Rocket Propulsion Elements: Total Impulse Problem

Question

I figured this is probably an okay place to post this question.

I managed to get (a), (c) and (d) (quite simply), but I have absolutely no idea why I can't work out (b).

I used $$ I_{sp}=\frac{I_t}{g_0\dot{m}\Delta t} $$

I assumed thrust was constant (is this wrong?). I did also check my units.

Thanks. :)

Answer 1

You got caught by those silly English units.

Your expression, $I_t = I_{sp}g_0 \dot m \Delta t$, works just fine when you use metric units. The mass flow rate, in metric units, is 127 kg/s. The total impulse, in metric units, is $$I_t = (217.5\,\text{s})(9.80665\,\text{m}/\text{s}^2)(127\, \text{kg}/\text{s})(65\,\text{s}) = 1.76\times 10^7\,\text{N}\,\text{s}$$ Converting this to customary units yields $3.96\times 10^6\,\text{lbf}\,\text{s}$.

Doing the exact same calculation, this time using customary units, yields $$I_t = (217.5\,\text{s})(32.17405\,\text{ft}/\text{s}^2)(280\, \text{lbm}/\text{s})(65\,\text{s}) = 1.2736\times 10^8\,\text{but in what units?}$$ The dimensional analysis says the units of this expression are lbm·ft/s. Those aren't the desired units; you want lbf·s. The metric F=ma becomes F=kma in customary units. Dividing by the numerical value of g₀ converts lbm·ft/s² to lbf. That same division here converts lbm·ft/s to lbf·s. And indeed, 1.2736×10⁸/32.17405 is 3.96×10⁶.

This suggests an alternate expression, $I_t = I_{sp} \dot m \Delta t$, where $I_{sp}$ is in seconds.This yields a numerical value of 217.5·280·65=3.96×10⁶. That's the right value (at least numerically). Note well: This alternate expression does not work in metric units. In metric, this calculation yields a value of 1.80×10⁶ kg·s. The correct value is 1.76×10⁷ N·s.

Strictly speaking, the above does not have the correct units; it has units of mass*time. Numerically, it yields the right value in customary units because dividing by the numerical value of g₀ cancels the explicit use of g₀ in the numerator.

This calculation is not correct in metric units. To get the right metric units from this calculation one needs to multiply the numerical value of that result (1.80×10⁶ kg·s) by the numerical value of g₀. Indeed, 1.80×10⁶ * 9.80665 = 1.76×10⁷.

Answer 2

You take the effective exhaust velocity and multiply it by the mass flow. That gives you the thrust. Multiply that by the operating time and you get the total impulse.