Calculating Pump Head for a Concentric Annulus Pipe

A 600 ft result on the calculation screen is a warning that velocity head, pressure, and pump power may have been mixed or that the wrong flow area was used. For a horizontal concentric annulus, calculate velocity from the annular area, friction from the hydraulic diameter, and pump duty from the complete system curve.

What is the 600 ft result telling you?

Head is energy per unit weight of liquid. For water, a pressure difference can be expressed as an equivalent water-column height, but head is not pressure itself and it is not motor horsepower. Darcy-Weisbach gives the friction head consumed by a flow path:

h_f = f(L/D_h)(V^2/(2g))

Multiplying that head by ρg gives pressure drop:

ΔP = ρg h_f = f(L/D_h)(ρV^2/2)

The acceleration term therefore appears in the head equation but cancels from the pressure equation. Omitting both g and density produces neither head nor pressure in valid units.

Calculation symptom	Likely cause	Correction
Hundreds of feet across a 0.15 m annulus	Wrong area, wrong diameter, unit mismatch, or pressure reported as head	Convert every dimension to one unit system and audit dimensions term by term
Velocity based on a 9 mm circular bore	Hydraulic diameter was incorrectly used as flow diameter	Use annular area for velocity and hydraulic diameter only in friction correlations
Head calculated from horsepower alone	Motor rating was treated as hydraulic output	Use the pump curve for head and include pump efficiency when calculating shaft power
Zero required head because the rods are horizontal	Static elevation was confused with friction loss	Set static elevation difference to zero but retain major and minor losses

Check: Confirm that the Darcy-Weisbach result carries units of length and that ρg h_f carries units of pressure before continuing.

Which annulus dimensions go into each equation?

For a concentric annulus with outer-pipe inside diameter D_o and inner-rod outside diameter D_i, the open flow area is:

A = (π/4)(D_o^2 - D_i^2)

The mean velocity is:

V = Q/A

The hydraulic diameter follows from D_h = 4A/P_w, where the wetted perimeter includes both surfaces:

P_w = π(D_o + D_i)

For this concentric geometry, that expression reduces to:

D_h = D_o - D_i

The two diameter calculations perform different jobs. The squared-diameter expression determines flow area and velocity. The diameter difference supplies the characteristic length in Reynolds-number and friction-loss calculations. Substituting D_h into the circular-pipe area formula is incorrect.

Configuration	Annular area	Hydraulic diameter	Velocity at 9 US gpm
D_o = 15 mm, D_i = 10 mm	98.17 mm^2	5 mm	5.78 m/s
D_o = 15 mm, D_i = 6 mm	148.44 mm^2	9 mm	3.83 m/s

These values use 9 US gpm = 0.0005678 m^3/s. The previously used 8.9 m/s velocity corresponds approximately to putting that flow through a 9 mm circular area; it does not represent the stated 15 mm by 6 mm annulus.

Check: Recalculate Q = AV with the displayed area and velocity. It must return 9 US gpm for each geometry.

How should the friction factor be selected?

Calculate Reynolds number with the mean annular velocity and hydraulic diameter:

Re = ρVD_h/μ

Read density ρ and dynamic viscosity μ at the operating water temperature. Then determine relative roughness from the wetted-surface roughness divided by D_h. Use those two quantities with an accepted friction-factor correlation or Moody chart.

The reported calculation used f ≈ 0.016 for glass at Re = 191,541. Another rough estimate used f = 0.03. These are Darcy friction factors only if they were obtained from a Darcy/Moody correlation. A Fanning friction factor is one quarter of the Darcy value, so mixing the definitions creates a factor-of-four error in calculated loss.

Changing the rod affects Reynolds number as well as velocity. With the same water properties, the Reynolds-number ratio is:

Re_new/Re_old = (V_new D_h,new)/(V_old D_h,old) ≈ 1.19

The larger passage therefore does not justify copying a friction factor solely because flow stayed at 9 US gpm. Recalculate it for each candidate diameter, especially when a geometry change moves operation toward a different flow regime.

Check: Label the selected value explicitly as a Darcy friction factor and record the Reynolds number and relative roughness used to obtain it.

How is major head loss calculated for each rod?

1. Convert Q, L, D_o, and D_i into a consistent unit system.
2. Calculate A = (π/4)(D_o^2-D_i^2).
3. Calculate V = Q/A.
4. Calculate D_h = D_o-D_i.
5. Calculate Reynolds number and select the Darcy friction factor.
6. Calculate h_f = f(L/D_h)(V^2/(2g)).
7. Convert to pressure only when required, using ΔP = ρg h_f.

For the 15 mm inside diameter, 6 mm rod, 0.15 m length, 9 US gpm case, applying the reported f = 0.016 gives:

h_f = 0.016(0.15/0.009)(3.83^2/(2g)) ≈ 0.199 m

That is approximately 0.65 ft of water across the straight annular segment. It is a conditional result: use it only when f = 0.016 matches the actual Reynolds number and roughness.

For design comparisons, the loss ratio is more useful than prematurely fixing a friction factor:

h_new/h_old = (f_new/f_old)(D_h,old/D_h,new)(V_new/V_old)^2

For the 10 mm-to-6 mm rod change, the geometric and velocity terms give approximately 0.243. If the two friction factors were equal, the new straight-section loss would be about 24.3% of the old loss. Apply the actual f_new/f_old ratio after calculating both Reynolds numbers.

Check: Increasing the open area from the 10 mm rod to the 6 mm rod must reduce velocity from about 5.78 to 3.83 m/s; a result that increases velocity has the area assignment reversed.

What other head must be added to the annulus loss?

Darcy-Weisbach for the 0.15 m straight section is only one term in total dynamic head. The horizontal arrangement makes the elevation term zero between points at the same height, but it does not remove pressure requirements, fitting losses, acceleration losses, or losses elsewhere in the circuit.

Calculate each local restriction with:

h_m = K(V_ref^2/(2g))

Take each loss coefficient K from the component data for its geometry. Use the velocity associated with the reference area specified for that coefficient. An annulus entrance, contraction, expansion, outlet, valve, connector, and measurement device can use different reference velocities; applying one velocity blindly to every term distorts the total.

System term	Location or source	Effect on pump head
Static elevation	Difference between suction and discharge reference elevations	Zero only when the selected endpoints are at the same elevation and have no other static-pressure requirement
Straight annulus friction	Each constant-geometry length	f(L/D_h)V^2/(2g)
Entrance and exit	Transitions into and out of the annulus	Add the applicable K V_ref^2/(2g)
Other circuit losses	Pipes, hoses, valves, fittings, and equipment	Calculate each segment at the same series flow
Required endpoint pressure	Difference between boundary pressures	Add ΔP/(ρg)

Check: Draw the complete path between the chosen suction and discharge reference points and account for every change in diameter, component, elevation, and boundary pressure.

How does the system calculation become a pump selection?

A centrifugal pump operates where its pump curve intersects the system curve. A pump does not independently impose both 9 US gpm and a selected head. The circuit resistance and pump characteristic establish the actual duty point.

1. Calculate total system head at several flow rates, including zero flow and points on both sides of 9 US gpm.
2. At each flow, recalculate velocity, Reynolds number, friction factor, straight losses, and minor losses.
3. Plot total system head against flow.
4. Overlay the pump curve for the actual pump configuration and operating speed.
5. Read the intersection and verify that it meets the required flow without exceeding the pump or motor operating limits shown by their data.

For a friction-dominated circuit with unchanged friction factors and no static-pressure term, head varies approximately with Q^2. That proportionality is useful for screening diameter alternatives, but the final curve must account for friction-factor changes and any static or pressure term.

Keep power separate from head. Hydraulic power is:

P_h = ρgQH

If η is confirmed as pump efficiency, required shaft power is P_shaft = ρgQH/η. The cited 0.8 value was described only as an estimated power correction factor. Identify whether it represents pump efficiency, motor efficiency, electrical power factor, or another correction before using it. Electrical power factor does not convert hydraulic power into pump shaft power, and a 1.5 hp motor rating does not prove that the pump can produce the required 9 US gpm duty point.

Check: The selected pump curve and calculated system curve must intersect at or beyond the required 9 US gpm point, while the corresponding shaft-power demand remains within the motor rating.

How is the result verified during commissioning?

1. Confirm the installed outer-pipe inside diameter, rod outside diameter, annular length, and concentric alignment.
2. Place pressure measurements at the same reference points used in the calculation. Account for any elevation difference between taps.
3. Start at a controlled condition and establish the target 9 US gpm using a suitable flow measurement.
4. Record suction pressure, discharge pressure, flow, water temperature, pump speed or operating state, and motor electrical loading.
5. Convert measured differential pressure to head with H = ΔP/(ρg), adding measured elevation and velocity-head differences when the endpoint areas differ.
6. Compare measured annulus pressure drop with the calculated straight and local losses, then compare the complete measured duty point with the pump and system curves.

A mismatch isolated to the annular segment points toward geometry, roughness, friction-factor, tap-location, or entrance-loss assumptions. A matching annulus drop with a mismatched total head points elsewhere in the circuit. A correct pressure rise but low flow points toward higher system resistance than modeled; correct flow with excessive motor loading requires a power and pump-condition review.

Check: Hold 9 US gpm steady, repeat the pressure readings, and verify that the measured flow-head point lies at the pump-curve/system-curve intersection without exceeding the motor rating.

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