Journal Bearing Sommerfeld Number

Calculate the dimensionless Sommerfeld hydrodynamic fluid film stability number for industrial sleeve and journal bearings per ISO 7902 and Machinery's Handbook.

Trades & Construction

Standard: ISO 7902 / DIN 31652 / Machinery's Handbook

Bearing & Fluid Parameters

Shaft Journal Radius (r)

Radial Clearance (c)

Kinematic & Operating Conditions

Shaft Speed (N)

RPM

Bearing Unit Load (P)

PSI

Lubricant Viscosity (μ)

✅Regime Assessment: Full Hydrodynamic Film

Sommerfeld number S = 0.5220 (≥ 0.15) indicates fully developed hydrodynamic fluid film. The rotating journal generates sufficient fluid wedge pressure to support the 500 PSI unit load with continuous surface separation.

Sommerfeld Number (S)

0.5220

Full Hydrodynamic Film

Clearance Ratio (r / c)

1,000

Typical: 500 to 1,500

Dynamic Viscosity

4.35e-6

Reyns (lb·s/in²)

Rotational Velocity (N)

60.0 rev/s

3,600 RPM

Interactive Journal Bearing Sommerfeld Derivation

Step-by-step mathematical substitution reflecting current bearing operating inputs

ISO 7902 / DIN 31652 / Machinery's Handbook

Design Scenario

Evaluating journal bearing with shaft radius r = 2 in, radial clearance c = 0.002 in, speed N = 3600 RPM, unit load P = 500 PSI, and dynamic viscosity μ = 30 cP.

Mathematical Solution

1Convert Absolute Viscosity to Imperial Reyns

1 Centipoise equals 1.45 × 10⁻⁷ Reyns (lb-sec/sq in) for Imperial engineering units.

\mu = 30 \times 1.45 \times 10^{-7} = 4.350e-6\text{ lb}\cdot\text{s/in}^2

2Convert Shaft Speed to Revolutions per Second

The Sommerfeld hydrodynamic equation strictly requires rotational frequency in cycles per second.

N = \frac{3600}{60} = 60.00\text{ rev/s}

3Calculate Clearance Ratio Squared

The ratio of journal radius to radial clearance characterizes fluid wedge geometry.

\left(\frac{2}{0.002}\right)^2 = (1000)^2 = 1,000,000

4Calculate Dimensionless Sommerfeld Number (S)

Values S ≥ 0.15 indicate full hydrodynamic film separation; S < 0.05 indicates boundary lubrication risk.

S = 1,000,000 \times \frac{4.350e-6 \times 60.00}{500} = 0.5220

S = 0.5220 (Full Hydrodynamic Film)

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What is Hydrodynamic Journal Bearing Lubrication and Sommerfeld Analysis?

The technical foundation and tribological equations governing fluid film formation in rotating machinery sleeve bearings.

Mathematical Foundation

Hydrodynamic Sommerfeld Equation (S)

S = \left(\frac{r}{c}\right)^2 \times \frac{\mu \times N}{P}

S

= Dimensionless Sommerfeld number for hydrodynamic film stability

r

= Physical journal radius of the rotating shaft (inches or mm)

c

= Radial clearance between shaft and journal bore (inches or mm)

\mu

= Absolute dynamic viscosity at operating temperature (Reyns: lb·s/in² or Pa·s)

N

= Rotational speed in revolutions per second (RPM / 60)

P

= Bearing unit projected load: W / (2r × L) in PSI or Pa

Laws & Principles

Boundary Lubrication Regime (S < 0.05): Hydrodynamic fluid wedge is insufficient to overcome applied unit load. Microscopic surface asperities directly contact, causing increased friction, elevated operating temperatures, and accelerated surface wear.
Mixed Lubrication Regime (0.05 ≤ S < 0.15): Partial fluid film separation where occasional peak asperity contact occurs, typically observed during machine start-up, stopping, or transient peak loading.
Full Hydrodynamic Regime (S ≥ 0.15): Fluid wedge generates sufficient hydrodynamic pressure to achieve complete continuous surface separation with zero steady-state metal-to-metal contact.

Step-by-Step Example Walkthrough

" A 4-inch diameter (2-inch radius) rotating journal operates at 3600 RPM inside a sleeve bearing with 0.002-inch radial clearance under a 500 PSI unit load, using oil with 30 cP operating viscosity. "

1. Convert 30 cP absolute viscosity to Imperial Reyns: 30 × 1.45 × 10⁻⁷ = 4.35 × 10⁻⁶ Reyns (lb·s/in²).
2. Convert 3600 RPM to Revolutions Per Second: N = 3600 / 60 = 60 rev/s.
3. Compute Clearance Ratio Squared: (2.0 / 0.002)² = (1000)² = 1,000,000.
4. Calculate Viscous Load Factor: (4.35 × 10⁻⁶ × 60) / 500 = 0.000000522.
5. Multiply Clearance Ratio Squared by Viscous Load Factor: 1,000,000 × 0.000000522 = 0.522.

Final Result: Calculated Sommerfeld Number is S = 0.5220, placing operation securely in the Full Hydrodynamic Regime (S ≥ 0.15) with stable fluid film separation.

Quick Reference: Journal Bearing Sommerfeld Number

The Sommerfeld Number (S = (r/c)² × (μ × N) / P) is the primary dimensionless characteristic parameter in hydrodynamic lubrication theory. Values of S ≥ 0.15 indicate a fully developed fluid film providing continuous shaft separation. Values of 0.05 ≤ S < 0.15 indicate mixed-film lubrication, while values below 0.05 signal boundary lubrication requiring operating speed, viscosity, or clearance adjustments.

Sommerfeld Lubrication Regime Reference

The Sommerfeld Number classifies bearing operating conditions into distinct tribological regimes per ISO 7902 and DIN 31652:

S Range	Regime	Physical Film State	Engineering Recommendation
S < 0.05	Boundary Lubrication	Insufficient fluid wedge; asperity contact occurs.	Increase shaft speed, switch to higher viscosity oil, or reduce unit load.
0.05 ≤ S < 0.15	Mixed / Thin Film	Partial fluid film; occasional asperity interaction.	Monitor bearing temperature and evaluate lubricant viscosity grade.
0.15 ≤ S ≤ 1.0	Full Hydrodynamic	Complete fluid film separation; continuous hydrodynamic pressure.	Optimal operating zone; maintain operating temperature and clean oil supply.
S > 1.0	High Viscous Shear	Thick fluid film with elevated parasitic viscous friction.	Evaluate potential energy loss from excessive oil viscosity or oversized clearance.

ISO VG Oil Viscosity Across Temperatures

Lubricant dynamic viscosity is strongly temperature-dependent. As temperature rises, viscosity drops rapidly, significantly decreasing the Sommerfeld number:

ISO VG Grade	Viscosity at 40°C (cP)	Viscosity at 60°C (cP)	Viscosity at 100°C (cP)	Viscosity Loss (40→100°C)
ISO VG 32	32 cP	~17 cP	~5.4 cP	-83%
ISO VG 46	46 cP	~24 cP	~6.8 cP	-85%
ISO VG 68	68 cP	~34 cP	~8.7 cP	-87%
ISO VG 100	100 cP	~46 cP	~11.0 cP	-89%

Field Engineering Directives

Best Practices

✓Convert dynamic viscosity to Reyns: Always convert Centipoise (cP) to Reyns (lb·s/in²) using the factor 1.45 × 10⁻⁷ before solving the Imperial Sommerfeld equation.
✓Evaluate at steady-state operating temperature: Ensure viscosity values reflect actual bearing sump temperature under maximum continuous load rather than nominal 40°C supplier data.

Common Oversights

✗Ignoring clearance square sensitivity: Doubling radial clearance decreases the clearance ratio squared by 75%, causing a 4-fold drop in the calculated Sommerfeld number.
✗Applying to rolling element bearings: Sommerfeld analysis applies exclusively to fluid film journal bearings; rolling element bearings must be evaluated per ISO 281 L10 fatigue life standards.

Frequently Asked Questions

How does a hydrodynamic journal bearing develop fluid wedge pressure?

As the journal rotates eccentrically within the bearing bore, viscous shear forces draw lubricant into the converging gap between the shaft and bearing surfaces. Because the lubricant is essentially incompressible, the converging geometry creates a continuous hydrodynamic pressure field that supports the external radial load without mechanical contact during steady-state rotation.

What is a Reyn and why is it used in the Imperial Sommerfeld equation?

A Reyn (lb·s/in²) is the Imperial unit of absolute dynamic viscosity, named after Osborne Reynolds. Because industrial unit load (P) is typically measured in PSI (lb/in²) and shaft dimensions in inches, dynamic viscosity must be expressed in Reyns so that units cancel into a dimensionless Sommerfeld number.

Why is the Sommerfeld Number so sensitive to radial clearance?

The Sommerfeld formula incorporates clearance as a squared ratio (r/c)². A small change in radial clearance—such as normal babbitt wear over time—causes a squared change in the clearance ratio, resulting in a substantial reduction in fluid film stability.

What operating conditions cause boundary lubrication in journal bearings?

Boundary lubrication (S < 0.05) occurs when rotational speed is too low to establish a hydrodynamic wedge (e.g., during machine start-up or shut-down), when lubricant viscosity drops significantly due to overheating, or when external unit loads exceed designed capacity.

Related Rotating Machinery Calculators

Calculation Provenance & Validation Record

Method

Dimensionless Sommerfeld Number for Journal Bearing Lubrication Regime

Formula

S = \left(\frac{r}{c}\right)^2 \times \frac{\mu \times N}{P}

Assumptions

Applies to full 360-degree cylindrical hydrodynamic journal bearings operating under steady-state radial load.
Dynamic viscosity is evaluated at the effective operating lubricant film temperature, converted to Reyns (lb·s/in²).
Shaft and bearing surfaces are rigid, parallel cylinders with minimal elastic or thermal bore distortion.
Laminar, Newtonian fluid flow assumption without cavitation or high-shear polymer thinning.

References

Hydrodynamic Plain Journal Bearings Under Steady-State Conditions (ISO 7902 / Machinery's Handbook) (2020 Edition) — ISO 7902-1 Section 4 & Machinery's Handbook Hydrodynamic Lubrication

Last substantive review: 2026-10-04

Automated test status: 4 golden test vectors passing (SN-01, SN-02, SN-03, SN-04)

Method & assumptions

Calculation Methodology

Trade estimation calculations derived from standard mechanical, electrical, and construction formulas.

Governing Standard 2020 Edition

Standard:ISO 7902 / DIN 31652 / Machinery's Handbook

Statutory building, electrical, and mechanical codes vary by jurisdiction. Confirm local municipality amendments before installation.

Key Assumptions & Constraints

Applies to full 360-degree cylindrical hydrodynamic journal bearings operating under steady-state radial load.
Dynamic viscosity is evaluated at the effective operating lubricant film temperature, converted to Reyns (lb·s/in²).
Shaft and bearing surfaces are rigid, parallel cylinders with minimal elastic or thermal bore distortion.
Laminar, Newtonian fluid flow assumption without cavitation or high-shear polymer thinning.

Field Trade Notice: For trade planning and engineering estimates. Final installations must conform to project blueprints, authority having jurisdiction (AHJ) code approvals, and site-specific inspections.