Rain Barrel Physics

During a recent trip to Ottawa, we purchased a quality rain barrel
from The ArbourShop.

It is fun to measure the fill-rate from the rain barrel's 
lower tap, with a full barrel and compare the measured result with a simple 
calculation using elementary Physics of fluid mechanics. 





The full tank holds about 200 litres (or 200 kg/440 lbs) of water. The water 
fills only to the overflow spout. The distance from the overflow spout and 
the lower tap-nozzle is about  0.75 meters.

---  (1)   Hydrostatic Pressure at bottom of full rain barrel ---------
The hydrostatic water pressure at the tap of the rain barrel corresponding 
to 0.75 meters of water is:
  Pstatic =  rho*g*h
where
   rho is the density of water (1gm/cc or 1kg/litre or 1000 kg/meter^3)
   g is acceleration of gravity  9.8 meter/sec^2
   h is the water height  .. here 0.75 meter

so Pstatic =  7350 Pascals (Newton/meter^2)  or equivalently 1.07 psi. This 
is the water pressure above atmospheric pressure or gauge pressure.
(by comparison, note that atmospheric pressure is about 101,000 Pascals or 
14.7 psi and that normal city-supply water pressure is 30-50 psi).


----  (2) Flow Rate from open rain barrel tap  -----
Using basic Physics (i.e. Bernoulli's equation which is just conservation of 
energy) we find that the water flow velocity at the tap for a height h of 
water above it is:
     v = Sqrt(2*g*h)     (Torricelli's law)
or   v = 3.83 meter/sec  (or 12.6 ft/sec  or  13.8 km/hr).
assuming that the rain barrel diameter is much greater than the tap diameter.

The volume flow rate from the tap (which has a circular diameter D of about 
1 cm when completely open) should be the flow-speed times the tap area:
  flow-rate = v*Pi*D^2/4
A simple well known correction factor for flow rate (so called "Vena 
Contracta") must be applied which is roughly 0.6. This is due to the steep 
convergent flow lines, which tend to make the exit area of the hole 
effectively smaller by this amount.
So the corrected result is:
 flow-rate = v*pi*D^2/4 * 0.6
 or flow-rate = 0.00018 meter^3/sec   or
 flow-rate = -.18 litre/sec.

------- Measurement  -----
The actual time to fill an 8 litre watering can is 45 sec:
Measured flow-rate = 8/45 = 0.18 litre/sec
which agrees nicely with the basic Physics prediction above.