Effect of transverse pitch on forced convective heat transfer over single row cylinder
 Kaprawi Sahim^{1}Email author
https://doi.org/10.1186/s4071201500387
© Sahim. 2015
Received: 16 March 2015
Accepted: 25 June 2015
Published: 28 July 2015
Abstract
Heat transfer problems are encountered in many engineering fields. Heat exchanger becomes important as the equipment for transfer heat. The tubes in heat exchanger consist of some tubes either inline or staggered arrangements. The heat transfer of three circular tubes arranged in single row is studied numerically to observe the effect of transverse pitch. The flow around the tube, the momentum equation in boundary layer is solved by finite volume method to obtain the velocity distribution. The energy equation is discretized by finite different methods and then solved by the iteration of relaxation method to obtain the temperature fields. The pressure term of the equation is replaced by the experimental data measured from wind tunnel tests. The calculation of velocity and temperature are carried out around the cylinder in the middle of the arrangement in which the number of nodes increase along longitudinal direction. The results show the local drag friction and the local Nusselt number at the tube surfaces change with transverse pitch. Starting from a certain value of transverse pitch, there is no influence on heat transfers and the tube arrays behave as a single tube. For small pitch, the heat transfer and drag coefficient variations are significant.
Keywords
Transverse pitch Heat transfer Circular cylinder Numerical methodBackground
Heat exchanger is one of the important apparatus in heat transfer that can be found in many processed industries. The arrangement of tubes in heat exchangers determines the heat transfer performance. Besides that, the distance between the tubes, called pitchs, plays also an important role. The pitchs make the geometrical shapes: rotated triangular pitch, triangular pitch, square pitch, and rotated square pitch [Brian Spalding and Taborek 1983]. All these pitchs are obtained from two main arrangements of tubes, inline or staggered.
The analytical models for the heat transfer of tube banks of inline and staggered arrangement are developed by Khan et al. [2006]. These models are developed in terms of longitudinal and transverse transverse pitch and Reynolds and Prandtl numbers. In this study, the authors explore the convection heat transfer associated with crossflow over the tubes for certain value of transverse pitch.
Minter Cheng 2004 has given numerical study of the influence of the distance (transverse pitch) of two circular cylinders on heat transfer. Equations of momentum in x and y directions are solved by finite volume method. In this case, the pressure distribution is guessed and then it is corrected until all variables converge. The heat transfer increase with transverse pitch ranging from 1 to 1.5 and heat transfer decrease with transverse pitch for transverse pitch greater than 1.5. The heat transfer equals to that of single cylinders if transverse pitch is equal to two. Heat transfer of each row of the heat exchanger is not the same, the smallest value is found in the first row, and it increases in the second and third rows as shown experimentally by Mehrabian 2007.
A study concerning with a complex flows and heat transfer in the laminar flow regime around a single row of tubes in a channel is given by Cho and Son 2008. The timedependent numerical approaches predicted the generation and evolution of vertical structures, wakes interactions, and their effects on the drag, lift, and heat transfer in the range of 20 < Re < 180.
Most heat transfer of flow around a circular cylinder occurs in the zone between the stagnation point and the separation point [Khan et al., 2005] Local variations are presented over the entire cylinder surface, including the zone beyond the separation points. At low Reynolds numbers, the heat transfer of the rear portion of a tube is at a minimum [AbdelRaouf et al. 2010; Buyruk, 2002; Kaptan et al., 2008; Rahmani et al., 2005]. The effects of longitudinal pitch on heat transfers have been also studied by Kim 2013 and Shinya et al. 1980.
A laminar boundary layer develops from the front stagnation point of a cylinder in crossflow and grows in thickness around the circular tube. The distributions of local heat transfer coefficients around cylinders are almost the same, except for the front half of the first row [Yanxing et al. 2000; Tahsee et al., 2013]. Transient numerical simulations of heat transfer were performed by Horvat and Mavko 2006 for heat exchanger segments with cylindrical and ellipsoidal tubes in the staggered arrangement. The value of Stanton number is lower for the ellipsoidal form in comparison to the cylindrical form of tube cross section.
From the literature survey, the effect of transverse pitch on heat transfers in tube bundles with single row is not yet established. There is no work that has been reported on the optimum limit of transverse pitch for a bank of tubes with single row. The purpose of this study is to investigate numerically the optimum limit of transverse pitch on heat transfers from single row of circular tube banks.
Governing equations
The only analytical model of heat transfer of tube bundle in a heat exchanger is given by Khan et al. (2006) which is valid for many rows of tubes, not for single row. In this study, the author tries to explore numerically the effect of transverse pitch by solving the mathematical equations of the fluid flow and the energy.
The governing equation of continuity, momentum, and energy equations in xdirection for steady incompressible flow of a Newtonian fluid with constant thermophysical properties, no heat generation, and negligible viscous dissipation are as follows:

u = v = 0, T = T _{S} at the cylinder surface

u = U _{ e }, T = T _{∞} far from cylinder
 1.
u ^{ + } = v ^{ + } = 0, T ^{ + } = 1 at the cylinder surface.
 2.
u ^{ + } = U _{ e } ^{ + }, T ^{+} = 0 far from cylinder surface.
It is noted that the boundary layer develops from the stagnation point of the cylinder surface.
Methods
By evaluating the integration of Eq. (8), the system of equations forms a Tridiagonal matrix that can be solved easily by the Cholesky method. The integration for all elements is carried out to obtain the velocity distribution of all nodes. The first term of the right hand of Eq. (5) is obtained by the measurement of the static pressure in a wind tunnel with a section test of 40 × 40 cm^{2}. The pressure measured at the surface of the middle cylinder then converts to velocity at the boundary layer.
Three circular cylinders of diameter 61 mm are placed, as the arrangement shown by Fig. 1, across the flow, and the pressures were measured at the surface of cylinder located in the middle. The calculation of Reynolds number is based on the maximum velocity between the cylinders. After calculation, we obtain maximum velocity U _{max} = 15 m/s for the cylinders with smallest transverse distance and the Reynolds number is 8.4 × 10^{4} which is laminar flow and it is less than the value of the transition Reynolds number [Lienhard et al. 2008].
The velocity distributions as the results of the solution of Eq. (9) are then injected into Eq. (10). We use Pr = 0.71 for all calculation in this study. The velocity distributions of the results of the solution to Eq. (9) are then injected into Eq. (10). We use Pr = 0.71 (air) for all calculations in this study.
Results and discussion
As shown by the dimensionless form of Eq. (6), the equation has no the diffusion term in x direction since the Reynolds number is sufficiently large to ensure the boundary layer structure to the flow. So the diffusion term is small compared with the other term of the equation since the term in this case is (∂^{2} u ^{+}/∂x ^{+2})/Re. Equation (2) or (6) is the momentum equation valid for steady laminar flow. As shown by many literatures, for example, Lienhard (2008), for the flow around a circular cylinder with, it means 100 < Re <3.10^{5}.
Angle of separation for different L/D
L/D  α (^{o}) 

1.17  129.23 
1.42  113.08 
1.67  100.38 
2.00  95.51 
2.33  95.07 
2.67  95.20 
3.00  94.31 
Local drag friction represents the velocity of flow near the surface of cylinder. For small distance of the cylinders, the flow tends to accelerate near α ≈ 90° as the minimum flow area. In this case, the flow characteristic behaves as the flow passes a nozzle and a diffuser where the cross sections change rapidly. The flow for α < 90° is a favorable gradient and never separates, the expanding area produces, α > 90°, low velocity and increasing pressure, an adverse gradient.
Conclusions
Numerical calculation for the forced convection heat transfer of a tube bank of single row in the air is performed in this work. The calculation is carried out to solve the momentum equation and energy equation. The velocity at the boundary layer is calculated by the results of pressure measurements. The numerical calculation is carried out by finite volume method to solve the momentum equation, and the iteration method is used to solve the energy equation. The results obtained are found that the effect of transverse pitch for L/D < 2 has a significant change of heat transfer. The heat transfer decreases with the decrease of L/D. The pitch has no influence on heat transfer for L/D ≥ 2. This study gives the limit of distance between cylinders in design of a tube bank in a heat exchanger.
Declarations
Authors’ Affiliations
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