quantity

mass flow rate

mass flow rate — from formula anatomy
10 related articles · 6 formulas

Description

Mass flow rate is the amount of mass passing through a given cross-section per unit time. It is a fundamental quantity in fluid dynamics and thermodynamics, describing how quickly matter moves through a system, such as a pipe, a jet engine, or a blood vessel. Unlike volume flow rate, which depends on the fluid's density, mass flow rate directly measures the actual material transported, making it the more physically meaningful quantity in conservation laws.

History

The concept emerged with the development of fluid mechanics in the 18th century, particularly through the work of Daniel Bernoulli and later the formulation of the continuity equation by Leonhard Euler. The explicit term 'mass flow rate' became standard with the rise of engineering thermodynamics in the 19th century.

How it works

Mathematically, mass flow rate is the product of fluid density, cross-sectional area, and flow velocity (ṁ = ρ·A·v). For compressible flows, density varies with pressure and temperature, so the calculation must account for local conditions. In practice, it is measured using devices like Coriolis flow meters, which detect the inertia of the moving fluid, or by combining volumetric flow measurements with density sensors.

In practice

It is essential in designing and monitoring pipelines, fuel injection systems, chemical reactors, and HVAC systems, where accurate mass transport is critical for efficiency and safety.

Fun fact

In a rocket engine, the mass flow rate of exhaust gases can exceed 1000 kilograms per second, which is why the thrust equation depends directly on this quantity.

Concept graph →

Formulas

$$r_H \propto \frac{\dot{M}}{\rho c_s}$$
Horizon radius scales with accretion rate over density and sound speed.
What is in the formula
r_HvariableAccretion horizon radius — radius metre
\dot{M}variableMass accretion rate — mass flow rate kilogram per second
\rhovariableMass density of accreting material — mass density kilogram per cubic metre
c_svariableSound speed in accreting material — speed metre per second
$$P = ackslash eta ackslash dot{M} c^2$$
Power released by accretion onto a compact object.
What is in the formula
Pvariableaccretion power — power watt
ηvariableefficiency of mass-to-energy conversion
Ṁvariablemass accretion rate — mass flow rate kilogram per second
cconstantSpeed of light in vacuum — speed of light = 299792458 metre per second
$$\dot{M}_{out} = \rho v A$$
Mass outflow rate is density times velocity times area.
What is in the formula
Ṁ_outvariableMass outflow rate — mass flow rate kilogram per second
ρvariableMass density of the outflowing material — mass density kilogram per cubic metre
vvariableFlow velocity perpendicular to the area — velocity metre per second
AvariableCross-sectional area through which material flows — area square metre
$$L_{\rm acc} = \frac{G M \dot{M}}{R}$$
Luminosity released by accretion onto a compact object.
What is in the formula
L_{\rm acc}variableaccretion luminosity — luminosity watt
Mvariablemass of the accreting object — mass kilogram
\dot{M}variablemass accretion rate — mass flow rate kilogram per second
Rvariableradius at which accretion energy is released — radius metre
GconstantConstant of universal gravitation. — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared
$$T(r) = \left( \frac{3 G M \dot{M}}{8 \pi \sigma r^3} \right)^{1/4}$$
Temperature radial profile for an accretion disk.
What is in the formula
T(r)variabledisk temperature at radius r — temperature kelvin
rvariableradial distance from the central object — radius metre
Mvariablemass of the central object — mass kilogram
\dot{M}variablemass accretion rate — mass flow rate kilogram per second
GconstantConstant of universal gravitation. — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared
\sigmaconstantConstant relating blackbody radiated power to temperature. — stefan boltzmann constant = 5.670374419e-8 watt per square metre per kelvin to the fourth
$$T_{\rm bb} \propto M^{-a} \dot{m}^{-b}$$
Blackbody temperature scales with mass and accretion rate.
What is in the formula
T_bbvariableBlackbody temperature — temperature kelvin
MvariableMass of the accreting object — mass kilogram
avariablePositive exponent for mass scaling
dot_mvariableMass accretion rate — mass flow rate kilogram per second
bvariablePositive exponent for accretion rate scaling
∝operatorProportional to — proportionality

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