Compressible fluid through a narrow tube.

[JonNash]

Homework Statement

For a flow of a compressible fluid through a narrow tube, the following statement is correct
a) the volume of the fluid crossing any section of the tube is constant.
b) density of the fluid is independent of pressure.
c) mass of the fluid crossing any section of the tube varies with time.
d) mass of the fluid crossing any section of the tube per unit time is constant.

The Attempt at a Solution

The confusion I'm having here is that if a certain volume is passing through the tube then a certain mass is also associated with it and since it is a compressible fluid I'm assuming a gas (picturing the nozzle of a tyre). Moreover since it is a narrow tube with a compressible gas there is obviously a constant flow, unless the tube (pictured just for easier understanding) is already deflated, then it stops quite abruptly. So I'm guessing a) or d) but unable to understand something basic here. Please help me out.

Thanks.

 

[Chestermiller]

Homework Statement

For a flow of a compressible fluid through a narrow tube, the following statement is correct
a) the volume of the fluid crossing any section of the tube is constant.
b) density of the fluid is independent of pressure.
c) mass of the fluid crossing any section of the tube varies with time.
d) mass of the fluid crossing any section of the tube per unit time is constant.

The Attempt at a Solution

The confusion I'm having here is that if a certain volume is passing through the tube then a certain mass is also associated with it and since it is a compressible fluid I'm assuming a gas (picturing the nozzle of a tyre). Moreover since it is a narrow tube with a compressible gas there is obviously a constant flow, unless the tube (pictured just for easier understanding) is already deflated, then it stops quite abruptly. So I'm guessing a) or d) but unable to understand something basic here. Please help me out.

Thanks.

The mass flow rate at steady state flow is constant with position along the tube. This is just conservation of mass between the inlet and any arbitrary cross section. The volumetric flow rate is equal to the mass flow rate divided by the density. If the pressure is decreasing axially along the tube, for a compressible fluid, the density is also decreasing, so the volumetric flow rate is increasing along the tube. If the pressure is increasing axially, the reverse is true. So the mass flow rate is constant, but the volumetric flow rate is not.

 

[JonNash]

If the pressure is decreasing axially along the tube, for a compressible fluid, the density is also decreasing, so the volumetric flow rate is increasing along the tube. If the pressure is increasing axially, the reverse is true. So the mass flow rate is constant, but the volumetric flow rate is not.

But the question does not specify whether or not the pressure is increasing or decreasing, so is it not safe to assume that the pressure is also a constant? If it is a constant then the volumetric flow rate along with the mass flow rate is also constant with time? In the end we have to choose whether mass or volume is constant with no specification on the pressure, since the mass flow rate is constant in all cases and volume is dependent on pressure, the answer to be chosen is just a matter of probability rather than an absolute answer. Am I right?

 

[AlephZero]

The question doesn't say steady flow through a tube with constant cross section.

I think you have to assume steady flow, otherwise you can only say that (b) is definitely false.

You don't need to assume the cross section is constant, though.

 

[Chestermiller]

But the question does not specify whether or not the pressure is increasing or decreasing, so is it not safe to assume that the pressure is also a constant? If it is a constant then the volumetric flow rate along with the mass flow rate is also constant with time? In the end we have to choose whether mass or volume is constant with no specification on the pressure, since the mass flow rate is constant in all cases and volume is dependent on pressure, the answer to be chosen is just a matter of probability rather than an absolute answer. Am I right?

If it's steady state flow, irrespective of whether the pressure is constant, the mass flow rate is constant. If it's steady flow and the pressure is not constant, the mass flow rate is still constant, but the volumetric flow rate isn't. Since it's impossible for the pressure to be exactly constant, if the flow is steady, the mass flow rate is constant. This is the proper choice (I think the problem statement meant to say that the flow is steady). In my judgement, it isn't a matter of probability.

Chet

 

[JonNash]

Alright then it is resolved, d) is the right choice. Thanks Chet and Aleph.

 

[lekhraj]

ρ1A1v1=ρ2A2V2 so the ans is d)