I am reading a textbook where an example of a one-dimensional binary diffusion is given. In a binary system composed of substances A and B, we write a relation for the substance A:

$\dot{m}_A'' = Y_A(\dot{m}_A'' + \dot{m}_B'') - \rho \mathcal{D}_{AB} \frac{dY_A}{dx}$

where the quantity $\dot{m}_A''$ is explained as the mass flow of substance A per unit area and its units are $\frac{kg}{s m^2}$. $Y_A$ is mass fraction of substance A, $\rho$ is density and $\mathcal{D}_{AB}$ is binary diffusivity. In words, the above equation states: mass flux of A = bulk transport of A + diffusion of A.

The problem seemed straightforward at first, but then I thought, what does it actually mean that the mass flow is per unit of area since the problem is assumed to be one-dimensional? The way I visualise this, all transport of mass in this example happens along direction $x$ only and there is no other direction for which it would make sense to express the area. When there is a mass flow from one point on the $x$ axis to another, this mass flows through a series of points on the $x$ axis, and not areas.

How should I treat this variable $\dot{m}_A''$ for a one-dimensional flow and how can I make sense of defining it as mass flow per unit of area?

  • $\begingroup$ The flow area is perpendicular to the x direction, and $\dot{m}"$ is the rate of mass flow through a unit area perpendicular to the x direction. $\endgroup$ Jan 14 '19 at 23:37

When a problem is 1D or 2D, it doesn't mean that it is fundamentally different from 3D. It just means the variation in one direction (or two directions) is significantly larger than the variation in the other directions. So if it's 1D, it just means the change in Y and Z are really small or zero. But those dimensions still exist.

You essentially have two choices. You can change the units of mass flux so it is some 1D equivalent. I don't recommend this. It means your units will change with dimensionality and makes it hard to compare with other systems.

Instead, you consider the mass flux to still be mass flowing through a unit area. You don't have an important second or third dimension, so you use a notional "unit" of area. This keeps all of your mass flux units the same regardless of dimensionality. This is the standard approach. You don't have an area, but you say "per unit of area" anyway.

Doing it this way means you can keep your physical constants the same and don't need to come up with dimensionality-specific constants for things like viscosity or diffusion coefficients.

This comes up often and with all kinds of different properties. See this question and answer as an example.

  • $\begingroup$ Thanks, this makes much more sense now. So I imagine keeping mass flux in 1D problem can be useful for instance in a system where there is flow through a channel in only one direction. The channel obviously has some cross-sectional area, so mass flow per unit area is a reasonable quantity that can tell us something about the flow rate. $\endgroup$ Jan 16 '19 at 14:48
  • $\begingroup$ @camillejr Exactly, which is true for simple things like pipe flow, and also holds just as well for things like rocket engines and jet engines. At least on the time-averaged sense. $\endgroup$
    – tpg2114
    Jan 16 '19 at 15:41

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