How to mathematically design stages

[GoSlash27]

I use a spreadsheet to design my stages before I build them so that I am assured they are the lightest and cheapest designs for the job. I've been asked to explain how the process works and have some free time this morning... so here goes.

I will demonstrate the process in 4 specific cases:

1) low thrust vacuum stage

2) medium thrust upper launch stage

3) high thrust liquid fuel booster stage

4) high thrust solid propellant booster stage

I will try to explain the unique considerations for each

This will take a while, so please bear with me...

a little background:

This method relies heavily on the reverse rocket equation , converting thrust to lifting capacity, and adjusting Isp & thrust for atmospheric density.

The rocket equation:

 

 DV= 9.81I_spln(R_WD)

where

DV= change in velocity in m/sec

9.81= g0; the standard conversion in m/sec²

I_sp= the engine's specific impulse in s

ln(x)= the natural log function

and

R_WD= the stage's wet- to- dry ratio; it's mass when fully fueled divided by it's mass when empty.

The reverse rocket equation:

R_WD= e^{( DV/9.81Isp)}

 

where e= Euler's number; approx. 2.718

Converting thrust to lifting capacity

T

_____= M

g0*g_r

where

T= thrust in kiloNewtons

g0= local surface gravity, 9.81 m/sec² on Kerbin

g_r = minimum acceleration requirement in gs

M = lifting capacity in tonnes

Adjusting thrust for surface atmospheric density:

SurfaceI_sp*T

___________ = surface T

VacuumI_sp

 

Adjusting Isp for a booster stage:

I always assume that the booster stage will work in an average atmospheric density of 50%, so...

(Ispsurf+Ispvac)

______________= Booster Isp

2

Converting Rwd to fuel fraction

(Rwd-1)/Rwd= fuel fraction

Figuring out tank mass in the final design

(Rwd-1)(NMe+Mp)

_______________ = Mt

(Rfe-Rwd)

where

N = number of engines

Me= mass of 1 engine

Rfe= the ratio of our fuel tank's mass when full to mass when empty (9 for LF&O tanks)

Mt= mass of empty tanks

a final note before I get into the demonstrations:

This process works fine by hand if you've already narrowed down the candidate engines (or have a limited set of options), but it would get tedious to do it by hand for all possible engines. This is why I use a spreadsheet to do the job. By comparing the results in mass and cost, I can quickly pick the best option for the stage.

Best,

-Slashy

Edited November 29, 2015 by GoSlash27
Lost formatting

[GoSlash27]

Case #1: The low thrust vacuum stage.

I have 10 tonnes of payload going to Duna and back. Assuming aerobraking at both ends and a 10% DV reserve, I want 2 km/sec DV total. Minimum acceleration is .5g

For the interplanetary stages, the most important considerations are stage mass, cost, and simplicity in that order.

Low mass here means the booster and injection stages will have less payload to lift to orbit. Cost is always a factor in career mode, and simplicity may give you a friendlier payload to shove into orbit.

Plugging theseinto my spreadsheet shows that the LV-N would be lightest and the Poodle would be cheapest, but for this demonstration I'll work out the LV-909 Terrier.

The process goes like this:

1) How much total mass can a single engine move at my required minimum G?

2) How much of this mass would need to be fuel in order to hit my DV goal?

3) How much payload would this theoretical rocket be able to handle?

4) How many engines do I actually need to handle my payload requirement?

5) How much fuel and tankage?

6) What does it cost?

Step 1

The LV-909 generates 60 kN of thrust in vacuum. at .5g minimum acceleration, this works out to

60/(9.81*.5)= 12.2 tonnes

Step 2

using the reverse rocket equation for 2km/sec DV and the LV-909's Isp of 345 sec,

2000/(9.81*345)= 0.591

e^(0.591) = 1.806 Wet to dry ratio

To convert this to fuel fraction, it's

(1.806-1)/1.806= .446.

44.6% of this example rocket would be fuel.

Step 3

Different fuels have different tanks to hold them, so you need to check the mass of your representative tanks when full to their mass when empty.

For LF&O tanks, they weigh 1/8 of their fuel.

Tank mass in this case would be .446/9=.0558. our fuel tanks comprise 5.58% of our total stage mass.

Adding the fuel and tanks yields .502; 50.2% of our total mass is fuel and tanks.

Multiplying this by our capacity yields

(12.2)*.502=6.12 tonnes of tank and fuel

and of course our engine is another .5t of mass

So a single engine's payload capacity would be

12.2t (lifting capacity) - 6.12t (mass of fuel and tankage) -.5t (mass of engine)= 5.58t payload.

Step 4

But of course we need to move 10 tonnes in our actual lifter, so we would need

10/5.58= 1.79 engines. And since there's no such thing as a partial engine, this rounds up to 2.

Step 5

So now that we have our number of engines and required wet to dry ratio, we can calculate the tank mass for our actual stage.

(Rwd-1)(NMe+Mp)

_______________ = Mt

(Rfe-Rwd)

where

N = number of engines

Me= mass of 1 engine

Rfe= the ratio of our fuel tank's mass when full to mass when empty (9 for LF&O tanks)

Mt= mass of empty tanks

(1.806-1)(2*.5+10)

_________________= Mt

(9-1.806)

.806(11)/7.194 = 1.23t

Our empty tanks weigh 1.23t.

Our fuel weighs 8 times as much; 9.84t.

Our engines weigh 2*.5t= 1t

Our payload weighs 10t

All together, our stage with payload is 22.1t

Plugging this into the rocket equation as a quick sanity check yields 2,031 m/sec DV

Step 5

We would need 22 FL-T100 tanks to hold this fuel, which is 2 FL-T800s, 1 FL-T400, and 1 FL-T200. Plus the cost of 2 Terriers. This would work out to about $4,080.

[von Ziegendorf]

In KSP (unlike real world) specific impulse conversion factor is equal to 9.82 (exactly). Small difference, I know. Please don't call me physics-....

[GoSlash27]

Case#2: The medium thrust upper launch stage

The procedure for this is exactly the same as for the low thrust vacuum stage with 2 minor adjustments: The DV requirement is 1800 m/sec and the g requirement is .7g. Otherwise, same priorities apply. Light is more important than cheap because excess mass here increases the workload of our booster stage.

Also, I like to make this stage recoverable in order to recoup some of the launch cost.

We have a 22.1t payload going to orbit, to which we've added a decoupler and reaction wheel (.15 tonnes) for a total of 22.3t.

acceleration requirement is .7g and DV is 1,800 m/sec

The aerospike would be lightest and the poodle would be cheapest, but I'll analyze the Skipper.

Step 1

The Skipper generates 650 kN of thrust in vacuum. at .7g minimum acceleration, this works out to

650/(9.81*.7)= 94.7 tonnes

Step 2

using the reverse rocket equation for 1800m/sec DV and the skipper's Isp of 320 sec,

1800/(9.81*320)= 0.573

e^(0.573) = 1.774 Wet to dry ratio

To convert this to fuel fraction, it's

(1.774-1)/1.774= .436.

43.6% of this example rocket would be fuel.

Step 3

Tank mass in this case would be .436/8=.0545. our fuel tanks comprise 5.45% of our total stage mass.

Adding the fuel and tanks yields .491; 49.1% of our total mass is fuel and tanks.

Multiplying this by our capacity yields

(94.7)*.491=46.5 tonnes of tank and fuel

and of course our engine is another 3t of mass

So a single engine's payload capacity would be

94.7t (lifting capacity) - 46.5t (mass of fuel and tankage) -3t (mass of engine)=45.2 t payload.

Step 4

This time we have the opposite problem; 1 engine is more than enough for this job. So we'll skip ahead to the next step and trim the unneeded fuel and tankage out of the design.

Step 5

So now that we have our number of engines and required wet to dry ratio, we can calculate the tank mass for our actual stage.

(1.774-1)(1*3+22.1)

_________________= Mt

(9-1.774)

.774(25.1)/7.226 = 2.69t

Our empty tanks weigh 2.69t.

Our fuel weighs 8 times as much; 21.5t.

Our engine weighs 3t

Our payload weighs 22.1t

All together, our stage with payload is 49.3t

Plugging this into the rocket equation as a quick sanity check yields 1,800 m/sec DV

Now... since I'm nowhere near the limit of what this stage can handle, I'll probably slap a control head on it and 6 parachutes (it weighs about 6 tonnes empty) and recover it while still barely suborbital to recoup some of the cost.

- - - Updated - - -

In KSP (unlike real world) specific impulse conversion factor is equal to 9.82 (exactly). Small difference, I know. Please don't call me physics-....

von Ziegendorf,

This is no longer true for KSP post- 1.0. That has been corrected and the correct g0 is now 9.81 m/sec^2.

Best,

-Slashy

- - - Updated - - -

I have to get ready for the Hawkeye game, so I'll pick this back up afterwards.

Best,

-Slashy

Edited October 10, 2015 by GoSlash27

[Right]

Wow, I hope to be able to use these one day. If I got comfortable with how and when to use them, it looks like a great time saver! Looking forward to the rest. Most of the time I try to award you rep for awesome information, the forums get mad at me and say I've been giving you too much.

[GoSlash27]

Case #3: The high thrust LF&O booster stage

The priorities get shifted when dealing with boosters. Light weight is no longer a priority since the weight can't cascade down the stack any further. Cost, however, is a big priority because this stage will be destroyed in a stock career launch.

Moreover, we are operating in atmosphere now. This adds a couple additional steps to our calculation. We need to figure the minimum acceleration in the worst- case scenario, which is sitting on the pad at sea- level. We also need to assume an average Isp at 50% atmosphere to get a good prediction of our stage size.

Aside from that, it's the same process we've used for vacuum stages.

Procedure:

1) How much thrust does this engine produce at sea level?

2) What is it's effective Isp over it's flight?

3) How much total mass can a single engine move at my required minimum G?

4) How much of this mass would need to be fuel in order to hit my DV goal?

5) How much payload would this theoretical rocket be able to handle?

6) How many engines do I actually need to handle my payload requirement?

7) How much fuel and tankage?

8) What does it cost?

For this example, we will assume a 50t payload. Minimum acceleration is 1.4G and required DV is 1,800 m/sec.

We will look at the Twin- Boar in this example.

This happens to be the ideal LF&O engine for this job in terms of cost, although the "Kickback" SRB could do it cheaper.

Step 1:

Thrust at sea level is T(vac)* Isp(atm)/Isp(vac)

2,000 kN* 280s/300s= 1,867 kN

The Twin Boar produces 1,857 kN thrust at sea level.

Step 2:

Average Isp over the duration of the flight is (Isp(atm)+Isp(vac))/2

280s+300s/2= 290s

The Twin Boar will average 290s Isp over the duration of it's flight.

From here on, it's the same procedure as before.

Step 3

The Twin Boar generates 1,857 kN of thrust at sea level. at 1.4g minimum acceleration, this works out to

1,857/(9.81*1.4)= 136 tonnes that a single engine can handle.

Step 4

using the reverse rocket equation for 1800 m/sec DV and the Twin Boar's average Isp of 290 sec,

1800/(9.81*290)= 0.633

e^(0.633) = 1.883 Wet to dry ratio

To convert this to fuel fraction, it's

(1.883-1)/1.883= .469.

46.9% of this example rocket would be fuel.

Step 5

Tank mass in this case would be .469/8=.059. our fuel tanks comprise 5.9% of our total stage mass.

Adding the fuel and tanks yields .528; 52.8% of our total mass is fuel and tanks.

Multiplying this by our capacity yields

(136)*.528=71.8 tonnes of tank and fuel

and of course our engine is another 6.5t of mass (since the Twin Boar has a free jumbo 64 attached, I subtract the mass and cost of that from the engine)

So a single engine's payload capacity would be

136t (lifting capacity) - 71.8t (mass of fuel and tankage) -6.5t (mass of engine)= 57.7t payload.

Step 6

This is more lifting capacity than we need, so we use 1 engine.

Step 7

So now that we have our number of engines and required wet to dry ratio, we can calculate the tank mass for our actual stage.

(1.883-1)(1*6.5+50)

_________________= Mt

(9-1.883)

.883(56.5)/7.117 = 7.01t

Our empty tanks weigh 7.01t.

Our fuel weighs 8 times as much; 56.1t.

Our engine weighs 6.5t.

Our payload weighs 50t

All together, our launch vehicle with payload is 119.6t

Plugging this into the rocket equation as a quick sanity check yields 1,801 m/sec DV

Step 8

Our engine already carries 32t of LF&O, so we need to add another 24t. That's another x200-32 and X-200-16.

Engine: $17,000

X200-32: $3,000

X200-16: $1,550

Stage cost: $21,550