# LOW REYNOLDS NUMBER CONVECTIVE HEAT TRANSFER ACROSS A CIRCULAR CYLINDER WITH UNIFORM VOLUMETRIC ENERGY DISSIPATION.

Byline: M. S. Kamran H. Ali F. Noor S. Imran A. Hussain and H. S. WangAbstract

Flow and heat transfer across a circular cylinder with uniform volumetric energy dissipation under unconfined condition was numerically investigated using Ansys Fluent 12.0. The fluid was water with Prandtl number 7. The Reynolds number varied from 10 to 40. The results were compared with constant surface temperature and constant heat flux at the surface of the cylinder. It was found that the average Nusselt numbers with uniform energy dissipation (UED) in the cylinder were about 11% lower than those with constant heat flux (CHF) boundary condition and approximately 4% higher than those with constant surface temperature (CST) boundary condition. The average Nusselt numbers under different boundary conditions were also correlated with the Reynolds numbers.

Key words: convective heat transfer cylinder uniform volumetric energy dissipation numerical simulation

INTRODUCTION

The demand for energy has led to a large global effort to develop and improve existing technologies and to find new ones. Magnetic refrigeration and heating are environmentally friendly emerging technologies with a realistic potential to replace the conventional vapour- compression refrigeration (Yu et al. 2010). Recently various reviews (Gschneidner and Pecharsky 2008; Kitanovski and Egolf 2010; Yu et al. 2010) reveal that in all up-to-date prototypes reported the pressure drop of the fluid in the regeneratoris significantly high and heat transfer rate is relatively low. This limits the performance of the refrigerators and heat pumps and has become a major obstacle.

Flow and heat transfer over circular cylinders are very common in many applications and have been extensively studied experimentally and numerically (Eckert and Soehngen 1953; Zukauskas et al. 1985; Williamson 1996; Hanafi et al. 2002; Nakamura and Igarashi 2004; Yoon et al. 2007; Henderson 1995; Bharti et al. 2007 and Park et al. 1998).The convective heat transfer over an array of cylinders under conjugate boundary condition i.e. uniform internal heating within each cylinder has been numerically studied for the Reynolds number(Re)ranging from 100 to 400 and Pr is 0.71(Wang and Georgidias 1996).

For active magnetic regenerator usually a very low mass flow rate of the working fluid yields optimal performance. The Reynolds number of the fluid flow in the prototypes reported so far is generally less than 50 (Nielson et al. 2009). During the magnetization and demagnetization processes under a magnetic field of the magnetic refrigeration cycle the energy dissipation occurs in the form of heat source and sink(conjugate boundary condition).Thus cylinders made of the magnetocaloric material (MCM) can be a possible alternative in the regenerative heat exchanger. In this study the flow and heat transfer over a single circular cylinder with uniform and constant volumetric energy dissipation is numerically investigated. The literature survey reveals that the flow and heat transfer characteristics for very low Re (below 100) have not been investigated. Thus the present simulation is performed for low Reynolds number range.

The present work investigates the influence of various boundary conditions for circular cylinder made of magnetocaloric material.

MATERIALS AND METHOD

Physical model and coordinate: The schematic diagram of the physical model and coordinate have been illustrated in figure (1). A circular cylinder of diameter d was placed in an unconfined fluid flow field having free stream velocity U8 and temperature T. The fluid was water. To simplify the following assumptions were made: (1) volumetric energy dissipation rate within the cylinder was taken to be uniform and constant; (2) the flow was laminar; (3) the flow was two dimensional (2D) incompressible and steady and (4) the properties of the cylinder and the fluid were constant. The Cartesian coordinate (x y) was taken from the centre of the cylinder. The angle was measured from the front stagnation point.

Governing equations:For two-dimensional incompressible steady laminar flow with constant properties the governing equations for the fluid region are as follows (Versteeg and Malalasekera 2007)

Mass conservation

Equations

Energy conservation

The boundary conditions are as below:

Equations

The mesh density was high for the regions of large gradients of velocity and temperature and mesh density was kept low where the gradients were small (Versteeg and Malalasekera 2007).

Equations

Table-1. A domain size with H/d LUS/d and LDS/d of 30 30 and 30 respectively was chosen.

The mesh size was determined by evaluating the values of CD and Nuav for different number of grid (Tabs

Equations

Numerical method: The finite volume method was adopted to discretise the governing equations. The semi implicit method for pressure linked equations (SIMPLE) algorithm was used for solving the pressure and velocity fields and the second order upwind difference scheme was used to discretise the convective terms in momentum and energy equations. The numerical simulation was carried out using the commercial software Ansys Fluent. The properties were taken at a reference temperature i.e. 20 oC. The Pr number of water was constant of 7 and the Renumber varied from 10 to 40.For comparison the calculation for the cases of constant surface temperature and constant heat flux boundary conditions were also performed (Tab 1).

Grid and domain size independence: The effect of domain size was examined by evaluating the drag coefficient CD and average Nusselt number Nuav for different upstream length LUS downstream length LDS and height H(tab. 1). The numerical conditions are given in 2-3). For the cases of CST and CHF boundary conditions a mesh size with grid number of 29600 was chosen. For the case of UED a mesh size with grid number of 33050 was chosen.

Table-1:Showing effect of domain size on CD and Nuavat Re = 40 and Pr = 7 for CST CHF and UED

###Nuav

###H/d###LUS/d###LDS/d###CD

###CST###CHF###UED

###10###10###40###1.617###7.279###8.512###7.617

###10###15###40###1.602###7.255###8.433###7.598

###10###15###50###1.602###7.257###8.505###7.595

###20###10###20###1.588###7246###8.487###7.568

###20###10###40###1.589###7.247###8.488###7.571

###20###15###20###1.557###7.200###8.452###7.554

###20###15###40###1.557###7.201###8.454###7.557

###20###20###20###1.545###7.182###8.419###7.517

###20###20###40###1.545###7.184###8.420###7.519

###30###10###20###1.587###7.245###8.376###7.582

###30###10###40###1.587###7.245###8.375###7.582

###30###15###20###1.538###7.196###8.433###7.943

###30###15###40###1.553###7.196###8.433###7.943

###30###20###20###1.538###7.175###8.393###7.507

###30###20###40###1.538###7.175###8.395###7.507

###30###30###20###1.526###7.158###8.387###7.487

###30###30###30###1.527###7.157###8.388###7.488

###40###40###40###1.518###7.145###8.374###7.475

###50###50###50###1.518###7.138###8.366###7.468

Table-2:Showing effect of mesh size on CD and Nuav under CST and CHF conditions at Re = 40 and Pr = 7

###Nuav

###No. of cells###CD

###CST###CHF

###19200###1.533###8.410###7.182

###29600###1.528###8.392###7.162

###40000###1.527###8.389###7.158

###58800###1.527###8.388###7.158

Table-3: Effect of mesh size on CD and Nuavunder UED conditions at Re = 40 andPr = 7

###No. of cells###Nuav

###CD

###UED

###20504###1.528###8.410

###33050###1.528###8.392

###46728###1.528###8.389

###69027###1.528###8.388

RESULTS AND DISCUSSION

To validate present numerical results the local Nusselt number and the drag coefficient were compared with the earlier numerical results by (Bharti et al. 2007) and experimental data by (Eckert et al. 1953). The local Nusselt number Nu along with the surface of the cylinder for both CST (a) and CHF (b) boundary conditions have been presented in figure (3). It was found that the results of the present study were in agreement with those of (Bharti et al. 2007). The numerical results agreed well with the experimental data in the region of less than about 80o and lower than the experimental data in the region of great than about 80o. The discrepancy may be due to the conditions between the measurements and numerical simulations. Comparison of drag coefficients between the present study and earlier results of (Park et al. 1998 and Henderson 1995) have been presented in Table 4.

Table-4: Comparison of drag coefficients at different Re

###CD

###Re

###Park et al. (1998)###Henderson (1995)###Present study

###10###2.78###-###2.836

###20###2.01###-###2.046

###30###-###1.73###1.721

###40###1.51###1.54###1.526

The flow field as shown in figure (4).Two steady symmetric vortices were found behind the cylinder and the wake region increased as the Renumber increased. The dimensionless temperature field across the cylinder for the condition of UED at four Renumbers for the diameter of 10.0 5.0 and 1.0 mm respectively as has been presented in figures (5-7)

It can be seen that when the diameter of the cylinder was reduced the temperature field in the wake region decreased.

Local Nusselt number Nualong the cylinder surface was obtained using Eq. (11) and has been shown in figure (8)for three boundary conditions and four Reynolds numbers. This was observed that Nu was a function of the angular position along the cylinder Re and the thermal boundary conditions. The Nu look the maximum value at front stagnation point and decreased till the separation point. The Nu increased again in the rear stagnation region due to flow circulation and reattachment but the value was considerably lower than that at the front stagnation point. The Nu also increased with increasing Re. The values of Nu were the highest for UED condition and slightly higher for UED than CST condition. Local Nusselt number was found to be independent of the cylinder size. Similar results have been reported by (Zukauskas et al. 1985 and Bharti et al. 2007).

The average Nusselt number Nuav was calculated by Eq.(13). Figure (9) shows the plot of Nuav against Re. For three boundary conditions the Nuav was found to be proportional to Re0.45 and as given in Eqs. (14) - (16) respectively. The Nuav with uniform volumetric energy dissipation in the cylinder were about 11% lower than those with constant heat flux boundary condition and about 4% higher than those with constant surface temperature boundary condition.

Equations

The results of present simulation were compared with the experimental data of (Park et al.1998 and Hendersen 1995) (Tab 4). A maximum variation of 4% from the experimental data was found. Constant heat flux boundary was seen to yield the highest local and average Nusselt number values.

Conclusions: A two-dimensional numerical analysis was carried out using Fluent for heat transfer from a circular cylinder with uniform volumetric energy dissipation (conjugate boundary condition) constant surface temperature and constant heat flux under an unconfined space. The average Nusselt number with uniform volumetric energy dissipation in the cylinder were about 11% lower than those with constant heat flux boundary condition and about 4% higher than those with constant surface temperature boundary condition. The average Nusselt numbers for three boundary conditions are well represented by correlations which can be useful for design.

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