Question

How long will it take for 500ml of water to flow through a 15cm long 1.5mm radius if the pressure differential across the pipe is 4kPa? The viscosity of water is 0.8 c poise.

Answer 1

Hint: Use Poiseuille’s law which states that the flow of liquid in a cylindrical pipe depends on four factors. Flow, directly proportional to the pressure differential, directly proportional to the four power of the radius of the pipe, inversely proportional to the length of the pipe, and inversely proportional to the viscosity of the liquid. After getting the flow of water from the above concept use the relation between the rate of flow and time taken by the liquid to flow.

Complete step-by-step answer:
Step 1: Express Poiseuille’s law mathematically.
$\therefore Q = \dfrac{{\Delta P\pi {r^4}}}{{8\eta l}}$
Where, $Q$ is the rate of flow of the water, $\Delta P$ is the pressure differential, $r$ is the radius of the pipe, $\eta $ is the viscosity of the liquid, and $l$ is the length of the pipe.
Step 2: Before putting the values of the variables convert all the values in the same unit.
$V = 500ml = 0.5{m^3}$
$l = 15cm = 0.15m$
$r = 1.5mm = 1.5 \times {10^{ - 3}}m$
$\Delta P = 4kPa = 4000Pa$
$\eta = 0.8centipoise = 0.8 \times {10^{ - 3}}Pa.s$
Step 3: substitute the values in the formula
$\therefore Q = \dfrac{{\Delta P\pi {r^4}}}{{8\eta l}} = \dfrac{{4000 \times 3.14 \times {{\left( {1.5 \times {{10}^{ - 3}}} \right)}^4}}}{{8 \times 0.8 \times {{10}^{ - 3}} \times 0.15}}$
\[ \Rightarrow Q = 6.624 \times {10^{ - 5}}{m^3}/s\]
Step 4: express the formula of the flow rate of liquid In terms of volume and the time that is taken by the liquid to flow.
$\therefore Q = \dfrac{V}{t}$
$ \Rightarrow t = \dfrac{V}{Q}$
Where, $V$ is the volume of the liquid and $t$ is the time taken by the liquid.
Substituting the values.
$\therefore t = \dfrac{{0.5}}{{6.624 \times {{10}^{ - 5}}}}$
$ \Rightarrow t = 7548.31s$
Therefore the water will take $7548.31s$ to flow through the pipe with given conditions.

Note:
The flow rate and the velocity of the liquid should not be confused. We can understand this difference with a very simple example, suppose you are standing on the bank of the river. Then the flow rate will tell us how much volume of water is flowing in a unit of time and the velocity of the water will be the velocity of the leaf floating and moving on the surface of the water. The flow rate depends on the size of the river and the viscosity of the water. And because of viscosity the water flows in layers, the upper layer flows fatter than the layer at the bottom.