△Mc2 here the definition of Vch(Equation(6) is now made using no instead of n. Once gain,only approximate expressions for Av/vch are feasible for the optimum c and mass fractions Normalizing all velocities by vch e obtain xn=√1+82-y (1 224√1+82 =1-21+82v+v2- 121+82 (13) M M-五+8224(1+ For 8=0, and neglecting the last term included in each case, we recover the simple expressions of Equation (7)and Figure 2. The main effect of the losses(8 can be seen to be: (a)An increase of the optimum C, seeking to take advantage of the higher efficiency thus obtained (b)a reduction of the maximum payload c)A reduction of the fuel fraction Both these last effects indicate a higher structural fraction due to the need to raise rated power to compensate for the efficiency loss. It is worth noting also that the losses are felt least in the trucking"mode(high Vch, i. e light engine or long duration). 2. The Optimum: Thrust Profile As was mentioned there is no a priori reason to operate an electric thruster at a constant thrust or specific impulse even if the power is indeed fixed We examine here a simple case to illustrate this point, namely, one with a constant efficiency as in the classical Stuhlinger optimization but allowing f, m and c to vary in time if this is advantageous. Of course these variations are linked by the constancy of the 16.522, Space Propulsion ecture 2 Prof. manuel martinez-Sanchez Page 6 of 19
16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 6 of 19 V 22 V - - L L c c so 2 o o ch M c +v M =e - - 1-e M M v ∆ ∆ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ (10) where the definition of vch (Equation (6)) is now made using η0 instead of η. Once again, only approximate expressions for ∆V vch are feasible for the optimum c and mass fractions. Normalizing all velocities by vch : L ch ch ch c v v x ; v = ; = vvv ∆ ≡ δ (11) we obtain 2 2 OPT 2 v v x = 1+ - - + ... 2 24 1+ δ δ (12) 3 L so 2 2 2 o o MAX M M v + = 1 - 2 1+ v + v - + ... M M 12 1+ ⎛ ⎞ ⎜ ⎟ δ ⎝ ⎠ δ (13) 3 P 2 2 o M v 1v = - + ... M 24 1+ 1+ ⎛ ⎞ ⎜ ⎟ δ δ ⎝ ⎠ (14) For δ = 0,and neglecting the last term included in each case, we recover the simple expressions of Equation (7) and Figure 2. The main effect of the losses ( ) δ can be seen to be: (a) An increase of the optimum c, seeking to take advantage of the higher efficiency thus obtained. (b) A reduction of the maximum payload, (c) A reduction of the fuel fraction. Both these last effects indicate a higher structural fraction, due to the need to raise rated power to compensate for the efficiency loss. It is worth noting also that the losses are felt least in the “trucking” mode (high vch , i.e. light engine or long duration). 2. The Optimum: Thrust Profile As was mentioned, there is no a priori reason to operate an electric thruster at a constant thrust or specific impulse, even if the power is indeed fixed. We examine here a simple case to illustrate this point, namely, one with a constant efficiency as in the classical Stuhlinger optimization, but allowing F, m i and c to vary in time if this is advantageous. Of course these variations are linked by the constancy of the power:
m(t)c(t)=5F(t)c(t) Consider the rate of change of the inverse mass with time 1】) dM m (16a) Multiplying and dividing by F2=m c (16b) M2 mc2 where a= F/M is the acceleration due to thrust Inte 111 (17) P On the other hand, the mission Av is △V=adt (18) and is a prescribed quantity. We wish to select the function a(t) which will give a naximum M(Equation 17)while preserving this value of Av. The problem reduces to finding the shape of a(t), whose square integrates to a minimum while its own value has a fixed integral. the solution(which can be found by various mathematical techniques but is intuitively clear) is that a should be a constant Using this condition, (17)and(18)integrate immediately. Eliminating a betweer these, we obtain M 1 (19) M M△V ntp The level of power is yet to be selected; it will determine the average specific impulse, and it is to be expected that an optimum will also exist U M=M+M d 16.522, Space Propulsion Prof. manuel martinez-Sanchez Page 7 of 19
16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 7 of 19 () () () () 1 1 2 P= m t c t = F t c t 2 2 η η i (15) Consider the rate of change of the inverse mass with time: 2 2 1 d M 1 dM m =- = dt dt M M ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ i (16a) Multiplying and dividing by 2 2 2 F =m c , i 2 2 2 2 1 d M F a = = dt 2 P M mc ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ η i (16b) where a=F M is the acceleration due to thrust. Integrating, t 2 f 0 0 11 1 - = a dt M M 2Pη ∫ (17) On the other hand, the mission ∆V is t 0 ∆V = a dt ∫ (18) and is a prescribed quantity. We wish to select the function a(t) which will give a maximum Mf (Equation 17) while preserving this value of ∆V . The problem reduces to finding the shape of a(t), whose square integrates to a minimum while its own value has a fixed integral. The solution (which can be found by various mathematical techniques, but is intuitively clear) is that a should be a constant. Using this condition, (17) and (18) integrate immediately. Eliminating a between these, we obtain f 2 0 0 M 1 = M M V 1+ 2 tP ∆ η (19) The level of power is yet to be selected; it will determine the average specific impulse, and it is to be expected that an optimum will also exist. Using M =M +M f Ls and