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Numerical and experimental study of natural convection heat transfer characteristics for vertical plate fin and tube heat exchangers with various tube diameters
International Journal of Heat and Mass Transfer 100 (2016) 320–331
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Numerical and experimental study of natural convection heat transfer characteristics for vertical plate fin and tube heat exchangers with various tube diameters Han-Taw Chen a,⇑, Yung-Shiang Lin a, Pin-Chun Chen a, Jiang-Ren Chang b a b
Department of Mechanical Engineering, National Cheng Kung University, Tainan 701, Taiwan Department of Systems Engineering and Naval Architecture, National Taiwan Ocean University, Keelung 202, Taiwan
a r t i c l e
i n f o
Article history: Received 17 February 2016 Received in revised form 11 April 2016 Accepted 13 April 2016 Available online 10 May 2016 Keywords: New correlations CFD Inverse method Natural convection Plate fin and tube
a b s t r a c t This study uses three-dimensional computational fluid dynamics commercial package along with experimental data and various flow models to investigate the natural convection heat transfer and fluid flow characteristics of a single-tube vertical plate fin and tube heat exchangers for various values of fin spacing and tube diameter. Temperature and velocity distributions of air between the two fins, fin temperature and heat transfer coefficient on the fins are determined using FLUENT along with various flow models. The inverse method in conjunction with the finite difference method and the experimental temperature data is applied to determine the fin temperature and heat transfer coefficient for the smaller tube. More accurate results can be obtained, if the heat transfer coefficient obtained is close to the inverse results and matches existing correlations. The numerical results of the fin temperature at the selected measurement locations also coincide with the experimental temperature data. The results show that RNG k–e turbulence model is more suitable for this problem than laminar flow model. Three proposed new correlations between the Nusselt number and the Rayleigh number are in good agreement with the inverse results and numerical results obtained. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Many experimental and numerical methods have been proposed to obtain the natural-convection heat transfer and fluid flow characteristics of plate-finned tube heat exchangers [4,7–9,18–22 ,27,28]. The air is heated at the tube and moves upward due to the density difference caused by the heating. The upwardmoving natural convection boundary layer flow is induced by two adjacent fins. The boundary layer begins to develop upward from the bottom of the heating horizontal tube. Its thickness increases along the circumference of the tube. After that, it is again heated by the heating tube. A plume of heated air over a heating tube can form a low velocity wake region. This means that there exhibits very complex three-dimensional heat transfer and fluid flow characteristics within the fins of the plate-finned tube heat exchangers due to a plume of heated air over a heating horizontal tube in natural convection. Therefore, it is necessary to pay attention to the interaction of heat transfer and fluid flow distributions in the ambient air region within the plate-finned tube heat ⇑ Corresponding author: Tel.: +886 6 2757575x62157; fax: +886 6 2352973. E-mail address: [email protected] (H.-T. Chen). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.04.039 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
exchanger. These complex flow patterns can be interesting to study the mechanism of heat transfer enhancement. The heat transfer coefficient is highest in the bottom region of the tube and is lowest in the top region of the tube. These phenomena may lead to local variations in the heat transfer coefficient on the fins. In other words, the heat transfer coefficient on the fins is very nonuniform. This means that the steady-state heat transfer coefficient on the fins should be a function of position. It is seen in Ref. [1] that the measurements of the local heat transfer coefficient on the plate fins under steady-state conditions are very difficult to perform because the local fin temperature and heat flux are required. Thus, the estimation of a more accurate heat transfer coefficient on the fins is an important task for the device of the high-performance heat exchangers. It is known that the physical quantity and the surface condition can be predicted using the measured temperature in the test material. Thus, the inverse heat conduction problems have become an interesting topic recently. So far, various inverse methods along with the measured temperature in the test material have been developed to analyze the inverse heat conduction problem [2,3]. Yildiz and Yuncu [4] experimentally explored the performance of annular fins on a horizontal cylinder in free convection with
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Nomenclature Af lateral surface area of the fin (m2) cl, c1e, c2e coefficients in turbulent model specific heat of air cp d0 outer diameter of the circular tube (mm) fp fin spacing (mm) gi gravitational acceleration components (m/s2) production of turbulent kinetic energy due to buoyancy Gb Gk production of turbulent kinetic energy due to the velocity gradient h local heat transfer coefficient on the fin (W/m2K) h average heat transfer coefficient on the fin (W/m2K) h heat transfer coefficient under the situation of T0 (W/ b m2K) h heat transfer coefficient in the jth sub-fin region (W/ j m2K) k turbulent kinetic energy ka thermal conductivity of air (W/m2K) L length and width of the fin (m) N number of sub-fin regions Nx, Ny, Nz number of grid points in x-, y- and z-directions Pr Prandtl number Prt turbulent Prandtl number Q total heat rate dissipated from the entire fin (W) S coefficient (21/2Sij) Sij mean strain rate tensor (2Sij = @ui/@xj + @uj/@xi) outer boundary surface of the circular tube St T fin temperature (K)
various fin spacings and diameters at low temperature differences. Huang et al. [5,6] applied the steepest descent method and a general purpose commercial code CFX4.4 to determine the local heat transfer coefficient for the plate finned-tube heat exchangers. Chen and Chou [7], Chen and Hsu [8], Chen et al. [9] and Chen and Lai [10] used the inverse method along with finite difference method, least squares method and experimental temperature data to predict the fin efficiency and average heat transfer coefficient on the fins of the plate fin and tube heat exchanger with a single tube and four tubes in staggered arrangements, respectively. It is found that the resulting estimate in Refs. [7–10] are in good agreement with those obtained from the correlation [11]. Recently, Chen et al. [12] applied three-dimensional computational fluid dynamics commercial package FLUENT [13] in conjunction with the experimental temperature data and inverse results [10] to estimate the heat transfer and fluid flow characteristics of the two-row plate fin and tube heat exchangers in a staggered arrangement with various air speeds and fin spacings for both laminar flow and RNG k–e turbulence models. It is found in Ref. [12] that more accurate results can be obtained, if the heat transfer coefficient obtained is close to the inverse results and matches existing correlations. Furthermore, the fin temperature measured at the selected locations also coincides with the experimental temperature data. Under such circumstances, the number of grid points may also need to change with fin spacing and air speed. Thus, this study applies the inverse method to estimate the heat transfer coefficient of this problem with a smaller tube diameter. A commercial package FLUENT [13] along with estimates obtained and appropriate flow model and grid points is used to determine the natural convection heat transfer and fluid flow characteristics of the plate fin and tube heat exchanger with a single tube. Mon and Gross [14] applied three-dimensional numerical study to investigate the effect of fin spacing on four-row annular-finned tube bundles in a staggered arrangement. Heat transfer and fluid
Ta Tj T0 T1 t ui x, y, z
air temperature (K) measured fin temperature at the jth measurement location (K) fin base temperature (K) ambient air temperature (K) fin thickness (m) air velocity in the i-direction (m/s) spatial coordinates (m)
Greek symbols ae, ak parameters in Eqs. (21) and (22) b volumetric thermal expansion coefficient bt parameter in RNG k–e model dij, dj2 Kronecker delta functions e viscous dissipation rate of turbulence kinetic energy g mean flow fields (Sk/e) g0 parameter m laminar kinematic viscosity (m2/s) meff effective kinematic viscosity (m2/s) mt turbulent kinematic viscosity (m2/s) q air density (kg/m3) Superscripts a in the air region f in the fin region mea measured data
flow characteristics are obtained using FLUENT in conjunction with RNG k–e turbulence model. However, the flow between the two fins is considered laminar. The other parts of the tube bundle are considered to be turbulent. They believed that the effect of the total number of grid points on the numerical results was small. 50,000–99,000 cells are used to discretize computational domain. Jang et al. [15,16] and Jang and Chen [17] obtained fluid flow and heat transfer characteristics of the fin and tube heat exchanger under the assumption of laminar flow between two fins. Kayansayan [18] studied the natural convection of air over an annular finned tube and observed the effects of the fin spacing, fin diameter to tube diameter ratio and Rayleigh number on the heat transfer. The finned-tube diameter is 24 mm and the tube length is 600 mm. From Ref. [18], the heat transfer coefficient increases with increasing ratio of diameter. Sajedi et al. [19] used experimental and numerical study on the optimal fin numbering in an external extended finned tube heat exchanger. Qiu et al. [20] applied three-dimensional numerical method to investigate the effects of fin angle, fin surface emissivity and tube wall temperature on heat transfer enhancement for a longitudinal externally-finned tube placed vertically in a small chamber. The numerical model is validated through comparison with experimental measurements and the appropriateness of general boundary conditions is examined. The results show that both convection and radiation heat transfer modes are important. Yaghoubi and Mahdavi [21] applied the control volume scheme to investigate natural convection heat transfer from a horizontal cooled finned tube. The thickness of the aluminum fins is 0.4 mm. Several tests are made between grid dimensions of 80-40-8, 120-80-16 and 170-130-20 nodes along the radial, tangential and axial directions, respectively. The results in terms of average convective heat transfer show that a 120-80-16 grid distribution is suitably fine to ensure a grid-independent solution with a difference less than 3% with the minimum grid size. Kumar et al. [22] used the software OpenFOAM-2.2 based on finite
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volume approach to investigate the transient 3D numerical simulations of natural convection of air around a circular finned tube with the annular plain fins kept in a small chimney. The fin spacing and fin diameter are varied from 2–12 mm and from 35–50 mm, respectively. The diameter and length of the circular tube are 24.9 mm and 610 mm, respectively. Perhaps, the change in temperature of the aluminum fins is very small so that the temperature measurements are not shown in the work of Yaghoubi and Mahdavi [21]. The walls of the box are specified as adiabatic boundary condition. The ceiling is set at a constant temperature of 300 K. The cylinder is assumed to be at a constant temperature of 340 K. The air in the enclosure is maintained at an initial temperature of 300 K. It is seen in Ref. [22] that the tube and fins are assumed to be at a constant temperature. For practical engineering problems, this assumption cannot be easy to do. Thus, Kumar et al. [22] did not carry out a comparison between numerical and experimental fin temperatures at the selected locations in order to validate the accuracy of the results obtained. The fin heat conduction equation is not solved. In other words, the problem studied by Kumar et al. [22] is not a conjugate problem. It is found in Refs. [14–21] that independence of grid points and laminar flow are assumed. It is known that quantitative studies of the heat transfer processes and the reliability are an important concept in the industrial applications. This implies that a more accurate prediction of the heat transfer coefficient is an important work for the device of the high-performance heat exchangers. Thus, this study applies the commercial package FLUENT [13] in conjunction with the experimental temperature data and inverse results to obtain free convection heat transfer and fluid flow characteristics of the present problem for two different tube diameters. The resulting heat transfer coefficient is compare with the correlation obtained from the current textbook [11] and the existing results. A comparison between the numerical and experimental fin temperatures is also made at the selected measurement locations. In order to understand the suitability of the grid independence assumption, the effect of grid points on the results obtained is studied for various values of the fin spacing and tube diameter. To the best of our knowledge, a few researchers used a commercial package FLUENT
Fig. 2. Schematic diagram of the physical model and the computational domain with dashed lines for d0 = 2 mm. (a) x–y plane, (b) y–z plane.
along with the inverse method to determine the heat transfer and fluid flow characteristics of this problem. 2. Formulation of inverse scheme
Fig. 1. Physical geometry of this study with various measurement locations and sub-fin regions for d0 = 2 mm.
Fig. 1 shows the physical geometry of the single-tube plate fin and tube heat exchanger. Parameters d0 and L represent the outer diameter of the tube and fin length, respectively. The center of the tube is located at (L/2, L/2). The fin temperature data at the selected locations and the ambient air temperature are measured from the present experimental apparatus for d0 = 2 mm. The twodimensional inverse heat conduction problem is introduced to
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obtain an estimate of the present study. The inverse method along with the finite difference method, the experimental temperature data and least squares method is applied to predict the unknown heat transfer coefficient on the fins of a single-tube plate fin and tube heat exchanger for d0 = 2 mm and various fin spacings. Unknown heat transfer coefficient is assumed to be nonuniformly distributed so that the entire fin is divided into several sub-fin regions. The heat transfer coefficient in each sub-fin region is assumed to be an unknown constant. The schematic diagram of this study with various measurement locations and the sub-fin regions can refer to Fig. 3 of Ref. [8] for d0 = 27.3 mm. The physical geometry with measurement locations and sub-fin regions is shown in Fig. 1 for d0 = 2 mm. To avoid duplication, a detailed diagram of the experimental apparatus and the experimental method can refer to Ref. [8]. Under the assumption of a thin fin, the temperature gradient in the z-direction (the fin thickness) can be disregarded. Thus, the fin temperature varies only in the x- and y-directions. In addition, the surface area of the fin is small when compared to the total fin surface area. This implies that the actual heat transfer rate dissipated through the fin tip is rather small compared to the total heat transfer rate drawn from the fin base. Thus, the boundary condition at the edge surface of the fin is
323
assumed to be insulated [5–10,12]. Under the assumption of steady state and constant thermal properties, the heat conduction equation for a thin fin can be expressed as
@ 2 T @ 2 T 2hðx; yÞ þ ¼ ðT T 1 Þ @x2 @y2 kf t
ð1Þ
Its corresponding boundary conditions are
@T ¼ 0 at x ¼ 0 and x ¼ L @x
ð2Þ
@T ¼ 0 at y ¼ 0 and y ¼ L @y
ð3Þ
T ¼ T 0 ðx; yÞ on St
ð4Þ
where x and y are the Cartesian coordinates. L and t denote the length and thickness of the square fin, respectively. h(x, y) is the unknown heat transfer coefficient. kf is the thermal conductivity of the fin. T0 is the fin base temperature or the outer surface temperature of the circular tube. T1 denotes the ambient air temperature. To predict the unknown heat transfer coefficient on the fins, the whole plate fin considered is divided into N sub-fin regions. The application of the finite difference method to Eq. (1) can produce the difference equation in the kth sub-fin region as
T iþ1;j 2T i;j þ T i1;j T i;jþ1 2T i;j þ T i;j1 þ ¼ mk T i;j ‘2 ‘2
for k ¼ 1; 2; . . . ; N ð5Þ
where ‘ is the distance between two neighboring nodes in the xk and y-directions and is defined as ‘ = L/(Nx 1) = L/(Ny 1). m denotes the unknown dimensionless parameter on the kth sub-fin =ðkftÞ. In the definition of ‘ k = 2L2 h region and is defined as m k k , Nx and Ny are the nodal numbers in the x and y directions, and m represents the average heat transfer coefficient in respectively. h k the kth sub-fin region. The difference equation for the node at the boundary surface, the interface of two adjacent sub-fin regions and in the vicinity of the tube is similar to that shown in Ref. [8]. To avoid repetition, their derivative process can be found in Ref. [8]. Rearrangement of Eq. (5) and the difference equation at the boundary surface, the interface between two adjacent sub-fin regions and in the vicinity of the tube yields the following matrix equation.
½A½T ¼ ½F
Fig. 3. Schematic diagram of the physical model and the computational domain with dashed lines for d0 = 27.3 mm. (a) x–y plane, (b) y–z plane.
ð6Þ
where [A] is a global conduction matrix. [T] is a matrix representing the nodal temperatures. [F] is a force matrix. The nodal temperatures are obtained from Eq. (6) using Gaussian elimination. Inverse analysis of this study can refer to Refs. [7–10]. In order to avoid repetition, its detailed procedures are not shown in this manuscript. The heat transfer behavior on the fins or local heat transfer and flow distributions between the two fins may be interfered significantly if an obtrusive thermocouple is used in the fin. This means that the fin temperature measurements can be inherently difficult in order not to interrupt the heat transfer behavior. Thus, T-type thermocouple is used to measure the fin temperature only at selected measurement locations [7–10]. The error range of T-type thermocouple is ±0.4%. The heat transfer rate dissipated from the jth sub-fin region Qj, average heat transfer coefficient on the fins heat transfer coefficient based on the fin base temperature h h, b and total heat transfer rate dissipated from the plate fin to the ambient Q can be obtained using these estimated results of the average heat transfer coefficient in each sub-fin region. The Qj, h, hb and Q values can be expressed as
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Q i 2Aj hj ðT mea T 1 Þ for j ¼ 1; 2; . . . ; N j h
N X hj Aj =Af
ð7Þ
ð8Þ
j¼1
and
Q¼
N X Q j ¼ 2Af hb ðT 0 T 1 Þ ¼ 2Af hðT av e T 1 Þ
ð9Þ
j¼1
where Af is the lateral surface area of the plate fin and is defined as Af = L2 – pd20/4. Tmea represents the experimental temperature of the j fin at the jth measurement location. Tave denotes the average of all the fin temperatures at the selected measurement locations. From is obtained from Eq. (8). This heat transfer Ref. [8], the value of h coefficient may include convective and radiative heat transfer coef and h in this study are obtained ficients. However, the values of h b from Eq. (9) for comparison with the results obtained using FLUENT. The fin efficiency gf is defined as the ratio of the actual total heat transfer rate from the fin to the dissipated heat from the fin maintained at the fin base temperature T0 and can be expressed as
gf ¼
Q 2Af ðT 0 T 1 Þh
¼
hb h
ð10Þ
3. Experimental apparatus The experiments have been carried out in a box with an upward opening, 550 mm long, 450 mm wide and 300 mm high by Chen and Chou [8] for d0 = 27.3 mm. The schematic diagram of this experimental apparatus can refer to Fig. 1 of Ref. [8]. In order to study the effect of tube diameter on the resulting estimates, experiments for d0 = 2 mm are performed in a box with an upward opening, 120 mm long, 120 mm wide and 120 mm high. The schematic diagram of the box with an upward opening for d0 = 2 mm is similar to that for d0 = 27.3 mm. Thus, a schematic diagram of an upwardly opening box made from acrylic-plastic sheets for d0 = 2 mm is not displayed in this manuscript. Test fins are made of AISI 304 stainless material. Their size is 0.1 m long, 0.1 m wide and 1 mm thick for d0 = 27.3 mm [8] and 0.05 m long, 0.05 m wide and 1 mm thick for d0 = 2 mm. It is found in Ref. [23] that the thermal conductivity of AISI 304 stainless material is 14.9 W/mK. The emissivity of the fin measured using FT-IR Spectrum 100 (Perkin Elmer Co., Ltd) is 0.18. The horizontal tube is placed on wooden supports, which is 98 mm above the bottom plate to prevent the ground effect. Thus, the distance between the top surface and the upper surface of the circular cylinder is 174.7 mm for d0 = 27.3 mm. The ambient air temperature and the fin temperature are measured using T-type thermocouples. Test fins are heated about 2 h using 200 W of power input. All the data signals are collected and converted by a data acquisition system (National Instruments NI SCXI-1000, 1102, 1300). The limit of error of the thermocouple is ±0.4%. The experimental methods and procedures are similar to those shown in Ref. [8]. Four thermocouples are fixed at four different positions of the interface between the fin and the circular tube using a satlon cyanoacrylate adhesive. A thermocouple is placed at 1.75 cm away from the test fins and y = 18 cm in order to measure the ambient air temperature T1 for d0 = 2 mm. Measurement of ambient air temperature T1 refers to Ref. [8] for d0 = 27.3 mm. The whole fin is divided into eight sub-fin regions, i.e. N = 8. Eight thermocouples are fixed at the selected locations of the fins in order to determine the fin temperature data, as shown in Fig. 1. The diameter of the spot size of the thermocouple is about 0.13 mm.
The least squares minimization technique is applied to minimize the sum of the squares of the deviations between the calculated and measured fin temperatures at selected locations of the j are found in Ref. fins. Details of estimating the unknown value h [8]. In order to avoid repetition, they are not shown in this manuscript. The computational procedures of this study are repeated mea inv T T until the values of j T mea j for j = 1, 2,. . ., N are all less than 105, j is the calculated temperature of the fin at the jth meawhere Tinv j valsurement location and is determined from Eq. (6). Once the h j h and g can ues for j = 1, 2,. . ., N are determined, the values of h, b
f
be obtained from Eqs. (8)-(10). 4. Three-dimensional numerical analysis The main purpose of this study is to use FLUENT along with two different flow models to determine the natural convection heat transfer and fluid flow characteristics of the single-tube plate fin and tube heat exchanger for two different diameters. It is seen in Ref. [24] that a comparison of the numerical predictions and experimental data for the hydrodynamic and thermal fields in a twochannel plate heat exchanger using laminar and two-equation turbulence models is made. The friction factors, Nusselt numbers, the outlet temperatures of the two streams and the temperature distributions on the channel end-plates are compared. The buoyancy effect is not considered taken into account. Their results show that realizable k–e model with non-equilibrium wall functions is used to obtain the closest result compared to the experimental data. Due to higher fin base temperature compared to ambient air temperature, the buoyancy effect is considered in this study. In order to validate the accuracy of the results obtained using the laminar flow and RNG k–e turbulence models, the fin temperature and heat transfer coefficient obtained are compared with the experimental temperature data at the selected measurement locations and inverse results, respectively. Under the assumptions of the steady-state and constant thermal properties, the threedimensional heat conduction equation for the thin fin can be expressed as
@2T ¼0 @x2i
ð11Þ
The boundary conditions at x = 0, y = 0, x = L, y = L and on the outer surface of the tube are the same as Eqs. (2)-(4). The boundary conditions at z = 0 and t/2 are given as
@T ¼ 0 at z ¼ 0 @z
ð12Þ
and
kf
@T ¼ hðx; yÞðT T 1 Þ at z ¼ t=2 @z
ð13Þ
where xi, i = 1, 2, 3, respectively indicate x, y and z. x, y and z are the Cartesian coordinates. h(x, y) denotes the local heat transfer coefficient on the fins. 4.1. Laminar flow model Thermal properties of the ambient air are assumed to be constant. The air flow for the entire flow region is assumed to be three-dimensional, incompressible, laminar and steady and to have no viscous dissipation. The continuity, momentum and energy equations can be expressed in tensor form as
@ui ¼0 @xi
ð14Þ
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uj
@ui 1 @p @ 2 ui ¼ þ m 2 þ g j bdj2 ðT a T 1 Þ @xj q @xj @xj
ð15Þ
and
c p uj
@T a ka @ 2 T a @ui @uj @ui ¼ þm þ @xj q @xj @xj @xj @xi @xj
ð16Þ
where ui, p, gj and Ta are the velocity components, pressure, gravitational acceleration components and the air temperature, respectively. b is the volumetric thermal expansion coefficient. dj2 is the Kronecker delta. q, m, cp and ka are the density, kinematic viscosity, specific heat and thermal conductivity of air, respectively. They are all assumed to be constant. 4.2. RNG k–e turbulence model The air flow for the entire flow region is assumed to be threedimensional, symmetric, turbulent and steady and to have no viscous dissipation. The continuity equation is the same as Eq. (14). The momentum and energy equations may be expressed as: [12,13,25]
@u0i u0j
@ui 1 @p @ @ui @uj uj ¼ þm þ þ g j bdj2 ðT a T 1 Þ @xj @xj @xi @xj q @xi @xj
4.3. Boundary conditions kin and ein are the intrinsic values of commercial package FLUENT and are given as kin = 0.5 and ein = 0.5 for RNG k–e turbulence model. The numerical results can be sensitive to the choice of kin and ein. The surface temperature of the tube T0 is assumed to be constant. Only the half-section of the fins is selected for all calculations. The computational domain is half of the box and is shown in Fig. 2 for d0 = 2 mm and Fig. 3 for d0 = 27.3 mm in dashed lines. The gradient of all dependent variables is set to zero at the side walls and a bottom wall. The boundary condition at the top of the box is set to the ambient temperature T1. At the solid surfaces, no-slip conditions are specified. The fin temperatures at various measurement locations, T0 and T1 are obtained from this experiment. The temperature and heat flux matching conditions at the fin-fluid interface are written as
T ¼ Ta
R
ð17Þ
c p uj
@u0j T 0a
@T a ka @ T a @ ¼ cp þ 2meff ðui Sij Þ @xj @xj q @x2j @xj
ð18Þ
2 u0i u0j ¼ 2mt Sij kdij 3
ð19Þ
and
ð20Þ
where dij is Kronecker delta function. Prt is the turbulent Prandtl number. The k and e equations in the RNG k–e turbulence model with buoyancy effects are expressed as:
@k @ @k þ Gk þ Gb qe ¼q ak meff @xj @xj @xj
ð21Þ
and
quj
@e @ @e e þ c1e ðGk þ c3e Gb Þ ¼q ae meff @xj @xj @xj k
qcl g3 ð1 g=g0 Þ e2 e2 c 2e q 3 1 þ bt g k k
ð24Þ
and
Z hðT T 1 ÞdA
ð25Þ
is obtained from Eq. (24) for FLUENT and is where the value of h and g used to compare with the approximate inverse result. h b f values are calculated from Eqs. (9) and (10) for FLUENT, respectively. In order to compare the present results with those obtained from the correlation, it is found from Ref. [11] that correlations of Nusselt number Nu and Rayleigh number Ra for square isothermal fins without tube, 0.2 < Ra < 4 104 and Pr = 0.71 and the annular circular isothermal fins with supporting cylinder, 3 < Ra < 104 and 1.67 < f = D/d0 < 1 are respectively expressed as
2
cp mt @T a u0j T 0a ¼ Prt @xj
ð23Þ
ðT T 1 Þhðx; yÞdA R ðT T 1 ÞdA Af
Af
where the effective kinematic viscosity meff is defined as meff = m + mt. The turbulent kinematic viscosity mt is defined as mt = clk2/e. The mean strain rate tensor Sij is defined as Sij = (əui/əxj + əuj/əxi)/2. The Reynolds stress tensor and turbulent heat fluxes in conjunction with Boussinesq approximation are given as
quj
Af
Q ¼2 2
@T @T a ¼ ka @z @z
and the value of Q for The average heat transfer coefficient h commercial software FLUENT can be approximated as
h¼
and
and kf
Ra0:89 Nu ¼ 4 18
31=2:7
!2:7 þ ð0:62Ra
0:25 2:7 5
Þ
for square fin
ð26Þ
and
Nu ¼
Ra ½2 expðC Þ expðb1 C Þ for annular circular fin 12p ð27Þ
fp=ka and Ra ¼ gbðT 0 T 1 Þfp3 ð fp Þ. where Nu and Ra are defined as Nu = h b L ma The parameter L⁄ is L for an isothermal square rectangular fin and D for an isothermal annular circular fin. Eq. (26) does not consider the effect of the tube diameter on Nu value. It is seen in Ref. [11] that h b
ð22Þ
where the parameters Gk, Gb and g are the production of turbulent kinetic energy due to the velocity gradient, the production of turbulent kinetic energy due to the buoyancy and the ratio of characteristic time scale of turbulence and the mean flow field, respectively. The definitions of Gk, Gb and g can refer to Refs. [12,13]. Parameters ak, ae, Prt, cl, c1e, c2e, g0 and bt are given as ak = ae = 1.393, Prt = 0.85, cl = 0.0845, c1e = 1.42, c2e = 1.68, g0 = 4.38 and bt = 0.012. The coefficient S is defined as S = 21/2Sij. Parameter c3e is 0 due to the air velocity perpendicular to the gravity.
value is calculated by Eq. (26) with an equivalent diameter D = 1.23L for square fins. Properties in Eqs. (26) and (27) are evaluated at the wall temperature. Parameters b1, C1 and C⁄ are defined as
b1 ¼ 0:17=f þ e4:8=f " # 1=2 4=3 23:7 1:1ð152=f2 Þ C1 ¼ 1 þ b1
ð28Þ ð29Þ
and
C ¼ ðC 1 =RaÞ3=4
ð30Þ
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5. Typical grid distribution An unstructured grid system containing a non-uniform distribution of the grid points is used to determine all of the numerical results. The grid density is controlled to ensure proper results with various values of the fin spacing. A finer mesh is needed n some regions where the temperature and velocity gradients are high, such as in the vicinity of the interface between the tube and fins. In some regions where the temperature and velocity gradients are small, a coarser mesh can be used to save computing time. The grid system used in this study is shown in Fig. 4. The number of grid points in the x, y and z directions of the fins, respectively, is taken from 14–15, 14 and 3–5 for d0 = 2 mm, and is 11, 8 and 3 for d0 = 27.3 mm. The number of grid points between the two fins is similar to that in the x and y direction of the fins for d0 = 2 mm and 27.3 mm. The number of grid points between the two fins in the z-direction is taken from 13–23 for d0 = 2 mm and from 7–10 for d0 = 27.3 mm. A coarser mesh is used in the remainder of this region to save computing time. The number of grid points outside the fins is 19 and 19 in the x and y directions, respectively, and is taken from 46–63 in the z direction for d0 = 2 mm. The number of grid points outside the fins is 23 and 12 in the x and y directions, respectively, and is taken from 36–38 in the z direction for d0 = 27.3 mm. The total number of grid points is from 21,339– 28,035 for d0 = 2 mm and from 9900–11,544 for d0 = 27.3 mm. Various grid points are to be tested for each case until the fin temperature and estimates obtained are as close to the experimental
temperature data, the inverse results and existing correlations as possible.
6. Results and discussion All physical properties are evaluated at the film temperature or the average of the fin base temperature and ambient temperature. The experimental temperature data are obtained from Refs. [8] and [26] for d0 = 2.73 cm. The commercial package FLUENT [13] is applied to determine the fluid flow and heat transfer characteristics of this problem. The measurement locations and sub-fin regions are set to 8 for L = 0.1 m and d0 = 27.3 mm and 6 for L = 0.05 m and d0 = 2 mm. Tables 1 shows a comparison between the results obtained using the laminar flow and RNG k–e turbulence models and the experimental temperature data and inverse results for fp = 15 mm and various d0 values. It is found from Table 1 that the fin temperatures at selected measurement locations and the average heat transfer coefficient obtained using RNG k–e turbulence model are closer to the experimental data and inverse results than those obtained using the laminar flow model for the different tube diameters. The L/d0 and Ra values are, respectively, 30.75 and 1301.1 for d0 = 2 mm and 5.68 and 1164.15 for d0 = 27.3 mm. The results show that RNG k–e turbulence model is more suitable for this problem than the laminar flow model even for L/d0 = 30.75 and Ra = 1301.1. Therefore, the following results are obtained using RNG k–e turbulence model. Based on the results obtained, the proposed correlations can be expressed as
2
Ra0:89 Nu ¼ 4 18
31=2:7
!2:7 þ ð0:62Ra
0:25 2:7 5
Þ
1
d0 L
ð31Þ
and
Nu ¼
Ra 0:65d0 ½2 expðC Þ expðb1 C Þ 1 12p L
ð32Þ
The proposed new correlation of Nub Ra1/3 can be expressed as
d0 Nu ¼ 0:255 þ 0:28Ra1=3 1 L
Fig. 4. Typical grid distribution in the computational domain. (a) d0 = 2 mm, (b) d0 = 27.3 mm.
ð33Þ
It is worth mentioning that the properties in Eqs. (31)-(33) are evaluated at the film temperature. It is seen that the relationship between Nu and Ra shown in Eq. (35) of Ref. [8] does not consider the effect of the tube diameter and is obtained from the inverse result. Thus, Eqs. (31)-(33) are different from the correlation given by Chen and Chou [8]. To further verify the reliability and availability of the present and h values between the results results, a comparison of the h b obtained using FLUENT and inverse results is made. Table 2 shows and h with various d and fp values. It is seen a comparison of h b 0 and h values obtained using FLUENT are in from Table 2 that h b good agreement with the inverse results for 10 mm 6 fp 6 20 mm values deviate from those and a single fin. However, the h b obtained from the correlations (26) and (27) with d0 = 2 mm, and are closer to those obtained from the correlation (27) than the correlation (26) with d0 = 27.3 mm. This means that the correlation (27) has better accuracy than the correlation (26) for this problem. It is found that the heat transfer coefficient given by Kayansayan [18] and Kumar et al. [22] is not compared with the correlation and h (27) in Ref. [11]. Table 2 shows the obtained results of h b increase with decreasing the tube diameter. This result is similar to that shown in Ref. [18].
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H.-T. Chen et al. / International Journal of Heat and Mass Transfer 100 (2016) 320–331 Table 1 Comparison of Tj with fp = 15 mm and various d0 values for various flow models. d0 = 2 mm, Ra = 1301.1 T0 = 315.82 K, T1 = 300.49 K
d0 = 27.3 mm, Ra = 1493.59 T0 = 336.86 K, T1 = 299.0 K
Fluent
Measured
Laminar
RNG
T1 = 309.06 T2 = 309.92 T3 = 309.12 T4 = 308.32 T5 = 309.02 T6 = 308.39
T1 = 304.16 T2 = 305.22 T3 = 304.22 T4 = 304.22 T5 = 305.40 T6 = 304.26
6.07
23.44
Fluent RNG
T1 = 305.23 T2 = 305.93 T3 = 305.50 T4 = 304.50 T5 = 304.97 T6 = 304.64
T1 = 318.08 T2 = 322.75 T3 = 318.09 T4 = 319.03 T5 = 319.06 T6 = 314.36 T7 = 317.07 T8 = 314.36
T1 = 312.45 T2 = 320.64 T3 = 312.47 T4 = 312.82 T5 = 312.89 T6 = 307.24 T7 = 311.41 T8 = 307.26
T1 = 314.24 T2 = 322.25 T3 = 314.08 T4 = 312.21 T5 = 311.94 T6 = 307.14 T7 = 309.65 T8 = 305.04
22.38 (Inv)
6.99
17.14
14.45 (Inv)
Tj (K)
h
Table 2 and h with various d and fp values. Comparison of h b 0 d0 (mm)
fp (mm)
Exp. data [25]
Laminar
Table 4 with various d and fp values. Comparison of h b 0
(W/m2K) h
(W/m2K) h b
Inv
FLUENT
Inv
FLUENT
Eq. (26)
Eq. (27)
d0 (mm)
2
10 15 20 1
18.1 21.95 22.96 23.06
16.83 21.4 25.16 27.07
6.26 6.64 6.72 6.73
5.16 5.71 6.13 6.32
6.59 6.66 6.71 –
5.51 5.91 6.05 –
27.3
10 15 20 1
11.68 14.05 16.56 17.89
9.33 13.06 17.48 19.93
4.45 4.85 5.18 5.34
3.93 4.66 5.04 5.4
6.84 6.89 7.02 –
5.15 5.54 5.77 –
Table 3 shows the effect of grid points on the fin temperature measurements at selected locations and the heat transfer coefficient for fp = 15 mm and various d0 values. It is seen that the effect of grid points on the heat transfer coefficient and fin temperature measurements cannot be ignored. Thus, various grid points are tested for each case until the fin temperature and heat transfer coefficient obtained are as close to the experimental temperature data and the inverse results as possible. for various values of d and Table 4 shows the comparison of h b 0 fp. An important finding is that hb values are in good agreement with those obtained from the proposed correlations (31)-(33) with d0 = 2 mm and 27.3 mm for 10 mm 6 fp 6 20 mm. Thus, the proposed correlations (31)-(33) can contribute to the design of values practical plate finned-tube heat exchanger. However, the h b obtained from the correlation (33) may be slightly closer to the inverse and FLUENT results than those obtained from the
fp (mm)
Ra
(W/m2K) h b Eq. (31)
Eq. (32)
Eq. (33)
Q (W)
gf
2
10 15 20
262.7 1301.1 4200.5
6.35 6.40 6.45
5.42 5.77 5.91
5.32 5.72 6.18
0.418 0.436 0.485
0.26 0.24 0.25
27.3
10 15 20
295.7 1431.9 4864.2
5.15 5.10 5.23
4.45 4.67 4.86
4.49 4.70 5.14
3.516 3.351 4.035
0.33 0.27 0.25
correlations (31) and (32). Table 4 also shows the variation of the fin efficiency gf with fp value for d0 = 2 mm and 27.3 mm. The gf value is determined based on the results obtained using FLUENT. It is found in Table 4 that the gf value decreases with increasing fin spacing. However, it can be improved with increasing tube diameter. The numerical results in Figs. 5–8 are obtained using the RNG k–e turbulence model. The air velocity pattern and the temperature contour in x–y plane at z = fp/2 away from the surface of the intermediate fin are, respectively, shown in Figs. 5 and 6 for fp = 15 mm and d0 = 2 mm and 27.3 mm. It is seen from Fig. 5 that the air inside the box is heated at the cylinder tube and moves upward due to the density difference caused by the heating. The upward-moving natural convection boundary layer flow is induced by two adjacent fins. It is seen from Fig. 5(b) boundary layer begins to develop upward from the bottom of the heating tube. Its thickness increases along the circumference of the tube. This is basically attributed to the up-moving boundary layer flow induced by fins.
Table 3 and h values for fp = 15 mm and various values of d . Effect of grid points on Tj, h b 0 Nxf Nyf Nzf Nza d0 = 2 mm
d0 = 27.3 mm
9 11 3 15
11 11 3 30
11 11 5 30
11 8 3 9
11 11 4 9
21 21 4 9
T1 T2 T3 T4 T5 T6 T7 T8 h
308.35 308.91 308.23 307.85 308.29 307.7 –
308.15 308.89 308.16 307.56 308.21 307.59 –
308.27 308.96 308.21 307.61 308.26 307.69 –
5.41
5.35
5.23
313.12 320.91 313.19 313.61 313.64 308.73 313.14 308.76 14.11
311.03 316.71 311.08 312.47 312.31 308.5 312.16 308.51 16.64
310.37 316.7 310.38 311.96 311.96 308.04 312 308.06 17.07
h b
3.15
3.06
2.98
4.29
4.77
4.76
328
H.-T. Chen et al. / International Journal of Heat and Mass Transfer 100 (2016) 320–331
Fig. 5. Velocity pattern in the middle between the two fins with fp = 15 mm. (a) d0 = 2 mm, (b) d0 = 27.3 mm.
Then, it moves along the side walls and reaches to the lower part of the box. After that, it is again heated by the heating circular tube. A plume of heated air over a heating tube will form a low velocity wake region. The two large natural re-circulating eddies are formed within the two adjacent fins in the x–y plane. Fig. 5 along with Fig. 3 of Ref. [8] and Fig. 1 shows that the two natural recirculations within two adjacent fins are located in the 1st and 3rd sub-fin regions for d0 = 27.3 mm and 4T = 37.86 K, where 4T denotes the difference between the ambient temperature and the fin base temperature. In other words, these two re-circulations occur in two adjacent fins and the gap between the top surface of the box and tube, and are located about 45° from the horizontal line for d0 = 27.3 mm. However, the two re-circulations are
observed in the vicinity of the left and right side of the horizontal tube for d0 = 2 mm with 4T = 15.33 K. The two re-circulations are symmetric on both sides of the circular tube and rotate in opposite directions for d0 = 2 mm and 27.3 mm. A similar phenomenon can be observed from Ref. [27]. The larger the value of 4T value, the stronger the buoyancy. Accordingly, the re-circulation can be moved upwardly due to the increase of the value of 4T. This means that the re-circulation position may depend on the value of 4T. The existence of such an oncoming flow is stronger and reinforces the buoyancy that is induced by the tube base itself. This strong circulation helps to transfer the heat from the neighboring fin surfaces to the ambient air through the top surface of the box. Thus, there exhibits very complex three-dimensional heat transfer and fluid
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329
Fig. 7. Heat transfer coefficient distribution on the fins with fp = 15 mm. (a) d0 = 2 mm, (b) d0 = 27.3 mm.
Fig. 6. Temperature contour in the middle between the two fins with fp = 15 mm. (a) d0 = 2 mm, (b) d0 = 27.3 mm.
flow characteristics within the fins of the plate-finned tube heat exchangers for this problem. It is seen in Fig. 6 that the temperature contour shows a nonuniform distribution within the two adjacent fins in the x–y plane. The low-temperature region is seen in the plume region. A higher temperature and temperature gradient are observed in the vicinity of the fin base. This indicates that there is a high rate of heat transfer in this region. Fig. 7 shows the distribution of the heat transfer coefficient on the fin with fp = 15 mm for d0 = 2 mm and 27.3 mm. From Fig. 3 of Ref. [8] and Figs. 1 and 5, the 6th and 8th sub-fin regions are located in the oncoming flow regions for d0 = 27.3 mm. The 4th–6th sub-fin regions are located in the oncoming flow regions for d0 = 2 mm. The 2nd sub-fin region is located in the plume region for d0 = 2 mm and 27.3 mm. It is found from Table 1 that the temperature measurements in 6th and 8th sub-fin regions are lower than those in other sub-fin regions for d0 = 27.3 mm. The temperature measurements in 4th and 6th sub-fin regions
are lower than other sub-fin regions for d0 = 2 mm. The temperature measurement in the 2nd sub-fin region is higher than that in other sub-fin regions with d0 = 2 mm and 27.3 mm. The temperature measurement in the downstream sub-fin regions is higher than that in the oncoming sub-fin regions for d0 = 27.3 mm. However, the difference of the fin temperature measurements are small for d0 = 2 mm because less heat is transferred to the fins. In other words, the fin temperature distribution for d0 = 2 mm is more uniform than that for d0 = 27.3 mm. This result indicates that the heat transfer coefficient on the fins is also a uniform distribution with d0 = 2 mm, as shown in Fig. 7a. It is seen that the difference between the fin temperature and ambient air temperature in the plume region is higher than that in other sub-fin regions with d0 = 2 mm and 27.3 mm. It is seen from Fig. 5(b) that plume region over a heating tube belongs to a low-speed wake region. Therefore, the heat transfer coefficient is highest in the bottom region of the tube and is lowest in the top region of the tube. As shown in Table 1 of Ref. [8] and Fig. 7(b), the heat transfer coefficient in the plume region is lower than that the oncoming flow region. These results indicate that the physical phenomenon of the obtained results is reasonable.
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points may need to vary with the fin spacing. This means that the effect of the total number of grid points on the numerical results obtained cannot be ignored. The commercial package, FLUENT, along with inverse results and the experimental temperature data can applied to obtain a more accurate heat transfer and fluid flow characteristics of the plate fin and tube heat exchangers. Acknowledgements The authors gratefully acknowledge the financial support provided by the National Science Council of the Republic of China under Grant No. NSC 102-2221-E-006-177-MY3. We would also like to thank Professor Chin-Hsiang Cheng at National Cheng Kung University for providing us access to the computational fluid dynamics software FLUENT. References
Fig. 8. Variation of Nu with Ra1/3 for d0 = 2 mm and 27.3 mm.
The variation of Nu with Ra1/3 is shown in Fig. 8 for d0 = 2 mm and 27.3 mm. From Fig. 8, Eq. (33) is consistent with the inverse results and numerical results obtained using FLUENT. Nu value increases with decreasing d0 value under the same Ra value. This value increases with decreasing the tube diammeans that the h b eter, as shown in Table 2. 7. Conclusions This study proposes three-dimensional computational fluid dynamics commercial package, FLUENT, along with the laminar flow and RNG k–e turbulence models, the experimental temperature data and inverse results to determine the natural convection heat transfer and fluid flow characteristics of a single circular tube plate finned-tube heat exchanger for various values of fp and d0. An important finding is that the RNG k–e turbulence model is more suitable than laminar flow for this problem. This conclusion is different from that shown in the previous works [14–21]. It is seen that the two re-circulations occur in the two adjacent fins and in the gap between the top surface of the box and the tube and are located about 45° from the horizontal line for d0 = 27.3 mm. However, the fluid re-circulations within the two adjacent fins are observed in the vicinity of the left and right of a horizontal tube with d0 = 2 mm. The recirculation position may depend on the difference between ambient temperature and the fin base tempera and h values obtained using FLUENT are in good ture. The h b agreement with the inverse results for 10 mm 6 fp 6 20 mm and value obtained is in good agreement with the a single fin. The h b existing correlations with d0 = 2 mm and 27.3 mm. However, the value obtained from the proposed correlations (31)-(33) is cloh b ser to the inverse result than that obtained from the existing correlations for 10 mm 6 fp 6 20 mm and a single fin. Thus, the proposed correlations have better accuracy than the existing correlations shown in Ref. [11]. In order to obtain a more accurate result, the selection of appropriate flow model along with appropriate grid points may be important. Meanwhile, the total number of grid
[1] R.L. Webb, Principles of Enhanced Heat Transfer, Wiley, New York, 1994, pp. 125–153. [2] K. Kurpisz, A.J. Nowak, Inverse Thermal Problems, Computational Mechanics Publications, Southampton, UK, 1995. [3] M.N. Özisik, Heat Conduction, 2nd ed., Wiley, New York, 1993. Chap. 14. [4] S. Yildiz, H. Yuncu, An experimental investigation on performance of annular fins on a horizontal cylinder in free convection heat transfer, Heat Mass Transfer 40 (2004) 239–251. [5] C.H. Huang, I.C. Yuan, H. Ay, A three-dimensional inverse problem in imaging the local heat transfer coefficients for plate finned-tube heat exchangers, Int. J. Heat Mass Transfer 46 (2003) 3629–3638. [6] C.H. Huang, I.C. Yuan, H. Ay, An experimental study in determining the local heat transfer coefficients for the plate finned-tube heat exchangers, Int. J. Heat Mass Transfer 52 (2009) 4883–4893. [7] H.T. Chen, J.P. Song, Y.T. Wang, Prediction of heat transfer coefficient on the fin inside one-tube plate finned-tube heat exchangers, Int. J. Heat Mass Transfer 48 (2005) 2697–2707. [8] H.T. Chen, Jui-Che Chou, Investigation of natural-convection heat transfer coefficient on a vertical square fin of finned-tube heat exchangers, Int. J. Heat Mass Transfer 49 (2006) 3034–3044. [9] H.T. Chen, W.L. Hsu, Estimation of heat transfer coefficient on the fin of annular-finned tube heat exchangers in natural convection for various fin spacings, Int. J. Heat Mass Transfer 50 (2007) 1750–1761. [10] H.T. Chen, J.R. Lai, Study of heat-transfer characteristics on the fin of two-row plate finned-tube heat exchangers, Int. J. Heat Mass Transfer 50 (2012) 45–57. [11] G.D. Raithby, K.G.T. Hollands, Natural convection, Chap. 4, in: W.M. Rohsenow, J.P. Hartnett, Y.I. Cho (Eds.), Handbook of Heat Transfer, third ed., McGrawHill, 1998. Chap. 4. [12] H.T. Chen, C.H. Lu, Y.S. Huang, K.C. Liu, Estimation of fluid flow and heat transfer characteristics for two-row plate-finned tube heat exchangers with experimental data, Heat Mass Transfer 52 (2016) 969–979. [13] FLUENT Dynamics Software, FLUENT, Lehanon, NH-USA, 2010. [14] M.S. Mon, U. Gross, Numerical study of fin-spacing effects in annular-finned tube heat exchangers, Int. J. Heat Mass Transfer 47 (2004) 1953–1964. [15] J.Y. Jang, M.C. Wu, W.J. Chang, Numerical and experimental studies of threedimensional plate-fin and tube heat changers, Int. J. Heat Mass Transfer 39 (1996) 3057–3990. [16] J.Y. Jang, L.F. Hsu, J.S. Leu, Optimization of the span angle and location of vortex generators in a plate-fin and tube heat exchanger, Int. J. Heat Mass Transfer 67 (2013) 432–444. [17] J.Y. Jang, L.K. Chen, Numerical analysis of heat transfer and fluid flow in a three-dimensional wavy-fin and tube heat changer, Int. J. Heat Mass Transfer 40 (1997). 3981-3066. [18] N. Kayansayan, Thermal characteristics of fin-and-tube heat exchanger cooled by natural convection, Exp. Therm. Fluid Sci. 7 (1993) 177–188. [19] R. Sajedi, M. Taghilou, M. Jafari, Experimental and numerical study on the optimal fin numbering in an external extended finned tube heat exchanger, Appl. Therm. Eng. 83 (2015) 139–146. [20] Y. Qiu, M. Tian, Z. Guo, Natural convection and radiation heat transfer of an externally-finned tube vertically placed in a chamber, Heat Mass Transfer 49 (2013) 405–412. [21] M. Yaghoubi, M. Mahdavi, An investigation of natural convection heat transfer from a horizontal cooled finned tube, Exp. Heat Transfer 26 (2013) 343–359. [22] A. Kumar, J.B. Joshi, A.K. Nayak, P.K. Vijayan, 3D CFD simulations of air cooled condenser-II: Natural draft around a single finned tube kept in a small chimney, Int. J. Heat Mass Transfer 92 (2016) 507–522. [23] V.S. Arpaci, S.H. Kao, A. Selamet, Introduction to Heat Transfer, Prentice-Hall, NJ, 1999, pp. 202–205. [24] E. Velayati, M. Yaghoubi, Numerical study of convective heat transfer from an array of parallel bluff plates, Int. J. Heat Fluid Flow 26 (2005) 80–91.
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Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Numerical and experimental study of natural convection heat transfer characteristics for vertical plate fin and tube heat exchangers with various tube diameters Han-Taw Chen a,⇑, Yung-Shiang Lin a, Pin-Chun Chen a, Jiang-Ren Chang b a b
Department of Mechanical Engineering, National Cheng Kung University, Tainan 701, Taiwan Department of Systems Engineering and Naval Architecture, National Taiwan Ocean University, Keelung 202, Taiwan
a r t i c l e
i n f o
Article history: Received 17 February 2016 Received in revised form 11 April 2016 Accepted 13 April 2016 Available online 10 May 2016 Keywords: New correlations CFD Inverse method Natural convection Plate fin and tube
a b s t r a c t This study uses three-dimensional computational fluid dynamics commercial package along with experimental data and various flow models to investigate the natural convection heat transfer and fluid flow characteristics of a single-tube vertical plate fin and tube heat exchangers for various values of fin spacing and tube diameter. Temperature and velocity distributions of air between the two fins, fin temperature and heat transfer coefficient on the fins are determined using FLUENT along with various flow models. The inverse method in conjunction with the finite difference method and the experimental temperature data is applied to determine the fin temperature and heat transfer coefficient for the smaller tube. More accurate results can be obtained, if the heat transfer coefficient obtained is close to the inverse results and matches existing correlations. The numerical results of the fin temperature at the selected measurement locations also coincide with the experimental temperature data. The results show that RNG k–e turbulence model is more suitable for this problem than laminar flow model. Three proposed new correlations between the Nusselt number and the Rayleigh number are in good agreement with the inverse results and numerical results obtained. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Many experimental and numerical methods have been proposed to obtain the natural-convection heat transfer and fluid flow characteristics of plate-finned tube heat exchangers [4,7–9,18–22 ,27,28]. The air is heated at the tube and moves upward due to the density difference caused by the heating. The upwardmoving natural convection boundary layer flow is induced by two adjacent fins. The boundary layer begins to develop upward from the bottom of the heating horizontal tube. Its thickness increases along the circumference of the tube. After that, it is again heated by the heating tube. A plume of heated air over a heating tube can form a low velocity wake region. This means that there exhibits very complex three-dimensional heat transfer and fluid flow characteristics within the fins of the plate-finned tube heat exchangers due to a plume of heated air over a heating horizontal tube in natural convection. Therefore, it is necessary to pay attention to the interaction of heat transfer and fluid flow distributions in the ambient air region within the plate-finned tube heat ⇑ Corresponding author: Tel.: +886 6 2757575x62157; fax: +886 6 2352973. E-mail address: [email protected] (H.-T. Chen). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.04.039 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
exchanger. These complex flow patterns can be interesting to study the mechanism of heat transfer enhancement. The heat transfer coefficient is highest in the bottom region of the tube and is lowest in the top region of the tube. These phenomena may lead to local variations in the heat transfer coefficient on the fins. In other words, the heat transfer coefficient on the fins is very nonuniform. This means that the steady-state heat transfer coefficient on the fins should be a function of position. It is seen in Ref. [1] that the measurements of the local heat transfer coefficient on the plate fins under steady-state conditions are very difficult to perform because the local fin temperature and heat flux are required. Thus, the estimation of a more accurate heat transfer coefficient on the fins is an important task for the device of the high-performance heat exchangers. It is known that the physical quantity and the surface condition can be predicted using the measured temperature in the test material. Thus, the inverse heat conduction problems have become an interesting topic recently. So far, various inverse methods along with the measured temperature in the test material have been developed to analyze the inverse heat conduction problem [2,3]. Yildiz and Yuncu [4] experimentally explored the performance of annular fins on a horizontal cylinder in free convection with
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321
Nomenclature Af lateral surface area of the fin (m2) cl, c1e, c2e coefficients in turbulent model specific heat of air cp d0 outer diameter of the circular tube (mm) fp fin spacing (mm) gi gravitational acceleration components (m/s2) production of turbulent kinetic energy due to buoyancy Gb Gk production of turbulent kinetic energy due to the velocity gradient h local heat transfer coefficient on the fin (W/m2K) h average heat transfer coefficient on the fin (W/m2K) h heat transfer coefficient under the situation of T0 (W/ b m2K) h heat transfer coefficient in the jth sub-fin region (W/ j m2K) k turbulent kinetic energy ka thermal conductivity of air (W/m2K) L length and width of the fin (m) N number of sub-fin regions Nx, Ny, Nz number of grid points in x-, y- and z-directions Pr Prandtl number Prt turbulent Prandtl number Q total heat rate dissipated from the entire fin (W) S coefficient (21/2Sij) Sij mean strain rate tensor (2Sij = @ui/@xj + @uj/@xi) outer boundary surface of the circular tube St T fin temperature (K)
various fin spacings and diameters at low temperature differences. Huang et al. [5,6] applied the steepest descent method and a general purpose commercial code CFX4.4 to determine the local heat transfer coefficient for the plate finned-tube heat exchangers. Chen and Chou [7], Chen and Hsu [8], Chen et al. [9] and Chen and Lai [10] used the inverse method along with finite difference method, least squares method and experimental temperature data to predict the fin efficiency and average heat transfer coefficient on the fins of the plate fin and tube heat exchanger with a single tube and four tubes in staggered arrangements, respectively. It is found that the resulting estimate in Refs. [7–10] are in good agreement with those obtained from the correlation [11]. Recently, Chen et al. [12] applied three-dimensional computational fluid dynamics commercial package FLUENT [13] in conjunction with the experimental temperature data and inverse results [10] to estimate the heat transfer and fluid flow characteristics of the two-row plate fin and tube heat exchangers in a staggered arrangement with various air speeds and fin spacings for both laminar flow and RNG k–e turbulence models. It is found in Ref. [12] that more accurate results can be obtained, if the heat transfer coefficient obtained is close to the inverse results and matches existing correlations. Furthermore, the fin temperature measured at the selected locations also coincides with the experimental temperature data. Under such circumstances, the number of grid points may also need to change with fin spacing and air speed. Thus, this study applies the inverse method to estimate the heat transfer coefficient of this problem with a smaller tube diameter. A commercial package FLUENT [13] along with estimates obtained and appropriate flow model and grid points is used to determine the natural convection heat transfer and fluid flow characteristics of the plate fin and tube heat exchanger with a single tube. Mon and Gross [14] applied three-dimensional numerical study to investigate the effect of fin spacing on four-row annular-finned tube bundles in a staggered arrangement. Heat transfer and fluid
Ta Tj T0 T1 t ui x, y, z
air temperature (K) measured fin temperature at the jth measurement location (K) fin base temperature (K) ambient air temperature (K) fin thickness (m) air velocity in the i-direction (m/s) spatial coordinates (m)
Greek symbols ae, ak parameters in Eqs. (21) and (22) b volumetric thermal expansion coefficient bt parameter in RNG k–e model dij, dj2 Kronecker delta functions e viscous dissipation rate of turbulence kinetic energy g mean flow fields (Sk/e) g0 parameter m laminar kinematic viscosity (m2/s) meff effective kinematic viscosity (m2/s) mt turbulent kinematic viscosity (m2/s) q air density (kg/m3) Superscripts a in the air region f in the fin region mea measured data
flow characteristics are obtained using FLUENT in conjunction with RNG k–e turbulence model. However, the flow between the two fins is considered laminar. The other parts of the tube bundle are considered to be turbulent. They believed that the effect of the total number of grid points on the numerical results was small. 50,000–99,000 cells are used to discretize computational domain. Jang et al. [15,16] and Jang and Chen [17] obtained fluid flow and heat transfer characteristics of the fin and tube heat exchanger under the assumption of laminar flow between two fins. Kayansayan [18] studied the natural convection of air over an annular finned tube and observed the effects of the fin spacing, fin diameter to tube diameter ratio and Rayleigh number on the heat transfer. The finned-tube diameter is 24 mm and the tube length is 600 mm. From Ref. [18], the heat transfer coefficient increases with increasing ratio of diameter. Sajedi et al. [19] used experimental and numerical study on the optimal fin numbering in an external extended finned tube heat exchanger. Qiu et al. [20] applied three-dimensional numerical method to investigate the effects of fin angle, fin surface emissivity and tube wall temperature on heat transfer enhancement for a longitudinal externally-finned tube placed vertically in a small chamber. The numerical model is validated through comparison with experimental measurements and the appropriateness of general boundary conditions is examined. The results show that both convection and radiation heat transfer modes are important. Yaghoubi and Mahdavi [21] applied the control volume scheme to investigate natural convection heat transfer from a horizontal cooled finned tube. The thickness of the aluminum fins is 0.4 mm. Several tests are made between grid dimensions of 80-40-8, 120-80-16 and 170-130-20 nodes along the radial, tangential and axial directions, respectively. The results in terms of average convective heat transfer show that a 120-80-16 grid distribution is suitably fine to ensure a grid-independent solution with a difference less than 3% with the minimum grid size. Kumar et al. [22] used the software OpenFOAM-2.2 based on finite
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volume approach to investigate the transient 3D numerical simulations of natural convection of air around a circular finned tube with the annular plain fins kept in a small chimney. The fin spacing and fin diameter are varied from 2–12 mm and from 35–50 mm, respectively. The diameter and length of the circular tube are 24.9 mm and 610 mm, respectively. Perhaps, the change in temperature of the aluminum fins is very small so that the temperature measurements are not shown in the work of Yaghoubi and Mahdavi [21]. The walls of the box are specified as adiabatic boundary condition. The ceiling is set at a constant temperature of 300 K. The cylinder is assumed to be at a constant temperature of 340 K. The air in the enclosure is maintained at an initial temperature of 300 K. It is seen in Ref. [22] that the tube and fins are assumed to be at a constant temperature. For practical engineering problems, this assumption cannot be easy to do. Thus, Kumar et al. [22] did not carry out a comparison between numerical and experimental fin temperatures at the selected locations in order to validate the accuracy of the results obtained. The fin heat conduction equation is not solved. In other words, the problem studied by Kumar et al. [22] is not a conjugate problem. It is found in Refs. [14–21] that independence of grid points and laminar flow are assumed. It is known that quantitative studies of the heat transfer processes and the reliability are an important concept in the industrial applications. This implies that a more accurate prediction of the heat transfer coefficient is an important work for the device of the high-performance heat exchangers. Thus, this study applies the commercial package FLUENT [13] in conjunction with the experimental temperature data and inverse results to obtain free convection heat transfer and fluid flow characteristics of the present problem for two different tube diameters. The resulting heat transfer coefficient is compare with the correlation obtained from the current textbook [11] and the existing results. A comparison between the numerical and experimental fin temperatures is also made at the selected measurement locations. In order to understand the suitability of the grid independence assumption, the effect of grid points on the results obtained is studied for various values of the fin spacing and tube diameter. To the best of our knowledge, a few researchers used a commercial package FLUENT
Fig. 2. Schematic diagram of the physical model and the computational domain with dashed lines for d0 = 2 mm. (a) x–y plane, (b) y–z plane.
along with the inverse method to determine the heat transfer and fluid flow characteristics of this problem. 2. Formulation of inverse scheme
Fig. 1. Physical geometry of this study with various measurement locations and sub-fin regions for d0 = 2 mm.
Fig. 1 shows the physical geometry of the single-tube plate fin and tube heat exchanger. Parameters d0 and L represent the outer diameter of the tube and fin length, respectively. The center of the tube is located at (L/2, L/2). The fin temperature data at the selected locations and the ambient air temperature are measured from the present experimental apparatus for d0 = 2 mm. The twodimensional inverse heat conduction problem is introduced to
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obtain an estimate of the present study. The inverse method along with the finite difference method, the experimental temperature data and least squares method is applied to predict the unknown heat transfer coefficient on the fins of a single-tube plate fin and tube heat exchanger for d0 = 2 mm and various fin spacings. Unknown heat transfer coefficient is assumed to be nonuniformly distributed so that the entire fin is divided into several sub-fin regions. The heat transfer coefficient in each sub-fin region is assumed to be an unknown constant. The schematic diagram of this study with various measurement locations and the sub-fin regions can refer to Fig. 3 of Ref. [8] for d0 = 27.3 mm. The physical geometry with measurement locations and sub-fin regions is shown in Fig. 1 for d0 = 2 mm. To avoid duplication, a detailed diagram of the experimental apparatus and the experimental method can refer to Ref. [8]. Under the assumption of a thin fin, the temperature gradient in the z-direction (the fin thickness) can be disregarded. Thus, the fin temperature varies only in the x- and y-directions. In addition, the surface area of the fin is small when compared to the total fin surface area. This implies that the actual heat transfer rate dissipated through the fin tip is rather small compared to the total heat transfer rate drawn from the fin base. Thus, the boundary condition at the edge surface of the fin is
323
assumed to be insulated [5–10,12]. Under the assumption of steady state and constant thermal properties, the heat conduction equation for a thin fin can be expressed as
@ 2 T @ 2 T 2hðx; yÞ þ ¼ ðT T 1 Þ @x2 @y2 kf t
ð1Þ
Its corresponding boundary conditions are
@T ¼ 0 at x ¼ 0 and x ¼ L @x
ð2Þ
@T ¼ 0 at y ¼ 0 and y ¼ L @y
ð3Þ
T ¼ T 0 ðx; yÞ on St
ð4Þ
where x and y are the Cartesian coordinates. L and t denote the length and thickness of the square fin, respectively. h(x, y) is the unknown heat transfer coefficient. kf is the thermal conductivity of the fin. T0 is the fin base temperature or the outer surface temperature of the circular tube. T1 denotes the ambient air temperature. To predict the unknown heat transfer coefficient on the fins, the whole plate fin considered is divided into N sub-fin regions. The application of the finite difference method to Eq. (1) can produce the difference equation in the kth sub-fin region as
T iþ1;j 2T i;j þ T i1;j T i;jþ1 2T i;j þ T i;j1 þ ¼ mk T i;j ‘2 ‘2
for k ¼ 1; 2; . . . ; N ð5Þ
where ‘ is the distance between two neighboring nodes in the xk and y-directions and is defined as ‘ = L/(Nx 1) = L/(Ny 1). m denotes the unknown dimensionless parameter on the kth sub-fin =ðkftÞ. In the definition of ‘ k = 2L2 h region and is defined as m k k , Nx and Ny are the nodal numbers in the x and y directions, and m represents the average heat transfer coefficient in respectively. h k the kth sub-fin region. The difference equation for the node at the boundary surface, the interface of two adjacent sub-fin regions and in the vicinity of the tube is similar to that shown in Ref. [8]. To avoid repetition, their derivative process can be found in Ref. [8]. Rearrangement of Eq. (5) and the difference equation at the boundary surface, the interface between two adjacent sub-fin regions and in the vicinity of the tube yields the following matrix equation.
½A½T ¼ ½F
Fig. 3. Schematic diagram of the physical model and the computational domain with dashed lines for d0 = 27.3 mm. (a) x–y plane, (b) y–z plane.
ð6Þ
where [A] is a global conduction matrix. [T] is a matrix representing the nodal temperatures. [F] is a force matrix. The nodal temperatures are obtained from Eq. (6) using Gaussian elimination. Inverse analysis of this study can refer to Refs. [7–10]. In order to avoid repetition, its detailed procedures are not shown in this manuscript. The heat transfer behavior on the fins or local heat transfer and flow distributions between the two fins may be interfered significantly if an obtrusive thermocouple is used in the fin. This means that the fin temperature measurements can be inherently difficult in order not to interrupt the heat transfer behavior. Thus, T-type thermocouple is used to measure the fin temperature only at selected measurement locations [7–10]. The error range of T-type thermocouple is ±0.4%. The heat transfer rate dissipated from the jth sub-fin region Qj, average heat transfer coefficient on the fins heat transfer coefficient based on the fin base temperature h h, b and total heat transfer rate dissipated from the plate fin to the ambient Q can be obtained using these estimated results of the average heat transfer coefficient in each sub-fin region. The Qj, h, hb and Q values can be expressed as
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Q i 2Aj hj ðT mea T 1 Þ for j ¼ 1; 2; . . . ; N j h
N X hj Aj =Af
ð7Þ
ð8Þ
j¼1
and
Q¼
N X Q j ¼ 2Af hb ðT 0 T 1 Þ ¼ 2Af hðT av e T 1 Þ
ð9Þ
j¼1
where Af is the lateral surface area of the plate fin and is defined as Af = L2 – pd20/4. Tmea represents the experimental temperature of the j fin at the jth measurement location. Tave denotes the average of all the fin temperatures at the selected measurement locations. From is obtained from Eq. (8). This heat transfer Ref. [8], the value of h coefficient may include convective and radiative heat transfer coef and h in this study are obtained ficients. However, the values of h b from Eq. (9) for comparison with the results obtained using FLUENT. The fin efficiency gf is defined as the ratio of the actual total heat transfer rate from the fin to the dissipated heat from the fin maintained at the fin base temperature T0 and can be expressed as
gf ¼
Q 2Af ðT 0 T 1 Þh
¼
hb h
ð10Þ
3. Experimental apparatus The experiments have been carried out in a box with an upward opening, 550 mm long, 450 mm wide and 300 mm high by Chen and Chou [8] for d0 = 27.3 mm. The schematic diagram of this experimental apparatus can refer to Fig. 1 of Ref. [8]. In order to study the effect of tube diameter on the resulting estimates, experiments for d0 = 2 mm are performed in a box with an upward opening, 120 mm long, 120 mm wide and 120 mm high. The schematic diagram of the box with an upward opening for d0 = 2 mm is similar to that for d0 = 27.3 mm. Thus, a schematic diagram of an upwardly opening box made from acrylic-plastic sheets for d0 = 2 mm is not displayed in this manuscript. Test fins are made of AISI 304 stainless material. Their size is 0.1 m long, 0.1 m wide and 1 mm thick for d0 = 27.3 mm [8] and 0.05 m long, 0.05 m wide and 1 mm thick for d0 = 2 mm. It is found in Ref. [23] that the thermal conductivity of AISI 304 stainless material is 14.9 W/mK. The emissivity of the fin measured using FT-IR Spectrum 100 (Perkin Elmer Co., Ltd) is 0.18. The horizontal tube is placed on wooden supports, which is 98 mm above the bottom plate to prevent the ground effect. Thus, the distance between the top surface and the upper surface of the circular cylinder is 174.7 mm for d0 = 27.3 mm. The ambient air temperature and the fin temperature are measured using T-type thermocouples. Test fins are heated about 2 h using 200 W of power input. All the data signals are collected and converted by a data acquisition system (National Instruments NI SCXI-1000, 1102, 1300). The limit of error of the thermocouple is ±0.4%. The experimental methods and procedures are similar to those shown in Ref. [8]. Four thermocouples are fixed at four different positions of the interface between the fin and the circular tube using a satlon cyanoacrylate adhesive. A thermocouple is placed at 1.75 cm away from the test fins and y = 18 cm in order to measure the ambient air temperature T1 for d0 = 2 mm. Measurement of ambient air temperature T1 refers to Ref. [8] for d0 = 27.3 mm. The whole fin is divided into eight sub-fin regions, i.e. N = 8. Eight thermocouples are fixed at the selected locations of the fins in order to determine the fin temperature data, as shown in Fig. 1. The diameter of the spot size of the thermocouple is about 0.13 mm.
The least squares minimization technique is applied to minimize the sum of the squares of the deviations between the calculated and measured fin temperatures at selected locations of the j are found in Ref. fins. Details of estimating the unknown value h [8]. In order to avoid repetition, they are not shown in this manuscript. The computational procedures of this study are repeated mea inv T T until the values of j T mea j for j = 1, 2,. . ., N are all less than 105, j is the calculated temperature of the fin at the jth meawhere Tinv j valsurement location and is determined from Eq. (6). Once the h j h and g can ues for j = 1, 2,. . ., N are determined, the values of h, b
f
be obtained from Eqs. (8)-(10). 4. Three-dimensional numerical analysis The main purpose of this study is to use FLUENT along with two different flow models to determine the natural convection heat transfer and fluid flow characteristics of the single-tube plate fin and tube heat exchanger for two different diameters. It is seen in Ref. [24] that a comparison of the numerical predictions and experimental data for the hydrodynamic and thermal fields in a twochannel plate heat exchanger using laminar and two-equation turbulence models is made. The friction factors, Nusselt numbers, the outlet temperatures of the two streams and the temperature distributions on the channel end-plates are compared. The buoyancy effect is not considered taken into account. Their results show that realizable k–e model with non-equilibrium wall functions is used to obtain the closest result compared to the experimental data. Due to higher fin base temperature compared to ambient air temperature, the buoyancy effect is considered in this study. In order to validate the accuracy of the results obtained using the laminar flow and RNG k–e turbulence models, the fin temperature and heat transfer coefficient obtained are compared with the experimental temperature data at the selected measurement locations and inverse results, respectively. Under the assumptions of the steady-state and constant thermal properties, the threedimensional heat conduction equation for the thin fin can be expressed as
@2T ¼0 @x2i
ð11Þ
The boundary conditions at x = 0, y = 0, x = L, y = L and on the outer surface of the tube are the same as Eqs. (2)-(4). The boundary conditions at z = 0 and t/2 are given as
@T ¼ 0 at z ¼ 0 @z
ð12Þ
and
kf
@T ¼ hðx; yÞðT T 1 Þ at z ¼ t=2 @z
ð13Þ
where xi, i = 1, 2, 3, respectively indicate x, y and z. x, y and z are the Cartesian coordinates. h(x, y) denotes the local heat transfer coefficient on the fins. 4.1. Laminar flow model Thermal properties of the ambient air are assumed to be constant. The air flow for the entire flow region is assumed to be three-dimensional, incompressible, laminar and steady and to have no viscous dissipation. The continuity, momentum and energy equations can be expressed in tensor form as
@ui ¼0 @xi
ð14Þ
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uj
@ui 1 @p @ 2 ui ¼ þ m 2 þ g j bdj2 ðT a T 1 Þ @xj q @xj @xj
ð15Þ
and
c p uj
@T a ka @ 2 T a @ui @uj @ui ¼ þm þ @xj q @xj @xj @xj @xi @xj
ð16Þ
where ui, p, gj and Ta are the velocity components, pressure, gravitational acceleration components and the air temperature, respectively. b is the volumetric thermal expansion coefficient. dj2 is the Kronecker delta. q, m, cp and ka are the density, kinematic viscosity, specific heat and thermal conductivity of air, respectively. They are all assumed to be constant. 4.2. RNG k–e turbulence model The air flow for the entire flow region is assumed to be threedimensional, symmetric, turbulent and steady and to have no viscous dissipation. The continuity equation is the same as Eq. (14). The momentum and energy equations may be expressed as: [12,13,25]
@u0i u0j
@ui 1 @p @ @ui @uj uj ¼ þm þ þ g j bdj2 ðT a T 1 Þ @xj @xj @xi @xj q @xi @xj
4.3. Boundary conditions kin and ein are the intrinsic values of commercial package FLUENT and are given as kin = 0.5 and ein = 0.5 for RNG k–e turbulence model. The numerical results can be sensitive to the choice of kin and ein. The surface temperature of the tube T0 is assumed to be constant. Only the half-section of the fins is selected for all calculations. The computational domain is half of the box and is shown in Fig. 2 for d0 = 2 mm and Fig. 3 for d0 = 27.3 mm in dashed lines. The gradient of all dependent variables is set to zero at the side walls and a bottom wall. The boundary condition at the top of the box is set to the ambient temperature T1. At the solid surfaces, no-slip conditions are specified. The fin temperatures at various measurement locations, T0 and T1 are obtained from this experiment. The temperature and heat flux matching conditions at the fin-fluid interface are written as
T ¼ Ta
R
ð17Þ
c p uj
@u0j T 0a
@T a ka @ T a @ ¼ cp þ 2meff ðui Sij Þ @xj @xj q @x2j @xj
ð18Þ
2 u0i u0j ¼ 2mt Sij kdij 3
ð19Þ
and
ð20Þ
where dij is Kronecker delta function. Prt is the turbulent Prandtl number. The k and e equations in the RNG k–e turbulence model with buoyancy effects are expressed as:
@k @ @k þ Gk þ Gb qe ¼q ak meff @xj @xj @xj
ð21Þ
and
quj
@e @ @e e þ c1e ðGk þ c3e Gb Þ ¼q ae meff @xj @xj @xj k
qcl g3 ð1 g=g0 Þ e2 e2 c 2e q 3 1 þ bt g k k
ð24Þ
and
Z hðT T 1 ÞdA
ð25Þ
is obtained from Eq. (24) for FLUENT and is where the value of h and g used to compare with the approximate inverse result. h b f values are calculated from Eqs. (9) and (10) for FLUENT, respectively. In order to compare the present results with those obtained from the correlation, it is found from Ref. [11] that correlations of Nusselt number Nu and Rayleigh number Ra for square isothermal fins without tube, 0.2 < Ra < 4 104 and Pr = 0.71 and the annular circular isothermal fins with supporting cylinder, 3 < Ra < 104 and 1.67 < f = D/d0 < 1 are respectively expressed as
2
cp mt @T a u0j T 0a ¼ Prt @xj
ð23Þ
ðT T 1 Þhðx; yÞdA R ðT T 1 ÞdA Af
Af
where the effective kinematic viscosity meff is defined as meff = m + mt. The turbulent kinematic viscosity mt is defined as mt = clk2/e. The mean strain rate tensor Sij is defined as Sij = (əui/əxj + əuj/əxi)/2. The Reynolds stress tensor and turbulent heat fluxes in conjunction with Boussinesq approximation are given as
quj
Af
Q ¼2 2
@T @T a ¼ ka @z @z
and the value of Q for The average heat transfer coefficient h commercial software FLUENT can be approximated as
h¼
and
and kf
Ra0:89 Nu ¼ 4 18
31=2:7
!2:7 þ ð0:62Ra
0:25 2:7 5
Þ
for square fin
ð26Þ
and
Nu ¼
Ra ½2 expðC Þ expðb1 C Þ for annular circular fin 12p ð27Þ
fp=ka and Ra ¼ gbðT 0 T 1 Þfp3 ð fp Þ. where Nu and Ra are defined as Nu = h b L ma The parameter L⁄ is L for an isothermal square rectangular fin and D for an isothermal annular circular fin. Eq. (26) does not consider the effect of the tube diameter on Nu value. It is seen in Ref. [11] that h b
ð22Þ
where the parameters Gk, Gb and g are the production of turbulent kinetic energy due to the velocity gradient, the production of turbulent kinetic energy due to the buoyancy and the ratio of characteristic time scale of turbulence and the mean flow field, respectively. The definitions of Gk, Gb and g can refer to Refs. [12,13]. Parameters ak, ae, Prt, cl, c1e, c2e, g0 and bt are given as ak = ae = 1.393, Prt = 0.85, cl = 0.0845, c1e = 1.42, c2e = 1.68, g0 = 4.38 and bt = 0.012. The coefficient S is defined as S = 21/2Sij. Parameter c3e is 0 due to the air velocity perpendicular to the gravity.
value is calculated by Eq. (26) with an equivalent diameter D = 1.23L for square fins. Properties in Eqs. (26) and (27) are evaluated at the wall temperature. Parameters b1, C1 and C⁄ are defined as
b1 ¼ 0:17=f þ e4:8=f " # 1=2 4=3 23:7 1:1ð152=f2 Þ C1 ¼ 1 þ b1
ð28Þ ð29Þ
and
C ¼ ðC 1 =RaÞ3=4
ð30Þ
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5. Typical grid distribution An unstructured grid system containing a non-uniform distribution of the grid points is used to determine all of the numerical results. The grid density is controlled to ensure proper results with various values of the fin spacing. A finer mesh is needed n some regions where the temperature and velocity gradients are high, such as in the vicinity of the interface between the tube and fins. In some regions where the temperature and velocity gradients are small, a coarser mesh can be used to save computing time. The grid system used in this study is shown in Fig. 4. The number of grid points in the x, y and z directions of the fins, respectively, is taken from 14–15, 14 and 3–5 for d0 = 2 mm, and is 11, 8 and 3 for d0 = 27.3 mm. The number of grid points between the two fins is similar to that in the x and y direction of the fins for d0 = 2 mm and 27.3 mm. The number of grid points between the two fins in the z-direction is taken from 13–23 for d0 = 2 mm and from 7–10 for d0 = 27.3 mm. A coarser mesh is used in the remainder of this region to save computing time. The number of grid points outside the fins is 19 and 19 in the x and y directions, respectively, and is taken from 46–63 in the z direction for d0 = 2 mm. The number of grid points outside the fins is 23 and 12 in the x and y directions, respectively, and is taken from 36–38 in the z direction for d0 = 27.3 mm. The total number of grid points is from 21,339– 28,035 for d0 = 2 mm and from 9900–11,544 for d0 = 27.3 mm. Various grid points are to be tested for each case until the fin temperature and estimates obtained are as close to the experimental
temperature data, the inverse results and existing correlations as possible.
6. Results and discussion All physical properties are evaluated at the film temperature or the average of the fin base temperature and ambient temperature. The experimental temperature data are obtained from Refs. [8] and [26] for d0 = 2.73 cm. The commercial package FLUENT [13] is applied to determine the fluid flow and heat transfer characteristics of this problem. The measurement locations and sub-fin regions are set to 8 for L = 0.1 m and d0 = 27.3 mm and 6 for L = 0.05 m and d0 = 2 mm. Tables 1 shows a comparison between the results obtained using the laminar flow and RNG k–e turbulence models and the experimental temperature data and inverse results for fp = 15 mm and various d0 values. It is found from Table 1 that the fin temperatures at selected measurement locations and the average heat transfer coefficient obtained using RNG k–e turbulence model are closer to the experimental data and inverse results than those obtained using the laminar flow model for the different tube diameters. The L/d0 and Ra values are, respectively, 30.75 and 1301.1 for d0 = 2 mm and 5.68 and 1164.15 for d0 = 27.3 mm. The results show that RNG k–e turbulence model is more suitable for this problem than the laminar flow model even for L/d0 = 30.75 and Ra = 1301.1. Therefore, the following results are obtained using RNG k–e turbulence model. Based on the results obtained, the proposed correlations can be expressed as
2
Ra0:89 Nu ¼ 4 18
31=2:7
!2:7 þ ð0:62Ra
0:25 2:7 5
Þ
1
d0 L
ð31Þ
and
Nu ¼
Ra 0:65d0 ½2 expðC Þ expðb1 C Þ 1 12p L
ð32Þ
The proposed new correlation of Nub Ra1/3 can be expressed as
d0 Nu ¼ 0:255 þ 0:28Ra1=3 1 L
Fig. 4. Typical grid distribution in the computational domain. (a) d0 = 2 mm, (b) d0 = 27.3 mm.
ð33Þ
It is worth mentioning that the properties in Eqs. (31)-(33) are evaluated at the film temperature. It is seen that the relationship between Nu and Ra shown in Eq. (35) of Ref. [8] does not consider the effect of the tube diameter and is obtained from the inverse result. Thus, Eqs. (31)-(33) are different from the correlation given by Chen and Chou [8]. To further verify the reliability and availability of the present and h values between the results results, a comparison of the h b obtained using FLUENT and inverse results is made. Table 2 shows and h with various d and fp values. It is seen a comparison of h b 0 and h values obtained using FLUENT are in from Table 2 that h b good agreement with the inverse results for 10 mm 6 fp 6 20 mm values deviate from those and a single fin. However, the h b obtained from the correlations (26) and (27) with d0 = 2 mm, and are closer to those obtained from the correlation (27) than the correlation (26) with d0 = 27.3 mm. This means that the correlation (27) has better accuracy than the correlation (26) for this problem. It is found that the heat transfer coefficient given by Kayansayan [18] and Kumar et al. [22] is not compared with the correlation and h (27) in Ref. [11]. Table 2 shows the obtained results of h b increase with decreasing the tube diameter. This result is similar to that shown in Ref. [18].
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H.-T. Chen et al. / International Journal of Heat and Mass Transfer 100 (2016) 320–331 Table 1 Comparison of Tj with fp = 15 mm and various d0 values for various flow models. d0 = 2 mm, Ra = 1301.1 T0 = 315.82 K, T1 = 300.49 K
d0 = 27.3 mm, Ra = 1493.59 T0 = 336.86 K, T1 = 299.0 K
Fluent
Measured
Laminar
RNG
T1 = 309.06 T2 = 309.92 T3 = 309.12 T4 = 308.32 T5 = 309.02 T6 = 308.39
T1 = 304.16 T2 = 305.22 T3 = 304.22 T4 = 304.22 T5 = 305.40 T6 = 304.26
6.07
23.44
Fluent RNG
T1 = 305.23 T2 = 305.93 T3 = 305.50 T4 = 304.50 T5 = 304.97 T6 = 304.64
T1 = 318.08 T2 = 322.75 T3 = 318.09 T4 = 319.03 T5 = 319.06 T6 = 314.36 T7 = 317.07 T8 = 314.36
T1 = 312.45 T2 = 320.64 T3 = 312.47 T4 = 312.82 T5 = 312.89 T6 = 307.24 T7 = 311.41 T8 = 307.26
T1 = 314.24 T2 = 322.25 T3 = 314.08 T4 = 312.21 T5 = 311.94 T6 = 307.14 T7 = 309.65 T8 = 305.04
22.38 (Inv)
6.99
17.14
14.45 (Inv)
Tj (K)
h
Table 2 and h with various d and fp values. Comparison of h b 0 d0 (mm)
fp (mm)
Exp. data [25]
Laminar
Table 4 with various d and fp values. Comparison of h b 0
(W/m2K) h
(W/m2K) h b
Inv
FLUENT
Inv
FLUENT
Eq. (26)
Eq. (27)
d0 (mm)
2
10 15 20 1
18.1 21.95 22.96 23.06
16.83 21.4 25.16 27.07
6.26 6.64 6.72 6.73
5.16 5.71 6.13 6.32
6.59 6.66 6.71 –
5.51 5.91 6.05 –
27.3
10 15 20 1
11.68 14.05 16.56 17.89
9.33 13.06 17.48 19.93
4.45 4.85 5.18 5.34
3.93 4.66 5.04 5.4
6.84 6.89 7.02 –
5.15 5.54 5.77 –
Table 3 shows the effect of grid points on the fin temperature measurements at selected locations and the heat transfer coefficient for fp = 15 mm and various d0 values. It is seen that the effect of grid points on the heat transfer coefficient and fin temperature measurements cannot be ignored. Thus, various grid points are tested for each case until the fin temperature and heat transfer coefficient obtained are as close to the experimental temperature data and the inverse results as possible. for various values of d and Table 4 shows the comparison of h b 0 fp. An important finding is that hb values are in good agreement with those obtained from the proposed correlations (31)-(33) with d0 = 2 mm and 27.3 mm for 10 mm 6 fp 6 20 mm. Thus, the proposed correlations (31)-(33) can contribute to the design of values practical plate finned-tube heat exchanger. However, the h b obtained from the correlation (33) may be slightly closer to the inverse and FLUENT results than those obtained from the
fp (mm)
Ra
(W/m2K) h b Eq. (31)
Eq. (32)
Eq. (33)
Q (W)
gf
2
10 15 20
262.7 1301.1 4200.5
6.35 6.40 6.45
5.42 5.77 5.91
5.32 5.72 6.18
0.418 0.436 0.485
0.26 0.24 0.25
27.3
10 15 20
295.7 1431.9 4864.2
5.15 5.10 5.23
4.45 4.67 4.86
4.49 4.70 5.14
3.516 3.351 4.035
0.33 0.27 0.25
correlations (31) and (32). Table 4 also shows the variation of the fin efficiency gf with fp value for d0 = 2 mm and 27.3 mm. The gf value is determined based on the results obtained using FLUENT. It is found in Table 4 that the gf value decreases with increasing fin spacing. However, it can be improved with increasing tube diameter. The numerical results in Figs. 5–8 are obtained using the RNG k–e turbulence model. The air velocity pattern and the temperature contour in x–y plane at z = fp/2 away from the surface of the intermediate fin are, respectively, shown in Figs. 5 and 6 for fp = 15 mm and d0 = 2 mm and 27.3 mm. It is seen from Fig. 5 that the air inside the box is heated at the cylinder tube and moves upward due to the density difference caused by the heating. The upward-moving natural convection boundary layer flow is induced by two adjacent fins. It is seen from Fig. 5(b) boundary layer begins to develop upward from the bottom of the heating tube. Its thickness increases along the circumference of the tube. This is basically attributed to the up-moving boundary layer flow induced by fins.
Table 3 and h values for fp = 15 mm and various values of d . Effect of grid points on Tj, h b 0 Nxf Nyf Nzf Nza d0 = 2 mm
d0 = 27.3 mm
9 11 3 15
11 11 3 30
11 11 5 30
11 8 3 9
11 11 4 9
21 21 4 9
T1 T2 T3 T4 T5 T6 T7 T8 h
308.35 308.91 308.23 307.85 308.29 307.7 –
308.15 308.89 308.16 307.56 308.21 307.59 –
308.27 308.96 308.21 307.61 308.26 307.69 –
5.41
5.35
5.23
313.12 320.91 313.19 313.61 313.64 308.73 313.14 308.76 14.11
311.03 316.71 311.08 312.47 312.31 308.5 312.16 308.51 16.64
310.37 316.7 310.38 311.96 311.96 308.04 312 308.06 17.07
h b
3.15
3.06
2.98
4.29
4.77
4.76
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Fig. 5. Velocity pattern in the middle between the two fins with fp = 15 mm. (a) d0 = 2 mm, (b) d0 = 27.3 mm.
Then, it moves along the side walls and reaches to the lower part of the box. After that, it is again heated by the heating circular tube. A plume of heated air over a heating tube will form a low velocity wake region. The two large natural re-circulating eddies are formed within the two adjacent fins in the x–y plane. Fig. 5 along with Fig. 3 of Ref. [8] and Fig. 1 shows that the two natural recirculations within two adjacent fins are located in the 1st and 3rd sub-fin regions for d0 = 27.3 mm and 4T = 37.86 K, where 4T denotes the difference between the ambient temperature and the fin base temperature. In other words, these two re-circulations occur in two adjacent fins and the gap between the top surface of the box and tube, and are located about 45° from the horizontal line for d0 = 27.3 mm. However, the two re-circulations are
observed in the vicinity of the left and right side of the horizontal tube for d0 = 2 mm with 4T = 15.33 K. The two re-circulations are symmetric on both sides of the circular tube and rotate in opposite directions for d0 = 2 mm and 27.3 mm. A similar phenomenon can be observed from Ref. [27]. The larger the value of 4T value, the stronger the buoyancy. Accordingly, the re-circulation can be moved upwardly due to the increase of the value of 4T. This means that the re-circulation position may depend on the value of 4T. The existence of such an oncoming flow is stronger and reinforces the buoyancy that is induced by the tube base itself. This strong circulation helps to transfer the heat from the neighboring fin surfaces to the ambient air through the top surface of the box. Thus, there exhibits very complex three-dimensional heat transfer and fluid
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Fig. 7. Heat transfer coefficient distribution on the fins with fp = 15 mm. (a) d0 = 2 mm, (b) d0 = 27.3 mm.
Fig. 6. Temperature contour in the middle between the two fins with fp = 15 mm. (a) d0 = 2 mm, (b) d0 = 27.3 mm.
flow characteristics within the fins of the plate-finned tube heat exchangers for this problem. It is seen in Fig. 6 that the temperature contour shows a nonuniform distribution within the two adjacent fins in the x–y plane. The low-temperature region is seen in the plume region. A higher temperature and temperature gradient are observed in the vicinity of the fin base. This indicates that there is a high rate of heat transfer in this region. Fig. 7 shows the distribution of the heat transfer coefficient on the fin with fp = 15 mm for d0 = 2 mm and 27.3 mm. From Fig. 3 of Ref. [8] and Figs. 1 and 5, the 6th and 8th sub-fin regions are located in the oncoming flow regions for d0 = 27.3 mm. The 4th–6th sub-fin regions are located in the oncoming flow regions for d0 = 2 mm. The 2nd sub-fin region is located in the plume region for d0 = 2 mm and 27.3 mm. It is found from Table 1 that the temperature measurements in 6th and 8th sub-fin regions are lower than those in other sub-fin regions for d0 = 27.3 mm. The temperature measurements in 4th and 6th sub-fin regions
are lower than other sub-fin regions for d0 = 2 mm. The temperature measurement in the 2nd sub-fin region is higher than that in other sub-fin regions with d0 = 2 mm and 27.3 mm. The temperature measurement in the downstream sub-fin regions is higher than that in the oncoming sub-fin regions for d0 = 27.3 mm. However, the difference of the fin temperature measurements are small for d0 = 2 mm because less heat is transferred to the fins. In other words, the fin temperature distribution for d0 = 2 mm is more uniform than that for d0 = 27.3 mm. This result indicates that the heat transfer coefficient on the fins is also a uniform distribution with d0 = 2 mm, as shown in Fig. 7a. It is seen that the difference between the fin temperature and ambient air temperature in the plume region is higher than that in other sub-fin regions with d0 = 2 mm and 27.3 mm. It is seen from Fig. 5(b) that plume region over a heating tube belongs to a low-speed wake region. Therefore, the heat transfer coefficient is highest in the bottom region of the tube and is lowest in the top region of the tube. As shown in Table 1 of Ref. [8] and Fig. 7(b), the heat transfer coefficient in the plume region is lower than that the oncoming flow region. These results indicate that the physical phenomenon of the obtained results is reasonable.
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points may need to vary with the fin spacing. This means that the effect of the total number of grid points on the numerical results obtained cannot be ignored. The commercial package, FLUENT, along with inverse results and the experimental temperature data can applied to obtain a more accurate heat transfer and fluid flow characteristics of the plate fin and tube heat exchangers. Acknowledgements The authors gratefully acknowledge the financial support provided by the National Science Council of the Republic of China under Grant No. NSC 102-2221-E-006-177-MY3. We would also like to thank Professor Chin-Hsiang Cheng at National Cheng Kung University for providing us access to the computational fluid dynamics software FLUENT. References
Fig. 8. Variation of Nu with Ra1/3 for d0 = 2 mm and 27.3 mm.
The variation of Nu with Ra1/3 is shown in Fig. 8 for d0 = 2 mm and 27.3 mm. From Fig. 8, Eq. (33) is consistent with the inverse results and numerical results obtained using FLUENT. Nu value increases with decreasing d0 value under the same Ra value. This value increases with decreasing the tube diammeans that the h b eter, as shown in Table 2. 7. Conclusions This study proposes three-dimensional computational fluid dynamics commercial package, FLUENT, along with the laminar flow and RNG k–e turbulence models, the experimental temperature data and inverse results to determine the natural convection heat transfer and fluid flow characteristics of a single circular tube plate finned-tube heat exchanger for various values of fp and d0. An important finding is that the RNG k–e turbulence model is more suitable than laminar flow for this problem. This conclusion is different from that shown in the previous works [14–21]. It is seen that the two re-circulations occur in the two adjacent fins and in the gap between the top surface of the box and the tube and are located about 45° from the horizontal line for d0 = 27.3 mm. However, the fluid re-circulations within the two adjacent fins are observed in the vicinity of the left and right of a horizontal tube with d0 = 2 mm. The recirculation position may depend on the difference between ambient temperature and the fin base tempera and h values obtained using FLUENT are in good ture. The h b agreement with the inverse results for 10 mm 6 fp 6 20 mm and value obtained is in good agreement with the a single fin. The h b existing correlations with d0 = 2 mm and 27.3 mm. However, the value obtained from the proposed correlations (31)-(33) is cloh b ser to the inverse result than that obtained from the existing correlations for 10 mm 6 fp 6 20 mm and a single fin. Thus, the proposed correlations have better accuracy than the existing correlations shown in Ref. [11]. In order to obtain a more accurate result, the selection of appropriate flow model along with appropriate grid points may be important. Meanwhile, the total number of grid
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