12.8 Hydraulics, Pressure & Unit Conversion Mathematics

12.8 Hydraulics, Pressure & Unit Conversion Mathematics

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1. Pressure and head

   1 psi  = 2.31 feet of water
   1 foot of water = 0.433 psi

Examples.

• An elevated tank with its water surface 145 feet above a customer's meter delivers 145 × 0.433 = 62.8 psi (before friction losses).
• A gauge reading 68 psi corresponds to 68 × 2.31 = 157 feet of head.
• A pump that must lift water 210 feet and deliver 40 psi at the top must produce 210 + (40 × 2.31) = 210 + 92.4 = 302.4 feet of total head, plus friction.

Head types: static head is elevation difference only; friction head is energy lost to pipe and fitting resistance; velocity head is V²/2g; and total dynamic head (TDH) is the sum the pump must actually produce.

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2. The continuity equation

$Q = A \times V$

where Q is flow, A is cross-sectional area, and V is velocity — in consistent units (cubic feet per second, square feet, feet per second).

Example. An 8-inch main carries 700 gpm. What is the velocity?

1. Area = 0.785 × (8 ÷ 12 ft)² = 0.785 × 0.4444 = 0.349 ft²
2. Flow in cfs = 700 gpm ÷ 448.8 gpm per cfs = 1.56 cfs
3. V = 1.56 ÷ 0.349 = 4.47 ft/sec — within the normal 2 to 5 ft/sec distribution range.

A useful shortcut for round pipe: V (ft/sec) = gpm ÷ (2.448 × D²) with D in inches. Checking: 700 ÷ (2.448 × 64) = 700 ÷ 156.7 = 4.47 ft/sec.

Grit channel check. The same equation verifies the 1.0 ft/sec grit channel target: if the channel is 2 ft wide with 1.2 ft of water depth and passes 2.2 cfs, V = 2.2 ÷ (2 × 1.2) = 0.92 ft/sec, slightly low — organics will begin settling with the grit.

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3. Friction loss

Friction loss rises steeply with velocity (roughly with the 1.85 power in the Hazen-Williams relationship) and falls steeply with diameter. Two operating consequences:

• Doubling the flow through a given pipe roughly triples to quadruples the friction loss.
• A tuberculated unlined cast iron main with a C-factor of 70 requires more than three times the pumping energy of a smooth main at C = 140 to move the same flow.

That is why leaving a partly closed valve in service, or deferring main replacement, shows up as an electric bill long before it shows up as a complaint.

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4. Horsepower and efficiency

  Water horsepower (WHP)  = (gpm x head in feet) / 3,960
  Brake horsepower (BHP)  = WHP / pump efficiency
  Motor horsepower (MHP)  = BHP / motor efficiency
  Wire-to-water efficiency = pump efficiency x motor efficiency

Worked example. 1,100 gpm against 96 feet TDH, pump efficiency 78 percent, motor efficiency 92 percent.

1. WHP = (1,100 × 96) ÷ 3,960 = 105,600 ÷ 3,960 = 26.7 hp
2. BHP = 26.7 ÷ 0.78 = 34.2 hp
3. MHP = 34.2 ÷ 0.92 = 37.2 hp — a 40 hp motor
4. Wire-to-water efficiency = 0.78 × 0.92 = 0.718, or 71.8 percent

Energy cost. kW = MHP × 0.746 = 37.2 × 0.746 = 27.8 kW. Running 18 hours a day at $0.105/kWh: 27.8 × 18 = 500 kWh/day × $0.105 = $52.50 per day, or roughly $19,160 per year. A five-point improvement in wire-to-water efficiency on that pump is worth about $1,300 a year — which is how an efficiency argument gets made to a budget committee.

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5. Areas and volumes

Example — pipe volume. How many gallons are in 1,500 feet of 12-inch main?

0.785 × (1.0 ft)² × 1,500 ft = 1,177.5 ft³ × 7.48 = 8,808 gallons — the number you need to size a disinfection dose or a flushing volume.

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6. The conversion table that prevents most errors

[!NOTE] Diagnose your own arithmetic. When an answer looks impossible, the error is nearly always a conversion: a factor of 8.34 (forgot the weight of water), 7.48 (cubic feet versus gallons), 1,440 (per day versus per minute), 1,000,000 (MGD versus gpd), or 12 (inches versus feet). Check those five before re-deriving the formula.