Mass flux explained

In physics and engineering, mass flux is the rate of mass flow per unit of area. Its SI units are kg m⁻² s⁻¹. The common symbols are j, J, q, Q, φ, or Φ (Greek lowercase or capital Phi), sometimes with subscript m to indicate mass is the flowing quantity. Mass flux can also refer to an alternate form of flux in Fick's law that includes the molecular mass, or in Darcy's law that includes the mass density.^[1]

Less commonly the defining equation for mass flux in this article is used interchangeably with the defining equation in mass flow rate. For example, Fluid Mechanics, Schaum's et al ^[2] uses the definition of mass flux as the equation in the mass flow rate article.

Definition

Mathematically, mass flux is defined as the limit$$j\_m\; =\; \backslash lim\_\; \backslash frac,$$where$$I\_m\; =\; \backslash lim\_\; \backslash frac\; =\; \backslash frac$$is the mass current (flow of mass per unit time) and is the area through which the mass flows.

For mass flux as a vector, the surface integral of it over a surface S, followed by an integral over the time duration to, gives the total amount of mass flowing through the surface in that time :$$m=\backslash int\_^\; \backslash iint\_S\; \backslash mathbf\_m\; \backslash cdot\backslash mathbf\; \backslash ,\; dA\; \backslash ,\; dt.$$

The area required to calculate the flux is real or imaginary, flat or curved, either as a cross-sectional area or a surface.

For example, 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,

\hat{n }. The relation is A = A \hat{n }.

If the mass flux passes through the area at an angle θ to the area normal

\hat{n }, then$$\backslash mathbf\_m\; \backslash cdot\; \backslash mathbf\; =\; j\_m\backslash cos\backslash theta$$where is the dot product of the unit vectors. That is, the component of mass flux passing through the surface (i.e. normal to it) is . While the component of mass flux passing tangential to the area is given by, there is no mass flux actually passing through the area in the tangential direction. The only component of mass flux passing normal to the area is the cosine component.

Example

Consider a pipe of flowing water. Suppose the pipe has a constant cross section and we consider a straight section of it (not at any bends/junctions), and the water is flowing steadily at a constant rate, under standard conditions. The area A is the cross-sectional area of the pipe. Suppose the pipe has radius . The area is then $$A\; =\; \backslash pi\; r^2.$$To calculate the mass flux (magnitude), we also need the amount of mass of water transferred through the area and the time taken. Suppose a volume passes through in time t = 2 s. Assuming the density of water is, we have:(since initial volume passing through the area was zero, final is, so corresponding mass is), so the mass flux is$$j\_m\; =\; \backslash frac\; =\; \backslash frac.$$

Substituting the numbers gives:$$j\_m\; =\; \backslash frac\; =\; \backslash frac\backslash times\; 10^4,$$which is approximately 596.8 kg s⁻¹ m⁻².

Equations for fluids

Alternative equation

Using the vector definition, mass flux is also equal to:^[3] $$\backslash mathbf\_\; =\; \backslash rho\; \backslash mathbf$$

where:

• = mass density,
• = velocity field of mass elements flowing (i.e. at each point in space the velocity of an element of matter is some velocity vector).

Sometimes this equation may be used to define as a vector.

Mass and molar fluxes for composite fluids

Mass fluxes

In the case fluid is not pure, i.e. is a mixture of substances (technically contains a number of component substances), the mass fluxes must be considered separately for each component of the mixture.

When describing fluid flow (i.e. flow of matter), mass flux is appropriate. When describing particle transport (movement of a large number of particles), it is useful to use an analogous quantity, called the molar flux.

Using mass, the mass flux of component i is$$\backslash mathbf\_\; =\; \backslash rho\_i\; \backslash mathbf\_i.$$

The barycentric mass flux of component i is$$\backslash mathbf\_\; =\; \backslash rho\; \backslash left\; (\backslash mathbf\_i\; -\; \backslash langle\; \backslash mathbf\; \backslash rangle\; \backslash right),$$where

\langle u \rangle is the average mass velocity of all the components in the mixture, given by$$\backslash langle\; \backslash mathbf\; \backslash rangle\; =\; \backslash frac\backslash sum\_i\; \backslash rho\_i\; \backslash mathbf\_i\; =\; \backslash frac\backslash sum\_i\; \backslash mathbf\_$$where

• = mass density of the entire mixture,
• = mass density of component i,
• = velocity of component i.

The average is taken over the velocities of the components.

Molar fluxes

If we replace density by the "molar density", concentration, we have the molar flux analogues.

The molar flux is the number of moles per unit time per unit area, generally:$$\backslash mathbf\_\; =\; c\; \backslash mathbf.$$

So the molar flux of component i is (number of moles per unit time per unit area):$$\backslash mathbf\_\; =\; c\_i\; \backslash mathbf\_i$$and the barycentric molar flux of component i is$$\backslash mathbf\_\; =\; c\; \backslash left\; (\backslash mathbf\_i\; -\; \backslash langle\; \backslash mathbf\; \backslash rangle\; \backslash right),$$where

\langle u \rangle this time is the average molar velocity of all the components in the mixture, given by:$$\backslash langle\; \backslash mathbf\; \backslash rangle\; =\; \backslash frac\backslash sum\_i\; c\_i\; \backslash mathbf\_i\; =\; \backslash frac\backslash sum\_i\; \backslash mathbf\_.$$

Usage

Mass flux appears in some equations in hydrodynamics, in particular the continuity equation:$$\backslash nabla\; \backslash cdot\; \backslash mathbf\_\; +\; \backslash frac\; =\; 0,$$which is a statement of the mass conservation of fluid. In hydrodynamics, mass can only flow from one place to another.

Molar flux occurs in Fick's first law of diffusion:$$\backslash nabla\; \backslash cdot\; \backslash mathbf\_\; =\; -\backslash nabla\; \backslash cdot\; D\; \backslash nabla\; n$$where is the diffusion coefficient.

See also

• Mass-flux fraction
• Flux
• Fick's law
• Darcy's law
• Wave mass flux and wave momentum
• Defining equation (physical chemistry)

Notes and References

1. Web site: Thesaurus: Mass flux . 2008-12-24.
2. Fluid Mechanics, M. Potter, D.C. Wiggart, Schuam's outlines, McGraw Hill (USA), 2008,
3. Vectors, Tensors, and the basic Equations of Fluid Mechanics, R. Aris, Dover Publications, 1989,