# Averaging Specific Impulse for combined propulsion

How would one go about calculating ISp of a rocket with combined propulsion? E.g. the Space Shuttle during the phase of launch, with both SRBs and SSME active?

Of course there's the simple "experimental" method, taking wet and dry mass, and speed at the beginning and end of acceleration, but that is not very good if we don't have the complete, working rocket at hand. For the planning stage, we'd know the exhaust speed of different engines, their thrust and mass flow - how then would we go about finding the ISp of the whole?

Considering that various engine contribute a different amount of thrust, how to assign weights to their ISp to calculate the value for the whole craft?

$$I_{sp}=\dfrac{I_{sp1}\dot{m}_1+I_{sp2}\dot{m}_2+{...}}{\dot{m}_1+\dot{m}_2+{...}}$$

So each $I_{sp}$ is simply weighted by its fraction of the total mass flow rate. This extends to any number of $I_{sp}$'s.

• Gee, i find that easier to understand than the formula I found. – kim holder May 31 '16 at 19:17
• @kimholder Yours is more general: it has summation whereas Mark's only adds two. – called2voyage May 31 '16 at 19:18
• @called2voyage Yes, but it also gets the thrust out of the issue, which was messing with my head. – kim holder May 31 '16 at 19:20
• @kimholder They are convertible. – called2voyage May 31 '16 at 19:21
• Both answers are very useful: thrust and mass flow are easier to measure than exhaust speed, so Kim's answer is better for "development from scratch". OTOH, for most existing engines, ISp is usually readily available, so Mark's equation comes in handy. – SF. Jun 1 '16 at 2:51

Well, what can I say - the Kerbal Space Program Wiki has a good answer to this.

$$g_n I_{sp}=\frac{\sum\limits_{i}F_{T_i}}{\sum_\limits{i}\overset{.}{m}}=\frac{{\sum\limits_{i}}F_{T_i}}{\sum\limits_{i}{\frac{F_{T_i}}{g_n I_{sp_i}}}}$$

Where:

• $I_{sp}$ is the specific impulse in seconds
• $I_{sp_i}$ is the specific impulse of each engine in seconds
• $F_{T_i}$ is the thrust of each engine in newtons
• $\overset{.}{m}$ is the fuel consumption in kilograms per second
• $g_n$ is the standard acceleration of gravity

When the fuel consumption is not used in this formula, it is only important that all thrust values have the same unit (e.g. kilonewtons) and the specific impulse have all the same unit (e.g. seconds). The result is then in the same unit as the specific impulses of the engines. If all engines have the same specific impulse the resulting specific impulse will be the same.

The result is equivalent to the weighted harmonic mean of the engines' specific impulses, weighted by each engine's thrust.

• I love that the wiki about a fictional universe in a video game is the most relevant source for real-world physics equations. – 0xDBFB7 May 31 '16 at 20:01
• @DC177E Actually, it had a relevant answer once Mark Adler fixed it's mistakes with some edits to this answer. But i know what you mean. The thing is, KSP was designed to teach users about real space flight. Yay KSP! – kim holder May 31 '16 at 20:10