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MHD Effects on Mixed Convection Flow Through a Diverging Channel with Circular Obstacle
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 90 (2014) 403 – 410
10th International Conference on Mechanical Engineering, ICME 2013
MHD effects on mixed convection flow through a diverging channel with circular obstacle a
b
Md. S. Alama,*, M.A.H. Khanb
Department of Mathematics, Jagannath University, Dhaka-1100, Bangladesh Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka-1000, Bangladesh
Abstract MHD mixed convection steady laminar heat and mass transfer flow through an open diverging channel with a circular obstacle is studied numerically in the present work. A circular obstacle is placed at the centre of the channel and the right wall of the channel is heated with temperature Th . The top and bottom walls are adiabatic. Using a set of appropriate transformations, the governing equations along with the initial and boundary conditions are transformed into non-dimensional form, which are then solved by employing Galerkin weighted residuals finite element scheme. MHD mixed convection fluid flow and heat transfer within the channel is governed by the inertia force, magnetic intensity and viscosity variation parameter namely Reynolds number (Re), Hartman number (Ha) and Prandtl number (Pr). The computation is carried out for a wide range of Hartmann number Ha, Prandtl number Pr and Reynolds number Re. The obtained results are presented in the form of streamlines, isotherms, average Nusselt number and average temperature of the fluid for various relevant dimensionless groups. © Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license ©2014 2014The The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and peer-review under responsibility of the Department of Mechanical Engineering, Bangladesh University of Selection and peer-review under responsibility of the Department of Mechanical Engineering, Bangladesh University of Engineering Engineering and Technology (BUET). and Technology (BUET)
Keywords: MHD mixed convection; diverging channel; circular obstacle; finite element method.
1. Introduction Mixed convection involves features from both forced and natural flow conditions. In mixed convection flows, the forced convection and free convection effects are comparable in magnitudes. Mixed convection problem has got its extensive applications in the field of engineering, for example cooling of electronic devices, furnaces, lubrication technologies, chemical processing equipment, drying technologies etc. * Corresponding author. Tel.: +88029582775; fax: +88027113752. E-mail address: [email protected]
1877-7058 © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and peer-review under responsibility of the Department of Mechanical Engineering, Bangladesh University of Engineering and Technology (BUET) doi:10.1016/j.proeng.2014.11.869
404
Md. S. Alam and M.A.H. Khan / Procedia Engineering 90 (2014) 403 – 410
Actual channels in practice are often found to have different shapes rather than rectangular one. Moreover, the channel may be in an ideal situation, such as a parallel-plate channel, but a convergent or a divergent channel. There are some situations, such as in electronic equipment cooling, one is not clear if the divergence or convergence of the channel would make the heat transfer better. Furthermore, MHD flow in diverging channels has important applications in MHD pumps and generators, liquid metal magnetohydrodynamics and physiological fluid flow. Comprehensive reviews have been conducted by Dennis et al. [1], Drazin [2]. Layek et al. [3] studied steady MHD flow in a diverging channel with suction or blowing, where they analyzed steady two-dimensional divergent flow of an electrically conducting incompressible viscous fluid in a channel formed by two non-parallel walls, caused by a source of fluid volume at the intersection of the walls. Magnetic fields are generally used to control the natural convection of semiconductor melts such as silicon or gallium arsenide to improve crystal quality in Hadid et al. [4]. Moreover, the steady flow of a viscous incompressible fluid in a linearly diverging asymmetrical channel was studied by Makinde [5]. He expanded the solution into a Taylor series with respect to the Reynolds number and performed a bifurcation study. Alam and Khan [6] show the critical analysis of the MHD flow in convergentdivergent channels. They analyzed the solution to perform the bifurcation of the parameters and the critical relationship of the parameters. However, studies on natural and mixed convection from converging or diverging channels are limited. A detailed survey of literature on mixed convection in internal flows has been presented in Aung [7]. Sparrow et al. [8] presented the results of an experimental and numerical study on natural convection from isothermal converging channels. The maximum half angle of inclination of the plates considered for their study was 150. Sparrow and Ruiz [9] carried out an experimental study of natural convection from a diverging channel and presented an universal correlation for converging, parallel and diverging channel, based on the maximum inter-wall spacing. Gau et al. [10] performed an experimental study of both buoyancy assisting and buoyancy opposing mixed convection heat transfer from a converging channel. One of the channel walls was placed vertically and was Alternation of heat transfer in uniformly heated and the other wall was adiabatic, with an inclination of 30. channels due to introduction of obstacles, partitions and fins attached to the wall(s) has received sustained attention recently. Billah et al. [11] executed heat transfer and flow characteristics for MHD mixed convection in a lid-driven cavity with heat generating obstacle. From a review of literature, it is clear that comprehensive studies on MHD mixed convection from diverging channel with circular obstacle is scarce. The objective of the present study is to investigate the effect of various flow and thermal configurations on MHD mixed convection for a wide-range pertinent controlling parameters in the diverging channel. These parameters include Prandtl number Pr, Hartman number Ha, Reynolds number Re and Richardson number Ri. Average Nusselt number, average temperature of the fluid are also presented. 2. Model Configuration The schematic diagram of the problem herein investigated is shown in Fig.1. The system consists of a horizontal diverging channel with sides of length L. A solid circular obstacle is located at the center of the channel. The coordinate system is considered Cartesian with origin at the end of the left inlet of the computational domain.
y x
ui
g
Th Lh
Ti
L Fig. 1 Physical model of the diverging channel
B0
405
Md. S. Alam and M.A.H. Khan / Procedia Engineering 90 (2014) 403 – 410
The right wall of the channel is hot with temperature Th and the height of the heated surface is Lh . All other solid walls of the channel are adiabatic. It is assumed that the incoming flow has a uniform velocity ui at the ambient temperature Ti.. The inlet opening is situated at the left end, whereas the outlet opening is at the top and bottom of the right vertical wall of the channel. The uniform magnetic field of strength B0 is applied to the right vertical wall in horizontal direction. 3. Mathematical Formulation A two-dimensional hydrodynamic, steady, laminar, incompressible, mixed convection flow is assumed within the channel and the fluid properties are considered to be constant. The radiation and joule heating effect is taken as negligible in this study. The dimensionless equations describing the flow under Boussinesq approximation are as follows: wU wV (1) 0 wX wY
wP 1 §¨ w 2U w 2U ·¸ wX Re ¨© wX 2 wY 2 ¸¹
U
wU wU V wX wY
U
wV wV V wX wY
U
wT wT V wX wY
w 2T 1 §¨ w 2T 2 ¨ Re Pr © wX wY 2
where Re
ui L
X
2 2 wP 1 § ¨w V w V 2 ¨ wY Re © wX wY 2
, Pr
P cp , Ri N
(2) 2 · ¸ RiT Ha V ¸ Re ¹
· ¸ ¸ ¹
(3)
(4)
gE'TL ui2
and Ha 2
VB0 2 L2 are Reynolds number, Prandtl number , Richardson P
number and square of Hartmann number respectively. The above equations are non dimensionalized by using the following dimensionless quantities X
x ,Y L
y ,U L
u ,V ui
v , P ui
p
U ui
2
T Ti , Th Ti
, and T
The boundary conditions for the present problem are specified as follows: At the inlet: U = 1, V = 0, T 0 . At the outlet: Convective boundary condition (CBC), P = 0 At circular obstacle boundaries: U 0, V 0, T 0 At the heated right vertical wall: T
1
wT 0 wN Where N is the non-dimensional distance acting normal to the surface. At top and bottom walls of the channel: U
0, V
0,
406
Md. S. Alam and M.A.H. Khan / Procedia Engineering 90 (2014) 403 – 410
The average Nusselt number at heated surface can be expressed as Nu
1 Lh
Lh
wT
³ wX dY , where Lh is the length of the right vertical heated wall. 0
The average temperature of the fluid is defined as T
³T
dV , where V is the channel volume. V
4. Computational procedure The governing equations along with the boundary conditions are solved numerically, employing Galerkin weighted residual finite element techniques. The equation of continuity has been used as constraint due to mass conservation. The basic unknowns for the governing differential equations are the velocity components U, V, the temperature T and the pressure P. The Galerkin finite element technique yields the subsequent nonlinear residual equations. Three points Gaussian quadrature is used to evaluate the integrals in these equations. The non-linear residual equations are solved using Newton-Raphson method to determine the coefficients of the expansions. 5. Grid refinement check An extensive mesh testing procedure is conducted to guarantee a grid-independent test for Ri 1.0, Re 100 and Pr 0.71, Ha 20 . The domain is divided into a set of non-overlapping regions called elements. Five different non-uniform grid systems shown in Table 1 with the number of elements within the resolution field are examined. The extreme values of Nu and T are used as a sensitivity measure of accuracy of the solution and are selected as the monitoring variables. Considering both the accuracy of numerical values and computational time, the present computations are performed with 4848 nodes and 3232 elements grid system. Table 1: Grid sensitivity check at Ri 1.0, Re 100 and Pr Nodes (elements) Nu T 1392(928) 5.814493 0.249295 1518(1012) 5.851071 0.250581 3444(2296) 5.923958 0.241458 4848(3232) 5.883601 0.24164 5232 (3488) 5.899925 0.241952
0.71, Ha
20 Time(s) 1.75 1.781 2.375 2.797 2.922
6. Code Validation A computational model is validated for mixed convection heat transfer by comparing the results on mixed convection in ventilated cavity with left heated wall performed by Raji and Hasnaoui [12]. In the present work numerical predictions have been obtained on the triangular mesh with 4848 nodes and 3232 elements for the same boundary condition of Raji and Hasnaoui [12]. The model of Raji and Hasnaoui [12] is reproduced at first and then the comparison is depicted in Fig. 2. It can be decided that the current code can be used to predict the flow field for the present problem. Present work Raji and Hasnaoui [12] Streamlines
Isotherms
Fig. 2 Streamline and isotherm comparison of present work with Raji and Hasnaoui [12] for Re
10 and Pr
0.71, Ha
0
Md. S. Alam and M.A.H. Khan / Procedia Engineering 90 (2014) 403 – 410
7. Results and Discussions
Ha 0
Ha 50
Ha 100
Ha
200
(a) Fig 3: Effect of Hartman number Ha at Ri 1.0, Re 100 and Pr
(b)
7.1 on (a) Streamlines and (b) Isotherms.
407
408
Md. S. Alam and M.A.H. Khan / Procedia Engineering 90 (2014) 403 – 410
(a)
(b)
Fig 4: Effect of Hartman number Ha on (a) average fluid temperature and (b) average Nusselt number in the channel while Re 100. Pr
(a)
7.1 .
(b)
Fig 5: Effect of Prandtl number Pr on (a) average fluid temperature and (b) average Nusselt number in the channel while Re 100, Ha
20 .
The characteristics of the MHD mixed convection flow and temperature fields through the divergent channel are examined by investigating the effect of the Reynolds number Re, Prandtl number Pr and Hartman number Ha. The fluid flow patterns inside the channel are presented in terms of streamlines and isotherms in Fig. 3(a)-(b) at the four different values of Hartman number Ha( 0,50,100,200) with Ri 1.0, Re 100 and Pr 7.1 . In mixed convection regimes, the velocity field contains no vortices near the circular obstacle and the streamlines become symmetrical along side walls of the channel with increasing values of Ha. This is because; the magnetic field acts against the flow and reduces the fluid velocity. Thermal boundary layer thickness varies as Ha increases and the isothermal lines become distorted at large values of Ha. Fig. 4 (a)-(b), represent the variation of average temperature T of the fluid and the average Nusselt number (Nu) at the heated surface in the diverging channel due to the rising
409
Md. S. Alam and M.A.H. Khan / Procedia Engineering 90 (2014) 403 – 410
values of Ha. It is clearly observed that as Ri increases, average fluid temperature T reduces for all values of Ha specially in absence of Ha more rapidly. On other hand, Nu increases gradually when Ri rises at all values of Ha and particularly in absence of Ha, Nu increases more swiftly. The average temperature of the fluid and the rate of heat transfer remain almost constant for large values of Ha 100. However, as Ha increases T increases, while the rate of heat transfer decreases. The variation of Nu at the heated surface, the average temperature T of the fluid in the channel for different Prandtl number along with Richardson number have been presented in Fig. 5(a)-(b). It is clearly seen that for 0 d Ri d 8 , T remains almost constant for lower values of Pr but T decreases while Pr increases. Nevertheless, Nu increases slowly as Ri increases for lesser values of Pr whereas Nu increases gradually with Ri due to the escalating values of Pr. The effect of Reynolds number Re on T and Nu is displayed in Fig. 6 (a)(b). It is observed that the average temperature T is decreasing sharply for 0 d Ri d 1.0 but decreases slowly beyond this value of Ri for the higher values of Re( 400, 500). However, the temperature T decreases uniformly with the increasing values of Ri when Re( 100, 200). Moreover, the average Nusselt number (Nu) is mounting leisurely for the minor values of Re( 100, 200) but increases significantly between 0 d Ri d 1.0 and then slow down for Ri t 1.0 at Re( 400, 500) . Maximum heat transfer rate is always found for the higher values of Re( 400, 500).
(a) (b) Fig 6: Effect of Reynolds number Re on (a) average fluid temperature and (b) average Nusselt number in the channel while Ha
20, Pr 7.1 .
Conclusion A computational study is performed to investigate the MHD mixed convection flow through a diverging channel with a circular obstacle. Effects of Reynolds number, Prandtl number and Hartman number in the channel with a circular obstacle are highlighted to study their impacts on flow structure and heat transfer characteristics. The effect of the magnetic parameter on velocity field is remarkable. On the other hand, the obstacle has a significant effect on streamlines. The variations of temperature field with mentioned parameters are notable. Magnetic field affects the mean bulk temperature and average heat transfer rate significantly. The heat transfer rate and the average temperature of the fluid changes rapidly against Ri with rising values of Re, Pr, Ha. The present investigation reveals that the opposing forced flow and thermal fields inside the channel strongly depend on the relevant dimensionless groups. References [1] S.C.R. Dennis, W. H. H. Banks, P. G. Drazin, M. B. Zaturska, Flow along a diverging channel, Journal of Fluid Mechanics. 336(1997) 183-202. [2] P. G. Drazin, Flow through a diverging channel: instability and bifurcation, Fluid Dynamics Research. 24(6)
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Md. S. Alam and M.A.H. Khan / Procedia Engineering 90 (2014) 403 – 410
(1999) 321-327. [3] G. C. Layek, S. G. Kryzhevich, A. S. Gupta, M. Reza, Steady magnetohydrodynamic flow in a diverging channel with suction or blowing, Z. Angew. Math. Phys. 64(1)(2013) 123-143. [4] H. Ben Hadid, R. Touihri, D. Henry, MHD damped convection under uniform magnetic fields, Adv. Space Res. 22(8)(1998) 1213-1216. [5] O.D. Makinde, Steady flow in a linearly diverging asymmetrical channel, Computer Assisted Mechanics and Engineering Sciences. 4(1997) 157-165. [6] M.S. Alam, M.A.H. Khan, Critical behaviour of the MHD flow in convergent-divergent channels, Journal of Naval Architecture and Marine Engineering. 7(2)(2010) 83 – 93. [7] W. Aung, Mixed convection in internal flows, Hand Book of Single Phase Convective Heat Transfer, John Wiley, New York, Chap. 15, 1987. [8] E. M. Sparrow, R. Ruiz, L. F. A. Azevedo, Experimental and numerical investigation of natural convection in convergent vertical channels, Int. J. Heat Mass Transfer. 31(5)(1988) 907-915. [9] E. M. Sparrow, R. Ruiz, Experiments on natural convection in divergent vertical channels and correlation of divergent, convergent, and parallel-channel Nusselt numbers, Int. J. Heat Mass Transfer. 31(11)(1988) 2197-2205. [10] C. Gau, T. M. Huang, W. Aung, Flow and Mixed Convection Heat Transfer in a Divergent Heated Vertical Channel, ASME J. of Heat Transfer. 118(3)(1996) 606-615. [11] M. M. Billah, M. M. Rahman, M. H. Kabir, U. M. Sharif, Heat transfer and flow characteristics for MHD mixed convection in a Lid-driven cavity with heat generating obstacle, Int. J. of Eng & Tech.3(32) (2011) 1-8. [12] A. Raji, M. Hasnaoui, Correlation on mixed convection in ventilated cavities, Revue Ge'n e' rale de Thermique. 37 (1998) 874-884.
ScienceDirect Procedia Engineering 90 (2014) 403 – 410
10th International Conference on Mechanical Engineering, ICME 2013
MHD effects on mixed convection flow through a diverging channel with circular obstacle a
b
Md. S. Alama,*, M.A.H. Khanb
Department of Mathematics, Jagannath University, Dhaka-1100, Bangladesh Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka-1000, Bangladesh
Abstract MHD mixed convection steady laminar heat and mass transfer flow through an open diverging channel with a circular obstacle is studied numerically in the present work. A circular obstacle is placed at the centre of the channel and the right wall of the channel is heated with temperature Th . The top and bottom walls are adiabatic. Using a set of appropriate transformations, the governing equations along with the initial and boundary conditions are transformed into non-dimensional form, which are then solved by employing Galerkin weighted residuals finite element scheme. MHD mixed convection fluid flow and heat transfer within the channel is governed by the inertia force, magnetic intensity and viscosity variation parameter namely Reynolds number (Re), Hartman number (Ha) and Prandtl number (Pr). The computation is carried out for a wide range of Hartmann number Ha, Prandtl number Pr and Reynolds number Re. The obtained results are presented in the form of streamlines, isotherms, average Nusselt number and average temperature of the fluid for various relevant dimensionless groups. © Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license ©2014 2014The The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and peer-review under responsibility of the Department of Mechanical Engineering, Bangladesh University of Selection and peer-review under responsibility of the Department of Mechanical Engineering, Bangladesh University of Engineering Engineering and Technology (BUET). and Technology (BUET)
Keywords: MHD mixed convection; diverging channel; circular obstacle; finite element method.
1. Introduction Mixed convection involves features from both forced and natural flow conditions. In mixed convection flows, the forced convection and free convection effects are comparable in magnitudes. Mixed convection problem has got its extensive applications in the field of engineering, for example cooling of electronic devices, furnaces, lubrication technologies, chemical processing equipment, drying technologies etc. * Corresponding author. Tel.: +88029582775; fax: +88027113752. E-mail address: [email protected]
1877-7058 © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and peer-review under responsibility of the Department of Mechanical Engineering, Bangladesh University of Engineering and Technology (BUET) doi:10.1016/j.proeng.2014.11.869
404
Md. S. Alam and M.A.H. Khan / Procedia Engineering 90 (2014) 403 – 410
Actual channels in practice are often found to have different shapes rather than rectangular one. Moreover, the channel may be in an ideal situation, such as a parallel-plate channel, but a convergent or a divergent channel. There are some situations, such as in electronic equipment cooling, one is not clear if the divergence or convergence of the channel would make the heat transfer better. Furthermore, MHD flow in diverging channels has important applications in MHD pumps and generators, liquid metal magnetohydrodynamics and physiological fluid flow. Comprehensive reviews have been conducted by Dennis et al. [1], Drazin [2]. Layek et al. [3] studied steady MHD flow in a diverging channel with suction or blowing, where they analyzed steady two-dimensional divergent flow of an electrically conducting incompressible viscous fluid in a channel formed by two non-parallel walls, caused by a source of fluid volume at the intersection of the walls. Magnetic fields are generally used to control the natural convection of semiconductor melts such as silicon or gallium arsenide to improve crystal quality in Hadid et al. [4]. Moreover, the steady flow of a viscous incompressible fluid in a linearly diverging asymmetrical channel was studied by Makinde [5]. He expanded the solution into a Taylor series with respect to the Reynolds number and performed a bifurcation study. Alam and Khan [6] show the critical analysis of the MHD flow in convergentdivergent channels. They analyzed the solution to perform the bifurcation of the parameters and the critical relationship of the parameters. However, studies on natural and mixed convection from converging or diverging channels are limited. A detailed survey of literature on mixed convection in internal flows has been presented in Aung [7]. Sparrow et al. [8] presented the results of an experimental and numerical study on natural convection from isothermal converging channels. The maximum half angle of inclination of the plates considered for their study was 150. Sparrow and Ruiz [9] carried out an experimental study of natural convection from a diverging channel and presented an universal correlation for converging, parallel and diverging channel, based on the maximum inter-wall spacing. Gau et al. [10] performed an experimental study of both buoyancy assisting and buoyancy opposing mixed convection heat transfer from a converging channel. One of the channel walls was placed vertically and was Alternation of heat transfer in uniformly heated and the other wall was adiabatic, with an inclination of 30. channels due to introduction of obstacles, partitions and fins attached to the wall(s) has received sustained attention recently. Billah et al. [11] executed heat transfer and flow characteristics for MHD mixed convection in a lid-driven cavity with heat generating obstacle. From a review of literature, it is clear that comprehensive studies on MHD mixed convection from diverging channel with circular obstacle is scarce. The objective of the present study is to investigate the effect of various flow and thermal configurations on MHD mixed convection for a wide-range pertinent controlling parameters in the diverging channel. These parameters include Prandtl number Pr, Hartman number Ha, Reynolds number Re and Richardson number Ri. Average Nusselt number, average temperature of the fluid are also presented. 2. Model Configuration The schematic diagram of the problem herein investigated is shown in Fig.1. The system consists of a horizontal diverging channel with sides of length L. A solid circular obstacle is located at the center of the channel. The coordinate system is considered Cartesian with origin at the end of the left inlet of the computational domain.
y x
ui
g
Th Lh
Ti
L Fig. 1 Physical model of the diverging channel
B0
405
Md. S. Alam and M.A.H. Khan / Procedia Engineering 90 (2014) 403 – 410
The right wall of the channel is hot with temperature Th and the height of the heated surface is Lh . All other solid walls of the channel are adiabatic. It is assumed that the incoming flow has a uniform velocity ui at the ambient temperature Ti.. The inlet opening is situated at the left end, whereas the outlet opening is at the top and bottom of the right vertical wall of the channel. The uniform magnetic field of strength B0 is applied to the right vertical wall in horizontal direction. 3. Mathematical Formulation A two-dimensional hydrodynamic, steady, laminar, incompressible, mixed convection flow is assumed within the channel and the fluid properties are considered to be constant. The radiation and joule heating effect is taken as negligible in this study. The dimensionless equations describing the flow under Boussinesq approximation are as follows: wU wV (1) 0 wX wY
wP 1 §¨ w 2U w 2U ·¸ wX Re ¨© wX 2 wY 2 ¸¹
U
wU wU V wX wY
U
wV wV V wX wY
U
wT wT V wX wY
w 2T 1 §¨ w 2T 2 ¨ Re Pr © wX wY 2
where Re
ui L
X
2 2 wP 1 § ¨w V w V 2 ¨ wY Re © wX wY 2
, Pr
P cp , Ri N
(2) 2 · ¸ RiT Ha V ¸ Re ¹
· ¸ ¸ ¹
(3)
(4)
gE'TL ui2
and Ha 2
VB0 2 L2 are Reynolds number, Prandtl number , Richardson P
number and square of Hartmann number respectively. The above equations are non dimensionalized by using the following dimensionless quantities X
x ,Y L
y ,U L
u ,V ui
v , P ui
p
U ui
2
T Ti , Th Ti
, and T
The boundary conditions for the present problem are specified as follows: At the inlet: U = 1, V = 0, T 0 . At the outlet: Convective boundary condition (CBC), P = 0 At circular obstacle boundaries: U 0, V 0, T 0 At the heated right vertical wall: T
1
wT 0 wN Where N is the non-dimensional distance acting normal to the surface. At top and bottom walls of the channel: U
0, V
0,
406
Md. S. Alam and M.A.H. Khan / Procedia Engineering 90 (2014) 403 – 410
The average Nusselt number at heated surface can be expressed as Nu
1 Lh
Lh
wT
³ wX dY , where Lh is the length of the right vertical heated wall. 0
The average temperature of the fluid is defined as T
³T
dV , where V is the channel volume. V
4. Computational procedure The governing equations along with the boundary conditions are solved numerically, employing Galerkin weighted residual finite element techniques. The equation of continuity has been used as constraint due to mass conservation. The basic unknowns for the governing differential equations are the velocity components U, V, the temperature T and the pressure P. The Galerkin finite element technique yields the subsequent nonlinear residual equations. Three points Gaussian quadrature is used to evaluate the integrals in these equations. The non-linear residual equations are solved using Newton-Raphson method to determine the coefficients of the expansions. 5. Grid refinement check An extensive mesh testing procedure is conducted to guarantee a grid-independent test for Ri 1.0, Re 100 and Pr 0.71, Ha 20 . The domain is divided into a set of non-overlapping regions called elements. Five different non-uniform grid systems shown in Table 1 with the number of elements within the resolution field are examined. The extreme values of Nu and T are used as a sensitivity measure of accuracy of the solution and are selected as the monitoring variables. Considering both the accuracy of numerical values and computational time, the present computations are performed with 4848 nodes and 3232 elements grid system. Table 1: Grid sensitivity check at Ri 1.0, Re 100 and Pr Nodes (elements) Nu T 1392(928) 5.814493 0.249295 1518(1012) 5.851071 0.250581 3444(2296) 5.923958 0.241458 4848(3232) 5.883601 0.24164 5232 (3488) 5.899925 0.241952
0.71, Ha
20 Time(s) 1.75 1.781 2.375 2.797 2.922
6. Code Validation A computational model is validated for mixed convection heat transfer by comparing the results on mixed convection in ventilated cavity with left heated wall performed by Raji and Hasnaoui [12]. In the present work numerical predictions have been obtained on the triangular mesh with 4848 nodes and 3232 elements for the same boundary condition of Raji and Hasnaoui [12]. The model of Raji and Hasnaoui [12] is reproduced at first and then the comparison is depicted in Fig. 2. It can be decided that the current code can be used to predict the flow field for the present problem. Present work Raji and Hasnaoui [12] Streamlines
Isotherms
Fig. 2 Streamline and isotherm comparison of present work with Raji and Hasnaoui [12] for Re
10 and Pr
0.71, Ha
0
Md. S. Alam and M.A.H. Khan / Procedia Engineering 90 (2014) 403 – 410
7. Results and Discussions
Ha 0
Ha 50
Ha 100
Ha
200
(a) Fig 3: Effect of Hartman number Ha at Ri 1.0, Re 100 and Pr
(b)
7.1 on (a) Streamlines and (b) Isotherms.
407
408
Md. S. Alam and M.A.H. Khan / Procedia Engineering 90 (2014) 403 – 410
(a)
(b)
Fig 4: Effect of Hartman number Ha on (a) average fluid temperature and (b) average Nusselt number in the channel while Re 100. Pr
(a)
7.1 .
(b)
Fig 5: Effect of Prandtl number Pr on (a) average fluid temperature and (b) average Nusselt number in the channel while Re 100, Ha
20 .
The characteristics of the MHD mixed convection flow and temperature fields through the divergent channel are examined by investigating the effect of the Reynolds number Re, Prandtl number Pr and Hartman number Ha. The fluid flow patterns inside the channel are presented in terms of streamlines and isotherms in Fig. 3(a)-(b) at the four different values of Hartman number Ha( 0,50,100,200) with Ri 1.0, Re 100 and Pr 7.1 . In mixed convection regimes, the velocity field contains no vortices near the circular obstacle and the streamlines become symmetrical along side walls of the channel with increasing values of Ha. This is because; the magnetic field acts against the flow and reduces the fluid velocity. Thermal boundary layer thickness varies as Ha increases and the isothermal lines become distorted at large values of Ha. Fig. 4 (a)-(b), represent the variation of average temperature T of the fluid and the average Nusselt number (Nu) at the heated surface in the diverging channel due to the rising
409
Md. S. Alam and M.A.H. Khan / Procedia Engineering 90 (2014) 403 – 410
values of Ha. It is clearly observed that as Ri increases, average fluid temperature T reduces for all values of Ha specially in absence of Ha more rapidly. On other hand, Nu increases gradually when Ri rises at all values of Ha and particularly in absence of Ha, Nu increases more swiftly. The average temperature of the fluid and the rate of heat transfer remain almost constant for large values of Ha 100. However, as Ha increases T increases, while the rate of heat transfer decreases. The variation of Nu at the heated surface, the average temperature T of the fluid in the channel for different Prandtl number along with Richardson number have been presented in Fig. 5(a)-(b). It is clearly seen that for 0 d Ri d 8 , T remains almost constant for lower values of Pr but T decreases while Pr increases. Nevertheless, Nu increases slowly as Ri increases for lesser values of Pr whereas Nu increases gradually with Ri due to the escalating values of Pr. The effect of Reynolds number Re on T and Nu is displayed in Fig. 6 (a)(b). It is observed that the average temperature T is decreasing sharply for 0 d Ri d 1.0 but decreases slowly beyond this value of Ri for the higher values of Re( 400, 500). However, the temperature T decreases uniformly with the increasing values of Ri when Re( 100, 200). Moreover, the average Nusselt number (Nu) is mounting leisurely for the minor values of Re( 100, 200) but increases significantly between 0 d Ri d 1.0 and then slow down for Ri t 1.0 at Re( 400, 500) . Maximum heat transfer rate is always found for the higher values of Re( 400, 500).
(a) (b) Fig 6: Effect of Reynolds number Re on (a) average fluid temperature and (b) average Nusselt number in the channel while Ha
20, Pr 7.1 .
Conclusion A computational study is performed to investigate the MHD mixed convection flow through a diverging channel with a circular obstacle. Effects of Reynolds number, Prandtl number and Hartman number in the channel with a circular obstacle are highlighted to study their impacts on flow structure and heat transfer characteristics. The effect of the magnetic parameter on velocity field is remarkable. On the other hand, the obstacle has a significant effect on streamlines. The variations of temperature field with mentioned parameters are notable. Magnetic field affects the mean bulk temperature and average heat transfer rate significantly. The heat transfer rate and the average temperature of the fluid changes rapidly against Ri with rising values of Re, Pr, Ha. The present investigation reveals that the opposing forced flow and thermal fields inside the channel strongly depend on the relevant dimensionless groups. References [1] S.C.R. Dennis, W. H. H. Banks, P. G. Drazin, M. B. Zaturska, Flow along a diverging channel, Journal of Fluid Mechanics. 336(1997) 183-202. [2] P. G. Drazin, Flow through a diverging channel: instability and bifurcation, Fluid Dynamics Research. 24(6)
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