Theoretical considerations on the dependence of naval magneto-hydrodynamic thruster's efficiency from seawater temperature and salinity.

Samoilescu, Gheorghe ; Grozeanu, Silvestru ; Constantinescu, Mircea 等

Abstract: The paper presents the dependence of the quality factor of a naval magneto hydrodynamic induction thruster from the penetration depth of electromagnetic field in seawater and the dependence of the penetration depth from salinity and water temperature. Also, there are presented a few theoretical considerations which explain the nature of this dependence.

Key words: magneto hydrodynamic propulsion, seawater, penetration depth

1. INTRODUCTION

In a short description, the magneto hydrodynamic (M.H.D.) naval thruster is made up of a canal, parallel with the ship's axis, with isolated walls in which it is found a pair of electrodes with the help of which it is injected an electric current in the sea water, perpendicular on the canal's axis.

Perpendicular in the canal, it is applied a magnetic field in the area where the electrodes are placed.

From the interaction of the current that travels in the sea water with this magnetic field it is generated a Laplace force which will evacuate the seawater through a nozzle.

The reactive force that appears pushes the ship forwards.

On the electrodes surface, which is injected by a current into sea water, will appear some electrochemical effects, which are not suitable for a good thruster working.

For that reason it will be tried to build-up an M.H.D. thruster based on the linear induction engine principle, where the moving armature is seawater.

Until now, it has not been built any M.H.D. thruster based on this principle, but many experimental and theoretical studies are conducted on this matter.

The efficiency of an M.H.D. induction thruster with seawater moving armature is very low because of seawater electrical conductivity which is very small.

Seawater electrical conductivity is approx. 107 times smaller than copper.

In linear induction engine theory, it is proved that a factor called by E.R.Laithwaite "quality factor", has a major role.

We can demonstrate that, if high frequency currents and very huge "polar steps" will be used, it is possible to gain a practical acceptable quality factor for an M.H.D. induction thruster. (Grozeanu&Dobref, 1998)

The authors demonstrate in another paper that the quality factor size depends on, the penetration depth of electromagnetic field" value in the seawater. (Grozeanu, 1997)

This penetration depth of electromagnetic field essentially depends on seawater electrical conductivity. Also seawater electrical conductivity depends on its temperature and salts and mineral concentrations from seawater.

Also, the penetration depth of electromagnetic field in a conductive environment has the next expression in case of a conductivity wichis big enough big:

[delta] [square root of 1/[pi]f[sigma][mu]] (1)

[sigma] is the electrical conductivity, f is the frequency and [mu] is the permeability of the environment.

Because the penetration depth of electromagnetic field in seawater for different domains is very important, we consider necessary to conduct a study on the penetration depth as a function of salinity and temperature of seawater.

2. RELATION BETWEEN QUALITY FACTOR AND PENETRATION DEPTH

The mathematic expression of quality factor of a linear induction machine, given by E.R. Laithwaite, is

Q = 2[[tau].sup.2.sub.p][mu]f[sigma]/[pi]d/[DELTA] (2)

[[tau].sub.p] represents the polar step of the machine, [DELTA] is the thickness of moving armature, d is the width of the air gap.

If [delta] represents the value of penetration depth and [lambda] = 2[[tau].sub.p], represents the wave length of the progressive magnetic wave in the air gap. Admitting that the entire air gap is filled with water, the expression of quality factor will be:

Q = 1/2[[pi].sup.2][([lambda]/[delta]).sup.2] (3)

In case of M.H.D. induction thruster width seawater as its moving armature, the velocity of seawater is variable. The maximum velocity is on the canal axis, and is 0 on the canal walls.

Using some approximations (Grozeanu, 2001) of small importance, the authors concluded that, in this case, the mathematic expression of the quality factor is:

Q = 1/2[[pi].sup.2][([lambda]/[delta]).sup.2] (1 - tanh Ha/Ha) (4)

where Ha is the Hartman number.

In case of magnetic fields of high intensity that are used on M.H.D. thruster, the Hartman number will be that big, so tanh Ha/Ha [[approximately equal to].sub.0] and the equation (3) can be used without major errors. Notice that, by increasing the penetration depth, the quality factor decrease, so thruster's efficiency is decreased.

A ship with M.H.D. propulsion can operate in different areas and at different depths.

In this case, temperature, salinity and dynamic viscosity suffer major variations. The modification of these parameters leads to the modification of seawater conductivity, and therefore to the modification of penetration depth.

3. THE DEPENDENCE OF ELECTROMAGNETIC FIELD PENETRATION DEPTH FROM SEAWATER TEMPERATURE AND SALINITY

If one plots the values of seawater electrical conductivity of seawater as a function of salinity S, for different temperatures using the experimental data, (Webster et al.2002) we get the diagram as in figure 1. If the values of penetration depth are plotted for a constant frequency as a function of salinity, at different temperatures, one obtain families of descending parabola, reversed proportionally with S which, at bigger concentrations, have the tendency to unite at high salinities as in figure 2.

Analyzing the graphics in figure 2, we notice that the dependence of temperature penetration depth is decreasing with the increase of salinity. The explanation we gave for this dependence is based on the relation between conductivity an electrolyte solution (like seawater), its dynamic viscosity c and salinity S. The link between these parameters is expressed with the relation of Waalden which has the mathematic expression. (Oniciu, 1986)

[sigma]/S [eta] = W = constant (5)

The dynamic viscosity of seawater depends on temperature and it is expressed with the equation of Poisseuille:

[eta] = [alpha]/1 + [beta][theta] + [gamma][[theta].sup.2] (6)

[theta] is the temperature in [degrees]C; and, [alpha] = 1,79 x [10.sup.-7] [Ns.sup.2]/m; [beta] = 0,037[([degrees]C).sup.-1]; [gamma] = 0,00022[([degrees]C).sup.-2]

Using the equations (5) and (6) to calculate the penetration depth, we obtain the next equation:

[delta] = [square root of [alpha]/[pi]f[[mu].sub.o]W] 1/[square root of 1 + [beta][theta] + [gamma][[theta].sup.2]] 1/[square root of S] (7)

Equation (7) expresse both the reversed dependence of the penetration depth of [square root of S], and the dependence from frequency and temperature.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

The variation of the penetration depth with the temperature, for a constant salinity and frequency, is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Equation (8) explains the decrease of the penetration depth when salinity is increasing:

If the known experimental data for the dependence of conductivity from temperature are used, the equation (8) is verified with remarkable precision.

If the dependence between dynamic viscosity and temperature is expressed with the equation Arhenius-Guzman (Grozeanu, 2003), the results are similar.

This confirms the correctness of the theoretical explanation given, even Waalden's rule has a small amount of approximation.

4. CONCLUSIONS

The efficiency of a naval M.H.D. induction thruster depend on its quality factor, and the quality factor depend reverse proportionally with the square of penetration depth. The penetration depth depends on the temperature and the salinity of seawater through its viscosity.

Viscosity strongly depends on temperature, influencing the mobility of hydrated ions and therefore, the conductivity. As the salinity increases, the dependence of conductivity from temperature decreases.

The dependence of the penetration depth on the experimental values of the sea water salinity may be represented by means of a family of descendent parables.

In point of parameters, the family of parables has got the sea water temperature and the frequency of the current. As much as the salinity increases, the dependence of the penetration depth on the sea water temperature decreases. This fact is explained by the authors through the dependence of conductivity on dynamics viscosity, expresses by means of Waalden's rule.

The rise of salinity and temperature increase the quality factor, and the decrease of these (for example, in polar water) will reduce the quality factor and the thruster's efficiency. Given the conditions of cold waters navigation, with low salinity, it is convenient that the thruster M.H.D. be made of a conductive M.H.D. thruster, assembled on the channel, upstream of the induction thruster.

The latter will used the water heated by the first and will have an increased efficiency.

5. REFERENCES

Oniciu, L., Chemistry, physics, electrochemistry, (1986) Didactics and Pedagogical Publishing House, Bucharest.

Grozeanu,S., (1997) On some energetic reports, in the induction liniary machine, Sielmec'97 Chisinau ISBN: 9975-910-22-X

Grozeanu,S.&Dobref,V.,(1998) Possibilites de mise en mouvement nonconvetionelle les equipements navales, 8-th, W.C.T.R., Antwerp, Belgium. ISBN : 0-08-044274-9

Grozeanu,S.,(2003) Theoretical considerations on the propulsive efficiency of the magneto-hydrodynamic naval propulsive.

The VII-th Session of Scientific Papers form the Field Forces Academy ISBN: 973-80088-85-2

Webster, F.,(2002)A.I.P. Physics Desk Reference.