CL: Mass/Molar Flux

Recall that in terms of mass transfer we are interested in either Ni (called the molar flux) or ni (mass flux) so that we are interesting in the rate at which mass traveles through a certain area in a given time. While we have written this as a flow dived by an area:

$\displaystyle{\mathbf{n} = \frac{\dot m}{A}}$,

we might instead (for something like a pipe flow) write that the mass flow rate is given by:

$\displaystyle{\dot m = \rho v A}$

where $\displaystyle{\rho}$ is the mass concentration (density), v is the (average) velocity in the pipe, and A is the cross section of the pipe so that we can then write the mass flux as:

$\displaystyle{\mathbf{n} = \frac{\dot m}{A} = \frac{\rho v A}{A} = \rho v}$

These simple arguments, in fact, can be applied to any mass flow to determine the total mass flux, so that, in general, the mass flux of component i is given by:

$\displaystyle{\mathbf{n_i} = \rho_i v_i}$

Similar arguments can give us an expression for the molar flux

$\displaystyle{\mathbf{N_i} = c_i v_i}$

where ci is the molar concentration of component i.

Unfortunately, the velocity in these expressions vi is not trivial to calculate (or measure!) without some further consideration. (Think, for example, of measuring the velocity of oxygen in the cool breeze on a summer afternoon instead of measuring the velocity of the air! Clearly, measuring the air velocity would be simple, but distinguishing the velocity of the oxygen from all the other gases that make up air would be quite troublesome.) So, let's think in more detail about the bulk velocity (like the air velocity).

Let's assume for a minute that we do actually know the velocities of each the individual gases that make up air. How might we calculate the bulk or average velocity?

Well, for our purposes there are essentially two methods: averaging by mass, or averaging by mole. If we wanted to calculate the bulk velocity of the air by mass we would add up all of the velocities of the individual gases, weighted by the mass of that particular gas and then divide the total by the mass of the air. In other words, we would calculate a weighted average (by mass) like this:

$\displaystyle{v = \frac{\sum_i \rho_i v_i}{\sum_i \rho_i}}$

We could do the exact same thing using moles

$\displaystyle{V = \frac{\sum_i c_i v_i}{\sum_i c_i}}$

where ci is the molar concentration of component i.

Now that we know how the actual velocity of each component in a mixture is related to the average (bulk) velocity (which we can measure). We can try to determine how to go in the other direction (in other words, how to calculate the actual component velocity from the bulk velocity.)

If we think of a crowd of people walking down the hallway, it is certainly possible that the individuals within the crowd are walking at different speeds. If we draw a box (centered) around the individuals at one time, and then again a little while later, it is clear that the crowd as a whole is moving at some average velocity, V (which is not necessarily equal to the velocity of any of the individuals). If we take the difference between the individual's velocity and the bulk velocity we get a quantitative measure of the individual's deviation (diffusion!) velocity.

$\displaystyle{v_{diff_i} = v_i - v}$ or $\displaystyle{v_{diff_i} = v_i - V}$

In exactly the same way that we defined a total mass (mole) flux using the component's total (actual) velocity, we can then define a component's diffusive flux ji (mass) or Ji (molar) to be:

$\displaystyle{j_i = \rho_i v_{diff_i} = \rho_i(v_i-v)}$
or
$\displaystyle{J_i = c_iv_{diff_i} = c_i(v_i-C)}$

We discussed earlier in the course that the actual origin of diffusive motion is the random fluctuations of molecules "swapping" their way down a concentration gradient. This explanation clearly has much in common with heat flow (down a temperature gradient) and thus it is not surprising that Fick's Law (proposed for binary mixtures) looks much like Fourier's Law

$\displaystyle{J_i = -D\nabla c_i}$

for the diffusive flux, where D is the binary diffusivity of component i in the system (species j. This expression shows that (just like temperature) concentration "flows downhill" (until there isn't a hill!) A more general equation to account for possible changes in overall concentration, ctotal (in non isobaric, isothermal systems), is

$\displaystyle{J_i = -c_{total}D\nabla y_i}$

where yi is the mole fraction.

NOTE:

While Fick's Law is formally true for only binary mixtures, it is often used for multi-component systems with a somewhat fudged value of D.

If we then plug this equation into our definition of the diffusive flux

$\displaystyle{J_i = c_i(v_i-V) = -c_{total}D\nabla y_i}$

we can solve for the component's actual velocity vi

$\displaystyle{v_i = \frac{-c_{total}D\nabla y_i}{c_i} +V}$

We can then plug this equation into our expression for the molar flux

$\displaystyle{N_i = c_iv_i = c_i\left ( \frac{-c_{total}D\nabla y_i}{c_i} +V \right ) = -c_{total}D\nabla y_i +c_iV}$

which (finally!) gives us the total molar flux in terms of things we can measure easily. Again, deriving the expression based on mass is not any more difficult

$\displaystyle{n_i = -\rho_{total}D\nabla \omega_i +\rho_i v}$

where $\displaystyle{\omega_i}$ is the mass fraction of component i.

OUTCOME:

Explain the difference between the total flux and the diffusive flux