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Frontiers in Aerospace Engineering (FAE) Volume 2 Issue 1, February 2013 www.fae-journal.org

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Second Law Analysis of Hydromagnetic Flowfrom a Stretching Rotating Disk: DTM-Padé Simulation of Novel Nuclear MHD PropulsionSystems Mohammad Mehdi Rashidi1 , Osman Anwar Bég*2 , Navid Freidooni Mehr1 , Behnam Rostami3 Mechanical Engineering Department, Engineering Faculty of Bu-Ali Sina University, Hamedan, IranDirector: Gort Engovation (Propulsion), 15 Southmere Avenue, Bradford, BD7 3NU, England, UKMechanical Engineering Department, Engineering Faculty of Bu-Ali Sina University, Hamedan, IranYoung Researchers & Elites Club, Hamedan Branch, Islamic Azad University, Hamedan, [email protected];*[email protected];[email protected];4 [email protected]

Abstract

In this paper, the second law analysis of hydromagneticconducting flow due to a stretching rotating disk with heattransfer is investigated using a semi-analytical/numericaltechnique termed DTM-Padé simulation. The study hasapplications in rotating magneto-hydrodynamic (MHD)energy generators for new space systems and also thermalconversion mechanisms for nuclear propulsion spacevehicles. The momentum and energy conservation equationsare non-dimensionalized using appropriate transformationsleading to a set of nonlinear, coupled, ordinary differentialequations for momentum in the radial, azimuthal andnormal directions and a temperature distribution equation.Using the appropriate velocity components and temperaturefield, the entropy generation equation is obtained. Theeffects of various parameters such as magnetic interactionparameter, rotation parameter, Eckert number, Brinkmannumber on the entropy generation and Bejan number areillustrated and described. The irreversibility mechanisms ofthe entropy generation for the emerging parameters are alsoinvestigated and suggestions for minimizing the entropy

generation are proposed. The DTM-Padé results are verifiedwith the numerical results and found to be in excellentagreement. The simulations also show the feasibility of usingmagnetic rotating disk drives in novel nuclear spacepropulsion engines.

Keywords

Second Law Analysis; Steady MHD Flow; Rotating Disk; RadialStretching; Heat Transfer; DTM-Padé Simulation; Bejan Number;Swirling Flow; Nuclear Space Propulsion; Energy Generators

Introduction

The second law of thermodynamics is employed byengineers to obtain the optimal design of thermal

systems via minimizing the irreversibility and entropygeneration, which can improve the efficiency ofindustrial systems. Entropy generation involvesthermodynamic irreversibilities, such as characteristicsof convective heat transfer, heat transfer across finitetemperature gradients, viscous dissipation effects andmagnetic field effects which arise in modern aerospaceheat transfer processes. The fluid flow and heat

transfer processes are intrinsically irreversible, whichleads to an increase entropy generation and usefulenergy destruction. Entropy generation has stimulatedsignificant interest in recent years in aerospace thermalsciences. Erbay et al. used second law analysis toinvestigate computationally the entropy generation ina channel with a finite volume method and SIMPLEalgorithm. Aïboud and Saouli applied second lawthermodynamics to analyze the entropy generation inmagneto-viscoelastic flow over a stretching surface.Al-Odat et al. simulated the effect of magnetic field onthe entropy generation due to laminar forcedconvection over a horizontal flat plate with an implicitfinite difference technique. Aïboud and Saouli furtherstudied second law thermodynamics and viscoelasticmagnetized flow from a stretching surface withdouble-diffusive convective heat and mass transfer,elaborating the influence of magnetic parameter,Reynolds number and Prandtl number on the entropygeneration number. Sahin described the effect of thevariable viscosity on the entropy generation in alaminar fluid flow, showing that the entropygeneration due to viscous friction became dominantfor low heat-flux terms. Ibáñez and Cuevas examined



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entropy generation minimization of a hydromagneticflow in a micro-channel. Makinde and Bég employed aperturbation expansion technique coupled with aspecial Hermite-Pade' approximations in the MAPLEprogram, to investigate volumetric entropy generation

numbers, irreversibility distribution ratio and theBejan number evolution in magneto-hydrodynamicchannel flow in a plasma propulsion duct. Arikoglu etal. reported on the effect of slip in entropy generationfrom rotating disk in hydromagnetic flow with adifferential transform method. Entropy generationanalysis was applied to modeling and optimization ofMHD induction devices by Salas et al.

Rotating Magneto-hydrodynamic (MHD) flowsexploit magnetic fields which can induce currents in a

movable conductive fluid. Liquid metals, plasmas andelectrolytes are all important examples of MHD fluids.MHD flows also utilize a Lorentzian drag force whichcan be used to regulate a variety of flow regimes.Many applications exist for rotating hydromagneticflows including propulsion systems, rotating MHDenergy generators, smart spacecraft landing gearsystems, hydrogen production with solar MHD plants,plasma fusion technology, nuclear thermal controlsystems, magneto-hydrodynamic chemical reactorprocessing and biomagnetic reactors. Many further

applications of rotating hydromagnetic flowsincluding swirling disk flows, flows from revolvingcones in liquid metal stirring etc, are documented inthe recent monograph by Bég et al.In order to describe many physical systems with amathematical model, nonlinear equations are used.Merical methods are frequently deployed to solvethese complex systems of differential equations, whichmay be multi-degree and often strongly coupled.Although many powerful numerical techniques existincluding finite element methods, network simulation,

finite difference methods and CFD codes such asFLUENT, computational expense is always a problem.Therefore semi-numerical/analytical methods, such asthe homotopy perturbation method (HPM),differential transform method (DTM) and homotopyanalysis method (HAM) have found increasingpopularity amongst aerospace fluid dynamicsresearchers. These techniques can successfully tacklenonlinear differential equations for a variety of boundary conditions, and therefore they holdsignificant promise in nuclear engineering sciences.

In this article, the second law analysis is employed tostudy the MHD fluid flow with heat transfer due to a

stretching rotating disk. This problem was firstlystudied by Turkyilmazoglu numerically, withoutconsidering entropy generation analysis. In this paper,these nonlinear swirling flow and heat conservationequations are solved via DTM-Padé simulation. As the

second law analysis is more reliable than the first lawanalysis for many aerospace and nuclear engineeringsystems, entropy generation analysis is also included.In the current study, the effect of variousthermophysical parameters, such as magneticinteraction parameter, rotation parameter, Eckertnumber, Prandtl number, Brinkman number andReynolds number on velocity and temperature fieldsand also on the entropy generation and Bejan numberare investigated in detail. The study has importantapplications in thermal optimization of novel MHD

coupled nuclear space propulsion systems usingrotating disks and hydromagnetics.

Mathematical Transport Model

The 3-dimensional steady magneto-hydrodynamiclaminar boundary-layer flow of a Newtonian,electrically-conducting, viscous fluid from a rotatingdisk in the presence of an externally applied axially-directed uniform magnetic field is considered. Thisregime is often also termed hydromagnetic VonKarman swirling flow. The governing equations; thecontinuity, Navier-Stokes (momentum conservation),generalized Ohm’s law and energy conservationequations can be presented, in the presence of Ohmic(Joule) heating and viscous dissipation, in vectorialform, as follows:

0. =∇ u (1)

( ) B J uuu ×+∇+−∇=∇⋅ 2µ ρ P (2)

[ ] Bu E J ×+= σ (3)

( ) σ µ ρ 22

J u+Φ+∇=∇⋅

T k T c p (4)where all parameters are defined in the nomenclature.The disc radius is much larger than the boundary layerthickness, so that edge effects can be neglected. Theapplied magnetic field is steady and sufficiently weakto neglect the induced magnetic field i.e. a lowmagnetic Reynolds number is assumed. This is areasonable assumption for the flow of certain workingfluids in novel nuclear-MHD propulsion systems andalso in nuclear engineering liquid metals, e.g., liquidsodium. The physical regime is depicted in Fig. 1 withreference to non-rotating cylindrical polar coordinateswhere all parameters are defined in the nomenclature.




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The disc radius is much larger than the boundary layerthickness, so that edge effects can be neglected. Theapplied magnetic field is steady and sufficiently weakto neglect the induced magnetic field i.e. a lowmagnetic Reynolds number is assumed. This is a

reasonable assumption for the flow of certain workingfluids in novel nuclear-MHD propulsion systems andalso in nuclear engineering liquid metals, e.g., liquidsodium. The physical regime is depicted in Fig. 1 withreference to non-rotating cylindrical polar coordinates( ) zr ,,θ .

FIG. 1 3-D SWIRLING VON KARMAN MAGNETO-HYDRODYNAMIC FLOW REGIME AND COORDINATE SYSTEM

The disk rotates with a constant angular velocity( )Ω and also stretches with a constant rate( )s in the radialdirection. The external uniform magnetic field B

applies transverse to the disk plane i.e. in the z-direction and possesses a constant magnetic fluxdensity B0. Furthermore all thermophysical and fluidproperties are assumed to be constant and radial andtangential electrical currents are neglected. The diskitself is electrically non-conducting, so that currentdensity vanishes both at the disk surface and in thefluid regime. The penultimate and final terms on theright hand side of eqn. (9) designate the viscousheating and Joule heating contributions, respectively,and have been shown to be significant in propulsion

applications. Following Turkyilmazoglu, the non-dimensional form for the mean flow velocities andtemperature distribution are provided by VonKarman’s classical transformations. With the aid of adimensionless normal distance from the wall,

, zR=η and an appropriate Reynolds numberincorporating the disk stretching rate, ( ) 21ν s R = wehave:

( ) ( ) ( ) ( ) ( ) ( )

−+= ∞∞ η θ η η η T T T H

RrsGrsF T wvu w,

1,,,,, (5)

where the dimensionless functionsF , G , H , and θ satisfy the following ordinary differential equations

defining the radial, azimuthal, axial momentumconservation and energy conservation in the regime:

02 =+′ F H (6)

022 =−′−+−′′ MF F H GF F (7)02 =−′−−′′ MGG H FGG (8)

( ) ( )0Pr 2222 =++′+′+′−′′ GF EcM GF Ec H θ θ (9)

The boundary conditions at the stretching disk surfaceand in the free stream, are prescribed respectively, as:

,as0

,0at,011

∞→=====−=−=−=

η θ

η θ ω

GF

GF H (10)

Here sΩ=ω denotes the “rotation strengthparameter” expressing the ratio of disk swirl to disk

radial stretch. The case of 0=ω corresponds to purestretching of the disk without rotation. The nonlinear,coupled ODEs defined in eqns. (6)–(9) under boundary conditions (10) constitute a robust two-point boundary value problem, which may be readily solvedwith a variety of numerical or semi-numericaltechniques. In the present study we implement acombination of the differential transform method(DTM) and Padé approximants, namelyDTM-Padé simulation, which is described in due course.

Entropy Generation Analysis

The volumetric rate of local entropy generation, for thepresent hydromagnetic swirling heat transfer problem,in the presence of the axial symmetry, can bepresented as follows:

[ ] ( ) ( )[ ] BV EQV J ×+⋅−+Φ+∇=′′′00

22

0

1

T T T

T

k S gen

µ (11)

where:

[ ]

∂∂

+

∂∂

+

∂∂

=∇

2222 1

zT T

r r T

T θ (12)

222

22

2

2

11

12

∂∂+

∂∂+

∂∂+

∂∂+

∂∂+

∂∂+

∂∂+

+∂∂+

∂∂=Φ

r v

zr

ur z

ur ww

r zv

zw

uv

r r u

θ θ

θ (13)

[ ] Bu E J ×+= σ (14)

It is assumed that electric force per unit charge isnegligible compared withV ×B in Eqs. (11) and (14)and furthermore that electric current is much greaterthan QV . With these assumptions, Eq. (11) can be




32

shown to take the form:

( )22

0

20

222

22

2

2

0

2

20

12

vuT B

r v

zr

zu

zv

z

wu

r r

u

T

zT

T

k S gen

++

∂∂+

∂∂+

∂∂

+

∂

∂++

∂

∂

+

∂∂=′′′

σ µ (15)

The right hand side of the above entropy generationequation consists of three parts. The first term is thelocal entropy generation due to heat transferirreversibility. The second larger part refers to thefluid friction irreversibility. The final part denotesmagnetic field effects. The dimensionless form of the

entropy generation rate, termed entropy generationnumber, defines the ratio between the actual entropygeneration rate genS ′′′ and the characteristic entropy

generation rate .0S ′′′ By using the Von Karmantransformation parameters, as defined in Eq. (5), theentropy generation number( )G N becomes, for thepresent problem:

( ) ( ) ( )

( ) ( )( ) ( ) ( )( )2222

222

Re2

Re4

η η η η

η η η θ α

GF BrM GF Br

H Ec

F Br

N G

++′+′+

′++′=(16)

Inspection of Eq. (16), indicates that with increasingvalue of α , the effect of heat transfer irreversibilityincreases. As the Brinkman number and Eckertnumber increase or the Reynolds number decreases,the entropy generation due to fluid frictionirreversibility increases. Irreversibility mechanismdomination is an important aspect of entropygeneration analysis since the entropy generationnumber does not provide any information regardingthis. The Bejan number and the irreversibilitydistribution ratio are introduced to overcome thisshortcoming. The Bejan number (Be) embodies theratio of entropy generation which is caused throughheat transfer to the total entropy generation. Theirreversibility distribution ratio (ϕ ) expresses the ratio between entropy generation due to fluid friction and joule dissipation in heat transfer. These twoparameters, for the present swirlingmagnetohydrodynamic thermal problem can beexpressed as:

( )( ) ( ) ( ) ( ) ( ) ( ) ( )

++

′+′+′++′

′= 2222222

2

Re

2

Re

4η η η η η η η θ α

η θ α

GF BrM GF Br H Ec

F Br Be (17)

( ) ( ) ( ) ( ) ( ) ( )

( )2

222222Re

2Re4

η θ α

η η η η η η φ

′

++

′+′+′+

=GF BrM GF Br H

EcF

Br

(18)

As illustrated by the definitions of the abovedimensionless parameters, the behavior of the Bejannumber and irreversibility distribution ratio arealmost the same. Thus we only examine the Bejannumber effects in this article. The Bejan number fallsin the range, 10 << Be . When 0= Be , fluid frictionirreversibility dominates. For 5.0= Be , heat transferand fluid friction irreversibility effects are the same.For 1= Be , the irreversibility mechanism is dominated by heat transfer effects.

DTM- Padé Simulation

In the current article, DTM is employed in conjunction

with Padé approximants to solve the governingequations (6) to (9), under boundary conditions (10).Zhou was the first one who developed DTM forelectrical circuit modeling. The principal attraction ofthis technique is that it can be applied directly tononlinear differential equations without requiringdiscretization and linearization, a strong advantageover purely numerical methods such as finite elementsand finite differences. DTM also does not require anyperturbation parameters and this offers a greatadvantage over asymptotic and perturbation

expansion techniques. This method has beensuccessfully implemented in a diverse array ofincluding nanofluid, hypersonic, swirl flow andgeothermic flow. In order to solve highly nonlineardifferential equation systems, arising frequently innuclear engineering fluid dynamics, it is pertinent tocombine DTM with Padé approximations and useDTM-Padé simulation. This method effectivelyincreases the convergence of DTM. Further details aregiven in ref. Numerous mathematical techniques existto increase the convergence radius of a given series.

The Padé approximant is a rational fraction. Details of

η

F (

η )

0 1 2 3 4 5-0.2

0

0.2

0.4

0.6

0.8

1

DTM, N=10DTM, N=20DTM, N=30DTM-Pade [5,5]DTM-Pade [10,10]Ref. [26]

FIG. 2 RADIAL VELOCITY COMPONENT,( )η F OBTAINED BY

DTM FOR DIFFERENTN AND ORDERS OF DTM-PADÉ INCOMPARISON WITH REF. [26] FOR .2,1 == M ω




33

such approximants are provided in Baker and Graves-Morris. The approximants are generated numericallyusing MATHEMATICA software. Stability, consistencyand convergence are successfully achieved by DTM-Padé simulation. DTM alone diverges (figure 2).

Results and Discussion

To verify the validity of the present computations,results have been benchmarked with the non-entropystudy of Turkyilmazoglu. Excellent correlation between the semi-numerical/analytical resultsobtained by DTM-Padé (Padé approximant of order)and the Chebsyhev spectral collocation numericalcomputations of Turkyilmazoglu is achieved, asobserved in Fig 3. Confidence in the present DTM-Padé solutions is therefore high. We note that the caseof equal rotational strength and disk stretchingcorresponds to ω = 1. The cases ofω = 0 i.e. purestretching of the disk without rotation, andω→∞ i.e.pure rotation of the disk without stretching areextreme scenarios and not considered in the presentstudy.

η0 0.5 1 1.5 2 2.5

0

1

2

3

4

5

F ( η )G ( η )− H ( η )θ ( η )Ref. [26]

FIG. 3 RADIAL, AZIMUTHAL AND AXIAL VELOCITY

FUNCTIONS AND TEMPERATURE FUNCTION OBTAINED BYDTM-PADÉ (PADÉ APPROXIMANTS [10, 10]) IN COMPARISON

WITH REF. [26] WITH 2,1Pr === M Ec AND 3=ω

Figures 4-7 present the velocity contours in alldirections. As the radial coordinate increases, theprimitive radial ( )u and azimuthal ( )v velocitycomponents clearly increase. These velocitycomponents are maximized near the surface of thedisk i.e. at low values of axial coordinate (bottom righthand corner of both Figs. 4 and 5). The velocity vectorsare shown in Fig. 7, in order to have a better grasp ofthe fluid flow. In both cases althoughω has beenprescribed as 1, the disk stretch rate is very high at 10.Conversely in Fig. 6 we observe that axial velocitycomponent is maximized near the disk surface for allvalues of radial coordinate (lower red band). The axialvelocity clearly decays as we depart from the surfaceand is minimized at greater distances from the disk( )40. z ~ . Figure 7 distinctly shows the convergence of

velocity vectors( )wu , towards the bottom right handcorner of the plot i.e. fluid is clearly drawn in a fan likemechanism outwards along the radial coordinate andin the negative axial direction. This pattern typifiesVon Karman swirling flow.

FIG. 4 RADIAL VELOCITY CONTOURS WITH ,1== ω M 6109,2Pr

−×=== ν Ec AND .10=s

FIG. 5 AZIMUTHAL VELOCITY CONTOURS WITH ,1== ω M 6109,2Pr

−×=== ν Ec AND .10=s

FIG. 5 AXIAL VELOCITY CONTOURS WITH ,1== ω M 6109,2Pr

−×=== ν Ec AND .10=s

r(m)

z ( m )

0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FIG. 7 VECTOR VARIABLES OFu AND w WITH ,1== ω M

6109,2Pr −×=== ν Ec AND .10=s

Figures 8-13 illustrate the influence of the emergingthermophysical parameters on entropy generationnumber, and Bejan number distributions, with radial




34

coordinate. In each of these graphs, dimensionlesstemperature difference is set to unity.

The entropy generation number and Bejan numbervariation with changing magnetic interactionparameter are presented in Figs. 8 and 9. As themagnetic interaction parameter increases, the amountof entropy generation increases considerably (it isminimized in the absence of a magnetic field, 0= M ).This implies that in order to control the entropy whichis generated from swirling flow on the disk surface,the value of magnetic interaction parameter should bereduced, an issue of interest in nuclear-MHD rotatingdisk propulsion. The fluid friction irreversibility isdominated mechanism of irreversibility, and is distantfrom of the disk surface for small values of magneticinteraction parameter. As the magnetic interactionparameter is elevated, the heat transfer mechanismdominates irreversibility. It should also be mentionedthat a minimum point arises for Bejan number near thedisk surface for any value of magnetic interactionparameter. At large values of M the Bejan numberescalates and after ascending very sharply from thedisk surface converges asymptotically to themaximum allowable value of unity, a trend sustainedinto the free stream. In essence when entropygeneration is maximized, the Bejan number isminimized and vice versa in the swirling flow regime.

η

N G

0 0.3 0.6 0.9 1.2 1.50

30

60

90

120

150

180M = 0.0M = 0.5M = 1.0M = 4.0M = 6.0

FIG. 8 THE VARIATION OF ENTROPY GENERATION NUMBER,

G N WITH MAGNETIC INTERACTION PARAMETER FOR8Re,4,2Pr ==== Br Ec AND .1=ω

η

B e

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

M = 0.0M = 0.5M = 1.0M = 4.0M = 6.0

Fig. 9 THE VARIATION OF BEJAN NUMBER, Be WITH

MAGNETIC INTERACTION PARAMETER FOR8Re,4,2Pr ==== Br Ec AND .1=ω

Figures 10, 11 illustrate the effect of rotation strength

parameter on the entropy generation and Bejannumber distributions with radial coordinate. Anincrease in ω generates a similar but morepronounced response to that of increasing magneticinteraction parameter i.e. it causes a considerable

increase in G N near the disk surface, leading to amaximum entropy generation value always at the disksurface itself. Further from the diskG N profiles decaysharply to vanish in the free stream. The Bejan numberis generally minimized closer to the disk andmaximized further from the disk, in direct contrast tothe entropy generation number. In all cases, the lowestvalue of Bejan number arises in the same vicinity asthe maximum value of entropy generation number i.e.near the disk surface. For larger values of rotationstrength parameter Bejan number achieves theasymptotic profile sooner. In the asymptotic statewhere the profiles all converge on unity in the freestream, the heat transfer irreversibility is dominant.

η

N G

0 0.4 0.8 1.2 1.6 20

500

1000

1500

2000

ω = 1ω = 2ω = 3ω = 4ω = 5

FIG. 10 THE VARIATION OF ENTROPY GENERATION NUMBER,G N WITH ROTATION STRENGTH PARAMETER FOR

3,2,1Pr ==== Br M Ec AND .6Re =

η

B e

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1ω = 1ω = 2ω = 3ω = 4ω = 5

FIG. 11 THE VARIATION OF BEJAN NUMBER, Be WITH

ROTATION STRENGTH PARAMETER FOR3,2,1Pr ==== Br M Ec AND .6Re =

Figures 12 and 13 depict the response of the entropynumbers to Eckert number. Increasing Eckert numbersignificantly increases the entropy generation number,

G N increases, as the Eckert number increases. Itshould also be noted that at a certain distance from thedisk surface ( ),45.0≅η the amount of generated

entropy converges on the same value for all Eckertnumbers. The influence of Eckert number on the Bejan




35

number ( ) Be profiles is similar to those induced by therotational strength parameter. Brinkman number,

T k r s Br ∆= 22µ represents the ratio of direct heatconduction from the disk surface to the viscous heatgenerated by shear in the boundary layer. It is distinctfrom Eckert number which symbolizes the kineticenergy of flow to the boundary layer enthalpydifference. It is in fact more appropriate for industrialflows, whereas Eckert number often arises in high-speed aerodynamic flows. Both parameters arequantifications of viscous dissipation effects. Bejannumber decays from a high value at the disk surface toa minimum near the disk surface and then ascendsrapidly to converge on unity in the free stream. A risein Ec clearly enhances Bejan number. In all cases wehave considered positive Ec values i.e. heat istransferred from the disk surface to the fluid byconvection currents. Further from the disk surfaceheat transfer irreversibility dominates. For largevalues of Prandtl number, the fluid frictionirreversibility dominates compared with heat transfer.Bejan number is clearly a maximum when entropygeneration number is a minimum.

η

N

G

0 0.4 0.8 1.2 1.6 20

250

500

750

1000

1250

Ec = 1Ec = 2Ec = 3Ec = 4Ec = 5

FIG. 12 THE VARIATION OF ENTROPY GENERATION NUMBER,

G N WITH ECKERT (VISCOUS HEATING) NUMBER FOR4,2,1Pr ==== Br M ω AND .6Re =

η

B e

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ec = 1Ec = 2Ec = 3Ec = 4Ec = 5

FIG. 13 THE VARIATION OF BEJAN NUMBER, Be WITH

ECKERT (VISCOUS HEATING) NUMBER FOR4,2,1Pr ==== Br M ω AND .6Re =

We further note that the correct trends for entropygeneration number and Bejan number have been

achieved by DTM-Padé simulation and have beenconcurred with the observations documented in other

studies, for example Aïboud and Saouli and Arikogluet al., the latter considering Von Karman swirlinghydromagnetic flow with slip effects. As the Brinkmannumber increases, the effect of fluid frictionirreversibility is amplified on the disk surface. Further

from the disk surface, the effect of heat transferirreversibility becomes larger as the Brinkman numberdecreases. For high Br , the effects of heat transfer andfluid friction irreversibilities are almost equal.Effectively, the present semi-numerical computationstestify that the contribution of Brinkman number, asoriginally proposed and elucidated by Bejan andfurther implemented in hydromagnetic heat transfer by Makinde and Bég, is therefore of significance insecond law thermodynamic analysis, and plays animportant potential role in nuclear MHD spacecraft

engine thermodynamic optimization.Conclusions

In the current investigation, second lawthermodynamic analysis has been utilized to obtainthe entropy generation equations for swirling VonKarman hydromagnetic flow with heat transfer from astretching rotating disk. A semi-numerical techniqueamalgamating the differential transform method(DTM) and Padé approximants i.e. DTM-Padésimulation, has also been applied to determinesolutions for the transformed momentum and energyconservation equations describing the swirling fluiddynamics, under appropriate boundary conditions.The nonlinear coupled multi-degree boundary valueproblem has been very efficiently solved with benchmarking to numerical shooting quadrature,demonstrating excellent correlation between bothmethods. The velocity contours in all directions andvelocity vectors have been visualized to illustrate theactual dynamics of the flow in primitive coordinates.Furthermore the influence of Brinkman number onentropy generation number and Bejan number, havealso been presented and analyzed. The powerfulcontributions of disk swirl and disk stretch(investigated via the rotational strength parameter),magnetic field retardation and viscous heating on theflow variables have clearly been identified in thecomputations. The fundamental objective of secondlaw thermodynamics analysis has been also achieved,namely, minimization of entropy in the swirling diskflow regime. The present computations have providedsome further insight into the thermodynamics andfluid mechanics of proposed rotating disk MHDsystems coupled with nuclear space propulsion




36

engines, although these require further evaluation, inparticular for transient and magnetic induction effects,aspects which are under consideration by the authors.DTM-Padé simulation holds excellent potential inanalyzing such aerospace problems.

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Anwar Bég, O., Sim, L., Zueco, J., and R. Bhargava.“Numerical Study of Magnetohydrodynamic ViscousPlasma Flow in Rotating Porous Media with HallCurrents and Inclined Magnetic Field Influence”Communications in Nonlinear Science and NumericalSimulation 15 (2010): 345-59.

Anwar Bég, O., Zueco, J., and L.M. López-Ochoa. “Networknumerical Analysis af Optically-Thick HydromagneticSlip Flow From a Porous Spinning Disk with RadiationFlux, Variable Thermophysical Properties and SurfaceInjection Effects” Chemical Engineering

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Entropy Generation in a Single Rotating Disk in MHDFlow” Applied Energy 85 (2008): 1225-36.

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of thermodynamic optimization of finite-size systemsand finite-time processes: CRC Press, Florida, 1996.

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Daud, H.A., Li, Q., Anwar Bég, O., and S.A.A. AbdulGhani.“Numerical Investigations of Wall-Bounded Turbulence”Proceedings of the Institution of Mechanical Engineers,Part C: Journal of Mechanical Engineering Science 225

(2011): 1163-74.Erbay, L., Ercan, M., Sülüs, B., and M. Yalçÿn. “Entropy

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Generation During Fluid Flow Between Two ParallelPlates with Moving Bottom Plate” Entropy 5 (2003): 506-18.

Fang, T. “Flow over a stretchable disk, Physics of Fluids” 19

(2007): 128105-10.Ghosh, S.K., Anwar Bég, O., Zueco J, and V.R. Prasad.

“Transient Hydromagnetic Flow in a Rotating ChannelPermeated by an Inclined Magnetic Field With MagneticInduction and Maxwell Displacement Current Effects”ZAMP: Journal of Applied Mathematics and Physics 61(2010): 147-69.

Hayat, T., Naza, R., Asghar, S., and S. Mesloub. “Soret–Dufour Effects on Three-Dimensional Flow of ThirdGrade Fluid” Nuclear Engineering and Design 243 (2012):1- 14.

Ibáñez, G., and S. Cuevas. “Entropy GenerationMinimization of a MHD (Magnetohydrodynamic) Flowin a Micro-Channel” Energy 35 (2010): 4149-55.

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Makinde, O.D., and O. Anwar Bég. “On InherentIrreversibility in a Reactive Hydromagnetic Channel

Flow” Journl of Thermal Science 19 (2010): 1-8.Pop, I, and V.M. Soundalgekar. “The Hall Effect on an

Unsteady Flow due to a Rotating Infinite Disc” NuclearEngineering and Design 44 (1977): 309-14.

Prasad, V.R., Vasu, B., Anwar Bég, O., and R. Parshad.“Unsteady Free Convection Heat and Mass Transfer in aWalters-B Viscoelastic Flow Past a Semi-Infinite VerticalPlate: a Numerical Study” Thermal Science 15 (2011):291-305.

Rashidi, M.M., Anwar Bég, O., and N. Rahimzadeh. “Ageneralized DTM for combined free and forced

convection flow about inclined surfaces in porousmedia” Chemical Engineering Communications 199(2012): 257-82.

Rashidi, M.M., Anwar Bég, O., Asadi, M., and M.T. Rastegari.

“DTM-Padé modeling of natural convective boundarylayer flow of a nanofluid past a vertical surface”International Journal of Thermal and Environmental 4(2012): 13-24.

Rashidi, M.M., Erfani, E., Anwar Bég, O., and R.S. Gorla.“Modified Differential Transform Method (DTM)simulation of hydromagnetic multi-physical flowphenomena from a rotating disk” World Journal ofMechanics 5 (2011): 217-30.

Rashidi, M.M., Keimanesh, M., Anwar Bég, O., and T.K.Hung. “MHD Bio-Rheological Transport Phenomena in aPorous Medium: a Simulation of Magnetic Blood FlowControl and Filtration”, International Journal forNumerical Methods in Biomedical Engineering 27 (2011):805-21.

Rawat, S., Bhargava, R., and O. Anwar Bég. “TransientMagneto-Micropolar Free Convection Heat and MassTransfer Through a Non-Darcy Porous Medium Channelwith Variable Thermal Conductivity and Heat Source

Effects” Proceedings of the Institution of MechanicalEngineers, Part C: Journal of Mechanical EngineeringScience 223 (2009): 2341-55.

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Induction Devices” Journal of Physics D: AppliedPhysics 32 (1999): 2605–8.

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Tripathi, D., Anwar Bég, O., and J. Curiel-Sosa. “HomotopySemi-Numerical Simulation of Peristaltic Flow ofGeneralised Oldroyd-B Fluids with Slip Effects”Computer Methods in Biomechanics and Biomedical

Engineering (2012): DOI:10.1080/10255842.2012.688109.Turkyilmazoglu, M., MHD Fluid Flow and Heat Transfer




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due to a Stretching Rotating Disk, International Journalof Thermal Sciences 51 (2012): 195-201.

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Zueco, J., and O. Anwar Bég. “Network Numerical Analysis

of Hydromagnetic Squeeze Film Flow DynamicsBetween Two Parallel Rotating Disks with InducedMagnetic Field Effects” Tribology International 43 (2010):532-43.

Zueco, J., Eguía, P., Granada, E., Míguez, J.L., and O. AnwarBég. “An Electrical Network for the Numerical Solutionof Transient MHD Couette Flow of a Dusty Fluid: Effectsof Variable Properties and Hall Current” InternationalCommunications in Heat and Mass Transfer 37 (2010):1432–9.

second law analysis of hydromagnetic flow from a stretching rotating disk: dtm-padé simulation of...

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