3.1. Primary MHD fluid

(3.1) \begin{align} &\boldsymbol{\nabla} \boldsymbol{\cdot} \left(n_{p}\it{v}_{p}\right)=0&&\quad \text{Mass},\nonumber\\[6pt] &m_{p}n_{p}\it{v}_{p}\boldsymbol{\cdot} \boldsymbol{\nabla} \it{v}_{p}=\boldsymbol{J}\times \boldsymbol{B}_{0}-\boldsymbol{\nabla} p_{p}&&\quad \text{Momentum},\nonumber\\[6pt] &\frac{3}{2}\boldsymbol{\nabla} \boldsymbol{\cdot} \left(p_{p}\it{v}_{p}\right)+p_{p}\boldsymbol{\nabla} \boldsymbol{\cdot} \it{v}_{p}=\frac{3}{2}\nu _{E}n_{e}\left(kT_{e}-kT_{p}\right)&&\quad \text{Energy} . \end{align}

3.2. Electron fluid

(3.2) \begin{align} &\frac{n_{e}^{2}}{n_{s}-n_{e}}=\left(\frac{2\pi m_{e}kT_{e}}{h^{2}}\right)^{3/2}\exp \left(-E_{I}/kT_{e}\right)&&\quad \text{Mass}-\text{Saha equation},\nonumber\\[6pt] &\boldsymbol{E}+\it{v}_{p}\times \boldsymbol{B}_{0}=\eta \boldsymbol{J}+\frac{\boldsymbol{J}\times \boldsymbol{B}_{0}}{en_{e}}&&\quad \text{Momentum}-\mathrm{Ohm}'\text{s law},\nonumber\\[6pt] &\frac{3}{2}\nu _{E}n_{e}\left(kT_{e}-kT_{p}\right)=\eta J^{2}&&\quad \text{Energy}-\text{Electron energy balance} .\end{align}

3.3. Maxwell

(3.3) \begin{align} &\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{J}=0&&\quad \mathrm{Ampere}\text{'s law},\nonumber\\ &\boldsymbol{\nabla} \times \boldsymbol{E}=0&&\quad \mathrm{Faraday}\text{'s law},\nonumber\\ &n_{i}=n_{e}&&\quad \mathrm{Poisson}\text{'s equation}-\text{charge neutrality}. \end{align}

Note that the primary and electron fluid variables are denoted by the subscript ‘p’ and ‘e’, respectively. The quantity $E_{I}$ is the ionisation potential of the seed gas, $n_{s}$ is the density of the initial unionised seed gas and $\boldsymbol{B}_{0}$ is the applied vacuum magnetic field. The resistivity $\eta$ and temperature equilibration frequency $\nu _{E}$ for a monatomic gas are related to the electron momentum exchange frequency $\nu _{M}$ in the usual way,

(3.4) \begin{align} \nu _{M}&=\nu _{ep}+\nu _{en}+\nu _{ei}\approx \nu _{ep},\nonumber\\[5pt] \eta &=\frac{m_{e}\nu _{M}}{n_{e}e^{2}},\nonumber\\[5pt] \nu _{E}&=2\frac{m_{e}}{m_{p}}\nu _{M}. \end{align}

Here, the separate contributions to $\nu _{M}$ represent electron collisions with (a) the primary gas ( $\nu _{ep}$ ), (b) the actual unionised seed particles ( $\nu _{en}$ ) and (c) the seed ions ( $\nu _{ei}$ ). Some discussion of the physics is warranted.

1. (i) The use of fluid equations assumes that all collision frequencies are large compared with the characteristic MHD time scale: $\nu _{E}\gg v_{p}/L$ with $L$ the macroscopic generator length scale. This condition is well satisfied in all regimes of interest.

2. (ii) With respect to momentum collisions, the electron–primary gas interactions dominate in the operational regime of interest: $\nu _{ep}\gg \nu _{en},\nu _{ei}$ . For mathematical simplicity, one thus assumes $\nu _{M}\approx \nu _{ep}$ in the following analysis. Even so, note that the analysis is generalised to include the full $\nu _{M}$ , and some results are included for comparison. Also, an expression for $\nu _{ep}$ is given shortly.

3. (iii) The additional unknown $n_{s}$ appears in the Saha equation, which as stated represents the initial, total unionised number density of the seed gas. The actual number density of unionised seed particles is $n_{n}=n_{s}-n_{i}=n_{s}-n_{e}$ . Now, because of the high collisionality, it follows that all species, except the very small mass electrons, have essentially the same velocity and temperature

(3.5) \begin{equation} v_{i}\approx v_{n}\approx v_{p},\quad T_{i}\approx T_{n}\approx T_{p} .\end{equation}

The neutral and ionised seed particles are dragged along and thermally equilibrated with the much denser primary gas. After a short calculation using the mass conservation equations (including ionisation and recombination) for the heavy species, it can be easily shown that the quantity $n_{s}$ satisfies $\boldsymbol{\nabla} \boldsymbol{\cdot} (n_{s}\it{v}_{p})=0$ . Therefore, assuming that all species have the same uniform cross-sectional profile at the generator inlet, it follows that

(3.6) \begin{equation} \frac{n_{s}}{n_{s0}}=\frac{n_{p}}{n_{p0}} ,\end{equation}

with a ‘0’ subscript denoting inlet value. This is the simple, required information relating $n_{s}$ to the basic unknown $n_{p}$ .

1. (i) The next point of interest is the assumption that only the vacuum magnetic field enters the analysis. This too is a very good approximation since simple scaling relations show that the induced magnetic field $B_{{\rm ind}}$ due to the generator currents is much smaller than the vacuum field: $B_{{\rm ind}}\ll B_{0}$ . One consequence of this approximation is that the only non-trivial information contained in Ampere’s law is the relation $\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{J}=0$ .

2. (ii) Another point to observe is that viscosity and thermal conduction are neglected in the model. This too is a good approximation since these effects are small in the core of the plasma where all profiles are essentially uniform across the cross-section. They are, however, important near the MHD channel walls where narrow boundary layers form. Since these represent energy loss mechanisms, they should be maintained when assessing the overall performance of an MHD generator, but they do not play an important role in the development of the ionisation instability in the core plasma.

3. (iii) A further key point of physics is that the electron energy equation describes a basic balance between preferential electron ohmic heating and the resulting temperature difference between electrons and the primary gas. Convection and compression effects can be shown to be small, and are therefore neglected.

4. (iv) Lastly, note that various similar, although not identical, forms of the starting equations have appeared in the literature (see for instance Mitchner & Kruger Reference Mitchner and Kruger1973; Rosa Reference Rosa1987; Messerle Reference Messerle1994) – there is nothing dramatically different from the present starting model. Still, it can be shown by a rigorous mathematical maximal ordering procedure that the model described above is entirely self-consistent and contains all the information required to calculate marginal stability of the ionisation instability.

The analysis will show that stability ultimately requires a relatively low fraction of seed gas to primary gas. Therefore, it should be a good approximation to assume that the momentum and energy exchange collision frequencies are dominated by electron–primary gas interactions. The validity of this approximation is tested at the end of the calculation. However, it is not explicitly employed during the analysis.

A final point is that while the primary fluid mass, momentum and energy relations have been listed as part of the starting equations, they actually do not enter the stability analysis. The reason, as stated, is that these relations are needed to determine the slow axial dependence of the equilibrium quantities, but are not directly required for the short wavelength stability analysis. The relations yield expressions for the perturbed fluid density, velocity and primary temperature, which when compared with other terms in the electron equations can be shown to be small by $1/k_{0}L\ll 1$ , where $k_{0}$ is the wavenumber of the unstable mode. These quantities can be calculated, if desired, after the analysis of the electron equations. Physically, the ionisation instability is predominantly an electron phenomenon involving ionisation and electron energy balance. The primary fluid does not play a significant role.

3.4. Equilibrium

The first step in the analysis is to examine the equilibrium properties of the plasma. In addition to expressing all quantities in terms of the input velocity and magnetic field, it will also be of use in the stability analysis to derive a relationship between the temperature difference, Hall parameter and ionisation fraction. As stated, the analysis focuses primarily on the behaviour of the electrons because of the $k_{0}L\gg 1$ assumption.

3.5. Electron Ohm’s law equilibrium properties

We begin by writing Ohm’s law in component form for a linear MHD Hall generator. The flow is along the $x$ axis and the magnetic field is uniform and in the $z$ direction. For a Hall generator, the electrodes are shorted out in the $y$ direction implying that $E_{y}=0$ . Also, no voltage or current flow occurs along the magnetic field direction, so that $E_{z}=J_{z}=0$ . Ohm’s law reduces to

(3.7) \begin{align} E_{x}&=\eta J_{x}+\frac{J_{y}B_{0}}{en_{e}},\nonumber\\[6pt] -v_{p}B&=\eta J_{y}-\frac{J_{x}B_{0}}{en_{e}},\nonumber\\[6pt] E_{z}&=0. \end{align}

As assumed, the fluid velocity, $\it{v}_{p}\approx v_{p}\boldsymbol{e}_{x}$ , of the primary gas will remain unchanged during the development of the instability.

Next, one introduces definitions of several basic parameters entering the analysis,

(3.8) \begin{align} \nu _{M}&=\nu _{ep}=n_{p}\overline{\sigma }_{ep}v_{Te}=n_{p}\overline{\sigma }_{ep}\left(\frac{2kT_{e}}{m_{e}}\right)^{1/2}&&\quad \text{Collision frequency},\quad \nonumber\\[6pt] \eta &=\frac{m_{e}\nu _{M}}{e^{2}n_{e}}&&\quad \text{Resistivity},\nonumber\\[6pt] \beta &=\frac{B_{0}}{en_{e}\eta }=\frac{\Omega _{e}}{\nu _{M}}&&\quad \text{Hall parameter},\nonumber\\[6pt] M&=\left(\frac{3}{5}\frac{m_{p}v_{p}^{2}}{kT_{p}}\right)^{1/2}&&\quad \text{Mach number}.\end{align}

Here, $\overline{\sigma }_{ep}$ is the known cross-section for momentum exchange collisions between the electrons and the primary gas. Also, the Mach number is important since the flow must be supersonic, $M\gt 1$ , for good energy conversion in a Hall MHD generator.

In addition, define an equilibrium load resistivity $\eta _{L}$ as

(3.9) \begin{equation} \eta _{L}\left(x\right)=-\frac{E_{x}}{J_{x}}, \end{equation}

which is then normalised as

(3.10) \begin{equation} Z\left(x\right)=\frac{\eta _{L}}{\eta }. \end{equation}

The quantity $Z(x)$ replaces $E_{x}(x)$ as one of the basic unknowns and is introduced to simplify the analysis.

Using these relations, one obtains, after a straightforward calculation, the desired expressions for the current densities and electric fields in terms of the fluid velocity,

(3.11) \begin{align} J_{x}&=\left[\frac{\beta ^{2}}{\beta ^{2}+1+Z}\right]en_{e}v_{p},\nonumber\\[6pt] J_{y}&=-\frac{1+Z}{\beta }J_{x}=-\left[\frac{\beta \left(1+Z\right)}{\beta ^{2}+1+Z}\right]en_{e}v_{p},\nonumber\\[6pt] \frac{E_{x}}{\eta }&=-ZJ_{x}=-\left[\frac{\beta ^{2}Z}{\beta ^{2}+1+Z}\right]en_{e}v_{p},\nonumber\\[6pt] \frac{E_{y}}{\eta }&=0. \end{align}

Also of interest are expressions for the power density to the load $S_{L}$ , the Ohmic power density $S_{\Omega }$ and the total input power density converted to electricity $S_{C}=S_{L}+S_{\Omega }$ :

(3.12) \begin{align} S_{L}&=-\boldsymbol{E}\boldsymbol{\cdot} \boldsymbol{J}=-E_{x}J_{x}=m_{e}n_{e}\nu _{M}v_{p}^{2}\frac{\beta ^{4}Z}{\left(\beta ^{2}+1+Z\right)^{2}},\nonumber\\[4pt] S_{\Omega }&=\eta J^{2}=\eta \left(J_{x}^{2}+J_{y}^{2}\right)=m_{e}n_{e}\nu _{M}v_{p}^{2}\frac{\beta ^{2}\left[\beta ^{2}+\left(1+Z\right)^{2}\right]}{\left(\beta ^{2}+1+Z\right)^{2}},\nonumber\\[4pt] S_{C}&=S_{L}+S_{\Omega }=m_{e}n_{e}\nu _{M}v_{p}^{2}\frac{\beta ^{2}\left(1+Z\right)}{\beta ^{2}+1+Z}. \end{align}

3.6. Electron energy balance

We now focus on calculating the temperature difference due to Ohmic heating. The desired relation is obtained by substituting into the energy balance relation given by (3.2) and repeated here for convenience:

(3.13) \begin{equation} \frac{3}{2}\nu _{E}n_{e}\left(kT_{e}-kT_{p}\right)=\eta J^{2}. \end{equation}

Substituting into this relation, one can evaluate the temperature ratio of interest $T_{e}/T_{p}$ . The result is

(3.14) \begin{equation} \frac{T_{e}}{T_{p}}-1=\frac{2\eta J^{2}}{3\nu _{E}n_{e}kT_{p}}=\left(\frac{5M^{2}}{9}\right)\frac{\beta ^{2}\left[\beta ^{2}+\left(1+Z\right)^{2}\right]}{\left(\beta ^{2}+1+Z\right)^{2}}. \end{equation}

Note that a large temperature ratio requires high values for the Mach number and Hall parameter.

This information, coupled with the Saha equation, defines all the equilibrium information required for the stability analysis.

4.1. The Saha equation

The Saha equation, given by (3.2), determines the relation between $\tilde{T}_{e}$ and $\tilde{n}_{e}$ . Linearisation of the Saha equation yields

(4.2) \begin{equation} \frac{\tilde{T}_{e}}{T_{e}}=2\alpha \frac{\tilde{n}_{e}}{n_{e}}, \end{equation}

where

(4.3) \begin{align} \alpha &=\frac{1}{2}\frac{N}{T_{e}\left(\mathrm{d}N/\mathrm{d}T_{e}\right)}=\frac{1}{2}\left(\frac{2kT_{e}}{3kT_{e}+2E_{I}}\right)\left(\frac{2-f_{I}}{1-f_{I}}\right)\nonumber\\[4pt]&\approx \left(\frac{kT_{e}}{2E_{I}}\right)\left(\frac{2-f_{I}}{1-f_{I}}\right)\propto \frac{1}{1-f_{I}} \end{align}

and $f_{I}=n_{e}/n_{s}$ is the equilibrium fraction of seed ionisation. The approximate expression makes use of the fact that $kT_{e}/E_{I}\ll 1$ . The value of $\alpha$ is critical in that it is a direct measure of the degree of ionisation of the seed gas. Small $\alpha \ll 1$ implies small to modest ionisation fraction: $f_{I}\lt 1$ . Moderate to large $\alpha \gt 1$ implies nearly full ionisation: $f_{I}\rightarrow 1$ . Also, as stated, the perturbed $\tilde{n}_{s}$ should be, and is, neglected.

4.2. The perturbed collision frequency

In the analysis, it is necessary to evaluate the perturbed collision frequency plus several related quantities. Consider first the momentum exchange collision frequency defined by $\nu _{M}=\nu _{ep}$ . A straightforward calculation leads to

(4.4) \begin{equation} \tilde{\nu }_{M}=\tilde{\nu }_{ep}=n_{p}\overline{\sigma }_{p}v_{Te}\left(\frac{1}{2}\frac{\tilde{T}_{e}}{T_{e}}\right)=\nu _{ep}\left(\frac{1}{2}\frac{\tilde{T}_{e}}{T_{e}}\right). \end{equation}

Next, $\tilde{T}_{e}/T_{e}$ from (4.2) is substituted, yielding the required relation between $\tilde{\nu }_{M}$ and $\tilde{n}_{e}$ ,

(4.5) \begin{equation} \frac{\tilde{\nu }_{M}}{\nu _{M}}=\alpha \frac{\tilde{n}_{e}}{n_{e}}. \end{equation}

From (4.5), it then follows that the perturbed Hall parameter is given by

(4.6) \begin{equation} \frac{\tilde{\beta }}{\beta }=\frac{\left(\Omega _{e}/\nu _{M}\right)\left(-\tilde{\nu }_{M}/\nu _{M}\right)}{\left(\Omega _{e}/\nu _{M}\right)}=-\frac{\tilde{\nu }_{M}}{\nu _{M}}=-\alpha \frac{\tilde{n}_{e}}{n_{e}}. \end{equation}

Similarly, one can easily derive the perturbed resistivity. Since the resistivity is defined as $\eta =m_{e}\nu _{M}/n_{e}e^{2}$ , this leads to

(4.7) \begin{equation} \frac{\tilde{\eta }}{\eta }=\frac{\tilde{\nu }_{M}}{\nu _{M}}-\frac{\tilde{n}_{e}}{n_{e}}=\left(\alpha -1\right)\frac{\tilde{n}_{e}}{n_{e}}. \end{equation}

Lastly, the perturbed energy equilibration time can be written as

(4.8) \begin{equation} \frac{\tilde{\nu }_{E}}{\nu _{E}}=\frac{\tilde{\nu }_{M}}{\nu _{M}}=\alpha \frac{\tilde{n}_{e}}{n_{e}}. \end{equation}

4.3. Maxwell’s equations

Maxwell’s equations lead to relationships between the components of the perturbed electric field and current density in terms of the wavenumbers. These relations are given by

(4.9) \begin{align} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{J} & =0\rightarrow k_{x}\tilde{J}_{x}+k_{y}\tilde{J}_{y}+k_{z}\tilde{J}_{z}=0,\nonumber\\ \boldsymbol{\nabla} \times \boldsymbol{E} & =0\rightarrow k_{x}\tilde{E}_{y}-k_{y}\tilde{E}_{x}=0\quad \textrm{and}\quad k_{x}\tilde{E}_{z}-k_{z}\tilde{E}_{x}=0 .\end{align}

Note that the term $\partial \boldsymbol{B}/\partial t$ has been neglected in Faraday’s law based on the well-satisfied assumption that the induced currents in both equilibrium and stability are small, plus the focus on marginal stability. The magnetic field is a pure static vacuum field.

4.4. The perturbed currents

The perturbed currents are found from the MHD Ohm’s law, neglecting variations in the primary gas velocity: $\tilde{v}_{p}=0$ . After some straightforward algebra using (4.9) plus the $z$ component of Ohm’s law, one finds that $\tilde{J}_{y},\tilde{J}_{z},\tilde{E}_{y},\tilde{E}_{z}$ can be expressed in terms of $\tilde{J}_{x},\tilde{E}_{x}$ as follows:

(4.10) \begin{align} \tilde{E}_{y}&=\frac{k_{y}}{k_{x}}\tilde{E}_{x},\nonumber\\ \tilde{E}_{z}&=\frac{k_{z}}{k_{x}}\tilde{E}_{x},\nonumber\\ \tilde{J}_{z}&=\frac{k_{z}}{k_{x}}\frac{\tilde{E}_{x}}{\eta },\nonumber\\ \tilde{J}_{y}&=-\frac{k_{x}}{k_{y}}\tilde{J}_{x}-\frac{k_{z}^{2}}{k_{x}k_{y}}\frac{\tilde{E}_{x}}{\eta }. \end{align}

Using these relations, one can write the $x$ and $y$ components of Ohm’s law in terms of $\tilde{J}_{x},\,\tilde{E}_{x}$ as

(4.11) \begin{align} \left(1-\beta \frac{k_{x}}{k_{y}}\right)\tilde{J}_{x}&-\left(1+\frac{\beta k_{z}^{2}}{k_{x}k_{y}}\right)\frac{\tilde{E}_{x}}{\eta }=\left[-\left(\alpha -1\right)J_{x}+\beta J_{y}\right]\frac{\tilde{n}_{e}}{n_{e}},\nonumber\\[3pt]\left(\frac{k_{x}}{k_{y}}+\beta \right)\tilde{J}_{x}&+\left(\frac{k_{y}}{k_{x}}+\frac{k_{z}^{2}}{k_{x}k_{y}}\right)\frac{\tilde{E}_{x}}{\eta }=\left[\left(\alpha -1\right)J_{y}+\beta J_{x}\right]\frac{\tilde{n}_{e}}{n_{e}} .\end{align}

Equation (4.11) can be solved for $\tilde{J}_{x}$ and $\tilde{E}_{x}$ . These results are then back substituted to obtain expressions for several other quantities of interest appearing in the stability analysis. After a slightly tedious calculation, one obtains the required results,

(4.12) \begin{align} \tilde{J}_{x}&=\frac{1}{1+\beta ^{2}\left(k_{z}^{2}/k_{0}^{2}\right)}\left\{\left(\alpha -1\right)\frac{k_{y}}{k_{0}}J_{\bot }+\beta \frac{k_{y}}{k_{0}}J_{\parallel }\vphantom{\frac{k_{z}^{2}}{k_{0}^{2}}}\right.\nonumber\\& \qquad\qquad\qquad\qquad\left. +\frac{k_{z}^{2}}{k_{0}^{2}}\left[\left(\beta ^{2}-\alpha +1\right)J_{x} +\alpha \beta J_{y}\right]\right\}\frac{\tilde{n}_{e}}{n_{e}},\nonumber\\[4pt]\frac{\tilde{E}_{x}}{\eta }&=-\frac{1}{1+\beta ^{2}\left(k_{z}^{2}/k_{0}^{2}\right)}\frac{k_{x}}{k_{0}}\left[\left(\beta ^{2}-\alpha +1\right)J_{\parallel }+\alpha \beta J_{\bot }\right]\frac{\tilde{n}_{e}}{n_{e}},\nonumber\\[4pt]\tilde{J}_{y}&=\frac{1}{1+\beta ^{2}\left(k_{z}^{2}/k_{0}^{2}\right)}\left\{-\left(\alpha -1\right)\frac{k_{x}}{k_{0}}J_{\bot }-\beta \frac{k_{x}}{k_{0}}J_{\parallel }\vphantom{\frac{k_{z}^{2}}{k_{0}^{2}}}\right.\nonumber\\& \qquad\qquad\qquad\qquad\left. +\frac{k_{z}^{2}}{k_{0}^{2}}\left[\left(\beta ^{2}-\alpha +1\right)J_{y}-\alpha \beta J_{x}\right]\right\}\frac{\tilde{n}_{e}}{n_{e}},\nonumber\\[4pt]J_{x}\tilde{J}_{x}+J_{y}\tilde{J}_{y}&=\frac{1}{1+\beta ^{2}\left(k_{z}^{2}/k_{0}^{2}\right)}\left[-\left(\alpha -1\right)J_{\bot }^{2}-\beta J_{\bot }J_{\parallel }+\frac{k_{z}^{2}}{k_{0}^{2}}\left(\beta ^{2}-\alpha +1\right)J^{2}\right]\frac{\tilde{n}_{e}}{n_{e}}. \end{align}

Here,

(4.13) \begin{align} k_{0}^{2}&=k_{x}^{2}+k_{y}^{2}+k_{z}^{2},\nonumber\\[2pt] J^{2}&=J_{x}^{2}+J_{y}^{2}=\left(J_{\bot }^{2}+J_{\parallel }^{2}\right)/\left(1-k_{z}^{2}/k_{0}^{2}\right),\nonumber\\[2pt] J_{z}&=0,\nonumber\\[2pt] J_{\bot }&=\frac{1}{k_{0}}\left(\boldsymbol{e}_{z}\boldsymbol{\cdot} \boldsymbol{k}\times \boldsymbol{J}\right)=\frac{1}{k_{0}}\left(k_{x}J_{y}-k_{y}J_{x}\right),\nonumber\\[2pt] J_{\parallel }&=\frac{1}{k_{0}}\left(\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{J}\right)=\frac{1}{k_{0}}\left(k_{x}J_{x}+k_{y}J_{y}\right). \end{align}

4.5. The energy balance equation

The last equation of interest is the energy balance equation. For convenience, this equation is repeated here, adding the standard time variation term on the left-hand side,

(4.14) \begin{equation} E_{I}\frac{\partial n_{e}}{\partial t}=\eta J^{2}-\frac{3}{2}\nu _{E}n_{e}\left(kT_{e}-kT_{p}\right) .\end{equation}

The time variation term vanishes in the limit of marginal stability, but is included to determine the correct sign for the stability boundary.

Linearising this equation leads to

(4.15) \begin{equation} \gamma E_{I}\tilde{n}_{e}=J^{2}\tilde{\eta }+2\eta \big(J_{x}\tilde{J}_{x}+J_{y}\tilde{J}_{y}\big)-\frac{3}{2}\left(kT_{e}-kT_{p}\right)\left(\tilde{\nu }_{E}n_{e}+\nu _{E}\tilde{n}_{e}\right)-\frac{3}{2}\nu _{E}n_{e}k\tilde{T}_{e} .\end{equation}

As before, neglect the variation of the primary gas temperature $\tilde{T}_{p}\ll \tilde{T}_{e}$ . Each of the terms in (4.15) can now be expressed in terms of $\tilde{n}_{e}$ . A short calculation yields

(4.16) \begin{align} &\left(\gamma n_{e}E_{I}\right)\frac{\tilde{n}_{e}}{n_{e}} =\left\{\eta J^{2}\left(\alpha -1+2K\right)-\frac{3}{2}n_{e}\nu _{E}\left[\left(3\alpha +1\right)\left(kT_{e}-kT_{p}\right)+2\alpha kT_{P}\right]\right\}\frac{\tilde{n}_{e}}{n_{e}}, \end{align}

with

(4.17) \begin{equation} K\left(k_{x},k_{y},k_{z}\right)=\frac{1}{1+\beta ^{2}\left(k_{z}^{2}/k_{0}^{2}\right)}\left[-\left(\alpha -1\right)\frac{J_{\bot }^{2}}{J^{2}}-\beta \frac{J_{\bot }J_{\parallel }}{J^{2}}+\frac{k_{z}^{2}}{k_{0}^{2}}\left(\beta ^{2}-\alpha +1\right)\right]. \end{equation}

Taking the limit $\gamma \rightarrow 0$ , one sees that the dispersion relation defining the condition for marginal stability is given by

(4.18) \begin{equation} \eta J^{2}\left(\alpha -1+2K\right)-\frac{3}{2}n_{e}\nu _{E}\left[\left(3\alpha +1\right)\left(kT_{e}-kT_{p}\right)+2\alpha kT_{P}\right]\leq 0\quad \text{for stability} .\end{equation}

This is the required relation with simplifications to follow.

Before proceeding with simplification, this subsection is completed by providing the justification for neglecting the perturbed quantities associated with the primary fluid. An important point to recognise is that the maximal ordering assumed in the starting equations requires that $\beta \sim \alpha \sim M^{2}\sim O(1)$ . With this assumption, it follows that all the electron perturbations just calculated are of the same order,

(4.19) \begin{equation} \frac{\tilde{n}_{e}}{n_{e}}\sim \frac{\tilde{T}_{e}}{T_{e}}\sim \frac{\tilde{J}_{x}}{J_{x}}\sim \frac{\tilde{J}_{y}}{J_{y}} .\end{equation}

If one now uses this ordering in the primary fluid mass, momentum and energy equations, it follows that

(4.20) \begin{equation} \frac{\tilde{n}_{p}}{n_{p}}\sim \frac{\tilde{v}_{p}}{v_{p}}\sim \frac{\tilde{T}_{p}}{T_{p}}\sim \frac{1}{k_{0}L}\frac{\tilde{n}_{e}}{n_{e}}. \end{equation}

We see that all primary perturbations are smaller by $1/k_{0}L$ than the electron perturbations. This is the justification for neglecting them in the derivation of the dispersion relation.

6.1. Open cycle fossil fuel generator

An open cycle fossil fuel generator is characterised by a weakly ionised seed gas that is at nearly the same temperature as the primary gas. To obtain a high electrical conductivity, the temperature must be relatively high ( $T_{e}\approx T_{p}\sim 2500\,\mathrm{K}$ ). The need for a high temperature can lead to material problems, which are one of the drawbacks of an open cycle fossil fuel system.

Keep in mind that the analysis so far has focused on closed cycle systems, which assume a monatomic gas. For a fossil fuel system, essentially all of the analysis still applies, with one exception. The relation between $\nu _{E}$ and $\nu _{M}$ is modified as follows:

(6.3) \begin{equation} \nu _{E}=2\frac{m_{e}}{m_{p}}\nu _{M}\rightarrow \nu _{E}=2\delta \frac{m_{e}}{m_{p}}\nu _{M} .\end{equation}

Here, $\delta$ is an energy equilibration enhancement factor whose value depends on the primary gas used in the generator (see for instance Rosa (Reference Rosa1987)). For a closed cycle system using a monatomic gas (e.g. Ar), then $\delta =1$ . For an open cycle system using a mixture of complex molecular gases (e.g. CO, CO2, NO), then $\delta \gt 200$ . When this effect is included in the analysis, the important place in which it appears is in the relationship between $\Delta T$ and $Z$ . Specifically, this relationship, given by (3.14), is now replaced by

(6.4) \begin{equation} \Delta T=\left(\frac{5M^{2}}{9}\right)\frac{\beta ^{2}\left[\beta ^{2}+\left(1+Z\right)^{2}\right]}{\left(\beta ^{2}+1+Z\right)^{2}}\rightarrow \Delta T=\left(\frac{5M^{2}}{9\delta }\right)\frac{\beta ^{2}\left[\beta ^{2}+\left(1+Z\right)^{2}\right]}{\left(\beta ^{2}+1+Z\right)^{2}}. \end{equation}

With this modification, consider the limit of an open cycle fossil fuel plant defined by $f_{I}\ll 1$ and $\delta \gg 1$ . The appropriate orderings and approximations are as follows. First, the assumption of a low seed gas ionisation fraction $f_{I}\ll 1$ , implies that

(6.5) \begin{equation} \alpha =\frac{kT_{e}}{2E_{I}}\frac{2-f_{I}}{1-f_{I}}\approx \frac{kT_{e}}{E_{I}}\ll 1. \end{equation}

Second, the energy balance relation between $\Delta T$ and $\beta ,\,Z$ , given in (6.4), shows that for modest to high Hall parameters, $\beta \sim 1$ to $\beta \gg 1$ , and typical Mach numbers $M\sim 1{-}2$ , the temperature difference is small for any $Z\boldsymbol{\lesssim }\beta$ ,

(6.6) \begin{equation} \Delta T=\left(\frac{5M^{2}}{9\delta }\right)\frac{\beta ^{2}\left[\beta ^{2}+\left(1+Z\right)^{2}\right]}{\left(\beta ^{2}+1+Z\right)^{2}}\sim \frac{1}{\delta }\ll 1. \end{equation}

The temperature difference is small as expected in an open cycle system with a complex molecular primary gas – energy transfer between electrons and primary gas is rapid when $\delta \gg 1$ .

Next, to apply the stability criterion, one needs to estimate $\alpha /\Delta T$ , the ratio of two small parameters. Since $\alpha \sim 1/20$ and $\delta \boldsymbol{\gtrsim }200$ , observe that

(6.7) \begin{equation} \frac{\alpha }{\Delta T}\sim \alpha \delta \gg 1. \end{equation}

This scaling is substituted into the expression for the stability $\beta$ limit leading to the simplified result,

(6.8) \begin{align} \beta ^{2}&\leq 4\alpha \left(2+\frac{1}{\Delta T}\right)\left[1+\alpha \left(1+\frac{1}{\Delta T}\right)\right]\nonumber\\ &\approx \left(\frac{2\alpha }{\Delta T}\right)^{2}\gg 1\rightarrow \beta \leq \frac{2\alpha }{\Delta T}. \end{align}

While the instability limits the value of $\beta$ , this limit is high; that is, it does not pose a serious constraint.

All quantities, except $Z$ have now been scaled. This last scaling is obtained by substituting into the first figure of merit and then minimising with respect to $Z$ . From (6.1), one sees that the figure of merit reduces to

(6.9) \begin{equation} \frac{S_{\Omega }}{S_{L}}=\frac{\left[\beta ^{2}+\left(1+Z\right)^{2}\right]}{\beta ^{2}Z}. \end{equation}

This function has a minimum with respect to $Z$ at

(6.10) \begin{equation} Z=\left(1+\beta ^{2}\right)^{1/2}\sim \beta, \end{equation}

which is consistent with the scaling assumed previously, and results in a figure of merit given by

(6.11) \begin{equation} \frac{S_{\Omega }}{S_{L}}=2\frac{1+\left(1+\beta ^{2}\right)^{1/2}}{\beta ^{2}}\sim \frac{1}{\beta }. \end{equation}

High $\beta$ , or equivalently high $B_{0}$ , leads to good performance.

Turning to the second figure of merit, one sees that it scales as

(6.12) \begin{equation} \sigma =\frac{e^{2}n_{e}}{m_{e}\nu _{M}}\sim \frac{n_{e}}{T_{e}^{1/2}} .\end{equation}

This form is misleading in that raising $n_{e}$ and lowering $T_{e}\approx T_{p}$ to increase $\sigma$ also increases the contributions of $\nu _{en}$ and $\nu _{ei}$ to $\nu _{M}$ . When these contributions are included, it has been shown (Rosa Reference Rosa1987) that there is actually an optimum density that maximises $\sigma$ . At the optimum, $n_{e}(T_{e})$ and $\sigma (T_{e})$ are given by

(6.13) \begin{align}n_{e}^{2}\left(T_{e}\right)&=\left(\frac{p_{p}\overline{\sigma }_{ep}}{\overline{\sigma }_{en}}\right)\left[\frac{N(T_{e})}{kT_{e}}\right],\nonumber\\[4pt] \sigma(T_{e})&=\frac{e^{2}n_{e}}{m_{e}}\frac{1}{\left[2(\nu _{ep}\nu _{en})^{1/2}+\nu _{ei}\right]}.\end{align}

The conductivity $\sigma _{e}$ is a rapidly increasing function of $T_{e}$ and independent of $B_{0}$ .

What are the main conclusions for an open cycle generator? There are three. (i) The ionisation instability poses a limit on the maximum allowable $\beta$ , but this limit is high and thus does not represent a serious constraint on performance. (ii) The first figure of merit $S_{\Omega }/S_{L}$ becomes smaller as $\beta$ increases. In other words, high $\beta$ , corresponding to high $B_{0}$ , is a good strategy to maximise performance. (iii) The second figure of merit $\sigma$ is independent of $B_{0}$ , but increases with $T_{e}\approx T_{p}$ . High temperatures, of the order of 2500 K, are achievable with fossil fuel power plants.

6.2. Standard closed cycle generator

The open cycle fossil fuel generator has to deal with high temperature materials problems, which, while difficult, are not insurmountable. Even so, there is less interest now in fossil fuel plants, particularly those powered by coal, because of $\mathrm{CO}_{2}$ emissions.

In contrast, because of basic physics and engineering constraints, nuclear power plants, both fission and fusion, cannot at present achieve comparably high temperatures in the primary coolant. Typically, $T_{p}\sim 1000{-}1300\,\mathrm{K}$ is closer to the upper limit. Because of the strong exponential $T_{e}$ dependence in the Saha equation, the electron density would be reduced by approximately 10 orders of magnitude when $T_{e}\approx T_{p}$ is reduced by a factor of approximately 2, from 2000 K to 1000 K! This has motivated the idea of using a closed cycle generator, with a monatomic primary gas such as argon (Kerrebrock Reference Kerrebrock1964; Kerrebrock & Hoffman Reference Kerrebrock and Hoffman1964; Sheindlin et al. Reference Sheindlin, Batenin and Asinovsky1964). The reasoning is that with the long energy equilibration time between argon and seed electrons, say potassium, plus the continuous Ohmic heating along the channel, it should be possible to maintain a finite temperature difference between the two species. ‘We can have our cake and eat it’. Lower temperature argon, easy to produce in a nuclear plant, combined with high temperature electrons needed for large electrical conductivity, should lead to highly efficient energy conversion.

Unfortunately, here is where the ionisation instability enters the picture. This instability is excited when there is a substantial temperature difference. The question then is how much impact does the maximum allowable stable temperature difference have on the energy conversion efficiency, as defined by the figures of merit? This question is now addressed for a standard closed cycle MHD generator defined by $f_{I}\ll 1$ and $\delta =1$ . The analysis proceeds as follows.

As noted previously, the weakly ionised seed gas assumption, $f_{I}\ll 1$ , leads to the ordering

(6.14) \begin{equation} \alpha =\frac{kT_{e}}{2E_{I}}\frac{2-f_{I}}{1-f_{I}}\approx \frac{kT_{e}}{E_{I}}\ll 1 .\end{equation}

Now, since the primary gas temperature is assumed to be low, one needs to operate in a regime where the temperature difference is no longer small, but is instead finite: $\Delta T\sim 1$ . However, this assumption leads to a contradiction. Specifically, for $\Delta T\sim 1$ , the stability condition, (6.8), requires that $\beta \sim \alpha ^{1/2}\ll 1$ . When substituted into the energy balance relation, (6.6), and setting $\delta =1$ for a monatomic gas, one sees that this implies $\Delta T\sim \beta ^{2}\ll 1$ , which violates the original assumption.

After some thought, note that for a weakly ionised seed gas in a monatomic primary gas, the self-consistent ordering assumption becomes

(6.15) \begin{align} \alpha &\ll 1,\nonumber\\ \Delta T\sim \alpha ^{1/2}&\ll 1,\nonumber\\ \alpha /\Delta T\sim \alpha ^{1/2}&\ll 1. \end{align}

These orderings imply that

(6.16) \begin{align} \Delta T&\approx \frac{5M^{2}}{9}\beta ^{2}\sim \alpha ^{1/2}\ll 1&&\quad \text{Energy balance},\nonumber\\ \beta ^{2}&\leq \frac{4\alpha }{\Delta T}\sim \alpha ^{1/2}\ll 1&&\quad \text{Stability condition}. \end{align}

The two figures of merit reduce to

(6.17) \begin{align} \frac{S_{\Omega }}{S_{L}}&\approx \frac{\left(1+Z\right)^{2}}{\beta ^{2}Z}\approx \frac{4}{\beta ^{2}}\sim \frac{1}{\alpha ^{1/2}}\gg 1&&\quad \text{Minimized figure of merit}\,(Z=1),\nonumber\\ \sigma \left(T_{e}\right)&=\frac{e^{2}n_{e}}{m_{e}}\frac{1}{\left[2\left(\nu _{ep}\nu _{en}\right)^{1/2}+\nu _{ei}\right]}&&\quad T_{e}\approx T_{p}. \end{align}

This corresponds to an unattractive mode of operation: (i) the temperature difference is small because of the instability (making it difficult to obtain a high $\sigma$ ); (ii) the Hall parameter is small (leading to an undesirably low Hall voltage); (iii) $S_{\Omega }/S_{L}$ is large (indicating that most of the converted electricity is going into Ohmic heating and not the load); and (iv) the conductivity is very small because of the low electron density implied by the low temperature in Saha’s equation.

The main conclusion from this analysis is that the ionisation instability imposes a strong constraint on the performance of a closed cycle MHD generator using a monatomic primary gas plus a weakly ionised seed current. This conclusion was borne out experimentally early in the programme (Velikhov & Dykhne Reference Velikhov and Dykhne1963; Velikhov et al. Reference Velikhov, Dykhne and Shipuk1965). The instability has largely been viewed as a ‘show stopper’ and much of the research in this area has been strongly curtailed.

6.3. Advanced closed cycle generator

In spite of the dire theoretical predictions and poor experimental performance, experimentalists did find a way to improve closed cycle operation. It was observed that when the seed gas became essentially fully ionised, the ionisation instability would be suppressed. There have been several contributions in the theoretical literature (Petit & Valensi Reference Petit and Valensi1969; Mitchner & Kruger Reference Mitchner and Kruger1973; Nakamura & Riedmuller Reference Nakamura and Riedmuller1974; Kien Reference Kien2016) that support this conclusion but, to the author’s knowledge, no sharply defined stability condition and corresponding scaling relations have explicitly appeared. This gap is filled in the present work and represents one important contribution. It is also worth noting that while performance could be improved experimentally, the gains were not sufficiently large so as to strongly regenerate interest in this area of research. A second important contribution of the present work is to investigate whether access to much higher magnetic fields, now possible because of REBCO superconductors, can lead to much larger gains in performance, perhaps sufficiently large to regenerate interest in closed cycle MHD energy conversion.

The advanced closed cycle MHD generator is defined by $f_{I}\rightarrow 1$ and $\delta =1$ . The analysis to predict performance again requires some thought with respect to the orderings of various quantities and the corresponding experimental consequences. The starting point is the assumption of near full ionisation of the seed gas: $f_{I}\rightarrow 1$ . In fact, for best performance, the seed gas must be very near full ionisation so that the following ordering holds for the ionisation parameter:

(6.18) \begin{equation} \alpha =\frac{kT_{e}}{2E_{I}}\frac{2-f_{I}}{1-f_{I}}\approx \frac{kT_{e}}{2E_{I}}\frac{1}{1-f_{I}}\gg 1. \end{equation}

The parameter has obviously switched from being very small to very large, which is the key mathematical insight needed to produce a high performance closed line MHD generator.

Best performance also requires operation in the regime of a large Hall parameter plus a large load impedance. Not so obviously, the relative size of these two parameters must satisfy

(6.19) \begin{equation} \beta ^{2}\gg Z\gg \beta \gg 1. \end{equation}

For example, $Z\sim \beta ^{3/2}$ . When this assumption is substituted into the energy equation, this leads to

(6.20) \begin{equation} \Delta T=\left(\frac{5M^{2}}{9}\right)\frac{\beta ^{2}\left[\beta ^{2}+\left(1+Z\right)^{2}\right]}{\left(\beta ^{2}+1+Z\right)^{2}}\approx \left(\frac{5M^{2}}{9}\right)\frac{Z^{2}}{\beta ^{2}}\gg 1. \end{equation}

The temperature difference is now large, one desirable goal. In the interesting operational regime, the ratio of the two large parameters $\alpha ,\,\Delta T$ is assumed to satisfy

(6.21) \begin{equation} \frac{\alpha }{\Delta T}\approx \left(\frac{9kT_{e}}{10E_{I}M^{2}}\right)\frac{\beta ^{2}}{Z^{2}}\frac{1}{1-f_{I}}\gg 1 ,\end{equation}

which sets the required level of ionisation. Next, substitution into the stability criterion leads to

(6.22) \begin{align} \beta ^{2} & \leq 4\alpha \left(2+\frac{1}{\Delta T}\right)\left[1+\alpha \left(1+\frac{1}{\Delta T}\right)\right]\nonumber\\[3pt]& \approx 8\alpha ^{2}\gg 1, \end{align}

which reduces to

(6.23) \begin{equation} \beta \leq 2^{1/2}\frac{kT_{e}}{E_{I}}\frac{1}{1-f_{I}}. \end{equation}

Equation (6.23) is an important result. It clearly shows the relation between the maximum stable $\beta$ and the degree of ionisation $1-f_{I}$ . Stability at high $\beta$ requires nearly full ionisation.

Before proceeding, it is worthwhile noting that so far, it has not been necessary to specify a precise scaling of $Z$ with $\beta$ , only the range given by (6.19). As such, a simple way to clarify the abovementioned tangle of orderings is to actually make two specific assumptions with respect to $Z\text{ and }kT_{e}/E_{I} : Z\sim \beta ^{\kappa }$ with $1\lt \kappa \lt 3/2$ and $kT_{e}/E_{I}\sim 1/\beta$ , both consistent with the abovementioned assumptions. Then, all relevant quantities can be scaled directly with $\beta$ , and simultaneously satisfy all the orderings previously discussed,

(6.24) \begin{align} \frac{kT_{e}}{E_{I}} & \sim \frac{1}{\beta }\ll 1,\nonumber\\ Z & \sim \beta ^{\kappa }\gg 1 \quad \textrm{and}\quad 1\lt \kappa \lt 3/2,\nonumber\\ \Delta T & \sim \beta ^{2\left(\kappa -1\right)}\gg 1,\nonumber\\ \alpha & \sim \beta \gg 1,\nonumber\\ \frac{\alpha }{\Delta T} & \sim \beta ^{3-2\kappa }\gg 1,\nonumber\\ 1-f_{I} & \sim \frac{1}{\beta ^{2}}\ll 1. \end{align}

We can now use this ordering scheme to evaluate the two figures of merit. Consider first $S_{\Omega }/S_{L}$ . A short calculation yields

(6.25) \begin{equation} \frac{S_{\Omega }}{S_{L}}=\frac{\left[\beta ^{2}+\left(1+Z\right)^{2}\right]}{\beta ^{2}Z}\approx \frac{Z}{\beta ^{2}}\sim \frac{1}{\beta ^{2-\kappa }}\ll 1. \end{equation}

If one wants to minimise $S_{\Omega }/S_{L}$ , then high $\beta$ (i.e. high $B_{0}$ ) is a good strategy.

The second figure of merit requires a little more work, which makes use of the relations $n_{e}\approx n_{s} , 1-f_{I}=n_{s}/N\approx n_{e}/N$ and $\beta =(eB_{0}/m_{e})/\nu _{M}=\Omega _{e}/\nu _{M}$ . The result is

(6.26) \begin{equation} \sigma =\frac{e^{2}n_{e}}{m_{e}\nu _{M}}=\left(\frac{e^{2}n_{e}}{m_{e}}\right)\left(\frac{NkT_{e}}{2E_{I}\alpha }\right)\leq 2^{1/2}\left(\frac{kT_{e}}{E_{I}}\right)\left(\frac{e^{2}N}{m_{e}\Omega _{e}}\right). \end{equation}

We see that if the goal is to achieve large $\sigma$ , then small $\Omega _{e}$ (i.e. small $B_{0}$ ) is the path to take.

As stated earlier, the two figures of merit have opposite requirements on $B_{0}$ to achieve good performance. Low $S_{\Omega }/S_{L}$ requires high $B_{0}$ , while high $\sigma$ requires low $B_{0}$ . The physical explanation for these opposing requirements is as follows. The conductivity is proportional to $n_{e}$ . Now, as $B_{0}$ increases, the need to ever more closely approach full ionisation for stability requires, by virtue of Saha’s equation, that the seed density $n_{s}$ decreases. Since $n_{e}\approx n_{s}$ , the implication is the electron density and, hence, the conductivity will also decrease as $B_{0}$ increases. In contrast, for the second figure of merit $S_{\Omega }/S_{L}$ , the electron density cancels when forming the ratio. The resulting $S_{\Omega }/S_{L}$ is only a function of $\beta \propto B_{0}/T_{e}^{1/2}$ with the $B_{0}$ dependence dominating, since the electron temperature only varies slightly because of its exponential dependence in Saha’s equation. The high performance associated with high field leads to the result that $S_{\Omega }/S_{L}$ will decrease as $B_{0}$ increases, which is a favourable result.

Resolving this dichotomy requires a global solution to MHD generator design, which is not possible using only a local stability criterion. The global analysis has been completed and will be reported on in the near future. As a preview, note that a global design is actually strongly influenced by engineering constraints which are not considered here. Including these constraints demonstrates that there is a high but optimum magnetic field that maximises performance.

We close this subsection by discussing one further major point – the significance of the precise scaling of the ionisation fraction $f_{I}$ . After all, it seems extremely difficult to measure or control the ionisation so accurately. This, however, is not the main concern. Instead, the nearness to full ionisation provides a strong constraint between the electron temperature and electron density via Saha’s equation. The constraint leads to the following relation:

(6.27) \begin{equation} n_{e}\left(T_{e}\right)=\left(\frac{kT_{e}}{2E_{I}}\right)\left(\frac{N}{\alpha }\right)\leq 2^{1/2}\left(\frac{kT_{e}}{2E_{I}}\right)\left(\frac{N}{\beta }\right), \end{equation}

which sets the maximum allowable density of the unionised seed gas. This is a critical design constraint for closed cycle MHD generators to be used as a topping cycle for a nuclear power plant.

What are the conclusions with respect to advanced closed cycle MHD Hall generators? Overall, they are positive: (i) the temperature difference is large, which is just what is needed to create a high conductivity plasma assuming a low primary gas temperature; (ii) the seed gas must be very close to full ionisation for stability, as observed experimentally; and (iii) the two figures of merit make conflicting predictions as to the value of high field. This is not necessarily bad, but more work on global designs is needed to resolve the conflict and determine the optimum magnetic field.

7.1. Input parameters

The first step in the analysis is to choose ‘furnace’ values corresponding to the Argon gas leaving the heat exchanger. These values are held fixed during all calculations. For the HTGR they are given by (NEA 2022):

1. (a) argon coolant input pressure to the Laval nozzle: $p_{in}=5\times 10^{6}\text{ Pa}\approx 50\text{ atm}$ ;

2. (b) argon coolant input temperature to the Laval nozzle: $T_{in}=1000\,\mathrm{K}$ ;

3. (c) argon coolant input velocity to the Laval nozzle: highly subsonic $v_{in}^{2}\ll 2kT_{in}/m_{p}$ ;

4. (d) argon coolant output Mach number from the Laval nozzle: $M=1.8$ .

The output properties of the Laval nozzle gas serve as the primary argon input quantities to the MHD generator.

\begin{align*} &\quad\!\!\left(\mathrm{a}\right)\text{Argon MHD inlet pressure}{\colon} \nonumber\\ &\qquad p_{p}=\left(\!1+\frac{\gamma -1}{2}M^2\!\right)^{-\frac{\gamma }{\gamma -1}}p_{in}=\left(\frac{3}{M^{2}+3}\right)^{5/2}p_{in}=0.801\times 10^{6}\text{ Pa}\approx 8.01\text{ atm}. \end{align*}

(b) Argon MHD inlet temperature:

\begin{align*} T_{p}=\left(\!1+\frac{\gamma -1}{2}M^2\!\right)^{-1}T_{in}=\left(\frac{3}{M^{2}+3}\right)T_{in}=481\,\mathrm{K}. \end{align*}

(c) Argon MHD inlet number density:

\begin{align*}n_{p}=\frac{p_{p}}{kT_{p}}=\left(1+\frac{\gamma -1}{2}M^2\right)^{-\frac{1}{\gamma -1}}\frac{p_{in}}{kT_{in}}=\left(\frac{3}{M^{2}+3}\right)^{3/2}\frac{p_{in}}{kT_{in}}=1.21\times 10^{26}\,\mathrm{m}^{-3}. \end{align*}

(d) Argon MHD inlet velocity:

\begin{align*} v_{p}=\left(\frac{\gamma kT_{p}M^{2}}{m_{p}}\right)^{1/2}=\left(\frac{5kT_{p}M^{2}}{3m_{p}}\right)^{1/2}=735\,\mathrm{m}\,\sec^{-1}. \end{align*}

Here, $\gamma$ is the ratio of specific heats (for a monatomic gas, $\gamma =5/3$ ). The abovementioned values are held fixed for all calculations. Observe that the MHD generator input temperature is 481 K, substantially reduced from the initial 1000 K because of the need for a Laval nozzle to produce a supersonic flow velocity.

Two other input quantities that are held fixed during a given calculation are the magnetic field and converted electric power density. Separate calculations scan the values of these two quantities, whose range covers:

1. (i) Magnetic field: $3\,\mathrm{T}\lt B_{0}\lt 20\,\mathrm{T}$ ;

2. (ii) Power density: $S_{C}=100,200,300\,\mathrm{MW}\,\mathrm{m}^{-3}$ .

The magnetic field is allowed to vary over a wide range. The lower value of 3 T corresponds to typical operation of MHD generators prior to the almost complete termination of the USA programme in the 1990s. The higher value of 20 T is at the limit of practical magnetic fields using the recently developed REBCO superconductors (Vieira et al. Reference Vieira2024). As discussed, the second scanning parameter of interest is the inlet converted electric power density and three plausible discrete values are chosen. Too low a value outside this range implies a large, uneconomical generator. Too high a value translates into serious material problems on the generator walls and electrodes.

7.2. How to obtain a solution

Assume now that all the input parameters have been specified, including values for $B_{0}$ and $S_{C}$ . It is shown now that obtaining values for all the physical quantities of interest requires the solution to a nonlinear algebraic equation for the temperature $T_{e}$ , a simple numerical calculation. The procedure requires making an initial guess for $T_{e}$ and then evaluating the following quantities in the order listed,

\begin{align*} &\Delta T=\frac{T_{e}}{T_{p}}-1&&\quad \text{Definition}\nonumber\\ &\nu_{M}=n_{p}\overline{\sigma }_{ep}\left(\frac{2kT_{e}}{m_{e}}\right)^{1/2}&&\quad \text{Definition},\nonumber\\ &\beta =\frac{eB_{0}}{m_{e}\nu _{M}}&&\quad \text{Definition},\nonumber\\ &Z=\xi +\left[\frac{\xi \left(\xi +1\right)}{\mu }\right]^{1/2}&&\quad \text{Energy balance},\nonumber\\ &\mu =\frac{9}{5M^{2}\beta ^{2}}\Delta T &&\quad \text{Energy balance},\nonumber\\ &\xi =\frac{\beta ^{2}\mu }{1-\mu }-1&&\quad \text{Energy balance},\nonumber \end{align*}

(7.1) \begin{align} &n_{e}=\frac{\left(\beta ^{2}+1+Z\right)}{\beta ^{2}\left(1+Z\right)}\frac{S_{C}}{m_{e}\nu _{M}v_{p}^{2}}&&\qquad\qquad \text{Power density},\nonumber\\ &1-f_{I}=\frac{n_{e}/N\left(T_{e}\right)}{1+n_{e}/N\left(T_{e}\right)}&&\qquad\qquad \text{Saha},\nonumber\\ &\alpha =\frac{1}{2}\left(\frac{2kT_{e}}{3kT_{e}+2E_{I}}\right)\left(\frac{2-f_{I}}{1-f_{I}}\right)&&\qquad\qquad \text{Definition} .\end{align}

For algebraic simplicity, $\gamma =5/3$ has substituted wherever appropriate in these expressions.

All the quantities of interest have now been evaluated for the given guess of $T_{e}$ . These values are now substituted into the marginal stability criterion for the ionisation instability, repeated here for convenience,

(7.2) \begin{equation} \beta ^{2}=4\alpha \left(2+\frac{1}{\Delta T}\right)\left[1+\alpha \left(1+\frac{1}{\Delta T}\right)\right]. \end{equation}

In general, this constraint will not be satisfied for the $T_{e}$ guess. It is here that a simple numerical iteration on $T_{e}$ is all that is required to satisfy the marginal stability constraint.

7.3. Results

Following the procedure just discussed, a large number of cases have been evaluated. The most important results are summarised in figures 2 and 3, where the figures of merit $S_{\Omega }/S_{L}$ and $\sigma$ have been plotted versus magnetic field $B_{0}$ for three values of $S_{C}$ .

Figure 2. Figure of merit $S_{\Omega }/S_{L}$ versus magnetic field for three values of $S_{C}\,(\mathrm{MW}\,\mathrm{m}^{-3})$ .

Figure 3. Figure of merit $\sigma$ versus magnetic field for three values of $S_{C}\,(\mathrm{MW}\,\mathrm{m}^{-3})$ .

Several conclusions can be drawn. First, $S_{\Omega }/S_{L}$ decreases rapidly with magnetic field, transforming from undesirable values greater than unity to attractive values much less than unity. In other words, high field is a potential winner for improving the efficiency of closed cycle MHD generators operating at the marginal stability boundary of the ionisation instability. Second, the figure of merit is almost independent of the required load power density over a reasonably wide range $100\lt S_{C}(\mathrm{MW}\,\mathrm{m}^{-3})\lt 300$ . This is not surprising since the scaling factor $S_{C}$ cancels when calculating the ratio $S_{\Omega }/S_{L}$ . Lastly, as shown by the dashed line, to keep $S_{\Omega }/S_{L}$ below 20 % requires a magnetic field of 14 T or greater. The 20 % value is a desirable practical goal implying that 80 % of the converted electrical power is supplied to the load and only 20 % to heat the electrons.

The conclusions with respect to the second figure of merit $\sigma$ , as shown in figure 3, are quite the opposite. Here, lower magnetic field leads to higher conductivity. Also, at any given magnetic field, the conductivity is approximately linearly proportional to the value of $S_{C}$ . Higher conductivity requires a higher electron density which, in turn, requires a higher converted power density. Also, in almost all cases, the conductivity is relatively high, greater than $10\text{ mho}\,\mathrm{m}^{-1}$ , which is the requirement for a high-quality plasma even though the primary gas temperature is only 481 K.

Additional useful information is illustrated in figure 4. Plotted here are (a) electron temperature $T_{e}$ , (b) electron density $n_{e}$ , (c) Hall parameter $\beta$ and (d) the fraction of unionised seed gas $1-f_{I}$ , all versus the magnetic field $B_{0}$ .

Figure 4. Curves of (a) $n_{e}$ , (b) $T_{e}$ , (c) $\beta$ and (d) $1-f_{I}$ , versus $B_{0}$ for three values of $S_{C}$ .

The following points are worth noting. The required temperature is, interestingly, almost independent of magnetic field and load power density. The reason can be traced back to the strong exponential dependence in the Saha equation. Even small changes in the electron temperature result in enormous changes in the electron density, which would lead to comparably large changes in the load power density.

The electron density decreases rapidly as the field increases. This is a consequence of needing ever lower densities to more closely approach full ionisation, as required by the marginal stability criterion. The density is also a rapidly increasing function of the power density to the load. This is not surprising – more load power requires more current, which in turn requires more electrons.

The Hall parameter increases linearly with $B_{0}$ , which is to be expected since $\beta \propto B_{0}$ . Values of $\beta$ of the order of 10–20, much larger than those in early experiments, are needed if high field is desirable. Also, the seed gas becomes progressively closer to full ionisation as $B_{0}$ increases. As stated, this is a consequence of the ionisation instability marginal criterion. Both quantities are almost independent of $S_{C}$ .

As a specific reference case, assume values for the critical generator parameters given by $M=1.8 , S_{C}=100\text{ MW}\,\mathrm{m}^{-3}$ and $B_{0}=8\mathrm{T}$ . It then follows that

(7.3) \begin{align} S_{\Omega }/S_{L}&=0.3720,\nonumber\\ \sigma &=12.35\text{ mho}\,\mathrm{m}^{-1},\nonumber\\ T_{e}&=4223\,\mathrm{K},\nonumber\\ T_{e}/T_{p}&=8.783,\nonumber\\ n_{e}&=7.478\times 10^{19}\mathrm{m}^{-3},\nonumber\\ n_{e}/n_{p}&=6.180\times 10^{-7},\nonumber\\ \beta &=8.249\nonumber\\ 1-f_{I}&=0.01679 .\end{align}

As compared with earlier, lower field generators, note the higher temperature ratio and higher Hall parameter.

The last point of interest involves a comparison of results for two other forms of the marginal stability boundary. Keep in mind that the abovementioned results correspond to an exact numerical solution of the stability criterion, but using the approximate form of collision frequency $\nu _{M}\approx \nu _{ep}$ . The first comparison involves the $\nu _{M}\approx \nu _{ep}$ approximation. As previously stated, but not proved, it has been assumed that the dominant collision mechanism is between electrons and the primary gas. One can now offer proof by recalculating the results presented in figures 2 and 3, i.e. the curves of $S_{\Omega }/S_{L}\text{ vs }B_{0}$ and $\sigma \text{ vs }B_{0}$ for fixed $S_{C}=100\text{ MW}\,\mathrm{m}^{-3}$ . The new curves are obtained by setting (see for instance Wesson Reference Wesson2011 for $\nu _{ei}$ )

(7.4) \begin{align} \nu _{M}&=\nu _{ep}+\nu _{en}+\nu _{ei},\nonumber\\[4pt] \nu _{ep}&=n_{p}\overline{\sigma }_{ep}v_{Te},\nonumber\\[4pt] \nu _{en}&=\left(n_{s}-n_{e}\right)\overline{\sigma }_{en}v_{Te},\nonumber\\[4pt] \nu _{ei}&=\frac{2^{1/2}}{12\pi ^{3/2}}\frac{n_{e}e^{4}ln\Lambda }{\varepsilon _{0}^{2}m_{e}^{1/2}\left(kT_{e}\right)^{3/2}}\quad\Lambda =4\pi \frac{\varepsilon _{0}^{3/2}\left(kT_{e}\right)^{3/2}}{e^{3}n_{e}^{1/2}} \end{align}

and solving the equations numerically. Results are presented shortly.

The second comparison involves the analytic approximation for the stability boundary. Here, one again assumes $\nu _{M}\approx \nu _{ep}$ and uses the approximate form of the stability boundary given by (6.22), which leads to

(7.5) \begin{align} \frac{S_{\Omega }}{S_{L}}&\approx \frac{Z}{\beta ^{2}},\nonumber\\ \sigma &\approx 2^{1/2}\left(\frac{kT_{e}}{E_{I}}\right)\left(\frac{e^{2}N}{m_{e}\Omega _{e}}\right). \end{align}

Solutions are obtained using the procedure described in (7.1) and (7.2),

(7.6) \begin{align} &\Delta T\approx \frac{T_{e}}{T_{p}}&&\quad \text{Definition},\nonumber\\[3pt] &\nu _{M}\approx n_{p}\overline{\sigma }_{ep}\left(\frac{2kT_{e}}{m_{e}}\right)^{1/2}&&\quad \text{Definition},\nonumber\\[3pt] &\beta =\frac{eB_{0}}{m_{e}\nu _{M}}&&\quad \text{Definition},\nonumber\\[3pt] &Z\approx \left(\frac{9\Delta T}{5M^{2}}\right)^{1/2}\beta &&\quad \text{Energy balance},\nonumber\\[3pt] &n_{e}\approx \frac{S_{C}}{m_{e}\nu _{M}v_{p}^{2}Z}&&\quad \text{Power density},\nonumber\\[3pt] &1-f_{I}\approx n_{e}/N\left(T_{e}\right)&&\quad \text{Saha},\nonumber\\[3pt] &\alpha \approx \frac{kT_{e}}{2E_{I}}\left(\frac{1}{1-f_{I}}\right)&&\quad \text{Definition},\nonumber\\[3pt] &\beta ^{2}\approx 8\alpha ^{2}&&\quad \text{Marginal stability}. \end{align}

These approximations greatly simplify the analysis resulting in a simple algebraic equation for the electron temperature. After a slightly tedious calculation, one obtains

(7.7) \begin{align} w^{-7/2}\exp \left(-w\right)&=C,\nonumber\\[4pt] w&=\frac{E_{I}}{kT_{e}}, \nonumber\\[4pt] C&=\left[\frac{1}{\left(160\pi ^{3}\right)^{1/2}}\frac{m_{p}h^{3}k^{1/2}}{m_{e}^{2}E_{I}^{5/2}\overline{\sigma }_{ep}}\right]\left(\frac{T_{p}^{1/2}S_{C}}{Mp_{p}}\right)=7.715\times 10^{-13}\left(\frac{T_{p}^{1/2}S_{C}}{Mp_{p}}\right). \end{align}

Observe that the marginally stable electron temperature is independent of $B_{0}$ . Also, because of the strong exponential dependence of $w$ , the resulting electron temperature is only a weak function of $M,\,S_{C},\,T_{p},\,p_{p}$ .

Assume now that $T_{e}$ (i.e. $w$ ) is known from a simple numerical solution of (7.7). This result is substituted into the expression for the figures of merit leading to

(7.8) \begin{align} \frac{S_{\Omega }}{S_{L}}&=\left[\left(\frac{18}{5}\right)^{1/2}\frac{m_{e}^{1/2}\overline{\sigma }_{ep}E_{I}}{ek^{3/2}}\right]\frac{p_{p}}{wT_{p}^{3/2}MB_{0}}=0.6050\frac{p_{p}}{wT_{p}^{3/2}MB_{0}},\nonumber\\[4pt] \sigma &=\left[4\pi ^{3/2}\frac{e\left(m_{e}E_{I}\right)^{3/2}}{h^{3}}\right]\frac{w^{-5/2}e^{-w}}{B_{0}}=6.184\frac{w^{-5/2}e^{-w}}{B_{0}}. \end{align}

Observe that $S_{\Omega }/S_{L}$ is inversely proportional to $B_{0}$ . High field is desirable to maximise the fraction of converted power that is delivered to the load as opposed to heating electrons. The conductivity is also inversely proportional to $B_{0}$ , but in this case, high field is undesirable. The conductivity decreases as the field increases.

The overall comparison of the three different theoretical models are illustrated in figures 5 and 6.

Figure 5. Comparisons of $S_{\Omega }/S_{L}$ for three different stability models.

Figure 6. Comparisons of $\sigma$ for three different stability models.

The basic stability model used to plot figures 2 and 3 is shown in blue. The more exact model, which includes electron-seed neutral and electron-ion collisions, is shown in orange. Both of these models are in good agreement for all magnetic fields, particularly at high field. The close comparison is the justification for focusing solely on electron–primary gas collisions in the analysis.

The analytic model is also in good agreement with both of the other models at high fields. It begins to deviate at lower fields where the assumption $\beta \gg 1$ starts to break down. The analytic model shows the linear inverse scaling of both figures of merit with $B_{0}$ .