Control volume finite element simulation of MHD forced and natural convection in a vertical channel with a heat-generating pipe

International Journal of Heat and Mass Transfer 55 (2012) 2813–2821

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Control volume finite element simulation of MHD forced and natural convection in a vertical channel with a heat-generating pipe Rehena Nasrin ⇑, M.A. Alim Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka 1000, Bangladesh

a r t i c l e

i n f o

Article history: Received 30 June 2011 Received in revised form 31 January 2012 Accepted 31 January 2012 Available online 22 March 2012 Keywords: Heat-generation Vertical channel Combined convection MHD Finite element method

a b s t r a c t Using the Galerkin weighted residual control volume finite element method, the effects of magnetic field and Joule heating on combined convection flow and heat transfer characteristics inside an octagonal vertical channel containing a heat-generating hollow circular pipe at the centre is performed. The flow enters at the bottom and exits from the top surface. All solid walls of the octagon are considered to be adiabatic. Graphical representation of streamlines, isotherms, average Nusselt number and maximum temperature of fluid for different combinations of Hartmann number (Ha), Joule heating parameter (J) and Richardson number (Ri) are displayed. The results indicate that the flow and thermal fields in the vertical channel depend markedly on the above mentioned parameters. In addition rate of heat transfer is obtained optimum in the absence of both MHD and Joule heating effects. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Mixed convection in channels is of great interest of the phenomenon in many technological processes, such as the design of solar collectors, thermal design of buildings, air conditioning and the cooling of electronic circuit boards. Alteration of heat transfer in channels due to introduction of obstacles, partitions and fins attached to the wall(s) has received sustained attention recently. A literature review on the subject shows that many authors have considered mixed convection in vented enclosures with obstacles, partitions and fins, thereby altering the convection flow phenomenon. Engineers employ MHD (magnetohydrodynamic) principles in the design of heat exchanger, pumps and flow meters, in space vehicle propulsion, control and re-entry in creating novel power generating systems and developing confinement schemes for controlled fusion. Joule heating in electronics and in physics refers to the increase in temperature of a conductor as a result of resistance to an electrical current flowing through it. A combined natural and forced convection flow of an electrically conducting fluid in a channel in the presence of magnetic field and Joule heating effects is of special technical significance because of its frequent occurrence in many industrial applications such as geothermal reservoirs, cooling of nuclear reactors, thermal insulations and petroleum reservoirs. These types of problems also arise in electronic packages, microelectronic devices during their operations. Some real applications of present geometry are noted mainly vertical channel pump, ⇑ Corresponding author. E-mail address: [email protected] (R. Nasrin). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2012.02.023

oil-filled pump, well submersible pump, low head up-flow bead filter, etc. Real enclosures in practice are often found to have different shapes rather than rectangular ones. Some examples of nonrectangular channels include various channels of constructions, panels of electronic equipment and solar energy collectors, etc. Several geometrical configurations, more or less complex, have been examined under theoretical, numerical or experimental approaches. Hung and Fu [1] studied the passive enhancement of mixed convection heat transfer in a horizontal channel with inner rectangular blocks by geometric modification. Calmidi and Mahajan [2] studied mixed convection in a partially divided rectangular enclosure over a wide range of Reynolds and Grashof numbers. Their findings were that the average Nusselt number and the dimensionless surface temperature depended on the location and height of the divider. Numerical study on mixed convection in a square cavity due to heat generating rectangular body was carried out by Shuja et al. [3]. They investigated the effect of exit port locations on the heat transfer characteristics and irreversibility generation in the cavity and showed that the normalized irreversibility increased as the exit port location number increased and the heat transfer from the solid body enhanced while the irreversibility reduced. Conjugate mixed convection arising from protruding heat generating ribs attached to substrates forming channel walls is numerically analyzed by Madhusudhana Rao and Narasimham [4]. The obtained outcome was that the natural convection induced mass flow rate in mixed convection was correlated in terms of the Grashof and Reynolds numbers. In pure natural convection it varied as 0.44 power of Grashof number. The maximum temperature

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Nomenclature magnetic induction (Wb m2) specific heat at constant pressure (J kg1 K) outer diameter of the cylinder (m) non-dimensional outer diameter of the pipe gravitational acceleration (ms2) Hartmann number Joule heating parameter thermal conductivity of fluid (Wm1 K1) thermal conductivity of solid (Wm1 K1) thermal conductivity ratio of the solid and fluid length of the each side of the octagon (m) Nusselt number dimensional pressure (Nm2) non-dimensional pressure Prandtl number generated heat per unit volume (Wm2) heat generating parameter Reynolds number Richardson number dimensional temperature (K)

was correlated in terms of pure natural convection and forced convection inlet velocity asymptotes and the heat transferred to the working fluid via substrate heat conduction was found to account for 41–47% of the heat removal from the ribs. Omri and Nasrallah [5] conducted mixed convection in an air-cooled cavity with differentially heated vertical isothermal sidewalls having inlet and exit ports by a control volume finite element method. They investigated two different placement configurations of the inlet and exit ports on the sidewalls. Best configuration was selected analyzing the cooling effectiveness of the cavity, which suggested that injecting air through the cold wall was more effective in heat removal and placing inlet near the bottom and exit near the top produced effective cooling. Later on, Singh and Sharif [6] extended their works by considering six placement configurations of the inlet and exit ports of a differentially heated rectangular enclosure whereas the previous work was limited only two different configurations of inlet and exit port. Unsteady mixed convection in a horizontal channel containing heated blocks on its lower wall was studied numerically by Najam et al. [7]. Tsay et al. [8] rigorously investigated the thermal and hydrodynamic interactions among the surface-mounted heated blocks and baffles in a duct flow mixed convection. They focused particularly on the effects of the height of baffle, distance between the heated blocks, baffle and number of baffles on the flow structure and heat transfer characteristics for the system at various Re and Gr/Re2. At the same time, a numerical analysis of laminar mixed convection in an open cavity with a heated wall bounded by a horizontally insulated plate was presented by Manca et al. [9], where three heating modes were considered: assisting flow, opposing flow and heating from below. Results were reported for Richardson number from 0.1 to 100, Reynolds numbers from 100 to 1000 and aspect ratio in the range 0.1–1.5. They showed that the maximum temperature values decreased as the Reynolds and the Richardson numbers increased. The effect of the ratio of channel height to the cavity height was found to play a significant role on streamline and isotherm patterns for different heating configurations. The investigation also indicated that opposing forced flow configuration had the highest thermal performance, in terms of both maximum temperature and average Nusselt number. Later, similar problem for the case of assisting forced flow configuration was tested experimentally by Manca et al. [10] and based on the

DT u, v U, V x, y X, Y

dimensional temperature difference (K) velocity components (ms1) non-dimensional velocity components Cartesian coordinates (m) non-dimensional Cartesian coordinates

Greek symbols a thermal diffusivity (m2 s1) b thermal expansion coefficient (K1) h non-dimensional temperature l dynamic viscosity of the fluid (m2 s1) t kinematic viscosity of the fluid (m2 s1) q density of the fluid (Kg m3) r fluid electrical conductivity (X1 m1) Subscripts max maximum h hot i inlet state s solid

flow visualization results, they pointed out that for Re = 1000 there were two nearly distinct fluid motions: a parallel forced flow in the channel and a recirculation flow inside the cavity and for Re = 100, the effect of a stronger buoyancy determined a penetration of thermal plume from the heated plate wall into the upper channel.

Th

y adiabatic

B0 Cp d D g Ha J kf ks K L Nu p P Pr q Q Re Ri T

x

g

B0

Ti , vi

Fig. 1. Physical model of the vertical channel.

Fig. 2. Mesh structure for vertical channel with a heat generating pipe.

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Fig. 3. Comparison between present numerical code and Saha et al. [15] using Ra = 104 and Pr = 0.71. Table 1 Grid sensitivity check at Pr = 1.73, Re = 10, Ri = 1, Q = 5, K = 5, Ha = 10, J = 1 and D = 0.3. 7224 (4816)

12982 (5784)

26538 (8992)

40295 (10936)

80524 (18080)

Nu hmax Time (s)

2.312753 1.232832 226.265

2.026821 1.091826 292.594

1.715743 1.069835 388.157

1.415496 1.0218254 421.328

1.415495 1.0218253 627.375

Ha = 0

Ha = 10

Ha = 30

Ha = 50

Nodes (elements)

Ri = 0.1

Ri = 1

Ri = 10

Fig. 4. Streamlines for various Ha and Ri with Pr = 1.73, K = 5, Re = 10, J = 1 and Q = 5.

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The results showed that the considered parameters have noticeable effect on the flow pattern and heat and mass transfer. Brown and Lai [14] numerically studied a horizontal channel with an open cavity and obtained correlations for combined heat and mass transfer which covered the entire convection regime from natural, mixed to forced convection. Saha et al. [15] performed natural convection heat transfer within octagonal enclosure. Their results showed that the effect of Ra on the convection heat transfer phenomenon inside the enclosure was significant for all values of Pr studied (0.71–50). It was also found that, Pr influenced natural convection inside the enclosure at high Ra (Ra >104). Rahman et al. [16] studied numerically the effect of Prandtl number on hydromagnetic mixed convection in a double-lid driven cavity with a heat-generating obstacle the opposing mixed convection in a vented enclosure. They found that the flow and temperature field were strongly depend on the above stated parameter for the values considered. The variation of the average Nusselt number, the average temperature of the fluid and the temperature at the body centre for various Richardson number had been presented. Couple effects of hydromagnetic and Joule heating on natural convective flow in an obstructed cavity was rigorously analyzed by Rehena

Ha = 0

Ha = 10

Ha = 30

Ha = 50

Bhoite et al. [11] studied numerically the problem of mixed convection flow and heat transfer in a shallow enclosure with a series of block-like heat generating component for a range of Reynolds and Grashof numbers and block-to-fluid thermal conductivity ratios. They showed that higher Reynolds number tend to create a recirculation region of increasing strength at the core region and the effect of buoyancy becomes insignificant beyond a Reynolds number of typically 600, and the thermal conductivity ratio had a negligible effect on the velocity fields. Barletta and Celli [12] studied mixed convection MHD flow in a vertical channel: effects of Joule heating and viscous dissipation where the local balance equations were written in a dimensionless form and solved both analytically by a power series method and numerically considering the effects of Joule heating and viscous dissipation. They found that with the increase of Reynolds and Richardson numbers the convective heat transfer became predominant over the conduction heat transfer and the rate of heat transfer from the heated wall was significantly depended on the position of the inlet port. Very recently Rahman et al. [13] developed the magnetic field and Joule heating effects on double-diffusive mixed convection in a horizontal channel with an open cavity.

Ri = 0.1

Ri = 1

Ri = 10

Fig. 5. Isotherms for various Ha and Ri with Pr = 1.73, K = 5, Re = 10, J = 1 and Q = 5.

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Nasrin [17] where the heat transfer rate grew up with rising Prandtl number while it devalued for increasing magnetic and Joule heating parameters. On the basis of the literature review, it appears that no work was reported on mixed convection in an octagonal channel with heat generating circular block. The present study addresses the effects of MHD, Joule heating and Richardson number on the thermal and flow fields for such geometry. The numerical computation covers a wide range of Hartmann number (0 6 Ha 6 50), Joule heating parameter (0 6 J 6 5) and Richardson number (0.1 6 Ri 6 10).

!

@V @V @P 1 @2V @2V Ha2 þ Rih  þV ¼ þ þ V @X @Y @Y Re @X 2 @Y 2 Re ! @h @h 1 @2h @2h þV ¼ þ U þ JV 2 @X @Y RePr @X 2 @Y 2

U

The geometry of the problem herein investigated is depicted in Fig. 1. The system consists of an octagonal channel with sides of length L, within which a heat generating hollow circular body with outer diameter d is centered. The hollow circular body has a thermal conductivity of ks and generates uniform heat q per unit volume. All solid walls of the octagon are considered to be adiabatic. It is assumed that the incoming flow has a uniform velocity, vi and temperature, Ti. The outer boundary of the hollow pipe is considered heated initially of temperature Th. The inlet opening is located at the bottom, whereas the outlet opening is at the top of the octagon.

ð9Þ

For hollow pipe the energy equation is

K @ 2 hs @ 2 hs þ RePr @X 2 @Y 2

!

þQ ¼0

where Re ¼ vmi L, Pr ¼ am , J ¼

2. Physical model

ð8Þ

rB20 Lv i qC p DT ,

ð10Þ 2

Ri ¼ gbvD2TL and Q ¼ kqL are Reynolds s DT i

number, Prandtl number, Joule heating parameter, Richardson number and heat generating parameter, respectively. Also magnetic rB2 L2 parameter (Ha) is defined as Ha2 ¼ l0 .

3. Mathematical simulation A two-dimensional, steady, laminar, incompressible, mixed convection flow is considered within the channel and the fluid properties are assumed to be constant. The radiation effects are taken as negligible. The dimensional equations describing the flow under Boussinesq approximation are as follows:

@u @ v þ ¼0 @x @y

ð1Þ

! @u @u 1 @p @2u @2u þv ¼ þm þ @x @y @x2 @y2 q @x ! @v @v 1 @p @2v @2v rB2 v u þm þ þv ¼ þ gbðT  T c Þ  0 2 2 @x @y q @y @x @y q ! @T @T kf @ 2 T @ 2 T rH20 2 þm ¼ þ 2 þ m u 2 @x @y qcp @x @y qC p

u

ð2Þ

ð3Þ

ð4Þ

For hollow cylinder the energy equation is

ks @ 2 T s @ 2 T s þ qcp @x2 @y2

! þq¼0

ð5Þ

The boundary conditions for the present problem can be written as follows: inlet: u = 0, v = vi, T = Ti outlet: convective boundary condition p = 0 cylinder boundary: u = 0, v = 0, Ts = Th walls of the octagon: u ¼ 0; v ¼ 0; @T ¼0    @n ks @T s ¼ at the fluid–solid interface: @T k @n solid @n fluid

at at at at

the the the the

f

The dimensionless equations describing the flow under Boussinesq approximation are as follows:

@U @V þ ¼0 @X @Y U

@U @U @P 1 @2U @2U þV ¼ þ þ @X @Y @X Re @X 2 @Y 2

ð6Þ ! ð7Þ

Fig. 6. Effect of Hartmann number on (i) average Nusselt number and (ii) maximum temperature of the fluid.

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The above equations are non-dimensionalized by using the following dimensionless quantities

x y d u v p ðT  T i Þ X ¼ ;Y ¼ ;D ¼ ;U ¼ ;V ¼ ;P ¼ ;h ¼ ; L L L ðT h  T i Þ vi vi qv 2i ðT s  T i Þ hs ¼ ðT h  T i Þ The boundary conditions for the present problem are specified as follows: at the inlet: U = 0, V = 1, h = 0 at the outlet: convective boundary condition P = 0 at all solid boundaries: U = 0, V = 0 @h at the walls of the octagon: @N ¼  0   @h s at the fluid–solid interface: @N ¼ K @h @N solid fluid where N is the non-dimensional distances either X or Y direction acting normal to the surface and K is the dimensionless ratio of the thermal conductivity (ks/kf). The average Nusselt number at the heat generating body may be expressed as

@h ds @n

The momentum and energy balance equations are the combinations of mixed elliptic–parabolic system of partial differential equations that have been solved by using the Galerkin weighted residual finite element technique. The continuity equation has been used as a constraint due to mass conservation. The basic unknowns for the above differential equations are the velocity components U, V the temperature, h and the pressure, P. The six node triangular element is used in this work for the development of the finite element equations. All six nodes are associated with velocities as well as temperature; only the corner nodes are associated with pressure. This means that a lower order polynomial is chosen for pressure and which is satisfied through continuity equation. The velocity component and the temperature distributions and linear interpolation for the pressure distribution according to their highest derivative orders in the differential Eqs. (6)– (10) as

J=5

0

Ls

4. Numerical procedure

J=3

Z

coordinate along the circular surface, respectively.

J=1

1 Ls

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   @h 2ffi @h 2 , Ls and S are the dimensionless length and þ @X @Y

J=0

Nu ¼ 

@h where @n ¼

Ri = 0.1

Ri = 1

Ri = 10

Fig. 7. Streamlines for various J and Ri with Pr = 1.73, K = 5, Re = 10, Ha = 10 and Q = 5.

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UðX; YÞ ¼ Nb U b ; ðVðX; YÞ ¼ Nb V b ; hðX; YÞ ¼ Nb hb ; hs ðX; YÞ

where the coefficients in element matrices are in the form of the integrals over the element area and along the element edges S0 and Sw as

¼ Nb hsb ; PðX; YÞ ¼ Hk Pk ; where b = 1, 2, . . . . . ., 6; k = 1–3. Substituting the element velocity component distributions, the temperature distribution and the pressure distribution from Eqs. (6)–(10) the finite element equations can be written in the form

K abx U b þ K aby V b ¼ 0

ð11Þ

K abcx U b U c þ K abcy V c U c þ M alx Pl þ

1 ðS xx þ Sabyy ÞU b ¼ Q au Re ab

K abcx U b V c þ K abcy V c V c þ M aly P l þ

1 ðS xx þ Sabyy ÞV b  RiK ab hb Re ab

þ

Ha2 K ab V b ¼ Q av Re

K abcx U b hc þ K abcy V b hc þ

ð12Þ

Z Z K abx ¼ Na Nb;x dA; K aby ¼ Na N b;y dA; K abcx A A Z Z ¼ Na N b Nc;x dA; K abcy ¼ Na Nb Nc;y dA; A A Z Z Na Nb dA; Sabxx ¼ Na;x Nb;x dA; Sabyy K ab ¼ A A Z Z ¼ Na;y Nb;y dA; Malx ¼ Ha Hl;x dA; A A Z Z Z Ha Hl;y dA; Q au ¼ Na Sx dS0 ; Q av ¼ Na Sy dS0 ; Q ah M aly ¼

ð13Þ ¼ 1 ðS xx þ Sabyy Þhb  JK ab V 2b ¼ Q ah RePr ab

K ðS xx þ Sabyy Þhb þ Q ¼ Q ahs RePr ab

A

Z Sw

ð14Þ

Z

Sw

N a q2w dSw :

The discretization equation for the average Nusselt number is R1 Nu ¼  L1s 0 K abcn hY dS.

J=3 J=1 J=0

Ri = 0.1

S0

Na q1w dSw ;

J=5

ð15Þ

Q ahs ¼

S0

Ri = 1

Ri = 10

Fig. 8. Isotherms for various J and Ri with Pr = 1.73, K = 5, Re = 10, Ha = 10 and Q = 5.

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The set of non-linear algebraic Eqs. (12)–(15) are solved using reduced integration technique [18,19] and Newton–Raphson method [20]. 4.1. Grid refinement check In order to determine the proper grid size for this study, a grid independence test is conducted with five types of mesh as shown in Table 1 for Pr = 1.73, Re = 10, Ri = 1, Q = 5, K = 5, Ha = 10, J = 1 and D = 0.3. The extreme values of Nu and hmax are used as a sensitivity measure of the accuracy of the solution and are selected as the monitoring variables. Considering both the accuracy of numerical values and computational time, the present calculations are performed with 40295 nodes and 10936 elements grid system Fig. 3.

and reduces the fluid velocity. The streamlines change radically due to the deviation of Ri. The size of spinning cells in the velocity field becomes larger due to buoyancy force for increasing Ri (=10) with compared to the previous stages (Ri = 0.1 and 1). In addition, the velocity field bifurcates near the circular pipe. Consequently, thermal boundary layer thickness reduces as Ha increases and the isothermal lines become denser at the adjacent area of the heat source. Also, the bend in isothermal lines appears due to the free and forced convective current inside the channel. The lines become more concentrated from the forced convection dominant region to free convection dominant regime for a particular Ha. For higher Ha, the isothermal lines take an onion shape from the circular body to exit port. This is due to confining the thermal boundary layer for the effect of higher magnetic field.

4.2. Mesh generation In finite element method, the mesh generation is the technique to subdivide a domain into a set of sub-domains, called finite elements, control volume, etc. The discrete locations are defined by the numerical grid, at which the variables are to be calculated. It is basically a discrete representation of the geometric domain on which the problem is to be solved. The computational domains with irregular geometries by a collection of finite elements make the method a valuable practical tool for the solution of boundary value problems arising in various fields of engineering. Fig. 2 displays the finite element mesh of the present physical domain. 4.3. Code validation A description of the code and its validation are available in Rehena and Salma [21] and are not repeated here. The existing code in [21] has been modified for the current problem. 4.4. Comparison The model comparison is an essential part of a mathematical investigation. Hence, the outcome of the present numerical code is benchmarked against the numerical result of Saha et al. [15] which was reported for natural convection flow and heat transfer within octagonal enclosure. The comparison is conducted while employing the dimensionless parameters Ra = 104 and Pr = 0.71. Present result for both the streamlines and isotherms is shown in Fig. 3, which is an outstanding agreement with those of Saha et al. [15]. This justification boosts the assurance in this numerical code to carry on with the above stated objective of the existing investigation. 5. Results and discussion The mixed convection phenomenon inside an octagonal vertical channel having a heat-generating circular body is influenced by different controlling parameters such as D, K, Q, Re, Pr, Ha, J and Ri. Analysis of the results is made through obtained streamlines, isotherms, average Nusselt number and maximum temperature of the fluid for three parameters varied as 0 6 Ha 6 50, 0 6 J 6 5 and 0.1 6 Ri 6 10, while the other parameters D, Re, Pr, K, Q are kept fixed at 0.3, 10, 1.73, 5, and 5, respectively. Figs. 4 and 5 provide the information about the influence of Ha at the three different values of Ri on streamlines as well as isotherms. The flow with the absence of magnetic field (Ha = 0) creates a couple of tiny recirculation regions near the heat generating hollow pipe as shown in Fig. 4. These spinning regions disappear sequentially and the streamlines cover all the area of the octagon by varying Ha as the magnetic field acts against the flow

Fig. 9. Effect of Joule heating parameter on (i) average Nusselt number and (ii) maximum temperature of the fluid.

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The variations of the average Nusselt number (Nu) at the heated surface and the maximum temperature of the fluid (hmax) for different magnetic parameter with Richardson numbers have been presented in Fig. 6. It is clearly seen from the figure that for a particular value of Ri, the heat transfer rate and maximum temperature of fluid are the lowest for the largest Hartmann number Ha = 50. For the reason that the fluid with the absence of magnetic field is capable to carry more heat away from the heat source and dissipated through the out flow opening. Moreover, Nu remains invariant for higher values of Hartmann number and it rises very unhurriedly for the lowest Ha due to rising Ri. On the other hand, at Ha = 0, hmax devalues sharply in the forced convection dominated region (Ri 6 1) and then it remains constant with mounting Ri. In addition, for the remaining Ha, it is decreases slowly for varying Ri. Figs. 7 and 8 provide the information about the influence of J at the three different values of Ri on streamlines as well as isotherms. The flow pattern remains almost similar at a fixed value of J due to the variation of Ri except the size of spinning cells. For the largest value of Joule heating parameter J, the size of vortex increase with compared to the remaining values of J for all convective regimes. Thermal boundary layer thickness decreases as J increases and the isothermal lines become denser at the adjacent area of the heat source in all convection regimes. For a fixed J, the isothermal lines change from bend to flatten with the effect of rising Ri. In addition, thermal current activities become stronger near the heat generating tube. The variation of Nu at the heated surface, hmax of the fluid in the octagon for different Joule heating parameter J along with Richardson number have been presented in Fig. 9. The average Nusselt number is taken as a function of Ri. It is clearly seen that for a particular value of Ri, the average rate of heat transfer and the maximum temperature of the fluid become lower and higher respectively for higher values of J. On the other hand, Nu remains constant at J = 5 but for other considered values of J, it rises slowly with increasing Ri. Also, hmax diminishes gradually due to the variation of Ri. 6. Conclusion A computational study is performed to investigate the influences of magnetic and Joule heating parameter on forced and natural convection in an octagonal channel with a heat-generating horizontal circular body. Results are obtained for wide ranges of Hartmann number (Ha), Richardson number (Ri) and Joule heating parameter (J). The following conclusions may be drawn from the present investigations:  The influence of the aforesaid parameters on velocity field is remarkable. Particularly, the size of vortex devalues due to the hindrance of imposed magnetic field at all convection regions. On the other hand, Joule heating plays a significant role on streamlines.  The changes of temperature field with the mentioned parameters are noteworthy. Mainly, the thermal boundary layer thickness reduces for mounting Ha. The isothermal lines move from

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the center of the heat source as J increases. The nature of thermal plume rise from the body changes radically with the escalating Ri.  The heat transfer rate decreases with rising Ha and J. The maximum temperature of the fluid goes down and goes up for rising values of Ha and J, respectively.

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