![]() The nozzle of a turbine engine is usually designed to make the exit pressure equal to free stream. There is a simplified version of the general thrust equation that can be used for gas turbine engines. These effects are described in detail on other pages at this site. If the exit velocity becomes very high, there are other physical processes which become important and affect the efficiency of the engine. A moderate amount of flow is accelerated to a high velocity in these engines. This is the design theory behind pure turbojets, turbojets with afterburners, and rockets. The other way to produce high thrust is to make the exit velocity very much greater than the incoming velocity. A large amount of air is processed each second, but the velocity is not changed very much. This is the design theory behind propeller aircraft and high-bypass turbofan engines. As long as the exit velocity is greater than the free stream, entrance velocity, a high engine flow will produce high thrust. One way is to make the engine flow rate (m dot) as high as possible. We see that there are two possible ways to produce high thrust. Let us look at this equation very carefully, for it has some interesting implications. Normally, the magnitude of the pressure-area term is small relative to the m dot-V terms. The general thrust equation is then given by: General Thrust Equation F = (m dot * V)e – (m dot * V)0 + (pe – p0) * Ae ![]() Across the exit area we may encounter an additional force term equal to the exit area Ae times the exit pressure minus the free stream pressure. If there is a net change of pressure in the flow there is an additional change in momentum. The fluid pressure is related to the momentum of the gas molecules and acts perpendicular to any boundary which we impose. There is an additional effect which we must account for if the exit pressure p is different from the free stream pressure. Then F = (m dot * V)e – (m dot * V)0Ī units check shows that on the right hand side of the equation: mass/time * length/time = mass * length / time^2 Pressure We will denote the exit of the device as station “e” and the free stream as station “0”. Since the mass flow rate already contains the time dependence (mass/time), we can express the change in momentum across the propulsion device as the change in the mass flow rate times the velocity. So “m dot” is not simply the mass of the fluid, but is the mass flow rate, the mass per unit time. For example, we can write Newton’s second law as either F = d(mv)/dt or F = (mv)dot Note: The “dot” notation is used a lot by mathematicians, scientists, and engineers as a symbol for “d/dt”, which means the variable changes with a change in time. Aerodynamicists denote this parameter as m dot (m with a little dot over the top). Its dimensions are mass/time (kg/sec, slug/sec, …) and it is equal to the density r times the velocity V times the area A. Mass flow rate is the amount of mass moving through a given plane over some amount of time. For a moving fluid, the important parameter is the mass flow rate. But if we are dealing with a fluid (liquid or gas) and particularly if we are dealing with a moving fluid, keeping track of the mass gets tricky. If we are dealing with a solid, keeping track of the mass is relatively easy the molecules of a solid are closely bound to each other and a solid retains its shape. If we keep the mass constant and just change the velocity with time we obtain the simple force equation – force equals mass time acceleration a F = m * a Mass Flow Rate So, between two times t1 and t2, the force is given by: F = ((m * V)2 – (m * V)1) / (t2 – t1) Force Momentum is the object’s mass m times the velocity V. Momentumįrom Newton’s second law of motion, we can define a force F to be the change in momentum of an object with a change in time. ![]() For right now, let us just think of the propulsion system as some machine which accelerates a gas. We will discuss the details of various propulsion systems on some other pages. To accelerate the gas, we need some kind of propulsion system. ![]() A gas or working fluid is accelerated to the rear and the engine and aircraft are accelerated in the opposite direction. Thrust is a mechanical force which is generated through the reaction of accelerating a mass of gas, as explained by Newton’s third law of motion. Thrust is generated by the propulsion system of the airplane. Thrust is the force which moves an aircraft through the air. Home > Beginners Guide to Aeronautics Thrust Equation ![]()
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