛𝑙𝑒𝑠𝑠 𝑡𝑢𝑏𝑒
These equations show that, the average values of ∆Q/Q and ∆Q/Mf are the critical value for determination of the optimum cases. The values of these two parameters are shown in Table 5, 6 and 7. Any case that follows these two criteria simultaneously, is a candidate for the optimum case. Results are shown in Tables 5, 6 and 7 for the Resh=7500, 15000 and 30000, respectively. The optimum cases (or cases) are highlighted in the tables. As can be seen, there are two optimum cases for Resh=7500 while, there is only one optimum case for Resh=15000 and 30000. The selection of each of these two optimum cases is depending on the design conditions. The designers should specify that maximum energy transfer is more important or minimum material consumption. If the “Maximum heat transfer” is more important, the cases with (ω=0.8 and σ =192), (ω=0.2 and σ =392) and (ω=0.2 and σ =392) are the optimum cases for Resh=7500, 15000 and 30000, respectively. While, if the “minimum material consumption” is more important, the case with (ω=0.2 and σ =392) is the optimum cases for all Reynolds numbers. This indicates 19
that, it is better to consume the material for construction of the fins in a way which involve lower height fins with higher number of fins instead of higher height fins with lower number of fins. For example in Table 6, the mass of fins for case #4, #6 and #9 is approximately equal to 60 g, while the value of ∆Q/Q for them is 17%,25% and 32%, respectively. This optimum case can be compared with the correlation proposed by Arslanturk [27] (which determine the optimum case based on the maximum heat transfer rate for a constant volume or mass of fin). It is as follows: 𝑑𝑡 +2𝐻𝑓
(
𝑑𝑡
)𝑜𝑝𝑡 = (0.5275 + 0.1795
8𝑣𝑓 𝜋𝑑𝑡3
− 0.0324 + (
8𝑣𝑓 2
𝜋𝑑𝑡3
ℎ𝑡𝑜𝑡 𝑑𝑡
) )[2.5894 − 0.1661 ln (
2𝑘
)+
ℎ𝑠ℎ 𝑑𝑡 2
0.0817 ln (
2𝑘
) ]
(42)
For the optimum case (i.e. the case with ω=0.2 and σ =392) the parameters of the right hand side of Equation (42) are as follows: vf=3.45E-8 (m3), dt=0.01 (m) and htot=2671 (W/m2K) (which has been obtained from Equation (30)). By substituting these values in Equation (42) the optimum fin height is obtained. It is equal to 1.8 (mm) that is not exactly equal with the optimum fin height of the present work (which is 1 mm) however; they are in a good agreement.
5. Conclusions
In this study, the heat transfer intensification in shell and helically coiled tube heat exchangers due to installation of the annular fins on the outer surface of the helical coil was numerically investigated. Realizable K-e was selected as the turbulence model. Furthermore, the second order upwind was used as discretization scheme for all governing equations and the SIMPLE algorithm was applied for the pressure and velocity coupling scheme. For validation of the model first the predicted heat transfer was compared with the approximate method which is based on the experimental correlations for both coil and shell side Nusselt numbers of shell and helically coiled finless tube heat exchangers and consideration of the E-NTU relations of these types of heat exchangers. Then the heat transfer coefficient of the shell side was compared with the experimental heat transfer coefficients of the previous works. An acceptable agreement has been observed in these comparisons. Results indicates: a) The difference between the predicted heat transfer rates due to changing of the turbulence model are negligible. 20
b) The effect of ω and σ on the heat transfer rate was obtained at various shell side Reynolds number. c) For an arbitrary Reynolds number in the range 7500≤ Resh≤ 30000, if the σ is held constant, the heat transfer rate generally increases with the increase of ω. The maximum increase of the heat transfer rate is 17.25%, 35.73% and 44.11% for σ =96, 192 and 392, respectively. d) For an arbitrary Reynolds number in the range 7500≤ Resh≤ 30000, if the ω is held constant, the heat transfer rate always increases with the increase of σ. The maximum increase of the heat transfer rate is 29.35%, 32.24%, 41.74% and 44.11% for ω equals to 0.2, 0.4, 0.6 and 0.8, respectively. e) By holding the ω and σ constant, the heat transfer rate always increases with the increase of Resh. f) The optimum cases which has the maximum heat transfer rate and minimum material consumption for construction of the fins are obtained. If the “minimum material consumption” is more important, the case with (ω=0.2 and σ =392) is the optimum cases for all Reynolds numbers. g) It is better to consume the material for construction of the fins in a way which involve lower height fins with higher number of fins instead of higher height fins with lower number of fins. h) If the σ and Resh are held constant, all heat transfer coefficients (i.e. hf, ht and htot) will generally increase with increase of the ω (or height of the coil). i) A diagram and some correlations has been obtained for prediction of heat transfer coefficient of the shell side.
21
References
[1] Ashkan Alimoradi, Farzad Veysi, Prediction of heat transfer coefficients of shell and coiled tube heat exchangers using numerical method and experimental validation, International Journal of Thermal Sciences 2016; 107:196-208. [2] M. Nitsche, R.O. Gbadamosi, A Practical Guide for Planning, Selecting and Designing of Shell and Tube Exchangers, CHAPTER 12 Finned Tube Heat Exchangers, Heat Exchanger Design Guide, 2016, Pages 247–264. [3] Ghorbani N, Taherian H, Gorji M, Mirgolbabaei H. Experimental study of mixed convection heat transfer in vertical helically coiled tube heat exchangers. Exp Therm Fluid Sci 2010, 34, 900-905. http://dx.doi.org/10.1016/ j.expthermflusci.2010.02.004. [4] L. P. Chauvet, D. J. Nevrala, S. D. Probert, Heat-Transfer Correlations for an Immersed Finned Heat-Exchanger Coil Transferring Heat from a Hot-Water Store, Applied Energy 44 (1993) 283-314. [5] Alireza Bahadori, Hari B.Vuthaluru, Predictive tool for estimation of convection heat transfer coefficients and efficiencies for finned tubular sections, International Journal of Thermal Sciences 49 (2010) 1477-1483. [6] Jong Min Choi, Yonghan Kim, Mooyeon Lee, Yongchan Kim, Air side heat transfer coefficients of discrete plate finned-tube heat exchangers with large fin pitch, Applied Thermal Engineering 30 (2010) 174–180. [7] Wenzhi Cui, Longjian Li, Mingdao Xin, Tien-Chien Jen, Qinghua Chen, Quan Liao, A heat transfer correlation of flow boiling in micro-finned helically coiled tube, International Journal of Heat and Mass Transfer 49 (2006) 2851–2858. [8] Prabhat Kumar Gupta, P.K. Kush, Ashesh Tiwari, Design and optimization of coil finnedtube heat exchangers for cryogenic applications, Cryogenics 47 (2007) 322–332. [9] Prabhat Kumar Gupta, P.K. Kush, Ashesh Tiwari, Experimental research on heat transfer coefficients for cryogenic cross-counter-flow coiled finned-tube heat exchangers, international journal of refrigeration 32 (2009) 960–972.
22
[10] Prabhat Kumar Gupta, P.K. Kush, Ashesh Tiwari, Experimental studies on pressure drop characteristics of cryogenic cross-counter flow coiled finned tube heat exchangers, Cryogenics 50 (2010) 257–265. [11] Diala Karmo, Salman Ajib, Ayman Al Khateeb, New method for designing an effective finned heat exchanger, Applied Thermal Engineering 51 (2013) 539-550. [12] Yonghan Kim, Yongchan Kim, Heat transfer characteristics of flat plate finned-tube heat exchangers with large fin pitch, International Journal of Refrigeration 28 (2005) 851–858. [13] Mooyeon Lee, Yonghan Kim, Hosung Lee, Yongchan Kim, Air-side heat transfer characteristics of flat plate finned-tube heat exchangers with large fin pitches under frosting conditions, International Journal of Heat and Mass Transfer 53 (2010) 2655–2661. [14] Longjian Li, Wenzhi Cui, Quan Liao, Xin Mingdao, Tien-Chien Jen, Qinghua Chen, International Journal of Heat and Mass Transfer 48 (2005) 1916–1925. [15] Chi-Wen Lu, Jeng-Min Huang, W.C. Nien, Chi-Chuan Wang, A numerical investigation of the geometric effects on the performance of plate finned-tube heat exchanger, Energy Conversion and Management 52 (2011) 1638–1643. [16] Paisarn Naphon, Thermal performance and pressure drop of the helical-coil heat exchangers with and without helically crimped fins, International Communications in Heat and Mass Transfer 34 (2007) 321–330. [17] Pongjet Promvonge, Sompol Skullong, Sutapat Kwankaomeng, Chinaruk Thiangpong, Heat transfer in square duct fitted diagonally with angle-finned tape—Part 1: Experimental study, International Communications in Heat and Mass Transfer 39 (2012) 617–624. [18] Lei Sun, Chun-Lu Zhang, Evaluation of elliptical finned-tube heat exchanger performance using CFD and response surface methodology, International Journal of Thermal Sciences 75 (2014) 45-53. [19] Daisuke Suzuki, Hiroyasu Mochizuki, Thermal–hydraulic analysis of air cooled finned heat transfer tubes, Annals of Nuclear Energy 95 (2016) 1–11. [20] Gregory J. Zdaniuk, Louay M. Chamra, Pedro J. Mago, Experimental determination of heat transfer and friction in helically-finned tubes, Experimental Thermal and Fluid Science 32 (2008) 761–775. [21] Balaram Kundu, Kwan-Soo Lee, Analytical tools for calculating the maximum heat transfer of annular stepped fins with internal heat generation and radiation effects, Energy (2014) 1-16. 23
[22] Jnana Ranjan Senapati, Sukanta Kumar Dash, Subhransu Roy, Numerical investigation of natural convection heat transfer from vertical cylinder with annular fins, International Journal of Thermal Sciences 111 (2017) 146-159. [23] Abdul Aziz, Tiegang Fang, Thermal analysis of an annular fin with (a) simultaneously imposed base temperature and base heat flux and (b) fixed base and tip temperatures, Energy Conversion and Management 52 (2011) 2467–2478. [24] B. Kundu, D. Barman, Analytical study on design analysis of annular fins under dehumidifying conditions with a polynomial relationship between humidity ratio and saturation temperature, International Journal of Heat and Fluid Flow 31 (2010) 722–733. [25] Cheng-Hung Huang, Yun-Lung Chung, An inverse problem in determining the optimum shapes for partially wet annular fins based on efficiency maximization, International Journal of Heat and Mass Transfer 90 (2015) 364–375. [26] Cihat Arslanturk, Simple correlation equations for optimum design of annular fins with uniform thickness, Applied Thermal Engineering 25 (2005) 2463–2468. [27] Cihat Arslanturk, Erratum to ‘‘Simple correlation equations for optimum design of annular fins with uniform thickness” [Applied Thermal Engineering 25 (2005) 2463–2468], Applied Thermal Engineering 29 (2009) 1271–1272. [28] ANSYS FLUENT 12.0 User's Guide, Modeling Turbulence. ANSYS Inc; 2013. pp. 695759. [29] B.K. Hardik, P.K. Baburajan, S.V. Prabhu, Local heat transfer coefficient in helical coils with single phase flow, International Journal of Heat and Mass Transfer 89 (2015) 522–538. [30] T. L. Bergman, A. S. Lavine, F. P. Incropera, D. P. Dewitt, Fundamentals of Heat and Mass Transfer, seventh edition, Wiley, pp. 160-172. [31] Ashkan Alimoradi, Study of thermal effectiveness and its relation with NTU in shell and helically coiled tube heat exchangers, Case Studies in Thermal Engineering 9 (2017) 100–107.
24
Figure captions Fig. 1: The heat exchanger with its finned tube coil. Fig 2: Dimensions of the heat exchanger and geometrical parameters of the fins. Fig 3: Generated mesh for: a) Coil and fins. b) Shell. Fig 4: Changes of mesh from: a) grid #1. To b) #3. Fig 5: The selected section. Fig 6: Comparison between predicted heat transfers rates using numerical and analytical method. Fig 7: Contour of the local heat transfer rate coefficient (W/m2K) for the case with σ=192 and ω=0.4 at Resh=30000. Fig 8: Heat transfer coefficients of fin side (hf), outer side of the tube (ht) and their total (htot) as function of Resh, ω and σ. Fig 9: Comparison of the Nusselt numbers of the present study with the previous works at a) Resh=7500. b) Resh=15000. c) Resh=30000. Fig 10: Contour of the temperature (K) for cases with with σ=192 and Resh=30000 and ω equals to: a) 0.2, b) 0.4, c) 0.6 and d) 0.8 Fig 11: Contour of the velocity (m/s) for cases with with σ=192 and Resh=30000 and ω equals to: a) 0.2, b) 0.4, c) 0.6 and d) 0.8 Fig 12: Effect of σ and ω on the heat transfer at: a) Resh=7500. b) Resh=15000. c) Resh=30000.
25
Fig. 1: The heat exchanger with its finned tube coil.
26
Fig. 2: Dimensions of the heat exchanger and geometrical parameters of the fins.
27
Fig 3: Generated mesh for: a) Coil and fins.
28
Fig 3: Generated mesh for: b) Shell.
29
Fig 4: Changes of mesh from: a) grid #1
30
Fig 4: Changes of mesh to grid: b) #3.
31
Fig 5: The selected section.
32
Fig 6: Comparison between predicted heat transfers rates using numerical and analytical method.
33
Fig 7: Contour of the local heat transfer rate coefficient (W/m2K) for the case with σ=192 and ω=0.4 at Resh=30000.
34
Fig 8: Heat transfer coefficients of fin side (hf), outer side of the tube (ht) and their total (htot) as function of Resh, ω and σ.
35
200 180 160 Present work, R² = 0.9347
140
Nush
120 Ghorbani et al [3], R² = 0.9764
100 80
Gupta et al [9], R² = 0.7482
60 40 20 0 0
1000
2000
3000 Redh
4000
5000
6000
Fig 9: Comparison of the Nusselt numbers of the present study with the previous works at a) Resh=7500.
36
250
200 Present work, R² = 0.9635
Nush
150
Ghorbani et al [3], R² = 0.99
100
Gupta et al [9], R² = 0.7482
50
0 0
2000
4000
6000
8000
10000
12000
Redh
Fig 9: Comparison of the Nusselt numbers of the present study with the previous works at b) Resh=15000.
37
400 350 300 Present work, R² = 0.9309
Nush
250 200
Ghorbani et al [3], R² = 0.4473
150 Gupta et al [9], R² = 0.7482
100 50 0 0
5000
10000
15000
20000
25000
Redh
Fig 9: Comparison of the Nusselt numbers of the present study with the previous works at c) Resh=30000.
38
Fig 10: Contour of the temperature (K) for cases with with σ=192 and Resh=30000 and ω equals to: a) 0.2
39
Fig 10: Contour of the temperature (K) for cases with with σ=192 and Resh=30000 and ω equals to: b) 0.4
40
Fig 10: Contour of the temperature (K) for cases with with σ=192 and Resh=30000 and ω equals to: c) 0.6
41
Fig 10: Contour of the temperature (K) for cases with with σ=192 and Resh=30000 and ω equals to: d) 0.8
42
Fig 11: Contour of the velocity (m/s) for cases with with σ=192 and Resh=30000 and ω equals to: a) 0.2
43
Fig 11: Contour of the velocity (m/s) for cases with with σ=192 and Resh=30000 and ω equals to: b) 0.4
44
Fig 11: Contour of the velocity (m/s) for cases with with σ=192 and Resh=30000 and ω equals to: c) 0.6
45
Fig 11: Contour of the velocity (m/s) for cases with with σ=192 and Resh=30000 and ω equals to: d) 0.8
46
Fig 12: Effect of σ and ω on the heat transfer at: a) Resh=7500
47
Fig 12: Effect of σ and ω on the heat transfer at: b) Resh=15000.
48
Fig 12: Effect of σ and ω on the heat transfer at: c) Resh=30000.
49
Table 1: Range of geometrical parameters of the fins.
Case No
Fin height (Hf) (mm)
Number of fins
σ (fin/m)
ω
0
0
0
0
0
1
1
96
96
0.2
2
2
96
96
0.4
3
3
96
96
0.6
4
4
96
96
0.8
5
1
192
192
0.2
6
2
192
192
0.4
7
3
192
192
0.6
8
4
192
192
0.8
9
1
392
392
0.2
10
2
392
392
0.4
11
3
392
392
0.6
12
4
392
392
0.8
Table 2: Physical properties of materials.
Materials
Water
ρ kg/m3
998.2
cp j/kgK
4182
k(T) W/mK
=1.5326e-8T3-2.261e5T2+0.010879T-1.0294
µ(T) Pa.s
Pr
=2.1897e-
3.21-6.75
11T42.1897e-11T43.055e11T3+1.6028e-11T20037524T+0.33158
Copper
8978
381
387.6
none
50
Table 3: Mesh independency analysis for case with (σ=392 fin/m and ω=0.8) at Resh=15000.
Grids
1
2
3
Fins number of cells
50176
931214
931214
Coil side number of cells
342568
342568
1025896
Shell side number of cells
2060902
2356895
3324661
Total number of cells
2453646
3630677
5281771
Q (j)
6298.079
6913.241
6887.249
%Q
-
9.76
-0.37
Tosh (K)
292.08
292.88
292.84
%Tosh
-
0.273
-0.015
Table 4: Comparison of the turbulence models. σ
Q (W) for
Q (W) for
Q (W) for
Maximum
(fin/m)
(K-e)
(K-ω)
(LES)
Difference %
0.8
192
5289.60
5059.49
5309.30
4.35
15000
0.4
96
5117.21
5057.38
5168.81
1.18
30000
0.2
392
7337.38
7189.26
7385.34
2.03
No
Resh
ω
1
7500
2 3
51
Table 5: Determination of optimum cases at Resh=7500.
N.O.
ω
σ (fin/m)
Mf (g)
%∆Q/Qfinless
∆Q/Mf (W/g)
1
0.2
96
14.215
6.18
16.525
2
0.4
96
29.784
5.946
7.588
3
0.6
96
46.707
5.867
4.774
4
0.8
96
64.984
19.916
11.649
5
0.2
192
28.43
9.561
12.781
6
0.4
192
59.569
12.049
7.688
7
0.6
192
93.415
19.074
7.761
8
0.8
192
129.969
38.227
11.179
9
0.2
392
58.638
25.546
16.559
10
0.4
392
122.861
24.651
7.626
11
0.6
392
192.669
38.963
7.686
12
0.8
392
268.062
39.463
5.595
18.725
10.165
Average
52
Table 6: Determination of optimum cases at Resh=15000.
N.O.
ω
σ (fin/m)
Mf (g)
%∆Q/Qfinless
∆Q/Mf (W/g)
1
0.2
96
14.215
11.078
35.673
2
0.4
96
29.784
11.257
17.301
3
0.6
96
46.707
9.471
9.282
4
0.8
96
64.984
17.836
12.563
5
0.2
192
28.43
15.821
25.472
6
0.4
192
59.569
25.31
19.449
7
0.6
192
93.415
32.563
15.956
8
0.8
192
129.969
36.809
12.964
9
0.2
392
58.638
32.391
25.285
10
0.4
392
122.861
34.136
12.718
11
0.6
392
192.669
45.802
10.881
12
0.8
392
268.062
49.45
8.444
26.827
17.166
Average
53
Table 7: Determination of optimum cases at Resh=30000.
N.O.
ω
σ (fin/m)
Mf (g)
%∆Q/Qfinless
∆Q/Mf (W/g)
1
96
0.2
14.215
10.462
41.082
2
96
0.4
29.784
12.164
22.796
3
96
0.6
46.707
9.96
11.903
4
96
0.8
64.984
13.991
12.018
5
192
0.2
28.43
20.243
39.745
6
192
0.4
59.569
26.388
24.727
7
192
0.6
93.415
35.22
21.045
8
192
0.8
129.969
32.139
13.803
9
392
0.2
58.638
30.114
28.667
10
392
0.4
122.861
37.945
17.239
11
392
0.6
192.669
40.465
11.723
12
392
0.8
268.062
43.423
9.042
26.043
21.149
Average
54
