Trouble with fluid thermodynamics and nuclear thermal rockets

[Robert DeVries]

Summary:: In need of help determining the exhaust velocity of a rocket nozzle given temperature and propellant molar mass

Greetings and salutations! My name is Robert DeVries, world builder extraordinaire. I have come with questions in search of answers.

So for the last few days I've been trying to figure something out: Why is the ISP I'm getting from this equation so much higher than it should be? I think some context is in order.

I like to write science fiction, and I'm currently working on my first short story. However before I start truly beginning to write I'd like to finish designing the basics of this interplanetary ship: a nuclear thermal rocket-propelled warship. The website Project Rho, specifically its sub-site Atomic Rockets, has been extremely helpful as far as back-of-the-envelope equations go, allowing me to put in enough detail to my creations to seem at least half-way realistic. However now I've hit a brick wall, their section on the solid-core NTR claims that the general exhaust velocity/specific impulse of a NTR - and solid-core rockets in particular - is proportional to this equation [1/sqrt(molar mass of propellant)]. They have provided a table that they claim was created using this equation with an internal core temperature if 3,200 K, however when I use this equation on those same propellants the ISP is always at least 20% higher than it should be, up to 60%. Here's my spread sheet using the equation on several propellants:

In order to try and figure out where I went wrong I started looking up various methods of determining the exhaust velocity of a rocket nozzle which led me here: https://www.grc.nasa.gov/WWW/K-12/rocket/rktthsum.html
And that is where the brick wall hit me again.

The first think that really stumped me was Gamma. According to the links that this page leads to Gamma is determined by YET ANOTHER massive wall of equations, which I cannot hope to figure out within the next day. If anyone has some kind of solution for me, beit in the form of a video lecture, detailed explanation, or (please gods) and alternative method of determining the exhaust velocity of a hot gas, I'd be eternally grateful.

Yours truly,
Robert DeVries

 

[member 656954]

Is your story rock hard science fiction, @Robert DeVries? If not, readers are unlikely to understand anything to do with exhaust velocity/specific impulse, and even fewer will be able to do the math to verify whatever figures you write up. And even if it is, a short story full of facts and figures may not be as engaging as you hope. So, unless the exhaust velocity is critical to the story, just pick an impressive sounding number of get on with the harder part of writing a compelling narrative.

 

[DrStupid]

They have provided a table that they claim was created using this equation with an internal core temperature if 3,200 K, however when I use this equation on those same propellants the ISP is always at least 20% higher than it should be, up to 60%.

Using

$v_e = \sqrt {\left( {f + 2} \right) \cdot \frac{{R \cdot T}}{M}}$

with the degrees of freedom $f$ (I set it to three times the number of atoms per molecule), gas constant $R$, temperature $T$ and molar mass $M$, I get slightly better results. However, such calculations will always give you an estimation only. But that's sufficient for a science fiction story. To be out by orders of magnitude could be a problem. But that doesn't seem to be the case.