1 Analytic Expressions for Minimum Energy Consumption in Multicomponent Distillation: A Revisit of the Underwood Equations by Ivar J. Halvorsen and Sigurd Skogestad Norwegian University of Science and Technology (NTNU) Department of Chemical Engineering Paper 221g presented at Separations Systems Synthesis Wednesday, November 3, 1999, at 10:00 AM in Obelisk A - Wyndham Anatole AIChE Annual Meeting, Dallas TX 31. Oct - 5. Nov 1999 Email: Sigurd.Skogestad@chembio.ntnu.no, Ivar.J.HAlvorsen@ecy.sintef.no Web: http://www.chembio.ntnu.no/users/skoge http://www.chembio.ntnu.no/users/ivarh NTNU Department of Chemical Engineering AIChE Annual Meeting 1999, Dallas TX, 31. October-5. November 2 Motivation 1. Multicomponent separation can be difficult to understand 2. Extend Underwood’s minimum reflux calculations to the entire operat- ing range of product splits D/F and feasible component distribution 3. This is needed for integrated column sequences, e.g. Petlyuk columns Main results: 1. Simple graphical visualization of minimum energy for all possible prod- uct splits, just based on feed data. 2. Obtain minimum energy for the Petlyuk column directly from the same diagram. NTNU Department of Chemical Engineering AIChE Annual Meeting 1999, Dallas TX, 31. October-5. November
3 Result 1: D Visualize V min =f(D,splits) x d, r d V/F Top section V L ABC AB F, z, q A C BC ABC A,B,C AB ABC ABC Bottom BC ABC section x b, r b ABC B D/F NTNU Department of Chemical Engineering AIChE Annual Meeting 1999, Dallas TX, 31. October-5. November 4 Result 2: A Find V min for the Petlyuk column directly from the diagram: 3 AB A C V/F BC - - - - - - - - - - - - - - 4 AB /C V min - - - - - - - - - - - - - - A /BC V min 1 ABC B 2 5 6 Petlyuk V min D/F C A / BC V min Petlyuk AB / C ( , ) V min = max V min NTNU Department of Chemical Engineering AIChE Annual Meeting 1999, Dallas TX, 31. October-5. November
5 Selected references: Classical references for multicomponent distillation • Underwood (1946, 1948a,b), Fractional distillation of multicomponent mixtures • Shiras (1950), Calculation of Minimum Reflux in Distillation Columns • Franklin, Forsyth (1953), The interpretation of minimum reflux conditions in mul- ticomponent distillation Minimum energy expressions for Petlyuk arrangements • Fidkowski, Krolikowski (1986), Thermally Coupled Columns: Optimization proc. • Carlberg, Westerberg (1989) Temperature-Heat Diagrams for Complex Col- umns. 3. Underwood’s Method for the Petlyuk Column. Books: • King (1980), Separation Processes. • Stichlmair (1998), Distillation: Principles and Practice. NTNU Department of Chemical Engineering AIChE Annual Meeting 1999, Dallas TX, 31. October-5. November 6 Important: The two product distillation column has only two degrees of freedom (DOF) • Binary mixtures: #DOFs == #components: ==> We may specify a product completely • Multicomponent: #DOFs < #components ==> We cannot specify all components in a product, just two! Nice Implication of just two DOFs: • Visualize the entire operating range in 2 dimensions. • We choose the D-V plane NTNU Department of Chemical Engineering AIChE Annual Meeting 1999, Dallas TX, 31. October-5. November
7 Revisit of Underwood’s Equations Starting points: 1. Net transport of a component through a stage (1) w i = V n y i n – L + x i n , , + 1 n 1 (w is defined positive upwards) y i,n+1 Stage n+1 x i,n+1 w i L n+1 V n w i is constant in a section: y i,n Stage n x i,n Assume: L n V n-1 -constant molar flows -constant relative volatility α i x i 2. Vapour liquid equilibrium (VLE): y i = - - - - - - - - - - - - - - - - - - ∑ α i x i i NTNU Department of Chemical Engineering AIChE Annual Meeting 1999, Dallas TX, 31. October-5. November 8 3. Divide the material balance with V, multiply with the “Underwood” factor α i ⁄ ( α i φ ) , and take the sum over all components: – 2 x i n α i , ∑ - - - - - - - - - - - - - - - - - - - ( α i φ ) α i w i α i x i n – 1 ∑ L ∑ , i + 1 (2) - - - - - - - - - - - - - - - - - - - - - - - = - - - - - - - - - - - - - - - - - - - - - - - - - - - – - - - - - - - - - - - - - - - - - - - - - - - - - - - ( α i φ ) ( α i φ ) ∑ V – V – α i x i n , i i i 4. The solutions for φ which sets the left-hand side equal to one defines the Underwood roots: α i w i ∑ Definition equation: , (3) V = - - - - - - - - - - - - - - - - - - - ( α i φ ) – i α i r i D α i x i D z i , , ∑ ∑ (for the top section we can also use V T ) = - - - - - - - - - - - - - - - - - - - - F = - - - - - - - - - - - - - - - - - - - D ( α i φ ) ( α i φ ) – – i i Note the relations: for the top section, and w i = w i T = w i D = Dx i D = r i D z i F , , , , the bottom section: , where the recoveries ( ) x i B ( ) z i F w i = w i Botoom = w i B = – B = – r i B , , , , Dx i D w i D Bx i B – w , , , , i B and , and we trivially have r i D = - - - - - - - - - - - - - - - = - - - - - - - - - - - r i B = - - - - - - - - - - - - - - = - - - - - - - - - - - - - - r i B + r i D = 1 , , , , Fz i Fz i Fz i Fz i NTNU Department of Chemical Engineering AIChE Annual Meeting 1999, Dallas TX, 31. October-5. November
9 5. Simplify to: φ E x n φ ( , ) α i x i n L ∑ , ( , φ ) where E x n φ ( , ) (4) - - - - E x n = - - - - - - - - - - - - - - - - - - - - - - - - = - - - - - - - - - - - - - - - - - - - ( α i φ ) + 1 ∑ V – α i x i n , i i 6. Can derive by division of equations : m E x n φ j ( , φ k ) φ k ( , ) E x n x D x B + m (compare to α N for binary) (5) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - = - - - - - - - - - - - - - - - ( , φ j ) φ j E x n φ k ( , ) E x n 1 – x D 1 – x B + m ∑ Together with , we have now N c equations in order to compute x n+m = 1 x i n , + m i Underwood equations can be used to relate the composition on one stage to a composition on another stage in a multicomponent separation. Minimum energy computations: → ∞ m NTNU Department of Chemical Engineering AIChE Annual Meeting 1999, Dallas TX, 31. October-5. November 10 Underwood equations for the two product column: α i w i T ∑ , Definition of UW-roots in the top section: (6) V T V T = - - - - - - - - - - - - - - - - - - - ( α i φ ) – i F z,q V B α i w i B ∑ , Definition of UW-roots in the bottom section: (7) V B = - - - - - - - - - - - - - - - - - - - - ( α i ψ ) – V T -V B =(1-q)F i Note that and normally (8) w i B = w i D – w i F w i F = z i F , , , , α 1 > φ 1 > α 2 > φ 3 > …… > α Nc > φ Nc For the roots obey: (9) > < w i D 0 and w i B 0 , , ψ 1 > α > ψ 2 > α 2 > ψ 3 > …ψ Nc > α Nc – 1 1 Note the difference between the roots in the top ( ) and in the bottom ( )! φ ψ NTNU Department of Chemical Engineering AIChE Annual Meeting 1999, Dallas TX, 31. October-5. November
11 Underwood roots in the top and bottom approach the common roots ( ϕ ) as vapour flow is reduced V φ Nc ψ 3 φ 2 ... ψ 1 ψ Nc ψ 2 φ 1 ... α Nc ϕ Nc-1 α 3 ϕ 2 α 2 ϕ 1 α 1 < < < < < < Minimum energy: → ⇔ φ i → ψ i → ϕ i V V min + 1 (Infinite energy: → ∞ ⇒ φ i → α i and ψ i → α i .) V NTNU Department of Chemical Engineering AIChE Annual Meeting 1999, Dallas TX, 31. October-5. November 12 Underwood’s minimum reflux result Underwood roots in the top and bottom sections are defined by: α i w i T α i w i B , , ∑ ∑ V T and V B where (10) = - - - - - - - - - - - - - - - - - - - = - - - - - - - - - - - - - - - - - - - - w i B = w i T – z i F , , ( α i φ ) ( α i ψ ) – – i i As the vapour flow is reduced, the roots in the top section decrease, and the roots in the bottom section increase. Minimum reflux occur when the roots coincide ⇔ φ i ψ i ϕ i (11) V min = = + 1 Recall that V T -V B =(1-q)F where q is the liquid fraction in the feed. By subtracting equation from we obtain the well known Underwood’s “feed” equation: α i z i ∑ ( ) (12) 1 – q = - - - - - - - - - - - - - - - - - - - ( α i ϕ ) – i N c -1 common roots obey: α 1 > ϕ 1 > α 2 > ϕ 2 > …ϕ Nc > α Nc – 1 Feed equation is only valid for the active common roots, but fortunately it can be solved once for all the N c -1 potential common roots, and these depend only on the feed properties: α z q , , . The actual active common roots depends on the operation. NTNU Department of Chemical Engineering AIChE Annual Meeting 1999, Dallas TX, 31. October-5. November
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