An incompressible electrically conducting fluid such as salt solution (seawater) is assumed to flow through an MHD generator duct, where the top and bottom plates are electrodes and the side plates are insulating walls. A magnetic field is imposed perpendicularly on the insulated surfaces. A Cartesian coordinate system is adopted for this study so that the working fluid flows along the x-direction through the channel. Electrodes are placed inclined to the channel to form a wedge-like shape. A magnetic field is then imposed along the z-direction. A heated salt solution with an even temperature of 0.5 lattice units (423.15 K) is injected into the channel of the MHD Generator. The temperature of the working fluid is later increased to study the effect of temperature on power generation. The geometric model of the MHD generator is shown below in Figure 1 .

The fluid flow geometry described above is mathematically modeled by the following governing equations: Continuity, Momentum, and Magnetic induction, from Equations (1) to (3) as in Foldes et al. [16] and Equation (4) which is the energy equation.

$\frac{\partial \rho }{\partial t}+\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0.$ (1)

$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{\partial P}{\partial x}+\nu \frac{{\partial }^{2}u}{\partial y^2}+s_u.$ (2)

$\frac{\partial B}{\partial t}+u\frac{\partial B}{\partial x}+v\frac{\partial B}{\partial y}=\eta \frac{{\partial }^2{B}_y}{\partial y^2}.$ (3)

$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}={\alpha }_0\frac{{\partial }^{2}T}{\partial y^2}+s_t.$ (4)

The above system of equations models the non-steady incompressible fluid flow in the MHD generator. In the system of equations, $\rho$ is density and $u$ and $v$ are velocities in the $x$ and $y$ directions respectively. Also, $B$ and $T$ are magnetic and temperature fields respectively. In addition, $\nu$, $\sigma$ and ${\alpha }_0$ represent kinematic viscosity, electrical conductivity, and thermal diffusion coefficient respectively. Also, $S_u$ and $S_t$ in Equations (2) and (4) are the momentum and energy source terms respectively. The effect of the imposed magnetic field on the fluid is modeled with the magnetic induction Equation (3) obtained from Ohm’s law and Maxwell’s equations. In this study $S_t=0$, however, the momentum source term $S_u$ is denoted by the Boussinesq approximation such that,

$S_u=g{\beta }_T\left(T-T_0\right).$ (5)

Equation (5) models the Buoyancy force due to thermal diffusion where ${\beta }_T$ is the coefficient of thermal expansion, $g$ is the acceleration due to gravity, $T$ is the temperature of the system and $T_0$ is the initial temperature of the working fluid.

At the starting point of the flow through the MHD generator, the variables dictating the flow assume the following values,

$\{\begin{array}{l}t=0:u\left(x,0\right)=U_0,v\left(x,0\right)=0,\\ B\left(x,0\right)=0,T\left(x,0\right)=T_0.\end{array}$ (6)

The boundary conditions of the system at the lower and upper walls (electrodes), where L is the length of the channel and h is the distance between the two electrodes, are stated below,

$0<x<L:u\left(x,y\right)=0,B\left(x,y\right)=0,T\left(x,y\right)=T_w,\text{at}y=0,$

$0<x<L:u\left(x,y\right)=0,B\left(x,y\right)=0,T\left(x,y\right)=T_w,\text{at}y=h.$ (7)

In Equations (7), $T_w$ is the temperature at the plate surface. The fluid flow variables across the MHD channel assume the following initial values,

$t>0:u=U_o,v=0,T=T_0,B=B_0\text{at}y=\pm h.$ (8)

From Equations (6) to (8), $U_0$, $T_0$, and $B_0$ denote initial velocity, temperature, and magnetic field respectively.

The Lattice Boltzmann method is then adopted to simulate the fluid flow in the MHD generator channel.

To model the momentum of the fluid flow, we discretize Equation (9). Let $\xi \equiv c_{\alpha }$ where $c_{\alpha }$ denote a discrete velocity set. Then the discretized Lattice Boltzmann momentum equation with the BGK collision operator without the force terms in Xiong [19] is shown as Equation (18) below,

$f_{\alpha }\left(x+c_{\alpha }\text{Δ}t,t+\text{Δ}t\right)-f_{\alpha }\left(x,t\right)=-\left(\frac{f_{\alpha }-f_{\alpha }^{eq}}{\tau }\right)$. (18)

The macroscopic parameters, such as the working fluid’s density and velocity, are obtained from the distribution function, as in Equations (13) and (14), respectively.

The Magnetic induction equation is represented in LBM as follows. We let $z_{\alpha }\left(x_{\alpha },t\right)$ be the magnetic distribution function, then the magnetic induction equation is written in LBM form as,

$z_{\alpha }\left(x+c_{\alpha }\text{Δ}t,t+\text{Δ}t\right)-z_{\alpha }\left(x,t\right)=\frac{1}{{\tau }_m}\left(z_{\alpha }\left(x,t\right)-z_{\alpha }^{eq}\left(x,t\right)\right),$ (19)

where ${\tau }_m$ is the magnetic relaxation time, $z_{\alpha }^{eq}$ is the magnetic equilibrium distribution function. The magnetic field density of the system is obtained as,

$B={\displaystyle {\sum }_{\alpha =0}^8{z}_{\alpha }}.$ (20)

The equilibrium distribution function of the magnetic field $z_{\alpha }^{eq}$ is given in Jamali et al. [20] as,

$z_{\alpha }^{eq}=w_{\alpha }\left(B_{\beta }+3c_{\alpha }\left(u_{\alpha }{B}_{\beta }-B_{\alpha }{u}_{\beta }\right)\right),$ (21)

where $w_{\alpha }$ are the lattice weights for the magnetic field.

The LBM for the momentum source term in Miyan [21] is modeled as,

$S_u=3w_{\alpha }{g}_x{\beta }_T\theta c_x+3w_{\alpha }{g}_y{\beta }_T\theta c_y.$ (22)

where $w_{\alpha }$ denote lattice weights, $g_x$ and $g_y$ are gravity components, ${\beta }_T$ is the thermal Buoyancy coefficient. Also, $\theta$ is the temperature of the system and $c_x$, $c_y$ are lattice directions.

The energy distribution function $h_{\alpha }\left(x,t\right)$ is defined as,

$h_{\alpha }\left(x+c_{\alpha }\text{Δ}t,t+\text{Δ}t\right)-h_{\alpha }\left(x,t\right)=-\frac{1}{T_T}\left(h_{\alpha }\left(x,t\right)-h_{\alpha }^{eq}\left(x,t\right)\right),$ (23)

where $T_T$ is the thermal relaxation time and $h_{\alpha }^{eq}$ is the thermal equilibrium distribution function. The thermal equilibrium distribution function of the energy equation $h_{\alpha }^{eq}$ is given in Krüger et al. [22] as,

$h_{\alpha }^{eq}=w_{\alpha }T\left(1+\frac{u\cdot c_{\alpha }}{c_s^2}+\frac{{\left(u{c}_{\alpha }\right)}^2}{2c_s^4}-\frac{u\cdot u}{2c_s^2}\right)$, (24)

where $W_{\alpha }$ is the weight of the thermal distribution direction ( $\alpha$) and the temperature $T$ of the system is estimated as,

$T={\displaystyle {\sum }_{\alpha =0}^8{h}_{\alpha }}.$ (25)

We employ a periodic boundary technique at the inlet and outlet of the MHD power generator system and a bounce-back boundary technique at the system’s electrodes (wall). Assuming periodicity along the $x$ direction, the inlet and outlet boundary conditions for the momentum equation are as in Equation (26) and Equation (27) respectively:

$f\left(0,:,i\right)=f^{eq}\left(0,:,i\right)+f\left(0,:,noslip\left(i\right)\right)-f^{eq}\left(0,:,noslip\left(i\right)\right)\text{for}i=1,5,8,$ (26)

and,

$f\left(-1,:,i\right)=f\left(-2,:,i\right)\text{for}i=3,6,7.$ (27)

The bottom and top momentum boundary conditions at the electrodes of the generator are,

$\begin{array}{l}f\left(bo{t}_{block},i\right)=Vinlet\ast \left(wFluid\left[i\right]+wFluid\left[noslip\left[i\right]\right]\right)\\ -f\left[bot\text{\_}block,noslip\left[i\right]\right]\text{for}i=2,5,6,\end{array}$ (28)

$\begin{array}{l}f\left[top\text{\_}block,i\right]=Vinlet\ast \left(wFluid\left[i\right]+wFluid\left[noslip\left[i\right]\right]\right)\\ -f\left[top\text{\_}block,noslip\left[i\right]\right]\text{for}i=4,7,8.\end{array}$ (29)

The magnetic boundary conditions are as shown below,

$z_y\left[0,:,i\right]=-z_y\left[0,:,noslip\left[i\right]\right]\text{for}i=1,5,8,$ (30)

$z_x\left[0,:,i\right]=-z_x\left[0,:,noslip\left[i\right]\right]\text{for}i=1,5,8,$ (31)

$z_x\left[-1,:,i\right]=-z_x\left[-1,:,noslip\left[i\right]\right]\text{for}i=3,6,7,$ (32)

$z_y\left[-1,:,i\right]=-z_y\left[-1,:,noslip\left[i\right]\right]\text{for}i=3,6,7.$ (33)

The thermal energy inlet and outlet boundary conditions are as in Equations (34) and (35),

$\begin{array}{l}h\left[0,:,i\right]=Tinlet\ast \left(wTemp\left[i\right]+wTemp\left[noslip\left[i\right]\right]\right)\\ -h\left[0,:,noslip\left[i\right]\right]\text{for}i=1,5,8,\end{array}$ (34)

$h\left[-1,:,i\left]=h\right[-2,:,i\right]\text{for}i=3,6,7.$ (35)

The bottom and top thermal boundary conditions are as in Equation (36) and (37),

$\begin{array}{l}h\left[bot\text{\_}block,i\right]=Tinlet\ast \left(wTemp\left[i\right]+wTemp\left[noslip\left[i\right]\right]\right)\\ -h\left[bot\text{\_}block,noslip\left[i\right]\right]\text{for}i=2,5,6,\end{array}$ (36)

$\begin{array}{l}h\left[top\text{\_}block,i\right]=Tinlet\ast \left(wTemp\left[i\right]+wTemp\left[noslip\left[i\right]\right]\right)\\ -h\left[top\text{\_}block,noslip\left[i\right]\right]\text{for}i=4,7,8.\end{array}$ (37)

Table 1 contains dimensionless parameters of the MHD generator, physical properties of the working fluid (salt solution/seawater) and other parameters associated with the Lattice Boltzmann Method in lattice units used for the simulation. Equations (18), (19), and (23) are simulated in Python with the specified boundary conditions from Equations (26) to (37) and the dimensionless parameter values presented in Table 1 .

The velocity profile across different sections of the MHD generator channel is shown below in Figure 4 .

Figure 4 shows that the velocity across the channel at one-fifth of the channel length is fully developed and high; however, it reduces at four-fifths of the channel towards the outlet. The reduction in velocity of the working fluid is due to viscous forces at the walls, Lorentz force in the bulk of the working fluid, and the large width of the outlet.

The temperature profile of the working fluid in the MHD generator is shown below in Figure 5 .

As can be observed in Figure 5 , the working fluid’s temperature declines gradually along the channel towards the exit. An initial inlet temperature of 0.5 lattice units gradually dropped to about 0.2 lattice units towards the outlet of the MHD generator channel, which denotes a 60% drop in inlet temperature. The reduced temperature of the working fluid was due to heat loss at the electrodes/walls.

The electric power, $\bar{P}$ per unit length from the MHD generator is obtained by using Equation (39) as presented by Miyan [21] and E-sparX [24] given as,

$\bar{P}=\frac{\sigma u{B}^2}{\rho }$ (39)

where $\sigma$ is the electrical conductivity of the working fluid, $u$ is the velocity, $B$ is the magnetic flux density of the system and $\rho$ is the density of the working fluid. In this study, the net magnet flux $B$ is the difference between the imposed magnetic flux $B_0$ and the induced magnetic flux due to Lorentz force, $B_{ind}=\sqrt{b_x^2+b_y^2}$. Therefore, the power $\bar{P}$ in this study is computed by using Equation (40),

$\bar{P}=\frac{\sigma u{\left(B_0-B_{ind}\right)}^2}{\rho }.$ (40)

Figures 6-8 show the electrical power profiles in the MHD generator as color-mapped images and line graphs extracted along and across the generator’s channel.

At a constant wall (electrode) temperature of 0.1 lattice units, the various Figures represent the power produced at three different inlet temperatures of the working fluid (salt solution). In Figure 6 , when the inlet temperature was 0.5, the electric power produced in the generator increased from 0.4 to peak at 0.7 in the first one-fifth of the channel. It then gradually dropped and stabilized to 0.2 towards the exit, as in Figure 6(b) and Figure 6(c) .

However, when the temperature of the working fluid was increased from 0.5 to 1.0 as in Figure 7 , the electric power rose from 0.5 to peak at 0.8 in the first one-fifth of the channel and declined gradually to stabilize at 0.2 towards the exit, as can be observed in Figure 7(b) and Figure 7(c) . Therefore, between Figure 6 and Figure 7 , the net increase in the peak power produced by a 100% increase in the inlet temperature is 0.1, representing a 14.3% increase in the peak power produced.

Also, in Figure 8 , an inlet temperature of 1.5 caused the power to increase from 0.5 to peak at 0.9 and declined gradually to stabilize at 0.2 towards the generator’s outlet, as in Figures 8(b) and Figure 8(c) . Therefore, comparing the power produced in Figure 6 and Figure 8 , it can be observed that a 200% increase in the inlet temperature of the working fluid, resulted in a 28.6% increase in the peak power.

In all these situations, the increased temperature enhanced the velocity and pressures at the inlet, which facilitates ionization and conductivity of the working fluid, resulting in appreciable electric power in three-fifths of the generator channel. However, towards the exit, a drop in temperature resulted in ion recombination at low temperatures towards the outlet. The low temperature decreased the ionization and conductivity of the working fluid towards the outlet, accounting for the decline in electric power generated in the last two-fifths of the generator channel.

The findings of this study provide insight into the appropriate sections of the MHD generator from which high power can be tapped. The electrical power produced by the heated salt solution (seawater) as working fluid in the MHD power generator would be cheap, renewable, and safe for domestic and industrial use.