Numerical and experimental study of flow and heat transfer around a tube in cross-flow at low Reynolds number

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Abstract

A numerical and experimental study of the laminar flow and heat transfer characteristics of a cylinder in cross-flow is presented. The computational technique used is a stream function-vorticity formulation of the laminar flow steady state incompressible Navier–Stokes and energy equations and uses a Gauss–Seidel over-relaxation technique to obtain stream function and temperature distributions. Calculations are presented for an isothermally heated single tube in a duct with different blockage ratios. The variation of local Nusselt number, pressure and also isotherm and streamline contours are predicted with Reynolds number of 120 and 390. For the Reynolds number of 390, the local Nusselt number distributions are shown to be similar to those obtained through measurement of the local heat flux from the surface of a tube using a micro-foil heat flow sensor.

Introduction

Heat exchangers with tube banks in cross-flow are of great practical interest in many thermal and chemical engineering processes. In order to examine the process of flow and heat transfer for a single tube and a tube in a bank and their response to changes of geometry or flow conditions, it is possible to undertake experimental and numerical studies on simple geometrical models. Local surface values and overall flow and temperature distributions can be used to determine the effect of each geometrical or flow parameter. Consequently a large number of experimental and numerical studies have already been carried out to determine heat transfer and flow structures for single tubes and tube banks under different conditions to consider the effects of blockage, of longitudinal or transverse pitch ratios, of Reynolds number and of other conditions. Buyruk et al. (1995) have studied experimentally the local Nusselt number distribution within a single tube row for different blockage ratios in cross-flow for Reynolds numbers between 7960 and 47770. Akilbayev et al. (reported by Zukauskas, 1972), Perkins and Leppert (1964), West and Apelt (1982) and Hiwada and Mabuchi (1980) have investigated the effect of blockage by either changing duct width or by altering tube diameter in the subcritical Reynolds number range (i.e. laminar boundary layer). Chen et al. (1986), Paolino et al. (1986), and Launder and Massey (1978) have predicted Nusselt number distributions for single tubes in large ducts (i.e. no blockage effects) and for a tube in a bank.

Although it is beneficial to operate heat exchangers in turbulent flow, it is not unusual for them to operate with low Reynolds number flow. From a review of the literature it is evident that only a few experimental studies have investigated the flow and heat transfer around a tube for low Reynolds numbers. Eckert and Soehngen (1953) used a Mach–Zehnder interferometer to obtain the temperature field in the wake of heated cylinders up to 38.1 mm diameter for low Reynolds numbers. Krall and Eckert (1973) investigated the local heat transfer coefficient around a 4.73 mm cylinder for Reynolds numbers between 7.4 and 4640 without specifying the effect of blockage on the distribution. Dennis et al. (1968) calculated the distribution of heat transfer coefficient around a cylinder in viscous flow with Reynolds number between 0.01 and 40. They were able to show good agreement with experimental data for the average heat transfer coefficient but were unable to make comparisons for the circumferential distribution of heat transfer coefficient due to the absence of experimental data.

For tube banks, the overall flow and heat transfer characteristics have been investigated by several authors. Omohundro et al. (1949) obtained results for the overall pressure drop and heat transfer with laminar flow through a staggered tube bank. Bergelin et al. (1949) undertook similar work for in-line and staggered tube bank geometries. Antonopolous (1985) computed the convection heat transfer in tube assemblies for laminar flow. He calculated both local and average heat transfer coefficients and again the lack of experimental data for the circumferential distribution of heat transfer was evident. Similar observations can be made regarding the study of Faghri and Rao (1987) who computed values for average Nusselt number in finned and unfinned tube banks. More recently Zdravistch et al. (1995) carried out a numerical study of both laminar and turbulent heat transfer in tube banks; again, because of the absence of detailed measurements in laminar flow, they had to resort to comparing overall heat transfer coefficients using the data of Bergelin et al. (1949).

There is a relative scarcity of data for heat transfer from tubes and tube banks at low Reynolds numbers and therefore the purpose of this paper is to present results of an investigation into the local variation of heat transfer coefficient around a cylinder in low Reynolds number flow and to consider the effect of varying blockage. The numerical and experimental study considers the effect of blockage on the flow and heat transfer characteristics of a single tube in a duct. The Reynolds numbers used are 120 and 390 and the blockage ratio is varied from 0.18 to 0.47. The computational technique uses the stream function-vorticity formulation to solve the laminar, steady-state Navier–Stokes equations and energy equation. Numerical results are obtained around half the cylinder including the wake zone beyond the separation point. The experimental work involved a heated copper cylinder of 50 mm diameter instrumented with a surface heat flux sensor which was used to obtain the circumferential distribution of heat transfer coefficient. Whilst the experimental data presented is limited in its ranges of Reynolds number and geometry, it nevertheless makes a contribution to this area where data is scarce.

Section snippets

Grid

Some flow problems can be solved using an orthogonal coordinate system (ξ,η) where ξ and η are known functions of x and y and the flow boundaries are lines of constant ξ or constant η. The grid is then simply made up of lines of constant ξ and constant η. A common example is the cylindrical co-ordinate system. In the present study, the grid generation is based on that of Thompson et al. (1974) as developed by Johnson (1990). The method generates orthogonal curvilinear grids by solving two

Experimental procedure

Experiments were carried out using a thick-walled copper cylinder which was heated internally by an electric cartridge heater and held between two PVC tubes to form a single tube assembly as shown in Fig. 3. The heat flux from the surface of the cylinder was measured using an RdFTM micro-foil heat flux sensor type 27034 attached to the surface with an epoxy resin adhesive. The voltage output from the sensor was connected to a digital multimeter and this reading was converted to heat flux using

Results and discussion

For a cylinder in cross-flow a laminar boundary layer develops from the front stagnation point and grows in thickness around the cylinder. Separation of the laminar boundary layer takes place when the low velocity fluid close to tube wall can no longer overcome the adverse pressure gradient over the rear portion of the tube and the flow stalls forming a region of reverse flow close to the surface. This reverse flow is confined to the region between the separation point and the rear stagnation

Conclusions

Numerical calculations have been carried out for laminar flow and heat transfer for flow past a single cylinder in a row of cylinders. The numerical investigation has considered the effect of blockage and Reynolds number on the heat transfer and flow characteristics. Calculations have been carried out for Reynolds numbers of 120 and 390 and blockage ratios between 0.18 and 0.47. The results were compared with experiment. The main conclusions are as follows:

  • Increased blockage causes the

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