Cooling tower · water balance priced at utility rates
A process cooling tower circulates 1,200 gpm on a 10 °F range, year-round, and the conductivity controller holds 4 cycles of concentration. The eliminators are rated at 0.005% drift. The municipality bills water at $4.00 per 1,000 gallons and sewer at $5.50 per 1,000 gallons, and — because the plant metered its makeup and blowdown — it grants a sewer credit for water the tower evaporates rather than drains.
Given
R = 1200 gpm — Recirculation rate
ΔT = 10 F° — Cooling range
COC = 4 cycles — Cycles of concentration
d = 0.005 % — Drift rate
t = 365 d — Operating period
p_w = 4 $/1000 gal — Water rate
p_s = 5.5 $/1000 gal — Sewer rate
Determine
(a)the evaporation rate
(b)the blowdown needed to hold 4 cycles
(c)the drift loss
(d)the total makeup rate
(e)the makeup and evaporated volumes over the year
(f)the net annual water and sewer bill
Step 1 of 7(a) · solve for Evaporation rate
Evaporation is where the heat actually leaves: 0.1% of recirculation per °F of range, so 1,200 gpm on a 10 °F range boils off 12 gpm. It drives everything below — the blowdown is sized from it, and it is also the only loss the sewer never sees, which is what part (f) turns into money.
Rearranged for E
E=0.001RΔT
Your values, in your units
E=0.001(1,200gpm)(10F∘)
Converted to base units
E=0.001(4,542.49L/min)(5.55556C∘)
Answer
E=45.425L/min
Carried onward at full precision, not this rounded figure.
Holding 4 cycles against 12 gpm of evaporation costs B = E/(COC − 1) = 4 gpm to drain. The minus one is the part people drop: E/COC gives 3 gpm, the cycles drift up, and the first symptom is scale on the hottest condenser tubes, not a controller alarm.
Rearranged for B
B=COC−1E
45.425 L/mincarried from step 1
Your values, in your units
B=(4)−1(0.000757082m3/s)
Converted to base units
B=(4)−1(45.4249L/min)
Answer
B=15.142L/min
Carried onward at full precision, not this rounded figure.
Drift is 0.005% of the 1,200 gpm recirculation — 0.06 gpm, droplets out the stack. On the water bill it is a rounding error; it is computed here because the makeup sum in part (d) must close, and because on the chemical ledger those droplets leave at full basin strength.
Rearranged for D
D=100dR
Your values, in your units
D=100(0.005%)(1,200gpm)
Converted to base units
D=100(0.005%)(4,542.49L/min)
Answer
D=227.12mL/min
Carried onward at full precision, not this rounded figure.
Water leaves through three doors and the float valve replaces all three: M = 12 + 4 + 0.06 = 16.06 gpm, 1.34% of recirculation — the sanity ratio to carry in your head. This is the flow the water meter will actually record.
A year of 16.06 gpm is 8.44 million gallons through the makeup meter — 31,953 m³, the top line of the utility's invoice. Rates are quoted per 1,000 gallons, so the volume is kept as 8,441.1 kgal for the money step.
Rearranged for V
V=Qt
60.794 L/mincarried from step 4
Your values, in your units
V=(0.00101323m3/s)(365d)
Converted to base units
V=(60.7937L/min)(31,536,000s)
Answer
V=25.905ac⋅ft
Carried onward at full precision, not this rounded figure.
The same year of 12 gpm evaporation is 6.31 million gallons that never reached a drain. This volume is the whole argument for the evaporation credit: without the meters to prove it, the utility bills sewer on every gallon of makeup as if the roof were a floor drain.
Rearranged for V
V=Qt
45.425 L/mincarried from step 1
Your values, in your units
V=(0.000757082m3/s)(365d)
Converted to base units
V=(45.4249L/min)(31,536,000s)
Answer
V=19.356ac⋅ft
Carried onward at full precision, not this rounded figure.
Step 7 of 7(f) · solve for Net water and sewer cost
Makeup pays the water rate on every gallon; sewer is charged only on what actually drains — makeup minus evaporation, 2.13 million gallons. $33,765 of water plus $11,737 of sewer is a $45,501 year; without the credit the sewer line alone would be $46,426, more than doubling the bill.
The tower evaporates 12 gpm, bleeds 4 gpm to hold 4 cycles, drifts 0.06 gpm and draws 16.06 gpm of makeup — 8.44 million gallons a year, of which 6.31 million evaporate — for a net annual water and sewer bill of about $45,501 once the evaporation credit is claimed.
Why this order
The three cooling-tower chains in the water-treatment section size the tower, the inhibitor and the biocide; this one prices the water itself, and the order is the same because the physics is: evaporation first, since it is set by heat load alone; blowdown second, since B = E/(COC − 1) is a choice about cycles, not a fact about the tower; drift third; and only then can makeup, volumes and dollars exist. The money step has one idea in it worth the whole chain: a sewer bill is a claim that water went down the drain, and 79% of this tower's makeup went up the stack instead. The credit is not a discount program — it is the utility agreeing to bill what happened. Cross-check the split in round numbers: 16.06 gpm splits 12 up, 4.06 down (4 bleed + 0.06 drift, which the sewer meter counts together), and 8,441 kgal × $4.00 + 2,134 kgal × $5.50 = $33,765 + $11,737 = $45,501 ✓.
Two levers and one trap live in these numbers. Lever one is cycles: at 3 cycles the bleed doubles to 6 gpm and the bill grows by about $8,900 a year; at 6 cycles it falls to 2.4 gpm and saves $4,500 — which is why the conductivity controller earns its keep and why the chemistry program that makes high cycles safe is priced against exactly this arithmetic. Lever two is the credit itself: it exists only where a deduct meter (or a metered blowdown) proves the evaporation, and the meter that unlocks it costs a few thousand dollars against $34,700 a year of avoided sewer charges here. The trap is treating the 0.001-per-°F evaporation rule as exact: it is a design-day figure that runs low in cold dry weather and high in humid heat, and a tower that winters on free cooling evaporates far less — which cuts the credit, not just the load. The rule prices the budget; the meters settle the invoice.
Carried values move at full precision, not the rounded figure shown — chaining rounded numbers compounds error.
