Mass flow rate

  1. Fluid dynamics
  2. Physical quantities


In Physics and Engineering, mass flux is the rate of mass flow. The common symbol is \(\dot m\) (pronounced "m-dot"), although sometimes μ (Greek lowercase mu) is used. The SI units are kg s-1.

Sometimes, mass flow rate is termed mass flux or mass current, see for example Fluid Mechanics, Schaum's et al [1]. In this article, the (more intuitive) definition is used.

Definition

Mass flow rate is defined by: [2] [3]

\[\dot{m} = \frac{\Delta m}{ \Delta t}\]

where:

  • \(\dot{m}\) = mass flow rate,
  • Δm = change in mass flowing through the area,
  • Δt = time interval of mass flow.

In the limit Δm and Δt become infinitesimally small, the algebraic definition becomes a calculus definition: \[\dot m = \frac{{\rm d}m}{{\rm d}t}\]

(hence the overdotted m - Newton's notation for a time derivative). Since this is only the time derivative of mass, a scalar quantity, the mass flow rate is also a scalar quantity. The change in mass is the amount that flows after crossing the boundary for some time duration, not simply the initial amount of mass at the boundary minus the final amount at the boundary, since the change in mass flowing through the area would be zero for steady flow.

Alternative equations

Mass flow rate can also be calculated by:

\[\dot m = \rho \bold{v} \cdot \bold{A} = \rho Q = \bold{j}_{\rm m} \cdot \bold{A} \]

where:

The above equation is only true for a flat, plane area. In general, including cases where the area is curved, the equation becomes a surface integral:

\[\dot m = \iint_A \rho \bold{v} \cdot {\rm d}\bold{A} = \iint_A \bold{j}_{\rm m} \cdot {\rm d}\bold{A} \]

The area required to calculate the mass flow rate is real or imaginary, flat or curved, either as a cross-sectional area or a surface. E.g. for substances passing through a filter or a membrane, the real surface is the (generally curved) surface area of the filter, macroscopically - ignoring the area spanned by the holes in the filter/membrane. The spaces would be cross-sectional areas. For liquids passing through a pipe, the area is the cross-section of the pipe, at the section considered. The vector area is a combination of the magnitude of the area through which the mass passes through, A, and a unit vector normal to the area, \(\bold{\hat{n}}\). The relation is \(\bold{A} = A \bold{\hat{n}}\).

The reason for the dot product is as follows. The only mass flowing through the cross-section is the amount normal to the area, i.e. parallel to the unit normal. This amount is: \[\dot m = \rho v A \cos\theta \] where θ is the angle between the unit normal \(\bold{\hat{n}}\) and the velocity of mass elements. The amount passing through the cross-section is reduced by the factor \(\cos\theta \), as θ increases less mass passes through. All mass which passes in tangential directions to the area, that is perpendicular to the unit normal, doesn't actually pass through the area, so the mass passing through the area is zero. This occurs when θ = π/2: \[\dot m = \rho v A \cos(\pi/2) = 0\] These results are equivalent to the equation containing the dot product. Sometimes these equations are used to define the mass flow rate.

Usage

In the elementary form of the continuity equation for mass, in Hydrodynamics: [4]

\[ \rho_1 \bold{v}_1 \cdot \bold{A}_1 = \rho_2 \bold{v}_2 \cdot \bold{A}_2 \]

Mass flow rate isn't just for Hydrodynamics. The time derivative of mass occurs in Newton's second law:

\[ \bold{F} = \frac{{\rm d}\bold{p}}{{\rm d}t} = \frac{{\rm d}(m\bold{v})}{{\rm d}t} \]

where:

  • F = resultant force acting on the body,
  • m = mass of body,
  • v = velocity of the centre of mass of the body,
  • p = momentum of the centre of mass of the body,

which can be expanded by the product rule of differentiation:

\[\begin{align} \bold{F} & = m\frac{{\rm d}\bold{v}}{{\rm d}t} + \bold{v}\frac{{\rm d}m}{{\rm d}t} \\ & = m \bold{a} + \dot{m}\bold{v} \\ \end{align}\]

where a is the acceleration vector of the centre of mass of the body. However, in this equation the time derivative of mass is only the rate of change of mass of the system, not necessarily due to mass flow, although one can intuitively think of mass "flowing into and/or out of a system" as the system changes mass. This is justified since mathematically the flow of mass in transport is equivalent to the system gaining or losing mass in any way. This form of Newton's 2nd law allows physical systems of variable mass to be solved for, such as the classical and elementary rocket problem, i.e. solving the equation of motion for a rocket. [5] In this situation, mass is combusted by the rocket engines, so mass is effectively flowing out of the system (rocket) as it is combusted. The result is the mass of the rocket decreases as a function of time, and the mass flow rate is an important parameter to account for this.

Analogous quantities

In hydrodynamics, mass flow rate the rate of flow of mass. In electricity, the rate of flow of charge is electric current.

See also

References

  1. ↑ Fluid Mechanics, M. Potter, D.C. Wiggart, Schuam's outlines, McGraw Hill (USA), 2008, ISBN 978-0-07-148781-8
  2. ↑ http://www.engineersedge.com/fluid_flow/mass_flow_rate.htm
  3. ↑ http://www.grc.nasa.gov/WWW/k-12/airplane/mflow.html
  4. ↑ Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray, ISBN 0-7195-3382-1
  5. ↑ Calculus, H. Neill, P. Abbott, Teach Yourself Series
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