# Why does a diffuser increase pressure?

There are many resources that give explanations for the behavior of nozzles and diffusers. But at the crux point they all say, "Velocity goes down and pressure goes up." As if that doesn't warrant any deeper explanation, or as if it were just a law of nature.

To me, the reason behind this energy transfer isn't obvious.

To maintain a constant mass flow rate, $\dot{m} = const.$ , through a small inlet and a large outlet you would need a deceleration of the flow. To decelerate a flow requires a force, $F=ma$. Forces interactions are equal and opposite so the fluid would exert a larger force, relative to the inlet, if forced to slow down. But what is the origin of this decelerating force? Why does this geometry increase the $\frac{F}{A}$ of the molecules that pass through it? How is this behavior explained in terms of force interactions?

It's like you asked how an electric motor worked and someone said, "By turning electric energy into rotational motion." and you said, "How?" And they just repeat, "By turning electric energy into rotational motion." Energy balance just isn't descriptive enough.

• Have you heard of the Venturi effect? I think but I'm unsure this is what you're looking for: en.wikipedia.org/wiki/Venturi_effect – Klodd Nov 14 '16 at 10:04
• I meant to tell you, if physicsSE does not give you an answer you are happy with, as long as you don't cross post, try AviationSE and EngineeringSE. Let me know if you are trying your question there and I will up vote it there. Best of luck with it. – user108787 Nov 14 '16 at 11:03
• Good for you, the only thing is keep asking questions, you learn a lot of other related stuff that helps you in the future. – user108787 Nov 14 '16 at 14:45
• I am not sure what force you are looking for. The only force involved in this problem is a pressure-area force at each subsequent station. The gradient of pressure is what drives the dynamics of the flow. The way you seem to be viewing the problem is that the dynamics of the flow must generate a pressure gradient, but that is not how this problem works. It is in fact the pressure and its gradient that contributes to a momentum flux at any given station. It would be incorrect to suggest the momentum flux at any given station drives the pressure gradient. – TRF Dec 21 '16 at 19:06
• Continued. Now if you are using a geometry to drive the pressure gradient that is a totally different depiction of the problem, but you must always have a pressure gradient to impart dynamic motion on a fluid element (assuming inviscid and steady flow). The geometry of a tube or pipe will cause changes in the mass flux at each subsequent station, simply from the conservation of mass, which in turn will cause a pressure gradient throughout the tube geometry. It is this pressure gradient that accelerates or decelerates the flow on the basis of a pressure-area force at each individual station. – TRF Dec 21 '16 at 19:11

## 1 Answer

This is due to the conservation of energy in the flow. This property of a steady, incompressible and inviscid flow is described by the Bernoulli's principle which states that, in this flow,

$$\frac{1}{2}\rho v^2 + \rho g z + p= \textrm{constant}.$$

As Klodd pointed in the comment, you can see this principle in action with the Venturi effect. The contraction (resp. enlargment) of the section in a pipe leads to an increase (resp. decrease) of velocity to conserve the mass flow.

In a diffuser, the section grows along the stream and so the flow will slow down. Assuming the height is the same ($z=cst$), the previous equation will lead to, with index $1$ for upstream and index $2$ for downstream,

$$\frac{1}{2}\rho v_1^2 + p_1= \frac{1}{2}\rho v_2^2 + p_2 \quad \textrm{and} \quad v_1 > v_2.$$ Also pressure has to increase to conserve energy, i. e. $p_2$ has to be higher than $p_1$.

PS: I can expand the explanation but you really should read the Wikipedia article about Bernoulli's principle first.

EDIT: I'll add a few things as you want to know more about the acceleration/deceleration origin. You said To decelerate a flow requires a force, $F=ma$. That force comes from the pressure, $F=P\times S$ with $S$ the section of the pipe and the acceleration/deceleration comes also from the change of pressure. A variation of the pressure means a variation of force and also a variation of velocity, i.e. an acceleration/deceleration.

Imagine a fluid parcel (a material element of fluid, not just one molecule) which flows through the pipe. In the changing section, the pressure gradient is not null. If $\vec{x}$ is the direction of the flow, in the diffuser we have $dp/dx>0$ as pressure increases. Now it means that the fluid parcel (you can imagine a small cube of fluid) will see a difference of pressure between its front face (higher pressure) and its rear face (lower pressure). This pressure difference means that the material element experiences a force directing in $-\vec{x}$. This force causes then a deceleration of the fluid parcel.

To finish, I would like to add one thing, related to your comment. You said that :

With the bernoulli equation it is just as possible that slowing down a flow could result in some spontaneous jump in height, per the pgz term, but we know this physically makes no sense. There is no force in that direction. So, excluding empiricism, how do we know the velocity is transformed into pressure.

I would recommend you to take a look at surge tank. In a few words, when a valve is closed a pressure wave can propagate in a pipe. This pressure wave can destroy the facility because it leads to a backward flow in the pipe. The idea here is to place an open reservoir, or at least a not full and sufficiently large one, close to the valve to guide the flow in it. It allows to stabilize the flow using a change of height in the tank. This is an example of an exchange from kinetic energy to potential energy. You can find it very well described in this video and see it in action in this one.

• Thank you @Lalylulelo for your answer. I am actually looking for a force description of why pressure and velocity are exchanged. Conservation equations don't describe how the energy is transferred from velocity to pressure, just that it can be. With the bernoulli equation it is just as possible that slowing down a flow could result in some spontaneous jump in height, per the pgz term, but we know this physically makes no sense. There is no force in that direction. So, excluding empiricism, how do we know the velocity is transformed into pressure. – BoddTaxter Nov 14 '16 at 13:59
• @BoddTaxter Your real question is "how is the Bernoulli equation obtained, starting from the balance of momentum?" If you neglect viscous forces, the differential force balance reduces to the Euler equation. To get the Bernoulli equation, you do the same thing that you do in developing ordinary mechanical energy balances: you dot the differential force balance (Euler equation) with the velocity vector. This leads directly to the Bernoulli equation. – Chet Miller Nov 14 '16 at 16:29
• @BoddTaxter You should take a look at this question. There is an explanation that might suits you. Actually, even it's mathematical equivalent, it's the increase of pressure that leads to a decrease of velocity. – Lalylulelo Nov 16 '16 at 16:26
• @BoddTaxter I added a few details for you. – Lalylulelo Dec 22 '16 at 9:56