# What does k denote in this equation?

Recently, I was attempting to compare the rate of flow of different liquids using the formulas given below:

$$P = \rho g h$$ $$\frac{dh}{dt}\varpropto h$$ $$\frac{dh}{dt} = -k h$$ $$\frac{dh}{h} = -k dt$$ $$\int \frac{dh}{h} = -k \int dt$$ $$\ln h = -k t + c$$

My question here is: What does $k$ denote? Is it a physical quantity like density?

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If the flow rate is dominated by the viscosity, so inertial effects can be neglected, the flow rate is given by the Hagen-Poiseuille equation:

$$\Delta P = \rho gh = \frac{8 \mu L Q}{\pi r^4}$$

The pressure difference is just the pressure at the orifice or $\rho gh$ as given in your first equation. $\mu$ is the viscosity of the liquid, $L$ and $r$ are the length of the outlet pipe and its radius, and $Q$ is the volume flow rate. Rearranging the equation we get:

$$Q = \frac{\pi r^4 \rho g}{8\mu L} h$$

and the rate of change of height in your experiment is the volume flow rate divided by the area of your burette (it's negative because the height is decreasing):

$$\frac{dh}{dt} = -\frac{Q}{A}$$

so:

$$\frac{dh}{dt} = -\frac{\pi r^4 \rho g}{8\mu L A} h$$

Comapring this with our expression you'll see that the constant $k$ is:

$$k = -\frac{\pi r^4 \rho g}{8\mu L A}$$

In parctice the dimensions of the liquid outlet are poorly defined. Poiseuille's equation really only applies when the outlet is a pipe that's long compared with its radius. So in real life we have a tendancy to just use an experimentally measured value for $k$.

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will this work using a burette? – fluuufffy Feb 14 '14 at 8:15
It should give a pretty good guide e.g. if you use different fluids the flow should be proportional to $\rho/\mu$. Bear in mind the burette outlet isn't a well defined tube - you have a constriction at the tap as well as at the burette tip, and the orifice at the tip isn't a pipe. However I'd guess you'll get qualitative agreement with the HP equation. Give it a try and let us know how you get on. – John Rennie Feb 14 '14 at 8:21
true! however the shape of a cylinder is almost cylindrical except near the tap, so this can be a part of evaluation, i guess. anyways, thanks alot – fluuufffy Feb 14 '14 at 8:35
sorry to bother but another question: when using the value of the radius in the above equations, is it referring to the radius of the orifice or the body of the burette? – fluuufffy Feb 14 '14 at 8:50
@fluuufffy: it's the radius of the orifice. Well, that assumes that the orifice is the narrowest part of the kit. If the restriction was in the tap it would be the radius of the tap channel. – John Rennie Feb 14 '14 at 9:04

It's just a proportionality constant.

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