Fluid Dynamics: Bernoulli Air Tunnel

Question

I have the following question as part of a lab intro to explain the more complicated stuff I'l actually have to submit in the lab. I put all the questions here just to show my thought pattern/what I'm supposed to be working through.

Given:

Measurement 1 (fan off):
Micrometer reading = -0.010 inches
Transducer reading = 3.000 volts
Measurement 2 (fan on):
Micrometer reading = +0.500 inches
Transducer reading = 3.586 volts

(a) The relationship between pressure and voltage is linear, calculate an equation which relates the transducer voltage (in volts) to the pressure difference (in Pa) - I understand this, as this is simply do a linear interpretation two points, etc.

(b) Port #8 in the test section is displayed as 3.450V on the meter. Use (a) equation to convert this value to a pressure differentialin Pascal. Calculate the air velocity at this point - Plug and chug the answer from part (a)

(c) The average velocity through the test section at Port #8 is equal to the point velocity in (b). What is the Mach number? Is this incompressible flow? - this I don't know how to find I know M = V/a though

(d) Apply continuity to determine theaverage velocity at Port #2. Based on the same assumption as (c) the average velocity through the test section at Port #8 is equal to the point velocity - and since I don't understand (c) I don't understand this either

Answer 1

eWizardII,

For question (c), the Mach number M as you rightly point out is $M = V/a$, where $a$ here is the speed of sound in the medium, in this case air. The speed of sound in air is given by

$a^{2} = \sqrt{(\gamma \frac{P}{\rho})}$,

where $P$ is the fluid pressure (air pressure), $\rho$ is the fluid density and $\gamma = c_{P}/c_{v}$ and is the ratio of specific heats.

To be as accurate as possible here you would calculate the speed of sound at your measurement location (Port #8). For this case, where the velocities of the flow are slow (such low speed flow can be assumed to be laminar or streamline flow), you can safely assume that the flow is incompressible. Thus, to calculate you speed of sound at Port #8, you can assume that the density of air is that at room temperature and atmospheric pressure (as air density does not vary very much with pressure in the temperature/pressure ranges you will be dealing with). The specific heats can also be calculated at room temperature for this case. So to calculate $a$ at Port #8, use the above values for $\rho$, $c_{P}$ and $c_{v}$ with the pressure you calculated in step (b). This will give a reasonable value for $a$ (probably close to 340 $m s^{-1}$). For this flow you can expect a small Mach number $M$ calculated using

$M = \frac{V}{a}$.

For question (d) assume incompressible flow (with the air assumed as inviscid - another very reasonable assumption) and use Bernoulli’s theorem.

Hope this helps.