A quick method for designing liquid fueled launchers

[OhioBob]

After developing some basic guidelines for designing liquid fueled launch vehicles, I noticed that it really all comes down to just a few easy to remember ratios.  Summarized, they are

Second stage propellant mass ≈ 1 × payload mass
Second stage dry mass ≈ 0.25 × propellant mass
Second stage TWR ≈ 1.0 to 1.3
First stage propellant mass ≈ 2 × payload mass
First stage dry mass ≈ 0.25 × propellant mass
First stage TWR ≈ 1.3 to 1.5

In this case, payload mass refers to the total mass that sits atop the second stage, which includes decouplers, fairings, etc.  This mass is frequently more than the "useful" payload mass.

Using these ratios, we should get a launch vehicle capable of producing at least the 3400 m/s needed to attain Kerbin orbit, often with a little to spare.

To demonstrate, we should look at an example.  Let's say we have

     Total payload mass = 24 tons

The second stage propellant mass should equal the payload mass,

     Second stage propellant mass = 24 tons

     Use (1) X200-32 Fuel Tank + (1) X200-16 Fuel Tank

The second stage dry mass (including engine) will be about 0.25 times the propellant mass,

     Estimated second stage dry mass = 0.25 × 24 = 6 tons

     Estimated total mass of payload + second stage = 24 + 24 + 6 = 54 tons

Let's assume a TWR of 1.2 and compute the second stage thrust,

     Second stage target thrust = 54 x 9.81 x 1.2 = 636 kN (vacuum)

     Use Skipper engine (vacuum thrust = 650 kN, vacuum Isp = 320 s)

Let's now determine our actual mass and TWR,

     Actual total mass of payload + second stage = 54 tons

     Actual second stage TWR = 650 / 9.81 / 54 = 1.23

The first stage propellant mass should equal twice the payload mass,

     First stage propellant mass = 24 × 2 = 48 tons

     Use (1) Jumbo-64 Fuel Tank + (1) X200-32 Fuel Tank

The first stage dry mass is about 0.25 times the propellant mass,

     Estimated first stage dry mass = 0.25 × 48 = 12 tons

     Estimated total launch mass = 54 + 48 + 12 = 114 tons

Let's assume a TWR of 1.4 and compute the first stage thrust,

     First stage target thrust = 114 x 9.81 x 1.4 = 1566 kN (sea level)

Here we have a bit of a quandary, there are no engines that produce our target thrust.  The Mainsail has a sea level thrust of 1379 kN and the Twin-Boar has a thrust of 1867 kN.  Let's give both a try.

Using the Mainsail, and adding an interstage decoupler and four AV-T1 Winglets, we have

     Actual total launch mass = 114.548 tons

     Actual first stage TWR = 1379 / 9.81 / 114.548 = 1.23

The Twin-Boar has an integral fuel tank with 32 tons of propellant, thus we can delete the Jumbo-64 fuel tank.  As before, we add mass for a decoupler and winglets,

     Actual total launch mass = 115.048 tons

     Actual first stage TWR = 1867 / 9.81 / 115.048 = 1.65

The Mainsail is below our desired TWR range, and the Twin-Boar is above it.  Although, for design purposes, I like to target something in 1.3-1.5 range, it's acceptable to go outside that range when we have a situation like this, where our engines choices leave us with no other option.  I consider the absolute minimum TWR to be 1.2, and the maximum about 2.  Both the Mainsail and Twin-Boar fall within this 1.2-2 TWR range.

So which engine do we use?  In this case, the Twin-Boar is actually less expense than the Mainsail + Jumbo-64.  Also, with a liftoff TWR of 1.23, the Mainsail is pretty much maxed out.  Using the Twin-Boar we have some room for growth.  The second stage's TWR is high enough that it can handle some additional payload.  And with the Twin-Boar's high TWR, it can handle some additional first stage propellant to get the bigger payload to orbit.  Therefore,

     Use Twin-Boar engine (sea level thrust = 1867 kN, vacuum Isp = 300 s)

Now that we have our rocket design, let’s compute the Δv,

     First stage Δv = 300 × 9.80665 × LN(115.048 / (115.048-48)) = 1589 m/s

     Second stage Δv = 320 × 9.80665 × LN(54 / (54-24)) = 1845 m/s

     Total Δv = 1589 + 1845 = 3434 m/s (vacuum)

So we see that we have a design that will get our 24-ton payload to Kerbin orbit.  (Note that had we used the Mainsail, the total Δv would be 3496 m/s.)

Let’s compute one more thing, the payload fraction,

     Payload fraction = 24 / 115.048 = 0.21

Payload fraction comes in handy for some quick calculations.  For example, suppose we want to know how much payload we can put into orbit using a "Mammoth" first stage engine.  If we assume a liftoff TWR of 1.4, then the Mammoth, with a sea level thrust of 3746 kN, can lift a total mass of,

     Total launch mass = 3746 / 9.81 / 1.4 = 273 tons

Therefore, the total payload that it can launch is approximately,

     Payload mass = 273 × 0.21 = 57 tons

Using the ratios cited at the beginning of this post doesn't always work out perfectly.  For instance, 2.5m fuel tanks come in propellant increments of 4 tons.  So if we have a payload mass of, say, 26 tons, there is no combination of 2.5m fuel tanks that will give us that mass of propellant.  The ratios are intended to be used as a general guideline to provide a starting place.  In many cases it is necessary to massage the figures to get things to work out.  Nonetheless, the ratios are easy to remember and, when they can be followed, almost always result in a well-designed rocket that will get your payload efficiently to orbit.

(edit)

If you design a rocket that comes very close to matching the recommended ratios, it should deliver a Δv approximately equal to eleven times the specific impulse.  When the Isp of the first and second stage are different, you can use an average.  So, for the example given above, we can quickly estimate the Δv as follows:

     Approximate Δv = 11 × 310 = 3410 m/s (vacuum)

 

Edited November 19, 2018 by OhioBob

[guitarxe]

Hm... am I doing something wrong?

My payload weighs 0.9t, including fairing, decoupler and struts. Let's make that a round 1t.

So second stage propellant mass is also 1t. That neatly fits the FL-T200 tank, with 1t propellant and 0.125t dry mass for a total of 1.125t.
So then second stage dry mass is 0.25 * 1 = 0.25t. So that means total mass for second stage is 1 + 1 + 0.25 = 2.25t
With a 1.2 TWR that is 2.25 * 9.81 * 1.2 = 26.4kN, which nothing at all fits, so I took the 48-7S Spark and used tweak scale to scale it up a bit until it had 26.087kN thrust. Close enough.

So now with the engine mass added in, the full mass of second stage and payload is 2.472t.
The actual TWR for it is then 26.087 / 9.81 / 2.472 = 1.07.
Hm... no where close. Probably because we have 0.25t to allocate for dry mass, but half of that is taken up by the tank, so that only leaves 0.125t for the engine, and the scaled up version of the spark for the above thrust weighs 0.145t

But let's ignore that for now, it should be fine enough for the upper stage anyway, and continue.

So first stage propellant mass is 2 * payload mass = 2t. That again neatly fits the FL-T400 tank, with 2t propellant and 0.25t dry mass for a total of 2.25t.
The first stage dry mass is 0.25 * propellant mass = 0.5t. And total mass for launch is then 1 + 1 + 0.25 + 2 + 0.5 = 4.75t. That's pretty close to actual weight with decouplers and struts and engines, which is at 4.792t.
So we need a TWR of 1.4, and that is a thrust of 4.75 * 9.81 * 1.4 = 65.24kN. Again, no such engine except the poodle has that thrust at sea level, and the poodle obviously isn't what we need here, so the Spark to the rescue again. I scale it up until my TWR reads 1.44. I can't scale it by thrust alone because the widget to show the thrust in the editor shows it for vacuum only.

So with all that assembled, I have a dV of 2811m/s. 

 

[OhioBob]

On ‎4‎/‎14‎/‎2016 at 10:43 PM, guitarxe said:

My payload weighs 0.9t, including fairing, decoupler and struts. Let's make that a round 1t.

When dealing with very small payloads, the stage dry masses are often a little higher than the "0.25 x propellant mass."  The small parts tend to be less mass efficient.  I based my ratios mainly on 2.5m parts.
 

Quote

So now with the engine mass added in, the full mass of second stage and payload is 2.472t.

Something doesn't sound right here.  You say the payload is 1t, the fuel tank is 1.125t, and the engine is 0.145t.  That adds up to, 1 + 1.125 + 0.145 = 2.27t.
 

Quote

The first stage dry mass is 0.25 * propellant mass = 0.5t. And total mass for launch is then 1 + 1 + 0.25 + 2 + 0.5 = 4.75t. That's pretty close to actual weight with decouplers and struts and engines, which is at 4.792t.
So we need a TWR of 1.4, and that is a thrust of 4.75 * 9.81 * 1.4 = 65.24kN.

Typically for this step I use the actual mass of the second stage.  The estimated mass is used to select the second stage engine, but now that we know what the second stage's actual mass is, we should use it to compute the first stage thrust.
 

Quote

So we need a TWR of 1.4, and that is a thrust of 4.75 * 9.81 * 1.4 = 65.24kN. Again, no such engine except the poodle has that thrust at sea level, and the poodle obviously isn't what we need here, so the Spark to the rescue again. I scale it up until my TWR reads 1.44. I can't scale it by thrust alone because the widget to show the thrust in the editor shows it for vacuum only.

You don't say what the mass of the first stage engine is but, based on what you've told me, I'm estimating 0.44t.  Adding up the masses of all the parts, I get

Payload = 1
Stage 2 propellant = 1
Stage 2 tank = 0.125
Stage 2 engine = 0.145
Decoupler = 0.05
Stage 1 propellant = 2
Stage 1 tank = 0.25
Stage 1 engine = 0.44
TOTAL = 5.01

With a stage 1 TWR of 1.44, the thrust is 5.01 x 1.44 x 9.81 = 70.8 kN.  This is about 4.4 times the sea level thrust of the Spark, which is why I estimate the scaled up mass to be 0.44t.

If these numbers are correct, then the delta-v should be,

Δv = 300*9.80665*LN(2.27/1.27) + 300*9.80665*LN(5.01/3.01) = 3208 m/s

That's a little lower than the target 3400 m/s for a couple reasons.  First, as I already said, the small parts are less mass efficient, leading to slightly higher dry masses.  The other reason is that the Spark is not a very good engine in terms of specific impulse.  Most other second stage engines have an ISP of 320 s or greater.
 

Quote

So with all that assembled, I have a dV of 2811m/s. 

This part I don't understand.  Based on the numbers you've given me, I don't see how the Δv can be that low.  Even if I use your mass of 2.472t for the second stage (rather than 2.27t), I still manage to compute a vacuum Δv of about 2950 m/s.

 

Edited April 16, 2016 by OhioBob

[guitarxe]

Hm, something's not right with what both KER and MechJeb show me in terms of dV (and they both show wildly different numbers). Maybe my staging is messed up or something with the addon, I'll have to check...
Thanks for going through my example, though, that helped!

[OhioBob]

Alternative Design Method - Parallel Staging

Another common design method is to use parallel staging instead of serial staging.  It turns our that we can use the same ratios here as well, it's just that the first stage mass is now distributed between the two outboard stages.  I've found that it works very nicely to use three identical stages, such as the real-life Delta-IV Heavy.  In this case the rule is very simple:

Propellant mass per each common stage ≈ 1 × payload mass (3 stages total)

The plan here is to ignite all three engines at liftoff and feed propellant from the outboard propellant tanks to the center propellant tank using external fuel ducts (onion staging).  I've found that, rather than designing from the payload down, it is usually easiest to select an engine and then design a launch vehicle around it.  For example,

     Use Mainsail engine (sea level thrust = 1379 kN, vacuum thrust = 1500, vacuum Isp = 310 s)

If we assume a liftoff TWR of 1.4, then our total launch mass will be approximately,

     Estimated launch mass = (1379 × 3) / (9.81 × 1.4) = 301 t

We learned from the previous example that we should have a payload fraction of about 0.21, therefore

     Estimated payload mass = 301 × 0.21 = 63 t

We want to put a mass of propellant in each of our three common stages that is equal to the payload mass.  Let's call it 64 tons and go with,

     Use (2) Jumbo-64 Fuel Tanks per stage

Let's now determine the actual mass of each of our stages.  For the two outboard stages we must add for a decoupler, nosecone, fuel duct, and struts.  Let's say all of those extras come to 0.8 t/each.

     Mass of center stage = 78 t

     Mass of outboard stages = 78.8 t × 2 each = 157.6 t total

If we slap on a 64-t payload, our total launch mass and our mass at staging (i.e. outboard jettison) becomes,

     Total launch mass = 78 + 157.6 + 64 = 299.6 t

     Total mass at staging = 78 + 64 = 142 t

Let's now check our TWR,

     TWR at liftoff = (1379 × 3) / (9.81 × 299.6) = 1.41

     TWR at staging = 1500 / (9.81 × 142) = 1.08

Our staging TWR is on the low side, but it's still OK.  Let's now compute the Δv,

     Δv before staging = 310 × 9.80665 × LN(299.6 / (299.6-128)) = 1694 m/s

     Δv after staging = 310 × 9.80665 × LN(142 / (142-64)) = 1821 m/s

     Total Δv = 1694 + 1821 = 3515 m/s (vacuum)

So we see that we have plenty of Δv to get our 64-ton payload to orbit.

If you need to adjust the amount of propellant that you're carrying (for instance, to match a slightly lighter or heavier payload) then, generally, you should do so as follows:

●  If you need to add propellant, do so to the outboard stages.
●  If you need to subtract propellant, do so from the center stage.

The reason for this is because of the high TWR of the outboard stages in comparison to the center stage.

 

Edited January 19, 2019 by OhioBob

[OhioBob]

Real Solar System

Using the same method as described in the opening post, I was curious to see how the ratios worked out when launching from an earth-sized planet.  I don't use Realism Overhaul, but I do use Real Solar System with ROMini and stock aerodynamics.  I don't know how the size and mass scaling in ROMini compares to full-blown RO, but I assume it is the same or similar.  By playing around with the numbers and launching some test rockets, I've come up with the following:

Second stage propellant mass ≈ 6 × payload mass
Second stage dry mass ≈ 0.05 × propellant mass
Second stage TWR ≈ 0.8 to 1.0
First stage propellant mass ≈ 24 × payload mass
First stage dry mass ≈ 0.05 × propellant mass
First stage TWR ≈ 1.2 to 1.5

These numbers seem to work pretty well.  As before, it's just a starting point and the figures need to be fined tuned to the specific situation.  No example is needed here because the method is identical to the OP, just with different ratios.

I'm sure that other players have their own preferences and guidelines, which may differ from mine.  If anybody wants to post their own numbers, I'm interested in seeing what you use.