# Diffusion of solute in a dynamic solvent

Consider this hypothetical equipment.

A pipe made of semi-permeable membrane (permeable only to the solute) has a streamline flow of pure water going from A to B ($P_A>P_B$). Midway, another static compartment with a high concentration of the solute (which is maintained at a constant level $C$ by replenishment) is brought in contact such that exchange is possible across the two compartments. Disregarding the effect of gravity, how would you model the solute flow in the pipe? Assume that the solutes are consumed at B.

It seems intuitive that the solute will flow in the direction of B due to the bulk flow of the liquid, as you would expect from the common experience. But when I tried to characterize it thermodynamically, using the chemical potential, I could not come up with the proper terms. The concentration gradient would favor diffusion equally towards A and B, representing a gain in entropy. But what other thermodynamic gains will preferentially push the solutes in the direction of B, if any? Is a steady state possible, and if it is, what factors will determine the concentration at A?

I couldn't factor in the effect of pressure. Although the pressure gradient makes intuitive sense for the flow of liquid (Bernoulli equation and potential energy), how does it explain it's effect on the solute? Would a solute flow, without concomitant solvent-flow allowed, from a compartment with water at a higher pressure, to a one with lower pressure, although the initial concentration in both the compartments was same?

Having taken the topic after a long gap, I apologize if I seem to have totally muddled up the fundamentals. Although the precise mathematical equations aren't required, it might help understand the stuff going on here. Let me know if it isn't suited to Physics SE.

• Would it be possible to reformulate the problem in a way which keeps the solvent pipe (its its contents) statically positioned, while the solute injector moves along the pipe? It might be easier to solve in that frame of reference, then you could transform the result back to the original frame. – Christopher Oicles Nov 30 '16 at 2:19
• First you must calculate the Peclet number: $Pe\equiv \frac{U d}{\alpha}$, where $\alpha$ is diffusivity of solute. If $Pe\gg 1$, then flow will carry the solutes towards B with none going to A. The problem is identical to heat transfer into a flow, with solute concentration replacing temperature. – Deep Nov 30 '16 at 5:22
• @Zero I do not know how to calculate Peclet number in my case. Being a beginner, can you also give the intuition behind this calculation? – Satwik Pasani Dec 2 '16 at 16:47
• If flow speed $U=0$, then solute will diffuse symmetrically in both directions. Once there is flow this symmetry is disturbed. Solute is still diffusing in the direction of $A$ but it is now embedded inside fluid moving towards $B$. So depending on how fast the fluid is moving, solute may never reach $A$. Think of shouting to a faraway person in the direction opposite to which wind is blowing, or the chances of flower's scent reaching you if wind is blowing in the 'wrong' direction. – Deep Dec 3 '16 at 6:37
• @Zero That makes an intuitive sense. Can we quantify the tendencies to go preferentially towards B over A in terms of the differences of their chemical potential for the solute. The solute being something like Glucose. – Satwik Pasani Dec 7 '16 at 9:27