# Seawater magnetohydrodynamic pumps and thrusters: experiment, model and upscaling

Yves Delannoy, Ivan Montero, Alexandre Zapata, Nathanaël Machicoane

► To cite this version:

Yves Delannoy, Ivan Montero, Alexandre Zapata, Nathanaël Machicoane. Seawater magnetohydrodynamic pumps and thrusters: experiment, model and upscaling. Magnetohydrodynamics c/c of Magnitnaia Gidrodinamika, 2025, 61, pp.55 - 66. 10.22364/mhd.61.1-2.6. hal-05279338

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https://hal.science/hal-05279338v1

Submitted on 23 Sep 2025

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# SEAWATER MAGNETOHYDRODYNAMIC PUMPS AND THRUSTERS: EXPERIMENT, MODEL AND UPSCALING

Yves DELANNOY¹*, Ivan MONTERO¹, Alexandre ZAPATA¹, Nathanaël MACHICOANE¹,

¹ Univ. Grenoble Alpes, CNRS, Grenoble INP, LEGI, 38000 Grenoble, France.

*Corresponding author: yves.delannoy@grenoble-inp.fr

**Abstract:** Naval magneto-hydrodynamic (MHD) propulsion is faced to the high level of Joule dissipation associated to the low electrical conductivity of seawater, reducing the efficiency of the pump at the heart of a MHD thruster. A MHD pump experiment is presented in this paper, together with its CFD modelization. A simple analytical model is also written for conduction MHD pumps, then for a boat equipped with such a pump working as a MHD thruster. The numerical and analytical results are compared to our pump experiment and to a small scale boat model. Upscaling, we show how strong should be the magnetic field, and how large the MHD channel, to obtain a given efficiency for the pump or a given cruise speed for the boat.

**Keywords:** MHD pump, MHD propulsion, CFD, experiment.

**Introduction.** Electromagnetic pumps using steady magnetic fields and conduction through the fluid require low maintenance, are leak-tight thanks to their rotor-less feature and remain silent. They have been developed intensively some decades ago as Magneto-Hydro-Dynamic (MHD) thrusters for naval applications, culminating with the 30-m Yamato boat that reached a speed of 5.5 knots with 4-T superconducting magnets [1]. Recent high-temperature superconducting ribbons [2] may broaden the possibilities if they are integrated onboard in large magnets providing strong magnetic fields in straight channels. This study aims at precising how strong the magnetic field should be to have an efficient pump, and how large the channels should be to have an efficient thruster. We use a simple hydraulic model and a fixed drag coefficient to represent the boat without the channel.

To assess this approach, a pump experiment was built inside a superconducting magnet, tested experimentally and described by a CFD model augmented by a MHD module developed in our institution [3]. Furthermore, a model boat was built using permanent magnets in place of superconducting ones, and tested in a pool of artificial seawater.

This paper presents the pump theory and experiment, a simple analytical model for MHD boats and its application to our boat experiments. Using this model for upscaling, some quantitative elements are provided to estimate the necessary magnetic field and size of such pumps and thrusters.

**2. Analytical model of a pump.** A conduction MHD pump is considered here as a straight channel guiding a fluid flow at velocity $U$, inserted in a magnet providing a cross flow magnetic field $B$ and equipped with electrodes forcing a current density $j$ in the direction of $-U \times B$. We assume that those vector fields are perfectly perpendicular and uniform in a brick-shaped domain (Fig 1), of dimensions $L_U, L_B$ and $L_j$ along each vector field. The Ohm's law gives $j=\sigma(\Delta\varphi/L_j-UB)$ where $\sigma \approx 5 \ \Omega^{-1}m^{-1}$ is the conductivity of seawater and $\Delta\varphi$ is the potential difference imposed to the solution by the electrodes. Because of electrolysis in the ionic solution, the anode (resp. cathode) is at a potential $E_a>0$ (resp. $E_c<0$) with respect to the solution nearby, and the voltage difference between the electrodes, $\Delta\varphi_{el}=\Delta\varphi+(E_a-E_c)$,

<figure><img src="image_2.png" /><figcaption>Figure 1. Geometry of the MHD channel</figcaption></figure>
contains an electrochemical term $E_a-E_c \sim 2 \text{ V} \rightarrow 6 \text{ V}$ for $j \sim 0 \rightarrow 1 \text{ kA/m}^2$ [4]. Multiplying by the total current $I=jL_U L_B$, we obtain the total electrical power $P_{el}$, its part $P_{chim}$ consumed by electrochemical reactions and the rest $P_{channel}$ transferred to the solution:

$$P_{el}=I \cdot \Delta \varphi_{el} ; \quad P_{chim}=I \cdot (E_a-E_c) ; \quad P_{channel}=I \cdot \Delta \varphi \quad (1)$$

Notice that the ratio $\eta_{chim}=P_{channel}/P_{el}$ is usually greater than 0.9 or higher for large size channels.

From the Ohm's law, we find that $\Delta\varphi=RI+UBL_j$, where $R=L_j/(\sigma L_U L_B)$ is the electrical resistance of the channel and $UBL_j$ the counter-electromotive force due to the fluid velocity. Multiplying by $I$, we find the Joule dissipation $RI^2$, i.e. the integral of $j^2/\sigma$, and the integral of $f_L U$ (where $f_L=jB$ is the Lorentz force), i.e. the mechanical power transferred to the fluid:

$$\Delta\varphi=RI+UBL_j \Rightarrow P_{channel}=P_{joule}+P_{mhd} ; \quad P_{joule}=RI^2 ; \quad P_{mhd}=IBUL_j \quad (2)$$

Of course $P_{joule}$ is lost for propulsion or pumping, and maximizing the MHD efficiency $\eta_{mhd}=P_{mhd}/P_{channel}$ is a keypoint of any design. As for any electric receiver, $\eta_{mhd}=UB/(\Delta\varphi/L_j)$ increases when the applied voltage $\Delta\varphi$ decreases towards the counter-electromotive force $UBL_j$, but then the current vanishes and the thrust or pump head vanishes also.

As in any hydraulic pump, the mechanical power $P_{mhd}$ is partly dissipated by friction inside the pump, and the external circuit may only use the remaining part, i.e. the hydraulic power $P_{hyd}=Q\Delta p_{t,pump}$, convected out of the pump by the flowrate $Q=UL_B L_j$ flowing in the pump. Here $\Delta p_{t,pump}$ is the total pressure (or head) supplied by the pump (mechanical energy per unit volume of fluid). This head $\Delta p_{t,pump}$ corresponds to a pressure rise in a straight channel as the one of Fig 1, but it may also include a variation of the kinetic energy per unit volume, $1/2\rho U^2$, if the section evolves along the channel. The head loss $\Delta p_{t,loss}$ is expressed using a head loss coefficient $k_{loss}$ (considered constant here), and $P_{mhd}$ is expressed from Eq. 2, so that:

$$P_{mhd}=P_{loss}+P_{hyd} ; \quad P_{hyd}=Q\Delta p_{t,pump} \Rightarrow \Delta p_{t,pump}=(B/L_B)I-(k_{loss}/2)\rho U^2 \quad (3)$$

Notice that $k_{loss}$ is an important parameter, containing regular losses $f_{L_U}/D_h$ where $D_h=4L_B L_j/(2L_B+2L_j)$ is the hydraulic diameter and $f$ is the friction factor, but also minor losses due to entry effects or to geometric imperfections. The friction factor depends on the wall rugosity divided by $D_h$, and possibly on the velocity except at very high Reynolds numbers. The hydraulic efficiency $\eta_{hyd}=P_{hyd}/P_{mhd}$ is therefore difficult to predict.

Eq. 3 represents the characteristic of the pump for a given current $I$. A dimensionless equivalent can be built using reference values based on $B$, $\sigma$, $\rho$, $L_U$, i.e. on the fixed design parameters. The dimensionless velocity $U_a=\rho U/(\sigma B^2 L_U)$ is the inverse of a Stuart number, and is also a dimensionless flowrate: $U_a=\rho Q/(\sigma B^2 L_U L_B L_j)$. The dimensionless current density (and dimensionless current) is $I_a=\rho j/(\sigma^2 B^3 L_U)=\rho I/(\sigma^2 B^3 L_U^2 L_B)$. Eq. 3 becomes:

$$\Delta p_{t,a}=I_a-1/2k_{loss}U_a^2 \quad \text{where} \quad U_a=\frac{\rho U}{\sigma B^2 L_U}, \quad \Delta p_{t,a}=\frac{\rho \Delta p_{t,pump}}{\sigma^2 B^4 L_U^2}, \quad I_a=\frac{\rho I}{\sigma^2 B^3 L_U^2 L_B} \quad (4)$$

The pump efficiency $\eta_{pump}=\eta_{mhd}\eta_{hyd}=P_{hyd}/P_{channel}$ may also be expressed using $U_a$ and $I_a$:

$$\eta_{pump}=\frac{U_a \cdot I_a-(k_{loss}/2)U_a^2}{I_a+U_a} \quad (5)$$
<figure><img src="image_3.png" /><figcaption>Figure 2. Dimensionless characteristics for $k_{loss}=1$. Dashed lines for the pump experiment (part 3)</figcaption></figure>

Eq. 4 and Eq. 5 are plotted in Fig 2 for a loss coefficient $k_{loss}=1$ representative of the MHD pump and model boat studied experimentally in our team. A lower value of $k_{loss}$ would, for each value of $I_a$, enlarge the range of flowrate available before the head drops, and increase the maximum efficiency, that is reached just before this dropout.

### 3. Experimental helical channel pump.
An existing superconducting magnet was equipped with a tank in its 45 cm-diameter bore, inside which an helical channel between two cylindrical concentric electrodes constituted a MHD pump, connected to an external hydraulic circuit (Figure 3). The vertical magnetic field $B_{ctr}$ reached up to 4 T at the center point of the spiral channel and four experimental campaigns were performed at $B_{ctr}=3$; 3.5; and 4T.

The tank features anti-vortex grids to damp the tangential flow running out of the helical channel. Electrodes, made of titanium for its resistance to corrosion, are platinum-coated for electrochemical properties. A regulating valve could be installed in

the external hydraulic loop to change its hydraulic resistance. For each campaign, four configurations were tested: without vane (SV), with the vane fully open (VO), with the vane closed at 60% (V11) and with a varying position of the vane ("Vanne Variable"). For this last configuration, the power supply was used at a fixed voltage $\Delta\varphi_{el}=10$ V, whereas for SV, VO and V11, $\Delta\varphi_{el}$ varied from 4 V to 20 V.

The flowrate $Q$ was measured using a venturi installed in the external hydraulic loop and connected to a differential pressure sensor. The helical channel is equipped with six pressure taps drilled through the inner electrode, connected by flexible tubes through the bottom of the tank to absolute pressure sensors. Those taps are spaced half a turn apart along the helical channel, with tap number 1 located a quarter turn from the inlet and tap 6 a quarter turn from the outlet. All the pressure sensors are located 3.5 m away from the magnet center to protect them from magnetic field influence.

The current $I$ supplied to the electrodes was measured with a clamp ammeter, and two potentiostats were used to measured the anode and cathode potentials ($E_a>0$ and $E_c<0$) with respect to a reference silver wire just protruding from the electrode into the salt water.

Starting from the Ohm's law, it can be shown that the voltage $\Delta\varphi=\Delta\varphi_{el}-(E_a-E_c)$ applied to the ionic solution should contain a resistive term and a counter-electromotive term due to the displacement of the conductor in the magnetic field (see part 2 for a detailed development in a straight channel):

$$ \Delta\varphi=cQB+RI \quad \Rightarrow \quad \Delta\varphi_{el}=(E_a^0-E_c^0)+cQB+(R+R_{ch})I \quad (6) $$

Notice that the "electrochemical voltage loss" $E_a-E_c$ has a minimal value $E_a^0-E_c^0$ at low current, related to the thermodynamic potentials of the species involved in the electrolysis reactions, and it

<figure><img src="image_4.png" /><figcaption>Figure 3. Geometry of the MHD channel</figcaption></figure>
grows with the current density because of kinetic effects. Assuming, to the first order, that this increase occurs at a constant slope (i.e., $d(E_a-E_c)/dI=C^{st}=R_{ch}$), we obtain the second form in Eq.6.

To highlight this behavior in our experiments, we adjusted the constants ($E_a^0-E_c^0=2.57$ V, $R+R_{ch}=0.0407$ Ω, $c=31.6$ m$^{-1}$). Excluding several cases perturbed by bubbles attached to the silver wires, the potentiostat gave several values of $E_a-E_c$. Subtracting to $\Delta\phi_{el}$, we have $\Delta\phi=RI+cUB$ (not shown here) that can be correlated by $R=0.0227$ Ω.

From the pressure measurements at taps #1 and #6 (5 half-turns apart), we can reconstruct their difference $\Delta p_{61}$ which is a large part of the pressure head of the pump (the pump has a total of 6 half-turns). Due to the Lorentz force, this head includes a positive term proportional to the current and to an appropriate average of the magnetic field, itself proportional to $B_{ctr}$. Because of the head loss inside the helical channel, it also includes a negative term proportional to $Q^2$:

<figure><img src="image_5.png" /><figcaption>Figure 4. Current versus voltage (left) and MHD pressure head versus I.B (right) for all experiments</figcaption></figure>

We suppose here that the Reynolds number (of order $10^5$) is sufficiently high to have a constant hydraulic resistance $K_{61}$ whatever the flowrate [5]. Other studies [6] also used $a_{61}$, the pressure gain achieved for each unit of I·B. Fitting of the MHD experiments provided the following coefficients:

$$a_{61}=0.1457 \text{ mbar/A/T} ; K_{61}=0.0273 \text{ mbar/(m}^3\text{/h)}^2$$

The fit is illustrated in Figure 4, where we have plotted $\Delta p_{mhd}=\Delta p_{61}+K_{61}Q^2$ as a function of IB. Notice that the head loss $K_{61}Q^2$ is a minor correction ($\le 17$ mbar) to the pressure head, and that it has been validated at $B=0$ using an auxiliary pump to provide the flowrate.

4. CFD model of the pump. The numerical model was created with ANSYS™ and the MHD steady 3D turbulent flow was solved with Fluent™ and MHD User Defined Functions (UDF) [3].
<figure><img src="image_6.png" /><figcaption>**Figure 5.** MHD helicoidal canal (blue) inside the vessel (grey) with vortex-breaking grids (yellow). The inner wall of the channel (green) and the outer wall (orange) are concentric electrodes.</figcaption></figure>

In the CFD model (Figure 5), the hydraulic circuit is replaced by an inlet into the tank, where the flow rate measured in the experiments is imposed, and by a pressure condition at the tank outlet. The electrical potential applied to the electrodes is adapted to reach the current measured in the experimental setup. The top part of the tank is a zero wall shear stress because it models a fixed free surface. The superconductor magnet, located around the vessel and centred on the MHD channel, is modelled separately using our UDF module as a 2D axisymmetric electromagnetic model, then the B-field is interpolated on the 3D mesh. Regarding pressure in the MHD channel, six pressure values have been extracted at the same location as pressure sensors in the experimental setup.

The mesh is polyhedral because it usually leads to better and faster convergence [7]. It is a one million cells mesh including a prism layer of 6 cells on each wall, with a first layer height of 150 µm, connected with continuity of size to the polyhedral volume mesh. With such a prism layer, the first discretisation point is at a dimensionless distance $y^+ \approx 30$ of the wall in most parts of the MHD channel.

CFD simulations were conducted at three levels of magnetic field and 9 values of flow rate for each magnetic field (corresponding to the experiments in the SV configuration, see part 3). The calculated values of $\Delta\varphi$ and $\Delta p_{61}$ are compared to experiments in Figure 6. Using the total joule power dissipated in the whole domain, we also obtain the resistance of the fluid $R=P_{joule}/I^2$. Its calculated value $R=0.0248 \, \Omega$, is close to the experimental value $R=0.0227 \, \Omega$.

<figure><img src="image_7.png" /><figcaption>**Figure 6.** (Left) Voltage applied to the solution versus the current through the electrodes. (Centre) Pressure difference $\Delta p_{61}$ between pressure taps 6 and 1 versus the volumetric flow rate. (Right) Evolution of the same pressure difference according to Laplace force term $I \cdot B_{ctr}$.</figcaption></figure>

Regarding the pressure rise in the MHD channel, the CFD calculations are in fair agreement with the experimental data; the CFD model can predict accurately the hydraulic behaviour of the
electromagnetic pump. Moreover, the pressure rise in the MHD channel is approximately proportional to IB, i.e. to the Laplace-Lorentz force, because the head loss term is small compared to that pressure rise. The calculated powers and losses in the pump are compared in Table 1 to the experiments for Δφ<sub>el</sub>=9V and two different magnetic fields.

<table><caption>Table 1. Calculated and experimental characteristics of the electromagnetic pump</caption><tr><td>Flow rate Q [m³/h]</td><td colspan="2">17.75</td><td colspan="2">19.31</td></tr><tr><td>Reynolds number</td><td colspan="4">8×10⁴ ~ 9×10⁴</td></tr><tr><td>Hartmann number</td><td colspan="2">21.5</td><td colspan="2">28.4</td></tr><tr><td>Magnetic field B<sub>ct</sub> [T]</td><td colspan="2">3</td><td colspan="2">4</td></tr><tr><th>Powers & Losses [W]</th><th>Exp.</th><th>CFD</th><th>Exp.</th><th>CFD</th></tr><tr><td>P<sub>el</sub></td><td>1362</td><td>Not calculated</td><td>1231</td><td>Not calculated</td></tr><tr><td>P<sub>chim</sub></td><td>756</td><td>759 (+0.45%)</td><td>770</td><td>715 (-7%)</td></tr><tr><td>P<sub>channel</sub></td><td>606</td><td>606 (+0.07%)</td><td>461</td><td>516 (+12%)</td></tr><tr><td>P<sub>joule</sub></td><td>568</td><td>566 (-0.39%)</td><td>414</td><td>464 (+12%)</td></tr><tr><td>P<sub>mhd</sub></td><td>38</td><td>41 (+7%)</td><td>47</td><td>52 (+11%)</td></tr><tr><td>P<sub>loss</sub></td><td>6</td><td>12 (+107%)</td><td>7</td><td>15 (+108%)</td></tr><tr><td>P<sub>hyd</sub></td><td>32</td><td>29 (-11%)</td><td>40</td><td>37 (-6%)</td></tr></table>

The numerical MHD modelling of the electromagnetic pump with the Fluent™ CFD solver is promising. It was able to match fairly the experimental data both in terms of electrical and hydrodynamic behaviour. Some flaws are to be mentioned such as the electrochemistry and the external hydraulic circuit that are only obtained from experiment.

**5. Theory for MHD boats.** A MHD boat (sketched in Fig 7) is equipped with a propulsion channel comprising a MHD pump (see Fig 7), an inlet section, often converging, and an outlet section

that may be slightly diverging (with the risk of detachment and high head loss). The figure also shows the flow tube passing through the thruster, and its upstream section where the fluid velocity V<sub>∞</sub> is equal to the boat speed, and the relative modified pressure p<sup>*</sup><sub>∞</sub> is 0. The areas A<sub>i</sub> and

<figure><img src="image_8.png" /><figcaption>Figure 7. Propulsion channel of a MHD boat</figcaption></figure>

A<sub>o</sub> of the inlet and outlet sections are known from their ratios α<sub>i</sub>=A<sub>i</sub>/A<sub>mhd</sub> and α<sub>o</sub>=A<sub>o</sub>/A<sub>mhd</sub> to the crossflow section A<sub>mhd</sub>=L<sub>B</sub>L<sub>J</sub> of the MHD channel. We use here α<sub>o</sub>=1 because a divergent nozzle will produce detachments and reduce the boat speed (see experiment in part 6), and p<sup>*</sup><sub>o</sub>=0 since the jet is in equilibrium with the external water flow, which is similar to a wake flow in this region. Then a momentum balance on the propeller (section between A<sub>i</sub> and A<sub>o</sub>) gives the thrust:

$$T = \left[ \rho U^2 \left( 1 - \frac{1}{\alpha_i} \right) - p_i^* \alpha_i \right] A_{mhd} \quad (8)$$

The thrust in Eq. 8 includes the reaction to the MHD force applied to water, but also the pressure and friction forces applied on the internal walls of the thruster (bolded lines on the figure), generally downstream (drag) and balancing a large part of the MHD reaction force.

When the boat speed is stable (cruise conditions), the thrust equilibrates the drag of the boat (excluding the inner drag of the channel), expressed here using a constant drag coefficient C<sub>D</sub> and the projected frontal area S of the boat. We suppose that the inlet is adapted to the cruise velocity, i.e. p<sup>*</sup><sub>i</sub>=0. This is obtained only for α<sub>i</sub>=(U/V<sub>∞</sub>)<sub>cruise</sub>, but other values will decrease the thrust but only weakly. The thrust to drag equilibrium becomes:
$$ \rho U^2 A_{mhd} \left(1 - \frac{V_{\infty}}{U}\right) = S C_D \frac{\rho V_{\infty}^2}{2} \Rightarrow \left(\frac{U}{V_{\infty}}\right)^2 - \frac{U}{V_{\infty}} = \frac{\alpha_{SCD}}{2} \Rightarrow \frac{U}{V_{\infty}} = \frac{1}{2} \left(1 + \sqrt{1 + 2\alpha_{SCD}}\right) \quad (9) $$

where $\alpha_{SCD}=S.C_D/A_{mhd}$ is a dimensionless drag section, increasing with the drag coefficient and decreasing when the part of the front section occupied by the thruster increases.

The channel velocity is given as a function of the pump regime (i.e. of IB), by the generalized Bernoulli equation from $A_\infty$ to $A_o$, including the pump head (Eq. 3) and additional losses due to the entry section and exit nozzle ($k_{tot}$ represent the hydraulic loss between $A_i$ and $A_o$):

$$ \frac{\rho U^2}{2} = \frac{\rho V_{\infty}^2}{2} - k_{tot} \frac{\rho U^2}{2} + \frac{IB}{L_B} \quad (10) $$

Eliminating $U$ between Eq. 9 and Eq.10, we obtain the boat cruise speed $V_\infty$ as a function of the pump regime. Dividing it by $\sigma B^2 L_U / \rho$ and $I$ by $\sigma^2 B^3 L_U^2 L_B / \rho$ (as in part 2) gives the dimensionless form:

$$ \frac{\rho V_{\infty}^2}{2} = \frac{4 I B / L_B}{(1 + k_{tot}) (1 + \sqrt{1 + 2 \alpha_{SCD}})^2 - 4} \Rightarrow V_{\infty, a}^2 = \frac{2 I_a}{(1 + k_{tot}) \beta^2 - 1} \quad \text{where} \quad \beta = \frac{1}{2} (1 + \sqrt{1 + 2 \alpha_{SCD}}) \quad (11) $$

Here, $\beta$ is a shorthand for the group giving $U/V_\infty$ (Eq. 9). Using the Ohm's law in $P_{channel}=I\Delta\phi$, and Eq.8 for the thrust $T$, we can write them in dimensionless form using the volume $V_{mhd}=A_{mhd}.L_U$:

$$ T_a = V_{\infty,a}^2 (\beta^2 - \beta), \quad P_{ch,a} = I_a (I_a + \beta V_{\infty,a}) \quad \text{where} \quad T_a = \frac{\rho T}{\sigma^2 B^4 L_U V_{mhd}} \quad \text{and} \quad P_{ch,a} = \frac{\rho^2 P_{channel}}{\sigma^3 B^6 L_U^2 V_{mhd}} \quad (12) $$

The thruster efficiency $\eta=TV_\infty/P_{channel}$ is constructed from the useful power $D.V_\infty$ (with $D=T$):

$$ \eta = \frac{\eta_{max}}{1 + V_{\infty,a} / V_{a,ref}} \quad \text{with} \quad \eta_{max} = \frac{2(\beta - 1)}{(1 + k_{tot}) \beta^2 - 1} \quad \text{and} \quad V_{a,ref} = \frac{2\beta}{(1 + k_{tot}) \beta^2 - 1} \quad (13) $$

In any reaction thruster, the efficiency is limited by $\eta_{max}$, that takes into account the internal hydraulic loss ($k_{tot}$) and the kinetic energy lost in the jet (from $\beta=U/V_\infty$). A MHD thruster then divides this efficiency by $1+V_\infty/V_{ref}$, where $V_{ref}=(\sigma B^2 L_U / \rho)2\beta/[(1+k_{tot})\beta^2-1]$ represents the boat velocity above which the efficiency is divided by more than 2. Since this reference velocity increases with $B^2L_U$, we can know estimate the size and magnetic field necessary to have a correct efficiency (see part 6).

**6. Validation and upscaling.** The main parameters of the pump experiment (parts 3 and 4) and the model boat developed in our laboratory are exposed in Table 1:

<table><caption>**Table 1.** Parameters of the experiments</caption><thead><tr><th></th><th>L<sub>U</sub> / mm</th><th>L<sub>B</sub> / mm</th><th>L<sub>J</sub> / mm</th><th>σ / (Ω<sup>-1</sup>m<sup>-1</sup>)</th><th>ρ / (kg.m<sup>-3</sup>)</th><th>B / T</th><th>S / m<sup>2</sup></th></tr></thead><tbody><tr><td>Pump exp.</td><td>3200</td><td>39</td><td>80</td><td>19.5</td><td>1163</td><td>3.51;4.09;4.68</td><td>-</td></tr><tr><td>Model boat</td><td>100</td><td>28</td><td>32.5</td><td>5.26</td><td>1025</td><td>0.57</td><td>0.0163</td></tr></tbody></table>

The salinity of water was 35 g/l for the boat and 200 g/l for the pump (to increase $\sigma$). The values of $B$ in the table are computed averages (on the MHD channel) of the component along $L_B$, and the three values for the pump correspond to $B_{ctr}=3$ T;3.5 T;4 T at the center of the magnet. The front section of the boat takes into account the real immersion of the hull.

For the pump (that has a 3-turns helical channel in place of a straight one as in Fig 1), we measured experimentally the channel voltage $\Delta\phi$ and the pressure gain $\Delta p_{t_{61}}$ produced by 2.5 turns of the helix. All experiments (for 30 A<i>I</i><i>&lt;</i>300 A and 2.5 m<sup>3</sup>/h<i><i>&lt;</i></i>Q<i>&lt;</i>25 m<sup>3</sup>/h) were fit by $\Delta\phi=R_{exp}I+cQB_{ctr}$ with $R_{exp}=40.7$ mΩ and $c=31.62$ m<sup>-1</sup>; and by $\Delta p_{t_{61}}=a_{61}IB_{ctr}-K_{61}Q^2$ with $a_{61}=14.57$ m<sup>-1</sup> and $K_{61}=3.54\times10^7$ Pa.m<sup>-6</sup>.s<sup>2</sup>. To add the sixth half turn, we multiply the first term of $\Delta p_{t_{61}}$ by 1.195 (from our magnetic field calculation) and the second by 1.2 (=6/5). We also take into account that $B_{ctr}=B/1.17$ and $Q=UL_JL_B$. Our experimental laws become:

$$ \left\{ \begin{array}{l} \Delta\phi = R_{exp} I + d U B \\ \Delta p_{t \text{ pump}} = a I B - (k_{\text{loss}}/2) \rho U^2 \end{array} \right. \quad \text{with} \quad \left\{ \begin{array}{l} R_{\text{exp}} = 40.7 \, \text{m}\Omega \quad \text{and} \quad d = 84.3 \, \text{mm} \\ a = 14.88 \, \text{m}^{-1} \quad \text{and} \quad k_{\text{loss}} = 0.711 \end{array} \right. \quad (14) $$
Notice that $a$ and $R_{exp}$ differ from the values $1/L_B=25.6 \text{ m}^{-1}$ and $L_J/(\sigma L_U L_B)=32.9 \text{ m}\Omega$ that would be obtained for a straight channel (part 2), because of the different distribution of the current density in the helical channel. However $d$ is close to the value $L_J=80 \text{ mm}$ obtained in part 2. The loss coefficient also differs from the regular loss coefficient $k_{th}=1.3$ calculated for a straight pump channel with the same dimensions, at a flowrate of $20 \text{ m}^3\text{/h}$.

Switching to dimensionless forms and constructing the pump efficiency, we obtain:

$$ \Delta p_{ta} = a L_B I_a - \frac{k_{loss}}{2} U_a^2, \quad \eta_{pump} = \frac{U_a}{I_a} \frac{a L_B I_a - (k_{loss}/2) U_a^2}{(R_{exp} \sigma L_U L_B / L_J) I_a + (d/L_J) U_a} \quad \text{for} \quad \begin{cases} 0.19 < U_a < 3.4 \\ 2.2 < I_a < 53 \end{cases} \qquad (15) $$

Those laws are drawn in Fig 5 together with the theoretical laws for straight channels, showing a qualitative agreement and also that the efficiency could only be improved with a lower current and a lower velocity (at the expense of a lower head produced by the pump!). The numerical approach (part 4) is closer to experiments, since it uses a spiral channel and the real distributions of $B$ and $j$.

On our model boat (Fig 8) a transparent plate on the rear part initially produced a diverging nozzle ($\alpha_o=3$), and was then removed ($\alpha_o=1$). The boat speed, reached $0.2 \text{ m/s}$ with the rear plate and $0.3 \text{ m/s}$ without. A CFD study showed strong recirculations in the divergent when equipped with the plate, and smooth mixing with the outer flow when this plate was removed. Therefore, we adopt a straight outlet ($\alpha_o=1$) to describe our model without plate. The loss coefficient of the channel (without diverging nozzle) was estimated to $k_{tot}=1$ (among which $k_{loss}=0.8$ for the mhd channel). With the current $I=21 \text{ A}$ measured in the experiment, our model (Eq. 11) gives the experimental cruise speed of $0.3 \text{ m/s}$ for $C_D=0.27$, what seems reasonable for such a hull. Then $\beta=2.14$, showing that the inlet section is almost adapted to the cruise speed ($\alpha_i=3$ is not far from $U/V_\infty=2.14$), and $\eta_{max}=28 \%$, what could be increased with a better hydraulic design (decrease $k_{tot}$ and/or $C_D$), or with a larger channel (decrease $\alpha_{SCD}$). However, at $V_\infty=0.3 \text{ m/s}$, $\eta=0.0083\%$ is far below $\eta_{max}$ since $V_{ref}=9 \times 10^{-5} \text{ m/s}$, showing that the product $B^2L_U$ is much too low for such a model boat. Considering the pump part, we have $U_a=3.7 \times 10^3$, $I_a=1.2 \times 10^7$, $\Delta p_{ta}=6.8 \times 10^6$ (see Eq. 15), and $\eta_{pump}=0.016 \%$, i.e. parameters far out of the interesting zone plotted in Fig 2.

<figure><img src="image_9.png" /><figcaption>Figure 8. Model boat, bottom view</figcaption></figure>

Upscaling is favourable, but insufficient since $U_a$ is only proportional to $L^{-1}$: A boat with $L_U=10 \text{ m}$, if $B$, $\alpha_{SCD}$ and $k_{tot}$ are unchanged, would reach $V_\infty=0.3 \text{ m/s}$ at $I_a=1240$ and $U_a=37$, which is still too high to have a reasonable pump (and boat) efficiency. Increasing $B$ is also needed, i.e. we have to use superconducting magnets. For example with $B=10 \text{ T}$, $L_U=5 \text{ m}$, $\alpha_{SCD}$ and $k_{tot}$ unchanged, we have $V_{ref}=1.4 \text{ m/s}$ and we can reach $V_\infty=3 \text{ m/s}$ with a boat efficiency $\eta=8.8\%$ (and $\eta_{pump}=17\%$). Furthermore, the loss and drag coefficient decrease for large scales (our model boat is penalized by a very strong relative rugosity of its channel). Dividing those coefficient by two, i.e. for $B=10 \text{ T}$, $L_U=5 \text{ m}$, $k_{tot}=0.5$ and $\alpha_{SCD}=2.45$, we reach $V_\infty=3 \text{ m/s}$ for $I_a=2.13$, with a thruster efficiency $\eta=20\%$ (and $\eta_{pump}=31 \%$ if $k_{loss}=0.4$). With that superconducting thruster, reduced drag and internal losses ($C_D=0.135$, $k_{tot}=0.5$), the boat scaled up by a factor 50 needs $P_{channel}=400 \text{ kW}$ to reach $V_\infty=3 \text{ m/s}$.

**7. Conclusion.** Starting from a pump experiment and its CFD detailed description, a simple model is proposed here where the MHD channel is characterized by a loss coefficient, and the running point of the pump by two other dimensionless parameters ($I_a$, $U_a$). Such a pump (equipped with
permanent magnets) was installed on a small experimental model boat, described by two additional dimensionless parameters (additional losses and $\alpha_{SCD}$). The resulting model can be used for upscaling, with quick turnaround time, in the pre-design phase of MHD pumps and thrusters

**Acknowledgements** We thank LEGI's support teams, who built the pump and propeller, and the students who worked on the boat during their projects or internships.

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