Payload Fraction Accounting for Gross-Mass-Sensitive Term

As I prepared for this post tonight, I realized that I wasn’t really modifying the rocket equation at all–I have been using the rocket equation and a summation of mass terms to find the payload fraction, which I consider an especially useful value to know.

Furthermore, if you read my previous post, you probably figured out pretty quickly that all of the dry mass in the rocket doesn’t correlate to the propellant mass. That’s a pretty good guess for very large rockets with large delta-V’s, but as you get smaller, the assumption really starts to fall apart.

One of the big masses in a rocket outside of the propellant tanks are the engines themselves, and they really don’t correlate to propellant mass at all. They correlate to the initial mass, because the engines are typically sized to give you some particular value of thrust-to-weight when you light them up. So I went ahead and extended the previous derivation of payload fraction to include this important feature.

So let’s define some terms. Lambda is the relationship between the vehicle dry mass that correlates to propellant, and phi is the relationship between vehicle dry mass that correlates to initial mass.

     \begin{displaymath} \lambda = \frac{m_{propellant-mass-dependent}}{m_{propellant}} \end{displaymath}

     \begin{displaymath} \phi = \frac{m_{initial-mass-dependent}}{m_{initial}} \end{displaymath}

We plug these definitions into an equation for initial mass, and I simplified things by just going ahead and combining the propellant-sensitive mass into (1+lambda)*propellant_mass. Then I went ahead and replaced propellant mass with the expression from the rocket equation that related propellant mass to initial mass, and now I had the equation entirely in terms of payload mass and initial mass, which is exactly what I was after.

     \begin{displaymath} m_{initial} = m_{payload} + \phi m_{initial} + (1 + \lambda) m_{propellant} \end{displaymath}

     \begin{displaymath} (1 - \phi) m_{initial} = m_{payload} + (1 + \lambda) m_{initial}\left(\frac{MR - 1}{MR}\right) \end{displaymath}

     \begin{displaymath} m_{payload} = (1 - \phi) m_{initial} - (1 + \lambda) m_{initial}\left(\frac{MR - 1}{MR}\right) \end{displaymath}

     \begin{displaymath} \frac{m_{payload}}{m_{initial}} = (1 - \phi) - \left(\frac{(1 + \lambda) (MR - 1)}{MR}\right) \end{displaymath}

Then it was a matter of doing the algebra to simplify things down until I had an expression for payload fraction like I did before.

     \begin{displaymath} \frac{m_{payload}}{m_{initial}} = \frac{MR (1 - \phi) - (1 + \lambda) (MR - 1)}{MR} \end{displaymath}

     \begin{displaymath} \frac{m_{payload}}{m_{initial}} = \frac{MR - \phi MR + (1 + \lambda - MR - \lambda MR)}{MR} \end{displaymath}

     \begin{displaymath} \frac{m_{payload}}{m_{initial}} = \frac{1 + \lambda - \lambda MR - \phi MR}{MR} \end{displaymath}

     \begin{displaymath} \frac{m_{payload}}{m_{initial}} = \frac{1 - \lambda (MR - 1) - \phi MR}{MR} \end{displaymath}

     \begin{displaymath} \frac{m_{payload}}{m_{initial}} = \left(\frac{1}{MR}\right) - \lambda\left(1 - \frac{1}{MR}\right) - \phi \end{displaymath}

Noting that 1/MR is the final mass fraction (FMF), and that 1 – 1/MR is the propellant mass fraction (PMF), things continue to simplify:

     \begin{displaymath} \frac{m_{payload}}{m_{initial}} = FMF - \lambda(PMF) - \phi \end{displaymath}

Take a look at the final equation. You’ll note a few things. The first one is that if phi is zero, then the expression is just the same as the one I derived yesterday. The second one is how lambda and phi impact the payload fraction differently. You can see that the effect of lambda is reduced somewhat by having the propellant mass fraction multiplied by it (and the PMF is always less than one) but that phi has no such reduction.

This is because in the case where you had a vanishing small impulse, MR would be very close to one. And one minus one would be zero and the impact of lambda would be almost eliminated. But because phi is multiplied by one, its effect is still present even if the impulse was very small. This is because any rocket whose engines were sized to deliver a particular thrust-to-weight at ignition would still pay a mass penalty for those engines, even if it just briefly turned them on and then turned them off.

In an upcoming post I will show how to calculate lambda and phi from other vehicle design parameters, like mixture ratio, tank mass per unit volume, propellant densities, thrust-to-weight ratios of engines and vehicles, and so forth.

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MS, nuclear engineering, University of Tennessee, 2014, Flibe Energy, president, 2011-present, Teledyne Brown Engineering, chief nuclear technologist, 2010-2011, NASA Marshall Space Flight Center, aerospace engineer, 2000-2010, MS, aerospace engineering, Georgia Tech, 1999

About Kirk Sorensen

MS, nuclear engineering, University of Tennessee, 2014, Flibe Energy, president, 2011-present, Teledyne Brown Engineering, chief nuclear technologist, 2010-2011, NASA Marshall Space Flight Center, aerospace engineer, 2000-2010, MS, aerospace engineering, Georgia Tech, 1999
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2 Responses to Payload Fraction Accounting for Gross-Mass-Sensitive Term

  1. Pete says:

    I remember a great analysis of wind turbine design where mass/cost components were broken down and grouped by what they scaled with. For example power scaled with diameter squared, torque scaled with diameter cubed (const tip speed), gearbox mass/cost scaled with torque, blade and tower mass scaled with diameter cubed, generator mass/cost scaled with power (diameter squared), etc. Very informative and it largely explains why wind turbines are not bigger than they are.

    I always wanted to do a similar parametric model for rocket vehicles, incrementally increasing the accuracy of various scaling relationships as required and including options for different rocket vehicle types, mission profiles, propellant choices, and so forth. Once such a cost and/or mass model is developed it could then be searched to find optimal designs, explore sensitivities, evaluate the benefit of different design tricks, etc. Ultimately one would still want to do a from scratch paper design or two to check the veracity of the final design.

    But unfortunately I never did this at more than a crude level. Developing the scale relationships for everything and estimating costs requires significant expertise.

  2. Ian Woollard says:

    @Pete To a fair approximation rockets are just a collection of pressure vessels. Pressure vessels are completely boring with regards scaling laws, except at the smallest scales (where minimum gauge bites) they scale perfectly linearly with volume.

    There is a small effect for pipes though- a bigger diameter pipe gets more flow because you get less viscosity losses down the walls, so big rockets are at an advantage. So turbopumps and pipes are more mass efficient at large scale. But for low accelerations none of that matters much.

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