# 1.5: Distillation

- Page ID
- 76362

## Introduction to Distillation

\(B\) = mass or molar flow rate of the bottoms stream leaving the systems (mass time^{-1} or mole time^{-1})

\(D\) = mass or molar flow rate of the distillate stream leaving the system (mass time^{-1} or mole time^{-1})

\(F\) = mass or molar flow rate of the feed stream entering the system (mass time^{-1} or mole time^{-1})

\(L\) = mass or molar flow rate of the liquid reflux returned to the column from the condenser (mass time^{-1} or mole time^{-1}); also generic flow rate of the liquid phase in the rectifying section

\(\overline L\) = mass or molar flow rate of the liquid leaving the bottom of the column and entering the reboiler (mass time^{-1} or mole time^{-1}); also generic flow rate of the liquid phase in the stripping section

\(n\) = generic stage number, stage 1 is at the top of the column

\(R\) = reflux ratio

\(V\) = mass or molar flow rate of vapor leaving the top of the column and entering the condenser (mass time^{-1} or mole time^{-1}); also generic flow rate of the vapor phase in the rectifying section

\(\overline V\) = mass or molar flow rate of the gaseous boilup returned to the column from the reboiler (mass time^{-1} or mole time^{-1}); also generic flow rate of the vapor phase in the stripping section

\(x\) = mass or mole fraction of the light key in a liquid stream

\(x_B\) = mass or mole fraction of the light key in the bottoms stream

\(x_D\) = mass or mole fraction of the light key in the distillate stream

\(x_n\) = mass or mole fraction of the light key in the liquid leaving stage \(n\)

\(y\) = mass or mole fraction of the light key in vapor stream

\(y_n\) = mass or mole fraction of the light key in the vapor leaving stage \(n\)

\(z_F\) = mass or mole fraction of the light key in the feed stream

Overall material balance

\[F = D + B \tag{20.1}\]

Material balance on light key

\[F z_F = x_DD + x_BB \tag{20.2} \]

Combination of material balances in Equations 20.1 and 20.2

\[D = F \left( \dfrac{z_F - x_B}{x_D - x_B} \right) \tag{20.3}\]

\[R=\frac{L}{D} \tag{20.4}\]

\[V_B=\frac{\overline V}{B} \tag{20.5}\]

Material balance on stages \(1-n\), the rectifying section of the column

\[y_{n+1}=\left(\frac{L}{V}\right)x_n+y_1-\left(\frac{L}{V}\right)x_0 \tag{20.6}\]

Rectifying section operating line

\[y_{n+1}=\left(\frac{R}{R+1}\right)x_n+\left(\frac{x_D}{R+1}\right) \tag{20.7}\]

Stripping section operating line

\[y_{n+1}=\left(\frac{V_B+1}{V_B}\right)x_n-\left(\frac{x_B}{V_B}\right) \tag{20.8}\]

## McCabe-Thiele Method for Finding N and Feed Stage Location

\(\Delta H^{\rm vap}\) = enthalpy change of vaporization of the feed stream at the column operating pressure (energy mole^{-1})

\(C_{P_L}\) = heat capacity of the liquid feed stream (energy mole^{-1} temperature^{-1})

\(C_{P_V}\) = heat capacity of the vapor feed stream (energy mole^{-1} temperature^{-1})

\(F\) = molar flow rate of the feed stream entering the system (mole time^{-1})

\(L\) = molar flow rate of the liquid phase in the rectifying section (mole time^{-1})

\(\overline L\) = molar flow rate of the liquid phase in the stripping section (mole time^{-1})

\(L_F\) = molar flow rate of the liquid portion of the feed stream (mole time^{-1})

\(n\) = generic stage number, stage 1 is at the top of the column

\(q\) = metric that reflects the physical state of the feed stream (unitless)

\(R\) = reflux ratio

\(T_b\) = bubble-point temperature of the feed stream at the column operating pressure (temperature)

\(T_d\) = dew-point temperature of the feed stream at the column operating pressure (temperature)

\(T_F\) = temperature of the feed stream (temperature)

\(V\) = molar flow rate of the vapor phase in the rectifying section (mole time^{-1})

\(\overline V\) = molar flow rate of the vapor phase in the stripping section (mole time^{-1})

\(V_F\) = molar flow rate of the vapor portion of the feed stream (mole time^{-1})

\(x_B\) = mole fraction of the light key in the bottoms stream

\(x_D\) = mole fraction of the light key in the distillate stream

\(x_n\) = mole fraction of the light key in the liquid leaving stage

\(z_F\) = mole fraction of the light key in the feed stream

### Finding the Theoretical Number of Stages from known Reflux Ratio, Boilup Ratio, Distillate Composition and Bottoms Composition

Rectifying section operating line

\[y_{n+1}=\left(\frac{R}{R+1}\right)x_n+\left(\frac{1}{R+1}\right)x_D \tag{21.1}\]

Stripping section operating line

\[y_{n+1}=\left(\frac{V_B+1}{V_B}\right)x_n-\left(\frac{1}{V_B}\right)x_B \tag{21.2}\]

### Plotting the q-line

\[F = L_F + V_F \tag{21.3}\]

\[q=\frac{(\overline L -L)}{F}=1+(\frac{\overline V-V}{F}) \tag{21.4}\]

For sub-cooled liquid, q > 1

\[q=1+\frac{C_{P_L}(T_b-T_F)}{\Delta H^{\rm vap}} \tag{21.5}\]

For a saturated liquid,

\[q=1 \tag{21.6}\]

For a mixture of liquid and vapor, 0 < q < 1

\[q=\frac{L_F}{F} \tag{21.7}\]

For a saturated vapor, q = 0

\[q = 0 \tag{21.8}\]

For sub-heated vapor, q < 0

\[q=\frac{C_{P_V}(T_d-T_F)}{\Delta H^{\rm vap}} \tag{21.9}\]

q-line

\[y=\left(\frac{q}{q-1}\right)x-\left(\frac{1}{q-1}\right)z_F \tag{21.10}\]

Watch this two-part video series from LearnChemE that demonstrates how to use the McCabe-Thiele graphical method to determine the number of equilibrium stages needed to meet a specified separation objective: McCabe-Thiele Graphical Method Example Part 1 (8:21): https://youtu.be/Cv4KjY2BJTA and McCabe-Thiele Graphical Method Example Part 2 (6:35): https://youtu.be/eIJk5uXmBRc

Watch this video from LearnChemE for a conceptual demonstration of how to relate stepping off stages to distillation column design: McCabe-Thiele Stepping Off Stages (7:02): https://youtu.be/rlg-ptQMAsg

## McCabe-Thiele Method for Finding the Minimum Number of Stages, the Minimum Reflux Ratio, and the Minimum Boilup Ratio

\(\alpha\) = relative volatility of the light key and the heavy key at a given temperature (unitless)

\(\alpha_F\) = relative volatility of the light key and the heavy key at the feed temperature (unitless)

\(\gamma_{HK}\) = activity coefficient of the heavy key; can be a function of \(x\) and/or \(T\); 1 for an ideal solution (unitless)

\(\gamma_{LK}\) = activity coefficient of the light key; can be a function of \(x\) and/or \(T\); 1 for an ideal solution (unitless)

\(B\) = molar flow rate of the bottoms leaving the system (mol time^{-1})

\(D\) = molar flow rate of the distillate leaving the system (mol time^{-1})

\(F\) = molar flow rate of the feed stream (mol time^{-1})

\(L\) = molar flow rate of liquid within the rectifying section, assumed constant in McCabe-Thiele model (mol time^{-1})

\(\overline L\) = molar flow rate of liquid within the stripping section, assumed constant in McCabe-Thiele model (mol time^{-1})

\(N_{t,\rm min}\) = minimum required number of theoretical stages for a given combination of equilibrium data, \(x_D\) and \(x_B\)

\(P_{HK}^{\rm sat}\) = saturated vapor pressure of the heavy key at a given temperature, i.e. by Antoine equation (pressure)

\(P_{LK}^{\rm sat}\) = saturated vapor pressure of the light key at a given temperature, i.e. by Antoine equation (pressure)

\(q\) = metric that indicates that physical state of the feed stream, i.e. \(q\) = 1 for saturated liquid (unitless)

\(R\) = reflux ratio = \(L/D\) (unitless)

\(R_{\rm min}\) = reflux ratio that requires an infinite number of stages in the rectifying section (unitless)

\(V\) = molar flow rate of vapor within the rectifying section, assumed constant in McCabe-Thiele model (mol time^{-1})

\(\overline V\) = molar flow rate of vapor within the stripping section, assumed constant in McCabe-Thiele model (mol time^{-1})

\(V_B\) = boilup ratio = \(\overline V/B\)

\(V_{B,\rm min}\) = boilup ratio that requires an infinite number of stages in the stripping section (unitless)

\(V_F\) = molar flow rate of the vapor component of the feed stream (mol time^{-1})

\(x_B\) = target mole fraction of the light key in the bottoms product

\(x_D\) = target mole fraction of the light key in the distillate product

\(x_{HK}\) = mole fraction of the heavy key in the liquid phase

\(x_{LK}\) = mole fraction of the light key in the liquid phase

\(y_{HK}\) = mole fraction of the heavy key in the vapor phase

\(y_{LK}\) = mole fraction of the light key in the vapor phase

\(z_F\) = mole fraction of the light key in the feed stream

\[R_{\rm min}=\frac{(L/V)_{\rm min}}{1-(L/V)_{\rm min}} \tag{22.1}\]

\((L/V)_{\rm min}\) = slope of the line that connects (\(x_D\), \(x_D\)) to the intersection of the q-line and the equilibrium curve

\[\alpha=\frac{y_{LK}/y_{HK}}{x_{LK}/x_{HK}}=\frac{\gamma_{LK}P_{LK}^{\rm sat}}{\gamma_{HK}P_{HK}^{\rm sat}} \tag{22.2}\]

\[V_{B,\rm min}=\frac{1}{(\overline L /\overline V)_{\rm max}-1} \tag{22.3}\]

\((\overline L/\overline V)_{\rm min}\) = slope of the line that connects (\(x_B\), \(x_B\)) to the intersection of the q-line and the equilibrium curve

\[V_{B}=\frac{L+D-V_F}{B}=\frac{D(R+1)-V_F}{B} \tag{22.4}\]

when \(q \leq 0\), \(V_F = F\); when \(0 < q < 1\), \(V_F = (1-q)F\); when \(q \geq 1\), \(V_F = 0\)

*we will use Eq 22.4 to calculate \(V_B\) as a function of our selected \(R\)

\(x_D = 0.80\), \(q = 0\), \(z_F = 0.25\)

\((L/V)_{\rm min}\) =

\(R_{\rm min}\) =

\(x_D = 0.90\), \(x_B = 0.20\)

\(N_{t,\rm min} =\)

## Distillation Energy Demand and Correlations for Efficiency

\(\alpha\) = relative volatility of the light key and heavy key (unitless). For equation 23.2, this is evaluated at the average column temperature.

\(\Delta H^{\rm vap}\) = average heat of vaporization for the stream entering the condenser or reboiler (energy mole^{-1})

\(\Delta H_S^{\rm vap}\) = average heat of vaporization for the steam entering the reboiler (energy mole^{-1})

\(\mu\) = liquid phase viscosity (cP). For eqs 23.1 and 23.2, this is the viscosity of the feed stream at the average column temperature.

\(B\) = bottoms flow rate (mole time^{-1})

\(C_{P,\rm H_2O}\) = heat capacity of liquid water (energy mole^{-1} temperature^{-1}) or (energy mass^{-1} temperature^{-1})

\(D\) = distillate flow rate (mole time^{-1})

\(E_O\) = stage efficiency (unitless)

\(L\) = liquid flow rate in the rectifying section (mole time^{-1})

\(\overline L\) = liquid flow rate in the stripping section (mole time^{-1})

\(m_{\rm cw}\) = flow rate of cooling water to condenser (mass time^{-1}) or (mole time^{-1})

\(m_s\) = flow rate of steam to reboiler (mass time^{-1}) or (mole time^{-1})

\(Q_C\) = energy demand (cooling) for the condenser (energy time^{-1})

\(Q_R\) = energy demand (heating) for the reboiler (energy time^{-1})

\(R\) = reflux ratio = \(L/D\) (unitless)

\(T_{\rm in}\) = temperature of cooling water entering the condenser (temperature)

\(T_{\rm out}\) = temperature of cooling water leaving the condenser (temperature)

\(V_B\) = boilup ratio = \(\overline V/B\) (unitless)

\(V_F\) = molar flow rate of the vapor portion of the feed (mole time^{-1})

### Correlations for Stage Efficiency

Drickamer and Bradford

\[E_O=13.3-68.8\log_{10}{\mu} \tag{23.1}\]

Restrictions on eq 23.1: \(\mu = 0.066 – 0.355\) cP, \(T = 157 – 420\)°F, \(P = 14.7 – 366\) psia, \(E_O = 41 – 88\)%

O’Connell

\[E_O=\frac{50.3}{(\alpha\mu)^{0.226}} \tag{23.2}\]

when \(0.1 \leq \alpha \mu \leq 1\), adjust \(E_O\) calculated by 23.2 with correction factor from Table 7.5

Restriction on eq 23.2: \(\alpha = 1.16 – 20.5\)

### Condenser and Reboiler Energy Demand

total condenser

\[Q_C=D(R+1)\Delta H^{\rm vap} \tag{23.3}\]

partial reboiler

\[Q_R=BV_B\Delta H^{\rm vap} \tag{23.4}\]

for partially vaporized feed \((0 < q < 1)\) and total condenser

\[Q_R=Q_C\left[1-\frac{V_F}{D(R+1)}\right] \tag{23.5}\]

\[m_{\rm cw}=\frac{Q_C}{C_{P,\rm H_2O}(T_{\rm out}-T_{\rm in})} \tag{23.6}\]

if using saturated steam for the reboiler

\[m_S=\frac{Q_R}{\Delta H_S^{\rm vap}} \tag{23.7}\]

## Distillation Packed Column Depth

\(\rm HETP\) = height of equivalent theoretical plates

\(H_{OG} = x\) (length)

\(\lambda\) = local slope of equilibrium curve/local slope of operating line

\[{\rm HETP}=H_{OG}\frac{\ln\lambda}{\lambda-1} \tag{24.1}\]

We aim to distill benzene and toluene to a distillate that contains 95 mol% benzene and a bottoms stream that contains 95% toluene. The feed stream is 100 kmol/hr of an equimolar mixture with q = 0.50. We will be operating at 1.0 atm, \(R/R_{\rm min}\) of 1.8 with a packed column containing 25-mm metal Bialecki rings. Assume operating at 70% of the flooding velocity. What depth of packing is needed to achieve this separation?

- For Antoine equation of the form \(\log_{10}{p^*} = A – B/(T+C)\), where \(T\) is in °C and \(p^*\) is in mmHg
- Benzene: \(A = 6.89\), \(B = 1204\), \(C = 220\)
- Toluene: \(A = 6.96\), \(B = 1350\), \(C = 220\)

- 25-mm metal Bialecki rings: \(a=210\), \(\epsilon = 0.956\), \(C_h =0.692\), \(C_p =0.891\), \(C_l =1.461\), \(C_v =0.331\), \(C_s =2.521\)
- Toluene: \({\rm MW} = 92.14\), \(\rho_L = 0.87\) g/mL, \(\mu_L = 0.590\) cP, \(\sigma_L = 27.73\) dyne/cm
- Benzene: \({\rm MW} = 78.11\), \(\rho_L = 0.88\) g/mL, \(\mu_L = 0.652\) cP, \(\sigma_L = 28.88\) dyne/cm
- \(D_L = 1.85*10^{-5}\) cm
^{2}/s (Table 3.4, Seader) - \(D_V = 0.0565\) cm
^{2}/s (estimated via eq 3-36, Seader) - \(\mu_V = 0.0133\) cP, estimated from online gas viscosity calculator (LMNO Engineering) as a function of T (94°C)