How Do I Calculate a Coupled-Tank Overflow Timeline?

Wrong fixes and why they fail

Tank 1 starts at 27 ft with a 40 ft overflow elevation, leaving 13 ft of storage. Its reported inflow is 7,500 gpm, versus 1,100 gpm into Tank 2. The number that matters is net storage flow: pump inflow minus flow through the connecting pipe and any unreported outlet. A 30 s calculation interval can represent that balance, but a small interval cannot correct uncertain pump flow, incorrect elevations, or a bad pipe-flow curve.

Hydraulic mechanism

The two vessels exchange water only when their hydraulic grades differ. For vented tanks containing water, the driving head is the difference between free-surface elevations referenced to one datum. Equal liquid depths represent equal hydraulic grades only when the tank bases share the same elevation. A riser, elevated connection, closed or throttled valve, trapped gas, check valve, or different base elevation changes the effective system.

For a bottom-to-bottom connection with both tanks at the same base elevation, Tank 1 initially has 3 ft more head. Flow therefore starts from Tank 1 toward Tank 2. The pipe flow reduces Tank 1's accumulation and increases Tank 2's accumulation by the same amount. It cannot create or remove water from the combined system.

The reported connection is a 16 in carbon-steel pipe approximately 300 ft long. Pipe diameter, length, roughness, fittings, valve position, entrance losses, and exit losses determine how much transfer a given head difference produces. As levels move, both transfer magnitude and possibly direction move with them.

Quantities and deciding measurements

Two closed-connection timing checks imply the tank geometry used in earlier calculations. Tank 1 rising 13 ft at 7,500 gpm for gives about 84,231 gal/ft. Tank 2 rising 16 ft at 1,100 gpm for gives about 84,563 gal/ft. These differ by about 0.4% and correspond to a cylindrical diameter near 120 ft. Treat that diameter as a reverse-calculated check, not a replacement for the tank drawing.

Governing mass and energy balances

For constant-area tanks, write both balances with one signed transfer flow:

A1 dh1/dt = Q1,in - Q12 - Q1,out
A2 dh2/dt = Q2,in + Q12 - Q2,out

Q12 is positive from Tank 1 to Tank 2. Adding the equations cancels the intertank flow. That cancellation supplies the strongest mass-balance check:

d(V1 + V2)/dt = Q1,in + Q2,in - Q1,out - Q2,out

Calculate the connection from Darcy-Weisbach with minor losses:

|H1 - H2| = [f(L/D) + sum(K)] v^2/(2g)
Q12 = sign(H1 - H2) Ap v
Re = rho v D / mu

The friction factor f depends on Reynolds number and relative roughness. A noniterative correlation valid across laminar and turbulent regimes can simplify each step, but its inputs still need the actual inside diameter, roughness, and fluid properties. Near zero head difference, preserve the sign explicitly and return zero flow when the grades are equal.

One proposed fitted relationship was Q_cfs = 0.0283 dH^2 + 0.0363 dH, with dH in feet. At the initial 3 ft difference it gives 0.3642 ft3/s, or approximately 163 gpm. Used with that fit, initial net flows are about 7,337 gpm into Tank 1 storage and 1,263 gpm into Tank 2 storage. The fit requires validation against the Darcy-Weisbach system curve over the entire modeled head range; its positive quadratic term makes extrapolation especially sensitive at larger head differences.

Time-step reconstruction procedure

1. Establish one elevation datum. Convert both level readings to free-surface elevations and confirm the tank bases, pipe connection elevations, overflow elevation, and any high points.

2. Obtain each tank's area-versus-level or volume-versus-level table. Equal diameter and height support equal geometry, but certified dimensions decide the volume calculation.

3. Build time-dependent boundary conditions for pump flow, outlet flow, and valve position. If only pump information exists, calculate the operating point from the pump curve and system head at each level rather than holding a nominal flow constant.

4. At the beginning of each 30 s interval, calculate . Solve the pipe equation for signed Q12, including straight-pipe and minor losses.

5. Calculate net flow for each tank. Convert gallons to volume units compatible with tank area, then update level with Delta h = Qnet Delta t/A or use the tank strapping table for nonconstant area.

6. Repeat until Tank 1 crosses 40 ft. Interpolate within the final interval between the last level below 40 ft and the first level at or above it.

7. Repeat with half the time interval. A materially different overflow time indicates that the original interval is too large or the flow solver is discontinuous near reversal.

With the connecting valve closed and no outlets, the model should reproduce approximately 2 h 26 min for Tank 1 and for Tank 2 using the geometry implied by those benchmarks. This isolates integration and unit errors before pipe transfer is introduced.

Flow-history and scenario corrections

Constant inflow is the largest investigative assumption. A centrifugal pump operating against a rising liquid level normally moves along its pump curve unless a flow-control system holds the commanded rate. Flow indication, pump speed, discharge pressure, valve position, and start-stop status separate a controlled constant-flow case from a changing operating point.

Reported scenarios for Tank 2 contain values that require different hidden assumptions. Raising Tank 2 from 24 ft to 30 ft in was associated with both 3,600 gpm and 5,922 gpm. A further estimate used 11,280 gpm to rise from 30 ft to 40 ft in one hour. Separate gross pump inflow, pipe transfer, outlet flow, and net storage flow before using any of these numbers.

An illustrative coupled-tank case used about 7,000 gpm into Tank 1 and 8,500 gpm into Tank 2 to place both overflows near . That case demonstrates the required direction of correction: Tank 2 needs substantially more water than the reported 1,100 gpm if both tanks approached overflow together. It does not establish the actual pump flows.

Verification against the event timeline

Corroborate the operating account with a band of timelines rather than one false-precision timestamp: a nominal case, credible minimum and maximum pump-flow cases, alternative valve states, and level-instrument uncertainty. The account is hydraulically possible when its claimed overflow time and intermediate observations fall within cases that also close the combined mass balance.

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