Optimization of Rockets With Variable Exhaust Velocity

1. Abstract

Almost all large-scale rockets today are chemical reaction-based, and so their total delta-v is limited primarily by their maximum exhaust velocity. However, if future rocket technologies can deliver a sufficiently high exhaust velocity, the total energy or total mass expended becomes the limiting factor. In this paper, we consider the optimal path of such a rocket through the mass-energy parameter space, and calculate the final delta-v as a function of the endpoints along such a path. We also consider Robert Zubrin’s nuclear salt water rocket design as a case study in theoretical limits on engine performance.

 

2. Background

Chemical-based rockets, which, until now, have been used for all Earth-based launches and practically all space travel, are limited in their capabilities primarily by how high an exhaust velocity an engine can achieve, which is in turn limited by the reaction energy of the chemicals powering the engine. As a simple example, consider the Saturn V rocket; the Saturn V’s three stages contained a total of kg of kerosene/LOX propellant and  kg of LH2/LOX propellant (Wade 2008). These fuels, if burned at the efficiency of the Saturn V’s engines, contain  J of energy. The Apollo spacecraft itself masses only  kg, and was accelerated to a velocity of  m/s, for a total energy of  J, or an efficiency of 22.8%.

Even assuming fixed exhaust velocity, this is far less than optimal. The reason why is obvious: for a fixed exhaust velocity rocket, the energy required for a given delta-v  is:

where  is the initial mass of the rocket and  is the exhaust velocity. Taking the derivative with respect to , we get:

The optimal value for  can then be solved for analytically; this turns out to be, for any value of , and the total energy required is equal to, or 54.4% more than the kinetic energy of the payload. It is clear, then, that the exhaust velocity produced by chemical rockets is significantly lower than that the optimal velocity for minimal-energy space missions even within our own solar system; the optimal delta-v for the Space Shuttle main engines, the highest exhaust velocity chemical engines currently in use, is only 6,570 m/s, lower than that required for LEO. This simple fact means that virtually all current rockets are optimized to give the highest possible exhaust velocity, rather than optimizing for energy or mass usage.

 

3. Variable exhaust velocity

We now consider the behavior of a hypothetical rocket in which the rate of reaction mass expenditure and the rate of energy expenditure are completely independent; that is, the source of reaction mass and the source of energy are separate, and arbitrarily high or low amounts of energy can be applied to each unit of reaction mass. Consider the two-dimensional parameter space , where  is the internal (potential) energy of the rocket, and  is the total mass of the rocket.

Before any thrust is applied, any given rocket will be at some point  in this parameter space. After all the available energy has been expended, the rocket moves to the point , where it is assumed that  and . The path from  to  can be expressed as the function , where f is continuous and monotonically increasing. The exhaust velocity of the rocket for any value of is, then, simply:

where  is the specific energy of the reaction mass. The conventional assumption that  is constant is, therefore, equivalent to assuming that  is a constant, or that the path along  from  to  is a straight line.

At a given point , define  to be the (infinitesimal) amount of energy expended at that point,  to be the amount of reaction mass expended, and  to be the amount of velocity gained. We know from classical mechanics that:

and so it is easy to show that:

We want to optimize the total change in velocity, , which can be calculated by simply integrating this quantity:

where  is just the first derivative of the function  defined earlier. If we define  to be the function  which is integrated over, then if  is optimal, the classical Euler-Lagrange equation must hold:

Since Q has no explicit dependence on f(m), this is equivalent to saying that:

Taking the derivative, this reduces to the simple first-order ordinary differential equation:

whose solution is:

where  and  are constants. If we plug in the beginning and end points of the rocket’s path through parameter space into this equation, we get the equations:

 

from which we can derive the function describing the optimal potential energy at any point:

 

 

To get the total delta-v, we plug this function back into the integral between  and , and integrate:

 

This equation, then, represents the end state of a rocket simultaneously optimized for both energy and mass; it describes the maximum velocity attainable by any engine given those initial parameters. Note that, if mass is removed as a constraint by taking the limit as  reduces to , which is just the velocity of a body with mass  and kinetic energy . Hence, if reaction mass is removed as a constraint, the rocket becomes, not just optimally efficient, but perfectly efficient at converting potential energy into kinetic energy, although this is only a theoretical maximum.

 

4. The nuclear salt-water rocket

As a case study, we can examine the proposed nuclear salt water rocket of Dr. Robert Zubrin. The nuclear salt water rocket (NSWR) works by using a solution of a soluble uranium compound in ordinary or heavy water to form a nuclear critical mass. This flashes the water into high-temperature steam, which is then expelled out of the rocket at extremely high velocity. Such a rocket could have an extremely high specific impulse, roughly in the range of  seconds.

Suppose that such a rocket, with a  mass ratio, were to be launched towards the outer solar system; using Tsilokovsky’s equation, we can calculate a theoretical final velocity of roughly  m/s. However, if we allow the exhaust velocity to vary- using a lower concentration of uranium at the beginning, and a higher concentration at the end- we get a theoretical final velocity of  m/s, a  improvement over the original.

 

5. References

Wade, Mark. “Saturn V.” Encyclopedia Astronautica. 2008. 25 Oct. 2009.

Analog Science Fiction and Fact Magazine 15 Dec. 1992. Center for Experimental Nuclear Physics and Astrophysics. University of Washington, 12 July 1996. Web. 25 Oct. 2009..

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