Added at top May 2016:
@Johann commented on May 20th :
Please note that the Question is talking about an Aquaponics system, not Hydroponics. The entire water system needs to completely recirculate at least once an hour due to fish/oxygen needs. Your pump will be a very large one or a multi staged set up.
This suggests that while my advice may be correct for a given flow rate, the desired flow rate is much higher than I assumed. The actualtime to circulate all fluid is an importnt parameter nd must be specified by the 'client'.
A quick update based on that, is that my power required formula still holds as an approximation. ie
So, if 1 hour circulation time is required.
Power = Gallons_per_hour / 16.5 Watts
For a 5000 gallon system
= 5000 /16.5 = 300 Watts or about 1/2 HP.
As the volume involved appears to be more than 5000 gallons (adding up the various tanks & beds) then perhaps a 750 Watt / 1 HP pump may be needed for once per hour circulation.
Engineers view :-).
Replace with answer by somebody with practical hydroponics experience when available.
You can determine desired flow rate needed from hydroponics lore.
Let's say the system = 5000 gallons or 20,000 litres (4:1 is close enough for this purpose).
To recirculate the total fluid in 1 week = 168 hours requires 5000/168 = 30 gallons per hour or about 120 litres per hour
(= 2 litres/minute or 33 cc/second - about 1/3 of a 100cc medicine measure per second.
Not very fast.)
According to the formula in part (2) below that would requires about 120/66 or about a 2 Watt pump.
That's very small.
Taking the other approach - a 50 Watt pump will supply about 50 x 66 = 3300 litres per hour. Total system fluid will recirculate in about 20,000l/3300 = 6 hours.
If 6 hours is acceptable then a 50 Watt pump will be adequate. That sizing was probably conservative so I'dd expect you'd actually get substantially more flow and quicker turnover than that if required.
Say flow wanted = L litres/hour.
Lift from top of sump if full to grow bed = 2 feet.
If you want to be able to pump with simp only half full = 3 feet from top, required lift = 5 feet. Say 1.5 metres. If you want to be able to pump sump from very bottom you need more power or be willing to accept a lower flow rate (probably acceptable).
So say Lift at full flow = 1.5 metres = height = head = h
(1) The easy way.
Pump manufacturers provide lift x volume charts and these can be used to select required power.
(2) The slightly harder way. In your case -
Read on ...
Pump power at 100% efficiency ~= head x litres/second x 10 Watts
litres/second = litres/hour/3600.
For system with less than 100% efficiency multiply power needed by K.
K = 100/efficiency %.
eg 50% efficiency -> k= 2.
25% efficiency K = 100/25 = 4 etc.
So Power = metres lift x litres / hour * 10/3600 * k
Assume k-5 for now = 20% efficient = easily achieved.
This includes pipeline head or pressure losses at typical flow rates.
Power = lift x litres /h x 50/3600 = h x l x 0.0139
Close enough to start:
Power = lift x litres per hour / 100
For 1.5m lift Power = litres per hour / 66
or litres per hour = 66 x Watts.