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Kaushal Patel 1

International Journal of Research in Mathematics

Simulation of Incompressible ElectricallyConducting Fluid through Cylindrical Duct using

Finite Difference MethodKaushal Patel

Abstract: In this paper we deal with the steady state flow ofincompressible and electrically conducting fluid through acylindrical channel. In this we consider the effect of magneticfield or Hartmann number on velocity profile. An externaluniform magnetic field is applied to the fluid which is directedin a co-circular cylinder perpendicular to the flow direction.The governing partial differential equations of cylindricalco-ordinate system are transform in Cartesian system and thensolve numerically using finite difference method. And finallywe discuss the effects of magnetic and Hartmann parameter onthe velocity.

Keywords: Cylindrical duct, Electro conducting fluid,Hartmann number, Navier - Stokes equation.

I. INTRODUCTION

The increasing number of technical utilization ofmagmatic fluid in industries, since the last century due to itssubstantial applications. For example, magneto-hydrodynamics (MHD) steam plants and MHD generators areused in the modern power plants. The basic concept of theMHD generator is to generate electrical energy from themotion of conductive fluid that is crossing a perpendicularmagnetic field. Carnot efficiency is improved by thepresence of MHD unit. Another example is the MHD pumpsand flow meters. In this type of pumps, the electrical energyis converted directly to a force which is applied on theworking fluid. MHD separation in metal casting withsuperconducting coils is another important application.

A very useful proposed application which involveselectromagnetic fluid is the lithium cooling blanket in anuclear fusion reactor. The high-temperature plasma ismaintained in the reactor by means of magnetic field. Theliquid-lithium circulation loops, which will be locatedbetween the plasma and magnetic windings, are calledlithium blankets. The lithium performs two functions: itabsorbs the thermal energy released by the reaction and itparticipates in nuclear reactions in which tritium is produced.The lithium blanket is thus a very important reactorcomponent. On other hand, the blanket will be acted upon byan extremely strong

Kaushal Patel, Department of Mathematics, Veer Narmad South GujaratUniversity, suratEmail: [email protected]

magnetic field. Consequently, to calculate the flow of liquidmetal in channels or pipes situated at different angles to themagnetic field, and to determine the required pressure drop,heat transfer, etc., knowledge of the appropriate MHDrelationships will be necessary.Williams published results of experiments with electrolytesflowing in insulated tubes. The tubes were placed betweenthe poles of a magnet, and the potential difference across theflow was measured using wires passed through the walls.Hartmann and Lazarus made some very comprehensivetheoretical and experimental studies of this subject. Theyperformed their experiments with mercury which has anelectrical conductivity 100,000 times greater than that of anelectrolyte. This made it possible to observe a wider range ofphenomena than in the experiments by Williams. Theyexamine the change in drag and the suppression of turbulencecaused by magnetic field. Hartmann obtained the exactsolution of the flow between two parallel, non-conductingwalls with the applied magnetic field normal to the walls.Shercliff [1, 2], in 1956, has solved the problem ofrectangular duct, from which he noticed that for highHartmann numbers M the velocity distribution consists of auniform core with a boundary layer near the walls. In 1962,Gold and Lykoudis [13] obtained an analytical solution forthe MFM flow in a circular tube with zero wall conductivitywhile in 1968, Gardner and Lykoudis [7, 10, 12, and 13] haveacquired experimentally some results for circular tube withand without heat transfer. The MFM flow is also examinednumerically by Al-Khawaja et al. for the case of circular tubewith heat transfer and for the case of uniform wall heat fluxwith and without free convection. The solution for MFMsquare duct flow is obtained using spectral method byAl-Khawaja and Selmi [9] for the case of uniform walltemperature. This result enabled to solve the problem for acircular pipe in an approximate manner for large M assumingwalls of zero conductivity and, subsequently, walls withsmall conductivity.

II. PROBLEM STATEMENT

The forced convection in a horizontal co-circular pipe ofradius ‘a’ and ‘b’ in a uniform transverse magnetic field B0

as shown in figure 1. A homogeneous, incompressible,viscous, electrically-conducting fluid flows through ahorizontal cylindrical pipe and is subjected to a uniformsurface temperature and a uniform surface heat flux. In

Kaushal Patel 2


conjunction with defining this problem, the followingassumptions are made:a) Assume to be the fluid is incompressible and change oftemperature of fluid does not affect the properties of fluid.b) The pipe is sufficiently long that it can be assumed theflow and heat transfer are fully developed and entrance orexit effects can be neglected.c) Only pressure and temperature vary linearly with axialdirection.c) The contributions of viscous and Joulean dissipation inthe energy equation are small and can be neglected. Thisassumption has been shown to be applicable to a similarproblem when no external electric field is imposed on theflow.d) The induced magnetic field produced as a result ofinteraction of applied field, B0, with either main orsecondary flow, will be assumed negligibly small comparedto B0.

Figure 1: MFM cylindrical duct flow

This assumption follows from the fact that themagnetic Reynolds number based on the flow is muchsmaller than unity under conditions found in typicalapplications.

The motion of an electrically conducting fluid inthe presence of a magnetic field obeys the well-knownequations of magnetohydrodynamics. The fluid is treated asa continuum, and the classical results of fluid dynamics andelectro dynamics are combined to express the phenomenon.For the steady flow of a viscous, incompressible fluid withconstant properties, the full magnetohydrodynamic systemcan be reduced to two equations involving the velocity,pressure, and magnetic field, i.e. the modified Navier-Stokesequation and the induction equation, along with thesolenoidal conditions on the two vector quantities:Here we consider an incompressible, Newtonian flow with a

velocity field ⃗ = ( , , ) and magnetic field ⃗ =(ℎ , ℎ , ℎ )0. For incompressible Newtonian liquid metalfluid and steady-state conditions, the modifiedNavier-Stokes equations under the effect of magnetic fieldbody force including induction and energy equations incylindrical coordinate system of vector forms are,respectively,

( ) + ( ) + ( ) = 0 (1)R-component:+ − + = − | | +ℎ + − + ℎ + − +− +(2)−Component:+ + + = − | | +ℎ + − + ℎ + −+ + + (3)z - Component:+ + = − | | + ℎ +− + ℎ + + +(4)The energy equation in cylindrical coordinates allows fornon-constant physical properties, energy generation, andconversion of mechanical to internal energy using viscousdissipation, which is expressed in terms of unspecifiedviscous stress–tensor components :+ + = ++ − ( ) + +− − + − −+ − + − ++ Φ + | |

(5)

+ += + + + 2 ++ + 2 + + ++ + + + Φ + | |(6)

Where and is the heat capacity at constant volume andpressure respectively, k the thermal conductivity, and Thelast two terms in the right hand of the energy equation,Equation 5, represent the viscous and Joulean dissipations,respectively. It can be applied to non- Newtonian fluids ifthe appropriate constitutive equation relating the viscousstress to the rate of strain is known.In addition to solenoidal conditions on the two vectors( ) + ( ) + ( ) = 0 And( ) + ( ) + ( ) = 0

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Now we transform cylindrical systems to Cartesian systems[6]. We convert the cylindrical system to Cartesian Systemby taking the transformation( , ) = cos( , ) = sinBy the simplification system of equations is( . ∇) + ∇ + | | = ∇ + ( . ∇) (7)∇ + [( . ∇) − ( . ∇) ] = 0 (8)( . ∇) = ∇ + + | |(9)And solenoidal conditions on the two vectors∇. = 0 And ∇. = 0 (10)For very small magnetic Reynolds number (i.e theinduced magnetic field produced as a result of interaction ofapplied field, , will be assumed negligibly smallcompared to ), The induced equation , equation 8, can bederived from Maxwell’s equation along with the twosolenoidal conditions, equation 10. The last two terms in theright hand of the energy equation, Equation 9, represent theviscous and Joule dissipations, respectively. That term canbe neglected compared to the other ones in the equation.

After simplifications by assuming fully developed flow, thatis two dimensional flows given in Figure 2, and since theflow is laminar due to the variations of turbulence in thepresence of magnetic field, the dimensionless governingequations for the flow become

Fig.2: MFM square duct flow

∇ − ∗ = 1 (11)∇ − ∗ = 1 (12)∇ + 4 = 0 (13)And ∇ − 4 = 0 (14)

Where the negative dimensionless pressure gradient γ isrelated to V by= ∫ ∫ ∗ ∗ (15)

From the force and energy balances one can show,respectively, that = −2 and = −1/ . Wherethe mean dimensionless temperature is given by= ∫ ∫ ∗ ∗∫ ∫ ∗ ∗ (16)

The boundary conditions are ∗ = 0 (from no-slipcondition), ∗ = 0 (from electrically insulated surface), and= 0 (for isothermal surface and constant surface heatflux).

III. NUMERICAL DIFFERENTIATIONS

In this paper, the MFM problem for two heat transfer limits;constant temperature and constant heat flux boundaryconditions, is investigated numerically for square duct(Figure 2). The modified dimensionless Navier-Stokesequations with uniform-temperature-condition havingenergy equation (Equation 7), anduniform-heat-flux-condition having energy equation(Equation 8) are transferred into finite-difference equations(using the central-difference scheme).,∗ + ,∗ + ,∗ + ,∗ − 4 ,∗ − ∆ ∗ ,∗ −,∗ ) = (∆ ∗) (17),∗ + ,∗ + ,∗ + ,∗ − 4 ,∗ −∆ ∗ ,∗ − ,∗ = 0(18) , + , + , + , − 4 , +4(∆ ∗) , , = 0 (19)And, + , + , + , − 4 , − 4(∆ ∗) , = 0

(20)With the following definitions of dimensionless pressuregradient and mean dimensionless temperature given,respectively, as= ∑ ∑ ,∗ (∆ ∗) (21)

And = ∑ ∑ , , (∆ ∗)∑ ∑ , (∆ ∗) (22)

IV. RESULTS

Some noticeable heat transfer results are obtained for theMFM co-circular duct flow with uniform temperature andheat flux boundary conditions. In resulting figures, theflattening of the axial velocity due to the presence of themagnetic field and the increase of the friction factor with thefield. The negative dimensionless temperature distributionsat the mid-plane either along or normal to the magnetic fieldalways decrease as the Hartmann number increases for bothboundary condition limits. This is because the temperaturedistributions are more homogenous as the magnetic field isturned on. This can be seen from the results presented in andis due to the fact that velocity profile becomes more flattenedas M increases, particularly along the direction of themagnetic field. Also we notice that the temperaturedistributions along and normal to the field are almost

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identical for any Hartmann number. The uniformity of thetemperature across the duct is greater for the former case butit is constant during the applied magnetics process.

Fig.3: Negative normalized axial velocity

Fig.4: Dimensionless axial velocity

Fig.5: Normalized induced axial magnetic field

Fig.6: Dimensional less induced axial magnetic field

Fig 7: Effect of Hartmann Number on velocity profileConclusion

The Study of co – Circular Cylindrical duct flow withelectrically conducting fluid and with two heat transferlimits has been studied. The problem is analyzednumerically when a uniform transverse magnetic field isapplied to the duct. The assumption of laminar flow ismostly valid in MFM flows since the turbulences will bedamped out due the opposing force induced in the flow. Thefluid mechanic part of this problem was consideredextensively and the results were shown using the spectralmethod. Also, the heat transfer results for only uniformtemperature boundary condition were shown. In the presentwork, we consider two heat transfer limits (uniform heat fluxand temperature boundary conditions) numerically usingiterative Gauss-Seidel method, and the software packageMatLab is utilized to achieve this approach. The resultsobtained for the case of constant temperature condition agreevery well with reference.

REFERENCES

[1] J. A. Shercilff, “Magneto-hydrodynamic Pipe Flow”,part 2 High Hartmann Number. Journal of Fluid Mech.,Vol. 13, 1962, p. 513.

[2] J. A. Shercliff, “The Flow of Conducting Fluids inCircular Pipes under Transverse Magnetic Field” J.Fluid Mech., Vol. 1, 1956, p. 644.

[3] J. D. Anderson, “Computational Fluid Dynamics: Thebasics with Applications”, McGraw-Hill, New York,1995.

[4] J. F. Thompson , B. K. Soni and N. P. Weatherill ,“Hand book of Grid Generation”, CRC Press, BocaRaton London, New York, Washington, 1998.

[5] J. F. Thompson, Z.U.A Warsi and W. Mastin,“Numerical Grid generation: Foundations andApplications”, North Holland, January 1985.

[6] K. B. Patel and M.C. Prajapati, “Simulation ofcylindrical heat diffusion problem using Cartesiansystem”, International Journal of Advanced EngineeringTechnology, Vol. V, Issue II, April-June,2014. Pp.36-40

[7] M. J. Al-Khawaja, R. A. Gardner and R. Agarwal,“Numerical study of magneto – fluid mechaniccombined free-and-forced convection Heat Transfer”.

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Int. J. Heat Mass Transfer, Vol. 42, 1999, pp. 467-475.

[8] M. J. Al-Khawaja , R. A. Gardner and R. Agarwal,“Numerical Study of magneto-fluidmechanics Forcedconvection Pipe Flow”, Engineering Journal of QatarUniversity, Vol.7, 1994, pp. 115-134.

[9] M. J. Al-Khawaja and M. Selmi, “Highly AccurateSolution of a Laminar Square Duct Flow in a TransverseMagnetic Field With Heat Transfer Using SpectralMethod”, Journal of Heat Transfer, Vol. 128, 2006, pp.413-417.

[10] N. M. Ozizik, Heat Transfer: A Basic Approach,McGraw-Hill, ISBN 0-07-047982-8, North CarolinaState University, pp. 289-291, Chap. 7.

[11] R. A. Gardner and P. S. Lykoudis,“Magneto-Fluid-Mechanic Pipe Flow in a TransverseMagnetic Field” Part Two. Heat Transfer. Journal ofFluid Mech., Vol. 48, Part 1, 1971, pp. 129-141.

[12] R. A. Gardner, “Laminar Pipe Flow in a TransverseMagnetic Field with Heat Transfer”, InternationalJournal of Heat Mass Transfer", Vol. 11, 1968, pp.1076-1081.

[13] R.Gold, Magnetohydrodynamic Pipe Flow. Journal ofFluid Mech., Part 1, Vol. 13, 1962, p. 505.

Kaushal Patel is working as anAssistant Professor at Department ofMathematics, Veer Narmad SouthGujarat University, Surat, Gujarat,India, since 9 years. His area of interestis Computational Fluid Dynamics andMathematical Modelling.

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