
Pipe sizing example:
A site has 50 meters of head and a flow of 5 liters per second, a very typical situation here on the coast. The pipeline will have to be 150 meters long, and is going to be polyethylene plastic, (poly pipe). We will accept a head loss of 10% at maximum flow, that is, 5 meters of height lost to pipe friction. The power line length is 50 meters to the battery.
Refer to the section on "Power Calculations" for more detailed explanations of these formula.
Step 1.
Determine the available power.
5 liters per second is 0.005 cubic meters per second )
Power = ( 1000 x Q x h x 9.8 ) x efficiency) watts
Power = ( 1000 x 0.005 x 50 x 9.8 ) x 0.5 ) watts
Power = ( 5 x 50 x 10 ) x 0.5 = 1250 watts
Note: the acceleration due to gravity value of "9.8" was rounded up to "10"
The final output will be somewhat less as the net head will be 45 meters at full flow, or around 1125 watts, 90 % of the last calculation. This is close enough, and we decide to proceed with the project.
Step 2.
Plot a flow of 300 liters per minute on a friction chart applicable to the pipe material you are using, so the flow velocity is less than 5 feet per second. Note the pipe size indicated, and draw lines for several sizes that are available. This will likely be 2 inch, 2.5 inch, 3 inch and 4 inch.
Pipe friction charts are available from pipe suppliers. Each type of pipe will have a different coefficient of friction, so a chart used for steel will not be accurate for plastic pipe.
From the chart ...
2 inch gives a velocity of 8.2 feet per second which is too fast.
2.5 inch results in 5.2 feet per second.
3 inch gives 3.7 feet per second.
Pipe inside diameters vary with pipe schedule, (wall thickness), so you will seldom get exactly the diameters listed. Adjust the line you drew on the graph to match the pipe diameter you will be using. This is extremely important if you are using small diameter pipe like 1 inch or 1.25 or 1.5 inch PVC. Some pipe is sold by schedule and some by 'DR' rating (dimension ratio). On long runs of 1000 feet, even the slight differences in inside diameter can make a big difference to head loss. Be careful when making calculations.

If you really want to know the PVC pipe loss, here is the standard 'HazenWilliams' formula that is used for pressure pipe friction loss. You can also Google it to find more information on the subject.
f= 0.2083 (100/C)^1.852 X Q^1.852 / id^4.865
where f = friction loss per 100 feet
C = coefficient of friction, usually 150 for PVC
Q = water flow in USGM
id = pipe inner diameter
For Poly pipe, C is often 140 meaning higher friction. Always try to find out the C for your pipe and the exact id.

Our allowable head loss is 5 meters in 150 meters of pipe length, or 3.33 meters per 100 meters of pipe. Charts are usually normalized to meters per 100 meters or feet per 100 feet, so you may want to convert at this point. In this example, 2.5 inch poly pipe will deliver the required maximum flow with 3.3 meters of head loss per 100 meters of pipe, and the flow velocity is right on 5 feet per second.
It is possible to run more water through the 2.5 inch pipe by increasing the nozzle size. This will increase the head loss beyond our limits, but it will produce more power. In fact we could drop to 70% of the gross head before the power curve would start to fall. However, wide ranges of pressure result in variable jet velocities which in turn determine turbine RPM. In DC systems this is often acceptable, but in AC systems, RPM must be maintained, and a low jet velocity results is a significant loss of efficiency. This is why hydro equipment is so site specific, what worked well at one site may perform badly at another.
Step 3.
Determine the correct nozzle diameter to accept a maximum of 5 liters per second. Often, several nozzles are cut and installed to match flow conditions. 'Stream Emgine' turbines used on some of the example sites featured here are equipped with 2, 3 or 4 nozzles. Each nozzle can have a control valve permitting a wide range of flow to suit conditions. In larger AC systems, spear nozzles are often used which permit continuous adjustments of flow by varying the position of a movable spear within the nozzle.
Refer to a nozzle flow table to determine the correct size to use, or start small and gradually increase the size while measuring the flow in a bucket or through a weir.
Since water is non compressible, the flow velocity and nozzle size can be calculated from scratch. One only needs these formulas. The flow will typically be 5% less than the jet area suggests, so cut it a fraction big. Also, jet velocities are often about 5% less than calculated due to inefficiencies in the nozzle and other complex factors.
Area = 3.1428 x R^2 (Radius)
Circumference = 3.1428 x D (Diameter)
Jet velocity = 8 x H ^ 1/2 ( 8 times the square root of the net height in feet)
Imagine a solid bar of water passing through the nozzle traveling at the jet velocity. One only needs to determine the bars volume per unit time as it passes by. This is usually in one second. The diameter and hence cross sectional area are found first, then simply multiply this by the 'bars' length passing per second to arrive at the volume. This is easy to do in cubic inches and linear inches, then convert to gallons or litres. See the conversion chart below.


Units of measure:
Any time we deal with the subject of energy, power, pressure or heat, various definitions are required to make meaningful calculations and comparisons. This section explains the meaning of the 'SI' or metric units, and compares them to the imperial or US units.
Every attempt has been made to ensure accuracy on these figures. In all your calculations, be sure the result makes sense, and that you are not getting a result that seems out of proportion to the input numbers. In dealing with energy transfer, be sure to factor in losses due to inefficiencies. This is very important when comparing costs from the BTU's produced from burning fossil fuels to KWh / BTU's achieved from electrical heat sources.
Definitions
Ampere

Unit of electrical current, a quantity of electron flow equal to 6 * 10^{18} electrons / second. Analogous to water volume or quantity flowing in a pipe.

Volt

Unit of potential difference or electromotive force. Analogous to water pressure in a pipe. In a house, normal wall voltage is 117 volts. Stoves and dryers use 240 volts.

Watt

Unit of power, the product of amps times volts. The rate of work done in a set unit of time. Equal to the energy spent by one amp flowing through one ohm for one second.

Kilowatt

Unit of power, one thousand watts

Kilowatt Hour

Unit of power, the rate of work done in a set unit of time. In this case one hour. If you run a 100 watt bulb for 10 hours, that equals 100 watts times 10 hours = 1000 watt hours, or one kilowatt hour. Electricity is usually bought and sold using block units of kilowatt hours, (1000 watts for a period of time). Normal utility billing is every month or two, so if you use say 15 kilowatts over a 24 hour period for 60 days, that will work out to 15 kW hours a day times 60 days = 900 kw hours. At 7 cents a kilowatt hours, that will cost you $ 63 dollars. It is important to note that you can use up the electricity at any rate you need during a day, the meter just counts the total used per billing period.

Ohm

The measure of electrical resistance equal to that resistance which dissipates energy at the rate of one watt from a current of one ampere.

Joules  kilowatts  BTU's
1 Joule
1 Joule
1 Joule
1 GJ
1 BTU
1 kW
1 kWh
1 kWh

1 watt flowing for one second
1 Newton / metre
0.000948 BTU
948,000 BTU
1054.6 Joule
3412 BTU / hr
3412 BTU
3.6 Mega Joule

Definition of a BTU
Energy required to heat one pound of water by 1 degree F.
10 BTU = 10 pounds (1 imperial gallon ) heated by 1 degree F.
Energy
1 Watt hour
1 kWh
1 Therm

3.413 BTU
3413 BTU
100000 BTU

Power and Heat flow
1 kW
1 kW
1 Ton
1 HP
1 hp

0.948 BTU/sec = 3413 BTU/hr
1.3415 HP = 738 ft lb/sec = 44,268 ft lb/min
12000 BTU/hr
0.764 kW = 2546 BTU/hr
0.7455 kW = 550 ft lb/sec = 33000 ft lb/min

Power and Heat flow
1 kW 
0.948 BTU/sec = 3413 BTU/hr 
1 kW 
1.3415 HP = 738 ft lb/sec = 44,268 ft lb/min 
1 Ton 
12000 BTU/hr 
1 hp 
0.764 kW = 2546 BTU/hr 
1 hp 
0.7455 kW = 550 ft lb/sec = 33000 ft lb/min 
Volumes and conversions.
litre
cubic ft
us gallon
imp gallon
cubic foot
cubic meter

= US gal * 0.26442
= litres * 28.313
= litre * 3.785
= litre * 4.5459
= US gallons * 7.49
= 1000 litres

Length
meters
feet
km
miles

= feet * 0.3048
= meters * 3.281
= miles * 1.609
= km * 0.621

Pressure
1 PSI = 6894.76 Newtons / sq. m = 6894.76 Pascal
Absolute Pressure = psia = Gauge pressure (psig) + 14.7
Chemical content of fuels
1 ft^3 of natural gas = 1020 BTU (chemical energy constant)
1 gallon (US) # 2 fuel oil = 140,000 BTU
1 gallon (US) propane = 91,200 BTU
Calorie (c)
Heat required to raise 1 gm. water by 1 degree C.
Kilocalorie (C) = 1000 calories. C = food calorie
 

Some practical fuel cost examples: Updated Dec 2015.
Natural gas
is sold by its heating value in 'Giga Joules', not by its volume because the heating ability per unit volume can vary. Since it is a commodity, its price can vary.
Electricity
is sold by the 'kilowatt hour'. In BC, the level one tier cost for 1 kwh is currently $0.08, December 2015 price, and $ 0.12 cents for the level two tier. That is 8.0 cents for 1,000 watts for one hour  or 100 watts for ten hours. So to run a typical 1500 watt electric heater for one hour it costs (1.5kW X 8.0 cents, or about 12 cents an hour)
Some definitions you have to understand for this to make sense:
 ' / ' means per or divided by. * or X mean to multiply.
 1 Joule = 1 watt for 1 second (a tiny amount of energy )
 1 Giga Joule = 1 billion Joules (a fixed amount of energy used for billing purposes)
 For the purpose of these examples, the cost of electricity is 8.0 cents per kW hour. This is ignoring the two tier price system since a careful consumer can keep the monthly use with in the first price tier. This would not apply to a home heated by electricity.
1 Giga Joule is the same amount of energy as 277 kilowatt hours. That is about the amount of electricity used in 15 days in an average home that is heated by natural gas. For an electrically heated home, this figure will be considerably higher in the heating season.
In BC, the Dec 2015 cost of 277 kW hours of electricity (1 GJ worth) = 277 X 8.0 cents/kwh = $ 22.16, taxes extra. (1,376 KWh is the price tier change)
The cost of 1 GJ of natural gas (277 kW worth) in Dec 2015 hovers about $ 3.56 per GJ, but with storage, delivery and PST, the price is $11.00 per GJ.
So you can see that heating with natural gas is presently about half the cost as heating with electricity.
Facts about Oil:
The New York spot price of one US gallon of heating oil at the end of Dec 2015 is US $ 01.08, or US $0.29 cents a liter. Oil, like natural gas, is traded as a commodity on the stock markets and with oil at an all time low in 2015, this price per gallon is much lower than in the past years. Since this price is so volitile of late, take what you read below with some doubt! It is likely a liter of oil would sell for Cdn 90 cents at this time, 2015 price.
A typical home oil tank holds 300 US gallons, or 1130 liters.
The heating value of 1 liter of # 2 heating oil is 37,000 BTU, or British Thermal Units. (An old, but useful measuring system  based on 140,000 BTU per USA gallon)
At 37,000 BTU per liter / 3,413 BTU per kW hour = 10.8 kW hour per liter of heating oil. This means that there is the same amount of energy in one liter of oil as that used by a 1000 watt electric heater running for 10.8 hours.
10.8 kW of electricity at $ 0.08 per kW = (10.8 X 8.0) = 86 cents
And as stated above, 1 liter of heating oil costs about 90 cents. Add to that the lower efficiency of some gas or oil furnaces and the real cost is somewhat more. If the furnace is 75 % efficient, then 25 % of the cost is going to waste. Electric heat on the other hand, although also quite expensive, is nearly 100 % efficient, so all the power used is turned to heat at the electric heater. If however you heated with electricity, then with tier two at 12 cents per KW, electricity would likely be more expensive.
 
