Tsiolkovsky rocket equation

Tsiolkovsky rocket equation

The Tsiolkovsky rocket equation, or the classical rocket equation, or ideal rocket equation is an equation that mathematically describes the relationship between the mass of a rocket including the fuel and the propellant being expelled to allow for the rocket to move. 

The lower the weight, the easier it is to escape the Earth’s gravitational pull. There is an interesting thought. The fuel is also part of the rocket weight that is being transported, which means that you need some fuel to carry fuel.

SpaceX Falcon Heavy. Source> wiki

It is credited to Konstantin Tsiolkovsky, who independently derived it and published it in 1903, although it had been independently derived and published already by William Moore in 1810, and later published in a separate book in 1813. Robert Goddard also developed the equation independently in 1912, and Hermann Oberth derived it independently about 1920.

Initially at time  t  = 0, the mass of the rocket, including fuel, is m₀. We suppose that the rocket is burning fuel at a rate of  b kg s^-1 so that, at time  t, the mass of the rocket plus remaining fuel is  m=m₀−bt. The rate of change of mass with time is dm/dt=−b  and is supposed to be constant with time. (The rate of change is, of course, negative.)

The maximum change of velocity of the vehicle, Δv (with no external forces acting) is

where:

• v_e=I_spg₀ is the effective exhaust velocity;
• I_sp is the specific impulse in dimension of time;
• g₀ is standard gravity;
• ln is the natural logarithm function;
• m₀ is the initial total mass, including propellant, a.k.a. wet mass;
• m_f is the final total mass without propellant, a.k.a. dry mass.

Given the effective exhaust velocity determined by the rocket motor's design, the desired delta-v (e.g., orbital speed or escape velocity), and a given dry mass m_f, the equation can be solved for the required propellant mass m₀−m_f: m₀=m_fe^Δv/v_e.

The necessary wet mass grows exponentially with the desired delta-v.

A rocket's required mass ratio as a function of effective exhaust velocity ratio; source: Wiki

Tsiolkovsky's theoretical rocket from t = 0 to t =Δt. Source: wiki

Rocket Equation: Derivation

 

The value m₀-m_f (or M_i-M_f) is the total working mass of propellant expended. ΔV (delta v) is the integration over time of the magnitude of the acceleration produced by using the rocket engine (what would be the actual acceleration if external forces were absent). In free space, for the case of acceleration in the direction of the velocity, this is the increase of the speed. In the case of an acceleration in opposite direction (deceleration) it is the decrease of the speed. Of course gravity and drag also accelerate the vehicle, and they can add or subtract to the change in velocity experienced by the vehicle. Hence delta-v may not always be the actual change in speed or velocity of the vehicle.

V_e=I_sp.g₀ where I_sp is the specific impulse in dimension of time (usually, seconds) and g₀ is the standard acceleration due to gravity in a vacuum near the surface of the Earth. Its dimension is distance per time squared and it is equal approximately to 9.81 m/s².

The specific impulse is a concept to the fuel efficiency in automobiles, which is measured in kilometers per liter of given fuel. In rocket or airplane engines the specific impulse represents the trust per unit propellant flow rate. In other words, it represents the force generated by the engine with respect to the fuel used in a given amount of time. It is the significant value that describes the performance of any rocket or jet engine and its propellant or fuel.

A higher specific impulse means better performance, that is, more power can be produced for a given weight. The higher the specific impulse, the less propellant is needed to produce a given thrust for a given time. The specific impulse of a rocket or aircraft engine is the total number of seconds that the engine can deliver thrust equal to the weight of the total propellant (for a rocket) or fuel (for an aircraft) mass under Earth’s gravitational acceleration g₀.

Delta-v or ΔV is a scalar quantity and it has the dimension of a speed. It is not the same as the physical change in velocity of the vehicle. Only if the thrust of the engine is applied in a constant direction (without changing of yaw and pitch of the spacecraft), Δv is simplified to the magnitude of the change in velocity.

A plot of Tsiolkovsky equation for different effective velocities; source: https://www.translatorscafe.com/

Until now, we have been talking about rocket in one piece, respectively single-stage rockets. What happens in case of multiple-stage rockets?

A single-stage rockets are very limited in the payload they can carry for example to the LEO (low Earth orbit). They would be very large and still the payload would be less than 1% of the gross takeoff mass of the system. So, it is very useful to create a system of multiple parts that can be thrown away on the way to space and then use smaller part of the rocker with smaller initial full mass. That's called staging. That can be either serial or parallel. 

Parallel and Serial staging; source: NASA

The most efficient  quantity of stages is between two and five. One advantage of the multistage design is that each stage can use a different type of engine that fits better for its particular operating conditions. For example, the lower stage motors or engines are designed to be used at atmospheric pressure, while the upper stages can use engines that are designed to work in a vacuum conditions.

For a multistage rocket consisting of n stages, the final velocity increment Δv_f is determined as

Quite a big part of the mass of the rocket is its fuel tank.  The more fuel we have, the more tank we need. But as we spent the fuel, the big tank remains, slowing down the rocket. So, rockets are most of the times designed as multi-stages rockets. That means that instead of one big tank, we have multiple smaller tanks. Each stage after burning out all the fuel, discard the stage, including the tank and the engine.  

Small note here: You can see the huge improvement of SpaceX reusing their first stage, as it lands back on an Autonomous Spaceport Drone Ship.

Rocket staging is a critical aspect of rocket technology, with each type providing unique advantages and challenges.

Single-Stage Rockets are straightforward to design and operate since they only have one engine or engine system. This simplicity makes them ideal for smaller payloads or missions requiring quick response times. As single-stage rockets do not have any separation or discard mechanism, their fuel reserves are therefore limited, which means that they can only reach a limited altitude before running out of fuel.

Two-Stage Rockets offer better efficiency than their single-stage rocket versions since once the first stage has depleted its fuel reserves, it is thrown away. By using staged combustion techniques, two-stage rockets can reach higher altitudes than single-stage vehicles.

:-)