Empirical Equations

Overview

Engineering reference data for Empirical Equations in miscellaneous.

Key Formulas

Unit Conversion

$y = x \cdot k$

Multiply by conversion factor.

Linear Interpolation

$y = y_1 + \frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1}$

Estimate between two known points.

Percentage

$p = \frac{\text{part}}{\text{whole}} \times 100\%$

Part as fraction of whole.

Variables

Common Examples

A key characteristic of empirical equations is that the units on the right-hand side of the equation do not always correspond to the units on the left-hand side. They are derived from fitting curves to experimental data, not from fundamental principles of dimensional analysis.

1. Inductance in an Air Filled Cylindrical Coil

The inductance $L$ (in henries, H) of a single-layer air-core solenoid can be approximated by Wheeler's formula, an empirical equation:

$$L \approx \frac{r^2 n^2}{9r + 10l}$$

where:

• $r$ is the coil radius (inches)
• $l$ is the coil length (inches)
• $n$ is the number of turns

Note: This formula mixes length dimensions directly in the denominator without explicit conversion factors, illustrating the typical unit-agnostic nature of an empirical constant.

2. Evaporation from a Water Surface

The evaporation rate from an open water surface can be estimated empirically, for example using the equation:

$$m = kA(p_s - p_v)$$

where:

• $m$ is the evaporation rate (kg/s)
• $k$ is an empirical coefficient dependent on air velocity and temperature
• $A$ is the water surface area (m²)
• $p_s$ is the saturation vapor pressure at water surface temperature (Pa)
• $p_v$ is the vapor pressure of the ambient air (Pa)

3. Compressed Air - Pressure Drop in Pipe Lines

For calculating the pressure drop ($\Delta p$) in a compressed air line, the empirical Darcy-Weisbach equation is often used with a friction factor derived from empirical charts (like Moody's diagram):

$$\Delta p = f \cdot \frac{L}{D} \cdot \frac{\rho v^2}{2}$$

where:

• $f$ is the Darcy friction factor (dimensionless, found empirically)
• $L$ is the pipe length (m)
• $D$ is the pipe internal diameter (m)
• $\rho$ is the air density (kg/m³)
• $v$ is the air velocity (m/s)

References

• Original Source