TDK Propulsion

C*

This post is about the propellant parameter, but I had to put a picture in of the world’s most awesome N motor, too, since it’s burning CTI’s C* propellant. (That’s in James Dougherty’s 1/2 scale Patriot — click through and scroll down for the video.) Continuing in the theme of previous theory posts, this one will be about the wonderful, beautiful term called “characteristic velocity”, or C*. C* is one of the terms that is extremely helpful in correlating theoretical and delivered performance, and in discussions I’ve had with lots of rocketeers over the past year or so, it also seems to be somewhat misunderstood. So let’s start at the beginning and see how useful it really is.

C* and $I_{sp}$ are related, and both are found in various forms of the equation for thrust:

$F_{th} = P_0A^*C_F = \dot m I_{sp} g = \dot m C^* C_F$

where $F_{th}$ is the thrust force, $P_0$ is the chamber pressure, $A^*$ is the throat area, $C_F$ is the thrust coefficient (provided by the nozzle), $\dot m$ is the propellant mass flow rate, and $g$ is the gravitational constant. Some quick rearranging shows that

$C^* = \frac{I_{sp}g}{C_F} = \frac{P_0A^*}{\dot m}$

which is also pretty handy, since we know all the terms in the rightmost equation, or at least can measure them directly. This means that we can calculate C* from test data – hey, another reason to build a test stand!

The true beauty of C* comes out, though, when we look at it from the other direction, ahead of time. As motor designers, we get to pick $P_0$ and $A^*$ , so all we need to calculate is $\dot m$ , and we can calculate C*. And $\dot m$ isn’t that bad to derive, either. So let’s give it a go, yes?

Everything in gas dynamics has something to do with the continuity equation:

$\dot m = \rho UA$

where $\rho$ is the gas density, $U$ is the gas velocity, and $A$ is the flow area in question. Since we don’t have a density meter to measure $\rho$ directly, it’s easiest to rewrite $\rho$ in terms of things we know, using the perfect gas law:

$P = \rho R_g T \rightarrow \rho =\frac{P}{R_gT}$

where $T$ is the temperature at the given location, and $R_g$ is the real gas constant (just the ideal gas constant divided by the molecular weight — $\frac{R}{\mathfrak{M}}$ ) for the gas in question. The only place that we know the flow velocity $U$ with certainty is at the throat, where we know it’s going exactly sonic speed. (All throat conditions are denoted by superscript *, by the way.) Leaning on one more important gasdynamic relation, we know that $U$ at the throat is simply

$U^* = \sqrt{\gamma R_g T^*}$

where $\gamma$ is the ratio of specific heats for the gas and $T^*$ is the temperature of the gas at the throat. Rewriting the continuity equation with our newly defined parameters, using conditions at the throat *:

$\dot m = \frac{P^*A^*}{R_gT^*}\sqrt{\gamma R_g T^*}$

The only problem with this equation (let’s call it Eq. (1)) is that everything is in terms of throat conditions — and just imagine how much of a pain it would be to measure stuff at the throat, where the heat flux has peaked and all your thermocouples and pressure tubes just melted. Yeah. Not fun. To make things easier, we can use the magic of the isentropic flow relations to move everything upstream into the chamber, where we can measure it (and consider it) with a bit more ease. Pause a minute to reflect and remember these old friends from the NASA Education page (or your favorite gas dynamics textbook):

$\frac{P_0}{P} = \left(1+\frac{\gamma-1}{2}M^2\right)^\frac{\gamma}{\gamma-1}\;\;\;\;\;\frac{T_0}{T} = \left(1+\frac{\gamma-1}{2}M^2\right)$

They give stagnation (chamber) pressure and temperature as a function of $\gamma$ and downstream Mach number $M$ . And since we’re dealing with throat conditions here, $M^*=1$ (oh, life is sweet), so these equations collapse into something even simpler. Rewrite and solve for $P = P^*$ and $T = T^*$ :

$P^* = P_0\left(\frac{2}{\gamma+1}\right)^\frac{\gamma}{\gamma-1}\;\;\;\;\;T^* = T_0\left(\frac{2}{\gamma+1}\right)$

and then substituting these into Eq. (1) gives us the lovely mess of

$\dot m = \frac{P_0\left(\frac{2}{\gamma+1}\right)^\frac{\gamma}{\gamma-1}A^*}{R_gT_0\left(\frac{2}{\gamma+1}\right)}\sqrt{\gamma R_g T_0 \left(\frac{2}{\gamma+1}\right)}$

Since I haven’t thrown up my hands in frustration and walked away from this post, you know it has to get simpler. So we massage all the $\frac{2}{\gamma+1}$ terms around

$\dot m = \frac{P_0\left(\frac{2}{\gamma+1}\right)^\frac{(\gamma+1)}{2(\gamma-1)}A^*}{R_gT_0}\sqrt{\gamma R_g T_0}$

and then push around $R_g$ and $T_0$ :

$\dot m = P_0A^*\left(\frac{2}{\gamma+1}\right)^\frac{(\gamma+1)}{2(\gamma-1)}\sqrt{\frac{\gamma}{R_gT_0}}$

and we’re done. It’s not that bad, when you look at it for a while, and we know everything in that equation from doing propellant combustion calculations ( $\gamma$ , $T_0$ , $R_g$ ) or motor design ( $P_0$ , $A^*$ ). But wait… we were starting on our quest to solve $C^*$ , so things get even simpler:

$C^* = \frac{P_0A^*}{\dot m} = \frac{P_0A^*}{P_0A^*\left(\frac{2}{\gamma+1}\right)^\frac{(\gamma+1)}{2(\gamma-1)}\sqrt{\frac{\gamma}{R_gT_0}}} = \sqrt{\frac{R_g T_0}{\gamma}}\left(\frac{\gamma+1}{2}\right)^\frac{(\gamma+1)}{2(\gamma-1)}$

or, massaging a little more,

$C^* = \sqrt{\frac{R_gT_0}{\gamma}\left(\frac{\gamma+1}{2}\right)^\frac{\gamma+1}{\gamma-1}}\;\;.$

At this point it should become apparent why C* is such a valuable term — in this form, it depends solely on the propellant’s combustion characteristics, so we can calculate it ahead of time with excellent accuracy. And because its other definition relies on things that we can measure with some simple instrumentation, we can measure it with decent accuracy as well. So C* is one of the most useful parameters to keep track of, because it quickly and easily tells us exactly how well our rocket motor combustion device is performing. If $P_c$ is lower than expected for a given throat area, C* must be lower than expected, indicating inefficiencies in the design.

Also, hopefully this exercise has taken some of the magic out of equilibrium programs like PEP or CEA that spit out a C* value from simple chemical formulation inputs. It’s not really any magic — just some math. And it’s only a short hop from there to $I_{sp}$ and performance calculation.

So learn C*. Calculate it. Measure it. Love it. It’s one more way to have fun with data from your test stand, and a good way to measure how well you’re doing as a research rocketeer.

Oh yeah, and the video from James Dougherty:

That’s his 7.5″ 1/2 scale Patriot on an N5800 C-Star. Wow. Can’t wait to see one of these in person.

Posted August 19, 2010 in Propulsion Theory. Add Comment / Trackback

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