# 1.3: Liquid-liquid Extraction


## Staged Liquid-Liquid Extraction and Hunter Nash Method

$$E_n$$ = extract leaving stage $$n$$. This could refer to the mass of the stream or the composition of the stream.

$$F$$ = solvent entering extractor stage 1. This could refer to the mass of the stream or the composition of the stream.

$$n$$ = generic stage number

$$N$$ = Final stage. This is where the fresh solvent S enters the system and the final raffinate $$R_N$$ leaves the system.

$$M$$ = Composition of the mixture representing the overall system. Points ($$F$$ and $$S$$) and ($$E_1$$ and $$R_N$$) must be connected by a straight line that passes through point $$M$$. $$M$$ will be located within the ternary phase diagram.

$$P$$ = Operating point. $$P$$ is determined by the intersection of the straight line connecting points ($$F$$, $$E_1$$) and the straight line connecting points ($$S$$, $$R_N$$). Every pair of passing streams must be connected by a straight line that passes through point $$P$$. $$P$$ is expected to be located outside of the ternary phase diagram.

$$R_n$$ = raffinate leaving stage $$n$$. This could refer to the mass of the stream or the composition of the stream.

$$S$$ = solvent entering extractor stage $$N$$. This could refer to the mass of the stream or the composition of the stream.

$$S/F$$ = mass ratio of solvent to feed

$$(x_i)_n$$ = Mass fraction of species $$i$$ in the raffinate leaving stage $$n$$

$$(y_i)_n$$ = Mass fraction of species $$i$$ in the extract leaving stage $$n$$

Process schematic for multistage liquid-liquid extraction.

Determining number of stages $$N$$ when (1) feed rate; (2) feed composition; (3) incoming solvent rate; (4) incoming solvent composition; and (5) outgoing raffinate composition have been specified/selected.

1. Locate points $$F$$ and $$S$$ on the ternary phase diagram. Connect with a straight line.
2. Do a material balance to find the composition of one species in the overall mixture. Use this composition to locate point $$M$$ along the straight line connection points $$F$$ and $$S$$. Note the position of point $$M$$.
3. Locate point $$R_N$$ on the ternary phase diagram. It will be on the equilibrium curve. Draw a straight line from $$R_N$$ to $$M$$ and extend to find the location of $$E_1$$ on the equilibrium curve.
4. On a fresh copy of the graph, with plenty of blank space on each side of the diagram, note the location of points $$F$$, $$S$$, and $$R_N$$ (specified/selected) and $$E_1$$ (determined in step 3).
5. Draw a straight line between $$F$$ and $$E_1$$. Extend to both sides of the diagram. Draw a second straight line between $$S$$ and $$R_N$$. Note the intersection of these two lines and label as “$$P$$”.
6. Determine the number of equilibrium stages required to achieve the desired separation with the selected solvent mass.

– Stream $$R_N$$ is in equilibrium with stream $$E_N$$. Follow the tie-lines from point $$R_N$$ to $$E_N$$.

– Stream $$E_N$$ passes stream $$R_{N-1}$$. Connect point $$E_N$$ to operating point $$P$$ with a straight line, mark the location of $$R_{N-1}$$.

– Stream $$R_{N-1}$$ is in equilibrium with stream $$E_{N-1}$$. Follow the tie-lines from stream $$R_{N-1}$$ to $$E_{N-1}$$.

– Stream $$E_{N-1}$$ passes stream $$R_{N-2}$$. Connect $$E_{N-1}$$ to operating point $$P$$ with a straight line, mark the location of $$R_{N-2}$$.

– Continue in this manner until the extract composition has reached or passed $$E_{1}$$. Count the number of equilibrium stages.

Watch this two-part series of videos from LearnChemE that shows how to use the Hunter Nash method to find the number of equilibrium stages required for a liquid-liquid extraction process.

##### Example

1000 kg/hr of a feed containing 30 wt% acetone, 70 wt% water. The solvent is pure MIBK. We intend that the raffinate contain no more than 5.0 wt% acetone. How many stages will be required for each proposed solvent to feed ratio in the table below?

 $$\textbf{S/F}$$ $$\textbf{S}$$ (kg/hr) $$\textbf{(x_A)_M}$$ target $$\textbf{(y_A)_1}$$ $$\textbf{N}$$ 1.0 2.0 0.2

## Hunter Nash Method for Finding Smin, Tank Sizing and Power Consumption for Mixer-Settler Units

### Staged LLE: Hunter-Nash Method for Finding the Minimum Solvent to Feed Ratio

$$E_n$$ = extract leaving stage $$n$$. This could refer to the mass of the stream or the composition of the stream.

$$F$$ = solvent entering extractor stage 1. This could refer to the mass of the stream or the composition of the stream.

$$n$$ = generic stage number

$$N$$ = Final stage. This is where the fresh solvent $$S$$ enters the system and the final raffinate $$R_N$$ leaves the system.

$$M$$ = Composition of the overall mixture. Points ($$F$$ and $$S$$) and ($$E_1$$ and $$R_N$$) are connected by a straight line passing through $$M$$.

$$P$$ = Operating point. Every pair of passing streams must be connected by a straight line that passes through $$P$$.

$$R_n$$ = raffinate leaving stage $$n$$. This could refer to the mass of the stream or the composition of the stream.

$$S$$ = solvent entering extractor stage $$N$$. This could refer to the mass of the stream or the composition of the stream.

$$S/F$$ = mass ratio of solvent to feed

$$S_{\rm min}/F$$ = Minimum feasible mass ratio to achieve the desired separation, assuming the use of an infinite number of stages.

$$(x_i)_n$$ = Mass fraction of species $$i$$ in the raffinate leaving stage $$n$$

$$(y_i)_n$$ = Mass fraction of species $$i$$ in the extract leaving stage $$n$$

$$P_{\rm min}$$ = Point associated with the minimum feasible $$S/F$$ for this feed, solvent and (raffinate or extract) composition. $$P_{\rm min}$$ is the intersection of the line connecting points ($$R_N$$, $$S$$) and the line that is an extension of the upper-most equilibrium tie-line.

Determining minimum feasible solvent mass ratio ($$S_{\rm min}/F$$) when (1) feed composition; (2) incoming solvent composition; and (3) outgoing raffinate composition have been specified/selected.

1. Locate points $$S$$ and $$R_N$$ on the phase diagram. Connect with a straight line.
2. Extend the upper-most tie-line in a line that connects with the line connecting points ($$S$$ and $$R_N$$). Label the intersection $$P_{\rm min}$$.
3. Find point $$F$$ on the diagram. Draw a line from $$P_{\rm min}$$ to F and extend to the other side of the equilibrium curve. Label $$E_1$$@$$S_{\rm min}$$.
4. On a fresh copy of the phase diagram, label points $$F$$, $$S$$, $$R_N$$ and $$E_1$$@$$S_{\rm min}$$. Draw one line connecting points $$S$$ and $$F$$ and another line connecting points $$E_1$$@$$S_{\rm min}$$
5. and $$R_N$$. The intersection of these two lines is mixing point $$M$$. Note the composition of species $$i$$ at this location.
6. Calculate

$\dfrac{S_{\rm min}}{F} = \dfrac{(x_i)_F - (x_i)_M}{(x_i)_M - (x_i)_S} \tag{5.1}$

##### Example

We have a 1000 kg/hr feed that contains 30 wt% acetone and 70 wt% water. We want our raffinate to contain no more than 5.0 wt% acetone. What is the minimum mass of pure MIBK required?

## Liquid-Liquid Extraction: Sizing Mixer-settler Units

$$\Phi_C$$ = volume fraction occupied by the continuous phase

$$\Phi_D$$ = volume fraction occupied by the dispersed phase

$$\mu_C$$ = viscosity of the continuous phase (mass time-1 length-1)

$$\mu_D$$ = viscosity of the dispersed phase (mass time-1 length-1)

$$\mu_M$$ = viscosity of the mixture (mass time-1 length-1)

$$\rho_C$$ = density of the continuous phase (mass volume-1)

$$\rho_D$$ = density of the dispersed phase (mass volume-1)

$$\rho_M$$ = average density of the mixture (mass volume-1)

$$D_i$$ = impeller diameter (length)

$$D_T$$ = vessel diameter (length)

$$H$$ = total height of mixer unit (length)

$$N$$ = rate of impeller rotation (time-1)

$$N_{\rm Po}$$ = impeller power number, read from Fig 8-36 or Perry’s 15-54 (below) based on value of $$N_{Re}$$ (unitless)

$$(N_{\rm Re})_C$$ = Reynold’s number in the continuous phase = inertial force/viscous force (unitless)

$$P$$ = agitator power (energy time-1)

$$Q_C$$ = volumetric flowrate, continuous phase (volume time-1)

$$Q_D$$ = volumetric flowrate, dispersed phase (volume time-1)

$$V$$ = vessel volume (volume)

Tank and impeller sizing

$\rm residence time = \dfrac{V}{Q_C + Q_D} \tag{5.2}$

Geometry of a cylinder

$V = \dfrac{\pi D_T^2H}{4} \tag{5.3}$

General guidelines

$\dfrac{H}{D_T} = 1 \tag{5.4}$

$\dfrac{D_i}{D_T} = \dfrac{1}{3} \tag{5.5}$

Impeller power consumption:

$P=N_{Po}N^3D_i^5{\rho}_m \tag{5.6}$

$N_{Re}=\frac{D_i^2N{\rho}_M}{{\mu}_M} \tag{5.7}$

${\rho}_M={\rho}_C{\Phi}_C+{\rho}_D{\Phi}_D \tag{5.8}$

${\mu}_M=\frac{{\mu}_C}{{\Phi}_C}\left[1+\frac{1.5{\mu}_D{\Phi}_D}{{\mu}_C+{\mu}_D}\right] \tag{5.9}$

## Modeling Mass Transfer in Mixer-Settler Units

$$\Delta\rho$$ = density difference (absolute value) between the continuous and dispersed phases (mass volume-1)

$$\phi_C$$ = volume fraction occupied by the continuous phase

$$\phi_D$$ = volume fraction occupied by the dispersed phase

$$\mu_C$$ = viscosity of the continuous phase (mass time-1 length-1)

$$\mu_D$$ = viscosity of the dispersed phase (mass time-1 length-1)

$$\mu_M$$ = viscosity of the mixture (mass time-1 length-1)

$$\rho_C$$ = density of the continuous phase (mass volume-1)

$$\rho_D$$ = density of the dispersed phase (mass volume-1)

$$\rho_M$$ = average density of the mixture (mass volume-1)

$$\sigma$$ = interfacial tension between the continuous and dispersed phases
(mass time-2)

$$a$$ = interfacial area between the two phases per unit volume (area volume-1)

$$c_{D,\rm in}$$, $$c_{D,\rm out}$$ = concentration of solute in the incoming or outgoing dispersed streams (mass volume-1)

$$c^*_D$$ = concentration of solute in the dispersed phase if in equilibrium with the outgoing continuous phase (mass volume-1)

$$D_C$$ = diffusivity of the solute in the continuous phase (area time-1)

$$D_D$$ = diffusivity of the solute in the dispersed phase (area time-1)

$$D_i$$ = impeller diameter (length)

$$D_T$$ = vessel diameter (length)

$$d_{vs}$$ = Sauter mean droplet diameter; actual drop size expected to range from $$0.3d_{vs}-3.0d_{vs}$$ (length)

$$E_{MD}$$ = Murphree dispersed-phase efficiency for extraction

$$g$$ = gravitational constant (length time-2)

$$H$$ = total height of mixer unit (length)

$$k_c$$ = mass transfer coefficient of the solute in the continuous phase (length time-1)

$$k_D$$ = mass transfer coefficient of the solute in the dispersed phase (length time-1)

$$K_{OD}$$ = overall mass transfer coefficient, given on the basis of the dispersed phase (length time-1)

$$m$$ = distribution coefficient of the solute, $$\Delta c_C/\Delta c_D$$ (unitless)

$$N$$ = rate of impeller rotation (time-1)

$$(N_{\rm Eo})_C$$ = Eotvos number = gravitational force/surface tension force (unitless)

$$(N_{\rm Fr})_C$$ = Froude number in the continuous phase = inertial force/gravitational force (unitless)

$$N_{\rm min}$$ = minimum impeller rotation rate required for complete dispersion of one liquid into another

$$(N_{\rm Re})_C$$ = Reynold’s number in the continuous phase = inertial force/viscous force (unitless)

$$(N_{\rm Sh})_C$$ = Sherwood number in the continuous phase = mass transfer rate/diffusion rate (unitless)

$$(N_{\rm Sc})_C$$ = Schmidt number in the continuous phase = momentum/mass diffusivity (unitless)

$$(N_{\rm We})_C$$ = Weber number = inertial force/surface tension (unitless)

$$Q_D$$ = volumetric flowrate of the dispersed phase (volume time-1)

$$V$$ = vessel volume (volume)

Calculating $$N_{\rm min}$$

$\dfrac{N_{\rm min}^2 \rho_M D_i}{g \Delta \rho} = 1.03 \left(\dfrac{D_T}{D_i}\right)^{2.76} (\phi_D)^{0.106} \left(\dfrac{\mu_M^2 \sigma}{D_i^5 \rho_M g^2 (\Delta \rho)^2} \right)^{0.084} \tag{6.1}$

${\rho}_M={\rho}_C{\phi}_C+{\rho}_D{\phi}_D \tag{6.2}$

${\mu}_M=\frac{{\mu}_C}{{\phi}_C}\left(1+\frac{1.5{\mu}_D{\phi}_D}{{\mu}_C+{\mu}_D}\right) \tag{6.3}$

Estimating Murphree efficiency for a proposed design

Sauter mean diameter

${\rm if}\;\; N_{\rm We} < 10,000,\; d_{vs}=0.052D_i(N_{\rm We})^{-0.6}\exp({4{\phi}_D}) \tag{6.4}$

${\rm if}\;\; N_{\rm We} >10,000,\; d_{vs}=0.39D_i(N_{\rm We})^{-0.6} \tag{6.5}$

$N_{\rm We}=\frac{D_i^3N^2{\rho}_C}{\sigma} \tag{6.6}$

mass transfer coefficient of the solute in each phase

$k_D=\frac{6.6D_D}{d_{vs}} \tag{6.7}$

$k_C=\frac{(N_{\rm Sh})_CD_c}{d_{vs}} \tag{6.8}$

$(N_{\rm Sh})_C = 1.237 \times 10^{-5} (N_{\rm Sc})_C^{1/3} (N_{\rm Re})_C^{2/3} (\phi_D)^{-1/2} \tag{6.9}$

$(N_{\rm Fr})_C^{5/12} \left( \dfrac{D_i}{d_{vs}} \right)^2 \left( \dfrac{d_{vs}}{D_T} \right)^{1/2} (N_{Eo})_C^{5/4} \tag{6.9}$

$(N_{\rm Sc})_C=\frac{{\mu}_C}{{\rho}_CD_C} \tag{6.10}$

$(N_{\rm Re})_C=\frac{D_i^2N{\rho}_C}{{\mu}_C} \tag{6.11}$

$(N_{\rm Fr})_C = \dfrac{D_i N^2}{g} \tag{6.12}$

$(N_{Eo})_C = \dfrac{\rho_D d_{vs}^2 g}{\sigma} \tag{6.13}$

Overall mass transfer coefficient for the solute

$\frac{1}{K_{OD}}=\frac{1}{k_D}+\frac{1}{mk_C} \tag{6.14}$

Murphree efficiency

$E_{MD}=\frac{K_{OD}aV}{Q_D}\left(1+{\frac{K_{OD}aV}{Q_D}}\right)^{-1} \tag{6.15}$

$a=\frac{6\phi_D}{d_{vs}} \tag{6.16}$

Experimental assessment of efficiency

$E_{MD}=\frac{c_{D,\rm in}-c_{D,\rm out}}{c_{D,\rm in}-c^*_D} \tag{6.17}$

##### Example

1000 kg/hr of 30 wt% acetone and 70 wt% water is to be extracted with 1000 kg/hr of pure MIBK. Assume that the extract is the continuous phase, a residence time of 5 minutes in the mixing vessel, standard sizing of the mixing vessel and impeller. Find the power consumption and Murphree efficiency if the system operates at $$N_{\rm min}$$, controlled at the level of 1 rev/s. Ignore the contribution of the solute and the co-solvent to the physical properties of each phase.

• MIBK
• density = 802 kg m-3
• viscosity = 0.58 cP
• diffusivity with acetone at 25°C = 2.90×10-9 m2 s-1
• Water
• density = 1000 kg m-3
• viscosity = 0.895 cP
• diffusivity with acetone at 25°C = 1.16×10-9 m2 s-1
• The interfacial tension of water and MIBK at 25°C = 0.0157 kg s-2. Use the ternary phase diagram to find $$m$$.

## Liquid-Liquid Extraction Columns

$$\Delta \rho$$ = density difference (absolute value) between the continuous and dispersed phases (mass volume-1)

$$\mu_C$$ = viscosity of the continuous phase (mass time-1 length-1)

$$\rho_C$$ = density of the continuous phase (mass volume-1)

$$\rho_D$$ = density of the dispersed phase (mass volume-1)

$$\sigma$$ = interfacial tension between the continuous and dispersed phases
(mass time-2)

$$D_T$$ = column diameter (length)

$$H$$ = total height of column (length)

$${\rm HETS}$$ = height of equilibrium transfer stage (length)

$$m^*_C$$ = mass flowrate of the entering continuous phase (mass time-1)

$$m^*_D$$ = mass flowrate of the entering dispersed phase (mass time-1)

$$N$$ = required number of equilibrium stages

$$u_0$$ = characteristic rise velocity of a droplet of the dispersed phase (length time-1)

$$U_i$$ = superficial velocity of phase $$i$$ (C = continuous, downward; D = dispersed, upward) (length time-1)

$$V^*_i$$ = volumetric flowrate of phase $$i$$ (volume time-1)

$U_i = \dfrac{4V_i^*}{\pi D_T^2} \tag{7.1}$

definition of superficial velocity

$\dfrac{U_D}{U_C} = \dfrac{m_D^*}{m_C^*} \left( \dfrac{\rho_C}{\rho_D} \right) \tag{7.2}$

$(U_D + U_C)_{\rm actual} = 0.50(U_D + U_C)_f \tag{7.3}$

for operation at 50% of flooding

$u_0 = \dfrac{0.01 \sigma \Delta \rho}{\mu_C \rho_C} \tag{7.4}$

for rotating-disk columns, $$D_T$$ = 8 to 42 inches, with one aqueous phase

$D_T = \left( \dfrac{4m_D^*}{\rho_D U_D \pi} \right)^{0.5} = \left( \dfrac{4m_C^*}{\rho_C U_C \pi} \right)^{0.5} \tag{7.5}$

$H = \rm HETS * N \tag{7.6}$

##### Example

1000 kg/hr of 30 wt% acetone and 70 wt% water is to be extracted with 1000 kg/hr of pure MIBK in a 2-stage column process. Assume that the extract is the dispersed phase. Ignoring the contribution of the solute and the co-solvent to the physical properties of each phase, find the required column diameter and height.

• MIBK
• density = 802 kg m-3
• viscosity = 0.58 cP
• Water
• density = 1000 kg m-3
• viscosity = 0.895 cP
• The interfacial tension of water and MIBK at 25°C = 0.0157 kg s-2.

This page titled 1.3: Liquid-liquid Extraction is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Monica Lamm and Laura Jarboe (Iowa State University Digital Press) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.