How it works — the dual-model deposition limit
A settling slurry in a horizontal pipe must run fast enough that turbulence and the moving bed sweep every particle along; slower than that, a stationary bed grows. Two classical models bracket the deposition limit velocity \(V_{sm}\), and this tool computes both and lets the larger govern — a conservative envelope for design:
Fine particles — grains buried in the viscous sublayer (Thomas-type correlation, coefficient 9.0 in any consistent unit system):
$$V_{sm}=9.0\,\bigl[g\,\nu\,(S_s-S_f)\bigr]^{0.371}\left(\frac{D}{\nu^{2}}\right)^{0.11}$$Ceiling — interface-friction analysis caps \(V_{sm}\) at the nominal ceiling of the classic deposition nomogram, with \(f_f\) the pure-fluid (Darcy) friction factor of the line:
$$\frac{(V_{sm})_{max}}{\sqrt{2gD\,(S_s-S_f)}}=\left(\frac{0.018}{f_f}\right)^{0.13}$$| Symbol | Meaning | Units |
|---|---|---|
| \(V_{sm}\) | critical deposition (limit stationary-deposit) velocity | m/s |
| \(V_{50}\) | velocity at which half the solids are contact-load supported | m/s |
| \(M\) | gradation exponent of the stratification power law | — |
| \(d_{50},\,d_{85}\) | particle sizes passing 50% / 85% by mass | mm |
| \(D\) | pipe inside diameter | m |
| \(S_s,\ S_f,\ S_m\) | relative density of solids, carrier fluid, mixture | — |
| \(C_v,\ C_w\) | solids concentration by volume / by weight | — |
| \(\nu,\ f_f\) | carrier kinematic viscosity; Darcy friction factor | m²/s, — |
| \(g\) | gravitational acceleration, 9.807 | m/s² |
Design practice then applies a margin: operate at Vm ≥ 1.1 × Vsm. Below Vsm a stationary bed forms (🔴); between Vsm and 1.1 × Vsm the line runs but without margin (🟡). Two useful quirks of the physics: the limit is worst for particles around 0.4–1.0 mm (the nomogram peak — when in doubt, design for that size), and an upward incline raises the limit, by about 50% near a 30° slope.
1 · Gradation index M and the V50 correlation
d50 in mm gives V50 in m/s (coefficient 12.9 for ft/s). The V50 form is calibrated for 0.15 mm ≤ d50 ≤ 1.4 mm; for coarser gradings the tool multiplies by \(\cosh(60\,d_{50}/D)\) (d50, D both in mm). M measures how wide the grading is: narrow sands sit near 1.7, very broad gradings toward 0.25 — outside that band M is clipped.
2 · Concentration converter
Example: sand at Ss = 2.65 pumped at Cw = 40 % by weight is Cv = 0.40 / (2.65 − 1.65 × 0.40) = 20.1 % by volume, and the mixture relative density is Sm = 1 + 1.65 × 0.201 = 1.332. The same identity in reverse takes Cv = 20 % to Cw = 39.8 %.
3 · Settling velocity and vertical runs
Industrial crushed particles (1–30 mm) settle in still water at roughly 50% of the smooth-sphere velocity — the tool computes the sphere value from the iterative drag law and halves it. Vertical lifts then need \(V_{all}\approx 4\!-\!5\) times that figure. Watch the particle-to-pipe ratio in verticals: larger than D/5 invites slugging (🟡), larger than D/3 can plug the line outright (🔴). In horizontal runs, d50/D > 0.018 (narrow gradings) signals fully stratified flow — a regime of extreme pressure gradient fit only for short distances.
4 · Service classes — abrasion severity drives every limit
The four service classes grade the duty from light (fine, dilute, low-SG slurries) to severe. Heavier classes tighten every operating limit; the tool checks your deposition-safe velocity against the discharge-nozzle limit of the selected class.
| Limit | Class 1 | Class 2 | Class 3 | Class 4 |
|---|---|---|---|---|
| Max discharge-nozzle velocity | 40 ft/s (12.2 m/s) | 32 ft/s (9.8 m/s) | 27 ft/s (8.2 m/s) | 20 ft/s (6.1 m/s) |
| Max throat velocity | 50 ft/s (15.2 m/s) | 40 ft/s (12.2 m/s) | 30 ft/s (9.1 m/s) | 20 ft/s (6.1 m/s) |
| Max impeller tip speed — all-metal | 8,500 sfpm (43.2 m/s) | 7,500 sfpm (38.1 m/s) | 6,500 sfpm (33.0 m/s) | 5,500 sfpm (27.9 m/s) |
| Max impeller tip speed — rubber-lined | 5,500 sfpm (27.9 m/s) | 5,000 sfpm (25.4 m/s) | 4,500 sfpm (22.9 m/s) | 4,000 sfpm (20.3 m/s) |
| BEP band — annular casing | 20–120% | 30–110% | 40–100% | 50–90% |
| BEP band — semivolute | 30–130% | 40–120% | 50–110% | 60–100% |
| BEP band — near volute | 50–110% | 60–130% | 70–120% | 80–110% |
| BEP band — annular/oblique neck | 10–110% | 20–100% | 30–90% | 40–80% |
Worked example (anchor)
Sand slurry, Ss = 2.65, d50 = 0.6 mm, d85 = 1.8 mm, pipe ID 300 mm, carrier water at 20 °C. The calculator reproduces, step for step:
| Quantity | Calculation | Value |
|---|---|---|
| M index | 1 / ln(1.8/0.6) = 1 / ln 3 | 0.910 |
| V50 | 3.93 × 0.60.35 × (1.65/1.65)0.45 | 3.29 m/s (10.8 ft/s) |
| Vsm fine-particle | 9.0 × (g·ν·1.65)0.371 × (D/ν²)0.11 | 2.75 m/s (9.0 ft/s) |
| Vsm ceiling | (0.018/ff)0.13 × √(2gD·1.65), ff ≈ 0.0143 | 3.21 m/s (10.5 ft/s) |
| Governing Vsm | max(2.75, 3.21) | 3.21 m/s |
| Recommended Vm | ≥ 1.1 × Vsm | 3.53 m/s (11.6 ft/s) |
| Concentration | Cv = 20 % → Cw, Sm | 39.8 %, 1.330 |
At Q = 950 m³/h the 300 mm line runs at Vm = 3.73 m/s ≥ 1.1 × Vsm — 🟢. And the same sand at Cw = 40 % by weight converts to Cv = 20.1 %, Sm = 1.332.
Engineering criteria applied
- Deposition traffic light — Vm < Vsm 🔴 stationary-bed risk; Vsm ≤ Vm < 1.1 × Vsm 🟡 no margin; ≥ 1.1 × Vsm 🟢.
- M clip — 0.25 ≤ M ≤ 1.7, flagged when the raw value is truncated.
- Stratification screen — d50/D > 0.018 🔴 fully stratified; 0.015–0.018 🟡 borderline.
- Vertical lifts — Vall ≈ 4–5 × crushed-particle settling velocity; d85/D > 1/5 🟡 slugging, > 1/3 🔴 plugging.
- Service class — the recommended velocity 1.1 × Vsm is checked against the class discharge-nozzle limit: ≤ 90 % 🟢, 90–100 % 🟡, over 🔴 (enlarge the pipe or step up the class).