General thrust equation

The general thrust equation is used to calculate the thrust force generated by neutral gas thrusters, including conventional rockets:

F_t = ˙m u_z,ex−u_z,0

+ (p_ex−p₀)A_ex (6.3)

wherem˙ is the mass flow rate of the propellant,A is the area, and the subscripts ‘ex’ and

‘0’ denote exit and the ambient free stream respectively. For the present purpose, the free stream velocityu_z,0 is zero, and the free stream pressure is defined to be equal to the outlet

pressure. The first term in (6.3) represents the component of the thrust force arising from the momentum of the ejected propellant, while the second term represents the pressure force difference between the exit area and the equivalent external area on the front of the thruster.

The general thrust equation (6.3) is visually represented in Figure 6.7a, showing the external pressure forces p_ex (red → and black ←/→ arrows) and p₀ (blue← arrow) acting on the thruster and the forcemu˙ z,ex imparted to the ejected propellant (green →arrows).

(a)

(b)

ṁu

p

p

F

p

p

Figure 6.7: (a) Force diagram for the general thrust equation. The thrust force is the sum of the external pressure forces p_ex (red → and black ←/→ arrows) and p₀ (blue ← arrow) acting on the thruster and the reaction to the force mu˙ z,ex = ρexu

2

z,ex imparted to the exiting propellant (green → arrows). (b) Force diagram for the internal forces method

(Section6.3.3). The thrust force is the sum of all the pressure forces within the propellant volume acting on the walls of the thruster (blue←andred→arrows) and on the background environment at the exit. This also includes the boundary layer friction force (magenta →),

which must be accounted for to avoid overestimating the thrust force.

In cases where p_ex ≈p₀ and u_z,ex ≈c_s across the exit, the equation F_t0 = ˙mc_s may be

used to give a rough estimation of the thrust force. For high pressure and large geometries as in conventional rocket nozzles, this approximation is valid since the static pressure p_ex,

axial velocityuz,ex, and mass flow rate densitym/A˙ ex are typically uniform in the main flow across most of the exit area. However, for the low operating pressures and small geometries of microthrusters, boundary layer effects near the wall are nontrivial and often significant

compared to the main flow, resulting in nonuniform profiles for all the three parametersp_ex, u_z,ex, and m˙. As seen in Figure 6.6, the sonic surfaces are typically not planar. Further- more, the position of the sonic surface does not coincide with the exit surface, and can vary significantly depending on the gas species and the specific choked flow conditions. Addi- tionally, the local sound speedc_s at the thruster exit is dependent on the local temperature

which, in most cases, is different from the initial temperature or stagnation temperature of the propellant upstream, and difficult to measure. In cases where p_ex 6=p0, and especially

for real case applications in space wherepex > p0, the pressure thrust force (second term in (6.3)) can be significant and its contribution must be taken into account with the addition of a term in the form ofp_exA_ex, though in practice p_ex cannot be measured easily.

Figure6.8usesPR-Cas an example, plotting the axial velocityu_z,ex(blue), static pressure pex (red), and mass density ρex (black) across the radius of the discharge chamber exit for thep0 = 0.349Torr (left, lines) and p0 = 0Torr (right, lines) cases. While the uz profile has a parabolic shape that is to some extent theoretically predictable, the p_ex and ρ_ex profiles are nontrivial. In the p₀ = 0.349Torr case in particular, overexpansion of the propellant to p_ex < p0 causes a dip in thepex profile (left, red line) in the middle of the discharge chamber exit. Additionally, the supersonicuz,ex cannot be calculated simply from first principles, as the boundary layer effectively reduces the exit area A_ex by an indefinite

amount depending on the local flow conditions.

_

_

Figure 6.8: Axial velocityu_z,ex (blue), static pressurep_ex (red), and mass densityρ_ex(black,

in units of[×10−2kg m3]on the right vertical axis) for the PR-Cp₀ = 0.349Torr case (left, lines) and p0= 0Torr case (right, lines). The profiles are generally nontrivial.

In these circumstances, calculation of the thrust force requires the integral form of the general thrust equation:

Ft= 2π Z R 0 r ρexu2z,ex+pex−p0 dr (6.4)

with the fluid density ρex, axial velocity uz,ex, and pressure pex profiles across the exit. The integration is performed beginning at the central axis at r = 0mm and ending at the discharge chamber wall at r = 2.1mm for PR-O and PR-C, and at r = 0.8mm for MiniPR respectively. To ascertain the accuracy of the thrust force calculations, a similar integration of the mass flow rate of the gas m˙ex = ρexuz,ex is performed across the exit area and compared to the supplied mass flow rate m˙ at the inlet. The error is found to be in the range−4.9 %≤∆ ˙m_ex≤ −0.4 %for thePR-OandPR-CCFD simulations, indicating that the results are reasonably precise. The error is significantly greater at −10.3 % on average for theMiniPRCFD simulations, which is most likely caused by having an insufficient number of cells across the radius of the discharge chamber (8 cells over r = 0.8mm) to accurately resolve the boundary layer and a smooth profile for the various fluid parameters. ∆ ˙m_ex is

taken into account by modifyingρ_ex in (6.4) to bringm˙_ex in line with the suppliedm˙. The computed cold gas thrust force for MiniPR is presented in the following section. Results from PR-O and PR-C are discussed later in Chapter 7 together with the computed thrust force during plasma operation.