Force, Torque, Pressure, Density, and Viscosity

Force and Torque Calibration

Force and torque are derived physical quantities requiring dynamic or static equilibrium against calibrated gravity masses or traceable transducers.

Force Standards and Instruments

1. Deadweight Force Machines (Primary Standard): Direct suspension of precision calibrated masses in the local gravitational field. Realizes force via Newton's second law:

$$F = m \cdot g_{\text{local}} \left( 1 - \frac{\rho_a}{\rho_m} \right)$$

Uncertainties can be better than $\pm 0.001\%$ ($10\text{ ppm}$). Requires local gravity determination via gravimeter survey. 2. Proving Rings (ASTM E74): Forged elastic alloy steel rings whose diametral deflection under compressive or tensile force is measured using an internal micrometer screw equipped with a vibrating reed. Deflection is correlated to force via a polynomial equation. 3. Strain Gage Load Cells: Elastic elements (shear web, column, or S-beam) bonded with four metallic foil strain gages configured in a full Wheatstone bridge. Under stress, resistance shifts alter the output signal ($e_o / e_i$ in $\text{mV/V}$). Key performance parameters include non-linearity, hysteresis, creep (signal decay over time under constant load), and zero return.

Torque Metrology

Torque (moment of force) is defined as the vector cross product $\vec{\tau} = \vec{r} \times \vec{F}$. In calibration systems, static torque is generated using precision deadweight torque arms:

$$\tau = m \cdot g_{\text{local}} \left( 1 - \frac{\rho_a}{\rho_m} \right) \cdot L$$

Where $L$ is the certified effective length of the horizontal lever arm, measured from the axis of rotation to the knife-edge suspension pivot.

Calibration Traps in Torque Calibration

• Pre-Calibration Exercising: Click-type torque wrenches must be exercised according to the applicable model and calibration procedure, for example three to five times to 100% capacity before taking calibration data to distribute grease across the internal toggle block and seat the spring.
• Storage setting: Follow the manufacturer’s guidance, commonly the minimum marked setting for adjustable click wrenches without going below it. Storage under spring load can contribute to change in some designs; do not assert a universal permanent shift of several percent. Verify actual as-found performance.
• Hand Load Position: The technician's hand must be centered precisely over the handle grip pivot marking. Off-axis pushing or pulling creates parasite bending moments.

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Pressure and Vacuum Calibration

Pressure is the normal force exerted by a fluid per unit surface area ($P = F / A$). The SI unit is the Pascal ($1\text{ Pa} = 1\text{ N/m}^2$; $1\text{ bar} = 100\text{ kPa}$; $1\text{ psi} \approx 6894.76\text{ Pa}$).

Deadweight Testers (Piston Gages)

A deadweight pressure balance establishes pressure using loaded mass, local gravity, effective piston area, and relevant corrections. Its standard role depends on how the effective area and other quantities were established: some are primary realizations, while others are calibrated by comparison. The instrument name alone does not establish primary status. Rotation can reduce static friction but does not eliminate all mechanical effects.

The generated pressure at the reference level is governed by:

$$P = \frac{m \cdot g_{\text{local}}}{A_e} \left( 1 - \frac{\rho_a}{\rho_m} \right) + \rho_f g_{\text{local}} h + \frac{\gamma C}{A_e}$$

Where:

• $A_e = A_0 [1 + (\alpha_p + \alpha_c)(T - 20^\circ\text{C})](1 + b P)$ = effective piston area corrected for temperature and elastic pressure distortion ($b$)
• $\rho_f g_{\text{local}} h$ = hydrostatic head correction due to height difference $h$, defined positive with the UUT below the DWT reference plane
• $\gamma C/A_e$ = pressure contribution from the surface-tension force of the hydraulic fluid around the piston circumference

Liquid Manometers

Hydrostatic column manometers determine differential pressure via fluid head height:

$$P = \rho_m \cdot g_{\text{local}} \cdot h$$

• Meniscus Reading Rule: In wetting liquids (water, manometer oil), read the lowest point of the concave meniscus at horizontal eye level. In non-wetting liquids (mercury), read the highest point of the convex meniscus.
• Inclined Manometers: Used for low draft differential pressures; the column is inclined at an angle $\theta$ from horizontal, amplifying meniscus travel by $1 / \sin\theta$.

Vacuum Measurement Technologies

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Density and Viscosity Metrology

Density Standards and Measurement

Density is volumetric mass ($\rho = m / V$, SI unit: $\text{kg/m}^3$).

• Hydrometers: Buoyancy floats with calibrated graduated stems (governed by ASTM E100 / ISO 387). The hydrometer sinks until displaced liquid weight equals total hydrometer weight. Surface tension effects must be corrected if measuring fluids differ from the calibration liquid.
• Pycnometers: Precision volumetric glass flasks (Gay-Lussac or Bingham types) with capillary-bore ground glass stoppers. Liquid density is determined gravimetrically: $\rho = (m_{\text{filled}} - m_{\text{empty}}) / V_{\text{calibrated}}$.
• Oscillating U-Tube Density Meters: Liquid fills a hollow U-shaped borosilicate glass tube that is electro-magnetically excited at resonance. Density is mathematically derived from the period of oscillation $T$:

$$\rho = A \cdot T^2 - B$$

The constants are determined for the actual calibrated system and relevant fluid conditions. Temperature stability and corrections must meet the method’s uncertainty requirement; ±0.01°C and Peltier control are possible implementations, not universal requirements.

Viscosity Metrology

Viscosity characterizes a fluid's internal resistance to flow.

• Dynamic Viscosity ($\eta$): Ratio of shear stress to shear rate (SI unit: $\text{Pa}\cdot\text{s}$; cgs unit: Poise, $1\text{ cP} = 1\text{ mPa}\cdot\text{s} = 10^{-3}\text{ Pa}\cdot\text{s}$).
• Kinematic Viscosity ($\nu$): Ratio of dynamic viscosity to density: $\nu = \eta / \rho$ (SI unit: $\text{m}^2/\text{s}$; cgs unit: Stokes, $1\text{ cSt} = 1\text{ mm}^2/\text{s} = 10^{-6}\text{ m}^2/\text{s}$).

Capillary Kinematic Viscometers (ASTM D445 / ISO 3104)

Glass capillary viscometers (Ubbelohde, Cannon-Fenske, Ostwald) measure the efflux time $t$ for a fixed fluid volume to flow under gravity through a precision capillary tube:

$$\nu = C \cdot t - \frac{E}{t^2}$$

Use the correction prescribed for the viscometer and method. Whether the kinetic-energy term is negligible depends on geometry, calibration, efflux time, and the required uncertainty; 200 seconds is not a universal threshold.

Rotational Viscometers

Rotational viscometers (e.g., Brookfield, cone-and-plate, Couette concentric cylinder) measure the viscous drag torque exerted on a spindle rotating at controlled angular velocity $\omega$. Ideal for detecting non-Newtonian behavior (shear-thinning pseudoplasticity, shear-thickening dilatancy, and thixotropic time-dependent thinning).