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## Calculating the Ideal Size of a Rocket Engine and the Decision to Combine Multiple Engines

Designing a rocket requires careful consideration of a number of variables, including the size and number of engines. So, how does one calculate the ideal size of a rocket engine and when does it make sense to combine several engines? Let’s delve into these fascinating aspects of rocket design.

### Calculating the Ideal Size of a Rocket Engine

The performance of a rocket engine is typically evaluated by its thrust. Calculating the thrust of a rocket engine involves several variables including the mass flow rate through the engine, the exit velocity of the exhaust, and the pressure at the nozzle exit. The mass flow rate, in particular, is determined by the throat area of the nozzle, the smallest cross-sectional area of the nozzle.

The mass flow rate (m dot) is given by the formula

$\stackrel{.}{m}=\frac{{A}^{*}{p}_{t}}{\sqrt{{T}_{t}}}\sqrt{\frac{\gamma }{R}}{\left(\frac{\gamma +1}{2}\right)}^{–\frac{\gamma +1}{2\left(\gamma –1\right)}}$

The area ratio from the throat to the exit (Ae) sets the exit Mach number. The formula for the area ratio is

$\frac{{A}_{e}}{{A}^{*}}={\left(\frac{\gamma +1}{2}\right)}^{–\frac{\gamma +1}{2\left(\gamma –1\right)}}{\frac{\left(1+\frac{\gamma –1}{2}{M}_{e}^{2}\right)}{{M}_{e}}}^{\frac{\gamma +1}{2\left(\gamma –1\right)}}$

Once we have the exit Mach number, we can calculate the exit pressure (pe) and exit temperature (Te) using the isentropic relations at the nozzle exit. The formulas for the exit pressure and temperature are

$\frac{{p}_{e}}{{p}_{t}}={\left(1+\frac{\gamma –1}{2}{M}_{e}^{2}\right)}^{–1}$ $\frac{Te}{{T}_{t}}={\left(1+\frac{\gamma –1}{2}{M}_{e}^{2}\right)}^{–\frac{\gamma }{\gamma –1}}$

Knowing the exit temperature, we can calculate the exit velocity (Ve) using the equation for the speed of sound and the definition of the Mach number. The formula for the exit velocity is

${V}_{e}={M}_{e}\sqrt{\gamma R{T}_{e}}$

Finally, we can calculate the thrust (F) of the rocket using the generalized thrust equation, which accounts for the fact that the exit pressure is only equal to free stream pressure at some design condition. The formula for the thrust is

$F=\stackrel{.}{m}{V}_{e}+\left({p}_{e}–{p}_{0}\right){A}_{e}$

### When to Combine Multiple Engines

Adding more engines or scaling up the size of existing engines are both valid ways of increasing thrust. However, the decision to use one approach over the other involves a careful balancing act.

On one hand, using a single large engine can lead to unstable exhausts where the combustion products ‘stick’ to one side of the nozzle, to a first degree of approximation similar to how a shower head when not turned on fully will run in a single stream rather than many small jets. The Rocketdyne F-1 engine used on the Saturn V is often considered the biggest practical size of engine. The Saturn V used 5 such engines, which meant that if one were to fail, there’d be very little in the way of backup.

On the other hand, using many smaller engines, while solving the problem of having a backup, presents its own set of challenges. The more engines you have, the higher the chances that having one fail catastrophically will result in the destruction of the entire craft. The plumbing alone can be a logistical nightmare to solve, as demonstrated by the N-1 rocket, which used 30 smaller engines in its first stage and flew four times, exploding on each occasion.

Moreover, it’s important to note that thrust is close to but not quite additive when multiple engines are involved. Plume-plume interactions can result in total thrust being slightly less than the sum of the thrusts from the individual engines.

Additionally, when multiple engines are used, the engines’ thrust can be vectored in different directions. However, the thrust available for acceleration of the spacecraft is reduced by the factor cos(θ), commonly referred to as “cosine losses”. This effect is due to the thrust not being aligned with the spacecraft’s center of mass, leading to a slight loss of linear acceleration.

### Conclusion

Designing a rocket engine involves complex mathematical calculations and engineering considerations. Choosing the right size and number of engines is a balancing act, involving trade-offs between thrust, stability, complexity, and risk of catastrophic failure. The ideal solution often lies somewhere in between the extremes of a single large engine and many smaller ones.