Poiseuille equation for water flow rate

In summary, the water pressure at the council tubby is around 1.5 bar, and using Poiseuille equation, one can estimate the flow rate at a typical kitchen tap. My assumption was that the pipe diameter was the same from the council tubby to where the water comes out of the tape, but when I substituted those values into Poiseuille's equation, I got an estimate of 0.0375 L/min. My order of magnitude calculation was off, and it would be better to use Bernoulli equation instead.
  • #1
ChiralSuperfields
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Homework Statement
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Relevant Equations
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For this problem,

The mains water pressure at the council tubby (just before it enters a house) is of the order of 1.5 bar. Using Poiseuille equation, estimate the flow rate in a typical home at the kitchen tap. You will need to make reasonable estimates on several parameters, clearly state these assumptions (note that this estimate does not need to agree with your measurement in Q1, but it shouldn’t be orders-of-magnitude different).

My assumption is that the pipe diameter is the same from the council tubby to where the water comes out of the tape.

##P_2 - P_1 ≈ 1 bar ≈ 100000 Pa##
##η ≈ 10^{-3} Pa \cdot s##
##L ≈ 10 m##
##π ≈ 3##
##r ≈ 1 cm ≈ 0.01 m##

However, when I substitute those values into Poiseuille's equation I get ##Q ≈ 0.0375 ≈ 0.04 \frac{m^3}{s}##. When I convert it to ##\frac{L}{min}## I get ##2400 \frac{L}{min}## However, Q should be in the order of ##10 \frac{L}{min}##. My order of magnitude calculation is orders of magnitude off.

I am not sure how to make the calculation more reasonable. If I increase the length to say 100m, it still dose not give a value in between ##10 \frac{L}{min}##.

I am wondering whether my assumption is wrong, that is, the pipe diameter varies from the council tubby to the where the water comes out. I think ##r## should be initially larger than what I said it to be, and there decreases to the value I measured when it reach's the tap cyclinder. However, I am not sure whether it possible to account for that in Poiseuille equation. Maybe I should use Bernoulli equation.

Any guidance would me much appreciated!

Many thanks!
 
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  • #2
Please quote the equation. Doesn't it have a term for the pipe radius? you don't show your value for that.
 
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  • #3
For a normal family home in USA, the diameter of the pipe connecting it to the city main via water meter is 1-inch or 3/4-inch.

domestic-water-piping-table3.jpg
 
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  • #4
haruspex said:
Please quote the equation. Doesn't it have a term for the pipe radius? you don't show your value for that.
Thank you for your reply @haruspex! The equation is ##Q = (P_2 - P_1)\frac{πr^4}{8ηL}##

Many thanks!
 
  • #5
Lnewqban said:
For a normal family home in USA, the diameter of the pipe connecting it to the city main via water meter is 1-inch or 3/4-inch.

View attachment 324058
Thank you for your reply @Lnewqban !

Sorry as @haruspex pointed out, I forgot to say ##r ≈ 1 cm ≈ 0.01 m##

Many thanks!
 
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  • #6
At 10 l/min, the Reynolds number is > 10000, and the flow is turbulent, you can’t use the poiseuille equation.
 
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  • #7
Actually, it's even worse because the flow leaves the tap at atmospheric pressure. The change in pressure from the mains to the tap is the full 1.5 bar (which is low). In my home its regulated down to about 50 psi from the mains (which are much higher).

Pipes snake through a home ( decreasing in diameter as they go in a trunk/branch system), so the length can add up, but 10 m doesn't seem unreasonable as a length. However, having a nearly 1 inch line at the sink tap is pretty out there. In my home lines for my sink are 3/8ths NPS ( I.D. ##\approx## 0.5 in. ). But even doing 10 meters of ##r = 0.25 \rm{in}## that equation predicts ##\approx 575\rm{ \frac{L}{min}}##.

As @Chestermiller points out its validity requires laminar flow ##Re = \frac{ 4 \rho Q}{\pi D \mu} < 2000 ## and it not satisfied for either scenario. So, if you are forced to use this its a poorly designed question.
 
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  • #8
Chestermiller said:
At 10 l/min, the Reynolds number is > 10000, and the flow is turbulent, you can’t use the poiseuille equation.
erobz said:
Actually, it's even worse because the flow leaves the tap at atmospheric pressure. The change in pressure from the mains to the tap is the full 1.5 bar (which is low). In my home its regulated down to about 50 psi from the mains (which are much higher).

Pipes snake through a home ( decreasing in diameter as they go in a trunk/branch system), so the length can add up, but 10 m doesn't seem unreasonable as a length. However, having a nearly 1 inch line at the sink tap is pretty out there. In my home lines for my sink are 3/8ths NPS ( I.D. ##\approx## 0.5 in. ). But even doing 10 meters of ##r = 0.25 \rm{in}## that equation predicts ##\approx 575\rm{ \frac{L}{min}}##.

As @Chestermiller points out its validity requires laminar flow ##Re = \frac{ 4 \rho Q}{\pi D \mu} < 2000 ## and it not satisfied for either scenario. So, if you are forced to use this its a poorly designed question.
Thank you for your replies @Chestermiller and @erobz !

I will refer to this thread when I email the profs on Monday. I will post what the outcome is.

Many thanks!
 
  • #9
I found the actual flow rate Q to be ##Q = \frac{1}{9.97} = 0.101 L/s (3 s.f.) = 6.03 L/min (3 s.f.) = 0.000101 m^3/s (3 s.f.)##
 
  • #12
Callumnc1 said:
Thank you for your reply @Lnewqban!

Was there a specific part that you wanted me to see?

Many thanks!
Mainly as a reference about typical pipe's diameters and water supply flows in a family home.
 
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  • #13
Lnewqban said:
Mainly as a reference about typical pipe's diameters and water supply flows in a family home.
Thank you for your reply @Lnewqban!

Yeah, it turns out the order of magnitude calculation is likely to be off due the bends in the pipe which decrease the fluid momentum that the Poiseuille equation dose not account for. I'm also right about the different diameter part.

Many thanks!
 
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  • #14
Callumnc1 said:
Thank you for your reply @Lnewqban!

Yeah, it turns out the order of magnitude calculation is likely to be off due the bends in the pipe which decrease the fluid momentum that the Poiseuille equation dose not account for. I'm also right about the different diameter part.

Many thanks!
Bends/fittings in the pipe are typically referred to as "minor head loss" terms, they can add up, but they aren't typically dominant. How about elevation head?
 
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  • #15
erobz said:
Bends in the pipe are typically referred to as "minor head loss", they can add up but they aren't typically dominant. How about elevation head?
Thank you for your reply @erobz! That is true you mention about elevation head. Where I was performing the experiment was 2 floors high and the house is about 200 m above sea level.

Many thanks!
 
  • #16
Callumnc1 said:
Thank you for your reply @erobz! That is true you mention about elevation head. Where I was performing the experiment was 2 floors high and the house is about 200 m above sea level.

Many thanks!
The mains typically enter in the basement. So, if you are doing this on the second floor you could have used 12 m of elevation head just getting to the height of the faucet. Also, your 10 m of pipe are probably also underestimated, and its not likely 1 " pipe, probably more like 1/2" pipe (or smaller).

What do you get if you take 12 m of elevation head out of ##\Delta P##, and change the length to 20 m of pipe at a radius of 7 mm?
 
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  • #17
erobz said:
The mains typically enter in the basement. So, if you are doing this on the second floor you could have used 12 m of elevation head just getting to the height of the faucet. Also, your 10 m of pipe are probably also underestimated, and its not likely 1 " pipe, probably more like 1/2" pipe (or smaller).

What do you get if you take 12 m of elevation head out of ##\Delta P##, and change the length to 20 m of pipe at a radius of 7 mm?
Thank you for your reply @erobz! I will do that problem once I have a bit more time from uni.

Many thanks!
 

1. What is the Poiseuille equation for water flow rate?

The Poiseuille equation for water flow rate, also known as the Hagen-Poiseuille equation, is a mathematical formula that describes the relationship between fluid flow rate, pressure, and viscosity in a cylindrical pipe. It is expressed as Q = (π * ΔP * r^4) / (8 * η * L), where Q is the flow rate, ΔP is the pressure difference, r is the radius of the pipe, η is the viscosity of the fluid, and L is the length of the pipe.

2. What is the significance of the Poiseuille equation in fluid dynamics?

The Poiseuille equation is a fundamental equation in fluid dynamics that is used to calculate the flow rate of a fluid through a pipe. It is important in understanding the behavior of fluids in pipes and is often used in engineering applications, such as in the design of pipes and pumps.

3. How does the Poiseuille equation differ from other equations for fluid flow rate?

The Poiseuille equation differs from other equations for fluid flow rate, such as the Bernoulli's equation, in that it takes into account the viscosity of the fluid. This makes it more accurate for calculating flow rates in viscous fluids, such as water.

4. Can the Poiseuille equation be used for non-Newtonian fluids?

No, the Poiseuille equation is only applicable to Newtonian fluids, which have a constant viscosity regardless of the applied shear stress. Non-Newtonian fluids, such as blood and ketchup, have varying viscosities and require different equations to calculate flow rates.

5. How can the Poiseuille equation be applied in practical situations?

The Poiseuille equation can be applied in various practical situations, such as in the design and optimization of pipes, pumps, and other fluid systems. It can also be used to calculate the flow rate of water in plumbing systems and to understand the behavior of fluids in medical procedures, such as blood flow in arteries.

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