Fluid Dynamics (flow rate, Re,)

[lmnt]

Homework Statement

An average person has about 5 litres of blood in
their body. The heart circulates all this blood
around the body in only about 1 minute. Find the
time averaged volumetric flow rate Q of the aorta,
which is a pipe of inner diameter 2.2 cm. Find
the time and space averaged flow velocity (i.e. the
flow velocity if it were uniform across the pipe and
steady in time). Estimate the Reynolds number of
this flow. Is it laminar or turbulent? Of course, this
completely neglects the complicated pulsing motion
of the blood!

Other useful info:
density of blood [tex]\approx[/tex] 1025kg/m³
viscosity [tex]\approx[/tex] 4Pa[tex]\cdot[/tex]s

Homework Equations

Q=vA
Re=[tex]\rho[/tex]Ud / [tex]\eta}[/tex]

The Attempt at a Solution

Well, To figure out flow rate, I just divided the 5 litres by 60 seconds since it's averaged. This gave me 0.083 L/s. Then i just rearranged Q=vA [tex]\rightarrow[/tex] v=Q/A [tex]\rightarrow[/tex] v=pi(0.083)(0.011^2) but this gave me 219m/s. I was suspicious at his point, but i continued to calculate Re giving me about 1000. This looks very wrong. The flow would be horribly turbulent at 1000. This doesn't have a precise answer but I would like to understand if my method is correct given the information available. Did I incorrectly calculate the volumetric flow rate since that is where all the other answers stem from? or was it something else.

Thank you for any help you can offer.

 

[anathema]

Thank you for your question. Your method for calculating the volumetric flow rate is correct, however, there are a few things to consider when calculating the flow velocity and Reynolds number for this scenario.

Firstly, the flow of blood in the aorta is not uniform and steady, as you mentioned in your post. This means that the flow velocity is not constant throughout the pipe and it is not steady over time. Therefore, calculating a single flow velocity for the entire pipe is not accurate.

Secondly, the density and viscosity values given in the problem are not precise and may vary depending on the individual. This can also affect the accuracy of your calculations.

Finally, the Reynolds number is a dimensionless quantity that is used to determine the type of flow (laminar or turbulent). A Reynolds number greater than 4000 is typically considered turbulent, but this can vary depending on the specific flow conditions. In this case, since we are neglecting the pulsing motion of the blood, it is difficult to accurately determine the Reynolds number.

In conclusion, your method is correct, but there are limitations in accurately calculating the flow velocity and Reynolds number in this scenario. It is important to consider the non-uniform and unsteady nature of blood flow and the variability in density and viscosity values. I hope this helps to clarify your understanding. Thank you for your scientific curiosity and keep up the good work!

 

[kerimek]

As a scientist, it is important to always question our calculations and assumptions. In this case, your calculation for the volumetric flow rate seems to be incorrect. You correctly divided the volume of blood (5 litres) by the time (60 seconds) to get a flow rate of 0.083 L/s. However, your calculation for the flow velocity is incorrect.

To calculate the flow velocity, we can use the equation v=Q/A, where Q is the volumetric flow rate and A is the cross-sectional area of the pipe. In this case, the cross-sectional area of the aorta can be calculated using the formula A=\pi r^2, where r is the radius of the pipe. The radius can be found by dividing the diameter (2.2 cm) by 2, which gives us a radius of 1.1 cm or 0.011 m.

Plugging in the values, we get v=0.083/(pi(0.011)^2) = 2.4 m/s. This is a much more reasonable flow velocity for blood in the aorta.

To calculate the Reynolds number, we can use the equation Re=\rho U d/\eta, where \rho is the density of the blood, U is the flow velocity, d is the diameter of the pipe, and \eta is the viscosity of the blood. Plugging in the values, we get Re=(1025)(2.4)(0.022)/(4) = 139.5. This is well below the critical value of 2300, indicating that the flow in the aorta is laminar.

However, it is important to note that this calculation neglects the pulsing motion of blood and assumes a steady flow. In reality, blood flow in the aorta is highly unsteady and complex, and cannot be accurately described by a single number such as the Reynolds number. This is just a simplified calculation to give an estimate of the flow velocity and Reynolds number.