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?
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 $k$.
It's just a proportionality constant.
