Rocket Equation

tahanson43206 · 2024-05-28 18:31:21

For SpaceNut ...

This forum contained no topics with both the words "rocket" and 'equation"

We have many posts that make use of or refer to the Rocket Equation.

This topic is offered for NewMars members who would like to show various scenarios that can be addressed with the Rocket Equation.

This topic leads off with a scenario posted by RGClark in one of his many topics about SSTO...

In post #3 of this topic, I will post a calculation done by ChatGPT4o.

We used the figures from RGClark's scenario, with the slight variation that instead of 200,000 kg for the entire vehicle, ChatGPT4o used 200,000 kg for the propellant, and came up with 25 tons and change for the airframe and payload delivered to orbit.

I am surprised by this figure, and look forward to analysis by our members.

(th)

tahanson43206 · 2024-05-28 18:31:55

This post is reserved for an index to posts that may be contributed by NewMars members over time.

(th)

tahanson43206 · 2024-05-28 18:34:41

I asked ChatGPT4o to compute a flight to orbit for a vehicle weighting 200 tons at launch, using the Rocket Equation.

ChatGPT4o used the 200,000 kg figure for propellant and it did not specify the propellant. Instead, it used a figure for exhaust velocity that i assume must be well established as a standard that can be achieved with chemical propellant.

[h1]Rocket Equation Analysis for Spacecraft Payload[/h1]
Introduction
The Tsiolkovsky rocket equation, or simply the rocket equation, is fundamental in understanding the relationship between a rocket's mass, the mass of its fuel, and the change in velocity (delta-v) it can achieve. This analysis will demonstrate the application of the rocket equation to determine if a spacecraft with an initial mass of 200 metric tons can place a 13-ton payload into orbit.
Rocket Equation
The rocket equation is given by:
Δv = v_e * ln(m_0 / m_f)
Where:
Δv is the change in velocity (delta-v).
v_e is the effective exhaust velocity.
m_0 is the initial total mass (including fuel).
m_f is the final mass (after the fuel is burned).
Initial Setup
Initial mass of the spacecraft (m_0): 200 metric tons (200,000 kg).
Desired payload to orbit: 13 metric tons (13,000 kg).
Calculation Example
Assumptions:
Effective exhaust velocity (v_e): 4,500 m/s (typical for chemical rockets).
Required Δv for Low Earth Orbit (LEO): 9,300 m/s.
Using the rocket equation:
9,300 = 4,500 * ln(200,000 / m_f)
Solving for m_f:
9,300 / 4,500 = ln(200,000 / m_f)
2.067 = ln(200,000 / m_f)
Exponentiate both sides to remove the natural logarithm:
e^2.067 = 200,000 / m_f
7.9 = 200,000 / m_f
m_f = 200,000 / 7.9 ≈ 25,316 kg
Therefore, the remaining mass after burning the fuel is approximately 25,316 kg.
Discussion
If the inert mass (dry mass without fuel) is set to zero, the payload mass would be close to the final mass.
In reality, the inert mass cannot be zero. However, with minimal inert mass, the payload mass will be slightly less than the final mass.
Achieving 13 tons of payload to orbit with 200 metric tons at liftoff is challenging but possible with higher specific impulse or more efficient propellant use.
Conclusion
By understanding and adjusting key parameters such as exhaust velocity and initial mass, we can better evaluate the feasibility of our payload goals. This analysis shows that placing a 13-ton payload into orbit with a 200-ton spacecraft is theoretically possible but requires careful consideration of the rocket's design and propellant efficiency.

I would greatly appreciate forum members running their own calculations and reporting the results.

(th)

SpaceNut · 2024-05-28 19:05:20

tsiolkovsky rocket equation
has been posted about in topics.

https://sentinelmission.org/rocketry-pr … t-equation

The equation is expressed as: Δv = Ve * ln (Mi/Mf)
Where:
Δv = velocity change of the rocket
Ve = exhaust velocity of the propellant
Mi = initial mass of the rocket (including propellant)
Mf = final mass of the rocket (after propellant is expelled)

To orbit 125 miles (200 kilometers) above Earth, a spacecraft must travel at a screaming 17,400 mph (28,000 km/h).

Your 33,480 km/hr is at a higher orbit

the x-33 was 129 mt with 2 engines of 1800 kN of trust each with LH2/LOX fuels and was to have a payload of 5.4 mT.

tahanson43206 · 2024-05-28 19:12:32

For SpaceNut ... in this topic I'm looking for members to show they can run the rocket equation.

You have shown that you can find a copy of it and paste it into a post, and that is an important skill.

However, it is not sufficient for ** this ** topic.

Please take the numbers quoted in post number 3, and see if you get the same result.

(th)

tahanson43206 · 2024-05-29 11:05:57

This is an experiment to see if GW Johnson's work (done in Microsoft Word) can be converted to bbcode for display here in the forum.

The document we will attempt to deliver is an answer to RGClark's SSTO scenario. I was surprised to see that the scenario would work at all, but it clearly does using GW's spreadsheet. The numbers for inert mass were set to zero, so that anything delivered to orbit shows up as payload. Obviously in real life, the inert mass would have value, and the payload would be set to zero. In that case, the ship itself ** is ** the payload.

Weight statement, metric tons
GWJ 5-29-2024
Payload: 13
Inert: 0 (ridiculous!)
Burnout: 13
Propellant: 200
Ignition: 213
Mass ratio = ign/bo = 213/13 = 16.3846, ln(MR) = 2.7963
Propulsion models (Vex = 9.80667*Isp/1000)
Isp, s
Vex, km/s
350
3.432
400
3.923
450
4.413
Rocket equation prediction at indicated mass ratio
dV = Vex ln(MR):
Isp, s
dV, km/s
350
9.597
400
10.970
450
12.340
Bear in mind these are ridiculous predictions:
It is NOT possible to build anything that has zero inert mass!
Look at the problem the other way around:
Mission (LEO eastward at low inclination, rendezvous, deorbit)
Required dV capability ~ 9.3 km/s
Propulsion
Isp, s
Vex, km/s
(dV/Vex)
350
3.432
2.70979
400
3.923
2.37063
450
4.413
2.10741
Rocket equation (reversed as MR = exp(dV/Vex))
Isp, s
MR
350
15.022
400
10.706
450
8.227
Weight statement-related:

Propellant mass fraction Wp/Wig = 1 – 1/MR
“allowance” = sum of payload and inert fractions = 1 – propellant mass fraction
Isp, s
Wp/Wig
allowance (= Wpay/Wig + Winert/Wig)
350
0.9334
0.0666 (everything that is “not propellant”)
400
0.9066
0.0934
450
0.8784
0.1216
Design considerations:

the “allowance” is your “design space”
Whatever you think your inert fraction is, subtract that from the allowance for payload fraction
Size the weight statement masses from the fractions and the payload you want to deliver
Size the thrust requirement using T/Wig = 1.5+
Size inert masses for engines, tanks, and equipment, then iterate until it matches inert fractions

The conversion was performed by ChatGPT4o, given the original text from a docx file, that had been rendered as odt format.

(th)

GW Johnson · 2024-05-29 13:25:46

The rocket equation as seen in all the books is quite simple: dV = Vex LN(MR) where MR = Wig/Wbo. It presumes a weight statement where Wpay + Winert = Wburnout, and W burnout + Wpropellant = Wignition. Vex is not the actual nozzle expansion velocity, but an effective exhaust velocity as if the exit plane expanded pressure were zero (so that only a momentum term appears in thrust). (In point of fact, Vex is the specific impulse value, if one uses dimensionally-consistent units of measure in thrust/flow rate.)

Everyone is taught to predict performance using the rocket equation "forward", with mass ratio MR and Vex as inputs. But for sizing designs, you need to use it backwards, in the reversed form MR = exp(dV/Vex). The mass ratio relates to the propellant mass fraction in a very simple way: Wpropellant/Wignition = 1 - 1/MR. So your mission sets the needed dV, and your propulsion sets the necessary value of Isp (and from it Vex = gc*Isp).

Use that to determine the propellant mass fraction required of the stage. Whatever is left after you subtract it from one is the sum of your inert and payload fractions, a sort of "allowance" for the non-propellant items. And THAT is the space in which you do your vehicle designs. What inert fractions those designs turn out to have, subtract from the allowance to determine payload fraction.

If your as-designed inert fraction exceeds the allowance, then payload fraction is inescapably negative, and your design is fundamentally infeasible, no matter what other requirements it might meet. The rocket equation feasibility requirement comes first and foremost.

GW

tahanson43206 · 2024-05-29 19:25:43

For GW Johnson... I gave your post to ChatGPT4o, and I am not surprised that it confirmed your observations. Here is the result:

The first try did not work... I'll try again.

Conclusion
The post by GW provides an accurate and practical explanation of the rocket equation and its application in rocket design. The focus on using the equation both forward for performance prediction and backward for design sizing is essential for practical rocketry. The method of determining propellant mass fraction and using it to allocate the remaining mass between inert structure and payload is a standard and effective approach in the industry.
Document 1: Review and Confirmation of GW's Work
Review of GW's Rocket Design Explanation
Conclusion: GW's explanation is accurate and practical. The focus on both forward and backward use of the rocket equation, along with determining propellant mass fraction and designing within constraints, aligns with industry standards in rocketry.
Document 2: Observations on Practical Rocket Design
Observations on Practical Rocket Design by GW
[]The remaining mass after accounting for propellant is allocated to inert structure and payload.
[]Ensuring the inert fraction is within allowable limits is critical for feasible design.
Conclusion: GW's approach to rocket design using the rocket equation is sound and reflects standard practices in the industry. Determining propellant mass fraction and designing within these constraints ensures practical and feasible rocket designs.

(th)

tahanson43206 · 2024-05-29 19:27:01

Here is another try:

Review of GW's Rocket Design Explanation
Key Points:
The rocket equation: \( \Delta V = V_{ex} \ln(\text{MR}) \)
Mass Ratio (MR) = \( \frac{W_{ig}}{W_{bo}} \)
Weight relationships: \( W_{pay} + W_{inert} = W_{burnout} \) and \( W_{burnout} + W_{propellant} = W_{ignition} \)
Effective exhaust velocity \(V_{ex}\) = \( I_{sp} \times g_0 \)
Forward and backward use of the rocket equation
Propellant mass fraction: \( \frac{W_{propellant}}{W_{ignition}} = 1 - \frac{1}{\text{MR}} \)
Designing within mass fraction constraints
Conclusion: GW's explanation is accurate and practical. The focus on both forward and backward use of the rocket equation, along with determining propellant mass fraction and designing within constraints, aligns with industry standards in rocketry.

and here is ** another ** try:

Review of GW's Rocket Design Explanation
Key Points:
The rocket equation: ΔV = Vex ln(MR)
Mass Ratio (MR) = Wig/Wbo
Weight relationships: Wpay + Winert = Wburnout and Wburnout + Wpropellant = Wignition
Effective exhaust velocity Vex = Isp * g0
Forward and backward use of the rocket equation
Propellant mass fraction: Wpropellant/Wignition = 1 - 1/MR
Designing within mass fraction constraints
Conclusion: GW's explanation is accurate and practical. The focus on both forward and backward use of the rocket equation, along with determining propellant mass fraction and designing within constraints, aligns with industry standards in rocketry.

(th)

tahanson43206 · 2024-05-29 19:27:55

And trying again for Document 2:

Observations on Practical Rocket Design by GW
Key Principles:
The rocket equation is fundamental in predicting performance and designing rockets.
Forward use of the rocket equation involves calculating \(\Delta V\) from known parameters.
Backward use involves determining the mass ratio from mission \(\Delta V\) and effective exhaust velocity.
Propellant mass fraction: \( \frac{W_{propellant}}{W_{ignition}} = 1 - \frac{1}{\text{MR}} \)
The remaining mass after accounting for propellant is allocated to inert structure and payload.
Ensuring the inert fraction is within allowable limits is critical for feasible design.
Conclusion: GW's approach to rocket design using the rocket equation is sound and reflects standard practices in the industry. Determining propellant mass fraction and designing within these constraints ensures practical and feasible rocket designs.

(th)

SpaceNut · 2024-06-01 14:51:09

Not sure if I have all of the numbers yet but wanted to see what I could do with stages that are already con structed.

The equation is expressed as: Δv = Ve * ln (Mi/Mf)

Natural log IN

Where:
Δv = velocity change of the rocket
Ve = exhaust velocity of the propellant
Mi = initial mass of the rocket (including propellant)
Mf = final mass of the rocket (after propellant is expelled)

Falcon 9 first stage modified for ssto.
Mass (without propellant)[39] 22,200 kg (48,900 lb)
Mass (with propellant) 433,100 kg (954,800 lb)
Liquid oxygen tank capacity 287,400 kg (633,600 lb)
Kerosene tank capacity 123,500 kg (272,300 lb)
Payload fairing 1,700 kg (3,700 lb)

Thrust (stage total)[4] 7,607 kN (1,710,000 lbf) (sea level)
Specific impulse Sea level: 282 seconds

Propulsion models (Vex = 9.80667*Isp/1000) 2.76548094
MR = 433,100 kg / 22,000 kg = 19.6863
Rocket equation (reversed as MR = exp(dV/Vex))
Δv = 2.76548094 * 2.97992296 = 8.24092

kbd512 · 2024-06-01 19:27:11

SpaceNut,

I ran numbers. Merlin-1D's Isp is too low. You either get no payload or very little, with 3,000kg being best case scenario. If the avg Isp is 338s (staged combustion), then you get 7,305kg to 17,799kg, with 12,060kg being the estimate. This is for an expendable SSTO, not reusable. I presume engine weight remains the same, Isp increases by using staged combustion, and CFRP tanks weigh 50% of what the Aluminum tanks weigh.

Falcon 9 expendable is 22,800kg.
Falcon 9 reusable is 17,500kg (requires landing the booster downrange on the drone ship).

SpaceNut · 2024-06-02 17:20:00

https://www.omnicalculator.com/physics/specific-impulse

Seems that the engine running on the RP/Lox are all about the same even the Russian models that we are building a replacement for.

Seems even methane / lox is not all that much better.

https://en.wikipedia.org/wiki/Compariso … et_engines

SpaceNut · 2024-06-23 19:23:28

Rocket Thrust Equations

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#1 2024-05-28 18:31:21

Rocket Equation

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Re: Rocket Equation

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#4 2024-05-28 19:05:20

Re: Rocket Equation

#5 2024-05-28 19:12:32

Re: Rocket Equation

#6 2024-05-29 11:05:57

Re: Rocket Equation

Weight statement, metric tons

Propulsion models (Vex = 9.80667*Isp/1000)

Rocket equation prediction at indicated mass ratio

Propulsion

Rocket equation (reversed as MR = exp(dV/Vex))

Weight statement-related:

Design considerations:

#7 2024-05-29 13:25:46

Re: Rocket Equation

#8 2024-05-29 19:25:43

Re: Rocket Equation

#9 2024-05-29 19:27:01

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#10 2024-05-29 19:27:55

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#11 2024-06-01 14:51:09

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#12 2024-06-01 19:27:11

Re: Rocket Equation

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