For a disk rotating in a laminar fluid, calculate the torque from the viscous boundary layer and then nondimensionalize it as a moment coefficient. For a disk wetted on both sides, the theoretical relationship is

[ c_M=\frac{3.87}{\sqrt{\mathrm{Re}}} ]

provided that

[ \mathrm{Re}=\frac{R^2\omega}{\nu}<3\times10^5. ]

Here, (R) is the disk radius, (\omega) is angular velocity, (\nu) is kinematic viscosity, and (c_M) is defined by (c_M=2M/(\rho\omega^2R^5)).

The central method is to evaluate the rotational Reynolds number, use the laminar-disk correlation (c_M=3.87/\sqrt{\mathrm{Re}}), and recover the driving torque from (M=c_M\rho\omega^2R^5/2). The resulting mechanical power is (P=M\omega), subject to corrections when the flow becomes transitional or turbulent.

Establish the Operating Regime

Calculate the Rotational Reynolds Number

The first step is to determine whether the disk operates within the theoretical laminar range:

[ \mathrm{Re}=\frac{R^2\omega}{\nu}. ]

The variables must use a consistent unit system. For example, (R) should be in metres, (\omega) in radians per second, and (\nu) in square metres per second.

Check the Laminar Limit

For the stated disk solution, the laminar correlation applies when

[ \mathrm{Re}<3\times10^5. ]

This criterion is specific to the rotating-disk flow model. It should not be confused with the standard Reynolds-number limits used for flow through circular pipes or other conduits.

Calculate the Moment Coefficient

Use the Two-Sided Disk Correlation

For a disk wetted on both sides, calculate the dimensionless moment coefficient from

[ \boxed{c_M=\frac{3.87}{\sqrt{\mathrm{Re}}}}. ]

The coefficient decreases as the rotational Reynolds number increases within the laminar regime because viscous torque grows more slowly than the reference scale (\rho\omega^2R^5).

Define the Coefficient Consistently

The moment coefficient is defined as

[ \boxed{c_M=\frac{2M}{\rho\omega^2R^5}}, ]

where:

(M) is the total driving torque on the disk,
(\rho) is fluid density,
(\omega) is angular velocity,
(R) is disk radius.

The factor of 2 is part of the stated convention. Other sources may define rotating-disk moment coefficients differently, so the definition must be checked before comparing correlations.

Recover the Torque

Calculate Total Torque

Once (c_M) is known, rearrange its definition:

[ \boxed{M=\frac{c_M\rho\omega^2R^5}{2}}. ]

Substituting the laminar correlation gives

[ M

\frac{3.87}{2} \frac{\rho\omega^2R^5}{\sqrt{\mathrm{Re}}}. ]

Because

[ \sqrt{\mathrm{Re}}=\frac{R\sqrt{\omega}}{\sqrt{\nu}}, ]

the torque can also be written as

[ \boxed{M\approx1.935,\rho R^4\sqrt{\nu\omega^3}}. ]

This is the total torque for both wetted disk faces under the stated convention.

Relate the Result to the Single-Sided Solution

For one side of the disk, the theoretical viscous torque is

[ M_{\text{one side}}

\frac{\pi}{2}\rho R^4\sqrt{\nu\omega^3}\left|G'(0)\right|, ]

where

[ -G'(0)\approx0.6159. ]

Therefore,

[ M_{\text{one side}} \approx 0.9675,\rho R^4\sqrt{\nu\omega^3}. ]

A disk wetted on both sides has approximately twice this torque:

[ M_{\text{two sides}} \approx 1.935,\rho R^4\sqrt{\nu\omega^3}. ]

This two-sided result is consistent with the coefficient (3.87/\sqrt{\mathrm{Re}}).

Convert Torque to Mechanical Power

Calculate Drive Power

The ideal mechanical power required to maintain rotation is

[ \boxed{P=M\omega}. ]

Using the coefficient form,

[ P=\frac{c_M\rho\omega^3R^5}{2}. ]

This represents the fluid-mechanical power associated with disk torque. The installed motor must also account for drive-train losses, bearings, seals, transmission efficiency, and any additional process loads.

Apply the Method to Pilot Equipment

In rotating membrane separators, contactors, and similar pilot-scale equipment, the calculated torque helps determine whether the selected drive can maintain the required speed.

The calculation should be performed at the most demanding expected condition, including changes in fluid viscosity, density, disk speed, and wetted geometry.

Distinguish Disk Shear from Conduit Friction

Use the Rotating-Disk Model for Disk Torque

The rotating-disk equation describes viscous shear generated by the disk boundary layers. Its governing length scale is the disk radius, leading to

[ \mathrm{Re}=\frac{R^2\omega}{\nu}. ]

Conduit friction factors should not be substituted into this disk-torque calculation.

Use Shape-Specific Friction Factors for Channels

If the pilot plant also contains flow through rectangular channels, annular spaces, or other non-circular passages, pressure-drop calculations require a separate friction-factor model.

For laminar conduit flow, the form is

[ \lambda=\frac{C}{\mathrm{Re}}, ]

where (C) depends on cross-sectional geometry. Representative values are:

Square duct: (C=57)
Equilateral triangular duct: (C=53)
Annular space: (C=96)
Rectangular duct with a 2:1 width-to-height ratio: (C=62)
Rectangular duct with a 4:1 width-to-height ratio: (C=73)

These coefficients address pressure loss in the passage, not torque generated by a rotating disk.

Understanding the Trade-offs

Account for Flow Transition

When

[ \mathrm{Re}\approx3\times10^5 ]

or higher, the flow begins transitioning from laminar toward turbulent behavior. The moment coefficient then deviates upward from

[ c_M=\frac{3.87}{\sqrt{\mathrm{Re}}}, ]

so the laminar formula can underpredict torque and motor power.

Avoid Extrapolating the Laminar Formula

The laminar correlation should not be extrapolated indefinitely with increasing speed or disk size. Turbulent torque requires an appropriate turbulent rotating-disk correlation or experimentally measured data.

Check the Actual Wetted Surfaces

The two-sided result assumes that both disk faces are wetted and experience comparable fluid conditions. If one side is dry, shielded, or exposed to a substantially different gap or flow condition, the total torque must be evaluated by summing the contributions from the individual surfaces.

Recognize Geometry and Clearance Effects

The ideal rotating-disk solution assumes a sufficiently open fluid region and a well-defined boundary-layer flow. Nearby stationary walls, narrow gaps, housings, seals, surface roughness, and throughflow can alter the torque substantially.

Making the Right Choice for Your Goal

Use the theoretical result as a baseline, then verify that the geometry and flow regime match the model.

If your primary focus is calculating laminar disk torque: Compute (\mathrm{Re}=R^2\omega/\nu), evaluate (c_M=3.87/\sqrt{\mathrm{Re}}), and use (M=c_M\rho\omega^2R^5/2) for a disk wetted on both sides.
If your primary focus is sizing the motor drive: Convert torque to power with (P=M\omega), then include mechanical losses and a margin for transitional or turbulent operation.
If your primary focus is pressure drop in pilot-plant passages: Use the geometry-specific relation (\lambda=C/\mathrm{Re}) rather than the rotating-disk moment coefficient.
If your primary focus is operation near or above the laminar limit: Use turbulent or experimentally validated torque data because the laminar relationship will tend to underpredict the required drive load.

With consistent definitions, the rotating-disk method provides a direct theoretical link between fluid properties, operating speed, disk size, torque, and motor power.

Summary Table:

Item	Key Information
Rotational Reynolds number	`Re = R^2ω/ν`
Laminar range	`Re < 3 × 10^5`
Two-sided moment coefficient	`cM = 3.87/√Re`
Moment coefficient definition	`cM = 2M/(ρω^2R^5)`
Two-sided disk torque	`M = 1.935ρR^4√(νω^3)`
Mechanical power	`P = Mω`
Above the laminar range	Use turbulent correlations or validated experimental data
Important corrections	Consider gaps, walls, seals, roughness, throughflow, and unequal wetted surfaces

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