Sizing a pump, product path and nozzle requires a defined duty and product measurements. These four steps distinguish preliminary calculations from the OEM checks and sample tests needed before final dimensions and drive limits are approved.
Step 1: Document Fluid Rheology, Specific Gravity, and Particulate Geometry
Before crunching mechanical numbers, we profile the exact physical behavior of the product under target factory floor temperatures:
- Dynamic Viscosity (cP): Record temperature, representative shear-rate range, instrument/geometry, method and sample history. One cP value does not fully describe a shear-thinning or thixotropic product. Tetra Pak’s measuring guidance explains why process-representative conditions matter.
- Density / Specific Gravity: Record density at process temperature and use it consistently for mass/volume and pressure/head conversions. Do not add bar directly to metres of liquid.
- Particulate Dimensions: Measure the largest rigid particle diameter along its longest axis. Record particulate softness, shear sensitivity, and volume percentage in the slurry.
Step 2: Calculate Target Flow Rate per Nozzle Based on Cycle Time and Fill Volume
Define the actual timing sequence. In an indexed, non-overlapping example, effective fill time is total cycle time minus handling overhead. Continuous or overlapping machines need their own timing model, and cylinder refill must also fit the sequence.
- Illustrative timing: Assume a 3.0 s cycle and 1.2 s non-overlapping handling overhead. Effective fill time is 3.0 − 1.2 = 1.8 s. These are chosen inputs, not recorded GDHP operating data.
- Flow per nozzle: Q = V/t. For an illustrative 500 mL dose, Q = 500/1.8 = 277.8 mL/s = 16.7 L/min while filling. Average delivered volume over the complete 3.0 s cycle is 10 L/min per head. Size shared feed systems for simultaneous heads, buffering and refill timing, not merely the single-head average.
Step 3: Compute Total Dynamic Head (TDH) and Line Friction Loss
For a Newtonian, incompressible liquid in steady, fully developed laminar flow through a straight circular pipe, use the Hagen–Poiseuille relationship. Re = ρvD/μ < 2,000 is a laminar-flow screening check for the Newtonian example, not a substitute for a non-Newtonian model. For shear-dependent foods, see Tetra Pak’s rheology and pressure-drop discussion.
ΔP = 128μLQ/(πD4)
Use μ in Pa·s, L and D in m, and Q in m³/s to obtain ΔP in Pa. Conversions: 1 cP = 0.001 Pa·s; 1 L/min = 1/60,000 m³/s; 1 bar = 100,000 Pa.
To avoid high line resistance: Calculate friction loss for each straight tubing run, elbow, manifold junction, and valve port. Convert fitting friction into equivalent pipe lengths. Add the static vertical lift head (elevation rise from supply reservoir to nozzle tip). Add backpressure generated across the nozzle exit orifice.
Keep head and pressure consistent: Convert pressure using H = ΔP/(ρg), g ≈ 9.81 m/s², before adding lift in metres. Include actual valve, bend, nozzle and inlet losses and transient operation. The Section 4 table gives straight-pipe results only, not a pressure rating or final selection.
Step 4: Select Pump Displacement and Nozzle Bore with Engineering Safety Margins
With target flow rate and pressure loss (Delta P) established, size the final hardware:
- Nozzle Bore Sizing: D = √(4Q/(πv)). Choosing an illustrative 1.5 m/s for a first calculation gives D ≈ 15.4 mm at the example flow. A 6 mm maximum particle dimension and the chosen 3:1 screening allowance instead give an 18 mm candidate, at about 1.09 m/s. Check actual nozzle and valve geometry, particle integrity and bottle opening; neither 1.5 m/s nor 3:1 is a universal acceptance limit.
- Pump Displacement and Cycle: Use Section 2’s swept-volume calculation, then verify intake, discharge and handling timing. Percentage of maximum cycle speed is not percentage of piston stroke. Use the actual operating window, not a generic 60–75% maximum-speed rule.
- Motor Torque and Drive Margin: Use the same supplier-approved duty and overload checks as Section 2. For piston drives, calculate F = ΔP × piston area, then include transmission geometry, friction and acceleration. For rotary shafts, use T = 9,550P/n. Do not apply a separate blanket 25% margin here.
Cycle-time sensitivity example, not a viscosity test: Increasing the illustrative 1.8 s effective fill time by 30–60% gives 2.34–2.88 s. With unchanged 1.2 s overhead, the total cycle becomes 3.54–4.08 s, up 18–36%. The theoretical single-head rate changes from 20.0 to 16.9–14.7 cycles/min. This shows why fill-time and total-cycle changes differ; it does not predict the behavior of a 50,000 cP paste.