{"id":58,"date":"2019-06-27T18:22:49","date_gmt":"2019-06-27T18:22:49","guid":{"rendered":"https:\/\/libraryresources.nse.org.ng\/chemicalengineeringseparations\/chapter\/distillation\/"},"modified":"2026-03-16T01:13:01","modified_gmt":"2026-03-16T01:13:01","slug":"distillation","status":"publish","type":"chapter","link":"https:\/\/libraryresources.nse.org.ng\/chemicalengineeringseparations\/chapter\/distillation\/","title":{"raw":"Distillation","rendered":"Distillation"},"content":{"raw":"

Introduction to Distillation<\/h2>\n$B$ = mass or molar flow rate of the bottoms stream leaving the systems (mass time-1<\/sup> or mole time-1<\/sup>)\n\n$D$ = mass or molar flow rate of the distillate stream leaving the system (mass time-1<\/sup> or mole time-1<\/sup>)\n\n$F$ = mass or molar flow rate of the feed stream entering the system (mass time-1<\/sup> or mole time-1<\/sup>)\n\n$L$ = mass or molar flow rate of the liquid reflux returned to the column from the condenser (mass time-1<\/sup> or mole time-1<\/sup>); also generic flow rate of the liquid phase in the rectifying section\n\n$\\overline L$ = mass or molar flow rate of the liquid leaving the bottom of the column and entering the reboiler (mass time-1<\/sup> or mole time-1<\/sup>); also generic flow rate of the liquid phase in the stripping section\n\n$n$ = generic stage number, stage 1 is at the top of the column\n\n$R$ = reflux ratio\n\n$V$ = mass or molar flow rate of vapor leaving the top of the column and entering the condenser (mass time-1<\/sup> or mole time-1<\/sup>); also generic flow rate of the vapor phase in the rectifying section\n\n$\\overline V$ = mass or molar flow rate of the gaseous boilup returned to the column from the reboiler (mass time-1<\/sup> or mole time-1<\/sup>); also generic flow rate of the vapor phase in the stripping section\n\n$x$ = mass or mole fraction of the light key in a liquid stream\n\n$x_B$ = mass or mole fraction of the light key in the bottoms stream\n\n$x_D$ = mass or mole fraction of the light key in the distillate stream\n\n$x_n$ = mass or mole fraction of the light key in the liquid leaving stage $n$\n\n$y$ = mass or mole fraction of the light key in vapor stream\n\n$y_n$ = mass or mole fraction of the light key in the vapor leaving stage $n$\n\n$z_F$\u00a0= mass or mole fraction of the light key in the feed stream\n\n \n\nOverall material balance\n\n\\begin{displaymath}\n

\\tag{20.1}<\/p>\n

F=D+B<\/p>\n\\end{displaymath}\n\nMaterial balance on light key\n\n\\begin{displaymath}\n\n\\tag{20.2}\n\nF z_F = x_D D + x_B B\n\n\\end{displaymath}\n\nCombination of material balances in Equations 20.1 and 20.2\n\n\\begin{displaymath}\n

\\tag{20.3}<\/p>\n

D=F\\left(\\frac{z_F-x_B}{x_D-x_B}\\right)<\/p>\n\\end{displaymath}\n\n\\begin{equation}\n

\\tag{20.4}<\/p>\n

R=\\frac{L}{D}<\/p>\n\\end{equation}\n\n\\begin{equation}\n

\\tag{20.5}<\/p>\n

V_B=\\frac{\\overline V}{B}<\/p>\n\\end{equation}\n\nMaterial balance on stages $1-n$, the rectifying section of the column\n\n\\begin{equation}\n

\\tag{20.6}<\/p>\n

y_{n+1}=\\left(\\frac{L}{V}\\right)x_n+y_1-\\left(\\frac{L}{V}\\right)x_0<\/p>\n\\end{equation}\n\nRectifying section operating line\n\n\\begin{equation}\n

\\tag{20.7}<\/p>\n

y_{n+1}=\\left(\\frac{R}{R+1}\\right)x_n+\\left(\\frac{x_D}{R+1}\\right)<\/p>\n\\end{equation}\n\nStripping section operating line\n\n\\begin{equation}\n

\\tag{20.8}<\/p>\n

y_{n+1}=\\left(\\frac{V_B+1}{V_B}\\right)x_n-\\left(\\frac{x_B}{V_B}\\right)<\/p>\n\\end{equation}\n\n \n\n\"\" \"\" \"\"\n

McCabe-Thiele Method for Finding N and Feed Stage Location<\/h2>\n$\\Delta H^{\\rm vap}$\u00a0= enthalpy change of vaporization of the feed stream at the column operating pressure (energy mole-1<\/sup>)\n\n$C_{P_L}$\u00a0= heat capacity of the liquid feed stream (energy mole-1<\/sup> temperature-1<\/sup>)\n\n$C_{P_V}$\u00a0= heat capacity of the vapor feed stream (energy mole-1<\/sup> temperature-1<\/sup>)\n\n$F$ = molar flow rate of the feed stream entering the system (mole time-1<\/sup>)\n\n$L$ = molar flow rate of the liquid phase in the rectifying section (mole time-1<\/sup>)\n\n$\\overline L$ = molar flow rate of the liquid phase in the stripping section (mole time-1<\/sup>)\n\n$L_F$\u00a0= molar flow rate of the liquid portion of the feed stream (mole time-1<\/sup>)\n\n$n$ = generic stage number, stage 1 is at the top of the column\n\n$q$ = metric that reflects the physical state of the feed stream (unitless)\n\n$R$ = reflux ratio\n\n$T_b$\u00a0= bubble-point temperature of the feed stream at the column operating pressure (temperature)\n\n$T_d$ = dew-point temperature of the feed stream at the column operating pressure (temperature)\n\n$T_F$ = temperature of the feed stream (temperature)\n\n$V$ = molar flow rate of the vapor phase in the rectifying section (mole time-1<\/sup>)\n\n$\\overline V$ = molar flow rate of the vapor phase in the stripping section (mole time-1<\/sup>)\n\n$V_F$\u00a0= molar flow rate of the vapor portion of the feed stream (mole time-1<\/sup>)\n\n$x_B$\u00a0= mole fraction of the light key in the bottoms stream\n\n$x_D$ = mole fraction of the light key in the distillate stream\n\n$x_n$ = mole fraction of the light key in the liquid leaving stage\n\n$z_F$\u00a0= mole fraction of the light key in the feed stream\n\n \n

Finding the Theoretical Number of Stages from known Reflux Ratio, Boilup Ratio, Distillate Composition and Bottoms Composition<\/strong><\/h3>\nRectifying section operating line\n\n\\begin{equation}\n

\\tag{21.1}<\/p>\n

y_{n+1}=\\left(\\frac{R}{R+1}\\right)x_n+\\left(\\frac{1}{R+1}\\right)x_D<\/p>\n\\end{equation}\n\nStripping section operating line\n\n\\begin{equation}\n

\\tag{21.2}<\/p>\n

y_{n+1}=\\left(\\frac{V_B+1}{V_B}\\right)x_n-\\left(\\frac{1}{V_B}\\right)x_B<\/p>\n\\end{equation}\n\n\"\" \"\" \"\"\n

Plotting the q-line\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><\/h3>\n\\begin{displaymath}\n

\\tag{21.3}<\/p>\n

F=L_F+V_F<\/p>\n\\end{displaymath}\n\n\\begin{equation}\n

\\tag{21.4}<\/p>\n

q=\\frac{(\\overline L -L)}{F}=1+(\\frac{\\overline V-V}{F})<\/p>\n\\end{equation}\n

For sub-cooled liquid, q > 1<\/p>\n\\begin{equation}\n

\\tag{21.5}<\/p>\n

q=1+\\frac{C_{P_L}(T_b-T_F)}{\\Delta H^{\\rm vap}}<\/p>\n\\end{equation}\n

For a saturated liquid,<\/p>\n\\begin{equation}\n

\\tag{21.6}<\/p>\n

q=1<\/p>\n\\end{equation}\n

For a mixture of liquid and vapor, 0 < q < 1<\/p>\n\\begin{equation}\n

\\tag{21.7}<\/p>\n

q=\\frac{L_F}{F}<\/p>\n\\end{equation}\n

For a saturated vapor, q = 0<\/p>\n\\begin{equation}\n

\\tag{21.8}<\/p>\n

q = 0<\/p>\n\\end{equation}\n

For sub-heated vapor, q < 0<\/p>\n\\begin{equation}\n

\\tag{21.9}<\/p>\n

q=\\frac{C_{P_V}(T_d-T_F)}{\\Delta H^{\\rm vap}}<\/p>\n\\end{equation}\n\nq-line\n\n\\begin{equation}\n

\\tag{21.10}<\/p>\n

y=\\left(\\frac{q}{q-1}\\right)x-\\left(\\frac{1}{q-1}\\right)z_F<\/p>\n\\end{equation}\n\nWatch this two-part video series from LearnChemE<\/a> 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<\/a> (8:21) and McCabe-Thiele Graphical Method Example Part 2<\/a> (6:35).\n\nWatch this video from LearnChemE<\/a> for a conceptual demonstration of how to relate stepping off stages to distillation column design: McCabe-Thiele Stepping Off Stages (7:02):\n\nhttps:\/\/youtu.be\/rlg-ptQMAsg\n\n \n\n \n\n\"\" \"\" \"\" \"\"\n

McCabe-Thiele Method for Finding the Minimum Number of Stages, the Minimum Reflux Ratio, and the Minimum Boilup Ratio<\/h2>\n$\\alpha$ = relative volatility of the light key and the heavy key at a given temperature (unitless)\n\n$\\alpha_F$ = relative volatility of the light key and the heavy key at the feed temperature (unitless)\n\n$\\gamma_{HK}$ = activity coefficient of the heavy key; can be a function of $x$ and\/or $T$; 1 for an ideal solution (unitless)\n\n$\\gamma_{LK}$ = activity coefficient of the light key; can be a function of $x$ and\/or $T$; 1 for an ideal solution (unitless)\n\n \n\n$B$ = molar flow rate of the bottoms leaving the system (mol time-1<\/sup>)\n\n$D$ = molar flow rate of the distillate leaving the system (mol time-1<\/sup>)\n\n$F$ = molar flow rate of the feed stream (mol time-1<\/sup>)\n\n$L$ = molar flow rate of liquid within the rectifying section, assumed constant in McCabe-Thiele model (mol time-1<\/sup>)\n\n$\\overline L$ = molar flow rate of liquid within the stripping section, assumed constant in McCabe-Thiele model (mol time-1<\/sup>)\n\n$N_{t,\\rm min}$ = minimum required number of theoretical stages for a given combination of equilibrium data, $x_D$ and $x_B$\n\n$P_{HK}^{\\rm sat}$\u00a0= saturated vapor pressure of the heavy key at a given temperature, i.e. by Antoine equation (pressure)\n\n$P_{LK}^{\\rm sat}$ = saturated vapor pressure of the light key at a given temperature, i.e. by Antoine equation (pressure)\n\n$q$ = metric that indicates that physical state of the feed stream, i.e. $q$ = 1 for saturated liquid (unitless)\n\n$R$ = reflux ratio = $L\/D$ (unitless)\n\n$R_{\\rm min}$\u00a0= reflux ratio that requires an infinite number of stages in the rectifying section (unitless)\n\n$V$ = molar flow rate of vapor within the rectifying section, assumed constant in McCabe-Thiele model (mol time-1<\/sup>)\n\n$\\overline V$ = molar flow rate of vapor within the stripping section, assumed constant in McCabe-Thiele model (mol time-1<\/sup>)\n\n$V_B$ = boilup ratio = $\\overline V\/B$\n\n$V_{B,\\rm min}$\u00a0= boilup ratio that requires an infinite number of stages in the stripping section (unitless)\n\n$V_F$\u00a0= molar flow rate of the vapor component of the feed stream (mol time-1<\/sup>)\n\n$x_B$\u00a0= target mole fraction of the light key in the bottoms product\n\n$x_D$ = target mole fraction of the light key in the distillate product\n\n$x_{HK}$ = mole fraction of the heavy key in the liquid phase\n\n$x_{LK}$ = mole fraction of the light key in the liquid phase\n\n$y_{HK}$ = mole fraction of the heavy key in the vapor phase\n\n$y_{LK}$ = mole fraction of the light key in the vapor phase\n\n$z_F$ = mole fraction of the light key in the feed stream\n\n \n\n\\begin{equation}\n

\\tag{22.1}<\/p>\n

R_{\\rm min}=\\frac{(L\/V)_{\\rm min}}{1-(L\/V)_{\\rm min}}<\/p>\n\\end{equation}\n\n$(L\/V)_{\\rm min}$ = slope of the line that connects ($x_D$, $x_D$) to the intersection of the q-line and the equilibrium curve\n\n\\begin{equation}\n

\\tag{22.2}<\/p>\n

\\alpha=\\frac{y_{LK}\/y_{HK}}{x_{LK}\/x_{HK}}=\\frac{\\gamma_{LK}P_{LK}^{\\rm sat}}{\\gamma_{HK}P_{HK}^{\\rm sat}}<\/p>\n\\end{equation}\n\n\\begin{equation}\n

\\tag{22.3}<\/p>\n

V_{B,\\rm min}=\\frac{1}{(\\overline L \/\\overline V)_{\\rm max}-1}<\/p>\n\\end{equation}\n

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

\\tag{22.4}<\/p>\n

V_{B}=\\frac{L+D-V_F}{B}=\\frac{D(R+1)-V_F}{B}<\/p>\n\\end{equation}\n

when $q \\leq 0$, $V_F = F$; when $0 < q < 1$, $V_F = (1-q)F$; when $q \\geq 1$, $V_F = 0$<\/p>\n

*we will use Eq 22.4 to calculate $V_B$ as a function of our selected $R$<\/p>\n \n

\n

Example<\/p>\n\n<\/header>\n

\n\n$x_D = 0.80$, $q = 0$, $z_F = 0.25$\n\n$(L\/V)_{\\rm min}$\u00a0=\n\n$R_{\\rm min}$\u00a0=\n\n\"\"\n\n \n\n<\/div>\n<\/div>\n \n
\n

Example<\/p>\n\n<\/header>\n

\n\n$x_D = 0.90$, $x_B = 0.20$\n\n$N_{t,\\rm min} =$\n\n\"\"\n\n \n\n<\/div>\n<\/div>\n

Distillation Energy Demand and Correlations for Efficiency<\/h2>\n$\\alpha$ = relative volatility of the light key and heavy key (unitless). For equation 23.2, this is evaluated at the average column temperature.\n\n$\\Delta H^{\\rm vap}$ = average heat of vaporization for the stream entering the condenser or reboiler (energy mole-1<\/sup>)\n\n$\\Delta H_S^{\\rm vap}$\u00a0= average heat of vaporization for the steam entering the reboiler (energy mole-1<\/sup>)\n\n$\\mu$ = liquid phase viscosity (cP). For eqs 23.1 and 23.2, this is the viscosity of the feed stream at the average column temperature.\n\n \n\n$B$ = bottoms flow rate (mole time-1<\/sup>)\n\n$C_{P,\\rm H_2O}$ = heat capacity of liquid water (energy mole-1<\/sup> temperature-1<\/sup>) or (energy mass-1<\/sup> temperature-1<\/sup>)\n\n$D$ = distillate flow rate (mole time-1<\/sup>)\n\n$E_O$ = stage efficiency (unitless)\n\n$L$ = liquid flow rate in the rectifying section (mole time-1<\/sup>)\n\n$\\overline L$ = liquid flow rate in the stripping section (mole time-1<\/sup>)\n\n$m_{\\rm cw}$ = flow rate of cooling water to condenser (mass time-1<\/sup>) or (mole time-1<\/sup>)\n\n$m_s$\u00a0= flow rate of steam to reboiler (mass time-1<\/sup>) or (mole time-1<\/sup>)\n\n$Q_C$\u00a0= energy demand (cooling) for the condenser (energy time-1<\/sup>)\n\n$Q_R$\u00a0= energy demand (heating) for the reboiler (energy time-1<\/sup>)\n\n$R$ = reflux ratio = $L\/D$ (unitless)\n\n$T_{\\rm in}$ = temperature of cooling water entering the condenser (temperature)\n\n$T_{\\rm out}$ = temperature of cooling water leaving the condenser (temperature)\n\n$V_B$ = boilup ratio = $\\overline V\/B$ (unitless)\n\n$V_F$\u00a0= molar flow rate of the vapor portion of the feed (mole time-1<\/sup>)\n\n \n

Correlations for Stage Efficiency<\/strong><\/h3>\nDrickamer and Bradford\n\n\\begin{equation}\n

\\tag{23.1}<\/p>\n

E_O=13.3-68.8\\log_{10}{\\mu}<\/p>\n\\end{equation}\n

Restrictions on eq 23.1: $\\mu = 0.066 - 0.355$ cP, $T = 157 \u2013 420$\u00b0F, $P = 14.7 \u2013 366$ psia, $E_O = 41 \u2013 88$%<\/p>\nO\u2019Connell\n\n\\begin{equation}\n

\\tag{23.2}<\/p>\n

E_O=\\frac{50.3}{(\\alpha\\mu)^{0.226}}<\/p>\n\\end{equation}\n

when $0.1 \\leq \\alpha \\mu \\leq 1$, adjust $E_O$ calculated by 23.2 with correction factor from Table 7.5<\/p>\n

Restriction on eq 23.2: $\\alpha = 1.16 \u2013 20.5$<\/p>\n\u00a0<\/strong>\n

Condenser and Reboiler Energy Demand<\/strong><\/h3>\ntotal condenser\n\n\\begin{equation}\n

\\tag{23.3}<\/p>\n

Q_C=D(R+1)\\Delta H^{\\rm vap}<\/p>\n\\end{equation}\n\npartial reboiler\n\n\\begin{equation}\n

\\tag{23.4}<\/p>\n

Q_R=BV_B\\Delta H^{\\rm vap}<\/p>\n\\end{equation}\n

for partially vaporized feed $(0 < q < 1)$ and total condenser<\/p>\n\\begin{equation}\n

\\tag{23.5}<\/p>\n

Q_R=Q_C\\left[1-\\frac{V_F}{D(R+1)}\\right]<\/p>\n\\end{equation}\n\n\\begin{equation}\n

\\tag{23.6}<\/p>\n

m_{\\rm cw}=\\frac{Q_C}{C_{P,\\rm H_2O}(T_{\\rm out}-T_{\\rm in})}<\/p>\n\\end{equation}\n\nif using saturated steam for the reboiler\n\n\\begin{equation}\n

\\tag{23.7}<\/p>\n

m_S=\\frac{Q_R}{\\Delta H_S^{\\rm vap}}<\/p>\n\\end{equation}\n

Distillation Packed Column Depth<\/h2>\n$\\rm HETP$ = height of equivalent theoretical plates\n\n$H_{OG} = x$ (length)\n\n$\\lambda$ = local slope of equilibrium curve\/local slope of operating line\n\n\\begin{equation}\n

\\tag{24.1}<\/p>\n

{\\rm HETP}=H_{OG}\\frac{\\ln\\lambda}{\\lambda-1}<\/p>\n\\end{equation}\n

\n

Example<\/p>\n\n<\/header>\n

\n\nWe 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, $R\/R_{\\rm min}$\u00a0of 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?\n