10.2 Hydrodynamic Journal Bearings & Lubrication Regimes

10.2 Hydrodynamic Journal Bearings & Lubrication Regimes

Hydrodynamic journal bearings (also termed sleeve or fluid-film bearings) are standard support elements in high-speed, heavy-duty rotating machinery such as steam and gas turbines, centrifugal compressors, automotive crankshafts, and electric power generators. Unlike rolling-element bearings that have finite fatigue lives, hydrodynamic bearings operate with complete physical separation between the journal and sleeve surfaces via a continuous, pressurized viscous fluid wedge. Under steady full-film operation, surface wear is virtually non-existent, and vibration damping is exceptionally high.

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Lubrication Regimes & The Stribeck Curve

The fundamental operating state of any lubricated sliding interface is characterized by the Stribeck curve, which plots the coefficient of friction $f$ as a function of the bearing characteristic parameter $\frac{\mu N}{P}$ (or $\frac{\mu \omega}{P}$), where $\mu$ is dynamic viscosity, $N$ is rotational speed, and $P$ is unit load:

   Friction
   Coeff (f)
     ^
0.15 |   +----------------+
     |   | Boundary       |\
0.10 |   | Lubrication    | \
     |   +----------------+  \
0.05 |                        \   +-------------+
     |                         \  | Mixed /     |   +---------------------+
     |                          \_| Thin-Film   |  /| Hydrodynamic (Full) |
0.00 |----------------------------+-------------+-/-+---------------------+--->
     0                                       (Knee)               mu * N / P

1. Boundary Lubrication

• Mechanics: Occurs at very low speeds, high unit loads, or low lubricant viscosity (e.g., during startup, shutdown, or shock overloads). The fluid film is thinner than the composite surface roughness ($h_0 < R_a$).
• Friction & Wear: High friction coefficient ($f \approx 0.08 - 0.15$). High metallic asperity contact occurs. Wear prevention relies entirely on chemical boundary additives (such as zinc dialkyldithiophosphate / ZDDP or sulfur-phosphorus extreme-pressure [EP] compounds) that form low-shear sacrificial films on metal surfaces.

2. Mixed (Thin-Film) Lubrication

• Mechanics: An intermediate transition zone where the hydrodynamic fluid pressure supports a substantial portion of the applied load, but isolated high asperity peaks still intermittently contact.
• Friction & Wear: The friction coefficient drops precipitously toward a minimum "knee point" ($f \approx 0.005 - 0.02$). Highly unstable; a minor reduction in speed or viscosity can trigger boundary wear.

3. Hydrodynamic (Full-Film / Thick-Film) Lubrication

• Mechanics: Occurs at high speeds, moderate unit loads, and adequate viscosity. A continuous viscous fluid wedge completely separates the journal and sleeve ($h_0 > 3 - 5 R_a$).
• Friction & Wear: Ultra-low friction ($f \approx 0.001 - 0.005$). Zero metal-to-metal contact; theoretical component life is infinite barring contamination or thermal breakdown. Friction is governed strictly by internal viscous shear stress in the lubricant ($\tau = \mu \, du/dy$). As $\mu N / P$ increases beyond the knee point, friction coefficient rises linearly due to viscous drag.

4. Elastohydrodynamic Lubrication (EHL)

• Mechanics: A specialized regime occurring in non-conformal concentrated contacts (e.g., ball/roller bearings, gear meshes, cam-follower pairs).
• Characteristics: Extreme localized contact pressures ($1 - 3\,\text{GPa}$) cause two phenomena simultaneously: (a) elastic deflection of the mating metallic surfaces, and (b) an exponential increase in lubricant viscosity by three to four orders of magnitude via the Barus equation ($\mu = \mu_0 e^{\alpha_p p}$). A microscopic fluid film ($0.1 - 1.0\,\mu\text{m}$) is preserved.

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Journal Bearing Geometry & Operational Parameters

Consider a journal of diameter $d$ (radius $r = d/2$) rotating inside a cylindrical bearing sleeve of internal diameter $D$ (radius $R = D/2$) with axial length $L$:

• Radial Clearance ($c$):

$$c = R - r$$

• Diametral Clearance ($c_d$):

$$c_d = D - d = 2c$$

Typical industrial clearance ratios range from $c/r = 0.001$ to $0.002$ ($1$ to $2\,\text{mil}$ of radial clearance per inch of journal diameter).

• Nominal Bearing Unit Pressure ($P$): Defined over the projected area $A_p = L \cdot d$ under steady radial load $W$:

$$P = \frac{W}{L \cdot d} = \frac{W}{2 r L}$$

• Length-to-Diameter Ratio ($L/d$): Standard bearings typically use $L/d$ between $0.5$ and $1.5$. High-speed modern turbomachinery designs favor compact ratios ($L/d = 0.5 - 0.8$) to reduce parasitic friction torque and minimize sensitivity to shaft slope/angular misalignment, at the expense of higher end-leakage flow.

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Petroff's Classical Friction Formulation

Petroff's equation provides a foundational baseline for friction torque by assuming a lightly loaded, concentric journal ($\epsilon = 0$) running at constant rotational frequency $n$ (in $\text{rev/s}$):

• Journal surface speed: $U = 2\pi r n$
• Velocity gradient across uniform radial clearance $c$: $\frac{du}{dy} = \frac{U}{c} = \frac{2\pi r n}{c}$
• Newton's law of viscosity yields shear stress: $\tau = \mu \frac{2\pi r n}{c}$
• Wetted surface area: $A = 2\pi r L$

The total viscous shear friction force $F_f$ is:

$$F_f = \tau A = \left( \mu \frac{2\pi r n}{c} \right) (2\pi r L) = \frac{4\pi^2 \mu n r^2 L}{c}$$

Friction torque $T_f$ about the journal rotational axis is:

$$T_f = F_f \cdot r = \frac{2\pi^2 \mu n r^3 L}{c}$$

Dividing friction force by normal load $W = P (2 r L)$ yields Petroff's coefficient of friction $f$:

$$f = \frac{F_f}{W} = 2\pi^2 \left( \frac{\mu n}{P} \right) \left( \frac{r}{c} \right)$$

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Hydrodynamic Lubrication Theory & Reynolds Equation

When a journal bearing supports a substantial radial load $W$, the journal shifts eccentrically within the clearance space by distance $e$ (center-to-center displacement). This eccentricity establishes a converging fluid wedge in the direction of journal rotation.

                 Bearing Sleeve (Center Ob)
                      /-------------\
                    /                 \
                   |      Journal      |
                   |     (Center Oj)   |
                   |        (->)       |
                    \    e /          /
                      \--/----------/
                        h_0 (Minimum Film)

Fluid drawn into this converging wedge is compressed, developing hydrodynamic pressure governed by the classical 2D Reynolds equation for incompressible thin-film flow:

$$\frac{\partial}{\partial x}\left( \frac{h^3}{\mu} \frac{\partial p}{\partial x} \right) + \frac{\partial}{\partial z}\left( \frac{h^3}{\mu} \frac{\partial p}{\partial z} \right) = 6 U \frac{\partial h}{\partial x}$$

where $x = r\theta$ is the circumferential coordinate, $z$ is the axial coordinate, and $h(\theta) = c(1 + \epsilon \cos\theta)$ is the local oil film thickness.

Key Hydrodynamic Variables

• Eccentricity Ratio ($\epsilon$):

$$\epsilon = \frac{e}{c} \quad (0 \le \epsilon < 1)$$

• Minimum Oil Film Thickness ($h_0$): Occurs at the line of centers downstream of the load line:

$$h_0 = c - e = c(1 - \epsilon)$$

• Attitude Angle ($\phi$): The angle between the applied radial load vector $W$ and the centerline connecting bearing and journal centers.

The Sommerfeld Number ($S$)

The Sommerfeld number (bearing characteristic number) is a dimensionless parameter that encapsulates all geometric, kinematic, and operational variables:

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

where $n$ is rotational frequency in $\text{rev/s}$ ($n = N_{\text{rpm}}/60$), $\mu$ is dynamic viscosity in $\text{Pa}\cdot\text{s}$ ($\text{N}\cdot\text{s/m}^2$), and $P = W/(2 r L)$ is unit load in $\text{Pa}$. In US customary units, $S = (r/c)^2 (\mu n / P)$ where $\mu$ is in $\text{reyns}$ ($\text{lbf}\cdot\text{s/in}^2$) and $P$ is in $\text{psi}$.

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Raimondi-Boyd Design Methodology

Because closed-form analytical solutions to the 2D Reynolds equation with axial side leakage do not exist, modern engineering design relies on numerical solutions developed by Raimondi and Boyd. These charts relate the Sommerfeld number $S$ and aspect ratio $L/d$ to four primary dimensionless performance variables:

1. Film Thickness Variable: $\frac{h_0}{c}$
2. Friction Variable: $\left( \frac{r}{c} \right) f$
3. Flow Variable: $\frac{Q}{r c n L}$
4. Side-Leakage Ratio: $\frac{Q_s}{Q}$

Representative Raimondi-Boyd Data for $L/d = 1$ (Full $360^\circ$ Journal Bearing)

(Raimondi and Boyd's finite-bearing solutions as tabulated in Shigley. At light loads the friction variable approaches Petroff's value $2\pi^2 S$, a quick check on any chart reading. Interpolate between rows or read the charts for intermediate $S$.)

Trumpler's Design Criteria for Safe Operation

To ensure structural integrity and prevent seizure under real-world industrial tolerances, design practice (per P.R. Trumpler) mandates:

1. Minimum Film Thickness: $h_0 \ge 0.0002\,\text{in} + 0.00004 d$ (or $h_0 \ge 5 - 10\,\mu\text{m}$ for precision shafts) to clear circulating particulate contamination.
2. Maximum Lubricant Temperature: Peak film temperature $T_{\max} \le 120^\circ\text{C}$ ($250^\circ\text{F}$) to prevent thermal oxidation and additive breakdown of mineral oils.
3. Overload Design Factor: $N_L \ge 2.0$, ensuring that doubling the operational load will not reduce $h_0$ below critical asperity contact.

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Thermal Equilibrium & Lubricant Heat Balance

Viscous shearing within the oil film generates continuous thermal energy that must be carried away by circulating oil flow. In steady-state operation, thermal equilibrium requires:

$$H_{\text{gen}} = H_{\text{diss}}$$

1. Heat Generation Rate ($H_{\text{gen}}$)

$$H_{\text{gen}} = 2\pi T_f n = W f U = W f (2\pi r n) \quad [\text{Watts}]$$

2. Heat Dissipation Rate ($H_{\text{diss}}$)

Lubricant pumped through the bearing carries thermal energy away via end leakage and circulating flow. Using the effective volumetric flow rate $Q_{\text{eff}}$:

$$Q_{\text{eff}} = Q \left( 1 - 0.5 \frac{Q_s}{Q} \right)$$ $$H_{\text{diss}} = \dot{m}_{\text{eff}} c_p \Delta T = \rho Q_{\text{eff}} c_p \Delta T$$

where $\rho$ is lubricant density (typically $850 - 900\,\text{kg/m}^3$), $c_p$ is specific heat capacity (typically $1,800 - 2,100\,\text{J/(kg}\cdot^\circ\text{C)}$), and $\Delta T = T_{\text{out}} - T_{\text{in}}$ is the lubricant temperature rise.

Equating $H_{\text{gen}} = H_{\text{diss}}$ yields the temperature rise formulation:

$$\Delta T = \frac{H_{\text{gen}}}{\rho Q_{\text{eff}} c_p}$$

Iterative Viscosity Convergence

Because dynamic viscosity $\mu$ decreases exponentially with temperature according to the ASTM D341 / Walther relationship, bearing analysis requires iterative convergence:

1. Assume an initial average film temperature: $T_{\text{avg}} = T_{\text{in}} + \Delta T / 2$.
2. Determine dynamic viscosity $\mu$ at $T_{\text{avg}}$ from the lubricant's viscosity-temperature chart.
3. Compute Sommerfeld number $S$, look up $(r/c)f$ and flow parameters, and compute resulting $\Delta T_{\text{calc}}$.
4. Update $T_{\text{avg,new}} = T_{\text{in}} + \Delta T_{\text{calc}} / 2$. Repeat until $T_{\text{avg}}$ converges within $\pm 0.5^\circ\text{C}$.

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Worked Engineering Example: Steam Turbine Governor Shaft Bearing

Problem Statement

A full hydrodynamic journal bearing ($360^\circ$) supports the governor rotor shaft of an industrial steam turbine under the following specifications:

• Journal diameter: $d = 50\,\text{mm}$ ($r = 25\,\text{mm} = 0.025\,\text{m}$)
• Bearing length: $L = 50\,\text{mm} = 0.050\,\text{m}$ ($L/d = 1.0$)
• Radial load: $W = 3,600\,\text{N}$ ($3.6\,\text{kN}$)
• Rotational speed: $N = 1,800\,\text{rpm} \implies n = 30\,\text{rev/s}$ ($U = 2\pi r n = 4.7124\,\text{m/s}$)
• Radial clearance: $c = 0.025\,\text{mm} = 25\,\mu\text{m}$ ($r/c = 1,000$)
• Lubricant: ISO VG 32 turbine oil. At the assumed average film temperature (about $50^\circ\text{C}$), dynamic viscosity is $\mu = 0.020\,\text{Pa}\cdot\text{s}$ ($20\,\text{cP}$)
• Oil properties: density $\rho = 860\,\text{kg/m}^3$, specific heat $c_p = 2,000\,\text{J/(kg}\cdot^\circ\text{C)}$
• Inlet oil supply temperature: $T_{\text{in}} = 35.0^\circ\text{C}$

Determine:

1. The bearing nominal unit pressure $P$.
2. The Sommerfeld number $S$.
3. The minimum oil film thickness $h_0$ and compare against Trumpler's minimum criterion.
4. The coefficient of friction $f$, friction torque $T_f$, and total heat generation rate $H_{\text{gen}}$.
5. The total volumetric oil flow rate $Q$, side-leakage flow $Q_s$, and lubricant temperature rise $\Delta T$.

Step-by-Step Solution

Step 1: Compute Nominal Bearing Pressure ($P$)

$$P = \frac{W}{L \cdot d} = \frac{3,600\,\text{N}}{(0.050\,\text{m})(0.050\,\text{m})} = \frac{3,600}{0.0025} = 1,440,000\,\text{Pa} = 1.44\,\text{MPa}$$

Step 2: Calculate the Sommerfeld Number ($S$)

$$S = \left( \frac{r}{c} \right)^2 \left( \frac{\mu n}{P} \right) = (1,000)^2 \cdot \frac{(0.020\,\text{Pa}\cdot\text{s})(30\,\text{s}^{-1})}{1.44 \times 10^6\,\text{Pa}} = 10^6 \cdot \frac{0.60}{1.44 \times 10^6} = 0.4167 \approx 0.417$$

Step 3: Evaluate Film Thickness & Trumpler's Criterion

$S = 0.417$ falls between the $\epsilon = 0.2$ row ($S = 0.631$) and the $\epsilon = 0.4$ row ($S = 0.264$). Interpolating linearly on $S$, the operating point lies a fraction $(0.417 - 0.264)/(0.631 - 0.264) = 0.416$ of the way toward the $\epsilon = 0.2$ row:

• Eccentricity ratio: $\epsilon \approx 0.4 - 0.416(0.2) = 0.32$
• Minimum film thickness variable: $h_0/c = 1 - \epsilon \approx 0.68$

Compute minimum film thickness $h_0$:

$$h_0 = c \cdot 0.68 = (25\,\mu\text{m})(0.68) = 17.0\,\mu\text{m} = 0.017\,\text{mm}$$

Check against Trumpler's criterion ($h_{0,\min} \ge 0.0002\text{ in} + 0.00004 d \approx 0.005\,\text{mm} + 0.00004 d$):

$$h_{0,\min} = 0.005 + 0.00004(50) = 0.005 + 0.002 = 0.007\,\text{mm} = 7.0\,\mu\text{m}$$

Since $h_0 = 17.0\,\mu\text{m} > 7.0\,\mu\text{m}$, the film comfortably separates the surfaces.

Step 4: Compute Friction, Torque, and Heat Generation

Interpolating the friction variable the same way:

$$\left( \frac{r}{c} \right) f \approx 5.79 + 0.416(12.8 - 5.79) = 8.7$$

(Petroff's concentric estimate, $2\pi^2 S = 8.2$, confirms the magnitude.) The coefficient of friction is:

$$f = \frac{8.7}{r/c} = \frac{8.7}{1{,}000} = 0.0087$$

Friction force and torque:

$$F_f = f \cdot W = (0.0087)(3{,}600\,\text{N}) = 31.3\,\text{N}$$ $$T_f = F_f \cdot r = (31.3\,\text{N})(0.025\,\text{m}) = 0.783\,\text{N}\cdot\text{m}$$

Friction heat generation rate:

$$H_{\text{gen}} = 2\pi T_f n = 2\pi (0.783\,\text{N}\cdot\text{m})(30\,\text{s}^{-1}) = 147.6\,\text{W}$$

(Alternatively: $H_{\text{gen}} = F_f \cdot U = 31.3 \times 4.7124 = 147.6\,\text{W}$.)

Step 5: Compute Oil Flow Rates and Temperature Rise

Interpolating the flow variables:

• Flow variable: $\dfrac{Q}{r c n L} \approx 3.99 - 0.416(0.40) = 3.82$
• Side-leakage ratio: $\dfrac{Q_s}{Q} \approx 0.497 - 0.416(0.217) = 0.41$

Total volumetric flow rate:

$$Q = 3.82 \times (0.025\,\text{m}) \times (2.5 \times 10^{-5}\,\text{m}) \times (30\,\text{s}^{-1}) \times (0.050\,\text{m}) = 3.58 \times 10^{-6}\,\text{m}^3\text{/s} \approx 0.215\,\text{L/min}$$

Side leakage flow:

$$Q_s = 0.41 \times 3.58 \times 10^{-6} = 1.47 \times 10^{-6}\,\text{m}^3\text{/s}$$

Effective circulating flow carrying heat (half of the side flow is assumed to leave at the inlet temperature):

$$Q_{\text{eff}} = Q \left( 1 - 0.5 \frac{Q_s}{Q} \right) = 3.58 \times 10^{-6} \times (1 - 0.205) = 2.85 \times 10^{-6}\,\text{m}^3\text{/s}$$

Mass flow rate:

$$\dot{m}_{\text{eff}} = \rho Q_{\text{eff}} = (860)(2.85 \times 10^{-6}) = 2.449 \times 10^{-3}\,\text{kg/s}$$

Lubricant temperature rise:

$$\Delta T = \frac{H_{\text{gen}}}{\dot{m}_{\text{eff}} c_p} = \frac{147.6\,\text{W}}{(2.449 \times 10^{-3})(2{,}000)} = \frac{147.6}{4.897} = 30.1^\circ\text{C}$$

The average film temperature is $T_{\text{avg}} = T_{\text{in}} + \Delta T / 2 = 35.0 + 15.1 = 50.1^\circ\text{C}$. ISO VG 32 oil is roughly $0.018$ to $0.020\,\text{Pa}\cdot\text{s}$ near $50^\circ\text{C}$, consistent with the assumed $\mu = 0.020\,\text{Pa}\cdot\text{s}$, so this trial is essentially converged. Had $T_{\text{avg}}$ come out noticeably different, the viscosity would be re-read at the new temperature and Steps 2 to 5 repeated.