# What is the link OR difference between change in momentum & pressure difference?

I ask in regards to the the thrust formula, which includes change in momentum, as well as difference in pressure between inlet & outlet of the engine to calculate generated thrust; this implies that momentum change & pressure difference are clearly two (2) different phenomenons.

But in reaction turbine blade cascades, we can observe a pressure distribution similar to that of an aircraft wing, producing lift which causes the turbine to spin, & a load component acting on the turbine bearings. Also, the inlet & outlet momentum change is seen to cause the turbine to spin. I would expect that a turbine's rotation can be explained by either of the explanations but not both together, since the pressure distribution over the blade profile is a consequence of the deflection of air by the blade, which also causes the change in momentum of the flow exiting the turbine. So the force acting upon a turbine blade can be calculated either by calculating momentum change, or by calculating pressure difference, but not as a sum of both.

Now, returning to the case of the engine, of course, there isn't a pressure difference vertically to the axis of an engine, so no force in this direction. But the pressure difference in line of the axis is a direct consequence of the acceleration of the flow (momentum change) through the engine:

So why is pressure difference & momentum change summed together to get thrust?

It's mr Bernouilli's signature discovery: total pressure is dynamic pressure (momentum change) plus static pressure. Both static and dynamic pressure are the result of the impact of air molecules upon the wing or turbine blade, which acts as the lifting surface. The main distinction between the two is that static pressure is the result of random velocity vectors averaging zero, while dynamic pressure has an average molecule velocity unequal to zero.

Bernouilli's law can also be written as

$$p_t = p_s + \dot{m} \cdot V$$

With $\dot{m}$ the mass of air that impacts the lifting surface per second, and $V$ its velocity. Since that is not a convenient way to make computations regarding aircraft structures, we usually see it in a format that uses air density and wing area.

There is an analogy with potential and kinetic energy. Both are forms of energy, and one form can be freely exchanged for another one. A falling object transforms $m\cdot \Delta h$ into $\frac {1}{2} \cdot \rho \cdot \Delta V^2$, but the total energy remains constant at any point in time.

The same goes for pressure, which is actually a measure of the energy of air molecules. A jet engine adds energy to the air molecules going through it. A turbine blade can be driven by either the dynamic pressure, or by the difference in static pressure before and behind the blade - there are different design blades according to the proportion of static & impulse. The exhaust shape of the engine can leave the outflowing air with a higher static pressure than surrounding air, or expand the higher static pressure completely into dynamic pressure.

Static pressure and impulse change are summed together because they are two forms of exactly the same entity: the energy added to the air stream.

• so the reaction of a turbine blade is the total of the pressure difference across the profile AND the momentum change across the blade? – Guha.Gubin Sep 29 '17 at 10:36
• Yes, accounting for efficiency, thermodynamic losses, friction etc. – Koyovis Sep 29 '17 at 12:06