Multiple engine specific impulse math

[Esoteric106]

I could use some help. I'm trying to figure out how to do the math to figure out my delta v for multI engine design with different isp I'm not getting the explanations.I can find.

No I don't want to use any mods

Thank you in advance 

[Dman979]

The standard dV formula is 9.8 * I_sp * Ln(Mass_full/Mass_empty).

 

For a single engine, that's simple- just plug in the Isp of the engine for the formula.

For multiple engines, it's not quite as complicated as you think. You need to take the average of all the different engine Isp's you'll be using in a given stage, and plug that in for the standard Isp.

In math form:

9.8 * ((Isp₁+Isp₂+Isp₃+Isp4...Isp_N)/N) * Ln(Mass_full/Mass_empty), where N is the total number of engines being used at one time.

Repeat that step once for each stage, then add the totals together to get the overall dV of your ship. Be sure to include all the mass above the stage in the calculations!

 

Hope that helps!

[Esoteric106]

so if I have 1 at 650 and 2 at 350 the equation would be 9.8*((650+350+350)/3)*ln (25.5/13.5) I would get  2 807.572 410 685 43 as my delta v correct?

made up numbers btw 

[Dman979]

Yes, you'd get ~2804.7 m/s of Delta-V.

As for units, g is in m/s², Isp is in seconds, and M ends up canceling itself out, so it could be any applicable unit of mass.

Also, welcome aboard!

[Esoteric106]

So now how would you equate srbs that would drop off mid burn of another engine? 

Thank you again by the way

Edited April 20, 2017 by Esoteric106

[Dman979]

Ah, you're welcome! Happy to help.

 

That's a little more complicated, and there's probably many ways to solve that. Here's mine.

I treat the SRB's in one stage, so I'd run the numbers twice.

Step 1: time how long the SRB's burn.

Step 2: calculate how much mass your other engines use in that time, and add that to the empty weight of the SRBs.

Step 3: subtract the total in step 2 from the mass of the rocket with the SRBs at the beginning of the burn. We'll refer to this as Mass_middle.

Step 4: count the SRB's Isp's in the average you did above, and use Mass_full/Mass_middle.

Step 5: calculate the Isp without the SRBs, and run that with Mass_middle/Mass_empty.

Step 6: Add the results of Steps 4 and 5 to find the Delta-V.

[Esoteric106]

Will try that thank you

[egoego]

Just wanted to add that those formulas are actually in the wiki: http://wiki.kerbalspaceprogram.com/wiki/Specific_impulse#Multiple_engines (the formula may look difficult if you have no mathematic background)

[monstah]

I just want to nitpick over something @Dman979 said on the I_sp average: yes, the rocket equation formula works for clusters of engines, and yes, the total I_sp is an average of the engines' I_sps... however, it's not the simple average (sum(...)/N) but a weighted average (TL;DR math: the formula for the entire ship is I_sp = sum(T_i) / sum(T_i/I_spi), where T_i and I_{sp i} are the thrust and I_sp of each individual engine).

If you're using just many engines of the same type, it makes no difference (total I_sp equals engine I_sp), and if your engines are consuming roughly the same amount of fuel each, then the math doesn't make much difference, either. But, since the Kraken is in the details:

To get there, you have to first use another formula, which defines I_sp: I_sp = T / (m' * g₀), where T is the total thrust (in N), m' is the propellant mass flow (in kg/s), g₀ is the surface gravity (in m/s²), and I_sp is in s (or you can use kilonewtons and metric tons instead of newtons and kilograms, which are the units the game shows you). That equation applies to either a single engine, or the entire ship. Figuring out the ship's total thrust is easy, just sum the thrust for each engine; the propellant mass flow is also just the sum for each engine, however, that value isn't shown in game. But, I_sp is, so you can calculate the engine's mass flow using the same equation: m' = T / (I_sp * g₀).

So, to for the entire ship, T = sum(T_i); m' = sum(m'_i) = sum(T_i / (I_spi * g₀)) = sum(T_i/I_spi) / g₀; and therefore I_sp = T / (m' * g₀) = sum(T_i) / sum(T_i/I_spi)

[Esoteric106]

Monstah thank you you have explained it perfectly and that what I was missing /having a hard time understanding. It clicked now I think I'll try the math when I get home on a rocket and see what I get. If you don't mind I'll post my results and se if I'm right.... (the only math background I have is 20 years of being a carpenter)

I know I am a very small minority being i want to learn the math and use no mods but it's way more enjoyable to me to plan out a mission calculate the math and burns like we did when space exploration still meant some thing

Again thank you all very much

[Reusables]

Sorry for being off-topic, but the average has a name: (weighted) harmonic mean. Don't know why such a thing has 'harmonic' in the name. (Probably pythagorian?)

 

Personally, I prefer to use inverse of ISP. It makes many calculations easier. (Including the average - still weighted tho) I think it also has the better meaning: consumption rate for certain dv. (Rocketry Inertia)

[Esoteric106]

I'm really slow with picking up math...inverse of isp?

[Reusables]

I mean, 1/ISP. Let's say L=1/ISP.

Then combined 'inertia' is L=Sum(Ti*Li)/Sum(Ti), and L*dv=ln (Mwet/Mdry). It may look more complex to get the actual dv, but actually more helpful to calculate mass ratios like fuel fraction and payload fraction.

[Esoteric106]

Ok I think I got the part I was missing I néed to convert the fuel/O2 flow to mass instead of kg/sec so th/(fuel mass) to get the Isp to average in to the entire stage/rocket? ? 

[Esoteric106]

ok so if i use this formula T/(m'*g0(9.81) on say a LV-t45 single engine stage it looks like this 167969/((6.166+7.536)*9.81)=1249.61481697725... should it not be 250? the only way i can get it to equal 250 is like this 167696/((6.166+7.536)*49)=250.177 986 827 485

This works for any Engine i have so far in career i have tried it on so far T30,T45,RE-15 they all work with 49 im really confused 

 

[Reusables]

3 minutes ago, Esoteric106 said:

ok so if i use this formula T/(m'*g0(9.81) on say a LV-t45 single engine stage it looks like this 167969/((6.166+7.536)*9.81)=1249.61481697725... should it not be 250? the only way i can get it to equal 250 is like this 167696/((6.166+7.536)*49)=250.177 986 827 485

This works for any Engine i have so far in career i have tried it on so far T30,T45,RE-15 they all work with 49 im really confused 

 

You almost got the math! You got one mistake: unit is not mass. 1 unit of liquid fuel or oxidizer weighs 5kg. That'll fix it.

[Esoteric106]

so the ((6.166+7.365)*49) should read (6.166+7.365)*5) ?

 

[Reusables]

1 hour ago, Esoteric106 said:

so the ((6.166+7.365)*49) should read (6.166+7.365)*5) ?

 

(6.166+7.365)*9.8*5. It's fuel consumption rate of the engine in kg/s.

[Esoteric106]

oh oh so i nee to add a multiple of 5

got you ok now that i have that down i will see if i can work out stages/rockets for multiple engines with different isps and see if i have it now
Thank you again for your time 

 

[Esoteric106]

ok another question....(sorry so slow) say i have a stage with 3 LV-T30 and 4 LV-T45s and 77.595/22.595Mass...  the formula for this would be ((167969*4)+(205161*3)/((45.979+56.196)*9.81*5)*ln(77.595/22.595)= 672027.519? 

Is this correct it seems stupid high

 

or do i just use one set of numbers for each engine ?

 

[Esoteric106]

or is this the proper formula ((167969*4)+(205161*3)/7)/((45.979+56.196)*9.81*5)*ln(77.595.22.595)=187.047794939213

 

[Esoteric106]

I figured it out.. one set of numbers per engine type my isp for this would be isp of 257.941 (258) witch gives me a dv of around 3200 

THANK YOU EVERYONE WHO HELPED ME

[Reusables]

6 hours ago, Esoteric106 said:

I figured it out.. one set of numbers per engine type my isp for this would be isp of 257.941 (258) witch gives me a dv of around 3200 

THANK YOU EVERYONE WHO HELPED ME

I got ISP of 256.9~257 with the configuration, how do you calculate it? 1 second is a bit big for calculation error.

[Esoteric106]

(167969+205161)/((13.271+16.220)*9.81*5)=257.947 698 482 348

Witch gives me a dv calculator of 9.81*258*ln (77.595/23.595)=3 122.658 202 490 95

I keep seeing you type 9.8 instead of 9.81 is that the difference? 

[Reusables]

No, I calculated on 3 LV-T30 and 4 LV-T45. In the case you should weight thrust & fuel consumption with the number to get the sum.

LV-T30 consumes 15.77/s while LV-T45 does 13.68/s.

So (167969*4+205161*3)/((13.68*4+15.77*3)*9.81*5)=257.2

Though you're right if you are using one of each.