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Journal Bearing Sommerfeld Number

Calculate the dimensionless Sommerfeld hydrodynamic fluid film stability score to prove industrial shaft levitation and prevent boundary lubrication failure.

Hydrodynamic Film Equation

Kinematic Operating Inputs

🟢 OPTIMAL (Hydrodynamic Regime): Fully developed, impenetrable fluid film. The metal shaft is completely floating on a high-pressure sheet of liquid oil. This bearing configuration possesses infinite fatigue life.

Sommerfeld Dimension (S)

0.5220
Absolute wedge stability score.

Absolute Film Viscosity

4.350e-6 Reyns
Equivalent to 30 cP.
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What is The Physics of SommerfeldNumber?

The technical foundation and mathematics behind the SommerfeldNumberCalc tool.

Mathematical Foundation

Hydrodynamic Sommerfeld Equation (S)
S=(rc)2×μ×NPS = \left(\frac{r}{c}\right)^2 \times \frac{\mu \times N}{P}
SS= Dimensionless Sommerfeld mathematical score for fluid wedge stability.
rr= Physical radius of the rotating steel shaft (inches).
cc= Radial clearance gap between the shaft and the stationary babbitt bearing (inches).
μ\mu= Absolute dynamic viscosity of the engine oil (converted from Centipoise to Reyns: lb-sec/in²).
NN= Rotational velocity of the shaft strictly expressed in Revolutions Per Second (RPM / 60).
PP= Physical bearing unit crushing load acting on the fluid wedge (PSI).

Laws & Principles

  • The Boundary Collapse Threshold (S < 0.05): If the shaft is spinning too slowly, or the oil is too thin (hot), the hydrodynamic wedge cannot build enough internal pressure to fight the crushing load. The oil film collapses entirely. The steel shaft violently grinds against the soft bearing babbitt. This represents catastrophic boundary lubrication failure.
  • The Mixed Film Marginal Zone (0.05 < S < 0.15): The fluid wedge is formed, but it is dangerously thin. The microscopic surface finish peaks (asperities) of the steel shaft are periodically slicing through the oil film and scraping the bearing shell. The oil viscosity must be increased or the crushing load removed to restore the gap.
  • The Infinite Life Hydrodynamic Plane (S > 0.15): A thick, impenetrable, high-pressure sheet of viscous oil is violently forced underneath the spinning shaft. The heavy steel spindle is literally levitating on liquid pressure. It does not physically touch the bearing wall. Wear drops to Absolute Zero.

Step-by-Step Example Walkthrough

" A 4-inch diameter (2-inch radius) industrial turbine shaft spinning at 3600 RPM sits in a journal bearing with a massive 500 PSI crushing load. The clearance gap is 0.002 inches. It is currently being flooded with 30 cP light viscosity oil. "

  1. 1. Convert 30 cP viscosity to Reyns: 30 * 1.45e-7 = 0.00000435 Reyns.
  2. 2. Convert 3600 RPM to Revolutions Per Second (N): 3600 / 60 = 60 RPS.
  3. 3. Calculate Clearance Ratio Squared: (2" Radius / 0.002" Clearance)^2 = 1,000,000.
  4. 4. Calculate Viscous Load Factor: (0.00000435 Reyns * 60 RPS) / 500 PSI = 0.000000522.
  5. 5. Multiply Clearance Ratio by Viscous Load Factor: 1,000,000 * 0.000000522 = 0.522 Sommerfeld Number.
Final Result: With a Sommerfeld score of 0.522, this turbine operates perfectly in the 'Optimal Hydrodynamic Regime'. The shaft is physically levitating on the 30 cP oil and experiencing zero metal-on-metal wear.

Quick Answer: How do you calculate the Sommerfeld Number for a journal bearing?

The Sommerfeld Number: S = (r/c)² × (μ × N) / P — r = shaft radius (in), c = radial clearance (in), μ = viscosity in Reyns (lb·s/in²), N = speed in rev/s, P = unit load (psi). Example: 4-in shaft, 3,600 RPM, 30 cP oil, 500 psi, c = 0.002 in: μ = 30 × 1.45×10−7 = 4.35×10−6 Reyns; N = 60 rps; (r/c)² = (2/0.002)² = 1,000,000. S = 1,000,000 × (4.35×10−6 × 60) / 500 = 0.522 — optimal hydrodynamic levitation. The shaft physically floats on pressurized oil. S < 0.05 = boundary collapse and metal-to-metal contact. S > 0.15 = full film separation and near-zero wear.

Sommerfeld Number Regime Classification

S is dimensionless and maps directly to lubrication regime. Actions and consequences listed for each zone.

S Value Regime Physical Condition Action
S < 0.05Boundary FAILUREFilm collapsed; shaft grinding babbitt. Catastrophic wear, temperature spike.SHUTDOWN. Increase viscosity, reduce load, or increase speed.
0.05 – 0.10Mixed film (marginal)Partial film only; asperity contact occurring. Rapid wear.Danger zone — increase oil grade or check for thermal thinning.
0.10 – 0.15Mixed/HD transitionFilm forming; occasional asperity contact at start/stop.Monitor temperature; verify lube supply pressure.
> 0.15Full hydrodynamicComplete film; shaft levitating on pressurized oil. Near-zero wear.Optimal — maintain conditions, monitor temperature creep.
> 1.0Excessive / over-designedVery thick film; high viscous friction, heat, and power loss.Consider reducing oil grade; accept parasitic friction loss.
S thresholds from Shigley's Mechanical Engineering Design and Raimondi-Boyd chart practice. Exact thresholds vary with eccentricity ratio and L/D ratio. Use for diagnostics alongside detailed Raimondi-Boyd analysis for design-critical applications.

Oil Viscosity vs. Temperature — Effect on S

Viscosity is the most thermally sensitive variable. As bearing temperature rises, μ drops and S collapses. Representative ISO VG viscosities:

Oil Grade 40°C (cP) 60°C (cP) 100°C (cP) S drop (40°→100°C)
ISO VG 3232 cP∼17 cP∼5.4 cP−83%
ISO VG 4646 cP∼24 cP∼6.8 cP−85%
ISO VG 6868 cP∼34 cP∼8.7 cP−87%
ISO VG 100100 cP∼46 cP∼11 cP−89%
ISO VG 150150 cP∼63 cP∼14 cP−91%
Convert cP to Reyns: multiply by 1.45×10−7. Example: 30 cP = 4.35×10−6 Reyns. Viscosity loss of 83–91% from 40°C to 100°C is — a bearing sized for cold-start may collapse to boundary film at operating temperature if oil grade is mismatched. Data: approximate ASTM D-341 Walther equation; use manufacturer data for design calculations.

Pro Tips & Sommerfeld Design Errors

Do This

  • ✓Always convert oil viscosity to Reyns before entering values — the Sommerfeld formula requires lb·s/in², not cP or cSt. Conversion: μ (Reyns) = μ (cP) × 1.45×10−7. Example: VG 68 at 60°C ∼34 cP → 34 × 1.45×10−7 = 4.93×10−6 Reyns. Using raw cP produces S values 6.9 million times too large, masking all boundary lubrication risk. Always use actual operating temperature viscosity, not 40°C catalog values — a 30 cP oil at 40°C may be only 8–10 cP at operating temperature.
  • ✓Calculate S at both cold-start AND steady-state operating temperature — both extremes require attention. At cold-start (high μ): S is large but high viscous drag generates heat that rapidly reduces μ. At steady-state (hot): S may drop into the marginal zone if oil grade is too light. Most dangerous: startup after extended shutdown. The shaft rests on the bearing at rest, oil drains from the contact zone, and for the first few seconds the bearing runs in boundary lubrication regardless of S. Large turbines use pre-lube oil pumps started before shaft rotation — flooding the bearing with oil before any load or speed develops.

Avoid This

  • ✗Don't underestimate (r/c)² — the clearance ratio squared typically dominates S by a factor of 10,000 to 1,000,000. For r = 2 in, c = 0.002 in: (r/c)² = 1,000,000. Babbitt wear increases c from 0.002 to 0.003 in → (r/c)² = 444,444 — S drops to 44% of original. Continued wear to c = 0.005 in pushes S into the marginal film zone. Measuring clearance at each overhaul and tracking the trend over time is the most accurate leading indicator of journal bearing health.
  • ✗Don't confuse journal bearing Sommerfeld analysis with ISO 281 rolling element bearing L10 life — they are completely different technologies. Journal bearings (babbitt, bronze, white metal) rely on hydrodynamic fluid film — analyzed by Sommerfeld Number. Rolling element bearings (ball, roller) rely on Hertz contact — analyzed by ISO 281 L10. Mixing methods gives meaningless results. Check the bearing specification drawing to confirm which type is installed before selecting an analysis method.

Frequently Asked Questions

How does a journal bearing shaft actually levitate on oil? What creates the pressure?

The shaft rotates eccentrically and drags viscous oil into a converging wedge gap on the loaded side. As oil flows into the narrowing space, it is compressed — since oil is incompressible, this creates a self-generated hydraulic pressure field (potentially thousands of psi). This pressure pushes the shaft away from the bearing wall until it reaches an operating position fully surrounded by pressurized oil — no metal contact. This is hydrodynamic lubrication — the pressure is generated by rotation itself, requiring no external pump for sustained operation. The Sommerfeld Number quantifies the strength of this pressure generation relative to applied load. S > 0.15 guarantees the wedge pressure exceeds the load — full levitation.

Why must I convert viscosity from cP to Reyns? What is a Reyn?

The Sommerfeld formula in Imperial units requires viscosity in Reyns (after Osborne Reynolds) — the Imperial unit: 1 Reyn = 1 lb·s/in². The conversion: 1 cP = 1.45×10−7 Reyns (a 6.9-million-fold difference). Oil suppliers always publish in cP or cSt; failing to convert gives S values millions of times too high, masking genuine boundary lubrication risk. If using kinematic viscosity (cSt): cP = cSt × oil density (g/cm³). For mineral oil at 40°C, density ≈ 0.875 g/cm³, so cP ≈ 0.875 × cSt. Unit conversion error in the Sommerfeld equation is the most common manual calculation failure.

Why does the Sommerfeld Number use (r/c)²? That makes S extremely sensitive to clearance.

Yes — deliberately. The (r/c)² term reflects Reynolds' equation: maximum oil film pressure scales with (r/c)² because a tighter clearance forces higher oil velocity gradients in the wedge, generating proportionally greater pressure. For r = 2 in, c = 0.001 in: (r/c)² = 4,000,000. At c = 0.002 in: (r/c)² = 1,000,000 — 4 times less. Doubling clearance reduces S by 4×. Babbitt wears over time, increasing clearance measurably at each overhaul. Measuring shaft diameter and bearing bore (clearance = (bore − shaft)/2) and plotting clearance trend over successive overhauls is the most reliable leading indicator of journal bearing end-of-life.

Why does S < 0.05 cause sudden catastrophic failure, not gradual wear?

Film collapse triggers a thermal runaway feedback loop: metal contact → heat generation → viscosity drops (μ falls) → S drops further → more contact → more heat. Babbitt melts at only 240–340°F depending on alloy. Once grinding begins, the contact zone can reach this temperature in seconds. The babbitt liquifies and flows, the bearing “wipes” — the shell may lock to the shaft and rotate with it. On a 10,000 HP turbine, a bearing wipe can cause shaft damage requiring weeks of repair and hundreds of thousands of dollars. This is why bearing temperature monitoring triggers automatic shutdown on temperature spikes — rising temperature is the early warning of increasing metal contact friction, before the thermal runaway completes.

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