New Design Method for Fully Thermally Coupled Distillation Column

Article pubs.acs.org/IECR

New Design Method for Fully Thermally Coupled Distillation Column Using Group and Approximate Methods Hosanna Uwitonze,† Sangil Han,‡ and Kyu Suk Hwang*,† †

Department of Chemical Engineering, Pusan National University, San 30 Jangjeon-dong, Kumjeong-gu, Pusan 609-735, Korea Department of Chemical Engineering, Changwon National University, Changwon, Gyeongsangnam-do 641-773, Korea

‡

S Supporting Information *

ABSTRACT: A new design method for determining the structure of a fully thermally coupled distillation column is proposed. The method involves approximate group method and the Fenske equation. To design the prefractionator, an approximate calculation procedure that relates compositions of multicomponent vapor and liquid streams is developed to formulate the expressions used to compute the number of stages. To investigate the usability and performance of this design method, different feed mixtures are tried. The results indicate that the method works well for a variety of process conditions; the results show that the structure design from group method leads to a process similar to a feasible actual process. The structure design results and operating conditions for the whole column system enable the examination of the optimal internal flows and their effects on the structure design equations. The results show that the proposed design method provides good initial values for rigorous simulation.

1. INTRODUCTION To design a fully thermally coupled distillation column (FTCDC), information about design requirements (operational conditions) is highly required. Information about operational conditions, such as the minimum reflux flow rate, has been widely investigated.1 Because the columns comprising a fully thermally coupled distillation column system are interlinked, prefractionator connected to main column, typical multicomponent design procedures are not directly applicable to the design of the column system when the information on the interlinking streams is not given. Despite the remarkable benefits of FTCDCs, its complex operation has prevented their commercial application, and its design method is still an area of ongoing research. Using only the operation conditions without the structural information leads to tedious iterative simulation to find a proper structure of the column. Triantafyllou and Smith2 proposed a FTCDC design method using a three-column model; they applied the conventional Fenske−Underwood−Gilliland shortcut design technique in each column. The method provides a good basis for investigating the degrees of freedom and the number of trays in an easy manner; however, it requires trial-and-error steps for matching the compositions of the interlinking streams. Fidkowski et al.1 studied the Petlyuk column at minimum reflux ratio, but column structure design was not taken into account. However, the results of their work are very interesting for the Petlyuk column design. The authors found an analytical solution of the Petlyuk column optimal operation, assuming ideal mixture, constant relative volatility, constant internal flow rates throughout the column, and separation into almost pure components. Amminudin et al.3 developed a semirigorous method for the initial design of an FTCDC based on the concept of equilibrium stage composition. In their study, the FTCDC was divided into two separate columns to eliminate interlinking © XXXX American Chemical Society

and obtain an optimal initial design that could be confirmed through rigorous simulation. Agrawal and Fidkowski4 have simplified the FTCDC structure design by eliminating one interconnection between prefractionator and main column. Subsequently, Kim 5 proposed a rigorous design procedure; the structure of the column system is based on liquid composition calculation in which the actual stage number is set to twice the minimum stage number. Later on, Kim6 proposed a semirigorous model based on the limiting requirements for minimum reflux and minimum number of stages; the procedure starts with feed and side product compositions and liquid flow rates in prefractionator and main column to establish the design parameters, such as column structure (number of stages), to meet the product specifications. Lee et al.7 proposed a design method from which internal sections of the column system are divided into six separate sections and matched to the sloppy arrangement with three conventional simple columns. Initially, the authors applied the shortcut design procedure. Long and Lee8 have used response surface to simultaneously design (find column structure) and optimize the column system. For the simplicity of the column construction, a divided wall structure is preferred and has been adopted in many studies.9−13 The divided wall column structure adds complexity in the design as the number of stages in the prefractionator has to be the same as or close to the number of stages in the midsection of the main column. With the aim of improving the operability of FTCDC systems, Lee et al.14 have suggested a modification of column Received: October 23, 2013 Revised: July 2, 2014 Accepted: July 10, 2014

A

dx.doi.org/10.1021/ie403575x | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

Article

system by splitting the main column into two columns from the side draw tray. Hwang et al.15 proposed another scheme of splitting the main column into three columns: the section between upper interlinking tray and the overhead, the midsection of the main column, and the section between the lower interlinking tray and the bottom. These modifications allow easy manipulation of interlinking streams. In this study, a structure design method is proposed for determining the structure of FTCDC, including the location of the feed tray, side draw tray, interlinking trays, and the total number of trays in each section of the main column and prefractionator. The proposed method combines the approximate group method and Fenske design equation. Prefractionator is considered as two column sections (rectifier and stripper) coupled around a feed tray. In this case study, focus is on developing an approximate calculation procedure that relates composition of multicomponent vapor and liquid streams entering and leaving the cascade to formulate an expression or formula to compute the number of stages for each column section. The performance of the proposed design method is tested using ten ternary feed systems classified according to characterization of mixtures through the easy separability index (ESI) and composition of key components. The article is organized as follows: Section 2 discusses the approximate group methods (Kremser group methods). Section 3 discusses column system design method, namely model design and assumptions, column operating conditions are detailed in here. Validation of the design method is undertaken in section 4. Section 5 is devoted to results and discussions, and conclusions are presented in section 6.

V1, iy1, i = VN + 1, iyN + 1, i ⌀A, i + L0, ix0, i(1 − ⌀S, i) ⌀A , i =

(1)

VN + 1HNV + 1 + L0H0L = V1H1V + LN HNL

(2)

Ae,Ni + 1

−1

;

⌀S, i =

Se, i − 1 Se,Ni + 1 − 1

(4)

Equation 1 was originally derived by Kremser for the design of absorbers where it was written only for the key component.16 The ⌀A,i and ⌀S,i in eq 3 denote the recovery fraction for species i for absorption and stripping, respectively, and are given in eq 4. Ae,i denotes effective absorption factor, and Se,i denotes effective stripping factor. To simplify calculations, the assumptions were made where pressure, oil rate, gas rate, and temperature were kept constant throughout the absorber. Kremser proposed the following three approximations: L1 ≈ L0

(5)

VN ≈ VN + 1

(6)

TN ≈

T0 + TN + 1 2

(7)

17

Edmister applied the Kremser group method for absorbers and strippers to distillation where two cascades are coupled to a condenser, reboiler and a feed stage. The recovery equation for each species in the enricher is given by eq 8, in which the component feeds to the enricher section are vapor (vF) from the feed stage. d is a stream passing up to the top of the cascade. Similarly, the recovery equation for each species in the exhauster is given by eq 9, in which the component fed to exhauster section are liquid (lF) from the feed stage. b is a stream passing down to the bottom.

2. KREMSER GROUP METHOD Countercurrent cascades, from which group methods are derived, are used extensively for vapor−liquid separation operations, including absorption, stripping, and distillation. For absorption and stripping, a single-section cascade is used to recover one selected component from the feed. For distillation, a two-section cascade is effective in achieving a separation between two selected components referred to as key components. For both cases, approximate calculation procedures relate compositions of components in vapor and liquid streams entering and exiting the cascade to the number of equilibrium stages required. These approximate methods provide only an overall treatment of the group of stages in the cascade without considering detailed changes in temperature, phase compositions, and flows from stage to stage. 2.1. Single-Section Cascade. Kremser presented a mathematical analysis of the relations of the oil absorption process.16 Group methods were originally devised for simple hand calculations that are performed in an iterative manner. The fundamental equations for group methods are the component mole balances for a vapor stream entering the bottom, a liquid stream entering the top, and liquid and vapor streams leaving (eq 1), and energy balance, eq 2. These equations are written around a complete set of trays forming a cascade. For a component appearing in entering streams, vapor, and liquid, the performance equation for absorber is given by eq 3. VN + 1, iyN + 1, i + L0, ix0, i = V1, iy1, i + LN , ixN , i

Ae, i − 1

(3)

A ⌀ +1 vF = C SE d ⌀AE

(8)

S ⌀ +1 lF = B AX b ⌀SX

(9)

Kamath et al.18 have investigated the potential of group methods for process simulation and optimization; they proposed further improvements specifically for modeling distillation columns in which they replaced Kremser approximations (eqs 5−7) by more realistic constraints based on physical insights underlying principles of distillation. As a result, the improved group method showed more accurate predictions. There are other group methods that can be used to obtain simple estimates and to check more rigorous computations.19 The group methods formulated for absorption, stripping, and distillation16−18 require that the number of stages should be known first; as we expect to find the number of stages based on stream compositions at both ends of prefractionator sections, the previous formulas as elaborated cannot be used directly. Hence, the necessity of new formulations. A brief schematic diagram of FTCDC is shown in Figure 1. Because a thermally coupled distillation column system has two interlinking streams between the prefractionator and main column, the conventional multicomponent column design procedure is not directly implemented unless the compositions of the interlinking streams are given. The design of the thermally coupled distillation column can be modeled through superstructures suitable for optimization procedures with mathematical programming techniques.20 However, the task is achieved at the expense of the significant computing times required and is likely to fail because of numerical issues. For a B



Article

energy consumption. Finally, the design is successful if it meets the product specifications as set by the designer. 3.1. Model Design and Assumptions. First, we consider that the column would not be necessarily symmetric (i.e., numbers of trays for prefractionator and midsection of the main column are not necessarily equal). To calculate the number of trays in the prefractionator, attention is paid to the composition of the feed and top and bottom products. The formulations are based on a component net flow model so that the composition in each section of the prefractionator can be calculated straightforwardly. Holland et al.21 presented all possible mole net flow models for divided wall columns (DWC) systems; later, Later Chu et al.22 proposed a retouch by evolving the total mole net flow model into a component net flow model. Figure 2 shows the component net flow model in which the

Figure 1. Schematic diagram of fully thermally coupled distillation column.

multicomponent mixture, the problem is clearly more complicated as the combinatorial nature of the system results in a superstructure that is significantly more complex to solve. In this work, the thermally coupled distillation column is divided into six column sections. The prefractionator that splits the feed into two products includes two column sections, 1_1 and 1_2, which are coupled around the feed stage. There are also four sections of the main column: column sections 2 and 3_1 are coupled around the upper interlinking stage and 3_2 and 4 are coupled around the lower interlinking stage. The structure of the prefractionator, open at both its ends (no condenser or reboiler attached to it), inspired us to figure out a new and suitable design method involving the entering and leaving streams.

3. DESIGN PROCEDURE For the sake of alleviating the complexity of the simultaneous solution of tray arrangement and energy consumption within a formal optimization algorithm, the design is approached as follows: (1) column structure design and (2) column system optimization. For the first step, it begins with the structure design of the column system based on the proposed procedure; then, simulation follows. Second, after the design of the column structure and simulation, the column system is optimized for the sake of minimizing heat duty supplied into the reboiler of the column system, taking into account the constraints imposed by the required purities of the product streams. At steady state, FTCDC has five degrees of freedom; after using the design specifications, the remaining degrees of freedom are utilized to obtain the proper values of the interconnecting vapor and liquid streams, which provide minimum energy consumption. The optimization of the interconnecting streams is exercised by formulating a cost function that includes column utility requirements for the reboiler and condenser. The search procedure provides an insight into how interconnecting streams affect the design equations and operability of the column system in terms of

Figure 2. Component net flow model for FTCDC.

column is divided into five sections. Section 1 corresponds to the prefractionator. Section 2 is a rectifying section, and section 4 is a stripping section. Section 3_1 is the upper midsection of the main column, and section 3_2 is the lower midsection of the main column. From the feed systems considered, A is the lightest component, B the middle component, and C the heaviest component. The top product in column section 1_1 is mostly A and B with a little C, and the bottom product is mostly B and C with a little A. Assumptions: it is assumed that the net f low of species C at the top of section 1 is equal to the f low of species C in the side draw; it is also assumed that the net f low of species A at the C



Article

Figure 3. Column system design procedure.

3.2. Column Operating Conditions. A brief schematic diagram of FTCDC is shown in Figure 1. Except the feed stream, output streams from the prefractionator feed the main column while the vapor and liquid flows going into the prefractionator are delivered from the main column. 3.2.1. Operating Requirements. Normally, the design of a distillation column begins with either the estimation of operational variables, such as liquid and vapor flow rates, or the computation of structural information. In this study, the operational variables are estimated first because we expect to use a method which relies on them for structural design of the prefractionator for which the internal flows have to be preestimated. The following equations serve to estimate the operating conditions of the prefractionator.1

bottom of section 1 is equal to the f low of species A in the side draw. Chu et al. used these assumptions for the design of DWC.22 If the feed flow rate (F), feed composition (zA, zB, zC), and product purity specifications (xA, xB, xC) are known, the aforementioned assumptions allow a designer to determine approximate values of the composition of key components at the ends of the prefractionator. Considering the schematic diagram shown in Figure 2, it is necessary to define the fraction of component B in the top product of the prefractionator. βP is the value of the recovered fraction of component B in the prefractionator top product while vapor boilup in the prefractionator is a minimum, which corresponds to minimal energy consumption.1 The following equations approximate the compositions at points D1 and B1. α − αC βP = B αA − αC (10) xA, D1 = x B, D1 =

D1

x B, B1 =

(11)

(12) (13)

SXA, S B1

(14)

(1 − βP)Z BF B1

xC, B1 = 1 − xA, B1 − x B, B1

(17)

(18)

where α’s are relative volatilities and ⌀’s are the solutions of the Underwood equation for saturated liquid feed; βP is the fraction of intermediate component in the top of the prefractionator. Minimum liquid flow and minimum vapor flow rates for prefractionator are given by eqs 17 and 18, respectively. For initialization of rigorous design of FTCDC, initial operating conditions are highly required; the minimum liquid flow rate in the main column is estimated with eq 19; this is the total liquid flow rate for the main column and prefractionator. Vapor boilup rate is found from the reflux flow rate and overhead product rate.1

βPZ BF D1

αCF αA − αC

⎤ ⎡ A⌀ αBB 2 VPmin = max⎢ + βP⎥ αB − ⌀2 ⎦ ⎣ αA − ⌀2

ZAF − SXA, S

xC, D1 = 1 − xA, D1 − xB , D1 xA, B1 =

L Pmin =

(15)

⎧ A ⌀1 αBB ⎫ A ⌀2 ⎬ + L Dmin = max⎨ , αB − ⌀2 ⎭ ⎩ αA − ⌀1 αA − ⌀2

(16)

From Figures 1 and 2, note that at the top of prefractionator, V2 and LP cross paths; thus, they are combined to form a net flow D1. VP and L3 are also combined to form net flow B1.

(19)

Column system structure design is discussed in the following section. D



Article

3.3. Column Structure Design. Structural design of FTCDC is undertaken in two steps: The structure design of the prefractionator is determined first. Second, the structure design of the main column is determined. As mentioned in section 3.1, for structure design of the prefractionator, possible combinations of the four design variables (the LK distribution into the prefractionator bottom, the HK distribution into prefractionator distillate, the LK specification into side product, and the HK specification into side product) were searched to fulfill the purity specifications of products A, B, and C. Therefore, without any restriction (specification) to prefractionator upper and bottom products, there exist hundreds of possible feasible specification sets to meet a given product specification and hundreds of corresponding column structure sets. Figure 3 summarizes the steps followed during column system design. 3.3.1. Prefractionator Column Structure Design. According to the envisaged design method, first we used eqs 10−16 to get an idea of the compositions of the streams leaving the prefractionator column, and through bubble and dew point calculations23 we calculated temperatures of the streams and the K-values for each component at the top and bottom of the prefractionator. The K-values are then used to calculate stripping and rectifying factors for the stripping and rectifying sections of the prefractionator. Figure 4a depicts the prefractionator sections, namely, the feed stage, two sections coupled around the feed stage, and

component vaporizes and diffuses from liquid phase to gas phase. Step 1. For section 1_1, the heavy key component material balance around the top stage, Figure 4b, “mth tray”, is VTyH, m + 1 + L TKH−1yH, m − 1 = L TKH−1yH, m + VTyH, m

yH, m + 1 =

L TKH−1 L K −1 yH, m + yH, m − T H yH, m − 1 VT VT

(20)

(21)

Repeated use of eq 21 for a group of M stages gives the following formula: 1 [y (aHM − 1 − 1) − yH, m − 1(aHM − 1 − aH)] yH, M = aH − 1 H, m (22)

The number of stage M is counted from the top toward the bottom of the column section. Equation 22 includes two consecutive vapor compositions of the two consecutive trays on the top of the prefractionator. From the same equation, yH,m is vapor composition of heavy key component exiting from the top stage and yH,m−1 is vapor composition of heavy component above the top stage which is in liquid vapor equilibrium with the liquid from the respective stage. yH,M is the vapor composition of heavy key component entering through the bottom (feed stage); aH is a rectifying factor corresponding to the heavy key component, aH = LTKH−1/VT > 1. Rearranging eq 22 gives 1 yH, m − 1 = M − 1 [y (aHM − 1 − 1) − yH, F (aH − 1)] aH − aH H, m (23)

Equation 23 principally implies rectifying factor, which in turn implies the flows flowing into the column section. The computation of yH,m−1 is explained in section S.1.2 of Supporting Information. For the case where two or more components are absorbed all along the absorption section, the component with the K-value that yields the smallest absorption factor is considered because when used in the design equation this yields the largest number of stages. During the evaluation of rectifying factor implied in eq 23, it is assumed that the Kvalue is constant all along the cascade. Further rearrangement of eq 23 gives

M=

⎡ (y − aHy ) + y (aH − 1) ⎤ ln⎢ H,m (y H,m−−1 y H,)F ⎥⎦ ⎣ H, m H, m − 1 ln aH

+1

(24)

Step 2. Likewise, the light component material balance around the bottom stage of section 1_2, Figure 4c, “nth tray” is L Bx L, n + 1 + VBKLx L, n − 1 = VBKLx L, n + L Bx L, n x L, n + 1 =

Figure 4. Schematic diagram of prefractionator: (a) prefractionator, (b) rectifying section, and (c) stripping section.

x L, N =

VBKL VK x L, n + x L, n − B L x L, n − 1 LB LB

(25)

(26)

1 [x L, n(s LN − 1 − 1) − x L, n − 1(s LN − 1 − s L)] sL − 1 (27)

entering and leaving streams for each section. The section above the feed stage refers to the rectifying section, whereas the section below refers to a stripping section. In column section 1_1, the components that do not appear in top product are absorbed; here, most of the heavy key component is condensed and diffuses from gas phase to liquid phase while the light key

Repeated use of eq 26 for a group of N stages gives eq 27. Equation 27 includes two consecutive liquid compositions of the two consecutive trays at the bottom of the prefractionator. From the same equation, xL,n refers to liquid composition for light key component leaving the bottom stage; xL,n−1 is the E



Article

operating efficiently, section 2 mainly deals with separation of light component A from middle component B. For section 2, a separation factor is defined and plugged into eq 32 along with relative volatility of the components to be separated in order to find a minimum number of stages. This Fenske expression does not assume constant molar flows and applies to separation between any two components with constant relative volatility, where the common rule of thumb is adopted to fix the actual number of stages.

liquid composition of light key component from a stage below the bottom stage, and xL,N is the liquid composition for the light component entering the column section through the top (feed stage). sL is a stripping factor corresponding to the light key component, and sL = VBKL/LB > 1. Rearranging eq 27 gives x L, n − 1 =

1 [x L, n(s LN − 1 − 1) − x L, F(s L − 1)] − sL

s LN − 1

(28)

The computation of xL,n−1 is explained in section S.1.4 of Supporting Information. Further rearrangement of eq 28 gives

N=

⎡ (x − s x ) + x (s − 1) ⎤ ln⎢ L,n (Lx L,n−−1 x L,)F L ⎥⎦ ⎣ L, n L, n − 1 ln s L

+1

(N2)min =

(N3_2 + N4)min =

ln S2,3_1 ln αij ln S3_2,4 ln αij

(32)

Likewise, for the section column between the lower interlinking stage and reboiler, a separation factor is defined and plugged into eq 33 along with relative volatility of the components to be separated in order to find the minimum number of stages.

(29)

If two or more components do not appear in the prefractionator bottom end product, the component with Kvalue that yields the smallest stripping factor is considered because when used in the design equation this yields the largest number of stages. From eqs 24 and 29, the number of equilibrium stages depends on the degree of separation of the key components, feed-phase condition, and their corresponding rectifying and stripping factors. These factors imply the K-values of the key components. On the basis of eq 20 or 25, one stage is equivalent to the equilibrium-flash equation. Equations 24 and 29 are quite reliable except when the K-values and liquid and vapor flows vary appreciably throughout the column section and/or when the mixture forms nonideal liquid solutions. Step 3. Feed tray location of prefractionator. From the discussions above, the rectifing and stripping cascades have a common and shared “feed tray”; the vapor leaving the feed tray goes in the rectifying while the liquid goes into the stripping section. Note that the prefractionator serves to prefractionate the feed components, with light and heavy key components. 3.3.2. Main Column Structure Design. The structure of the main column is designed by applying the Fenske design equation.19 From Figure 1, sections 2 and 3_1 constitute a column section that separates the light component A from the middle component B and heavy component C, which are sent in the main column through the top of the prefractionator. To design this column section, sections 2 and 3_1 are combined; apparently their actual tray number should be calculated individually because their respective reflux flows are different. From Figure 1, sections 3_2 and 4 separate middle component B from heavy component C; apparently their actual tray number should be calculated individually because their respective vapor and liquid flows are different. Equations 30 and 31 serve to compute the minimum number of trays wherein one can use the rule of thumb to calculate the actual tray numbers. (N2 + N3_1)min =

ln S2 i , j = {A, B} ln αij

(N4)min =

ln S4 i , j = {B, C} ln αij

(33)

Note that the interlinking stages may not be optimum; hence, their optimization is necessary. The composition difference between two interlinking trays of the prefractionator and main column has been made as small as possible; this minimizes the mixing that can take place on the interlinking stage. Otherwise the mixing lowers the efficiency of the column.

4. DESIGN METHOD VALIDATION To validate the proposed design procedure, it is applied to ten multicomponent feed systems with different ease of separability index (ESI) in order to investigate its usability. 4.1. Example Systems. Different feed mixtures are considered to analyze the effects of feed compositions and relative volatilities on the design equations. The feed systems are picked up and grouped to make mixtures with different ESIs. By definition, if ESI < 1, the A/B split is harder than the B/C split; if ESI > 1, the A/B split is easier than the B/C split.24 The Peng−Robinson EOS is used to perform equilibrium computations. The characterizations of nine feed systems are summarized in Table 1; the tenth is an equimolar alcohol feed system. Table 1. Feed Mixtures Used in Simulation Case Study feed

mixture

F1

F2

F3

M1, ESI = 1.04

A: propane B: n-butane C: n-pentane A: n-butane B: i-pentane C: n-hexane A: propane B: n-butane C: i-pentane

A: 0.4402 B: 0.3542 C: 0.2056 A: 0.3333 B: 0.3333 C: 0.3334 A: 0.5000 B: 0.2500 C: 0.2500

A: 0.250 B: 0.500 C: 0.250 A: 0.250 B: 0.500 C: 0.250 A: 0.250 B: 0.500 C: 0.250

A: 0.250 B: 0.250 C: 0.500 A: 0.250 B: 0.250 C: 0.500 A: 0.250 B: 0.250 C: 0.500

M2, ESI = 0.70

M3, ESI = 1.2

i , j = {A, B} (30)

4.2. Separation Objective. Usually, every operation should have an objective; for the case of the separation process, the objective is intended to achieve the specified product purities. The design specification for the products is arbitrarily set to 0.990 mole fraction of the lightest component in the overhead product, 0.98 mole fraction of intermediate component in side draw product, and 0.99 mole fraction of the

i , j = {B, C} (31)

The section column between the upper interlinking stage and condenser is mainly to eliminate the heavy component from the overhead product (distillate) while separating light component A from middle component B. For a column F



Article

Figure 5. Algorithm for liquid composition calculation.5

material balance equation (eq 34) where Mn is liquid held up in the nth tray.

heaviest component in bottom product. The mole fractions of light and heavy components in the middle product stream (side product) are set to 1% of each. 4.3. Simulation Procedure. In general, rigorous design has been carried out in four steps. For the first step, the numbers of trays are computed for all sections of the column system. Second, the initial operation variables (operational design or minimum flows) are computed. For the third step, the variables obtained from steps one and two are implemented in a simulation using a rigorous process model in order to find the tray liquid composition; Figure 5 illustrates the algorithm for liquid composition calculation. Lastly, the calculated liquid compositions are compared with product specifications to check whether they match; if not, the operational variables are updated and the simulation is repeated until convergence is reached. The minimum reflux flow rates for the prefractionator and the main column are implemented in computation of the liquid split ratio. The vapor split ratio is calculated using the minimum liquid flow and eq 10. The initial liquid flow rate (Ln) and vapor flow rate (Vn) are assumed from the reflux flow rate (LD), and vapor boil-up flow rate (VB) on the basis of equimolar overflow assumption; the tray liquid composition xn,i is evaluated from

Mn, i

dxn , i dt

= Ln − 1xn − 1, i + Vn + 1yn + 1, i + Fzn , i − Lnxn , i − Vnyn , i (34)

The commercial design software HYSYS is used to compute the equilibrium constant (Kn,i) with the help of the Peng− Robinson equation of state. The material balance equations are solved separately for each component by a tridiagonal matrix technique. With the new liquid composition, the tray equilibrium is computed again and the vapor and liquid flow rates are renewed (Figure 5). This procedure is iteratively continued until the sum of the absolute variation of tray temperatures is less than a given limit, 0.0001 times the total number of trays.23 The computed liquid compositions are compared with the product specifications; if they do not match, different reflux flow and vapor boil-up rates are tried until product specifications are yielded. The liquid split ratio needs to be checked to determine whether the ratio gives the optimum liquid flow rate in the prefractionator. G



Article

5. RESULTS AND DISCUSSION The operating conditions of the prefractionator (liquid and vapor flows) used in the structure design equations formulation serve to compute the number of stages of the cascades making the prefractionator. After the preliminary design variables (operating conditions and column system structures) are determined and the example cases are simulated, search optimization methodology is employed to analyze how the design variables (operating conditions) affect the design equations. Optimization of the column system is carried out considering column utility requirements for the reboiler and condenser. The structure results can be found in Supporting Information along with the search method used to investigate the effect of internal flow distribution on the prefractionator structure design equations. The feed composition has an effect on the structure of the prefractionator and main column as well. Within the rectifying section of the prefractionator, C component concentration reduces and becomes less and less as it approaches the top stage. At the same time, in the stripping section, the concentration of light component becomes less and less as it approaches the bottom stage. Component C is considered for the structural design of the rectifying section while component A is considered for the structural design of the stripping section. According to Tedder and Rudd,24 for a case with ESI < 1, the A/B split is harder than the B/C split because the fixed operating conditions to split C component from B component requires comparatively fewer stages when the B/C split is harder. This observation is not in accordance with the outcome of the prefractionator design equations. On the contrary, regardless of feed system characteristics, whether ESI is greater or less than unity, when the feed system is rich in light component, the stripping section of the prefractionator tends to have many equilibrium stages; likewise, when the feed system is rich in heavy component, the rectifying section of the prefractionator tends to have many equilibrium stages. Theses observations are obvious because the quantities of heavy or light components to be either absorbed or stripped are considerably greater (see Tables S.2.1, S.2.2 and S.2.3 in Supporting Information). The absorption factor shortcut method of the Edmister model results for M1-F1, M2-F1, and M3-F1 are compared with their respective rigorous models carried out through HYSYS. The results are summarized in Table 2. The operating conditions of the prefractionator and numbers of stages are used to initialize the rigorous simulation; these operating conditions are also used as internal flows for the Edmister absorption factor shortcut method model. From Table 2, it is noticed that most of the output variables for the absorption factor shortcut method of Edmister and the rigorous model match within one to three significant digits. To improve column system performance, it was advised to optimize the interlinking streams. The profit function/cost function requires calculating a net profit for a column system. The profit function is a function of the revenue generated from products within the limit of product purity constraints; the operating costs are associated with the column utility requirements for the reboiler and condenser. Formulation of cost function and results of optimization are summarized in section S.3 of Supporting Information. Internal flow distribution is the most dominant design variable influencing energy efficiency of the FTCDC. The

Table 2. Results of Simulation Case Study feed M1-F1

M2-F1

M3-F1

top product flow rate (kg mol/h) propane n-butane n-pentane bottom product flow rate (kg mol/h) propane n-butane n-pentane top product flow rate (kg mol/h) n-butane i-pentane n-hexane bottom product flow rate (kg mol/h) n-butane i-pentane n-hexane top product flow rate (kg mol/h) propane n-butane n-pentane bottom product flow rate (kg mol/h) n-butane i-pentane n-hexane

Edmister method

rigorous model

137.7185 0.7991 0.2008 0.0001

137.7 0.7992 0.2008 0.0000

112.2815 0.0001 0.5423 0.4576

112.3 0.0000 0.5423 0.4577

116.1178 0.7171 0.2825 0.0004

116.10 0.7174 0.2826 0.0000

133.8872 0.0004 0.3774 0.6222

133.9 0.0001 0.3773 0.6226

140.6036 0.8890 0.1109 0.0001

140.60 0.8891 0.1107 0.0002

109.3964 0.0001 0.4287 0.5712

108.00 0.0000 0.4289 0.5710

searching method used to analyze the effect of internal flow distribution on column efficiency provides a domain from which the optimal values of interconnecting streams are located; this method does not consider any requirement about utility cost, a reason why the resulting optimal interconnecting streams are different from those obtained through optimization. The searching method invoked in the previous paragraph leads to a response surface technique used to analyze the effect or impact that the changes in interlinking streams have on the operation of the column system (see Figures S4.1−S4.10 of Supporting Information). To establish the effects of changes in interlinking streams on the design equations, the equilibrium stage numbers corresponding to optimal values of interlinking streams are computed and compared to the numbers of equilibrium stages issued from minimum interlinking streams. It was noticed that changes in interlinking streams affect the design equations; the increase of interlinking streams increases the rectifying and stripping factors, and the number of stages changes accordingly. For all considered cases, it was noticed that as interlinking streamflow rates increase, the resulting number of equilibrium stages decreases. The ratios L/V or V/L of rectifying and stripping factors change according to changes recorded in interlinking stream rates (see Tables S5.1−S5.4 of Supporting Information).

6. CONCLUSION This work provides a new design method that can be used for the design of FTCDCs. The component net flow is performed H



Article

H = vapor enthalpy (kcal/mol) K = equilibrium constant M = number of stages N = number of stages s = stripping factor S = separation factor S = stripping factor x = liquid composition [mole fraction] y = vapor composition [mole fraction] z = feed composition [mole fraction] ⌀ = solutions of Underwood equation α = relative volatility

to understand the separation mechanism within the column system and to provide the approximate compositions of the end stream products of the prefractionator that are used for determining its respective structure using group method. For structural design of the prefractionator, an approximate calculation procedure relating the compositions of multicomponent vapor and liquid streams entering and leaving the cascade is developed to formulate an expression that can be used to compute the number of stages of each section. For the design of the main column, the Fenske equation is used. For the design of the overall column system, the results show that the proposed design method provides a configuration close to the optimal structure. For either stripping or rectifying sections of the prefractionator, the design equations can be generalized to all components of the feed system. If two or more components do not appear in the prefractionator upper end product, the component with the K-value that yields the smallest rectifying factor is considered because when used in the design equation, it yields the largest number of stages. On the basis of the formulation of the design equations, one stage is equivalent to the equilibrium-flash equation. The design equations are reliable in the case when operating conditions or K-values throughout the column sections do not vary considerably and/or when the mixture forms nonideal liquid solutions. The design equations formulated for the design of the prefractionator have some advantages over the Fenske−Underwood−Gilliland method; note that the task of performing Gilliland correlation for actual reflux ratio and theoretical stages and computation of the location of the feed stage are no longer required. The design method provides the interlinking stages; this gets rid of the trial-and-error steps for matching the compositions of the interlinking streams that was noticed when the Fenske−Underwood−Gilliland method was used. The design equations formulated for the design of the prefractionator can be used to design any column section of a distillation column once the operating conditions are known.

■

Subscripts

■

REFERENCES

(1) Fidkowski, Z. T.; Królikowski, L. Thermally coupled system of distillation columns: Optimization procedure. AIChE J. 1986, 32, 537−546. (2) Triantafyllou, C.; Smith, R. The design and optimization of fully thermally coupled distillation columns. Chem. Eng. Res. Des. 1992, 70, 118−132. (3) Amminundin, K. A.; Smith, R.; Thong, D. Y. C.; Towler, G. P. Design and optimization of fully thermally coupled distillation columns Part 1: Preliminary Design and Optimization Methodology. Chem. Eng. Res. Des. 2001, 79, 701−715. (4) Agrawal, R.; Fidkowski, Z. T. New thermally coupled schemes for ternary distillation. AIChE J. 1999, 45, 485−496. (5) Kim, Y. H. Rigorous design of fully thermally coupled distillation column. J. Chem. Eng. Jpn. 2001, 34, 236−243. (6) Kim, Y. H. Structural design and operation of a fully thermally coupled distillation column. Chem. Eng. J. (Amsterdam, Neth.) 2002, 85, 289−301. (7) Lee, S. H.; Shamsuzzoha, M.; Han, M.; Kim, H. Y. Study of the structural characteristics of a divided wall column using the sloppy distillation arrangement. Korean J. Chem. Eng. 2011, 28, 348−356. (8) Long, N. V. D.; Lee, M. Dividing wall column structure design using response surface methodology. Comput. Chem. Eng. 2012, 37, 119−124. (9) Halvorsen, I. J.; Skogestad, S. Optimizing control of Petlyuk distillation: Understanding the steady-state behavior. Comput. Chem. Eng. 1997, 21, 249−254. (10) Abdul Mutalib, M. I.; Smith, R. Operation and control of dividing wall distillation columns. Part 1: Degrees of freedom and dynamic simulation. Chem. Eng. Res. Des. 1998, 76, 308−318. (11) Abdul Mutalib, M. I.; Zeglam, A. O.; Smith, R. Operation and control of dividing wall distillation columns. Part 2: Simulation and pilot plant studies using temperature control. Chem. Eng. Res. Des. 1998, 76, 319−334. (12) Annakou, O.; Mizsey, P. Rigorous comparative study of energyintegrated distillation schemes. Ind. Eng. Chem. Res. 1996, 35, 1877− 1885. (13) Ramírez-Coronaa, N.; Jiménez-Gutiérreza, A.; Castro-Agüeroc, C.; Rico-Ramíreza, V. Optimum design of Petlyuk and divided-wall distillation systems using a shortcut model. Chem. Eng. Res. Des. 2010, 88, 1405−1418.

ASSOCIATED CONTENT

S Supporting Information *

Additional information about the application of this method to FTCDC systems, as well as further information about the simulation results of the case considered; summary of structure design results for ternary systems tried while validating the design method (Tables S2.1−S2.4); summary of optimal interlinking flow rates for ternary systems undertaken (Tables S3.1−S3.4). This material is available free of charge via the Internet at http://pubs.acs.org.

■

A = component A B = component B C = component C E = enriching section H = heavy key component L = light key component X = exhausting section i = component j = component 1 = solution number of Underwood equation 2 = solution number of Underwood equation

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Fax: +82-51-512-8563. Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS This work was supported by a 2-Year Research Grant of Pusan National University. ABBREVIATIONS a = rectifying factor A = absorption factor h = liquid enthalpy (kcal/mol) I



Article

(14) Lee, M. Y.; Jeong, S. Y.; Kim, Y. H. Application of a thermally coupled distillation column with separated main columns to gas concentration process. Korean J. Chem. Eng. 2008, 25, 1245−1251. (15) Hwang, K. S.; Kim, B. C.; Kim, Y. H. Design and Control of a Fully Thermally Coupled Distillation Column Modified from a Conventional System. Chem. Eng. Technol. 2011, 34, 273−281. (16) Kremser, A. Theoretical analysis of absorption process. Natural Petroleum News 1930, 22, 43−49. (17) Edmister, W. C. Absorption and stripping-factor functions for distillation calculation by manual- and digital-computer methods. AIChE J. 1957, 3, 165−171. (18) Kamath, R. S.; Biegler, L. T.; Grossmann, I. E. Aggregate models based on improved group methods for simulation and optimization of distillation systems. Comput. Chem. Eng. 2010, 34, 1312−1319. (19) Halvorsen, I. J.; Skogestad, S. Distillation Theory. Norwegian University of Science and Technology: Norway, 2000. (20) Caballero, J. A.; Grossmann, I. E. Generalized Disjunctive Programming Models for the Optimal Synthesis of Thermally Linked Distillation Columns. Ind. Eng. Chem. Res. 2001, 40, 2260−2274. (21) Holland, S. T.; Abbas, R.; Hildebrandt, D.; Glasser, D. Complex Column Design by Application of Column Profile Map Techniques: Sharp-Split Petlyuk Column Design. Ind. Eng. Chem. Res. 2010, 49, 327−349. (22) Chu, K. T.; Cadoret, L.; Yu, C. C.; Ward, J. D. A New Shortcut Design Method and Economic Analysis of Divided Wall Columns. Ind. Eng. Chem. Res. 2011, 50, 9221−9235. (23) Seader, J. D.; Henley, E. J. Separation Process Principles. John Wiley & Sons, Inc.: New York, 1998; pp 531−538. (24) Tedder, D. W.; Rudd, D. F. Parametric Studies in Industrial Distillation: Part I. Design Comparisons. AIChE J. 1978, 24, 303−315.

J


New Design Method for Fully Thermally Coupled Distillation Column

Recommend Documents