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Modeling of heat transfer for flow across tube banks
Chemical Engineering and Processing 39 (2000) 1 – 14 www.elsevier.com/locate/cep
Modeling of heat transfer for flow across tube banks A. Safwat Wilson, M. Khalil Bassiouny * Mechanical Power Department, Faculty of Engineering, Menoufia Uni6ersity, Shibin El-Kom, Egypt Received 30 March 1999; received in revised form 4 May 1999; accepted 4 May 1999
Abstract A calculation procedure for two-dimensional elliptic flow is applied to predict the pressure drop and heat transfer characteristics of laminar and turbulent flow of air across tube banks. The turbulence model used involves the solution of two partial differential equations, one for the kinetic energy of the turbulence and the other for its dissipation rate. These differential equations are solved simultaneously with those for the conservation equations of mass, momentum and energy using an implicit finite volume procedure. The numerical methodology utilizes the stepped boundary technique to approximate the tube surface which is kept at constant temperature. The computations are extended to cover the case of two rows of tubes undergoing cross flow with in-line and staggered tube arrangements besides the case of a single row. Thereby, Reynolds number (Re) as well as the normal and parallel tube spacing-to-diameter ratios are varied. Effects of the flow and the geometry parameters on the friction factor and the local and global Nusselt number are presented. Moreover, velocity vector diagrams and temperature contours as well as axial flow velocity and turbulence kinetic energy profiles along the flow field upstream, over and downstream the tubes are also given. The theoretical results of the present model are compared with previously published experimental data of different authors. Satisfactory agreement is demonstrated. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Mathematical model; Numerical computations; Tube banks; Cross flow; Heat transfer
1. Introduction Because many heat exchanger arrangements involve multiple rows of tubes in a fluid cross flow, the heat transfer characteristics for tube banks are of important practical interest. The gas flowing on the outside impinges perpendicularly on these tubes. With constant fluid velocity, this type of flow gives rise to an increase in turbulence compared with the case where the gas flows along the tubes and parallel to them. It can thus deduced that the heat transfer coefficient at the outer surface of a tube is higher for cross flow than for parallel flow. The distribution of velocity around a tube in cross flow will not be uniform. In the same way the rate of heat flow around a hot pipe across which air is passed is not uniform also. Depending on application and design criteria, there are many possible tube layout in an array. One of the greatest difficulties with this system is that the area for flow is continually changing.
Dedicated to Prof em. Dr.-Ing. E.-U. Schlu¨nder on the occasion of his 70th birthday. * Corresponding author. Fax: +20-48-235695.
Moreover, the degree of turbulence is considerably higher for a staggered bank of tubes than for in-line one. With the small bundles, which are common in the chemical industry, the selection of the true mean area for flow is further complicated by the changing in number of tubes in the rows. Most of the earlier studies were experimental in nature. A number of worker like Reiher [1], Hilpert [2], Griffiths and Awbery [3], Jakob [4], Fand [5] and Churchill and Bernstein [6] have studied the flow of a fluid past a single cylinder varying from a thin wire to a tube of 150 mm diameter with different Reynolds numbers (Re). Gnielinski [7] proposed correlations for calculating the global Nusselt number in case of a single row of tubes as will as tube bundles with multiple rows based on the correlation of Krischer [8] for a single tube in cross flow by introducing the mean velocity instead of the free stream velocity in the definition of Re. His method takes into account the arrangement of tubes and the number of tube rows in the flow direction. A research work into heat transfer and pressure drop outside tubes and tube assemblies in cross flow has been carried out by Niggeschmidt [9]. The heat
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transfer values measured by him are presented in empirical equations. The test measurements of Pierson [10] and Huge [11] were correlated by Grimison [12] for in-line and staggered arrangements. His empirical relation pertain to tube banks having ten or more rows of tubes in the flow direction. For fewer rows, Kays and London [13] gave numerical values for the ratio of Nu for n rows deep to that of ten rows. Hausen [14] modified slightly the correlation of Grimison and set out an empirical formula for the tubes arrangement factor ( fa) instead of the graphical representation by
Fig. 1. Heat flux to the cells adjacent to the tube wall.
Fig. 2. Grid system used in the present computational domain.
Grimison. An excellent review of experimental investigations for heat transfer from tube banks in cross flow is given in Zukauskas [15]. Results of experimental investigation on the influence of turbulence intensity, transversal and longitudinal pitch-to-diameter ratio on heat transfer and pressure drop in tube bundles of in-line arrangement are presented by Beziel and Stephan [16]. The same experiments have been performed by Li et al. [17] but for staggered arrangement of spirally corrugated tubes. Zhang and Chen [18] carried out experiments on in-line tube banks to study the effect of gap width between tube layers on the heat transfer performance. Experiments were performed by Moreno and Sparrow [19] to investigate the influence of yaw angle on heat transfer, pressure distribution and flow visualization in cases of staggered and in-line tube banks. The heat transfer characteristics of tube bundles subject to dilute particle laden flow were studied experimentally by Murray [20]. Numerical methods have also been used to simulate the heat transfer and fluid flow in tube banks with cross flow of pure gases [21–23] and with particles suspended in gas flow without heat transfer [24,25]. The complex geometry of the flow configuration poses an obstacle to the use of numerical methods. Different methods have been tried such as conformal mapping technique of Thom and Apelt [26], and the hybrid polar—Cartesian grid approach of Launder and Massey [27] and Fujii and Fujii [28], besides the finite volume method of Faghri and Rao [29] for the computation of flow and heat transfer in tube banks. The last one is restricted to laminar flow across in-line arrangements only with the assumption of a periodic, fully developed flow which deviates from the practical case. This paper is an extension of and a complement to the earlier works in this field. The numerical solutions were performed for both laminar and turbulent flow across tube banks of single and double rows in the flow direction in cases of in-line as well as staggered tube arrangements. The friction factor and the local and global Nu calculations are carried out for different values of Re as well as normal and parallel spacing-todiameter ratios.
2. Mathematical model
Fig. 3. Comparison between the present work and different empirical correlations for Nusselt number (single row).
The flow of air through the tube banks is simulated mathematically. The tubes are arranged in a single row or double rows in flow direction. Two arrays of tube banks namely, in-line and staggered arrangements are considered. In this, a single phase, incompressible fluid (air), developed, non-periodic and time independent flow is considered. Both laminar and turbulent flow regimes are taken into account. According to these assumptions, the conservation equations of mass, mo-
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Fig. 4. Comparison between the present work and different empirical correlations for Nusselt number (in-line and staggered arrangements).
mentum and energy are used in Cartesian two-dimensional domain as follows:
Where, Gh = G+ Gt =
2.1. Continuity equation
m m + t Pr Prt
The Prandtl number and turbulence Prandtl number for the present flow regime are taken to be 0.7 and 0.9, respectively.
( ( (ru)+ (r6)= 0 (x (y
2.2. Momentum equation in axial direction (x)
( (u (u ( ruu−me + ru6 −me (x (x (y (y =−
(u (6 (p ( ( + me + me (x (x (x (y (x
Where, me = m +mt and mt = Cmr
k2 o
2.3. Momentum equation in 6ertical direction (y)
( (6 ( (6 ru6 −me + r66 − me (x (x (y (y =−
(p ( (u ( (6 + m + m (y (x e (y (y e (y
2.4. Energy equation
( (h ( (h ruh−Gh + r6h −Gh =0 (x (x (y (y
Fig. 5. Comparison between the present work and empirical correlations for friction factor (in-line and staggered arrangements).
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Fig. 6. Variation of friction factor with Reynolds number for different Sn /D (single row).
Fig. 7. Variation of friction factor with normal tube spacing ration for different Re (single row).
Fig. 8. Effect of Reynolds number on Nusselt number with different Sn /D (single row).
Fig. 9. Effect of normal tube spacing ratio on Nusselt number with different Re (single row).
2.5. Turbulence model The eddy diffusivity appeared in the time averaged momentum equation is obtained by solving the two equations of turbulence model (k −o). The standard model is modified by Launder and Spalding [30] for incompressible flow. The differential equation of turbulence kinetic energy (k) may be written in the following form:
Fig. 10. Velocity vector diagram (single row).
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( me (o ( me (o o ruo − + r6o − = (C1Gk −C2ro) (x so (x so (y (y k The k− o model constants may be recasted as follows: C1 1.44
C2 1.92
Cm 0.09
CD 1
sk 1
so 1.3
2.6. Boundary conditions
Fig. 11. Dimensionless axial velocity profiles (single row).
The computational domain is bounded by inlet and outlet sections, two axes of symmetry and outer tube wall. The treatment of air adjacent to the domain boundaries is as follows:
Fig. 12. Dimensionless turbulence kinetic energy profiles (single row).
Fig. 14. Local Nusselt number around the tube for different Re (single row).
Fig. 13. Temperature contours (single row).
( m (k ( m (k ruk − e + r6k − e =Gk −CDro (x sk (x (y sk (y Where,
Gk = me 2
(u (x
2
+2
(6 (y
2
+
(u (6 + (y (x
n 2
and, k=
u%2 + 6%2 2
The energy dissipation rate (o) is calculated from the following equation:
Fig. 15. Local Nusselt number around the tube for different Sn /D (single row).
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6
c= 1−
Re=
p S 4 n D
umdh n
The other inlet conditions are 6= 0,
p=pin,
T= Tin,
k= I · u 2in o=
k 3/2 l · Sn /2
Fig. 16. Variation of friction factor with Reynolds number for different Sp /D (in-line arrangement).
Fig. 18. Effect of Reynolds number on Nusselt number with different Sp /D (in-line arrangement).
Fig. 17. Variation of friction with parallel tube spacing ratio for different Re (in-line arrangement).
2.6.1. Inlet section All the fluid parameters have uniform distributions along the inlet section. The initial velocity is calculated from the value of Re according to the following relation: u =uin =
cnRe dh
Where, Fig. 19. Effect of parallel tube spacing ratio on Nusselt number with different Re (in-line arrangement).
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Fig. 20. Velocity vector diagram (in-line arrangement).
The values of the initial turbulence intensity I and the length scale of turbulence eddies l are taken to be 0.03 and 0.005, respectively.
2.6.2. Axis of symmetry The flow at the axis of symmetry is considered as one dimensional, axial flow. The gradient of the flow parameters at the normal direction are equal to zero. The flow parameters at the axis of symmetry can be stated as follows: (f =0, (y
qw =
l(Tw − Tf) for y+ 5 5, yp
rutcp(Tw − Tf) for y+ \ 5. 1 Prt ln Ey+ + Pf k The heat transfer coefficient from the tube wall to the flowing air is determined from the following relation: qw =
(f =u, 6, p, h, k and o)
and, 6 = 0.
2.6.3. Exit section The computational domain length behind the tube is selected long enough to ensure fully developed flow condition at the exit section. The exit conditions are 6 = 0, p= pout, T=Tout
Fig. 21. Dimensionless axial velocity profiles (in-line arrangement).
2.6.4. Tube wall The tube wall temperature is considered constant and is taken to be 60°C above the fluid inlet temperature. All the other remaining flow parameters are set to zero at the tube surface. The standard wall function is used to modify the fluid parameters at the cells adjacent to the tube surface. In literature many forms of the Pfunction are available. One of them is that of Spalding and Jayatillaka [33], which is valid for incompressible flow past impermeable smooth surface:
Pf = 9.24
Pr Prt
0.75
−1
n
1 + 0.28e
− 0.007
Pr Prt
n
Fig. 22. Dimensionless turbulence kinetic energy profiles (in-line arrangement).
For constant tube wall temperature, the local Nu is calculated using the following equation: Nuu =
au · D l
Where, au =
qw Tw − Tf
The heat flux from the wall is calculated according to the thickness of the viscous sublayer y+.
Fig. 23. Temperature contours (in-line arrangement).
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Fig. 24. Local Nusselt number around the tubes for different Re (in-line arrangement).
a=
mcp(Tout − Tin) A(Tw −Tb)
The global Nusselt number along the entire domain is obtained by Nu =
aD l
Finally, the friction factor is defined as: f=
Dp dh 2 n D r·u 2m
3. Results and discussion For the sake of verifying the validity of the present computational model, a comparison study between the theoretical results and previous experimental data of different authors is performed for both heat transfer and flow resistance in tube banks undergoing cross flow. Global Nu results are compared with the correlations of Gnielinski [7], Niggeschmidt [9] and Hausen [14] in Fig. 3 for a single row of tubes with a normal spacing-to-diameter ratio of 1.5 and at different Re. The same comparison is carried out in Fig. 4 but for double rows of tubes in flow direction for both cases of
2.7. Numerical solution The finite volume technique [31] is used to solve the governing differential equations. The power scheme is employed to descritize the nonlinear elliptic partial differential equations. These equations are solved together iteratively until the residuals along the domain become B 5×10 − 3. The grid system used in the present numerical computations is Cartesian. The distribution of grids in the flow direction is constant along the computational domain length, while the vertical distribution is non-uniform. The grids are condensed over the tubes to ensure accurate simulation for the fluid inside the high gradient zone. Because of the nature of the Cartesian coordinates, the smooth curved surfaces of the tubes are transformed into series of steps, as indicated in Fig. 1. In order to reduce the steps size, the reasonable number of grids is selected. Therefore, the grid system is chosen to be 66×33. Fig. 2 shows the geometry parameters and the grid system employed in this work.
Fig. 25. Variation of friction factor with Reynolds number for different Sp /D (staggered arrangement).
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Fig. 26. Variation of friction factor with parallel tube spacing ratio for different Re (staggered arrangement).
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Fig. 28. Effect of parallel tube spacing ratio on Nusselt number with different Re (staggered arrangement).
Fig. 29. Velocity vector diagram (staggered arrangement).
Fig. 27. Effect of Reynolds number on Nusselt number with different Sp /D (staggered arrangement).
in-line and staggered tube arrangements with Sn /D= 1.5 and Sp /D=2. The figures show a fair agreement specially in the turbulent flow region. The theoretical work indicates the different flow zones namely: laminar, transient and turbulent with different slopes while the correlations have a fixed form for the whole regions. Fig. 5 represents a comparison between the calculated results and the measured data of Pierson [10] and Huge [11] for the pressure drop which are replicated by Jakob [4] in form of friction factor. The correlation of
Fig. 30. Dimensionless axial velocity profiles (staggered arrangement).
Fig. 31. Dimensionless turbulence kinetic energy profiles (staggered arrangement).
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Fig. 32. Temperature contours (staggered arrangement).
Kast [32] for staggered arrangement is used also in the present comparison. This correlation is independent of the geometry parameters. A noticeable deviation is observed between his results and that of Jakob. A satisfactory agreement between the present theoretical results for the friction factor and Jakob’s empirical relations can be observed for both cases of in-line and staggered arrangement of tubes.
3.1. Single row The variation of friction factor with Re at different values of the normal spacing-to- diameter ratio is indicated in Fig. 6. The figure shows that the friction factor decreases steeply in the laminar flow region and slightly in the turbulent flow one until it reaches a minimum value at Re=10 000 then it increases slightly once more. Fig. 7 represents the friction factor as a function of Sn /D with varying Re as a parameter. It can be easily seen that f exhibits a minimum at a normal spacing of
Fig. 34. Comparision between staggered and in-line arrangements for friction factor.
about 1.5 the tube outer diameter, above which the friction factor increases steeply. Minimum values of f may be obtained in the range of Re between 5000 and 100 000 below which f increases steeply too. The effect of varying Re number as well as Sn /D on the global Nu for a single row of tubes are illustrated in Figs. 8 and 9. It is clear that Nu increases always with Re. This increase has different slopes in laminar and turbulent flow regions. The normal spacing ratio has a significant influence on Nu. With increasing Sn /D, Nus-
Fig. 33. Local Nusselt number around the tubes for different Re (staggered arrangement).
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selt number decreases. This is owing to the nozzle effect between the tubes. The decrease becomes steeper for values of Sn /D between 1.25 and 2. This behaviour may be attributed to the sharp reduction in the separation zone with increasing Sn /D in the considered range. The velocity vector diagram upstream, over and downstream the tubes is plotted in Fig. 10 for Re= 1000 and Sn /D =1.5. The figure indicates the development of the flow velocity along the computational domain. The location of the flow separation from the tube surface is clear at an angle of about 145° where the velocity gradient vanishes. Also, the reversed flow region, where the velocity gradient is negative, and the redeveloped flow zone, where the velocity gradient is positive, are remarkable downstream. Fig. 11 presents the dimensionless axial velocity profiles at constant normal spacing ratio Sn /D= 0.5 and at two values of Re namely 250 and 5000. The velocity profiles are plotted at different locations upstream, downstream and over the tubes. The figure shows clearly the development of the axial velocity specially the recirculation zone due to the flow separation. Corresponding profiles are given in Fig. 12 for dimensionless turbulence kinetic energy. The curves illustrates also the development of k in the separation region, where a peak value of k takes place. The isothermal lines are drawn in Fig. 13 for laminar flow (Re=250) as well as turbulent flow (Re=5000) with Sn /D=1.5. The effect of the flow velocity and the separation zone on the temperature contours are remarkable in the plots. The behaviour of the local Nu around the tubes at different Re as well as at different Sn /D is represented
Fig. 35. Comparision between staggered and in-line arrangements for Nusselt number.
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in Figs. 14 and 15, respectively. The minimum value of Nuu at u equals about 145° may be corresponding to the flow separation from the tube surface and the beginning of the recirculation zone.
3.2. In-line arrangement The model is used also to study the in-line arrangement of tube banks. In this case, two rows are considered. The flow and heat transfer characteristics in this case depend on the normal and parallel tube spacing. Because the effect of the normal spacing Sn /D is studied in case of a single row, the effect of parallel tube pitch is taken into account. The results show a noticeable reduction in friction factor with increasing Re of the flowing air, as shown in Fig. 16. The trend of the curves show different slopes according to the transition in the flow regime from laminar to turbulent flow. The parallel tube spacing Sp /D has a great effect on the friction factor. The results show that increasing the tube pitch leads to an increase in pressure drop across the tube banks a condition which reflects on an increase in the friction factor as shown in Fig. 17. It is observed also that, at high Re the gradient of the friction factor approaches zero and the only effect on the friction factor in this case is the parallel tube spacing. This can be explained as follows: at relatively high Re the increase in the flow velocity leads to a corresponding increase in the pressure drop. On the contrary, it is found that the great effect on Nu comes from increasing Re but not the geometry parameter Sp /D, as shown in Fig. 18. When increasing Re, the separation zone becomes several times the tube diameter and hence the flow doesn’t reach the recovery point before entering the second tube zone whatever the value of Sp /D is. For this reason, it is found that at low Re, the parallel tube spacing has a significant effect. Fig. 19 represents the influence of Sp /D on Nu for low range of Re. It is clear from the figure that, the increase of Sp /D gives the flow behind the first row the chance to approach the reattachment point and enhance the heat transfer coefficient and consequently Nu. Further increase in Sp /D leads to a decrease in Nu. This is because the second row of tubes will lie, in this case, outside the recirculation zone resulting from the flow separation of the first row. At Re= 250 the peak value is around Sp /D=3. Increasing Re, the peak value is shifted to higher Sp /D gradually until it vanishes. In all cases the variation in Nu with respect to Sp /D lie in a very narrow band. Fig. 20 shows the velocity vector diagram at Re=1000, Sn /D=1.5 and Sp /D=2. The diagram indicates the development of the velocity vector inside the two recirculation zones behind each tube row. The axial velocity profiles at different locations inside the flow field are represented in Fig. 21. The figure shows the effect of Re on the velocity profiles specially at the minimum flow
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area and in the recirculation zones. It can be seen that the velocity profiles at the minimum flow area for high Re is close to the confined turbulent flow profiles. On the other hand, when Re increases, the turbulence kinetic energy increases too, as shown in Fig. 22. This can be attributed to the corresponding increase in the eddy diffusivity. The air temperature contours around the tubes are given in Fig. 23. When increasing Re the fluid velocity increases a condition which leads to reduce the temperature of air leaving the tube banks. When the cold flow passes through the first tube row, the same pattern of the local Nu in case of single row is obtained. This is because of the constancy of the initial conditions in both cases, while the local Nu around the second tube row is slightly increased. This can be attributed to the increase in air velocity attacking the second row because of the nozzle effect formed by the first row of tubes. On the other hand, the turbulence kinetic energy (which is considered as a measure of turbulence intensity) is increased behind the first row due to the recirculation zone. This increase in the turbulence intensity enhancing the heat transfer rate and consequently Nu. The behaviour of the local Nu at different Re is shown in Fig. 24. The figure indicates the increase in Nu with increasing Re in both rows.
3.3. Staggered arrangement In the case of staggered tube banks, the air leaves the first row as jets which possessing a great amount of momentum. The tubes of the second row lie in the axis of these jets a condition which leads to the impingement between the air jet and these tubes. For Sp /Sn B 1 (i.e. Sp /D B1.5), it is found that the second row of tubes blocks the flow and consequently reduces the impingement velocity which in turn reduces the friction factor. By increasing Sp /Sn more than one (i.e. Sp /D\ 1.5), the minimum flow area will lie between the tubes of the first row. Therefore the impingement flow velocity (which is the governing factor on the friction) on the second row of tubes decreases a condition which leads to a decrease in the friction factor as indicated in Figs. 25 and 26. It is found that the staggered arrangement of the tube banks leads to increase the local turbulence kinetic energy along the computational domain. This in turn enhances the coefficient of heat transfer from the tubes to the flowing air and consequently increases Nu with respect to Re, as shown in Fig. 27. At low Re, i.e. in the laminar flow range, it is found that increasing the parallel tube pitch leads to decrease the impingement velocity of the air stream on the second row of tubes. This leads to decrease the coefficient of heat transfer as shown in Fig. 28. The velocity vector represents the development of air flow field as plotted in Fig. 29. In this plot, the effect of the second tube row on the redirection of the flow field
is observed. It reduces the recirculation zone behind the first row. For this reason high velocity gradient between the two rows is obtained as indicated in Fig. 30. The corresponding increase in local turbulence kinetic energy is observed also in Fig. 31. Because of the increase of local air velocity in case of staggered tube arrangement compared with the in-line one, the flow convection overcomes the thermal penetration from the tube surface. So the temperature gradient is relatively higher. This behaviour is represented in Fig. 32. According to the air pattern described before, the local Nu around the second tube row differs from the first one. The high air velocity attacking the second row raises the coefficient of heat transfer at the tube front as shown in Fig. 33. The friction factor and the Nusselt number in both cases of in-line and staggered arrangements are compared together in Figs. 34 and 35, respectively. Both the friction factor and Nusselt number are higher in case of staggered arrangement for low range of Sp /D.
4. Conclusion In this work, a mathematical model is developed to simulate the laminar and turbulent flow fields inside tube banks. Different tube arrays are considered. The in-line and staggered tube arrangements as well as the single row of tubes are studied. The 2D elliptic partial differential conservation equations of mass, momentum and energy besides the two equations of turbulence model are solved simultaneously. It can be concluded from the theoretical results that: 1. In case of a single row of tubes, both the minimum friction factor and maximum Nusselt number can be obtained at Sn /D between 1.5 and 1.75 and Re between 5000 and 100 000. 2. In case of in-line arrangement, it is found that the highest Nu is obtained when Sp /D around 3. On the other hand, with increasing Sp /D, the pressure drop and consequently the friction factor increases. For these reasons it is recommended to use Sp /D53 to obtain the best performance and to achieve high degree of compactness. 3. For staggered tube banks when Sp /D is more than 1.5, no significant effect on the friction factor and Nusselt number is obtained. It is recommended to use Sp /D51.5 in order to reduce f and enhance Nu. 4. Nusselt number in case of staggered arrangement is somewhat higher than that of the in-line one specially in low range of Sp /D. 5. The local axial velocity and turbulence kinetic energy around the second row of tubes are somewhat higher in case of staggered arrangement than the in-line array.
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5. Nomenclature A C1 C2 CD Cm cp D dh E f Gk h I k m; Nu Nuu n Pf Pr Prt p q Re Sn Sp T u u% um ut 6 6% x y yp y+
Greek symbols a
Heat transfer area (m2) Coefficient in turbulent transport equations Coefficient in turbulent transport equations Coefficient in turbulent transport equations Coefficient in turbulent transport equations Specific heat at constant pressure (J kg−1 K−1) Tube outer diameter (m) Hydraulic diameter (m) Constant in near-wall description of velocity profile (E = 9.793) Friction factor Volumetric generation rate of turbulent kinetic energy (m2 s−2 s−1) Specific enthalpy (J kg−1) Turbulence intensity Turbulence kinetic energy (m2 s−2) Mass flow rate of air (kg s−1) Global Nusselt number Local Nusselt number Number of tube rows in flow direction P-function Prandtl number Turbulent Prandtl number Pressure (Pa) Heat flux (W m−2) Reynolds number Normal tube spacing (m) Parallel spacing (m) Temperature (K) Fluid velocity in x-direction (m s−1) Fluctuating component of velocity in x-direction Mean flow velocity, m s−1 Shear velocity (ut = t/r) (m s−1) Fluid velocity in y-direction (m s−1) Fluctuating component of velocity in y-direction Coordinate parallel to the main flow Coordinate normal to the main flow Distance from the tube wall to the cell pole Dimensionless normal distance from tube wall
Convective heat transfer film coefficient (W m−2 K−1)
G o k l m mt n c r sk
t
Coefficient of diffusivity (kg m−1 s−1) Energy dissipation rate (m2 s−2 s−1) von Karman’s constant Thermal conductivity (W m−1 K−1) Dynamic viscosity (Pa s) Turbulent viscosity (Pa s) Kinematics viscosity (m2 s−1) Void fraction of tube arrangement Density (kg m−3) Effective Schmidt number for transport of k Effective Schmidt number for transport of o Shear stress (N m−2)
Subscripts b e f in m out t w
Bulk Effective Film Inlet Mean Outlet Turbulent Wall
so
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[13] W.M. Kays, R.K. London, Basic heat transfer and flow friction data for gas flow normal to banks of staggered tubes: use of transient technique, Stanford Univ. Tech. Rep. 15 (1952) 1. [14] H. Hausen, Heat Transfer from Tubes in Counter Flow, Parallel Flow and Cross Flow, McGraw-Hill, USA, 1983, p. 54. [15] A. Zukauskas, Heat transfer from tubes in cross flow, Adv. Heat Transf. 8 (1972) 93 –160. [16] M. Beziel, K. Stephan, Heat transfer and pressure drop in tube banks at high turbulence intensity, Inst. Chem. Eng. Sympos. Ser. 2 (129) (1992) 891–897. [17] J. Li, Q. Zhou, S. Ye, N. Gu, Heat transfer and draft loss performance of air flowing across staggered banks of spirally corrugated tubes, Inst. Chem. Eng. Symp. Ser. 2 (129) (1992) 757 – 963. [18] Y. Zhang, Z. Chen, Effect of gap between layers on the heat transfer performance of aligned tube banks, Heat Transf. Eng. 13 (2) (1992) 33 – 41. [19] A.A. Moreno, E.M. Sparrow, Heat transfer, pressure drop, and fluid flow patterns in yawed tube banks, Int. J. Heat Mass Transf. 30 (10) (1987) 1979–1995. [20] D. Murray, Comparison of heat transfer in staggered and in-line tube banks with gas-particle cross flow, Exp. Ther. Fluid Sci. 6 (2) (1993) 177. [21] F. Zdravistch, C. Fletcher, M. Behnia, Numerical laminar and turbulent fluid flow and heat transfer predictions in tube banks, Int. J. Num. Meth. Heat Fluid Flow 5 (8) (1995) 717–733. [22] D. Yu, R. Barron, T. Ameel, R. Warrington, Mixed convection from horizontal tube banks between two vertical parallel plates, Num. Heat Transf. 27 (4) (1995) 473–486.
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[23] M. Tierney, CFD predictions of the hydraulic performance of heat exchanger, Inst. Chem. Eng. Symp. Ser. 1 (129) (1992) 397 – 408. [24] Y. Jun, W. Tabakoff, Numerical simulation of a dilute particulate flow (laminar) over tube banks, J. Fluids Eng. Trans. ASME 116 (4) (1994) 770 – 777. [25] J. Fan, D. Zhou, Q. Hua, K. Cen, Numerical computation of particle laden gas flows past staggered tube banks undergoing erosion, Powder Tech. 80 (1) (1994) 1 – 10. [26] A. Thom, C. Apelt, Fluid Computations in Engineering and Physics, Van Nostrand, London, 1961. [27] B. Launder, T. Massey, The numerical prediction of viscous flow and heat transfer in tube banks, J. Heat Transf. 100 (1978) 565 – 571. [28] M. Fujii, T. Fujii, A numerical analysis of laminar flow and heat transfer of air in an in-line tube bank, Num. Heat Transf. 7 (1984) 89 – 102. [29] M. Faghri, N. Rao, Numerical computation of flow and heat transfer in finned and unfinned tube banks, Int. J. Heat Mass Transf. 30 (2) (1987) 363 – 372. [30] B. Launder, D. Spalding, The numerical computation of turbulent flow, Comp. Meth. Appl. Mech. Eng. 3 (1974) 269–289. [31] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, McGraw Hill, New York, 1980. [32] W. Kast, Druckverlust bei der Stro¨mung quer zu Rohrbu¨ndeln, VDI-Wa¨rme-atlas, 2. Auflage, Abschnitt Ge, 1974. [33] D. Spalding, C. Jayatillaka, A survey of the theoretical and experimental information on the resistance of the laminar sublayer to heat and mass transfer, in: Proceedings of the Second All Soviet Union Conference on HMT Minsk, USSR, 1964.
Modeling of heat transfer for flow across tube banks A. Safwat Wilson, M. Khalil Bassiouny * Mechanical Power Department, Faculty of Engineering, Menoufia Uni6ersity, Shibin El-Kom, Egypt Received 30 March 1999; received in revised form 4 May 1999; accepted 4 May 1999
Abstract A calculation procedure for two-dimensional elliptic flow is applied to predict the pressure drop and heat transfer characteristics of laminar and turbulent flow of air across tube banks. The turbulence model used involves the solution of two partial differential equations, one for the kinetic energy of the turbulence and the other for its dissipation rate. These differential equations are solved simultaneously with those for the conservation equations of mass, momentum and energy using an implicit finite volume procedure. The numerical methodology utilizes the stepped boundary technique to approximate the tube surface which is kept at constant temperature. The computations are extended to cover the case of two rows of tubes undergoing cross flow with in-line and staggered tube arrangements besides the case of a single row. Thereby, Reynolds number (Re) as well as the normal and parallel tube spacing-to-diameter ratios are varied. Effects of the flow and the geometry parameters on the friction factor and the local and global Nusselt number are presented. Moreover, velocity vector diagrams and temperature contours as well as axial flow velocity and turbulence kinetic energy profiles along the flow field upstream, over and downstream the tubes are also given. The theoretical results of the present model are compared with previously published experimental data of different authors. Satisfactory agreement is demonstrated. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Mathematical model; Numerical computations; Tube banks; Cross flow; Heat transfer
1. Introduction Because many heat exchanger arrangements involve multiple rows of tubes in a fluid cross flow, the heat transfer characteristics for tube banks are of important practical interest. The gas flowing on the outside impinges perpendicularly on these tubes. With constant fluid velocity, this type of flow gives rise to an increase in turbulence compared with the case where the gas flows along the tubes and parallel to them. It can thus deduced that the heat transfer coefficient at the outer surface of a tube is higher for cross flow than for parallel flow. The distribution of velocity around a tube in cross flow will not be uniform. In the same way the rate of heat flow around a hot pipe across which air is passed is not uniform also. Depending on application and design criteria, there are many possible tube layout in an array. One of the greatest difficulties with this system is that the area for flow is continually changing.
Dedicated to Prof em. Dr.-Ing. E.-U. Schlu¨nder on the occasion of his 70th birthday. * Corresponding author. Fax: +20-48-235695.
Moreover, the degree of turbulence is considerably higher for a staggered bank of tubes than for in-line one. With the small bundles, which are common in the chemical industry, the selection of the true mean area for flow is further complicated by the changing in number of tubes in the rows. Most of the earlier studies were experimental in nature. A number of worker like Reiher [1], Hilpert [2], Griffiths and Awbery [3], Jakob [4], Fand [5] and Churchill and Bernstein [6] have studied the flow of a fluid past a single cylinder varying from a thin wire to a tube of 150 mm diameter with different Reynolds numbers (Re). Gnielinski [7] proposed correlations for calculating the global Nusselt number in case of a single row of tubes as will as tube bundles with multiple rows based on the correlation of Krischer [8] for a single tube in cross flow by introducing the mean velocity instead of the free stream velocity in the definition of Re. His method takes into account the arrangement of tubes and the number of tube rows in the flow direction. A research work into heat transfer and pressure drop outside tubes and tube assemblies in cross flow has been carried out by Niggeschmidt [9]. The heat
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transfer values measured by him are presented in empirical equations. The test measurements of Pierson [10] and Huge [11] were correlated by Grimison [12] for in-line and staggered arrangements. His empirical relation pertain to tube banks having ten or more rows of tubes in the flow direction. For fewer rows, Kays and London [13] gave numerical values for the ratio of Nu for n rows deep to that of ten rows. Hausen [14] modified slightly the correlation of Grimison and set out an empirical formula for the tubes arrangement factor ( fa) instead of the graphical representation by
Fig. 1. Heat flux to the cells adjacent to the tube wall.
Fig. 2. Grid system used in the present computational domain.
Grimison. An excellent review of experimental investigations for heat transfer from tube banks in cross flow is given in Zukauskas [15]. Results of experimental investigation on the influence of turbulence intensity, transversal and longitudinal pitch-to-diameter ratio on heat transfer and pressure drop in tube bundles of in-line arrangement are presented by Beziel and Stephan [16]. The same experiments have been performed by Li et al. [17] but for staggered arrangement of spirally corrugated tubes. Zhang and Chen [18] carried out experiments on in-line tube banks to study the effect of gap width between tube layers on the heat transfer performance. Experiments were performed by Moreno and Sparrow [19] to investigate the influence of yaw angle on heat transfer, pressure distribution and flow visualization in cases of staggered and in-line tube banks. The heat transfer characteristics of tube bundles subject to dilute particle laden flow were studied experimentally by Murray [20]. Numerical methods have also been used to simulate the heat transfer and fluid flow in tube banks with cross flow of pure gases [21–23] and with particles suspended in gas flow without heat transfer [24,25]. The complex geometry of the flow configuration poses an obstacle to the use of numerical methods. Different methods have been tried such as conformal mapping technique of Thom and Apelt [26], and the hybrid polar—Cartesian grid approach of Launder and Massey [27] and Fujii and Fujii [28], besides the finite volume method of Faghri and Rao [29] for the computation of flow and heat transfer in tube banks. The last one is restricted to laminar flow across in-line arrangements only with the assumption of a periodic, fully developed flow which deviates from the practical case. This paper is an extension of and a complement to the earlier works in this field. The numerical solutions were performed for both laminar and turbulent flow across tube banks of single and double rows in the flow direction in cases of in-line as well as staggered tube arrangements. The friction factor and the local and global Nu calculations are carried out for different values of Re as well as normal and parallel spacing-todiameter ratios.
2. Mathematical model
Fig. 3. Comparison between the present work and different empirical correlations for Nusselt number (single row).
The flow of air through the tube banks is simulated mathematically. The tubes are arranged in a single row or double rows in flow direction. Two arrays of tube banks namely, in-line and staggered arrangements are considered. In this, a single phase, incompressible fluid (air), developed, non-periodic and time independent flow is considered. Both laminar and turbulent flow regimes are taken into account. According to these assumptions, the conservation equations of mass, mo-
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3
Fig. 4. Comparison between the present work and different empirical correlations for Nusselt number (in-line and staggered arrangements).
mentum and energy are used in Cartesian two-dimensional domain as follows:
Where, Gh = G+ Gt =
2.1. Continuity equation
m m + t Pr Prt
The Prandtl number and turbulence Prandtl number for the present flow regime are taken to be 0.7 and 0.9, respectively.
( ( (ru)+ (r6)= 0 (x (y
2.2. Momentum equation in axial direction (x)
( (u (u ( ruu−me + ru6 −me (x (x (y (y =−
(u (6 (p ( ( + me + me (x (x (x (y (x
Where, me = m +mt and mt = Cmr
k2 o
2.3. Momentum equation in 6ertical direction (y)
( (6 ( (6 ru6 −me + r66 − me (x (x (y (y =−
(p ( (u ( (6 + m + m (y (x e (y (y e (y
2.4. Energy equation
( (h ( (h ruh−Gh + r6h −Gh =0 (x (x (y (y
Fig. 5. Comparison between the present work and empirical correlations for friction factor (in-line and staggered arrangements).
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A. Safwat Wilson, M. Khalil Bassiouny / Chemical Engineering and Processing 39 (2000) 1–14
Fig. 6. Variation of friction factor with Reynolds number for different Sn /D (single row).
Fig. 7. Variation of friction factor with normal tube spacing ration for different Re (single row).
Fig. 8. Effect of Reynolds number on Nusselt number with different Sn /D (single row).
Fig. 9. Effect of normal tube spacing ratio on Nusselt number with different Re (single row).
2.5. Turbulence model The eddy diffusivity appeared in the time averaged momentum equation is obtained by solving the two equations of turbulence model (k −o). The standard model is modified by Launder and Spalding [30] for incompressible flow. The differential equation of turbulence kinetic energy (k) may be written in the following form:
Fig. 10. Velocity vector diagram (single row).
A. Safwat Wilson, M. Khalil Bassiouny / Chemical Engineering and Processing 39 (2000) 1–14
5
( me (o ( me (o o ruo − + r6o − = (C1Gk −C2ro) (x so (x so (y (y k The k− o model constants may be recasted as follows: C1 1.44
C2 1.92
Cm 0.09
CD 1
sk 1
so 1.3
2.6. Boundary conditions
Fig. 11. Dimensionless axial velocity profiles (single row).
The computational domain is bounded by inlet and outlet sections, two axes of symmetry and outer tube wall. The treatment of air adjacent to the domain boundaries is as follows:
Fig. 12. Dimensionless turbulence kinetic energy profiles (single row).
Fig. 14. Local Nusselt number around the tube for different Re (single row).
Fig. 13. Temperature contours (single row).
( m (k ( m (k ruk − e + r6k − e =Gk −CDro (x sk (x (y sk (y Where,
Gk = me 2
(u (x
2
+2
(6 (y
2
+
(u (6 + (y (x
n 2
and, k=
u%2 + 6%2 2
The energy dissipation rate (o) is calculated from the following equation:
Fig. 15. Local Nusselt number around the tube for different Sn /D (single row).
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6
c= 1−
Re=
p S 4 n D
umdh n
The other inlet conditions are 6= 0,
p=pin,
T= Tin,
k= I · u 2in o=
k 3/2 l · Sn /2
Fig. 16. Variation of friction factor with Reynolds number for different Sp /D (in-line arrangement).
Fig. 18. Effect of Reynolds number on Nusselt number with different Sp /D (in-line arrangement).
Fig. 17. Variation of friction with parallel tube spacing ratio for different Re (in-line arrangement).
2.6.1. Inlet section All the fluid parameters have uniform distributions along the inlet section. The initial velocity is calculated from the value of Re according to the following relation: u =uin =
cnRe dh
Where, Fig. 19. Effect of parallel tube spacing ratio on Nusselt number with different Re (in-line arrangement).
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Fig. 20. Velocity vector diagram (in-line arrangement).
The values of the initial turbulence intensity I and the length scale of turbulence eddies l are taken to be 0.03 and 0.005, respectively.
2.6.2. Axis of symmetry The flow at the axis of symmetry is considered as one dimensional, axial flow. The gradient of the flow parameters at the normal direction are equal to zero. The flow parameters at the axis of symmetry can be stated as follows: (f =0, (y
qw =
l(Tw − Tf) for y+ 5 5, yp
rutcp(Tw − Tf) for y+ \ 5. 1 Prt ln Ey+ + Pf k The heat transfer coefficient from the tube wall to the flowing air is determined from the following relation: qw =
(f =u, 6, p, h, k and o)
and, 6 = 0.
2.6.3. Exit section The computational domain length behind the tube is selected long enough to ensure fully developed flow condition at the exit section. The exit conditions are 6 = 0, p= pout, T=Tout
Fig. 21. Dimensionless axial velocity profiles (in-line arrangement).
2.6.4. Tube wall The tube wall temperature is considered constant and is taken to be 60°C above the fluid inlet temperature. All the other remaining flow parameters are set to zero at the tube surface. The standard wall function is used to modify the fluid parameters at the cells adjacent to the tube surface. In literature many forms of the Pfunction are available. One of them is that of Spalding and Jayatillaka [33], which is valid for incompressible flow past impermeable smooth surface:
Pf = 9.24
Pr Prt
0.75
−1
n
1 + 0.28e
− 0.007
Pr Prt
n
Fig. 22. Dimensionless turbulence kinetic energy profiles (in-line arrangement).
For constant tube wall temperature, the local Nu is calculated using the following equation: Nuu =
au · D l
Where, au =
qw Tw − Tf
The heat flux from the wall is calculated according to the thickness of the viscous sublayer y+.
Fig. 23. Temperature contours (in-line arrangement).
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Fig. 24. Local Nusselt number around the tubes for different Re (in-line arrangement).
a=
mcp(Tout − Tin) A(Tw −Tb)
The global Nusselt number along the entire domain is obtained by Nu =
aD l
Finally, the friction factor is defined as: f=
Dp dh 2 n D r·u 2m
3. Results and discussion For the sake of verifying the validity of the present computational model, a comparison study between the theoretical results and previous experimental data of different authors is performed for both heat transfer and flow resistance in tube banks undergoing cross flow. Global Nu results are compared with the correlations of Gnielinski [7], Niggeschmidt [9] and Hausen [14] in Fig. 3 for a single row of tubes with a normal spacing-to-diameter ratio of 1.5 and at different Re. The same comparison is carried out in Fig. 4 but for double rows of tubes in flow direction for both cases of
2.7. Numerical solution The finite volume technique [31] is used to solve the governing differential equations. The power scheme is employed to descritize the nonlinear elliptic partial differential equations. These equations are solved together iteratively until the residuals along the domain become B 5×10 − 3. The grid system used in the present numerical computations is Cartesian. The distribution of grids in the flow direction is constant along the computational domain length, while the vertical distribution is non-uniform. The grids are condensed over the tubes to ensure accurate simulation for the fluid inside the high gradient zone. Because of the nature of the Cartesian coordinates, the smooth curved surfaces of the tubes are transformed into series of steps, as indicated in Fig. 1. In order to reduce the steps size, the reasonable number of grids is selected. Therefore, the grid system is chosen to be 66×33. Fig. 2 shows the geometry parameters and the grid system employed in this work.
Fig. 25. Variation of friction factor with Reynolds number for different Sp /D (staggered arrangement).
A. Safwat Wilson, M. Khalil Bassiouny / Chemical Engineering and Processing 39 (2000) 1–14
Fig. 26. Variation of friction factor with parallel tube spacing ratio for different Re (staggered arrangement).
9
Fig. 28. Effect of parallel tube spacing ratio on Nusselt number with different Re (staggered arrangement).
Fig. 29. Velocity vector diagram (staggered arrangement).
Fig. 27. Effect of Reynolds number on Nusselt number with different Sp /D (staggered arrangement).
in-line and staggered tube arrangements with Sn /D= 1.5 and Sp /D=2. The figures show a fair agreement specially in the turbulent flow region. The theoretical work indicates the different flow zones namely: laminar, transient and turbulent with different slopes while the correlations have a fixed form for the whole regions. Fig. 5 represents a comparison between the calculated results and the measured data of Pierson [10] and Huge [11] for the pressure drop which are replicated by Jakob [4] in form of friction factor. The correlation of
Fig. 30. Dimensionless axial velocity profiles (staggered arrangement).
Fig. 31. Dimensionless turbulence kinetic energy profiles (staggered arrangement).
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Fig. 32. Temperature contours (staggered arrangement).
Kast [32] for staggered arrangement is used also in the present comparison. This correlation is independent of the geometry parameters. A noticeable deviation is observed between his results and that of Jakob. A satisfactory agreement between the present theoretical results for the friction factor and Jakob’s empirical relations can be observed for both cases of in-line and staggered arrangement of tubes.
3.1. Single row The variation of friction factor with Re at different values of the normal spacing-to- diameter ratio is indicated in Fig. 6. The figure shows that the friction factor decreases steeply in the laminar flow region and slightly in the turbulent flow one until it reaches a minimum value at Re=10 000 then it increases slightly once more. Fig. 7 represents the friction factor as a function of Sn /D with varying Re as a parameter. It can be easily seen that f exhibits a minimum at a normal spacing of
Fig. 34. Comparision between staggered and in-line arrangements for friction factor.
about 1.5 the tube outer diameter, above which the friction factor increases steeply. Minimum values of f may be obtained in the range of Re between 5000 and 100 000 below which f increases steeply too. The effect of varying Re number as well as Sn /D on the global Nu for a single row of tubes are illustrated in Figs. 8 and 9. It is clear that Nu increases always with Re. This increase has different slopes in laminar and turbulent flow regions. The normal spacing ratio has a significant influence on Nu. With increasing Sn /D, Nus-
Fig. 33. Local Nusselt number around the tubes for different Re (staggered arrangement).
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selt number decreases. This is owing to the nozzle effect between the tubes. The decrease becomes steeper for values of Sn /D between 1.25 and 2. This behaviour may be attributed to the sharp reduction in the separation zone with increasing Sn /D in the considered range. The velocity vector diagram upstream, over and downstream the tubes is plotted in Fig. 10 for Re= 1000 and Sn /D =1.5. The figure indicates the development of the flow velocity along the computational domain. The location of the flow separation from the tube surface is clear at an angle of about 145° where the velocity gradient vanishes. Also, the reversed flow region, where the velocity gradient is negative, and the redeveloped flow zone, where the velocity gradient is positive, are remarkable downstream. Fig. 11 presents the dimensionless axial velocity profiles at constant normal spacing ratio Sn /D= 0.5 and at two values of Re namely 250 and 5000. The velocity profiles are plotted at different locations upstream, downstream and over the tubes. The figure shows clearly the development of the axial velocity specially the recirculation zone due to the flow separation. Corresponding profiles are given in Fig. 12 for dimensionless turbulence kinetic energy. The curves illustrates also the development of k in the separation region, where a peak value of k takes place. The isothermal lines are drawn in Fig. 13 for laminar flow (Re=250) as well as turbulent flow (Re=5000) with Sn /D=1.5. The effect of the flow velocity and the separation zone on the temperature contours are remarkable in the plots. The behaviour of the local Nu around the tubes at different Re as well as at different Sn /D is represented
Fig. 35. Comparision between staggered and in-line arrangements for Nusselt number.
11
in Figs. 14 and 15, respectively. The minimum value of Nuu at u equals about 145° may be corresponding to the flow separation from the tube surface and the beginning of the recirculation zone.
3.2. In-line arrangement The model is used also to study the in-line arrangement of tube banks. In this case, two rows are considered. The flow and heat transfer characteristics in this case depend on the normal and parallel tube spacing. Because the effect of the normal spacing Sn /D is studied in case of a single row, the effect of parallel tube pitch is taken into account. The results show a noticeable reduction in friction factor with increasing Re of the flowing air, as shown in Fig. 16. The trend of the curves show different slopes according to the transition in the flow regime from laminar to turbulent flow. The parallel tube spacing Sp /D has a great effect on the friction factor. The results show that increasing the tube pitch leads to an increase in pressure drop across the tube banks a condition which reflects on an increase in the friction factor as shown in Fig. 17. It is observed also that, at high Re the gradient of the friction factor approaches zero and the only effect on the friction factor in this case is the parallel tube spacing. This can be explained as follows: at relatively high Re the increase in the flow velocity leads to a corresponding increase in the pressure drop. On the contrary, it is found that the great effect on Nu comes from increasing Re but not the geometry parameter Sp /D, as shown in Fig. 18. When increasing Re, the separation zone becomes several times the tube diameter and hence the flow doesn’t reach the recovery point before entering the second tube zone whatever the value of Sp /D is. For this reason, it is found that at low Re, the parallel tube spacing has a significant effect. Fig. 19 represents the influence of Sp /D on Nu for low range of Re. It is clear from the figure that, the increase of Sp /D gives the flow behind the first row the chance to approach the reattachment point and enhance the heat transfer coefficient and consequently Nu. Further increase in Sp /D leads to a decrease in Nu. This is because the second row of tubes will lie, in this case, outside the recirculation zone resulting from the flow separation of the first row. At Re= 250 the peak value is around Sp /D=3. Increasing Re, the peak value is shifted to higher Sp /D gradually until it vanishes. In all cases the variation in Nu with respect to Sp /D lie in a very narrow band. Fig. 20 shows the velocity vector diagram at Re=1000, Sn /D=1.5 and Sp /D=2. The diagram indicates the development of the velocity vector inside the two recirculation zones behind each tube row. The axial velocity profiles at different locations inside the flow field are represented in Fig. 21. The figure shows the effect of Re on the velocity profiles specially at the minimum flow
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area and in the recirculation zones. It can be seen that the velocity profiles at the minimum flow area for high Re is close to the confined turbulent flow profiles. On the other hand, when Re increases, the turbulence kinetic energy increases too, as shown in Fig. 22. This can be attributed to the corresponding increase in the eddy diffusivity. The air temperature contours around the tubes are given in Fig. 23. When increasing Re the fluid velocity increases a condition which leads to reduce the temperature of air leaving the tube banks. When the cold flow passes through the first tube row, the same pattern of the local Nu in case of single row is obtained. This is because of the constancy of the initial conditions in both cases, while the local Nu around the second tube row is slightly increased. This can be attributed to the increase in air velocity attacking the second row because of the nozzle effect formed by the first row of tubes. On the other hand, the turbulence kinetic energy (which is considered as a measure of turbulence intensity) is increased behind the first row due to the recirculation zone. This increase in the turbulence intensity enhancing the heat transfer rate and consequently Nu. The behaviour of the local Nu at different Re is shown in Fig. 24. The figure indicates the increase in Nu with increasing Re in both rows.
3.3. Staggered arrangement In the case of staggered tube banks, the air leaves the first row as jets which possessing a great amount of momentum. The tubes of the second row lie in the axis of these jets a condition which leads to the impingement between the air jet and these tubes. For Sp /Sn B 1 (i.e. Sp /D B1.5), it is found that the second row of tubes blocks the flow and consequently reduces the impingement velocity which in turn reduces the friction factor. By increasing Sp /Sn more than one (i.e. Sp /D\ 1.5), the minimum flow area will lie between the tubes of the first row. Therefore the impingement flow velocity (which is the governing factor on the friction) on the second row of tubes decreases a condition which leads to a decrease in the friction factor as indicated in Figs. 25 and 26. It is found that the staggered arrangement of the tube banks leads to increase the local turbulence kinetic energy along the computational domain. This in turn enhances the coefficient of heat transfer from the tubes to the flowing air and consequently increases Nu with respect to Re, as shown in Fig. 27. At low Re, i.e. in the laminar flow range, it is found that increasing the parallel tube pitch leads to decrease the impingement velocity of the air stream on the second row of tubes. This leads to decrease the coefficient of heat transfer as shown in Fig. 28. The velocity vector represents the development of air flow field as plotted in Fig. 29. In this plot, the effect of the second tube row on the redirection of the flow field
is observed. It reduces the recirculation zone behind the first row. For this reason high velocity gradient between the two rows is obtained as indicated in Fig. 30. The corresponding increase in local turbulence kinetic energy is observed also in Fig. 31. Because of the increase of local air velocity in case of staggered tube arrangement compared with the in-line one, the flow convection overcomes the thermal penetration from the tube surface. So the temperature gradient is relatively higher. This behaviour is represented in Fig. 32. According to the air pattern described before, the local Nu around the second tube row differs from the first one. The high air velocity attacking the second row raises the coefficient of heat transfer at the tube front as shown in Fig. 33. The friction factor and the Nusselt number in both cases of in-line and staggered arrangements are compared together in Figs. 34 and 35, respectively. Both the friction factor and Nusselt number are higher in case of staggered arrangement for low range of Sp /D.
4. Conclusion In this work, a mathematical model is developed to simulate the laminar and turbulent flow fields inside tube banks. Different tube arrays are considered. The in-line and staggered tube arrangements as well as the single row of tubes are studied. The 2D elliptic partial differential conservation equations of mass, momentum and energy besides the two equations of turbulence model are solved simultaneously. It can be concluded from the theoretical results that: 1. In case of a single row of tubes, both the minimum friction factor and maximum Nusselt number can be obtained at Sn /D between 1.5 and 1.75 and Re between 5000 and 100 000. 2. In case of in-line arrangement, it is found that the highest Nu is obtained when Sp /D around 3. On the other hand, with increasing Sp /D, the pressure drop and consequently the friction factor increases. For these reasons it is recommended to use Sp /D53 to obtain the best performance and to achieve high degree of compactness. 3. For staggered tube banks when Sp /D is more than 1.5, no significant effect on the friction factor and Nusselt number is obtained. It is recommended to use Sp /D51.5 in order to reduce f and enhance Nu. 4. Nusselt number in case of staggered arrangement is somewhat higher than that of the in-line one specially in low range of Sp /D. 5. The local axial velocity and turbulence kinetic energy around the second row of tubes are somewhat higher in case of staggered arrangement than the in-line array.
A. Safwat Wilson, M. Khalil Bassiouny / Chemical Engineering and Processing 39 (2000) 1–14
5. Nomenclature A C1 C2 CD Cm cp D dh E f Gk h I k m; Nu Nuu n Pf Pr Prt p q Re Sn Sp T u u% um ut 6 6% x y yp y+
Greek symbols a
Heat transfer area (m2) Coefficient in turbulent transport equations Coefficient in turbulent transport equations Coefficient in turbulent transport equations Coefficient in turbulent transport equations Specific heat at constant pressure (J kg−1 K−1) Tube outer diameter (m) Hydraulic diameter (m) Constant in near-wall description of velocity profile (E = 9.793) Friction factor Volumetric generation rate of turbulent kinetic energy (m2 s−2 s−1) Specific enthalpy (J kg−1) Turbulence intensity Turbulence kinetic energy (m2 s−2) Mass flow rate of air (kg s−1) Global Nusselt number Local Nusselt number Number of tube rows in flow direction P-function Prandtl number Turbulent Prandtl number Pressure (Pa) Heat flux (W m−2) Reynolds number Normal tube spacing (m) Parallel spacing (m) Temperature (K) Fluid velocity in x-direction (m s−1) Fluctuating component of velocity in x-direction Mean flow velocity, m s−1 Shear velocity (ut = t/r) (m s−1) Fluid velocity in y-direction (m s−1) Fluctuating component of velocity in y-direction Coordinate parallel to the main flow Coordinate normal to the main flow Distance from the tube wall to the cell pole Dimensionless normal distance from tube wall
Convective heat transfer film coefficient (W m−2 K−1)
G o k l m mt n c r sk
t
Coefficient of diffusivity (kg m−1 s−1) Energy dissipation rate (m2 s−2 s−1) von Karman’s constant Thermal conductivity (W m−1 K−1) Dynamic viscosity (Pa s) Turbulent viscosity (Pa s) Kinematics viscosity (m2 s−1) Void fraction of tube arrangement Density (kg m−3) Effective Schmidt number for transport of k Effective Schmidt number for transport of o Shear stress (N m−2)
Subscripts b e f in m out t w
Bulk Effective Film Inlet Mean Outlet Turbulent Wall
so
13
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