Jet and rocket engines, simply

Internal combustion engines (ICE) are the most common form of heat engines, as they are used in vehicles, ships, airplanes, etc. The name is such because the combustion of fuel and an oxidizer occurs in  a certain space called chamber.  Simply say, fuel is ignited and work is done inside the engine. The same fuel and oxidizer, which is typically air, is emitted as an exhaust. That can happen either by using piston or turbine. 

Internal combustion heat engines work on the principle of the ideal gas law (pV=nRT), which claims that raising the temperature of a gas increases the pressure that makes the gas to expand. An internal combustion engine has a chamber, which has fuel added to it which ignites in order to raise the temperature of the gas. When the heat is added, the gas expands. Piston raises in case of piston engines. In case of turbine, the hot air is forced through the chamber and turns the turbine. Piston engine works in a cycle, so it is intermittent type of combustion engine, whereas the turbine is continuous type of combustion engine as it continuously exhausts the gas. 

A diagram of a turbine engine.

But only part of the energy is able to convert to useful work.  Heat engine efficiency, or thermal efficiency, is determined by the ratio of temperatures reached in the engine to that exhausted at the nozzle, and the upper limits is given by the second law of thermodynamics, also known as the Carnot efficiency. 

The energy efficiency of jet engines installed in aircraft or any other vehicle has two main components: propulsive efficiency and thermal efficiency. We have mentioned the thermal efficiency before. The other component tells how much energy ends up in the body of the vehicle rather than as kinetic energy of the jet. The propulsive efficiency is highest as the exhaust jet velocity gets closer to the vehicle speed as this gives the smallest residual kinetic energy. 

The formula for air-breathing engines moving at speed v with an exhaust velocity ve, is

For jet

For rocket

Note: Both equations are copy pasted.

Dependence of propulsion efficiency (η) upon the vehicle speed/exhaust velocity ratio (v/ve) for air-breathing jet and rocket engines:

Power-to-weight ratio or also called specific power is another measure of an engine performance. It is defined as a power output (W) divided by its mass (kg). Gas turbines for aircrafts have typically the largest specific power. 

Turbine has superior power-to-weight ratio compared to a piston engine. Turbines are more reliable for continuous high outputs. Turbine work better at high altitudes than the piston engines. So that is also a reason turbines are the engine choice for aircrafts. But turbines are also used at powerplants for the electricity generation. 

Some engine examples:

Another important measure is thrust-to-weight ratio. It is dimensionless ratio of thrust-to-weight of jet engine, rocket, or propeller engine, etc. The ratio varies during the operation since the fuel consumption, and also due to the gravity gradient especially for rockets. 

For the aircraft weight-to-thrust ratio plus the wing loading belongs to two very important parameters. Wind loading or wing surface area is the total mass of the aircraft divided by the area of its wing. Low wind loading has larger wing area relative to its mass, as compared to an aircraft with high wing loading. The thrust-to-weight ratio varies continually during the flight. Thrust varies with throttle setting, airspeed, altitude and air temperature. Weight varies with the fuel burn and payload changes,  For aircraft the weight to thrust ratio is typically the maximum static thrust at sea level divided by the maximum take off weight. 

In cruising flight, the thrust-to-weight ratio of an aircraft is the inverse of the lift-to-drag ratio because thrust is the opposite of drag (D), and weight is the opposite of lift (L). So, it means that T/W (in cruise flight) = D/L(in cruise flight). 

Note: I will later write in more detail about aircraft engines, talking about by-pass ratio as well. 

The thrust-to-weight ratio of a rocket is an indicator of its acceleration expressed in sense of gravitational acceleration g. The thrust-to-weight ratio is usually calculated from initial gross weight at sea level on Earth.

The small example: If  RD-180 rocket engine (which powers Atlas V) produces 3,820 kN of sea level thrust and has a dry mass of 5,307 kg, the sea level thrust-to-weight ratio is following, using  9.81 m/s²: 

T/W = 3 820 000 N / (5307kg . 9.81 m/s²)  = 74 (no dimensions) .

Many factors influence the thrust-to-weight ratio. The main factors are variation in speed and altitude, along with the weight change, due to the fuel being continuously burn our. As the altitude increases, so the temperature, pressure and density change. According to the use the gravitational field strength decreases with the increasing altitude. See, the following picture: Altitude in km, pressure in kg/m3, pressure in Pa, temperature in K. 

As we already said, the thrust-to-weight ratio of rocket varies as the fuel is burned. If the thrust is constant, the maximum trust-to-weight occurs just before the fuel is all consumed. Each rocket has its own thrust-to-weigh ratio curve. It can be also found as acceleration curve. During the take off, rocket is using thrust, and not aerodynamic lift. So the ratio has to be greater than 1. 

And not only that, watch the following figure:

When rocket lifts off, the rocket’s velocity increases with the altitude. After lift off rocket consumes thousands of gallons of fuel, significantly reducing its mass. As altitude increases, the air gets thinner, less drag force is acting against it and that causes a greater acceleration, g decreases with altitude acting in W = m g. Remember that at 0 km g = 9.81 m/s^2, and in official start of the space at 100 km g = 9.5 m/s^2. Even over the same thrust of the rocket, the resultant force is getting stronger because resultant force = Thrust – (Drag + Weight). If there is no thrust, engines stops working for any reason, momentum keeps rocket moving upwards, but speed decreases until it start to fall down due to the gravity.