The Rayleigh Equation

The Rayleigh Equation is useful in the analysis of simple distillation, as it shows how the concentration and quantity are related.

As the process is unsteady state in nature, the derivation is based on a differential approach to changes in concentration with time. The equation to be derived (known as the Rayleigh Equation) shows the relationship between total moles remaining in the still and the mole fraction of the more volatile component in the still.

[Similar equation can be obtained for the relationship between the total moles of distillate and the mole fraction of the more volatile component in the condensate receiver]

Material Balance for the still: see the Figure below

L1 = initial moles of liquid originally in still

L2 = final moles of liquid remained in still

x1 = initial liquid composition in still (mole fraction of A)

x2 = final liquid composition in still (mole fraction A)

At any time t, the amount of liquid in the still is L, with mole fraction of A in the liquid being x.

After a small differential time (t + dt) , a small amount of vapour dL is produced, and the composition of A in the vapour is y (mole fraction). The vapour is assumed to be in equilibrium with the residue liquid.The amount of liquid in the still is thus reduced from L to (L - dL), while the liquid composition changed from x to (x - dx). See the Figure below:

Performing a material balance on A:

Initial amount in still = Amount left in still + Amount vaporized

We have,

xL = (x - dx) (L - dL) + y dL

xL = xL - x dL - L dx + dx dL + y dL

Neglecting the term dx dL, the equation reduces to:

L dx = y dL - x dL

Re-arranging gives the following:

Integrating from L1 to L2, and from x1 to x2, we obtain the Rayleigh Equation: