Modes of Operation: Cross-Current

Cross-current mode is mostly used in batch operation.  Batch extractors have traditionally been used in low capacity, multi-product plants typically found in the pharmaceutical and agrochemical industries.   For washing and neutralization operations that require very few stages, cross-current operation is particularly practical and economical and offers a great deal of flexibility.  The extraction equipment usually includes an agitated tank that may also be used for the reaction steps.  In these tanks, the solvent is first added to the feed, the contents are mixed, settled, and then separated.   Single stage extraction is used when the extraction is fairly simple and can be achieved without a high amount of solvent.  If more than one stage is required, multiple solvent-washes will be required.

Though operation in cross-current mode a great deal of flexibility, it must be examined carefully due to the high solvent volume requirements and low extraction yields.  The following illustration gives a quick method to calculate solvent requirements for cross-current extraction.

A single-stage extractor can be represented as:

F = Feed quantity / rate, mass

R = Raffinate quantity / rate, mass

S = Solvent quantity / rate, mass

E = Extract quantity / rate, mass

X_f , X _r, Y _s, and Y_e are the weight fractions of solute in the feed, raffinate, solvent and extract, respectively.

Partition coefficient ‘m’ is defined as the ratio of Y_e to X_r at equilibrium conditions

The flows and concentrations are represented in solute-free basis as such a representation leads to simplification of equations. 

For example, for a 100 kg/hr feed containing 10% weight acetic acid, F = 100-10 = 90 kg/hr, X_r = 0.1/(1-0.1) = 0.111

The component mass balance can be represented as:

 F X_f + S Y_s = R X_r + E Y_e

Assuming (i) immiscibility of feed and solvent and (ii) the initial solvent is free of solute, i.e., F = R, S = E and Y_s = 0 and using the equilibrium relation of Y_e = m X_r, this equation simplifies to

 S = F/m (X_f /X_r–1)
Or
Reduction ratio, X_f /X_r = 1+ m S/F

For multi-stage crosscurrent operation:

Assuming that the partition coefficient (m) is constant over the concentration range and the solvent quantity in each of the ‘n’ stages is the same, i.e., S₁ = S₂ =…..=S _n = S/n,

Solvent Requirement is

 S = n * F/m [(X_f /X_r)^1/n - 1]
Or
Reduction ratio X_f /X_r = (1+mS/nF)ⁿ
It can be proved mathematically that the total solvent quantity would be minimum if the solvent were distributed equally between washes.

Reference: Optimize Liquid-Liquid Extraction, R. Madhaven, Cheresources.com

Modes of Operation: Counter-Current