Solid rocket Isp derivation from grain properties

3.3.38 · Physics › Rocket Propulsion

WHAT are we deriving?

HOW the grain feeds the exhaust — building $\dot{m}$

Yahan se kyun shuru karein? Thrust ko mass flow chahiye, aur mass flow grain ke jalne se manufacture hoti hai. Geometry sirf yahaan aati hai.

Grain apni surface ke normal direction mein $r$ rate se recede karta hai. Time $dt$ mein $A_b$ area par $rdt$ thickness ka ek shell gas banta hai:

$dm={\rho }_p{A}_b(rdt)$

toh mass generation rate hai $\boxed{\dot{m}={\rho }_p{A}_{b}r}$

• ${\rho }_p$ = solid propellant density
• $A_b$ = instantaneous burning area
• $r$ = linear burn rate

Ye step kyun? Volume burned per second = area × recession speed; density se multiply karo toh kg/s milti hai. Pure geometry + kinematics.

Burn rate law (Saint-Robert / Vieille)

Empirically burn rate chamber pressure $p_c$ ke saath badhta hai: $r=a{p}_c^n$ $a$ = temperature-dependent coefficient, $n$ = pressure exponent (typically $0.2$–$0.5$; stable operation ke liye $<1$ hona chahiye).

HOW the gas leaves — building thrust

Rocket nozzle se thrust (control volume par momentum conservation se derive kiya gaya): $F=\dot{m}{v}_e+(p_e-p_a)A_e$

Exhaust velocity ek hot gas ki isentropic expansion ke liye chamber temperature $T_c$ se (energy conservation $h_c=h_e+\frac{1}{2}{v}_e^2$ aur $h=c_{p}T$ se derive karo):

$v_e=\sqrt{\frac{2\gamma }{\gamma -1}\frac{R{T}_c}{M}\left[1-{\left(\frac{p_e}{p_c}\right)}^{\frac{\gamma -1}{\gamma }}\right]}$

Ye form kyun? Chamber mein available saari enthalpy $c_p{T}_c$ kinetic energy mein convert hoti hai; bracket wo fraction hai jo $p_c$ se $p_e$ tak expand karke actually extract hoti hai. Yahaan $R$ = universal gas constant, $M$ = exhaust ka molar mass, $\gamma =c_p/c_v$.

Putting it together → $I_{sp}$

Steady operation ke liye effective exhaust velocity $c=F/\dot{m}$ define karein. Toh definition se:

$\boxed{I_{sp}=\frac{c}{g_0}=\frac{F}{\dot{m}{g}_0}}$

$F=\dot{m}{v}_e+(p_e-p_a)A_e$ substitute karein:

$I_{sp}=\frac{1}{g_0}\left[v_e+\frac{(p_e-p_a)A_e}{\dot{m}}\right]$

Ek ideal (optimum-expanded, $p_e=p_a$) motor ke liye pressure term zero ho jaata hai:

$\boxed{I_{sp}=\frac{v_e}{g_0}=\frac{1}{g_0}\sqrt{\frac{2\gamma }{\gamma -1}\frac{R{T}_c}{M}\left[1-{\left(\frac{p_e}{p_c}\right)}^{\frac{\gamma -1}{\gamma }}\right]}}$

The characteristic velocity $c^{\ast }$ (jahaan grain does chamber pressure ko touch karta hai)

Grain geometry chamber pressure set karta hai, throat ke zariye. Mass in = mass out through throat $A_t$: ${\rho }_p{A}_{b}r=\frac{p_c{A}_t}{c^{\ast }},c^{\ast }=\frac{1}{\Gamma }\sqrt{\frac{R{T}_c}{M}},\Gamma =\sqrt{\gamma }{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{2(\gamma -1)}}$

$r=a{p}_c^n$ use karo aur equilibrium pressure ke liye solve karo: $p_c={\left(\frac{{\rho }_{p}a{c}^{\ast }{A}_b}{A_t}\right)}^{\frac{1}{1-n}}$

$n<1$ kyun matter karta hai: exponent $\frac{1}{1-n}$, $n\to 1$ jaate hi blow up ho jaata hai. Agar $n\ge 1$ hai toh ek chhoti si pressure rise burn rate ko itna badhaa deti hai ki throat vent nahin kar sakta → runaway explosion. Ye hai "why not use fast-burning high-$n$ propellant" ka steel-man.

Worked Examples

Flashcards

Connections

• Tsiolkovsky Rocket Equation — jahaan $I_{sp}$, $\Delta v=I_{sp}{g}_0\ln (m_0/m_f)$ mein feed hota hai.
• Nozzle Isentropic Expansion — $v_e$ formula ka origin.
• Characteristic Velocity c-star — chamber pressure mein chemistry factor.
• Grain Geometry and Thrust Profiles — kaise $A_b(t)$ thrust shape karta hai (neutral/progressive/regressive).
• Saint-Robert Burn Rate Law — pressure dependence aur stability.

Concept Map