Fluid Density Temperature Pressure

Overview

Engineering reference data for Fluid Density Temperature Pressure in fluid mechanics.

Key Formulas

Reynolds Number

$Re = \frac{\rho v D}{\mu}$

Ratio of inertial to viscous forces — determines flow regime.

Bernoulli's Equation

$P + \frac{1}{2}\rho v^2 + \rho g h = \text{const}$

Conservation of energy for steady, inviscid, incompressible flow.

Continuity Equation

$A_1 v_1 = A_2 v_2$

Conservation of mass for incompressible flow.

Darcy-Weisbach

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

Pressure drop due to friction in a pipe.

Variables

Symbol	Description	Unit
$Re$	Reynolds number	—
$\rho$	Fluid density	kg/m³
$v$	Flow velocity	m/s
$D$	Characteristic dimension	m
$\mu$	Dynamic viscosity	Pa·s
$P$	Pressure	Pa
$f$	Darcy friction factor	—

Volumetric Temperature Coefficients

The volumetric temperature expansion coefficient (β) varies with temperature and material. The table below lists average values for common fluids.

Bulk Modulus (Fluid Elasticity)

The bulk modulus (E) measures a fluid's resistance to uniform compression. Values are approximate and can vary with pressure and temperature.

Combined Temperature and Pressure Effect

When both temperature and pressure change, the final density can be calculated by combining the effects:

$$\rho_1 = \frac{\rho_0}{(1 + \beta (t_1 - t_0)) \cdot (1 - \frac{p_1 - p_0}{E})}$$

Where:

• ρ₁ = final density (kg/m³)
• ρ₀ = initial density (kg/m³)
• β = volumetric temperature expansion coefficient (m³/m³ °C)
• t₁ - t₀ = change in temperature (°C)
• p₁ - p₀ = change in pressure (Pa or N/m²)
• E = bulk modulus (Pa or N/m²)

Example Calculation

Problem: Calculate the density of water at 100 bar (10,000,000 Pa) and 20°C.

Given:

• Initial density of water at 0°C, ρ₀ = 999.8 kg/m³
• Average expansion coefficient for water (0-20°C), β = 0.000088 m³/m³ °C
• Bulk modulus of water, E = 2.15 × 10⁹ N/m²
• Initial pressure, p₀ = 1 bar (1 × 10⁵ Pa)
• Final pressure, p₁ = 100 bar (100 × 10⁵ Pa)

Calculation using the combined formula: ρ₁ = 999.8 / [ (1 + 0.000088 * (20 - 0)) * (1 - ((100e5 - 1e5) / 2.15e9)) ] ρ₁ ≈ 1002.7 kg/m³

Interactive Charts

References

• Original Source