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ENERGY ADDITION METHOD FOR THE ELECTROTHERMAL RAMJET
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Acta Astronautica Vol. 47, No. 11, pp. 789–797, 2000 ? 2001 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0094-5765(00)00130-2 0094-5765/01/$ - see front matter
ENERGY ADDITION METHOD FOR THE ELECTROTHERMAL RAMJET B. D. SHAW† MAE Department, University of California, Davis, CA 95616, USA (Received 18 January 2000)
Abstract—The electrothermal ramjet is a ramjet-in-tube device that has been proposed for accelerating masses to high speeds for applications such as direct space launch and hypersonic projectile studies. The electrothermal ramjet is similar to the ram accelerator, though one di erence is that heat is to be added to the propellant electrically rather than via combustion. This article presents a method for adding heat to the propellant in the electrothermal ramjet con guration via magnetically balanced traveling arc discharges. Calculations indicate that current and magnetic eld requirements for this method are within attainable ranges and that electrical heat addition should provide advantages over combustion heat release under certain conditions. ? 2001 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
Ramjet-type accelerators are under consideration as a method of accelerating masses to high speeds (up to about 104 m=s) for applications such as direct space launch and hypersonic projectile studies. With such accelerators, a stationary tube would be lled with gaseous propellant. A projectile would enter one end of the tube at a high (supersonic) speed and would travel down the tube. In most concepts, the projectile would e ectively act as the centerbody of a conventional ramjet, with the tube wall acting as the outer cowling. Wilbur et al. published studies on a particular class of these devices which they termed the “electrothermal ramjet” [1,2]. In their work, which was theoretical, they considered a gaseous hydrogen propellant and postulated that heat could be added to the propellant using electrical discharges or perhaps beamed electromagnetic radiation. The details of heat addition were not considered—only the thermodynamics and uid mechanics of the device were evaluated. Their work showed that ramjet-type accelerators appear feasible for accelerating masses to high speeds (¿15; 000 m=s) with accelerations of 300; 000 m=s2 or more, as long as sucient heat is added to the propellant. A geometry considered by Wilbur et al. is shown in Fig. 1, where the instantaneous frame of reference is moving with the projectile; for clarity, shock waves are † Tel.: +1-916-752-4130; fax: +1-916-752-4158. E-mail address: [email protected] (B. D. Shaw). 789
not shown in this gure. The ows entering the diffuser and exiting the nozzle are supersonic, while heat addition occurs subsonically (note that in Fig. 1 and throughout this paper, “M ” represents Mach number). The Mach number is unity at the nozzle throat. If sucient heat is added to the propellant, the projectile will accelerate down the tube. Heat addition is required to overcome shock losses, viscous losses and heat transfer losses and to increase the kinetic energy of the projectile. With devices of this type, viscous losses and heat transfer losses are typically small relative to shock losses. Hertzberg et al. developed the concept of ramjet-in-tube accelerators powered by chemical reactions (combustion) in premixed gases [3]; they termed these devices “ram accelerators” (in this paper, the term “ram accelerator” will refer to ramjet-in-tube accelerators powered only by chemical reactions, i.e., combustion, and the term “electrothermal ramjet” will refer to ramjet-in-tube accelerators powered by external electrical sources). In the ram accelerator con guration, the launch tube is lled with a combustible gas mixture. As the projectile travels down the tube, ow deceleration in the di user region raises the static temperature of the propellant, inducing ignition and allowing combustion heat release; ideally, the heat release occurs downstream of the di user. If suf cient heat is released per unit mass of propellant consumed and the heat is released in the correct location, the projectile is accelerated down the tube. Various geometries and operational modes have been considered for the ram accelerator. At lower
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Fig. 1. Schematic of an electrothermal ramjet con guration.
Fig. 2. Schematic of a thermally choked ram accelerator con guration.
speeds, it has been suggested that a particularly applicable mode of operation is the “thermally choked” mode [3]. In this mode, the ow entering the di user is supersonic and the ow after the diffuser is subsonic. Heat release occurs subsonically behind the projectile such that the ow becomes thermally choked downstream of the projectile. A schematic of a ram accelerator operating in the thermally choked mode is shown in Fig. 2, where the instantaneous frame of reference is moving with the projectile (for clarity, shock waves are not shown in Fig. 2). Another subsonic combustion mode studied by Hertzberg et al. utilized the basic projectile geometry shown in Fig. 1. Methods of powering ram accelerators with detonation waves have also been considered [3]. A factor that has been postulated to limit ram accelerator operation is structural failure of the projectiles. This failure has been associated with chemical reactions between the projectile and the propellant gas [4,5]. In addition, there are sometimes problems with unstarts, where a shock wave propagates ahead of the projectile. Some unstarts have been linked to projectile structural failure while others have been associated with having mixtures that are too energetic [4,5]. Unfortunately, there is no control of a projectile once it has been injected into a ram accelerator launch tube. Control of the projectile is desirable in case an unfavorable operating regime is encountered, e.g., a regime where unstarts are promoted. It is also desirable to control the launch process so that projectile accelera-
tion pro les can be tailored and ecient operation over the launch cycle can be attained. In the work presented here, an electrical energy addition scheme is presented for an electrothermal ramjet operating in the thermally choked mode. This energy addition scheme involves adding heat to the propellant via magnetically balanced traveling arc discharges that are stabilized in a location behind the projectile via externally applied magnetic elds. As described later, this scheme should provide advantages over combustion heat release under certain conditions. For example, this method should allow use of monatomic gases that will not react chemically with the projectiles. In addition, using electrical heat addition should allow heat release locations and pro les to be tailored, allowing control of projectile acceleration, attainment of high eciencies over the launch cycle and minimized launch tube lengths. The thermally choked mode is selected for study here because of the ease with which it may be modeled with reasonable accuracy and also because thermally choked operation of the ram accelerator has been experimentally demonstrated. It is expected that the basic concepts described here for energy addition via magnetically balanced arcs may be used in other con gurations, e.g., the geometry shown in Fig. 1 or where supersonic heat addition is employed. In the following, a methodology of adding heat via magnetically balanced traveling arcs is described. Following this, one-dimensional models of ramjet ow elds are developed, calculations for accelerator pro les are presented, and conclusions are then drawn. 2. ENERGY ADDITION VIA MAGNETICALLY BALANCED TRAVELING ARCS
A basic requirement of successful operation of a ramjet-in-tube device is that energy release occurs at proper locations relative to the projectile. For thermally choked operation, the energy release needs to occur such that choking occurs downstream of the projectile. It is proposed that traveling arc discharges can be used for this purpose with the electrothermal ramjet. Schematic drawings illustrating relevant concepts are shown in Figs. 3a and b (for clarity, shock waves are not shown in these gures). Figure 3a is a side view, while Fig. 3b is a front view with the projectile velocity out of the paper. In these gures, a two-dimensional projectile con guration with a wedge-type di user has been selected. This con guration appears to be particularly amenable to operation with traveling arc discharges because it allows easy introduc-
Electrothermal ramjet
Fig. 3. Schematic of energy addition method using magnetically stabilized thermionic arcs. Schematic (a) is a side view while (b) is a front view.
tion of nearly uniform magnetic elds, which are needed for control of the location of the heat release zone, as described below. The basic concepts described here could also be utilized with other projectile geometries, e.g., projectiles with circular cross sections. It is to be noted that operation of two-dimensional ram accelerators in the thermally choked mode has been demonstrated [6]. As Fig. 3a indicates, the projectile can be located between guide rails to ensure proper placement within the launch tube, as has been done in ram accelerator experiments (e.g., [6,7]). Cathode and anode rails would be placed on the top and bottom of the launch tube, and one or more arcs would exist between these rails. The arc(s) could initially be formed behind the projectile using, e.g., a high-voltage breakdown spark discharge with UV laser pre-ionization of the spark path. A magnetic ux density vector B (vectors are denoted using boldface) would be imposed perpendicular to the projectile velocity vector (note that the magnitude of the component of B normal to the wall is continuous across the wall and on either side of the wall [8]). This magnetic ux density produces the Lorentz force F = (IxB)L on each arc channel, where I is the current vector (for each arc), L the arc length, and x represents the vector
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cross product. The magnetic eld causes the arc to travel along the launch tube behind the projectile. In order to promote stable arc operation, the cathode rail could be heated to allow thermionic emission. Using thermionic emission to stabilize magnetically controlled arcs propagating between rails was demonstrated by Bond and Potillo [9]. They showed experimentally that highly-stable arc operation was possible at subsonic speeds as well as at supersonic speeds (where the arc speed is measured relative to the gas through which it moves) provided that the cathode rail was heated to allow thermionic emission. Bond and Potillo also showed that heating of the anode rail promoted arc stability. When thermionic emission was not used, the arcs could still exist but could become unstable and experience spatial uctuations. Based on these results, it is expected that heating of the cathode rail would be used here as well. Such heating could be accomplished, for example, by passing current through the cathode rail prior to injection of the projectile or by using an electrical resistance heater in contact with this rail. Experiments have shown that magnetically stabilized thermionic arcs of the type suggested for use here will be slanted at an angle with respect to the oncoming gas stream [9,10], as depicted schematically in Fig. 3a. The slant angle relative to the tube wall is very close to the Mach angle when the arc moves at supersonic speed relative to the oncoming gas. As the arc speed relative to the oncoming gas decreases, the angle decreases. The drag characteristics of these-type arcs have been found to be similar to those for solid cylinders of the same geometric con guration [10]. The drag coecient CD for a magnetically stabilized arc placed in a cross ow has been shown to be fairly well correlated by a relation of the form CD =
IB ; (1=2)v2 d
(1)
where CD is close to drag coecient values for solid cylinders [10] (CD is typically close to unity). In eqn. (1), I is the arc current, B the applied magnetic ux density magnitude, the density of the oncoming gas ow, v the ow velocity component normal to the arc, and d a characteristic dimension for the arc width normal to the ow. Equation (1) will be used later in this paper to calculate B values required to maintain an arc in position. The rate of energy dissipation E˙ in an arc is given by E˙ = I 2 R where R is the arc electrical resistance. For this analysis we will calculate R using R = L=(D2 =4), where is the arc electrical conductivity and D a characteristic arc diameter.
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Combining these relations yields the following equation: E˙ =
I 2L : (=4)D2
(2)
The rate Q˙ that heat is supplied to the propellant is given by the relation ˙ arc : Q˙ = E
(3)
In eqn. (3), arc is the fraction of heat generated in the arc that is transferred to the propellant gas ow, where it is recognized that the arc may lose heat, e.g., via radiation, which will not be absorbed by the propellant. Using eqn. (3) in eqn. (2) yields, after rearrangement, the following equation for the arc current I . 2 1=2 ˙ QD : (4) I= 4arc L To employ these relations it is necessary that the
ow eld conditions be speci ed. A model to predict ramjet ow conditions is developed in the next section. 3. THEORETICAL MODEL FOR THERMALLY CHOKED ELECTROTHERMAL RAMJET OPERATION
The ramjet con guration studied here is di erent from the thermally choked ram accelerator in that an external force (from a magnetic eld) is needed to hold the arc in place behind the projectile. This force is accounted for in the development of the electrothermal ramjet gas dynamic model, as described below. In the model, the gas ow is assumed to be one-dimensional, quasisteady and frictionless (though drag forces on the arc are allowed), and the propellant is treated as an ideal gas. These assumptions have been shown to lead to reasonably good results for thermally choked ram accelerator operation powered by combustion [3–5,11], and it is expected that they will work well for electrothermal heat addition. For the con guration shown in Fig. 3a, the equations for conservation of mass, momentum and energy take the following forms: 1 V1 = 3 V3 ; fp fa 1 V12 3 V32 + p1 + = + p3 + ; A 2 A 2
(5) (6)
V12 V2 (7) + q = h3 + 3 : 2 2 In eqns (5) – (7), fp is the thrust force on the projectile, fa the external (Lorentz) force required to hold the arc in place behind the projectile, p static h1 +
pressure, density, A the local tube cross-sectional area, V velocity, h enthalpy, q the amount of heat added per unit mass of propellant, and numerical subscripts denote locations in Fig. 3a. The reference frame is moving with the projectile. In the analyses presented here, monatomic gases (e.g., argon) will be considered as propellant candidates. Estimates indicate that over the pressure and temperature ranges expected at locations 1–3 in Fig. 3a, that monatomic gases of interest can reasonably be considered ideal with constant speci c heats. Using these assumptions, eqns. (5) – (7) can be combined to yield the relation 1=2
−1 2 Fp = Fa + M1 2(1 + ) 1 + M1 + Q 2 −(1 + M12 );
(8)
where Fp = fp =(p1 A) is a dimensionless thrust, Fa =fa =(p1 A) the dimensionless external force acting on the traveling arc, Q = q=(Cp T1 ) a dimensionless heat release parameter, Cp is speci c heat at constant pressure, T1 the static temperature upstream of the di user and is the speci c heat ratio. In developing eqn. (8), the thermal choking relation M3 = 1 has been used. Equation (8) can be rearranged to yield the following relation: (1 + M12 + Fp − Fa )2
−1 2 M1 : − 1+ Q= 2 2(1 + )M12 (9) Equation (9) allows dimensionless heat release values to be calculated for prescribed values of
; M1 ; Fp and Fa . As noted previously, drag characteristics of magnetically balanced arcs have been found to be similar to those for solid cylinders of the same geometric con guration. The force fa that acts on an arc to hold it in place is given by an equation of the form fa = 12 2 V22 CD Aa ;
(10)
where CD is the arc drag coecient and Aa the frontal area of the arc. Using eqns. (9) and (10), the dimensionless force acting on the arc is given by the following equation: Fa = 12 M22 CD
p2 Aa ; p1 A
(11)
where p2 =p1 is the static pressure ratio between locations 1 and 2. To evaluate eqn. (11), values of the Mach number M2 , i.e., the Mach number after the projectile but prior to the heat addition zone, are needed as are values for p2 =p1 . To obtain these M2 values, the equations for conservation of mass, momentum
Electrothermal ramjet
and energy can be written for the ow entering at location 1 and exiting at location 2 in Fig. 3a, as shown below. 1 V1 = 2 V2 ;
(12)
fp 1 V12 2 V22 + p1 + = p2 + ; A 2 2
(13)
V12 V2 = h2 + 2 : (14) 2 2 These relations can be combined to yield the following equation: 1=2 1 + M12 + Fp M1 1 + [( − 1)=2]M12 = : M2 1 + [( − 1)=2]M22 1 + M22 h1 +
This equation can be solved for M2 , i.e., 1=2 2 (b + 4a )1=2 − b ; M2 = 2a
(15)
where a=
−1 − 2 ; 2
M12 (1 + [( − 1)=2]M12 ) : (1 + M12 + Fp )2
It can also be shown that 1 + [( − 1)=2]M12 + Fp p2 = : p1 1 + [( − 1)=2]M22
For estimates of the order-of-magnitude of Fa we will use the characteristic values = 5=3 (for monatomic gases), M2 = 0:3 (which is a characteristic value for subsonic ow at location 2), CD = 1 [9], Fp = 5; M1 = 4 and Aa =A = 0:125; which is estimated to be applicable here. Using these values in eqns (11) and (16) yields Fa ≈ 1. This value is large enough that it provides a non-negligible contribution to the projectile thrust force, as shown by eqn. (8). The ballistic eciency is the fraction of energy deposition that increases the kinetic energy of the projectile. The ballistic eciency B is de ned by eqn. (17). B =
(16)
These equations show that M2 and p2 =p1 depend on M1 ; and Fp . It is interesting to note that Fp cannot be varied arbitrarily. For example, if Fp is prescribed to be too large, the speci c entropy change across the projectile, s2 − s1 , becomes negative (where s is speci c entropy), violating the second law of thermodynamics. For this analysis, a simpli ed model of ow around the projectile similar to what has been used for modeling thermally choked ram accelerators will be employed [4,5]. In this model, Fp is constrained such that the speci c entropy change s2 − s1 lies within the following upper and lower limits. The lower limit corresponds to having the inlet ow compress isentropically in the di user region with a normal shock existing at the minimum di user ow area. The ow after this normal shock expands isentropically and subsonically to location 2. The upper limit assumes that the ow passes by the projectile isentropically, with a normal shock existing downstream of the projectile. The lower limit produces the maximum attainable dimensionless thrust force Fp , while the upper limit produces Fp = 0.
fp V : Q˙
(17)
It is noted that conditions where Fp = B = 0 de ne the top operational speed (Mach number) of the projectile for a given heat addition rate. For the case of an ideal gas with constant speci c heats, eqn. (17) can be expressed as B =
b = 1 − 2 ; =
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Fp − 1 : Q
(18)
The value of Q that maximizes the ballistic eciency of the ram accelerator, denoted here as Qopt , can be found by setting the derivative d(B )=dQ equal to zero (where Fp is given by eqn. (8) with Fa =0) and solving for Qopt . When this is done, it is found that for supersonic inlet ows, Qopt =M12 −1. The maximum possible value of B for the ram accelerator, i.e., B; max , can be calculated by setting Q = Qopt , yielding B; max = ( − 1)= . The equation to predict M1 values that optimize the ballistic eciency for the electrothermal ramjet, i.e., for the case where Fa ¿ 0, is complex and is not listed here. From eqn. (8), however, it is obvious that higher values of B are possible when Fa ¿ 0 because of the contribution of the arc force to the overall momentum balance. For = 1:4 (which applies to typical diatomic gases), B; max =0:29 while for = 1:67 (which applies to monatomic gases), B; max = 0:40. There is thus an advantage, in terms of maximizing ballistic eciency, to working with monatomic gases with the ram accelerator. This advantage also applies to the electrothermal ramjet con guration considered here. 4. CALCULATIONS OF RAMJET OPERATING CONDITIONS
During operation in the thermally choked mode, it is essential that the inlet Mach number (i.e., M1 ) be suciently large so that the di user is “started”. For comparison with results from typical ram
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Fig. 4. Plots of Fp versus M1 for cases 1–3.
Fig. 5. Plots of B versus M1 for cases 1–3.
accelerator operation [3–5], an inlet Mach number range of 3– 6 is selected for study here. For the analysis we will consider three di erent operating conditions, which are denoted in the following as cases 1–3. Case 1 is for when Q is held constant over the launch cycle, which also corresponds to operation of a thermally choked ram accelerator operating with the same propellant mixture everywhere along the launch tube. Case 2 corresponds to varying Q so that a constant thrust force Fp is achieved on a projectile, and case 3 is where Q is varied so that ballistic eciencies are held constant. The initial conditions for each case are the same in that the Fp value at M1 = 3 is selected so that the ballistic eciency has the value B = 0:4 at M1 = 3. In making these calculations, the ratio Aa =A = 0:125 was used. Data are shown in Figs. 4 – 6 for Fp ; B , and Q versus M1 for cases 1–3. Figures 4 and 5 show that when Q is held constant (case 1) the projectile thrust force and the ballistic eciency drop rapidly as the projectile Mach number is increased. For
Fig. 6. Plots of Q versus M1 for cases 1–3.
case 1, the maximum achievable inlet Mach number is M1 = 5:7 (i.e., Fp = B = 0 at this Mach number), and as a result data are not shown for case 1 for M1 ¿ 5:7. This trend is expected, as this is typical behavior for combustion-powered ram accelerator operation. When Q is allowed to vary with M1 , however, markedly increased performance can be achieved. Varying Q such that Fp is held constant yields higher values of ballistic eciency (case 2 in Fig. 5), and selecting Q so that B is optimized over the launch cycle yields an increasing dimensionless projectile thrust as M1 increases (see case 3 in Fig. 4). Dimensionless heat release pro les are shown in Fig. 6, where it is apparent that Q values do not have to vary by large amounts in order for these di erent cases to be realized. Knowing the Q versus M1 pro les allows estimates to be made of requirements for arc current and magnetic ux magnitudes for magnetically balanced arcs. For predicting arc currents, eqn. (4) can be expressed in the following manner: 1=2 p1 Ac1 D2 (M1 Q)1=2 : (19) I= arc − 1 4L 1 =W )1=2 is the speed of In eqn (19), c1 = ( RT sound in the propellant gas upstream of the projectile where R is the universal gas constant and W is the propellant molecular weight. Equation (19) shows explicitly how the arc current depends on the inlet mach number M1 , the propellant molecular weight W , and the heat release parameter Q, i.e., with all other parameters held constant, I ∼ (M1 Q)1=2 =W 1=4 . The magnetic ux density magnitude B required to hold an arc in place behind an accelerating projectile is found by rearranging eqn. (1) and expressing appropriate variables in terms of the conditions at location 2 in Fig. 3a. This procedure
Electrothermal ramjet
Fig. 7. Plots of arc current I versus M1 for cases 1–3. The propellant is argon initially at 300 K and 9.19 bar.
yields eqn. (20). 1 2 V22 dCD : (20) 2 I In developing this equation, it has been assumed that the arc exists downstream of any recirculation zones behind the projectile and that v = V2 . Equations (19) and (20) will be used to provide estimates for I and B for a particular con guration of projectile mass and acceleration that is similar to ram accelerator conditions sometimes investigated [3–5]. We will consider the acceleration of a projectile with mass m=0:075 kg from an initial velocity of 970 m=s to a nal velocity of 1940 m=s. The projectile acceleration a is speci ed to be 105 m=s2 at the initial Mach number M1 = 3. Argon is selected as the propellant, which yields the initial and nal inlet Mach numbers M1 = 3 and 6. We will use the initial dimensionless thrust value Fp = 5:1 which provides B = 0:4 at M1 = 3. The launch tube cross section is square with a cross-sectional area A = 0:0016 m2 . The thrust force acting on the projectile is fp = ma = 7500 N, and the initial tube pressure is then p1 = fp =(Fp A) = 919 kPa. For the purpose of making estimates we will use the representative values L = A1=2 = 0:04 m, arc = 0:5 [12], d=D=0:005 m [13,14], =3000 1= m (which is a representative value for high-pressure argon arcs [14]), and CD = 1 [10,13]. Using these relations in eqns. (19) and (20), as well as data from Figs. 4 – 6 yields results for the arc current I and applied magnetic ux density B versus the inlet Mach number M1 ; these results are plotted in Figs. 7 and 8, respectively. Figure 7 shows that for cases 2 and 3, arc currents are in the range 7–22 kA while Fig. 8 shows that magnetic ux densities are in the range 1–2.4 T for these same cases. The electrical energy that is required to achieve an acceleration pro le can be calculated by noting that the in nitesimal amount of energy dE B=
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Fig. 8. Plots of applied magnetic ux density magnitude B versus M1 for cases 1–3. The propellant is argon initially at 300 K and 9.19 bar.
delivered over the time increment dt is given by dE = E˙ dt. This equation can be expressed as ˙ dV = (Ec ˙ 1 =a) dM1 , E˙ dt = E˙ dV=(dV=dt) = (E=a) which gives the amount of electrical energy delivered over the inlet Mach number increment dM1 . Using previous relations, it can be shown that ˙ 1 =a = mc12 M1 =(B arc ). The amount of electrical Ec energy E required to accelerate the projectile from the tube inlet Mach number M1; in to the Mach number M1 is then given by the following integral: Z mc12 M1 M1 dM1 : (21) E= arc M1;in B Using similar arguments, it can be shown that the time increment t and tube length x required to accelerate a projectile from M1; in to M1 are given by the following integrals: Z mc1 M1 dM1 ; (22) t= P1 A M1;in Fp x=
mc12 P1 A
Z
M1
M1;in
M1 dM1 : Fp
(23)
Equations (21) – (23) are plotted in Figs. 9 –11, respectively. These plots were generated by numerical integration using results contained in Figs. 4 – 6. Figure 9 shows that the energy requirements for case 1 are signi cantly larger than for cases 2 and 3, with cases 2 and 3 not being too di erent. For acceleration to M1 = 6 it is found that E = 718 kJ for case 2 and E = 529 kJ for case 3 (recall that the projectile cannot be accelerated to M1; max = 6 for case 1). Figures 10 and 11 show that the acceleration times and launch tube lengths are substantially smaller for cases 2 and 3 than for case 1. This is to be expected, as the ballistic eciency for case
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Fig. 9. Plots of the electrical energy E required to accelerate a projectile from M1; in = 3 to M1 for cases 1–3.
Fig. 10. Plots of the time t required to accelerate a projectile from M1; in = 3 to M1 for cases 1–3.
Fig. 12. Electrical power addition pro les required to accelerate a projectile from M1; in =3 to M1 for cases 1–3.
values are lower for case 1, which is to be expected because of the lower currents. The current, energy and magnetic ux density values estimated above are in ranges that are attainable with presently available equipment, e.g., equipment that has been developed for railgun applications. For example, Ref. [15] discusses operation of 60 MJ railgun energy supplies that can deliver electrical energy at a rate of 900 MW and Ref. [16] discusses switching devices and 60 MJ railgun energy supplies comprised of 2 MJ capacitor banks. In addition, B eld magnitudes up to 10 T or higher are easily attainable with electromagnets. Because the magnitude of the component of B normal to the wall is continuous across the wall and on either side of the wall [8], introduction of B elds can be readily accomplished by using planar metal walls, as shown in Fig. 3a. If the tube was made of a material such as Kevlar, a circular tube cross section could instead be used. 5. CONCLUSIONS
Fig. 11. Plots of the launch tube length x required to accelerate a projectile from M1; in =3 to M1 for cases 1–3.
1 goes to zero while for cases 2 and 3 it does not approach zero. ˙ can also be calArc energy deposition rates (E) culated using the following relation E˙ = I 2 R = I 2 L=(D2 =4)
(24)
where currents are to be evaluated using results from Fig. 9. As shown in Fig. 12, calculations yield the result that for cases 2 and 3, arc energy dissipation rates are in the ranges 33–130 MW and 33–330 MW, respectively, for 3 ¡ M1 ¡ 6. The E˙
This article has presented the magnetically balanced traveling arc as a method of adding heat to a propellant for operation of the electrothermal ramjet operating in the thermally choked mode. Calculations indicate that for acceleration of a 0.075 kg projectile from 970 to 1940 m=s, the required current, energy and magnetic ux density values are in ranges that are attainable with presently available equipment. In addition, this method should provide certain advantages over combustion. For example, the traveling arc method should allow control of projectile acceleration and achievement of high eciencies during the acceleration history. It is stressed, however, that the calculations presented invoked simplifying assumptions that allowed overall trends and approximate values to be calculated. Based on the results in this paper, it is worthwhile to perform more detailed studies to further assess the viability of this system. For
Electrothermal ramjet
example, more accurate experimental data are needed on arc diameter values that should be used for the ow conditions that would exist in the launch tube, and multidimensional CFD calculations could be used to predict detailed ramjet
ow elds and dynamic arc behaviors. It is noted that this energy addition method could potentially be used to accelerate projectiles to higher velocities, e.g., by adding more launch tube stages. For example, a projectile could exit an argon- lled launch tube and enter a neon- lled launch tube. Because neon has a lower atomic weight than argon, the projectile Mach number would be reduced and the projectile could again be accelerated up to a maximum Mach number (e.g., M1 = 6, which yields an exit velocity of about 2900 m=s). This process could be repeated by then utilizing a helium- lled launch tube, yielding an exit velocity of about 6100 m=s for M1 = 6. To achieve higher velocities, H2 could be used as a propellant, or higher terminal Mach numbers could simply be tolerated. The highest Mach numbers that a projectile can withstand will depend on the projectile materials and the residence time of the projectile in the launch tube. It should be mentioned that the arc thermal conductivity could be tailored somewhat by adding small amounts of easily ionized materials such as cesium. This might be necessary with a propellant such as helium, which has high ionization energies. It is also worth mentioning that in addition to varying magnetic ux densities (B) temporally, it would also be useful to vary B values spatially along the launch tube. This would be accomplished so that at any instant of time, the B eld would increase in magnitude with increasing distance downstream of the projectile. The arc could then be held at a desired location behind the projectile where the drag force acting on the arc matched the force the B eld exerted on the arc. Movement of the arc downstream of this position would increase the force of the B eld, which would move the arc closer to the projectile. If the arc was too close to the projectile, the weaker B eld in the projectile vicinity would be insucient to hold the arc against the ow exiting the projectile, and the arc would be driven back towards the desired equilibrium position relative to the projectile. The location of the heat release zone relative to the projectile could be controlled in this manner. This is desirable, for example, so that the projectile would not be subjected to excessive heating from the hot gases in the heat addition zone. Acknowledgements—The support of the California Space Institute is gratefully acknowledged. In addition,
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the e orts of students in B. D. Shaw’s research group (in particular Dr. I. Aharon) are gratefully acknowledged. REFERENCES
1. Wilbur, P. J., Mitchell, C. E. and Shaw, B. D., The electrothermal ramjet. Journal of Spacecraft and Rockets, 1983, 20, 603– 610. 2. Shaw, B. D., Mitchell, C. E. and Wilbur, P. J., The annular ow, electrothermal plug ramjet. Journal of Propulsion and Power, 1985, 1, 417– 424. 3. Hertzberg, A., Bruckner, A. P. and Bogdano , D.W., Ram accelerator: a new chemical method for accelerating projectiles to ultrahigh velocities. AIAA Journal, 1988, 26, 195 –203. 4. Higgins, A. J., Knowlen, C. and Brucker, A. P., Ram accelerator operating limits, Part 1: identi cation of limits. Journal of Propulsion and Power, 1998, 14, 951–958. 5. Higgins, A. J. Knowlen, C. and Brucker, A. P., Ram accelerator operating limits, Part 2: nature of observed limits. Journal of Propulsion and Power, 1998, 14, 959 –966. 6. Chang, X., Shimomura, K. and Taki, S., Preliminary tests of a rectangular bore ram accelerator. Proceedings of RAMAC II—The Second International Workshop on Ram Accelerators, University of Washington, July 17–20, 1995. 7. Seiler, F., Patz, G., Smeets, G. and Srulijes, J., The rail tube in ram acceleration: feasibility study with ISL’S RAMAC 30. Proceedings of RAMAC II—The Second International Workshop on Ram Accelerators, University of Washington, July 17– 20, 1995. 8. Kraus, J. D. and Fleisch, D. A., Electromagnetics With Applications, 5th edn. Mc-Graw Hill, New York, 1999, p. 412. 9. Bond, C. E. and Potillo, R. W., Stability and slanting of the electric arc in a thermionic rail accelerator. AIAA Journal, 1968, 6, 1565 –1567. 10. Bond, C. E., and Wickersheim, D. N., Convective electric arcs at Mach numbers up to 6.5. AIAA Journal, 1970, 8, 1748–1751. 11. Bauer, P., Knowlen, C. and Bruckner, A., Real gas e ects on the prediction of ram accelerator performance. Shock Waves, 1998, 8, 113–118. 12. Edels, H., Properties of the high pressure ultra high current arc, Eleventh International Conference on Phenomena in Ionized Gases, Prague, Czechoslovakia, 1973. Invited Papers, 9 – 59. 13. Nicolai, L. M. and Huethe, A. M., Properties of magnetically balanced arcs. The Physics of Fluids, 1969, 12, 2072–2082. 14. Petrenko, V. I. and Mitin, R. V., Investigation of internal structure of high- and superhigh-pressure pulsed discharges. Soviet Physics—Technical Physics, 1976, 20, 765 –769. 15. Bray, R. W., Weldon, J. M., Wehrlen, B. and Spann, M., Final commissioning of the balcones 60 MJ pulsed power supply. IEEE Transactions on Magnetics, 1991, 27, 433– 437. 16. McNab, I. R., Fletcher, J. S., Sucov, E. W., Rustan, P. L. and Huebschman, M.L., Thunderbolt. IEEE Transactions on Magnetics, 1991, 27, 130 –135.
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Acta Astronautica Vol. 47, No. 11, pp. 789–797, 2000 ? 2001 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0094-5765(00)00130-2 0094-5765/01/$ - see front matter
ENERGY ADDITION METHOD FOR THE ELECTROTHERMAL RAMJET B. D. SHAW† MAE Department, University of California, Davis, CA 95616, USA (Received 18 January 2000)
Abstract—The electrothermal ramjet is a ramjet-in-tube device that has been proposed for accelerating masses to high speeds for applications such as direct space launch and hypersonic projectile studies. The electrothermal ramjet is similar to the ram accelerator, though one di erence is that heat is to be added to the propellant electrically rather than via combustion. This article presents a method for adding heat to the propellant in the electrothermal ramjet con guration via magnetically balanced traveling arc discharges. Calculations indicate that current and magnetic eld requirements for this method are within attainable ranges and that electrical heat addition should provide advantages over combustion heat release under certain conditions. ? 2001 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
Ramjet-type accelerators are under consideration as a method of accelerating masses to high speeds (up to about 104 m=s) for applications such as direct space launch and hypersonic projectile studies. With such accelerators, a stationary tube would be lled with gaseous propellant. A projectile would enter one end of the tube at a high (supersonic) speed and would travel down the tube. In most concepts, the projectile would e ectively act as the centerbody of a conventional ramjet, with the tube wall acting as the outer cowling. Wilbur et al. published studies on a particular class of these devices which they termed the “electrothermal ramjet” [1,2]. In their work, which was theoretical, they considered a gaseous hydrogen propellant and postulated that heat could be added to the propellant using electrical discharges or perhaps beamed electromagnetic radiation. The details of heat addition were not considered—only the thermodynamics and uid mechanics of the device were evaluated. Their work showed that ramjet-type accelerators appear feasible for accelerating masses to high speeds (¿15; 000 m=s) with accelerations of 300; 000 m=s2 or more, as long as sucient heat is added to the propellant. A geometry considered by Wilbur et al. is shown in Fig. 1, where the instantaneous frame of reference is moving with the projectile; for clarity, shock waves are † Tel.: +1-916-752-4130; fax: +1-916-752-4158. E-mail address: [email protected] (B. D. Shaw). 789
not shown in this gure. The ows entering the diffuser and exiting the nozzle are supersonic, while heat addition occurs subsonically (note that in Fig. 1 and throughout this paper, “M ” represents Mach number). The Mach number is unity at the nozzle throat. If sucient heat is added to the propellant, the projectile will accelerate down the tube. Heat addition is required to overcome shock losses, viscous losses and heat transfer losses and to increase the kinetic energy of the projectile. With devices of this type, viscous losses and heat transfer losses are typically small relative to shock losses. Hertzberg et al. developed the concept of ramjet-in-tube accelerators powered by chemical reactions (combustion) in premixed gases [3]; they termed these devices “ram accelerators” (in this paper, the term “ram accelerator” will refer to ramjet-in-tube accelerators powered only by chemical reactions, i.e., combustion, and the term “electrothermal ramjet” will refer to ramjet-in-tube accelerators powered by external electrical sources). In the ram accelerator con guration, the launch tube is lled with a combustible gas mixture. As the projectile travels down the tube, ow deceleration in the di user region raises the static temperature of the propellant, inducing ignition and allowing combustion heat release; ideally, the heat release occurs downstream of the di user. If suf cient heat is released per unit mass of propellant consumed and the heat is released in the correct location, the projectile is accelerated down the tube. Various geometries and operational modes have been considered for the ram accelerator. At lower
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Fig. 1. Schematic of an electrothermal ramjet con guration.
Fig. 2. Schematic of a thermally choked ram accelerator con guration.
speeds, it has been suggested that a particularly applicable mode of operation is the “thermally choked” mode [3]. In this mode, the ow entering the di user is supersonic and the ow after the diffuser is subsonic. Heat release occurs subsonically behind the projectile such that the ow becomes thermally choked downstream of the projectile. A schematic of a ram accelerator operating in the thermally choked mode is shown in Fig. 2, where the instantaneous frame of reference is moving with the projectile (for clarity, shock waves are not shown in Fig. 2). Another subsonic combustion mode studied by Hertzberg et al. utilized the basic projectile geometry shown in Fig. 1. Methods of powering ram accelerators with detonation waves have also been considered [3]. A factor that has been postulated to limit ram accelerator operation is structural failure of the projectiles. This failure has been associated with chemical reactions between the projectile and the propellant gas [4,5]. In addition, there are sometimes problems with unstarts, where a shock wave propagates ahead of the projectile. Some unstarts have been linked to projectile structural failure while others have been associated with having mixtures that are too energetic [4,5]. Unfortunately, there is no control of a projectile once it has been injected into a ram accelerator launch tube. Control of the projectile is desirable in case an unfavorable operating regime is encountered, e.g., a regime where unstarts are promoted. It is also desirable to control the launch process so that projectile accelera-
tion pro les can be tailored and ecient operation over the launch cycle can be attained. In the work presented here, an electrical energy addition scheme is presented for an electrothermal ramjet operating in the thermally choked mode. This energy addition scheme involves adding heat to the propellant via magnetically balanced traveling arc discharges that are stabilized in a location behind the projectile via externally applied magnetic elds. As described later, this scheme should provide advantages over combustion heat release under certain conditions. For example, this method should allow use of monatomic gases that will not react chemically with the projectiles. In addition, using electrical heat addition should allow heat release locations and pro les to be tailored, allowing control of projectile acceleration, attainment of high eciencies over the launch cycle and minimized launch tube lengths. The thermally choked mode is selected for study here because of the ease with which it may be modeled with reasonable accuracy and also because thermally choked operation of the ram accelerator has been experimentally demonstrated. It is expected that the basic concepts described here for energy addition via magnetically balanced arcs may be used in other con gurations, e.g., the geometry shown in Fig. 1 or where supersonic heat addition is employed. In the following, a methodology of adding heat via magnetically balanced traveling arcs is described. Following this, one-dimensional models of ramjet ow elds are developed, calculations for accelerator pro les are presented, and conclusions are then drawn. 2. ENERGY ADDITION VIA MAGNETICALLY BALANCED TRAVELING ARCS
A basic requirement of successful operation of a ramjet-in-tube device is that energy release occurs at proper locations relative to the projectile. For thermally choked operation, the energy release needs to occur such that choking occurs downstream of the projectile. It is proposed that traveling arc discharges can be used for this purpose with the electrothermal ramjet. Schematic drawings illustrating relevant concepts are shown in Figs. 3a and b (for clarity, shock waves are not shown in these gures). Figure 3a is a side view, while Fig. 3b is a front view with the projectile velocity out of the paper. In these gures, a two-dimensional projectile con guration with a wedge-type di user has been selected. This con guration appears to be particularly amenable to operation with traveling arc discharges because it allows easy introduc-
Electrothermal ramjet
Fig. 3. Schematic of energy addition method using magnetically stabilized thermionic arcs. Schematic (a) is a side view while (b) is a front view.
tion of nearly uniform magnetic elds, which are needed for control of the location of the heat release zone, as described below. The basic concepts described here could also be utilized with other projectile geometries, e.g., projectiles with circular cross sections. It is to be noted that operation of two-dimensional ram accelerators in the thermally choked mode has been demonstrated [6]. As Fig. 3a indicates, the projectile can be located between guide rails to ensure proper placement within the launch tube, as has been done in ram accelerator experiments (e.g., [6,7]). Cathode and anode rails would be placed on the top and bottom of the launch tube, and one or more arcs would exist between these rails. The arc(s) could initially be formed behind the projectile using, e.g., a high-voltage breakdown spark discharge with UV laser pre-ionization of the spark path. A magnetic ux density vector B (vectors are denoted using boldface) would be imposed perpendicular to the projectile velocity vector (note that the magnitude of the component of B normal to the wall is continuous across the wall and on either side of the wall [8]). This magnetic ux density produces the Lorentz force F = (IxB)L on each arc channel, where I is the current vector (for each arc), L the arc length, and x represents the vector
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cross product. The magnetic eld causes the arc to travel along the launch tube behind the projectile. In order to promote stable arc operation, the cathode rail could be heated to allow thermionic emission. Using thermionic emission to stabilize magnetically controlled arcs propagating between rails was demonstrated by Bond and Potillo [9]. They showed experimentally that highly-stable arc operation was possible at subsonic speeds as well as at supersonic speeds (where the arc speed is measured relative to the gas through which it moves) provided that the cathode rail was heated to allow thermionic emission. Bond and Potillo also showed that heating of the anode rail promoted arc stability. When thermionic emission was not used, the arcs could still exist but could become unstable and experience spatial uctuations. Based on these results, it is expected that heating of the cathode rail would be used here as well. Such heating could be accomplished, for example, by passing current through the cathode rail prior to injection of the projectile or by using an electrical resistance heater in contact with this rail. Experiments have shown that magnetically stabilized thermionic arcs of the type suggested for use here will be slanted at an angle with respect to the oncoming gas stream [9,10], as depicted schematically in Fig. 3a. The slant angle relative to the tube wall is very close to the Mach angle when the arc moves at supersonic speed relative to the oncoming gas. As the arc speed relative to the oncoming gas decreases, the angle decreases. The drag characteristics of these-type arcs have been found to be similar to those for solid cylinders of the same geometric con guration [10]. The drag coecient CD for a magnetically stabilized arc placed in a cross ow has been shown to be fairly well correlated by a relation of the form CD =
IB ; (1=2)v2 d
(1)
where CD is close to drag coecient values for solid cylinders [10] (CD is typically close to unity). In eqn. (1), I is the arc current, B the applied magnetic ux density magnitude, the density of the oncoming gas ow, v the ow velocity component normal to the arc, and d a characteristic dimension for the arc width normal to the ow. Equation (1) will be used later in this paper to calculate B values required to maintain an arc in position. The rate of energy dissipation E˙ in an arc is given by E˙ = I 2 R where R is the arc electrical resistance. For this analysis we will calculate R using R = L=(D2 =4), where is the arc electrical conductivity and D a characteristic arc diameter.
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Combining these relations yields the following equation: E˙ =
I 2L : (=4)D2
(2)
The rate Q˙ that heat is supplied to the propellant is given by the relation ˙ arc : Q˙ = E
(3)
In eqn. (3), arc is the fraction of heat generated in the arc that is transferred to the propellant gas ow, where it is recognized that the arc may lose heat, e.g., via radiation, which will not be absorbed by the propellant. Using eqn. (3) in eqn. (2) yields, after rearrangement, the following equation for the arc current I . 2 1=2 ˙ QD : (4) I= 4arc L To employ these relations it is necessary that the
ow eld conditions be speci ed. A model to predict ramjet ow conditions is developed in the next section. 3. THEORETICAL MODEL FOR THERMALLY CHOKED ELECTROTHERMAL RAMJET OPERATION
The ramjet con guration studied here is di erent from the thermally choked ram accelerator in that an external force (from a magnetic eld) is needed to hold the arc in place behind the projectile. This force is accounted for in the development of the electrothermal ramjet gas dynamic model, as described below. In the model, the gas ow is assumed to be one-dimensional, quasisteady and frictionless (though drag forces on the arc are allowed), and the propellant is treated as an ideal gas. These assumptions have been shown to lead to reasonably good results for thermally choked ram accelerator operation powered by combustion [3–5,11], and it is expected that they will work well for electrothermal heat addition. For the con guration shown in Fig. 3a, the equations for conservation of mass, momentum and energy take the following forms: 1 V1 = 3 V3 ; fp fa 1 V12 3 V32 + p1 + = + p3 + ; A 2 A 2
(5) (6)
V12 V2 (7) + q = h3 + 3 : 2 2 In eqns (5) – (7), fp is the thrust force on the projectile, fa the external (Lorentz) force required to hold the arc in place behind the projectile, p static h1 +
pressure, density, A the local tube cross-sectional area, V velocity, h enthalpy, q the amount of heat added per unit mass of propellant, and numerical subscripts denote locations in Fig. 3a. The reference frame is moving with the projectile. In the analyses presented here, monatomic gases (e.g., argon) will be considered as propellant candidates. Estimates indicate that over the pressure and temperature ranges expected at locations 1–3 in Fig. 3a, that monatomic gases of interest can reasonably be considered ideal with constant speci c heats. Using these assumptions, eqns. (5) – (7) can be combined to yield the relation 1=2
−1 2 Fp = Fa + M1 2(1 + ) 1 + M1 + Q 2 −(1 + M12 );
(8)
where Fp = fp =(p1 A) is a dimensionless thrust, Fa =fa =(p1 A) the dimensionless external force acting on the traveling arc, Q = q=(Cp T1 ) a dimensionless heat release parameter, Cp is speci c heat at constant pressure, T1 the static temperature upstream of the di user and is the speci c heat ratio. In developing eqn. (8), the thermal choking relation M3 = 1 has been used. Equation (8) can be rearranged to yield the following relation: (1 + M12 + Fp − Fa )2
−1 2 M1 : − 1+ Q= 2 2(1 + )M12 (9) Equation (9) allows dimensionless heat release values to be calculated for prescribed values of
; M1 ; Fp and Fa . As noted previously, drag characteristics of magnetically balanced arcs have been found to be similar to those for solid cylinders of the same geometric con guration. The force fa that acts on an arc to hold it in place is given by an equation of the form fa = 12 2 V22 CD Aa ;
(10)
where CD is the arc drag coecient and Aa the frontal area of the arc. Using eqns. (9) and (10), the dimensionless force acting on the arc is given by the following equation: Fa = 12 M22 CD
p2 Aa ; p1 A
(11)
where p2 =p1 is the static pressure ratio between locations 1 and 2. To evaluate eqn. (11), values of the Mach number M2 , i.e., the Mach number after the projectile but prior to the heat addition zone, are needed as are values for p2 =p1 . To obtain these M2 values, the equations for conservation of mass, momentum
Electrothermal ramjet
and energy can be written for the ow entering at location 1 and exiting at location 2 in Fig. 3a, as shown below. 1 V1 = 2 V2 ;
(12)
fp 1 V12 2 V22 + p1 + = p2 + ; A 2 2
(13)
V12 V2 = h2 + 2 : (14) 2 2 These relations can be combined to yield the following equation: 1=2 1 + M12 + Fp M1 1 + [( − 1)=2]M12 = : M2 1 + [( − 1)=2]M22 1 + M22 h1 +
This equation can be solved for M2 , i.e., 1=2 2 (b + 4a )1=2 − b ; M2 = 2a
(15)
where a=
−1 − 2 ; 2
M12 (1 + [( − 1)=2]M12 ) : (1 + M12 + Fp )2
It can also be shown that 1 + [( − 1)=2]M12 + Fp p2 = : p1 1 + [( − 1)=2]M22
For estimates of the order-of-magnitude of Fa we will use the characteristic values = 5=3 (for monatomic gases), M2 = 0:3 (which is a characteristic value for subsonic ow at location 2), CD = 1 [9], Fp = 5; M1 = 4 and Aa =A = 0:125; which is estimated to be applicable here. Using these values in eqns (11) and (16) yields Fa ≈ 1. This value is large enough that it provides a non-negligible contribution to the projectile thrust force, as shown by eqn. (8). The ballistic eciency is the fraction of energy deposition that increases the kinetic energy of the projectile. The ballistic eciency B is de ned by eqn. (17). B =
(16)
These equations show that M2 and p2 =p1 depend on M1 ; and Fp . It is interesting to note that Fp cannot be varied arbitrarily. For example, if Fp is prescribed to be too large, the speci c entropy change across the projectile, s2 − s1 , becomes negative (where s is speci c entropy), violating the second law of thermodynamics. For this analysis, a simpli ed model of ow around the projectile similar to what has been used for modeling thermally choked ram accelerators will be employed [4,5]. In this model, Fp is constrained such that the speci c entropy change s2 − s1 lies within the following upper and lower limits. The lower limit corresponds to having the inlet ow compress isentropically in the di user region with a normal shock existing at the minimum di user ow area. The ow after this normal shock expands isentropically and subsonically to location 2. The upper limit assumes that the ow passes by the projectile isentropically, with a normal shock existing downstream of the projectile. The lower limit produces the maximum attainable dimensionless thrust force Fp , while the upper limit produces Fp = 0.
fp V : Q˙
(17)
It is noted that conditions where Fp = B = 0 de ne the top operational speed (Mach number) of the projectile for a given heat addition rate. For the case of an ideal gas with constant speci c heats, eqn. (17) can be expressed as B =
b = 1 − 2 ; =
793
Fp − 1 : Q
(18)
The value of Q that maximizes the ballistic eciency of the ram accelerator, denoted here as Qopt , can be found by setting the derivative d(B )=dQ equal to zero (where Fp is given by eqn. (8) with Fa =0) and solving for Qopt . When this is done, it is found that for supersonic inlet ows, Qopt =M12 −1. The maximum possible value of B for the ram accelerator, i.e., B; max , can be calculated by setting Q = Qopt , yielding B; max = ( − 1)= . The equation to predict M1 values that optimize the ballistic eciency for the electrothermal ramjet, i.e., for the case where Fa ¿ 0, is complex and is not listed here. From eqn. (8), however, it is obvious that higher values of B are possible when Fa ¿ 0 because of the contribution of the arc force to the overall momentum balance. For = 1:4 (which applies to typical diatomic gases), B; max =0:29 while for = 1:67 (which applies to monatomic gases), B; max = 0:40. There is thus an advantage, in terms of maximizing ballistic eciency, to working with monatomic gases with the ram accelerator. This advantage also applies to the electrothermal ramjet con guration considered here. 4. CALCULATIONS OF RAMJET OPERATING CONDITIONS
During operation in the thermally choked mode, it is essential that the inlet Mach number (i.e., M1 ) be suciently large so that the di user is “started”. For comparison with results from typical ram
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Fig. 4. Plots of Fp versus M1 for cases 1–3.
Fig. 5. Plots of B versus M1 for cases 1–3.
accelerator operation [3–5], an inlet Mach number range of 3– 6 is selected for study here. For the analysis we will consider three di erent operating conditions, which are denoted in the following as cases 1–3. Case 1 is for when Q is held constant over the launch cycle, which also corresponds to operation of a thermally choked ram accelerator operating with the same propellant mixture everywhere along the launch tube. Case 2 corresponds to varying Q so that a constant thrust force Fp is achieved on a projectile, and case 3 is where Q is varied so that ballistic eciencies are held constant. The initial conditions for each case are the same in that the Fp value at M1 = 3 is selected so that the ballistic eciency has the value B = 0:4 at M1 = 3. In making these calculations, the ratio Aa =A = 0:125 was used. Data are shown in Figs. 4 – 6 for Fp ; B , and Q versus M1 for cases 1–3. Figures 4 and 5 show that when Q is held constant (case 1) the projectile thrust force and the ballistic eciency drop rapidly as the projectile Mach number is increased. For
Fig. 6. Plots of Q versus M1 for cases 1–3.
case 1, the maximum achievable inlet Mach number is M1 = 5:7 (i.e., Fp = B = 0 at this Mach number), and as a result data are not shown for case 1 for M1 ¿ 5:7. This trend is expected, as this is typical behavior for combustion-powered ram accelerator operation. When Q is allowed to vary with M1 , however, markedly increased performance can be achieved. Varying Q such that Fp is held constant yields higher values of ballistic eciency (case 2 in Fig. 5), and selecting Q so that B is optimized over the launch cycle yields an increasing dimensionless projectile thrust as M1 increases (see case 3 in Fig. 4). Dimensionless heat release pro les are shown in Fig. 6, where it is apparent that Q values do not have to vary by large amounts in order for these di erent cases to be realized. Knowing the Q versus M1 pro les allows estimates to be made of requirements for arc current and magnetic ux magnitudes for magnetically balanced arcs. For predicting arc currents, eqn. (4) can be expressed in the following manner: 1=2 p1 Ac1 D2 (M1 Q)1=2 : (19) I= arc − 1 4L 1 =W )1=2 is the speed of In eqn (19), c1 = ( RT sound in the propellant gas upstream of the projectile where R is the universal gas constant and W is the propellant molecular weight. Equation (19) shows explicitly how the arc current depends on the inlet mach number M1 , the propellant molecular weight W , and the heat release parameter Q, i.e., with all other parameters held constant, I ∼ (M1 Q)1=2 =W 1=4 . The magnetic ux density magnitude B required to hold an arc in place behind an accelerating projectile is found by rearranging eqn. (1) and expressing appropriate variables in terms of the conditions at location 2 in Fig. 3a. This procedure
Electrothermal ramjet
Fig. 7. Plots of arc current I versus M1 for cases 1–3. The propellant is argon initially at 300 K and 9.19 bar.
yields eqn. (20). 1 2 V22 dCD : (20) 2 I In developing this equation, it has been assumed that the arc exists downstream of any recirculation zones behind the projectile and that v = V2 . Equations (19) and (20) will be used to provide estimates for I and B for a particular con guration of projectile mass and acceleration that is similar to ram accelerator conditions sometimes investigated [3–5]. We will consider the acceleration of a projectile with mass m=0:075 kg from an initial velocity of 970 m=s to a nal velocity of 1940 m=s. The projectile acceleration a is speci ed to be 105 m=s2 at the initial Mach number M1 = 3. Argon is selected as the propellant, which yields the initial and nal inlet Mach numbers M1 = 3 and 6. We will use the initial dimensionless thrust value Fp = 5:1 which provides B = 0:4 at M1 = 3. The launch tube cross section is square with a cross-sectional area A = 0:0016 m2 . The thrust force acting on the projectile is fp = ma = 7500 N, and the initial tube pressure is then p1 = fp =(Fp A) = 919 kPa. For the purpose of making estimates we will use the representative values L = A1=2 = 0:04 m, arc = 0:5 [12], d=D=0:005 m [13,14], =3000 1= m (which is a representative value for high-pressure argon arcs [14]), and CD = 1 [10,13]. Using these relations in eqns. (19) and (20), as well as data from Figs. 4 – 6 yields results for the arc current I and applied magnetic ux density B versus the inlet Mach number M1 ; these results are plotted in Figs. 7 and 8, respectively. Figure 7 shows that for cases 2 and 3, arc currents are in the range 7–22 kA while Fig. 8 shows that magnetic ux densities are in the range 1–2.4 T for these same cases. The electrical energy that is required to achieve an acceleration pro le can be calculated by noting that the in nitesimal amount of energy dE B=
795
Fig. 8. Plots of applied magnetic ux density magnitude B versus M1 for cases 1–3. The propellant is argon initially at 300 K and 9.19 bar.
delivered over the time increment dt is given by dE = E˙ dt. This equation can be expressed as ˙ dV = (Ec ˙ 1 =a) dM1 , E˙ dt = E˙ dV=(dV=dt) = (E=a) which gives the amount of electrical energy delivered over the inlet Mach number increment dM1 . Using previous relations, it can be shown that ˙ 1 =a = mc12 M1 =(B arc ). The amount of electrical Ec energy E required to accelerate the projectile from the tube inlet Mach number M1; in to the Mach number M1 is then given by the following integral: Z mc12 M1 M1 dM1 : (21) E= arc M1;in B Using similar arguments, it can be shown that the time increment t and tube length x required to accelerate a projectile from M1; in to M1 are given by the following integrals: Z mc1 M1 dM1 ; (22) t= P1 A M1;in Fp x=
mc12 P1 A
Z
M1
M1;in
M1 dM1 : Fp
(23)
Equations (21) – (23) are plotted in Figs. 9 –11, respectively. These plots were generated by numerical integration using results contained in Figs. 4 – 6. Figure 9 shows that the energy requirements for case 1 are signi cantly larger than for cases 2 and 3, with cases 2 and 3 not being too di erent. For acceleration to M1 = 6 it is found that E = 718 kJ for case 2 and E = 529 kJ for case 3 (recall that the projectile cannot be accelerated to M1; max = 6 for case 1). Figures 10 and 11 show that the acceleration times and launch tube lengths are substantially smaller for cases 2 and 3 than for case 1. This is to be expected, as the ballistic eciency for case
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B. D. Shaw
Fig. 9. Plots of the electrical energy E required to accelerate a projectile from M1; in = 3 to M1 for cases 1–3.
Fig. 10. Plots of the time t required to accelerate a projectile from M1; in = 3 to M1 for cases 1–3.
Fig. 12. Electrical power addition pro les required to accelerate a projectile from M1; in =3 to M1 for cases 1–3.
values are lower for case 1, which is to be expected because of the lower currents. The current, energy and magnetic ux density values estimated above are in ranges that are attainable with presently available equipment, e.g., equipment that has been developed for railgun applications. For example, Ref. [15] discusses operation of 60 MJ railgun energy supplies that can deliver electrical energy at a rate of 900 MW and Ref. [16] discusses switching devices and 60 MJ railgun energy supplies comprised of 2 MJ capacitor banks. In addition, B eld magnitudes up to 10 T or higher are easily attainable with electromagnets. Because the magnitude of the component of B normal to the wall is continuous across the wall and on either side of the wall [8], introduction of B elds can be readily accomplished by using planar metal walls, as shown in Fig. 3a. If the tube was made of a material such as Kevlar, a circular tube cross section could instead be used. 5. CONCLUSIONS
Fig. 11. Plots of the launch tube length x required to accelerate a projectile from M1; in =3 to M1 for cases 1–3.
1 goes to zero while for cases 2 and 3 it does not approach zero. ˙ can also be calArc energy deposition rates (E) culated using the following relation E˙ = I 2 R = I 2 L=(D2 =4)
(24)
where currents are to be evaluated using results from Fig. 9. As shown in Fig. 12, calculations yield the result that for cases 2 and 3, arc energy dissipation rates are in the ranges 33–130 MW and 33–330 MW, respectively, for 3 ¡ M1 ¡ 6. The E˙
This article has presented the magnetically balanced traveling arc as a method of adding heat to a propellant for operation of the electrothermal ramjet operating in the thermally choked mode. Calculations indicate that for acceleration of a 0.075 kg projectile from 970 to 1940 m=s, the required current, energy and magnetic ux density values are in ranges that are attainable with presently available equipment. In addition, this method should provide certain advantages over combustion. For example, the traveling arc method should allow control of projectile acceleration and achievement of high eciencies during the acceleration history. It is stressed, however, that the calculations presented invoked simplifying assumptions that allowed overall trends and approximate values to be calculated. Based on the results in this paper, it is worthwhile to perform more detailed studies to further assess the viability of this system. For
Electrothermal ramjet
example, more accurate experimental data are needed on arc diameter values that should be used for the ow conditions that would exist in the launch tube, and multidimensional CFD calculations could be used to predict detailed ramjet
ow elds and dynamic arc behaviors. It is noted that this energy addition method could potentially be used to accelerate projectiles to higher velocities, e.g., by adding more launch tube stages. For example, a projectile could exit an argon- lled launch tube and enter a neon- lled launch tube. Because neon has a lower atomic weight than argon, the projectile Mach number would be reduced and the projectile could again be accelerated up to a maximum Mach number (e.g., M1 = 6, which yields an exit velocity of about 2900 m=s). This process could be repeated by then utilizing a helium- lled launch tube, yielding an exit velocity of about 6100 m=s for M1 = 6. To achieve higher velocities, H2 could be used as a propellant, or higher terminal Mach numbers could simply be tolerated. The highest Mach numbers that a projectile can withstand will depend on the projectile materials and the residence time of the projectile in the launch tube. It should be mentioned that the arc thermal conductivity could be tailored somewhat by adding small amounts of easily ionized materials such as cesium. This might be necessary with a propellant such as helium, which has high ionization energies. It is also worth mentioning that in addition to varying magnetic ux densities (B) temporally, it would also be useful to vary B values spatially along the launch tube. This would be accomplished so that at any instant of time, the B eld would increase in magnitude with increasing distance downstream of the projectile. The arc could then be held at a desired location behind the projectile where the drag force acting on the arc matched the force the B eld exerted on the arc. Movement of the arc downstream of this position would increase the force of the B eld, which would move the arc closer to the projectile. If the arc was too close to the projectile, the weaker B eld in the projectile vicinity would be insucient to hold the arc against the ow exiting the projectile, and the arc would be driven back towards the desired equilibrium position relative to the projectile. The location of the heat release zone relative to the projectile could be controlled in this manner. This is desirable, for example, so that the projectile would not be subjected to excessive heating from the hot gases in the heat addition zone. Acknowledgements—The support of the California Space Institute is gratefully acknowledged. In addition,
797
the e orts of students in B. D. Shaw’s research group (in particular Dr. I. Aharon) are gratefully acknowledged. REFERENCES
1. Wilbur, P. J., Mitchell, C. E. and Shaw, B. D., The electrothermal ramjet. Journal of Spacecraft and Rockets, 1983, 20, 603– 610. 2. Shaw, B. D., Mitchell, C. E. and Wilbur, P. J., The annular ow, electrothermal plug ramjet. Journal of Propulsion and Power, 1985, 1, 417– 424. 3. Hertzberg, A., Bruckner, A. P. and Bogdano , D.W., Ram accelerator: a new chemical method for accelerating projectiles to ultrahigh velocities. AIAA Journal, 1988, 26, 195 –203. 4. Higgins, A. J., Knowlen, C. and Brucker, A. P., Ram accelerator operating limits, Part 1: identi cation of limits. Journal of Propulsion and Power, 1998, 14, 951–958. 5. Higgins, A. J. Knowlen, C. and Brucker, A. P., Ram accelerator operating limits, Part 2: nature of observed limits. Journal of Propulsion and Power, 1998, 14, 959 –966. 6. Chang, X., Shimomura, K. and Taki, S., Preliminary tests of a rectangular bore ram accelerator. Proceedings of RAMAC II—The Second International Workshop on Ram Accelerators, University of Washington, July 17–20, 1995. 7. Seiler, F., Patz, G., Smeets, G. and Srulijes, J., The rail tube in ram acceleration: feasibility study with ISL’S RAMAC 30. Proceedings of RAMAC II—The Second International Workshop on Ram Accelerators, University of Washington, July 17– 20, 1995. 8. Kraus, J. D. and Fleisch, D. A., Electromagnetics With Applications, 5th edn. Mc-Graw Hill, New York, 1999, p. 412. 9. Bond, C. E. and Potillo, R. W., Stability and slanting of the electric arc in a thermionic rail accelerator. AIAA Journal, 1968, 6, 1565 –1567. 10. Bond, C. E., and Wickersheim, D. N., Convective electric arcs at Mach numbers up to 6.5. AIAA Journal, 1970, 8, 1748–1751. 11. Bauer, P., Knowlen, C. and Bruckner, A., Real gas e ects on the prediction of ram accelerator performance. Shock Waves, 1998, 8, 113–118. 12. Edels, H., Properties of the high pressure ultra high current arc, Eleventh International Conference on Phenomena in Ionized Gases, Prague, Czechoslovakia, 1973. Invited Papers, 9 – 59. 13. Nicolai, L. M. and Huethe, A. M., Properties of magnetically balanced arcs. The Physics of Fluids, 1969, 12, 2072–2082. 14. Petrenko, V. I. and Mitin, R. V., Investigation of internal structure of high- and superhigh-pressure pulsed discharges. Soviet Physics—Technical Physics, 1976, 20, 765 –769. 15. Bray, R. W., Weldon, J. M., Wehrlen, B. and Spann, M., Final commissioning of the balcones 60 MJ pulsed power supply. IEEE Transactions on Magnetics, 1991, 27, 433– 437. 16. McNab, I. R., Fletcher, J. S., Sucov, E. W., Rustan, P. L. and Huebschman, M.L., Thunderbolt. IEEE Transactions on Magnetics, 1991, 27, 130 –135.