Mass flux

Inphysicsandengineering,massfluxis therate of mass flowper unit of area. ItsSI unitsare kg m⁻²s⁻¹.The common symbols arej,J,q,Q,φ,or Φ (Greeklowercase or capitalPhi), sometimes with subscriptmto indicate mass is the flowing quantity. Mass flux can also refer to an alternate form of flux inFick's lawthat includes themolecular mass,or inDarcy's lawthat includes the massdensity.

Less commonly the defining equation for mass flux in this article is used interchangeably with the defining equation inmass flow rate.For example,Fluid Mechanics, Schaum's et aluses the definition of mass flux as the equation in the mass flow rate article.

Definition

Mathematically, mass flux is defined as thelimit $$j_{m}=\lim _{A\to 0}{\frac {I_{m}}{A}},$$ where $$I_{m}=\lim _{\Delta t\to 0}{\frac {\Delta m}{\Delta t}}={\frac {dm}{dt}}$$ is the mass current (flow of massmper unit timet) andAis the area through which the mass flows.

For mass flux as a vectorj_m,thesurface integralof it over asurfaceS,followed by an integral over the time durationt₁tot₂,gives the total amount of mass flowing through the surface in that time (t₂−t₁): $$m=\int _{t_{1}}^{t_{2}}\iint _{S}\mathbf {j} _{m}\cdot \mathbf {\hat {n}} \,dA\,dt.$$

Thearearequired 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 afilteror amembrane,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.

Thevector areais a combination of the magnitude of the area through which the mass passes through,A,and aunit vectornormal to the area,$\mathbf {\hat {n}}$.The relation is$\mathbf {A} =A\mathbf {\hat {n}}$.

If the mass fluxj_mpasses through the area at an angle θ to the area normal$\mathbf {\hat {n}}$,then $$\mathbf {j} _{m}\cdot \mathbf {\hat {n}} =j_{m}\cos \theta$$ where·is thedot productof the unit vectors. That is, the component of mass flux passing through the surface (i.e. normal to it) isj_mcosθ.While the component of mass flux passing tangential to the area is given byj_msinθ,there isnomass flux actually passingthroughthe area in the tangential direction. Theonlycomponent of mass flux passing normal to the area is the cosine component.

Example

Consider a pipe of flowingwater.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, understandard conditions.The areaAis the cross-sectional area of the pipe. Suppose the pipe has radiusr= 2 cm = 2 × 10⁻²m.The area is then $$A=\pi r^{2}.$$ To calculate the mass fluxj_m(magnitude), we also need the amount of mass of water transferred through the area and the time taken. Suppose a volumeV= 1.5 L = 1.5 × 10⁻³m³passes through in timet= 2 s. Assuming thedensity of waterisρ= 1000 kg m⁻³,we have: $${\begin{aligned}\Delta m&=\rho \Delta V\\m_{2}-m_{1}&=\rho (V_{2}-V_{1})\\m&=\rho V\\\end{aligned}}$$ (since initial volume passing through the area was zero, final isV,so corresponding mass ism), so the mass flux is $$j_{m}={\frac {\Delta m}{A\Delta t}}={\frac {\rho V}{\pi r^{2}t}}.$$

Substituting the numbers gives: $$j_{m}={\frac {1000\times \left(1.5\times 10^{-3}\right)}{\pi \times \left(2\times 10^{-2}\right)^{2}\times 2}}={\frac {3}{16\pi }}\times 10^{4},$$ which is approximately 596.8 kg s⁻¹m⁻².

Equations for fluids

Alternative equation

Using the vector definition, mass flux is also equal to: $$\mathbf {j} _{\rm {m}}=\rho \mathbf {u}$$

where:

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

Sometimes this equation may be used to definej_mas a vector.

Mass and molar fluxes for composite fluids

Mass fluxes

In the case fluid is not pure, i.e. is amixtureof 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 themolar flux.

Using mass, the mass flux of componentiis $$\mathbf {j} _{{\rm {m}},\,i}=\rho _{i}\mathbf {u} _{i}.$$

Thebarycentric mass fluxof componentiis $$\mathbf {j} _{{\rm {m}},\,i}=\rho \left(\mathbf {u} _{i}-\langle \mathbf {u} \rangle \right),$$ where$\langle \mathbf {u} \rangle$is theaveragemass velocityof all the components in the mixture, given by $$\langle \mathbf {u} \rangle ={\frac {1}{\rho }}\sum _{i}\rho _{i}\mathbf {u} _{i}={\frac {1}{\rho }}\sum _{i}\mathbf {j} _{{\rm {m}},\,i}$$ where

• ρ= mass density of the entire mixture,
• ρ_i= mass density of componenti,
• u_i= velocity of componenti.

The average is taken over the velocities of the components.

Molar fluxes

If we replace densityρby the "molar density",concentrationc,we have themolar fluxanalogues.

The molar flux is the number of moles per unit time per unit area, generally: $$\mathbf {j} _{\rm {n}}=c\mathbf {u}.$$

So the molar flux of componentiis (number of moles per unit time per unit area): $$\mathbf {j} _{{\rm {n}},\,i}=c_{i}\mathbf {u} _{i}$$ and thebarycentric molar fluxof componentiis $$\mathbf {j} _{{\rm {n}},\,i}=c\left(\mathbf {u} _{i}-\langle \mathbf {u} \rangle \right),$$ where$\langle \mathbf {u} \rangle$this time is theaveragemolar velocityof all the components in the mixture, given by: $$\langle \mathbf {u} \rangle ={\frac {1}{n}}\sum _{i}c_{i}\mathbf {u} _{i}={\frac {1}{c}}\sum _{i}\mathbf {j} _{{\rm {n}},\,i}.$$

Usage

Mass flux appears in some equations inhydrodynamics,in particular thecontinuity equation: $$\nabla \cdot \mathbf {j} _{\rm {m}}+{\frac {\partial \rho }{\partial t}}=0,$$ which is a statement of the mass conservation of fluid. In hydrodynamics, mass can only flow from one place to another.

Molar flux occurs inFick's first lawofdiffusion: $$\nabla \cdot \mathbf {j} _{\rm {n}}=-\nabla \cdot D\nabla n$$ whereDis thediffusion coefficient.

See also

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