- Email: [email protected]
CFD Supported Performance Estimation of an Internally Finned Tube Heat Exchanger Under Mixed Convection Flow
Available online at www.sciencedirect.com
Procedia Engineering 38 (2012) 585 – 597
International Conference on Modelling, Optimisation and Computing (ICMOC 2012), April 10 - 11, 2012
CFD supported performance estimation of an internally finned tube heat exchanger under mixed convection flow S.K.Routa, D. N. Thatoia, A.K. Acharyaa, D. P. Mishraa,* a
S’O’A University, I. T. E.R., Jagamara, Bhubaneswar – 751 030, Odisha, India.
Abstract Results of a numerical analysis using Finite Volume Method of an internally finned axi-symmetric tube heat exchanger have been presented in this paper. The parametric study has been done using a computational fluid dynamics (CFD) program named FLUENT to estimate the performance of the heat exchanger with different fin shapes, sizes and numbers. The results obtained from the study for a steady and laminar flow of fluid under mixed flow convection heat transfer condition shows that there exists an optimum number for fins to keep the pipe wall temperature at a minimum. The wall temperature optimises at a definite fin height beyond which it is insensitive to any height variation. Moreover, amongst the three different shapes considered for fin, results show that wall temperature is least for triangular shaped fins, compared to rectangular and T-shaped fins. In addition to study of thermal characteristics, the pressure drop caused by presence of fins has also been studied.
© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Noorul Islam Centre for Higher Education ,NICHE, Kumarakoil – 629180 Tamil Nadu, India. Keywords: Internally finned tube; Wall temperature; Fin height; finite volume method; Fin profile
* Corresponding author. Tel.: +91-674-2350181; fax: +91-2351880. Email: [email protected]
1. Introduction Internally finned tubes commonly used in engineering applications as effective and efficient means to improve convective heat transfer perform differently depending on whether the flow is laminar or turbulent. For laminar flow and heat transfer, comprehensive experimental and numerical investigations have been performed for variable fluid properties, mixed convection and fin geometry. Heat transfer
1877-7058 © 2012 Published by Elsevier Ltd. doi:10.1016/j.proeng.2012.06.073
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augmentation techniques play a vital role here since heat transfer coefficients are generally low for laminar flow in plain tubes. Designing a tubular heat exchanger with fins having different shapes and sizes is one such augmentation technique discussed in this paper. Nomenclature Cp specific heat at constant pressure D tube diameter Dh hydraulic diameter Gr Grashof number g acceleration of gravity Hf height of fin h average heat transfer coefficient hx local heat transfer coefficient k thermal conductivity L length of test section Lf length of fin Lx axial distance m Mass flow rate of air Nux local Nusselt number based hydraulic tube diameter n fin number p pressure P perimeter Pr Prandlt number q Uniform heat flux Re Reynolds number
Ri Tcx Tmi Tmx Tsavg Tsx U Vin V Wf
on
Richardson number local centerline fluid temperature inlet fluid temperature mean fluid temperature tube surface average temperature local tube surface temperature Velocity inlet velocity volume of tube width / thickness of fin
Greek Symbols fluid density ambient fluid density dynamic viscosity kinematic viscosity thermal diffusivity wall position Subscripts f fin in inlet
When an array of fins is used to enhance heat transfer, the prime focus is to optimise geometry of fins which will maximise the heat transfer rate under space and cost constraints. Extensive work has been carried out by different researchers to analyze heat transfer rate and pressure drop characteristics from tubes having fins of various shapes (rectangular, triangular, T-sectional, twisted). Experimental investigations have been performed by different researchers to study the friction factor, heat transfer rate, and pressure drop and temperature distribution of tube wall. Kern and Kraus [1] have identified longitudinal, radial and pin fins with straight profile as main fin geometries in external finned heat exchanger. Kundu and Das [2] later proposed straight taper fins with better heat transfer coefficients to weight ratio. Micro fins offer effective heat transfer and are more suitable as better heat exchanging medium than twisted tubes. Smit and Meyer [3] performed an experiment to compare three different (micro fins, twisted tapes and high fins) heat transfer enhancement methods to that of smooth tubes using the geotropic refrigerant mixture R-22/R-142b. The results corroborated the effectiveness of micro fins over twisted tubes. For non symmetric convective boundary conditions, two dimensional heat transfer equation is solved analytically [4] to obtain temperature profile in a finned tube. For a given volume, the optimized variables like the fin thickness and ratio of outer radius to inner radius of a fin have been determined. Malekzadeh et al. [5] optimized the shape of non-symmetric annular fins based on two nonlinear dimensional heat transfer analysis. They compared the results obtained through both differential quadrature method and finite difference method. Brien and Sohal [6] investigated experimentally forced convection heat transfer in a narrow rectangular duct fitted with a circular tube and/or a delta-winglet pair. They obtained a
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comparison of local and average heat transfer distributions for the circular tube with and without winglets and found at higher Reynolds numbers the enhancement level is close to 50%. Moreover, the combined effect of heat transfer rate and pressure drop depends on the enhancement of baseline winglet pair. Aziz and Fang [7] proposed an alternate solution for the energy equations for one-dimensional steady conduction in the longitudinal fins of three different cross sections namely rectangular, trapezoidal, and concave parabolic profile. The temperature and the heat flux is specified at the base of the fin and the temperature distribution in the fins are provided for these conditions. While fins offer solutions to augment heat exchange problems, they also result in pressure drop of flow. Experimental investigations show that the heat transfer characteristics and flow friction is greatly influenced by the fin spacing, size and shape of the fin. Bilir [8] investigated effects of vortex generators on heat transfer and pressure drop characteristics on fin-and-tube heat exchanger with three different types of vortex generators. The cumulative effect of vortex generators offers a better heat exchange solution with moderate drop in pressure. Hussein and Salman [9] conducted an experiment to study pressure drop and heat transfer characteristics of water flow in a horizontal tube with or without longitudinal fins and found that heat transfer enhancement is 16 times more and friction factor is 4.5 times more of a finned tube than a bare tube having Re < 4000. The fins existing in annuli influence the flow pattern, temperature distribution and heat transfer rate. Zhu [10] focussed on model and simulation of four types of basic fins (plain fin, strip fin, offset fin, perforated fin, wavy fin) considering fin thickness, thermal entry effect and end effect for 132.3
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investigation is started from matching the present CFD results with the existing analytical results for the case of two dimensional axi-symmetric smooth tubes. Then numerical analysis has been performed to determine the wall temperature distribution and average heat transfer coefficient of rectangular longitudinal fins mounted at the inner periphery of the tube. The flow of air through the tube is considered to be laminar with density taken as the function of temperature according to ideal gas relation and the wall is subjected to constant heat flux. Then the investigation is carried forward by changing the spacing between the fins, shape and size to analyse its heat transfer and fluid flow characteristics. 2. Mathematical modeling 2.1. Mathematical Formulation Fig. 1 gives the schematic representation of the internally finned tube for which the computational investigation is carried out .The tube has diameter D ( neglecting thickness) and length L. Air enters into the tube at one end where as the other end is exposed to the surrounding atmosphere. Longitudinal fins are placed symmetrically around inner periphery of the tube. The investigation is initiated with the rectangular fins of length Lf, width Wf and height, Hf as shown in Fig. 1 and subsequently the cross sectional area of the fin has been changed to triangular and T shapes. The flow field in the domain would be computed by using three-dimensional, incompressible Navier-Stokes equations (2D axi-symmetric model for simple tube) along with the energy equations. The fluid used in the simulation is air treated to be incompressible, enters the tube at with an inlet velocity of 0.264 m/s at a .temperature ranging from of 300 to 700 K.
Fig. 1. Schematic diagram of computational domain and the boundary condition applied to it
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2.2 Governing equations The governing equations for the above analysis can be written as: Continuity Equation
.( v ) 0
(1)
Momentum Equation D
Ui Dt
p xi
Ui xj
xj
Uj xi
g
(2)
The density is taken to be a function of temperature according to ideal gas law while laminar viscosity and thermal conductivity are kept constant. The Boussinesq approximation is not adopted in the model since the variation of with temperature is very high in the range of operating parameters. Energy Equation
D
T Dt
xi
T Pr xi
(3)
0
(4)
Conduction Equation
xi
k
T xi
2.3 Boundary Conditions The boundary conditions are mentioned in Fig. 1. The tube wall and fin are solid and have been given a no-slip boundary condition. Pressure outlet boundary condition has been imposed at the outlet of the tube and constant heat flux condition is applied on the wall of the tube. The velocity inlet boundary condition has been employed at the inlet face which supplies air into the tube. 2-D Axi-symmetric model has been used in case of fin less tube as in Fig. 2.At the pressure outlet boundary, the velocity will be computed from the local pressure field so as to satisfy the continuity but the scalar variable such as temperature T is computed from the zero gradient condition, Dash [19].
Fig. 2. Smooth tube 2-D axi-symmetric model
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2.4. Computations of some important heat transfer and fluid flow parameters The local Nusselt number for the finned heated tube is computed as
hx Dh kair
Nux
(5)
And the local heat transfer coefficient has been computed as given below
hx
q
(6)
Tsx Tcx
Where hx, Dh and kair are the local heat transfer coefficient, hydraulic diameter and thermal conductivity of air respectively. Tsx and Tcx are the local tube wall temperature and local centreline fluid temperature. The dimensionless number affecting the heat transfer and fluid flow are, The Reynolds number
Re
Vin Dh
(7)
air
The Grashof number Gr Gr
g
Tout Tin Dh3 (
2
air
)
(8)
The Richardson number:
Ri
Gr Re 2
(9)
The hydraulic diameter of the fin channel is defined as
Dh
4P Ac
(10)
Where Ac is the minimum heat exchanger flow area, A is the total heat transfer area, and L is the length of tube. In case of plane tube without fin, the calculation of mean fluid temperature and the surface temperature [20] can be expressed as:
Tmx
Tmi
qP x mc p
(11)
Tsx
Tmx
q h
(12)
Where P is the surface perimeter which, in the considered analysis, is D, Cp is the specific heat of air and is taken as constant value of 1.007 kJ / kg-K.
m = vA
(13)
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And the local heat transfer coefficient under constant heat flux at the wall of a smooth tube is given as
hx
q Tsx Tmx
(14)
3. Numerical solution procedure The finite volume models have been constructed using Gambit (Version 2.1.6) and subsequent analysis has been done by exporting the models to a commercial finite volume analysis package, Fluent 12.1. The governing equations solved by Fluent are the Navier-Stokes equations combined with the continuity equation, the thermal equation, and constitutive property relationships. Once the analyses are completed, the resulting data can be easily evaluated by the Fluent postprocessor. SIMPLE algorithm with a PRESTO scheme for the pressure velocity coupling was used for the pressure correction equation and the cell face values of pressure could be obtained from simple arithmetic averaging of centroid values. However, it is to be noted as the density becomes a function of temperature then the SIMPLE algorithm for pressure velocity coupling will not work properly because the pressure variation from cell to cell will not be smooth due to the presence of source term in the momentum equation. So in order to obtain the cell face pressure new pressure interpolation technique is needed which is available through body force weighted or PRESTO (Pressure Staggered Option) scheme in Fluent. The body-force-weighted scheme computes the face pressure by assuming that the normal gradient of the difference between pressure and body forces is constant. This works well if the body forces are known a priori in the momentum equations (e.g., buoyancy and axi-symmetric swirl calculations). The PRESTO (Pressure Staggering Option) scheme uses the discrete continuity balance for a "staggered'' control volume about the face to compute the "staggered'' (i.e., face) pressure. This procedure is similar in spirit to the staggered-grid schemes used with structured meshes. Note that for triangular, tetrahedral, hybrid, and polyhedral meshes, comparable accuracy is obtained using a similar algorithm. The PRESTO scheme for pressure interpolation is available for all meshes in Fluent. Under relaxation factors of 0.3 for pressure, 0.6 for momentum and 0.8 for temperature were used for the better convergence of all the variables. Hexahedral cells were used for the entire computational domain. Convergence of the discretized equations are said to have been achieved when the whole field residual for all the variables fell below 10-3 for u, v, w and p whereas for energy the residual level was kept at 10-6. 4. Results and discussion 4.1. Matching with analytical solution For both analytical and numerical solutions, following flow parameters are considered: tube length (L) 5 m, diameter (D) 0.07 m and the flow is considered to be laminar of Re = 1200. The tube wall is subjected to a constant wall heat flux of 200 W/m2. The analytical results (solution to equation 11, 12 and 14) for axial mean temperature, surface temperature distribution, and surface heat transfer coefficient have been compared with the CFD results (Fig.3, Fig. 4 and Fig. 5).
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500 480
Analytical Present CFD
460 440
TS (K)
Tm (K)
420 400 380 360 340 320 300 0.0
2
L = 5 m, D = 0.07 m, q = 200 W/m , Re = 1200 0.2
0.4
0.6
0.8
620 600 580 560 540 520 500 480 460 440 420 400 380 360 340 320 300 280
1.0
Analytical Presente CFD
2
2
Q=200 W / m , Re = 1200, L = 5 m, D = 0.07m 0.0
0.2
0.4
0.6
x/L
x/L
Fig. 3.
Fig. 4.
0.8
1.0
Fig. 3. Smooth tube mean fluid temperature distribution: a comparison of present CFD results with computed analytical solution. Fig. 4. Smooth tube surface temperature distribution: a comparison of present CFD results with computed analytical solution
6
P re s e n te d C F D A n a lytic a l
4
L = 5 m , D = 0 .0 7 m , 2 q = 2 0 0 W /m , R e = 1 2 0 0
x
2
h ( W \ m K)
5
3
2
1
0 0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
x /L Fig. 5. Smooth tube surface heat transfer coefficient: a comparison of present CFD results with computed analytical solution
It can be seen from the plot the present CFD results match well with the analytical results. It is evident from the Fig. 4 that the wall temperature distribution is almost linear after a tube length around 3 m (corresponding to x/L = 0.6 ) as the flow is thermally developed fully after this length and the present CFD results for wall temperature distribution matched well with the analytical solution.
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4.2. Effect of the fin number on surface temperature distribution Numerical analysis of predicting the surface temperature distribution as a function of fin number has been done under following flow parameters: total length (finned tube length+ air flow inlet section) = 5.1 m having 0.1 m inlet section, D = 0.07 m, fin length (Lf) =5 m, height (Hf) = 0.0125 m and thickness (Wf) = 0.008 m. The wall temperature distribution for fin less tube and tube having different fin number has been shown in Fig. 6. It can be seen from the plot lower temperature of the wall for fined tube compared to fin less tube due to higher heat transfer rate to the air flows inside the tube. As the fin number is increased the average wall temperature is found to be minimum (around 470 K) for a fin number 10 and after that it shoots up to 480 K when the fin number is increased to 12. It clearly indicates that the average heat transfer coefficient is maximum when the number of fins are attached to the internal surface of the tube is 10.
650
S m o o th tu b e n= 4 n = 6 n = 8 n = 10 n = 12
600 550
482
450
480 478
TSavg (K)
Ts (K)
500
400
476 474 472
L = 5 m , D = 0.07 m 470 2 q = 200 W /m , R e = 1200 4 6 8 H f =0.0125 m , W f = 0.008 m
350 300 0.0
0.2
0.4
0.6
4
6
8
10
12
F in n u m b e r
0.8
1.0
x/L Fig. 6. Tube surface temperature as a function of fin number
4.3. Effect of fin height on surface temperature distribution The surface temperature distribution as a function of fin height has been presented in Fig. 7. Increasing the depth of the fin results into decrease in the surface temperature and reaches minimum for a depth 0.026 m and after that the wall temperature does not fall any more ( TSavg is around 415 K) even we increase fin height. Increase of depth beyond 0.03 m is not possible due to dimension limitations of the tube.
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S.K. Rout et al. / Procedia Engineering 38 (2012) 585 – 597 g 600 0 .0 1 2 5 0 .0 1 7 0 0 .0 2 1 5 0 .0 2 6 0 0 .0 3 0 5
550
T (K) s
500
g
(
)
L = 5 .1 m , D = 0 .0 7 m 2 q = 2 0 0 W /m , R e = 1 2 0 0 n = 4 , W f = 0 .0 0 8 m
450 400 350 300 0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
x /L Fig.7. Wall surface temperature distribution as a function of fin height
4.4. The Effect of fin Profile on surface temperature distribution The shape of the fin profile plays a crucial role on wall temperature distribution which has been shown in Fig. 8. For the present numerical analysis three different types of fins (keeping fin volume constant) have been chosen having cross sectional area of rectangular, triangular and T-shaped. It is evident from the figure the wall temperature is minimum for triangular shape fin compared to other shape. This indicates that the heat transfer is highest in case of triangular fin. It seems that the average heat transfer coefficient is higher for triangular fin compared to other sections. 600
T s h a p e d fin T ria n g u la r s h a p e d fin R e c ta n g u la r s h a p e fin
550
Ts (K)
500
450
400
2
L = 5 m , q = 2 0 0 W /m , n = 4 , R e = 1 2 0 0 , D = 0 .0 7 m
350
300 0 .0
0 .2
0 .4
0 .6
x /L Fig. 8. Wall surface temperature distribution as a function of fin shape
0 .8
1 .0
S.K. Rout et al. / Procedia Engineering 38 (2012) 585 – 597
4.5. Effect of buoyancy on surface temperature distribution The effect of buoyancy on surface temperature has been done under same flow parameters as cited in para 4.3. The results have been presented in graphical representation in Fig.9. 600
T o p w a ll te m p e r a tu r e B a s e w a ll te m p e r a tu r e
550
450
s
T (K)
500
400
350
L = 5 m , D = 0 .0 7 m , n = 4 , 2 q = 200 W / m , R e = 1200
300 0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
x /L
Fig. 9. Buoyancy effect on the wall temperature distribution
It can be seen from Fig. 9 that there is a remarkable difference of the temperature distribution of top wall (corresponding to = 900) and bottom wall ( = 2700) of the tube. This probably occurs due to the differential buoyancy effect of air which gets more heated as it moves forward. Under the considered flow condition having Re as 1200, Grashof number (Gr) is computed as 7182781 and Richardson number works out to Ri (Gr/Re2) > 1, which clearly indicates the flow is dominated by the buoyancy effect. It is also seen from the figure the difference of wall temperature distribution is becoming more and more prominent as we proceed in the downstream of flow. So from the present investigation it can be concluded that due to the effect of the buoyancy the hot fluid moves to the top wall of the tube which causes higher wall temperature compared to bottom wall. 4.6. Variation of Local Nusselt number distribution along the downstream of flow Nusselt number variation as a function of axial distance of a finned tube has been shown in Fig. 10. It can be seen from the Fig. 10 that the local Nusselt number is initially very high at the inlet of the tube, signifying the dominance of convective heat transfer over conductive transfer and it decreases continuously along the downstream of flow and reaches a minimum value at around 4 m from the entrance of the tube and increases further towards the end of the tube. Due to the growth of thermal boundary layer, the Nusselt number decreases up to a certain distance suggesting the dominance of conduction heat transfer. However, after 4 m the Nusselt number increases again signifies the effect of convective heat transfer is becoming more pronounced due to the buoyancy effect. Even with variable thermal conductivity, the Nusselt number variation along the downstream of the finned tube shows the minimum value exists at a distance of 4 m from the entrance of the tube. Higher value of Nusselt number at the entrance of the finned tube for the case of variable thermal conductivity compared to average thermal conductivity shows that conduction heat transfer does not play any significant role at the beginning of the tube. However, towards the end of the tube the variable conductivity Nusselt number is lower compared to average conductivity Nusselt number which signifies the dominance of conductive transfer.
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L = 5 m , D = 0 .0 7 m , R e = 1 2 0 0 , q = 2 0 0 W / m n = 8 , H f = 0 .0 1 2 5 m , W f = 0 .0 0 8 m
5 .5
2
5 .0
K avg K v a r ia b le
Nux
4 .5 4 .0 3 .5 3 .0 2 .5 0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
x /L Fig. 10. Variation of local Nusselt number as a function of axial distance: A comparison of Nusselt number for average thermal conductivity and variable thermal conductivity
4.7. Variation of pressure drop along the downstream of the flow for different fin number The axial pressure distribution as shown in the Fig. 11 varies linearly along the length of the tube. The numerical investigation has been performed for finless tube and also for finned tube having fin number 4, 8 and 12. It can be clearly visualised from the plot for the same Reynolds number as the fin number increases the pressure drop is also becoming more and more. This signifies as the number of fins increases more and more pumping power is required to deliver same amount of fluid. 30 2
L = 5 m, D = 0.07 m, q =200 W/m , Re = 1200
27 24 21
n = 12
P/pv
2
18 15
n=8
12
n=4
9 6
smooth tube
3 0 0.0
0.2
0.4
x/L
0.6
0.8
1.0
Fig. 11. Variation of pressure drop as a function of axial distance for different fin number
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5. Conclusions The study concentrates on numerical investigation of wall temperature distribution of an internally finned tube and their validation with existing analytical results of axi-symmetry plain tube for axial temperature distribution, surface temperature distribution and the surface heat transfer coefficient. The following conclusions can be arrived from the present investigation: There exists an optimum fin number for which the wall temperature is minimum and the heat transfer is maximum and in the present case the optimum number of fin is found to be 10. It is also seen from the present investigation the top wall ( = 900) temperature distribution is higher compared to the bottom wall ( = 2700) due to buoyancy effect. There exists an optimum fin height of 0.026 m, where the wall temperature is found to be minimum and the rate of heat transfer is maximum after which the rate of heat transfer remains unchanged. For same volume it is seen that the wall temperature distribution is minimum for triangular shaped fin as compared to rectangular and T- shaped fin. The dominance of conduction heat transfer is found to be maximum at a distance of 4 m from the entrance of the finned tube. References [1] [2]
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
Kern QD, Kraus DA. Extended Surface Heat Transfer, McGraw-Hill, New York, 1972. Kundu BK, Das PK . Performance analysis and optimization of straight taper fins with variable heat transfer coefficient. International Journal of Heat and Mass Transfer 2002, 45: 4739–4751. Smit FJ, Meyer JP .R-22 and Zeotropic R-22/R-142b Mixture Condensation in Microfin, High-fin and twisted tape Insert tubes. ASME Journal of Heat Transfer 2002, 12:912-921. Arslanturk C .Performance analysis and optimization of a thermally non-symmetric annular fin. Int. Comm. Heat Mass Transfer 2004 , 31(8):1143-1153. Malekzadeh P, Rahideh H, Setoodeh AR .Optimization of non-symmetric convective-radiative annular fins by differential quadrature method. Energy Conversion and Management 2007, 48: 1671-1677. Brien O, Sohal JM .Heat Transfer Enhancement for Finned-Tube Heat Exchangers with winglets. ASME Journal of Heat Transfer 2005, 127:171-178. Aziz A, Fang .Alternative solutions for longitudinal fins of rectangular, trapezoidal, and concave parabolic profiles. Energy Conversion Management 2010, 51(11): 2188-2194. Bilir L, Ozerdem B, Erek A, Ilken Z .Heat Transfer and Pressure Drop Characteristics of Fin-Tube Heat Exchangers with Different Types of Vortex Generator Configurations.Journal of Enhanced heat transfer 2010, 17(3):243-256. Hussein A and Salman K .Free and force convection heat transfer in the thermal entry region for laminar flow inside a circular cylinder horizontally oriented. International Communications in Heat and Mass Transfer 2007, 48:2185-2194. Zhu Y. Simulation on the Laminar Flow and Heat Transfer in Four Basic Fins of Plate-Fin Heat Exchangers. ASME Journal of Heat Transfer 1996, 130(4):1-8. Ha MY, Kim JG .Numerical Simulation of Natural Convection in Annuli with Internal Fins. KSME International Journal 2004, 18(4):718—730. Tatsumi K, Yamaguchi M, Nishino Y, Nakabe K .Flow and Heat Transfer Characteristics of a Channel with Cut Fins. J. enhanced heat transfer 2010, 17(2):153-168. Sakalis VD, Hatzikonstantinou PM .Laminar Heat transfer in the Entrance Region of Internally Finned Ducts. ASME Journal of Heat Transfer 2001, 123:1030-1034. Haldar SC, Kochhar GS, Manohar K, Sahoo RK .Numerical study of laminar free convection about a horizontal cylinder with longitudinal ¿ns of ¿nite thickness. International Journal of Thermal Sciences 2007, 46:692–698. Duplain E, Baliga BR .Computational optimization of the thermal performance of internally finned ducts. International Journal of Heat and Mass Transfer 2009, 52:3929–3942. Liu X , Jensen MK. Geometry Effects on Turbulent Flow and Heat Transfer in internally finned tubes. ASME Journal of Heat Transfer 2001, 123:1036-1042. Jayakumara JS, Mahajani SM, Mandal JC, Vijayan P , Bhoi R. Experimental and CFD estimation of heat transfer in helically coiled heat exchangers. Chemical Engineering Research and Design 2008, 86(3): 221-232. Park K, Kim BS, Lim H-Jae, Han JiW, Park KO, Lee J, Yu K-Y. Performance Improvement in Internally Finned Tube by Shape Optimization. World Academy of Science, Engineering and Technolo 2007, 28:25-30. Dash SK. Heatline visualization in turbulent flow. Int. J. Numerical Methods for Heat and Fluid Flow 1996, 6(4), 37-46. Incropera P, Dwiet. Fundamental of Heat and Mass transfer, John Wiley & Sons, fifth edition, 2006.
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Procedia Engineering 38 (2012) 585 – 597
International Conference on Modelling, Optimisation and Computing (ICMOC 2012), April 10 - 11, 2012
CFD supported performance estimation of an internally finned tube heat exchanger under mixed convection flow S.K.Routa, D. N. Thatoia, A.K. Acharyaa, D. P. Mishraa,* a
S’O’A University, I. T. E.R., Jagamara, Bhubaneswar – 751 030, Odisha, India.
Abstract Results of a numerical analysis using Finite Volume Method of an internally finned axi-symmetric tube heat exchanger have been presented in this paper. The parametric study has been done using a computational fluid dynamics (CFD) program named FLUENT to estimate the performance of the heat exchanger with different fin shapes, sizes and numbers. The results obtained from the study for a steady and laminar flow of fluid under mixed flow convection heat transfer condition shows that there exists an optimum number for fins to keep the pipe wall temperature at a minimum. The wall temperature optimises at a definite fin height beyond which it is insensitive to any height variation. Moreover, amongst the three different shapes considered for fin, results show that wall temperature is least for triangular shaped fins, compared to rectangular and T-shaped fins. In addition to study of thermal characteristics, the pressure drop caused by presence of fins has also been studied.
© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Noorul Islam Centre for Higher Education ,NICHE, Kumarakoil – 629180 Tamil Nadu, India. Keywords: Internally finned tube; Wall temperature; Fin height; finite volume method; Fin profile
* Corresponding author. Tel.: +91-674-2350181; fax: +91-2351880. Email: [email protected]
1. Introduction Internally finned tubes commonly used in engineering applications as effective and efficient means to improve convective heat transfer perform differently depending on whether the flow is laminar or turbulent. For laminar flow and heat transfer, comprehensive experimental and numerical investigations have been performed for variable fluid properties, mixed convection and fin geometry. Heat transfer
1877-7058 © 2012 Published by Elsevier Ltd. doi:10.1016/j.proeng.2012.06.073
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S.K. Rout et al. / Procedia Engineering 38 (2012) 585 – 597
augmentation techniques play a vital role here since heat transfer coefficients are generally low for laminar flow in plain tubes. Designing a tubular heat exchanger with fins having different shapes and sizes is one such augmentation technique discussed in this paper. Nomenclature Cp specific heat at constant pressure D tube diameter Dh hydraulic diameter Gr Grashof number g acceleration of gravity Hf height of fin h average heat transfer coefficient hx local heat transfer coefficient k thermal conductivity L length of test section Lf length of fin Lx axial distance m Mass flow rate of air Nux local Nusselt number based hydraulic tube diameter n fin number p pressure P perimeter Pr Prandlt number q Uniform heat flux Re Reynolds number
Ri Tcx Tmi Tmx Tsavg Tsx U Vin V Wf
on
Richardson number local centerline fluid temperature inlet fluid temperature mean fluid temperature tube surface average temperature local tube surface temperature Velocity inlet velocity volume of tube width / thickness of fin
Greek Symbols fluid density ambient fluid density dynamic viscosity kinematic viscosity thermal diffusivity wall position Subscripts f fin in inlet
When an array of fins is used to enhance heat transfer, the prime focus is to optimise geometry of fins which will maximise the heat transfer rate under space and cost constraints. Extensive work has been carried out by different researchers to analyze heat transfer rate and pressure drop characteristics from tubes having fins of various shapes (rectangular, triangular, T-sectional, twisted). Experimental investigations have been performed by different researchers to study the friction factor, heat transfer rate, and pressure drop and temperature distribution of tube wall. Kern and Kraus [1] have identified longitudinal, radial and pin fins with straight profile as main fin geometries in external finned heat exchanger. Kundu and Das [2] later proposed straight taper fins with better heat transfer coefficients to weight ratio. Micro fins offer effective heat transfer and are more suitable as better heat exchanging medium than twisted tubes. Smit and Meyer [3] performed an experiment to compare three different (micro fins, twisted tapes and high fins) heat transfer enhancement methods to that of smooth tubes using the geotropic refrigerant mixture R-22/R-142b. The results corroborated the effectiveness of micro fins over twisted tubes. For non symmetric convective boundary conditions, two dimensional heat transfer equation is solved analytically [4] to obtain temperature profile in a finned tube. For a given volume, the optimized variables like the fin thickness and ratio of outer radius to inner radius of a fin have been determined. Malekzadeh et al. [5] optimized the shape of non-symmetric annular fins based on two nonlinear dimensional heat transfer analysis. They compared the results obtained through both differential quadrature method and finite difference method. Brien and Sohal [6] investigated experimentally forced convection heat transfer in a narrow rectangular duct fitted with a circular tube and/or a delta-winglet pair. They obtained a
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comparison of local and average heat transfer distributions for the circular tube with and without winglets and found at higher Reynolds numbers the enhancement level is close to 50%. Moreover, the combined effect of heat transfer rate and pressure drop depends on the enhancement of baseline winglet pair. Aziz and Fang [7] proposed an alternate solution for the energy equations for one-dimensional steady conduction in the longitudinal fins of three different cross sections namely rectangular, trapezoidal, and concave parabolic profile. The temperature and the heat flux is specified at the base of the fin and the temperature distribution in the fins are provided for these conditions. While fins offer solutions to augment heat exchange problems, they also result in pressure drop of flow. Experimental investigations show that the heat transfer characteristics and flow friction is greatly influenced by the fin spacing, size and shape of the fin. Bilir [8] investigated effects of vortex generators on heat transfer and pressure drop characteristics on fin-and-tube heat exchanger with three different types of vortex generators. The cumulative effect of vortex generators offers a better heat exchange solution with moderate drop in pressure. Hussein and Salman [9] conducted an experiment to study pressure drop and heat transfer characteristics of water flow in a horizontal tube with or without longitudinal fins and found that heat transfer enhancement is 16 times more and friction factor is 4.5 times more of a finned tube than a bare tube having Re < 4000. The fins existing in annuli influence the flow pattern, temperature distribution and heat transfer rate. Zhu [10] focussed on model and simulation of four types of basic fins (plain fin, strip fin, offset fin, perforated fin, wavy fin) considering fin thickness, thermal entry effect and end effect for 132.3
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investigation is started from matching the present CFD results with the existing analytical results for the case of two dimensional axi-symmetric smooth tubes. Then numerical analysis has been performed to determine the wall temperature distribution and average heat transfer coefficient of rectangular longitudinal fins mounted at the inner periphery of the tube. The flow of air through the tube is considered to be laminar with density taken as the function of temperature according to ideal gas relation and the wall is subjected to constant heat flux. Then the investigation is carried forward by changing the spacing between the fins, shape and size to analyse its heat transfer and fluid flow characteristics. 2. Mathematical modeling 2.1. Mathematical Formulation Fig. 1 gives the schematic representation of the internally finned tube for which the computational investigation is carried out .The tube has diameter D ( neglecting thickness) and length L. Air enters into the tube at one end where as the other end is exposed to the surrounding atmosphere. Longitudinal fins are placed symmetrically around inner periphery of the tube. The investigation is initiated with the rectangular fins of length Lf, width Wf and height, Hf as shown in Fig. 1 and subsequently the cross sectional area of the fin has been changed to triangular and T shapes. The flow field in the domain would be computed by using three-dimensional, incompressible Navier-Stokes equations (2D axi-symmetric model for simple tube) along with the energy equations. The fluid used in the simulation is air treated to be incompressible, enters the tube at with an inlet velocity of 0.264 m/s at a .temperature ranging from of 300 to 700 K.
Fig. 1. Schematic diagram of computational domain and the boundary condition applied to it
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2.2 Governing equations The governing equations for the above analysis can be written as: Continuity Equation
.( v ) 0
(1)
Momentum Equation D
Ui Dt
p xi
Ui xj
xj
Uj xi
g
(2)
The density is taken to be a function of temperature according to ideal gas law while laminar viscosity and thermal conductivity are kept constant. The Boussinesq approximation is not adopted in the model since the variation of with temperature is very high in the range of operating parameters. Energy Equation
D
T Dt
xi
T Pr xi
(3)
0
(4)
Conduction Equation
xi
k
T xi
2.3 Boundary Conditions The boundary conditions are mentioned in Fig. 1. The tube wall and fin are solid and have been given a no-slip boundary condition. Pressure outlet boundary condition has been imposed at the outlet of the tube and constant heat flux condition is applied on the wall of the tube. The velocity inlet boundary condition has been employed at the inlet face which supplies air into the tube. 2-D Axi-symmetric model has been used in case of fin less tube as in Fig. 2.At the pressure outlet boundary, the velocity will be computed from the local pressure field so as to satisfy the continuity but the scalar variable such as temperature T is computed from the zero gradient condition, Dash [19].
Fig. 2. Smooth tube 2-D axi-symmetric model
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2.4. Computations of some important heat transfer and fluid flow parameters The local Nusselt number for the finned heated tube is computed as
hx Dh kair
Nux
(5)
And the local heat transfer coefficient has been computed as given below
hx
q
(6)
Tsx Tcx
Where hx, Dh and kair are the local heat transfer coefficient, hydraulic diameter and thermal conductivity of air respectively. Tsx and Tcx are the local tube wall temperature and local centreline fluid temperature. The dimensionless number affecting the heat transfer and fluid flow are, The Reynolds number
Re
Vin Dh
(7)
air
The Grashof number Gr Gr
g
Tout Tin Dh3 (
2
air
)
(8)
The Richardson number:
Ri
Gr Re 2
(9)
The hydraulic diameter of the fin channel is defined as
Dh
4P Ac
(10)
Where Ac is the minimum heat exchanger flow area, A is the total heat transfer area, and L is the length of tube. In case of plane tube without fin, the calculation of mean fluid temperature and the surface temperature [20] can be expressed as:
Tmx
Tmi
qP x mc p
(11)
Tsx
Tmx
q h
(12)
Where P is the surface perimeter which, in the considered analysis, is D, Cp is the specific heat of air and is taken as constant value of 1.007 kJ / kg-K.
m = vA
(13)
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And the local heat transfer coefficient under constant heat flux at the wall of a smooth tube is given as
hx
q Tsx Tmx
(14)
3. Numerical solution procedure The finite volume models have been constructed using Gambit (Version 2.1.6) and subsequent analysis has been done by exporting the models to a commercial finite volume analysis package, Fluent 12.1. The governing equations solved by Fluent are the Navier-Stokes equations combined with the continuity equation, the thermal equation, and constitutive property relationships. Once the analyses are completed, the resulting data can be easily evaluated by the Fluent postprocessor. SIMPLE algorithm with a PRESTO scheme for the pressure velocity coupling was used for the pressure correction equation and the cell face values of pressure could be obtained from simple arithmetic averaging of centroid values. However, it is to be noted as the density becomes a function of temperature then the SIMPLE algorithm for pressure velocity coupling will not work properly because the pressure variation from cell to cell will not be smooth due to the presence of source term in the momentum equation. So in order to obtain the cell face pressure new pressure interpolation technique is needed which is available through body force weighted or PRESTO (Pressure Staggered Option) scheme in Fluent. The body-force-weighted scheme computes the face pressure by assuming that the normal gradient of the difference between pressure and body forces is constant. This works well if the body forces are known a priori in the momentum equations (e.g., buoyancy and axi-symmetric swirl calculations). The PRESTO (Pressure Staggering Option) scheme uses the discrete continuity balance for a "staggered'' control volume about the face to compute the "staggered'' (i.e., face) pressure. This procedure is similar in spirit to the staggered-grid schemes used with structured meshes. Note that for triangular, tetrahedral, hybrid, and polyhedral meshes, comparable accuracy is obtained using a similar algorithm. The PRESTO scheme for pressure interpolation is available for all meshes in Fluent. Under relaxation factors of 0.3 for pressure, 0.6 for momentum and 0.8 for temperature were used for the better convergence of all the variables. Hexahedral cells were used for the entire computational domain. Convergence of the discretized equations are said to have been achieved when the whole field residual for all the variables fell below 10-3 for u, v, w and p whereas for energy the residual level was kept at 10-6. 4. Results and discussion 4.1. Matching with analytical solution For both analytical and numerical solutions, following flow parameters are considered: tube length (L) 5 m, diameter (D) 0.07 m and the flow is considered to be laminar of Re = 1200. The tube wall is subjected to a constant wall heat flux of 200 W/m2. The analytical results (solution to equation 11, 12 and 14) for axial mean temperature, surface temperature distribution, and surface heat transfer coefficient have been compared with the CFD results (Fig.3, Fig. 4 and Fig. 5).
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500 480
Analytical Present CFD
460 440
TS (K)
Tm (K)
420 400 380 360 340 320 300 0.0
2
L = 5 m, D = 0.07 m, q = 200 W/m , Re = 1200 0.2
0.4
0.6
0.8
620 600 580 560 540 520 500 480 460 440 420 400 380 360 340 320 300 280
1.0
Analytical Presente CFD
2
2
Q=200 W / m , Re = 1200, L = 5 m, D = 0.07m 0.0
0.2
0.4
0.6
x/L
x/L
Fig. 3.
Fig. 4.
0.8
1.0
Fig. 3. Smooth tube mean fluid temperature distribution: a comparison of present CFD results with computed analytical solution. Fig. 4. Smooth tube surface temperature distribution: a comparison of present CFD results with computed analytical solution
6
P re s e n te d C F D A n a lytic a l
4
L = 5 m , D = 0 .0 7 m , 2 q = 2 0 0 W /m , R e = 1 2 0 0
x
2
h ( W \ m K)
5
3
2
1
0 0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
x /L Fig. 5. Smooth tube surface heat transfer coefficient: a comparison of present CFD results with computed analytical solution
It can be seen from the plot the present CFD results match well with the analytical results. It is evident from the Fig. 4 that the wall temperature distribution is almost linear after a tube length around 3 m (corresponding to x/L = 0.6 ) as the flow is thermally developed fully after this length and the present CFD results for wall temperature distribution matched well with the analytical solution.
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4.2. Effect of the fin number on surface temperature distribution Numerical analysis of predicting the surface temperature distribution as a function of fin number has been done under following flow parameters: total length (finned tube length+ air flow inlet section) = 5.1 m having 0.1 m inlet section, D = 0.07 m, fin length (Lf) =5 m, height (Hf) = 0.0125 m and thickness (Wf) = 0.008 m. The wall temperature distribution for fin less tube and tube having different fin number has been shown in Fig. 6. It can be seen from the plot lower temperature of the wall for fined tube compared to fin less tube due to higher heat transfer rate to the air flows inside the tube. As the fin number is increased the average wall temperature is found to be minimum (around 470 K) for a fin number 10 and after that it shoots up to 480 K when the fin number is increased to 12. It clearly indicates that the average heat transfer coefficient is maximum when the number of fins are attached to the internal surface of the tube is 10.
650
S m o o th tu b e n= 4 n = 6 n = 8 n = 10 n = 12
600 550
482
450
480 478
TSavg (K)
Ts (K)
500
400
476 474 472
L = 5 m , D = 0.07 m 470 2 q = 200 W /m , R e = 1200 4 6 8 H f =0.0125 m , W f = 0.008 m
350 300 0.0
0.2
0.4
0.6
4
6
8
10
12
F in n u m b e r
0.8
1.0
x/L Fig. 6. Tube surface temperature as a function of fin number
4.3. Effect of fin height on surface temperature distribution The surface temperature distribution as a function of fin height has been presented in Fig. 7. Increasing the depth of the fin results into decrease in the surface temperature and reaches minimum for a depth 0.026 m and after that the wall temperature does not fall any more ( TSavg is around 415 K) even we increase fin height. Increase of depth beyond 0.03 m is not possible due to dimension limitations of the tube.
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550
T (K) s
500
g
(
)
L = 5 .1 m , D = 0 .0 7 m 2 q = 2 0 0 W /m , R e = 1 2 0 0 n = 4 , W f = 0 .0 0 8 m
450 400 350 300 0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
x /L Fig.7. Wall surface temperature distribution as a function of fin height
4.4. The Effect of fin Profile on surface temperature distribution The shape of the fin profile plays a crucial role on wall temperature distribution which has been shown in Fig. 8. For the present numerical analysis three different types of fins (keeping fin volume constant) have been chosen having cross sectional area of rectangular, triangular and T-shaped. It is evident from the figure the wall temperature is minimum for triangular shape fin compared to other shape. This indicates that the heat transfer is highest in case of triangular fin. It seems that the average heat transfer coefficient is higher for triangular fin compared to other sections. 600
T s h a p e d fin T ria n g u la r s h a p e d fin R e c ta n g u la r s h a p e fin
550
Ts (K)
500
450
400
2
L = 5 m , q = 2 0 0 W /m , n = 4 , R e = 1 2 0 0 , D = 0 .0 7 m
350
300 0 .0
0 .2
0 .4
0 .6
x /L Fig. 8. Wall surface temperature distribution as a function of fin shape
0 .8
1 .0
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4.5. Effect of buoyancy on surface temperature distribution The effect of buoyancy on surface temperature has been done under same flow parameters as cited in para 4.3. The results have been presented in graphical representation in Fig.9. 600
T o p w a ll te m p e r a tu r e B a s e w a ll te m p e r a tu r e
550
450
s
T (K)
500
400
350
L = 5 m , D = 0 .0 7 m , n = 4 , 2 q = 200 W / m , R e = 1200
300 0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
x /L
Fig. 9. Buoyancy effect on the wall temperature distribution
It can be seen from Fig. 9 that there is a remarkable difference of the temperature distribution of top wall (corresponding to = 900) and bottom wall ( = 2700) of the tube. This probably occurs due to the differential buoyancy effect of air which gets more heated as it moves forward. Under the considered flow condition having Re as 1200, Grashof number (Gr) is computed as 7182781 and Richardson number works out to Ri (Gr/Re2) > 1, which clearly indicates the flow is dominated by the buoyancy effect. It is also seen from the figure the difference of wall temperature distribution is becoming more and more prominent as we proceed in the downstream of flow. So from the present investigation it can be concluded that due to the effect of the buoyancy the hot fluid moves to the top wall of the tube which causes higher wall temperature compared to bottom wall. 4.6. Variation of Local Nusselt number distribution along the downstream of flow Nusselt number variation as a function of axial distance of a finned tube has been shown in Fig. 10. It can be seen from the Fig. 10 that the local Nusselt number is initially very high at the inlet of the tube, signifying the dominance of convective heat transfer over conductive transfer and it decreases continuously along the downstream of flow and reaches a minimum value at around 4 m from the entrance of the tube and increases further towards the end of the tube. Due to the growth of thermal boundary layer, the Nusselt number decreases up to a certain distance suggesting the dominance of conduction heat transfer. However, after 4 m the Nusselt number increases again signifies the effect of convective heat transfer is becoming more pronounced due to the buoyancy effect. Even with variable thermal conductivity, the Nusselt number variation along the downstream of the finned tube shows the minimum value exists at a distance of 4 m from the entrance of the tube. Higher value of Nusselt number at the entrance of the finned tube for the case of variable thermal conductivity compared to average thermal conductivity shows that conduction heat transfer does not play any significant role at the beginning of the tube. However, towards the end of the tube the variable conductivity Nusselt number is lower compared to average conductivity Nusselt number which signifies the dominance of conductive transfer.
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L = 5 m , D = 0 .0 7 m , R e = 1 2 0 0 , q = 2 0 0 W / m n = 8 , H f = 0 .0 1 2 5 m , W f = 0 .0 0 8 m
5 .5
2
5 .0
K avg K v a r ia b le
Nux
4 .5 4 .0 3 .5 3 .0 2 .5 0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
x /L Fig. 10. Variation of local Nusselt number as a function of axial distance: A comparison of Nusselt number for average thermal conductivity and variable thermal conductivity
4.7. Variation of pressure drop along the downstream of the flow for different fin number The axial pressure distribution as shown in the Fig. 11 varies linearly along the length of the tube. The numerical investigation has been performed for finless tube and also for finned tube having fin number 4, 8 and 12. It can be clearly visualised from the plot for the same Reynolds number as the fin number increases the pressure drop is also becoming more and more. This signifies as the number of fins increases more and more pumping power is required to deliver same amount of fluid. 30 2
L = 5 m, D = 0.07 m, q =200 W/m , Re = 1200
27 24 21
n = 12
P/pv
2
18 15
n=8
12
n=4
9 6
smooth tube
3 0 0.0
0.2
0.4
x/L
0.6
0.8
1.0
Fig. 11. Variation of pressure drop as a function of axial distance for different fin number
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5. Conclusions The study concentrates on numerical investigation of wall temperature distribution of an internally finned tube and their validation with existing analytical results of axi-symmetry plain tube for axial temperature distribution, surface temperature distribution and the surface heat transfer coefficient. The following conclusions can be arrived from the present investigation: There exists an optimum fin number for which the wall temperature is minimum and the heat transfer is maximum and in the present case the optimum number of fin is found to be 10. It is also seen from the present investigation the top wall ( = 900) temperature distribution is higher compared to the bottom wall ( = 2700) due to buoyancy effect. There exists an optimum fin height of 0.026 m, where the wall temperature is found to be minimum and the rate of heat transfer is maximum after which the rate of heat transfer remains unchanged. For same volume it is seen that the wall temperature distribution is minimum for triangular shaped fin as compared to rectangular and T- shaped fin. The dominance of conduction heat transfer is found to be maximum at a distance of 4 m from the entrance of the finned tube. References [1] [2]
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
Kern QD, Kraus DA. Extended Surface Heat Transfer, McGraw-Hill, New York, 1972. Kundu BK, Das PK . Performance analysis and optimization of straight taper fins with variable heat transfer coefficient. International Journal of Heat and Mass Transfer 2002, 45: 4739–4751. Smit FJ, Meyer JP .R-22 and Zeotropic R-22/R-142b Mixture Condensation in Microfin, High-fin and twisted tape Insert tubes. ASME Journal of Heat Transfer 2002, 12:912-921. Arslanturk C .Performance analysis and optimization of a thermally non-symmetric annular fin. Int. Comm. Heat Mass Transfer 2004 , 31(8):1143-1153. Malekzadeh P, Rahideh H, Setoodeh AR .Optimization of non-symmetric convective-radiative annular fins by differential quadrature method. Energy Conversion and Management 2007, 48: 1671-1677. Brien O, Sohal JM .Heat Transfer Enhancement for Finned-Tube Heat Exchangers with winglets. ASME Journal of Heat Transfer 2005, 127:171-178. Aziz A, Fang .Alternative solutions for longitudinal fins of rectangular, trapezoidal, and concave parabolic profiles. Energy Conversion Management 2010, 51(11): 2188-2194. Bilir L, Ozerdem B, Erek A, Ilken Z .Heat Transfer and Pressure Drop Characteristics of Fin-Tube Heat Exchangers with Different Types of Vortex Generator Configurations.Journal of Enhanced heat transfer 2010, 17(3):243-256. Hussein A and Salman K .Free and force convection heat transfer in the thermal entry region for laminar flow inside a circular cylinder horizontally oriented. International Communications in Heat and Mass Transfer 2007, 48:2185-2194. Zhu Y. Simulation on the Laminar Flow and Heat Transfer in Four Basic Fins of Plate-Fin Heat Exchangers. ASME Journal of Heat Transfer 1996, 130(4):1-8. Ha MY, Kim JG .Numerical Simulation of Natural Convection in Annuli with Internal Fins. KSME International Journal 2004, 18(4):718—730. Tatsumi K, Yamaguchi M, Nishino Y, Nakabe K .Flow and Heat Transfer Characteristics of a Channel with Cut Fins. J. enhanced heat transfer 2010, 17(2):153-168. Sakalis VD, Hatzikonstantinou PM .Laminar Heat transfer in the Entrance Region of Internally Finned Ducts. ASME Journal of Heat Transfer 2001, 123:1030-1034. Haldar SC, Kochhar GS, Manohar K, Sahoo RK .Numerical study of laminar free convection about a horizontal cylinder with longitudinal ¿ns of ¿nite thickness. International Journal of Thermal Sciences 2007, 46:692–698. Duplain E, Baliga BR .Computational optimization of the thermal performance of internally finned ducts. International Journal of Heat and Mass Transfer 2009, 52:3929–3942. Liu X , Jensen MK. Geometry Effects on Turbulent Flow and Heat Transfer in internally finned tubes. ASME Journal of Heat Transfer 2001, 123:1036-1042. Jayakumara JS, Mahajani SM, Mandal JC, Vijayan P , Bhoi R. Experimental and CFD estimation of heat transfer in helically coiled heat exchangers. Chemical Engineering Research and Design 2008, 86(3): 221-232. Park K, Kim BS, Lim H-Jae, Han JiW, Park KO, Lee J, Yu K-Y. Performance Improvement in Internally Finned Tube by Shape Optimization. World Academy of Science, Engineering and Technolo 2007, 28:25-30. Dash SK. Heatline visualization in turbulent flow. Int. J. Numerical Methods for Heat and Fluid Flow 1996, 6(4), 37-46. Incropera P, Dwiet. Fundamental of Heat and Mass transfer, John Wiley & Sons, fifth edition, 2006.
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