Author: Kiyasatfar, M
Date published: January 1, 2011
(ProQuest: ... denotes formulae omitted.)
Magneto hydrodynamic micro pumps have recently attraction a wide attention in micro fluidic research. The pumping source in MHD micro-pumps is the Lorentz forces, which is produced due to consequence of the interaction between the magnetic and electric fields. MHD pump with having no moving part can be generating continuous flow and suitable to pump biological fluid.
The MHD phenomenon was first observed by W. Richie [I]. Winowich et al. , Ramos and Winowich  applied both the Galerkin-scheme finite element method (FEM) and the finite difference method (FDM) formulations to analyze MHD channel flows and determine the influences of the Reynolds number, Hartmann number and the wall conductivity on the electric and hydrodynamic fields. Their results predicted that the axial velocity profiles are distorted into M-shape due to the applied magnetic field.
Recently theoretical and experimental studies on Dc and Ac MHD micro-pumps were investigated. Lemoff et al.  and Lem off and Lee  constructed a practical Ac MHD micro-pump in which the Lorentz force is used to propel an electrolytic solution along a micro-channel etched in silicon. Experimental measurements were conducted on various concentrations of sodium chloride- ( NaCl ) solutions to determine the maximum current allowed in the micro-channels before gas bubbles were observed. The experimental results indicated that the electrolysis phenomenon was greatly reduced when the frequency of the applied current was sufficiently high. Jang and Lee  presented a Dc micro-pump, and obtained the performance of the MHD micro-pump in single phase and using simple model. Eijkel et al.  developed and fabricated an Ac MHD micro pump for chromatographic application. Zhong et al.  used the MHD to circulate fluids in conduits fabricated with ceramic tapes. To avoid gas bubbles, Homsy et al.  described the operation of Dc MHD micro-pump with high current densities with out introducing gas bubbles into the pumping channel. Pei-Jen Wang et al.  replicated a simplified MHD flow model based upon steady state, incompressible and fully developed two-dimensional laminar flow to take account of the frictional effects on the channel side walls of the MHD pump and in their study, the Lorentz forces are converted in to the hydrostatic pressure grathent in the momentum equation. H. Duwairi and M. Abdullah  presented a transient fully developed laminar flow and temperature distribution in a MHD micro pump. They studied the effect of different parameters on the transient velocity and temperature, such as aspect ratio, Hartman number, Prandtl number and Eckert number. Ho  described a static balance between the Lorentz force, body force and surface tension of liquid by measuring the hydraulic head with the applied voltage. From his experiment results, a relevant conclusion told that the effect of bubbles would influence the maximum hydraulic head and low down % 50 peak value approximately. Ho  in his research, focused on the prediction of pumping performance in MHD flow. To reach it, an analytic model based on the steady state, incompressible and fully developed laminar flow theory provided to analyze the flow characteristic with different scalar dimensions in duct channel.
Mohammed, Q. Al-Odat et al.  numerically investigated on Magnetic field effect on local entropy generation due to steady two-dimensional laminar forced convection flow past a horizontal plate. In their study focused on the entropy generation characteristics and its dependency on various dimensionless parameters. Aytac Arikoglu et al.  researched on the effect of slip on entropy generation in a single rotating disk in MHD flow. The essential purpose of their study is to allow the designer to use the second law of thermodynamic in efficiency calculations of rotating fluidic system.
In this paper, we studied numerically the steady state, incompressible and fully developed laminar flow and showed the effects of magnetic flux density and current on flow velocity, temperature and local entropy generation rate distribution in MHD pump.
A schematic view of the MHD pump and the coordinate set are illustrated in Fig. 1 . The MHD pump dimensions and other parameters are shown in Table 1.
The physiological saline (NaCl solution) used in MHD pump as a conducting fluid. The properties of the working medium are given in Table 2.
The following suppositions are used to derive the governing equations: constant magnetic, electric and mechanical fluid properties. While the Reynolds number in micro-channel is assumed to be small, the flow field in the MHD micro-pump is treated as steady state, incompressible, and fully developed laminar flow condition. Since scalar dimensions of the channel in y and z direction are much small compared to that in flow direction x, the flow velocity in y component, V and in z direction cw' assumed to be zero. The effect of surface tension is neglected because the channel filled with fluid. Based upon the above assumption, the axial flow velocity u(y, z) is invariant along in x direction. Moreover, heat loss in the flow channel is considered insignificant.
Ohm's law and Lorentz force:
where J, B and F are the current density, the magnetic density and the Lorentz force vectors. ...s are the electric field intensity and the flow velocity vectors and the electrical conductivity of the fluid. From the above suppositions, the governing equations for the simplified flow field can be written as follow:
Where P the pressure and µ is the dynamic viscosity of the fluid. In the MHD pump, the Lorentz forces acting on the fluid particles are considered as hydrostatic pressure head uniformly distributed over the channel region. Hence
where ^sup Δ^p is the pressure head along the channel with length L, caused by interaction between magnetic and electric fields. That is:
where ... is the length of electrode and ... is the loss pressure.
Combining (1), (6)-(5) outcomes to (4) :
The equation (7) obtains the velocity field. The volumetric flow rate (Q) is given by eq. (8).
where ... are the density, specific heat, temperature and the thermal conductivity respectively. With respect to the above assumptions the energy equation could be written into eq. (10).
The boundary conditions for solving the governing equations are:
at all boundaries the no-slip boundary condition is : ...
the electrode and magnets are supposed to be in room temperature ( 25 °C)
For solving the nonlinear differential equations, the code upon finite difference method is provided and utilized.
Equation of entropy generation:
The volumetric rate of local entropy generation can be expressed in the following form:
where ... and ... represent the entropy generation rates due to heat transfer and fluid friction respectively, and are defined as:
The total entropy generation rate over the volume ( §en ) can be calculated as follows:
Bejan number (Be), which is the ratio of heat transfer irreversibility to the total entropy generation, and mathematically can be expressed as
Comparison between the numerical solution and experimental results.
Fig. 2 shows numerical solution and experimental results for the volumetric flow rate as a function of the width of channel. It is shown that there is a good agreement between them so that their maximum relative error is not more than 8 % at the width 1-5x10 m with respect to the volumetric flow rate of the order of 13 ml/s.
The numerical solution (0.003 m/s~0.036 m/s) and experiment results (0.003 m/s~0.034 m/s) for average flow velocity are presented in Fig. 3. As shown average flow velocity increases linearly with the applied current at an interval of 0.1 A to 1.3 A.
3. Results and Discussions
In this section, we presented the effect of magnetic flux density and current on a maximum velocity, temperature and entropy generation in a MHD pump.
Fig. 4 shows the effect of magnetic flux density on the maximum velocity. As it seen, with increasing the magnetic flux density, the maximum velocity increases in a linearly fashion and indicates that while as the other properties are remained constant, the magnetic flux density have a direct and potent effect on the velocity of the fluid flow. The effect of magnetic flux density on the maximum temperature is shown in Fig. 5. As shown the maximum temperature remains constant with increasing the magnetic flux density. This good result is useful in designing especially in biological goals because the MHD pump can achieve higher volumetric flow rate without any change in thermal properties.
The effect of the current on the maximum velocity and maximum temperature is shown in Figs. 6 and 7. As shown with increasing applied current the maximum velocity varies linearly and maximum temperature increase exponentionally.
In this research we also studied the second law of thermodynamic aspects of fluid flow and heat transfer inside of the MHD pump. According to our aforementioned the above results the interactions between the magnetic and electric fields generate velocity which increases temperature in the working medium. By solving the entropy generation rate equations [eqs. (11), (12) and (13)] the effect of current and magnetic flux density on the distribution of local volumetric entropy generation, integrated entropy generation and Bejan number are specified.
Figs. 8, 9 and 10 exhibit the volumetric local entropy generation rate distributions within the MHD pump for the various applied current and magnetic flux density. The contours show that the highest volumetric local entropy generation rate occurs near the channel wall, particularly near the electrodes. Increasing the current significantly affects on the distribution of the volumetric local entropy generation rate so that causes increasing in their magnitude.
The distribution of volumetric local entropy generation rate is not so much affected by increasing of the magnetic flux density.
The effect of magnetic flux density on the entropy generation is illustrated in Fig. 11. It is observed that total entropy generation rate remain constant as magnetic flux density increasing.
Fig. 12 depict the entropy generation rate for different values of the current. It is seen that with increasing the applied current the entropy generation rate increase exponentionally.
The calculation bring out that in all investigated cases, the entropy generation rate due to fluid friction are quite lower than those due to the heat transfer. In other words, the Bejan number is very close to one, and it means that irreversibility due to the heat transfer dominates.
The specific conclusions derived from this study can be listed as follows:
The implicit finite difference method is used for conducting numerical simulations.
The obtained numerical results are in good agreement with experimental results.
The maximum velocity increase linearly by increasing magnetic flux density.
The maximum temperature and entropy generation rate almost remain constant with increasing magnetic flux density
By increasing applied current the maximum velocity increases linearly.
With increasing applied current the maximum temperature and entropy generation rate increase exponentially.
In the MHD pump increasing the magnetic flux density for achieving the higher velocity don't changes the thermal properties.
In the MHD pump increasing the current to attain higher velocity changes the thermal properties.
The higher volumetric local entropy generation rate happens near the channel walls, particularly, near the electrode.
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1 M. Kiyasatfar, 2 N. Pourmahmoud, 3 M. M. Golzan, 4M. Eskandarzade
1 CFD Research center, Mechanical Engineering Dept, Urmia University, West Azerbayjan, Iran,
2 CFD Research center, Mechanical Engineering Dept., Urmia University, West Azerbayjan, Iran
3 Physics Department faculty of science, Urmia University, West Azerbaijan, Iran
4 Dept. of Mech. Eng., Urmia University of Technology, West Azerbaijan, Urmia, Iran
Received: 12 October 2010 /Accepted: 24 January 2011 /Published: 28 January 2011