Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 185 (2017) 239 – 245
6th Russian-German Conference on Electric Propulsion and Their Application
Optimization of Design Parameters of Spacecraft Equipped with Electro Rocket Low-Thrust Engine and Calculation its Applying Area at Low Earth Orbit Sergey A. Ishkov*a a
Samara University, 34, Moskovskoye shosse, Samara, 443086, Russia
Abstract The problem of electro rocket engine type and its main design parameters optimization for spacecraft located at low circular Earth orbit is studied. The efficiency of chemical (impulse) and electro rocket engine of low-thrust is analyzed. Introduced criteria of efficiency – payload mass maximization, introduced assumption that engine exhaust velocity is constant. In accordance with criteria and assumption, the analytical solution for spacecraft design parameters of spacecraft is obtained. Also, analytical solution for area of efficiency of electro rocket low-thrust engine is obtained. Calculation of spacecraft design parameters equipped with electro rocket low-thrust engine and area of its efficiency applying is carried out. © 2017 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license © 2017 The Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of RGCEP – 2016. Peer-review under responsibility of the scientific committee of the 6th Russian-German Conference on Electric Propulsion and Their Application Keywords: Spacecraft; electro rocket engine of low-thrust; specific payload mass; characteristic velocity; area to mass ratio; design parameters; total resource;
1. Introduction Nowadays, electro rocket engine (ERE) of low-thrust is increasingly used in near Earth space. Achieved technological breakthrough in area of ERE and its power plants designing, allow providing solution of transport problems in near Earth space. ERE of low-thrust provide increasing efficiency of orbital transfers, efficiency of Earth remove sensing spacecraft station keeping, since ERE have high exhaust velocity [1]. However, there are some factors that limited ERE of low thrust application for above tasks. Main factor is necessity to power reserve for ERE operation. For this purpose, spacecraft equipped with additional solar panels, which provide spacecraft mass increasing and additional atmospheric resistance, since area of spacecraft middle section increase. Another limited factor is necessity to switch-off the payload equippment during ERE operation. Thereby, the problem of ERE efficiency area determine is important. Research in this area [2, 3] carried out in area of control programs for station keeping, but problem of ERE efficiency compared with another engines does not studied.
* Corresponding author. Tel.: +7-960-825-0782. E-mail address:
[email protected]
1877-7058 © 2017 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the 6th Russian-German Conference on Electric Propulsion and Their Application
doi:10.1016/j.proeng.2017.03.306
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Sergey A. Ishkov / Procedia Engineering 185 (2017) 239 – 245
The problem of ERE of low-thrust efficiency area calculation for low altitude spacecraft keeping is studied. Atmospheric resistance considered as perturbing factor. Efficiency analyzed by comparison of initial mass of two spacecraft. First spacecraft equipped with chemical (impulse) engine (SC1) and second equipped with ERE of low thrust (SC2). Pay load mass of doth spacecraft is constant. 2. Problem statement Take into account simple model of station keeping during all spacecraft life time. Both spacecraft operate in a range of heights between maximal and minimal. When SC1 go down to minimal, its engine takes impulse and increase apogee. In apogee engine takes second impulse and correct eccentricity. SC2 correct orbit similarly, but correction duration is greater and multiple to orbital period, with purpose to keep orbit eccentricity equal to zero. Introduce variable – specific keeping height H * , determine from condition: T
2 ³ Vav U H V H dt
Vav U H * V 2 H * T ,
0
(1)
0,5 S Cx M 1 average value of area to mass ratio, U H atmospheric density, V H spacecraft
where Vav
speed, S – area of middle section, Cx coefficient of atmospheric resistance force, M – spacecraft mass, T – total spacecraft resource. All future calculations will carried out for specific keeping height H * . Introduce next assumptions: 1 SC1 fuel consumption and SC2 working fluid consumption to station keeping are small, and all calculations are carried out for average masses. 2 Characteristic velocity gains for station keeping determine in accordance with total compensation of atmospheric deceleration:
aT H * T Vxm ,
Vx
where
aD H *
(2)
,
P Vср U H * RE H *
(3)
where aD atmospheric deceleration at height H * , RE average Earth radius, Vxm characteristic velocity reserve for purpose design equipment operation. 3. Mass model of spacecraft, equipped with chemical (impulse) engine Write mass model of SC1: M10 M SC M F1 M E1 ,
(4)
where M10 SC1 initial mass, M SC SC1 mass, without fuel and engine mass (payload mass), M F1 fuel mass, necessary to station keeping and for purpose design equipment operation, M E1
M F1 K1 engine mass with mass of
fuel tanks, K1 specific engine mass. In accordance with assumptions, fuel mass takes the form:
M F1
aD1 H * T Vxm Isp
M1av ,
where Isp specific impulse, M1av SC1 average mass. Let SC1 average mass equal to: M M1av M 01 F1 , 2 after transformation, we obtain:
(5)
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Sergey A. Ishkov / Procedia Engineering 185 (2017) 239 – 245
M F1
M 01 Vx , Vx · § I sp ¨ 2 ¸¹ ©
(6)
aD H * T Vxm . Substitute solution (6) in (4), divide left and right side of mass model of SC1 to M10 ,
where Vx
we obtain solution for specific payload mass of SC1: Vx PPL1 1 1 K1 . Vx · § I sp ¨ 2 ¸¹ ©
(7)
4. Mass model of spacecraft, equipped with ERE engine of low-thrust The mass model of SC2 takes the form: M 20 MSC M PP M WF M E2 , where M PP
(8) P C J PP mass of power plant, J PP specific mass of power plant, C – exhaust velocity, 2 Ke
Ke coefficient of efficiency, P – thrust of engines, M WF mass of working fluid, M E2 engine with working fluid tanks. In a similar way (6), the mass of the working fluid can be determined:
аD2 Н * Т Vxm KG
М WF
С
M 2av ,
M WF K 2 mass of
(9)
K where G coefficient of gravitational wasters during maneuvering. By substituting (9) to (8) and dividing the left and right side to M20, we obtain:
aD2 H * T Vxm K G a 02 C P PL2 1 J PP 1 K 2 , (10) Vx · 2 Ke § C ¨ 2 ¸¹ © P initial acceleration from ERE. where a 02 M 02 The ERE thrust is determined from the following considerations: 1 The acceleration from the thrust is significantly greater than atmospheric deceleration, under maximal solar activity level (SAL), at minimal spacecraft operation altitude. 2 Acceleration from the thrust must provide orbit correction without significant decreasing of productivity for the purpose of design equipment. Atmospheric decelerations aD1 and aD2 , in general case, are different, since SC2 is equipped with additional solar panels. Let us sssume:
aD2 H *
aD1 H * K V .
(11)
where
KV
V2 V1
1
'S , S
P C , 2 Ke Kel where ' S additional area of solar panels, Kel specific electric power of solar panels. By substituting (13) in (12) and taking into account area to mass ratio, we obtain: a C Cx K V 1 02 . 8 K el V1 Ke 'S
During KV calculation, we assume: 1 Area of middle section for average area to mass ratio calculate as one fourth of surface area:
(12) (13)
(14)
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Sergey A. Ishkov / Procedia Engineering 185 (2017) 239 – 245
'S (15) , 2 where S1 and S2 SC1 and SC2 are the middle section area respectively. 2 SC1 and SC2 masses are close. Value of exhaust velocity of SC2 ERE C is determined value – parameter of problem. Let us determine approximately Сopt as maximum of function PPL2 (10). Neglecting the decrease in spacecraft S2
S1
mass C !! Vx : 2 aD1 T Vxm KG 1 KG2
Сopt
a02 JРР
where JPP
(16)
,
J PP adduced mass of power plant. Ke
5. Determination of the area of ERE efficiency Let us determine the area of efficiency of ERE compared with chemical engine in space of T and H * . In order to do so it is necessary to determine the payoff function: 'P
P PL 2 P PL
aD1 T * Vxm 1 K a02 C JPP aD1 T * KV Vxm KG 1 K , Isp
1
Vx1 2
2
Isp
1
Vx 2 2
(17)
where T * limit value of resource T for predetermined 'P . Calculations show, that all practical important range of altitudes (220 km and higher) satisfy inequalities Vx Isp and Vx С. This means that we can neglect for Vx in numerator of (17).
Determine T H * from (17):
'P T*
V 1 К1 Vxm K G 1 Vxm a02 C J РР xm 2 Isp С § 1 К1 1 К 2 · aD1 ¨ KV ¸ ¨ I sp ¸ C © ¹
.
(18)
6. Numeric calculation Let us calculate area of SC2 efficiency in accordance with (18). Some specific coefficients are listed in table 1. Numeric values corresponds to real spacecraft with ERE of low-thrust and ERE characteristics. The calculations are carried out for circular orbits of a height of 250 – 500 km. Table 1. Specific coefficients of SC2 and its ERE Parameter Coefficient
V1
Coefficient
'P
Adduced mass of power plant
Dimension
kg/m J PP
2
Numeric value 0,005
dimensionless
0,05
kg/W
0,03
2
m /W
100
m/s
3500
Coefficient K1
dimensionless
0,02
Coefficient K2
dimensionless
0,02
m/s
100
Specific electric power of solar panels Specific impulse of SC1 engine
Characteristic velocity gains Vxm
I sp
K el
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Sergey A. Ishkov / Procedia Engineering 185 (2017) 239 – 245
Acceleration from thrust at high altitudes is equal to a02
104 m/s2 , at low altitudes it satisfies the condition
a02 ! 10 aD H * K V . It guarantees that acceleration from ERE is greater than atmospheric deceleration and that the duration of orbit correction is enough. ERE exhaust velocity C is a parameter of the problem. It is determined numerically from (18): T * C o min . In a general case, we can determine C approximately from (16), in any case for T.
A graph of T * C for H = 300 km and the average SAL plot are shown in fig. 1. A graph of Copt H * for border value of lifetime T * a plot is shown in fig. 2. In accordance with problem statement, SC2 area to mass ratio is greater
than SC1. A graph of K V H * for border value of lifetime T * and C
Coopt plot is shown in fig. 3. The plot of the
area of ERE efficiency, obtained during numerical calculation, for maximal, minimal and average SAL is shown in fig. 4.
Fig. 1. Border value of resource T* as function of ERE exhaust velocity C at circular orbit (height 350 km) for average solar activity level
Fig. 2. Value of optimal ERE exhaust velocity C̃ opt at border height H* for border life time T* for average SAL
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Fig. 3. Coefficient Kσ, as function of orbit height for border life time T* for C = Copt for average SAL
F
65
W m 2 Hz
F
175
W m 2 Hz
F
275
W m 2 Hz
Fig. 4. Area of ERE of low thrust efficiency for maximal, minimal and average SAL 7. Discussion Numerical calculation results show, that ERE exhaust velocity for high altitudes (greater than 300 km) is constant and that at altitudes less than 300 km it increases (fig. 2). A constriction of ERE efficiency area at the left side (fig. 4) is explained by a significant increase of area to mass ratio of SC2 at low altitudes, since acceleration a02 is increased, furthermore the mass of power unit and the area of the spacecraft middle section also increase. The constriction of ERE efficiency area at the right side (fig. 4) is explained by decreasing atmospheric acceleration and decreasing of the possible gain of ERE during its low operational time. An increasing in an atmospheric density, due-to an increase in the level of solar activity, expands the area of ERE efficiency, a decrease in atmospheric density constricts it. Furthermore, an increase of the characteristic velocity Vxm shifted down the area of ERE efficiency and shifted it up during its decrease. 8. Conclusion
Sergey A. Ishkov / Procedia Engineering 185 (2017) 239 – 245
The results obtained substantiate the suitability of ERE type thrusters for problems of orbit keeping for low altitude spacecraft. References [1] V.V. Salmin, S.A. Ishkov, O.L. Starionava Solution methods for variational problems of low thrust space flight mechanics. European Academy of Natural Science Press, 2014. [2] Khodnenko V.P., Khromov A.V. Selecting design parameters for earth remote sensing spacecraft orbit correction system. Questions of electro mechanic, Vol. 121, no. 2 (2011), 15-22. [3] V.V. Salmin, V.V. Volotsuev, S.V. Shikhanov Spacecraft preset orbital parameters control by means of thrusters. Vestnik of the Samara University. 4(2013). 248-254.
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