Simple calculator

[laukejas]

Hello all,

My head hurts for 2 days now as I try to make a simple spreadsheet with Excel to use in Orbiter, which calculates dV.

I use Tsiolkovsky equation, but now I want to make a reversed thing: find maximum payload.

These values are known:
Vessel dry weight;
Fuel mass;
ISP;
dV budget;
dV limit.

I need to make a formula which calculates what is the maximum payload to stay within dV limit.

Example:

Dry weight is 5kg, fuel mass is 11kg, ISP is 16739, and that results in 19093m/s dV. However, trip requires only 16000m/s dV, so vessel can be added with some payload, until dV budget and dV limit match. Question is, how much?

I know from practice it is around 1,665kg, but I can't make a formula which calculates this from data I wrote before.

It should be simple, but I just can't make it. Can anyone help?

 

[Phil Smith]

Here we go -
deltaV = Isp*LN((5+11+x)/(5+x))
exp(deltaV/Isp) = (5+11+x)/(5+x)
where x is your payload mass, 5 and 11 - your given masses.
just solve this equation for x.

x = (5+11-5*exp(deltaV/Isp))/(exp(deltaV/Isp)-1)

 

[laukejas]

Thanks, but are you sure about this? I'm getting -5kg with your equation, which can't be right.

 

[blixel]

Hello all,

My head hurts for 2 days now as I try to make a simple spreadsheet with Excel to use in Orbiter, which calculates dV.

I know from practice it is around 1,665kg, but I can't make a formula which calculates this from data I wrote before.

It should be simple, but I just can't make it. Can anyone help?

Here's a copy of a dV calculator I made back in October. This is a copy, so you're welcome to edit and do what you want with it. The green fields are where you provide your input. The red fields are where calculations are done. (So you don't want to type anything in the red fields.)

 

[laukejas]

Thanks, blixel, but I've already got that... I'm trying to make advanced one, with reverse calculation.

Can someone please help me with reversing that formula?

 

[blixel]

Thanks, blixel, but I've already got that... I'm trying to make advanced one, with reverse calculation.

Can someone please help me with reversing that formula?

Oh, I guess I didn't understand your question. I thought you were just wanting to plug in some additional payload and see what that would do to your overall dV.

But you're saying that if you know your trip is going to cost 16000 m/s, and you know you have a total of 19093 m/s, then how much extra mass can you carry given that you have 3093 m/s worth of left over dV.

Good question. I'll be interested in the responses myself.

 

[indy91]

When I'm using Phil's formula I get about 1.871kg, which seems good. Maybe you need to correct the brackets in your calculation.

 

[laukejas]

1,87kg ain't good... That's more than 300m/s dV off the target. Correct should be 1,665kg (or similar).

Where did you put the brackets?

 

[MattBaker]

I think we've got a confusion resulting on decimal points/commas.
In the US (the whole world?) a comma is about making it easier to read, for example "The Moon is 386,000 kilometers away". A decimal point is about showing, well, decimal numbers like "An inch equals 2.54 centimeters".
While in Germany it's the other way round, the Moon is 386.000 kilometers away and an inch is 2,54 centimeters.

So I'm quite sure indy spoke about one-thousand eight-hundred and seventy-one kilograms, but got the usual trouble with commas, decimal points and stuff like that.:tiphat:

 

[indy91]

Actually just as Phil posted it, but I have a lot of experience in very stupid typing errors with calculators.
I also checked your first dv assumption and dV = 16739*ln((5+11)/5)) = 19469.98 m/s . This is also not correct, so perhaps your initial numbers aren't right or missing digits? At least I think Phil's forumula is correct.

P.S. I use . as decimal point.

 

[dgatsoulis]

I agree with Phil and indy91. The result of the payload mass for 16 km/s total ΔV is 1.871 kg (dot is decimal)

Here is the calculation I made (I know, the loooong way around)
Me is emptymass Mf is fuelmass and Mp is payload mass.

Here is a link for a calculator I made of that equation. You enter fuel mass, empty mass, ΔV and exhaust velocity (ISP) and it returns you the payload mass.

payload mass calculator

Don't touch the red text, that's the calculation.

 

[laukejas]

dgatsoulis, didn't you forget to convert ISP to exhaust velocity (x9.80665)? Because in Wikipedia, formula is with effective exhaust velocity, while 16739 is ISP.

 

[dgatsoulis]

dgatsoulis, didn't you forget to convert ISP to exhaust velocity (x9.80665)? Because in Wikipedia, formula is with effective exhaust velocity, while 16739 is ISP.

You mean that the value of the ISP you posted is in seconds!?

And how is that supposed to give you ~19 km/s of total ΔV?

if we use 16739x9.80665 = 164153.514 m/s as exhaust velocity (what kind of ship has 164 km/s exhaust velocity?)
then the total ΔV without any payload is 190935.293 m/s (190 km/s not 19 km/s)

You can see that that is the source of the confusion. Next time please post the units for the ISP also. Some use seconds and some use m/s.

 

[laukejas]

:facepalm: My god, I can't believe that I missed such a simple thing. Forgot that ISP has two measurements! No wonder that I had these strange numbers.

Thank you all very much, especially dgatsoulis. When I'll finish that calculator, I'll upload it to Orbiter Hangar or somewhere for everybody to use.

Eh... :facepalm:

 

[dgatsoulis]

:facepalm: My god, I can't believe that I missed such a simple thing. Forgot that ISP has two measurements! No wonder that I had these strange numbers.

Thank you all very much, especially dgatsoulis. When I'll finish that calculator, I'll upload it to Orbiter Hangar or somewhere for everybody to use.

Eh... :facepalm:

No worries, these things happen. The link for the calculator above works and it will return you the payload mass in kilograms as long as you input the emptymass and the fuelmass in kg and the exhaust velocity and ΔV in m/s.

 

[Phil Smith]

:facepalm: My god, I can't believe that I missed such a simple thing. Forgot that ISP has two measurements! No wonder that I had these strange numbers.

Thank you all very much, especially dgatsoulis. When I'll finish that calculator, I'll upload it to Orbiter Hangar or somewhere for everybody to use.

Eh... :facepalm:

you're welcome man!
and one more by the way - what a great specific impulse! what kinda propellant do ya use?

 

[laukejas]

I started this calculation with XR-5, I believe... But now as I look at it, it has ISP of 20915m/s. So probably something else. I'm not sure really

 

[AlfalfaQc]

I did the math for the equations to find the payload and I get this equation:
mp=(mf/(e^(dV/ISP)-1))-me

I get the same answer as phil's equation, I just find it simpler to compute (less chance of error).

 

[Phil Smith]

for minimazing chance of error you can also use funtion "goal seek" in excel.
for example: you type in A1 cell "0" and in A2: "=Isp*LN((5+11+A1)/(5+A1))-deltaV"
by the goal seek command (get in "A2" "zero" by changing "A1") you'll get your payload mass in the cell A1.
PS this is a very useful command for solvin equations. One thing - it finds out just one root.

 

[laukejas]

Thank you again. I'm back with one more question.

I find it difficult to write a calculator which would tell how much fuel would be needed to launch to orbit on planet with atmosphere, such as Earth. For example, launching from Moon surface to orbit requires about 2200m/s dV (found this on various sources) although orbital velocity is around 1700m/s. It's easy to calculate needed fuel amount.

However, in atmospheric launch, varying ISP and ascent profile make standard calculation inaccurate. For example, typical launch with XR5 includes taking off with main engines, reaching 20km height and around 1200 TAS, then engaging SCRAM engines, and finishing orbit insertion with main engines again.
How would I determine how much fuel will be used in such case (with varying payload weight)?

Or with some other multi-stage vessels, such as Energia launchers, which include solid fuel rockets - what formulas to use in that case?