# Cause-effect relation between pressure & velocity

Summary:
Is there a way to change pressure in a duct that has a uniform cross sectional area regarding Bernoulli and continuity equations?
For a steady, non-viscous and incompressible flow, one can apply both Bernoulli's principle (no potentials) as

$$p+\frac{\rho v^2}{2} = p_t$$

where ##p##, ##\rho,##, ##v##, and ##p_t## are static pressure, density, flow velocity, and total pressure, respectively,

and continuitiy principle as

$$\rho vA=\dot m=Flow Rate$$

where ##A## is cross sectional area of the interest in the duct or stream tube.

From both of the equations above one can drive

$$p+\frac {k}{A^2}=cst$$

where ##k## is defined by ##k=\frac {\dot m^2}{2\rho}##.

From the last equation, it is implied that the only way of changing pressure is changing the cross sectional area. However, I worry that if there is an alternative way of changing the pressure without cross sectional area. In other words, is there a way to change pressure in a duct that has a uniform cross sectional area? (##A=cst##)

Gold Member
What does your intuition tell you here? You have an equation that implies one thing, but do you feel like there should be a way to change the pressure without an area change for a steady, inviscid, incompressible flow? What does this have to do with causality?

It is understanding fundamental concepts from physics point of view. For example we can consider a cylinder-piston system with an infinite length linear cylinder and a limited length piston. When we move the piston towards one direction, then it is possible do change velocity of the flow. From Bernoulli's principle, this should imply a reduction in the pressure. Then, cause is changing the velocity (without changing the cross sectional area) and effect is a change in pressure. This seems to be a method to modify pressure. However, I want to be sure that this reasoning is correct.

FactChecker
Gold Member
Ignoring gravity and any potentials, the logic that I am familiar with is about the effect that a change in the cross-sectional area in a steady, constant-energy flow. I can not see any way that the Bernoulli principle would imply a change in velocity if there is no associated change in the cross-sectional area.

russ_watters
Mentor
From the last equation, it is implied that the only way of changing pressure is changing the cross sectional area. However, I worry that if there is an alternative way of changing the pressure without cross sectional area. In other words, is there a way to change pressure in a duct that has a uniform cross sectional area? (##A=cst##)
I'm not clear what your goals or constraints are, but I can think of a couple options:
-Increase airflow

hmmm27
I'm not clear what your goals or constraints are, but I can think of a couple options:
-Increase airflow
How to increase airflow in a constant diameter pipe? (i.e. total pressure) Adding a variable speed fan at one end?

russ_watters
Mentor
How to increase airflow in a constant diameter pipe? (i.e. total pressure) Adding a variable speed fan at one end?
Yes, that would work.

rcgldr
Homework Helper
is there a way to change pressure in a duct that has a uniform cross sectional area?
In a real world situation that takes more into account than an idealized Bernoulli situation, friction between the air and the duct walls results in a decrease in pressure as the air flows through the duct. One way to reduce pressure without changing cross sectional area is to make the duct longer.

russ_watters
Gold Member
It is understanding fundamental concepts from physics point of view. For example we can consider a cylinder-piston system with an infinite length linear cylinder and a limited length piston. When we move the piston towards one direction, then it is possible do change velocity of the flow. From Bernoulli's principle, this should imply a reduction in the pressure. Then, cause is changing the velocity (without changing the cross sectional area) and effect is a change in pressure. This seems to be a method to modify pressure. However, I want to be sure that this reasoning is correct.

The piston problem is a poor example here. First, it is an unsteady problem that is not solvable using Bernoulli's equation. The total pressure changes when you start the piston moving. Second, an impulsively-started piston in a tube is a classical problem in compressible flows because it produces a traveling normal shock, which is also not tractable using Bernoulli's equation.

Bernoulli's equation only applies to incompressible, steady, inviscid flows. There are really two ways to look at this: conservation of energy and a force balance (conservation of momentum). Bernoulli can be derived either way (from the energy equation or the Euler equation).

Energy: Ultimately, I think this is probably the most intuitive approach. Bernoulli's equiation is a relationship between kinetic and potential (pressure) energy in the flow (ignoring gravitational potential energy for simplicity). If you increase kinetic energy (velocity), the potential energy falls and vice versa.

Force/Momentum: You could also argue from Newton that any change in velocity (acceleration) requires a force to enact that change, and the only force (again ignoring gravity) in an inviscid flow is pressure. So in order to increase velocity, you need a pressure force pointing in the direction of increase, meaning the pressure gradient and velocity gradient have opposite signs. (Example: if pressure decreases from left to right, it results in a net force field pointing in the right direction and therefore acceleration to the right.) This is not as explicitly obvious in the equation but all of that is contained within it.

Neither of these need to be causal relationships. It is simply an example of conservation laws where two quantities vary with one another in specific ways.

You could have a pipe flowing steadily and then do something to increase the upstream pressure. Now the pressure gradient is providing a larger force and the flow rate increases. A change in pressure caused that change in velocity. You could also validly argue that if you place a constriction in the flow, conservation of mass dictates that the flow rate must increase through the smaller area and a pressure gradient must develop to support that. A change in velocity caused that change in pressure.

So you see, there is no one correct direction of causality. They are simply related through conserved quantities.

I'm not clear what your goals or constraints are, but I can think of a couple options:
-Increase airflow

Both of these would violate his original constraint, though. Increasing air flow would change the total pressure of the system. Adding an obstruction would change the pressure along with the velocity. The former violates the assumptions underpinning Bernoulli's equation and the second fits neatly within its constraints. In a duct, adding an obstruction would be changing the area.

In a real world situation that takes more into account than an idealized Bernoulli situation, friction between the air and the duct walls results in a decrease in pressure as the air flows through the duct. One way to reduce pressure without changing cross sectional area is to make the duct longer.

This is somewhat incomplete. Generally speaking, the addition of friction more directly impacts total pressure, not static pressure. Friction is due to viscosity, which dissipates energy causing total pressure to decrease. It's precise relationship with velocity and pressure is a bit more complicated. If the flow is incompressible and the pipe has a constant area, then you know ##u_1 = u_2## (average velocities) to conserve mass, which means the pressure decreases. If the flow is compressible, things get wacky (see: Fanno flow).

russ_watters
Mentor
Both of these would violate his original constraint, though. Increasing air flow would change the total pressure of the system. Adding an obstruction would change the pressure along with the velocity. The former violates the assumptions underpinning Bernoulli's equation and the second fits neatly within its constraints.
What constraint? Also, no, it wouldn't necessarily change velocity. See next answer:

In a duct, adding an obstruction would be changing the area.
Not in places where the obstruction isn't. I'm envisioning an HVAC system with dampers at the outlets, and an adjustable speed fan. To change the pressure in the duct without changing the velocity, you close the dampers and speed up the fan. As I said to the OP, the system hasn't been described to us, so I have no idea if my vision matches what he is after. Maybe it does, per his suggestion in post #6.

Gold Member
What constraint?

He was going about this with analysis via Bernoulli. Doing what you suggested would violate the assumptions inherent in Bernoulli's equation. Admittedly, I misspoke. The obstruction would not, but changing the air flow rate would. You couldn't readily compare the two air flow cases.

Not in places where the obstruction isn't. I'm envisioning an HVAC system with dampers at the outlets, and an adjustable speed fan. To change the pressure in the duct without changing the velocity, you close the dampers and speed up the fan. As I said to the OP, the system hasn't been described to us, so I have no idea if my vision matches what he is after. Maybe it does, per his suggestion in post #6.

Placing dampers at the outlet will change the outlet area. Changing the fan speed changes total pressure. The former breaks his no area change assumption and the later breaks Bernoulli.

Thanks for the discussion. I think I wasn't clear enough about my question. I wrote a constant diameter pipe in paritcular to free the discussion from conservation of mass. Flow in the pipe is steady, incompressible, and inviscid. So, yes, pretty idealized thought experiment here.

With a constant diameter pipe/duct changing velocity would change the mass flow rate (mass conservation). (In a continuum, is this possible? If yes) It would also change total pressure (momentum conservation). Therefore, my question here becomes how to change the velocity? (In the argument with russ_watters, we concluded that a variable speed fan would work resulting a change in the total prussure.)

However, another question incurs here. In the same pipe, without changing the total pressure, it seems impossible to change the velocity (thus pressure) or pressure (thus velocity) except forcing the flow to stop at a point (i.e. blocking completely downstream of the pipe).

In a constant diameter pipe, if we would change the total pressure, we have a way of doing it: using a fan. If we would decrease the velocity down to the zero, again we have a way of doing it: blocking the flow. However, I couldn't imagine a way to increase velocity in a fixed total pressure setting.

russ_watters
Mentor
Thanks for the discussion. I think I wasn't clear enough about my question. I wrote a constant diameter pipe in particular to free the discussion from conservation of mass. Flow in the pipe is steady, incompressible, and inviscid. So, yes, pretty idealized thought experiment here.
This still isn't a complete description of a physical system, nor what we are allowed to change in the system. The math is not stand-alone - it exists to provide analysis of a physical system. Unless you describe it, I feel free to pick my own system that does what you ask. That's an inefficient approach though because you may have an idea in your head that you aren't telling us about what the system setup looks like.

With a constant diameter pipe/duct changing velocity would change the mass flow rate (mass conservation).
That's true. That's a hard constraint of the math/physics.
(In a continuum, is this possible? If yes) It would also change total pressure (momentum conservation). Therefore, my question here becomes how to change the velocity? (In the argument with russ_watters, we concluded that a variable speed fan would work resulting a change in the total prussure.)

However, another question incurs here. In the same pipe, without changing the total pressure, it seems impossible to change the velocity (thus pressure) or pressure (thus velocity) except forcing the flow to stop at a point (i.e. blocking completely downstream of the pipe).

In a constant diameter pipe, if we would change the total pressure, we have a way of doing it: using a fan. If we would decrease the velocity down to the zero, again we have a way of doing it: blocking the flow. However, I couldn't imagine a way to increase velocity in a fixed total pressure setting.
These are opposite sides of the same coin. If I'm allowed to design the system with a variable speed fan and a damper at the end of the duct, I can change the velocity and pressure (static or total) in the duct up or down, independent of each other.

russ_watters
Mentor
He was going about this with analysis via Bernoulli. Doing what you suggested would violate the assumptions inherent in Bernoulli's equation.
I don't see how. What I'm describing is basically a standard VAV HVAC system and these systems of course obey Bernoulli's principle/equation.
Placing dampers at the outlet will change the outlet area. Changing the fan speed changes total pressure. The former breaks his no area change assumption...
The issue here is in the description of the system by the OP. He didn't say "outlet area", he said "duct". The way I describe such systems, these are independent parts of the system. I'm not trying to be argumentative here: these systems are designed specifically to control/change the duct pressure and airflow independently of each other while in operation. Not telling us what is on each end of the duct (or what we are allowed to put on each end of the duct) is exactly what's wrong with his description.
...the later breaks Bernoulli.
How? If you're saying that these different flow scenarios are required to have the same total energy, I don't see that the OP is saying that. But again; the OP is the one who needs to tell us what we can change and what we can't. There's no value in debating/guessing it.

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These are opposite sides of the same coin. If I'm allowed to design the system with a variable speed fan and a damper at the end of the duct, I can change the velocity and pressure (static or total) in the duct up or down, independent of each other.
That's true, as long as the pipe has a constant diameter, no problem, I was investigating a way of manipulating velocity and pressure for steady, incompressible, and inviscid flow setting.