Question

To show how the R^4 dependence of the Poiseuille law affects flow rate, calculate the bulk fllow through a 12 in. long fire hose nozzle with an inner diameter of 2 in. delivering water with n= 0.01 = 0.01 poise from a pressue of 100 psi and exiting to a pressure 14.7 psi. Assume there is a pump which can provide the necessary volume and give the answer in gallons/min. Please show work. Thanks!

Answer 1

Hagen Poiseuille’s law

P1 - P2 = 32 x n x u x L / D²   .......... Eq1

Velocity = volumetric flow rate/area

u = Q/A

u = Q/[(3.14/4)*D² ]

u = 4Q/3.14*D²

Put the value of u into eq1

P1 - P2 = 32 x n x 4 x Q x L / 3.14 x D² x D²  

D = 2R

P1 - P2 = 32 x n x 4 x Q x L / 3.14 x 4 x R² x 4 x R²  

P1 - P2 = (2.547 x n x Q x L) / (R⁴)

Pressure drop is inversely proportional to R⁴

Pressure drop decreases when R increases.

Inlet pressure

P1 = 100 psi x 101325 Pa/14.7psia = 689285.71 Pa

Outlet pressure

P2 = 14.7 psi x 101325 Pa/14.7psia = 101325 Pa

Viscosity

n = 0.01 poise x 0.1 Pa-s/poise = 0.001 Pa-s

Diameter

D = 2 in x 1m/39.37in = 0.0508 m

Length L = 12 in x 1m/39.37in = 0.3048 m

u = velocity =?

P1 - P2 = 32 x n x u x L / D²

689285.71 - 101325 = 32 x 0.001 x u x 0.3048 / 0.0508²

587960.71 = 3.7795 u

u = 155564.60 m/s

Volumetric flow rate

Q = Area x u

= (3.14/4)*0.0508² x 155564.60

= 315.14 m3/s x 264.172 gallon/m3

= 83252 gallon/s x 60s/min

= 4995119.62 gallon/min