A Paper about the Slope of the Equilibrium Line

Transcription

1 A publication of CHEMICAL ENGINEERING TRANSACTIONS VOL. 69, 208 Guet Editor: Eliabetta Brunazz Eva Sorenen Copyright 208, AIIC Servizi S.r.l. ISBN ; ISSN The Italian Aociation of Chemical Engineering Online at A Paper about the Slope of the Equilibrium Line Ro Taylor a,*, Marku u b a Clarkon Univerity, Potdam, New York, USA b Sulzer Chemtech Ltd, Winterthur, Switzerland taylor@clarkon.edu There i no imple definition of the lope of the equilibrium line multicomponent ytem becaue vapor liquid equilibrium i repreented by a urface. Thi paper decribe a method of approximating the lope in multicomponent ytem baed on the new concept the deign component, defined a that compound we eek to eliminate from the mixture in a particular ection of the column. Unlike traditional method, thi new approach can alo be applied to aborber and tripper column. Two analytical approximate method of calculating the deign lope are decribed and hown to be in good agreement with the rigorou deign flah calculation even in cae where the relative volatility i far from contant. Thee imple approximation, which need nothing beyond what i readily available from any column imulation program, appear to be adequate for deign and can be eaily programmed in any preadheet.. Background In the tandard model for point, tray, and ection efficiencie for ditillation in tray column due to Lewi (936 the ection efficiency i related to the tray efficiency by: η ln = ( + η ( λ Tray Section ( ln( λ with the Murphree vapor tray efficiency obtained from: λ η Point e η = λ The point efficiency i found from Tray (2 η Point = e N OG (3 where the overall number of tranfer unit for the ga, N OG, can be expreed a follow: λ = + NOG NG N with m λ = (4 L L / G where i the tripping factor and i the lope of the equilibrium line. Inpection of Equation (-4 how that the important parameter are the number of tranfer unit for ga and liquid phae and the tripping factor. The way in which the efficiency depend on thee parameter i hown in Figure a reproduced from u and Taylor, 208. The performance of packed column often i be expreed uing the HETP which i related to the height of packing by HETP / = H N EQM (5

2 Where N EQM i the number of equilibrium tage needed to accomplih the ame eparation poible in a real packed column of height H. The equilibrium tage model of ditillation and aborption i decribed in a number of textbook (ee, e.g., Seader and Henley, If the equilibrium line may be aumed traight, the HETP may be related to the height of a tranfer unit by ( λ /( HETP = H OG ln λ with HOG = HG + λ HL (6 Figure b how how HETP varie with the tripping factor. In thi figure the value of and are adjuted o that each line ha the ame maximum value of efficiency and minimum value of HETP. Figure : ependence of (a ection efficiency (left and (b HETP (right on the tripping factor. Number and height of tranfer unit adjuted o the maximum or minimum i the ame for each line in the repective plot. Method for etimating the lope of the equilibrium line in multicomponent ytem are baed on adaptation of method propoed for binary ytem, which ue the two key component in a peudo-binary mixture (ee, for example, Lockett, 986. Thee method uffer from one or more drawback:. They ignore the impact of other compound on the thermodynamic behaviour of the mixture, 2. They can lead to improper etimate of the lope (e.g. negative value, 3. They apply only to ditillation, and cannot be ued for aborption or tripping operation. The method we propoe now avoid all of thee drawback. 2. The eign Component 2. efinition and Slope Calculation The deign component i defined to be that component which we eek to eliminate from the column ection for which it i defined. The concept of the deign component can be interpreted a an extenion of the HK/LK methodology to multicomponent ytem with multiple product tream. The lope of the deign component i obtained a follow, where i the equilibrium concentration: m = y * ( x + Δx y Δx * ( x (7 A rigorou procedure to etimate the lope i a follow:. Carry out a bubble point calculation uing, a a bai, the temperature and the compoition of the liquid on the tray in quetion. Thi will provide the equilibrium vapor compoition 2. Add a mall amount Δ to the mole fraction of the deign component Δ 3. Normalize the liquid fraction 4. Carry out a econd bubble point calculation to get Δ 5. Compute the lope from Equation (7 It hould be noted that the econd flah calculation in tep 4 of the above procedure i to be carried out at the ame temperature ued for the flah calculation in tep. Thi method i eaily applied to aborption and tripping where two key component can be difficult or impoible to identify.

3 2.2 Analytical Approximation to eign Component Flah In thi ection we preent two analytical approximation that are eaily programmed in a preadheet, that are capable of good to excellent etimation of the deign component lope and that require no more information than i routinely available from a flowheet imulation package. The vapor liquid equilibrium urface for a multicomponent ytem can be repreented by * i / y c = α i xi S with S = = j j x α j and α i = K i / K (8 It i important to recognize that thi equation i valid at all point on the equilibrium urface regardle of the contancy of the relative volatilitie. The lope of the VLE urface in the primary direction taken by the deign component, m, i given approximately by: m = mα, Δm (9 where, i the lope for ytem where the relative volatilitie are aumed contant, and where i a correction for compoition dependent relative volatilitie. Thee two term are defined by: m x S α, = S 2 (0 S x Δ x = c α k m k x 2 = S xk k ( A complete derivation of thee reult i provided by Taylor and u (208. From the definition of the relative volatilitie we find: α i = x K K x i K αi x (2 To make ue the approximation derived above we need the mole fraction derivative of the K-value. Our propoal i to approximate the mole fraction derivative uing mole fraction and K-value difference. For example: K x i ( K ( x, K x +, + ( K ( x, K x, ( K ( x, K x +, + (3 where i the tage index. In other word, we ue the difference between the K-value and mole fraction on two adjacent tage or, in the third approximation, the two tage either ide of the one of immediate interet denoted by. For the topmot tage it i neceary to ue the downward difference. Converely, for the bottom tage it i neceary to ue the upward difference. The identity of the deign component may change at feed tage and at intermediate product take-off point. It i, however, eential not to change the identity of the component deignated by the ubcript when uing Eq(8 for uch a tage. Other circumtance that require ome care in taking the derivative will be dicued in Taylor and u ( Cae Studie A part of thi invetigation we have looked at a wide range of column deign: a 74 valve-tray C4 plitter, a 54 valve-tray BTX column, a C4-C5 plitter, multiple column from a light end equence (Luyben 203, a idetream column, a imple aborber, a more complicated aborber with multiple feed, a reboiled aborber, and a reboiled tripper. In the latter we have non-ideal ditillation, extractive ditillation (Holland, 98, and azeotropic ditillation procee (Seader and Henley, The model conidered are identified in the dicuion and on the figure by the following acronym: eign Flah: Rigorou method. Eq(7. CRV: Contant relative volatility model, Eq(0. XVRV: ApproXimate Variable Relative Volatility model, Eq(9-3. AVRV: Accurate Variable Relative Volatility model, Eq(9-,2.

4 Here accurate refer to the computation of the K-value derivative, not to the model itelf (which may not alway be better than the other method lited. We illutrate the implication for deign by looking at the impact of the lope on ection efficiency uing Eq(-4. The number of tranfer unit were obtained uing the Chan and Fair correlation and typical value are 2 for the ga phae and 3-6 for the liquid. The ection efficiency ha been etimated auming the number of tranfer unit to remain contant in each column ection (but are not necearily the ame in all ection of a column. The efficiency difference plot (Figure 2 to 4, how the deviation of the outcome of analytical approximation relative to the deign flah lope. 3. epropanizer Our firt example i the depropanizer from the ditillation train decribed by Luyben (203. The pecification were elected to obtain a bottom product that contain no more than 0.% propane and an overhead that ha no more than 0.02% n-butane. The deign component in the rectifying ection i n-butane and for the tripping ection, the deign component i propane. The lope of the equilibrium line computed from the method we are conidering i hown in Figure 2. The CRV model i, in fact, cloer to the deign flah method than either of the approximation that conider K- value derivative. Efficiency dicrepancie alo are hown in Figure 2 (right. For thi column the difference do not exceed 3.5% (and often are much le. Thu, any of the approximate method i adequate for deign purpoe; the CRV model being implet i uggeted here. Figure 2: (a Slope profile (left, and (b efficiency difference (right for depropanizer from Luyben (203. eign component are n-butane in the rectifying ection and propane in the tripping ection. 3.2 Ternary Nonideal itillation Thi ignificantly more challenging example i taken from oherty and Malone (200 who refer to it a the e Roier problem. The tak i to deign a ditillation column to eparate a feed of 20 mol/ methanol, 0 mol/ iopropanol, and 20 mol/ water. The bottom product i to contain no more than 0.5 mol% methanol and the ditillate i to contain at leat 99 mol% methanol but no more than 50 ppm water. Following oherty and Malone, the NRTL model wa ued for the activity coefficient. They etimate the minimum reflux a 5; we ued a value of 8 in thi example and pecified the bottom product rate at 30 mol/. We found that 99 equilibrium tage with the feed to tage 20 were needed to get the ditillate product below 50 ppm water. Since thi i the key pecification, the deign component in the top ection of the column mut be water (and iopropanol in the bottom ection. Slope profile and efficiency difference relative to the deign flah method are hown in Figure 3. The difference between efficiencie etimated with the XVRV model i le than.5%. The AVRV model i barely ditinguihable from the XVRV model (which i why it cannot be een in the figure. The efficiency dicrepancy for the CRV model i le than 2% except for a few tage near the feed where it i between 4 and 5.5%.

5 Figure 3: (a Equilibrium lope profile (left, and (b efficiency difference relative to deign flah in nonideal ditillation proce. eign component are water in the top ection, and methanol below the feed. 3.3 Extractive itillation Extractive ditillation column are more difficult to analye becaue they are likely have multiple ection with correponding change in deign component. The purpoe of thi example, from Holland (98 i to eparate the azeotropic mixture acetone methanol uing water a an entrainer. The deign component are water in the top ection, methanol in the ection between the feed and acetone in the bottom ection. The lope of the equilibrium line and efficiency difference (relative to the deign flah value are hown in Figure 4. With the exception of the CRV model around the feed tage, none of the lope model ha an efficiency delta of more than 5%. The method that take relative volatility derivative into account (XVRV and AVRV rarely differ by more than 2% from the deign flah method. Figure 4: (a Equilibrium lope profile (left, and (b efficiency difference relative to deign flah in extractive ditillation proce. eign component are water in the top ection, methanol between the feed, and acetone in the bottom ection.

6 4. Concluion and Recommendation In thi paper we have propoed a new method for etimating the lope of the equilibrium line for multicomponent ytem baed on differentiation of the phae equilibrium urface with repect to the mole fraction of the deign component. That component i defined a that which we eek to eliminate from the mixture in a particular ection of the column. There i only one deign compound per column ection. The concept can be eaily applied to aborber and tripper column. We have provided two imple approximate analytical formulae that will allow the eay calculation of the deign lope from the reult provided by any column imulation program. The CRV (Locally Contant Relative Volatility model work reaonably well for hydrocarbon ytem (with or without light gae. More interetingly, it alo eem to work well for ome non-ideal ytem where the relative volatility i definitely not independent of compoition. For uch cae the XVRV method generally i uperior a long a we pay attention to the approximation of the K-value derivative. ifference between efficiencie etimated uing the implet analytical approximation and the rigorou deign flah method are rarely more than 5%, and often are le than 2%. We therefore appear to have a traightforward method for etimating the lope that require no pecially programmed flah calculation and which i eaily implemented in any preadheet uing only information available from a tandard flowheet imulation, i.e. equilibrium liquid and vapor compoition of each tage. The deign flah concept ha been ued for the deign of packed column by Sulzer for many year and ha proven to be ucceful. The implified method preented here allow other eaily to ue the method. Acknowledgement We thank Sulzer Chemtech and Clarkon Univerity for having made poible a abbatical of Marku u in 206. Thi paper i baed in part on work carried out during thi abbatical at Clarkon Univerity. All imulation carried out pecifically for thi paper were performed uing ChemSep ( Many of thoe imulation carried out for an early phae of thi work were done by Emily Capek. Therea Scott aited with the calculation for thi paper. Symbol c (- Total number of component H G, H L (m Height of tranfer unit in ga and liquid phae H OG (m Overall height of tranfer unit HETP (m Height equivalent to a theoretical plate m (- Slope of the equilibrium line N G, N L (- Number of tranfer unit in ga and liquid phae N OG (- Number of tranfer unit overall ga x i, y i (- Mole fraction in the liquid, vapour α (- Relative volatility Δm α, (- Slope correction for compoition dependency η (- Efficiency λ (- Stripping factor = m / (L/G Subcript/Supercript Pertaining to deign component j,k,a,b Compound indice Point, Tray, Section enoting type of efficiency * enoting equilibrium Reference oherty M., Malone M.F., 200, Conceptual eign of itillation Sytem, McGraw-Hill, New York u. M, Taylor R., 208, Hidden tie An explanation for O Connell ucce, CEP, in pre Holland C.., 98, Fundamental of Multicomponent itillation, McGraw-Hill, New York Lewi W.K., 936, Rectification of Binary mixture, Ind. Eng. Chem., 28 (4, pp Lockett M.J., 986, itillation tray fundamental, Cambridge Univerity, New York, NY, Luyben W.L., 203, Control of a Train of itillation Column for the Separation of Natural Ga Liquid, Ind. Eng. Chem. Re., 52, Seader J.., Henley E.J., 2006, Separation Proce Principle, 2 nd Ed., Wiley, New York Taylor R., u M., 208, A Paper about the Slope of the Equilibrium Line, Manucript in preparation.