Article pubs.acs.org/IECR
Pressure Control in Distillation Columns: A Model-Based Analysis Miguel Mauricio-Iglesias,*,† Thomas Bisgaard,† Henrik Kristensen,‡ Krist V. Gernaey,† Jens Abildskov,† and Jakob K. Huusom† †
CAPEC-PROCESS, Department of Chemical and Biochemical Engineering, Technical University of Denmark, Søltofts Plads, Building 229, DK-2800 Lyngby, Denmark ‡ CP Kelco ApS, Ved Banen 16, 4623 Lille Skensved, Denmark ABSTRACT: A comprehensive assessment of pressure control in distillation columns is presented, including the consequences for composition control and energy consumption. Two types of representative control structures are modeled, analyzed, and benchmarked. A detailed simulation test, based on a real industrial distillation column, is used to assess the differences between the two control structures and to demonstrate the benefits of pressure control in the operation. In the second part of the article, a thermodynamic analysis is carried out to establish the influence of pressure on relative volatility for (pseudo)binary mixtures. A simple criterion is found, based on the difference in the scaled heats of vaporization of the light and heavy compounds: A large difference indicates that relative volatility is sensitive to pressure changes, whereas no a priori conclusion can be made for small differences. Depending on the sensitivity of relative volatility to pressure, it is shown that controlling the bottom-tray pressure instead of the top-tray pressure leads to operation at the minimum possible average column pressure, so that significant energy savings can be achieved.
1. INTRODUCTION In both the academic and professional literature on process control, it is often assumed that a distillation column operates at constant pressure. The benefits of operating at constant pressure are the minimization of the need for compensation for temperature control, prevention of column flooding and weeping, better use of the column capacity, and stability downstream. Additionally, Kister1 reported problems with pressure control related to hot-vapor bypasses, cooling-water throttling, and condensate accumulation, among others. In contrast to these claims, the information available in the open literature on pressure control design is scarce and sometimes contradictory. As an example, Buckley et al.2 advised against tight pressure control because a rapid change in pressure could lead to flooding or weeping in the column. On the contrary, Shinskey3 advocated tight pressure control, particularly when temperature measurements are used to infer the composition, which is an approach that is widely applied in practice. A rigorous assessment of the effect of dynamic pressure variations by simulating distillation performance requires a very detailed model. Common assumptions such as constant molar overflow, negligible enthalpy holdup in the vapor phase, and infinitely fast vapor flow dynamics do not hold in such models.4 Otherwise, the effects of pressure variations on the vapor flow dynamics and, as a consequence, on the energy inventory of each section of the column are not properly represented. The need for a comprehensive model might be a cause for the shortage of mathematical analyses of pressure control in distillation columns in the literature. Important questions that have not been tackled or have been considered only qualitatively, include the following: How should the manipulated variable be selected? What is the influence of the condenser structure on the speed and/or magnitude of the control action? Which pressure measurement should be used for the control? In this work, we carry out a critical assessment © 2014 American Chemical Society
of the design of pressure control in distillation columns. To do so, we benchmark two different design structures aimed at pressure control for distillation with a total condenser, both theoretically and as a solution for a real industrial case study. Based on the results and some ideas already presented by Li et al.,5 we derive the thermodynamic conditions that indicate in which tray the pressure should be controlled. This article is organized as follows: First, the models of the distillation column and the condenser in the two control structure examples are presented and analyzed. Then, we introduce the case study: recovery of 2-propanol from an aqueous solution. The two control structures are tested for this case study using real data corresponding to 311 h of operation and benchmarked against indicators of process and control performance. In the last section, the dependence of the relative volatility on pressure is analyzed, and its implications for pressure control are discussed.
2. MODELS OF DISTILLATION COLUMN AND CONDENSERS 2.1. Distillation Column Model. Let j = 1, 2, ..., NC, denote components in a binary or multicomponent mixture, and let i = 1, 2, ..., NS, denote column stages counted from the top (Figure 1). Conservation of mass can thus be expressed in moles as Received: Revised: Accepted: Published: 14776
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nonrandom two-liquid (NRTL) equation was used]. As a consequence of eq 3, the pressure in a tray can be calculated as NC
Pi =
∑ xi ,jγi ,jPisat,j (4)
j=1
In a trayed column, liquid enters a tray from the downcomer and leaves from the weir outlet. Furthermore, vapor enters a tray in the bottom and passes through the liquid, as it partly condenses, until it enters the vapor phase of the tray. From the vapor phase, a flow, caused by a pressure gradient, leaves the tray. Liquid flow can be described by the Francis weir formula,6 in which the volumetric liquid flow rate is proportional to the amount of liquid over a weir to the power of 2/3; hence
Figure 1. Distillation column diagram with labels for internal and external streams.
d M1, j = Rx D, j + V2y2, j − L1x1, j − V1y1, j dt d Mi , j = Li − 1xi − 1, j + Vi + 1yi + 1, j + Fz i i , j − Li xi , j − Vy i i,j , dt i = 2, ..., NS − 1
(5)
hoW, i = hcl, i − hW
(6)
hcl, i =
M T, i MWiL ρi L A t, i
(7)
CLi
is a constant that depends on the liquid loading of the individual trays, ρL is the liquid-phase density, MWL is the liquid-phase molecular weight, At,i is the active tray area at stage i, hW is the weir height, hcl,i is the clear liquid height (any froth is ignored), and hoW,i is the liquid height above the weir. Both At,i and hW are dimensional parameters of the column. The vapor flow through perforated plates can be described as suggested by Kolodzie and Van Winkle.7 In this work, the volumetric flow rate is simplified to be proportional to the square root of the pressure gradient in terms of liquid height; thus
d M NS , j = L NS− 1x NS− 1, j + FNSz NS , j − Bx NS , j − VNSyN , j S dt (1)
where Mi,j is the holdup of component j in stage i and R, L, V, F, and B are the flow rates of reflux, liquid, vapor, feed, and bottoms, respectively. The mole fractions of the liquid, vapor, and feed are x, y, and z, respectively. Assuming that the molar vapor holdup is negligible compared to the liquid holdup and that the liquids are incompressible, the internal energy of the tray can be approximated by its enthalpy,4 and the conservation of energy can be expressed as
Vi =
CiV
(ρi V )0.5 MWiV
(ΔPi − ΔPs, i − 1)0.5
ΔPi = Pi − Pi − 1 ΔPs, i − 1 = ρi L− 1ghcl, i − 1
d L V (M T,1h1L) = RhDL + V2h2V + Fh 1 F,1 + Q 1 − L1h1 − V1h1 dt d L (M T, ihiL) = Li − 1hiL− 1 + Vi + 1hiV+ 1 + Fh i F, i + Q i − Lihi dt V − Vh i = 2, ..., NS − 1 i i ,
(8) (9) (10)
CVi
is a constant that depends on the vapor loading of the individual trays; g is the gravitational constant; and ΔPs,i is the static pressure drop caused by the liquid height, hcl,i. The set of equations is solved sequentially as described by Gani et al.8 Given the liquid-phase component holdups and temperature for each tray and time step, the algebraic eqs 3−10 are solved, and the mass and energy balances (eq 1 and 2) are formulated for the next time step. The set of ordinary differential equations (ODEs) was solved using the stiff solver ode15s available in MATLAB. 2.2. Model and Analysis of the Condenser Structure. Two of the control structures assessed in this study were taken from Sloley9 and represent, from the control point of view, two different approaches: using the cooling-fluid flow as the manipulated variable (MV) or manipulating the level of condensate in a partially flooded condenser. Control structure A (Figure 2) consists of manipulating the flow of cooling water to the condenser to vary the cooling duty and, hence, the degree of subcooling of the condensate. The cooling-water flow
d (M T, NSh NLS) = L NS− 1h NLS− 1 + FNShF, NS + Q N − Bh NLS − VNSh NVS S dt (2)
The subscript T refers to the total holdup; the superscripts L and V correspond to the liquid phase and vapor phase, respectively; h is the enthalpy; and Q is the heat transfer. Assuming an ideal vapor phase and a negligible contribution from the Poynting correction, the modified Raoult’s law states yi , j = xi , jγi , jPisat , j / Pi
⎧ ρL ⎪ ⎪CiL i L hoW, i1.5 , hoW, i > 0 Li = ⎨ MWi ⎪ ⎪ 0, hoW, i ≤ 0 ⎩
(3)
where the vapor pressure of a liquid, Pi,jsat, is calculated by Antoine’s equation and the activity coefficients, γi, are determined by an activity coefficient model [in this work, the 14777
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significantly lower than that in the area occupied by the condensing vapor. Therefore, varying the heat-transfer area is an effective means to manipulate the cooling duty in the condenser. As the control action depends on the level of condensate, the sizing of the valve is a key design parameter to ensure fast action. Another key element in design is to ensure that the condensate can flow by gravity flow at the desired rate. Commonly, a certain overdesign would be needed if pressure were to be efficiently controlled, which would imply operating at high water flow rates. An advantage of this structure is that the water flow rate is constant and its temperature in the outlet is relatively low. Also, as a consequence of the high flow rate, variations in the outlet water temperature are dampened. However, one of the problems with this structure is that it leads to level variations in the reflux drum that can be propagated to the downstream units by the distillate flow, depending on the level controller tuning. 2.3. Model of Control Structure A. Control structure A (Figure 2) is modeled considering both the condenser and the reflux drums within the bounds of the control volume. As a modification of the general model of a distillation column, the cooling duty to the exchanger is calculated as
Figure 2. Diagram of control structure A and similar, where the cooling-fluid flow is the manipulated variable.
will be the minimum that can guarantee the pressure set point, and as a consequence, the water temperature in the outlet will be the maximum possible. Advantages of this structure are that heat recovery can be enhanced because the highest temperature level is reached and that its design and instrumentation are simple. On the contrary, working with warm water can cause fouling problems that are aggravated by operating at the lowest water flow rate. As a consequence, it is only useful for a limited range of pressures and cooling duties, for which the weather conditions along the year must be checked. Other structures based on control structure A include circulating a water flow to which cooling water is added at the same rate at which hot water is removed. Thus, the fouling is mitigated because the water circulates at a higher velocity. If the amount of water circulating is very high, it can have nonnegligible inertia and slow dynamics; however, the fundamental control aspects of its operation can be covered by the analysis of control structure A. Control structure B (Figure 3) is based on operating at a constant cooling-water flow and manipulating the level of condensate within the condenser to vary the heat-transfer area. The heat transfer in the area flooded with condensate is
Q cnd = εmC ̇ pΔT
(11)
where Cp is the specific heat capacity of water, ṁ is the mass flow of cooling water, and ΔT is the maximum temperature difference (at the vapor and water inlet).10 The efficiency (ε) of the heat exchanger where a phase change takes place is determined as
ε = 1 − exp( −NTU)
(12)
where NTU is the number of transfer units, which is defined as the ratio between the overall heat-exchange coefficient (U), the exchange area (A), and the minimum of the heat capacities (ṁ Cp) NTU =
UA mC ̇ p
(13)
It must be noted that a fluid undergoing a phase change has an infinite heat capacity because it can exchange heat without changing its temperature; therefore, the cooling water will always have a lower capacity than the condensing vapor, regardless of the mass flow used in the condenser. The overall heat-transfer coefficient, U, is the inverse of the sum of three resistances to heat transfer between the vapor side and the cooling-water side, namely, that between the vapor and the condenser wall, that along the thickness of the condenser wall (e), and (iii) that between the condenser wall and the cooling water 1 1 e 1 = + + UA h1A1 kA m h 2A 2
(14)
In a well-designed heat exchanger, the resistance to transfer by conduction is negligible. Additionally, the heat-transfer coefficient associated with condensation is very high. Hence, U can be approximated by the heat-transfer coefficient on the cooling-water side, which is the inverse of the limiting resistance
Figure 3. Diagram of control structure B and similar, where a valve manipulates the level in a partially flooded condenser. Veq and Lc represent the vapor flow in the equalization line and the liquid flow between the condenser and the reflux drum, respectively. Both flows can be bidirectional.
U ≈ h2
(15)
Because h2 is the heat-transfer coefficient on the cooling-water side, it depends on the velocity of the water flow. Classical 14778
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correlations for internal turbulent flow (e.g., Dittus−Boetler and Sieder−Tate)10 predict the Nusselt number to be proportional to the Reynolds number to the power 4/5. Assuming that the properties of water do not vary in the range of operation of the condenser, this dependency can be written as ⎛ ṁ ⎞ Nu ∝ Re 4/5 ⇒ U ≈ h2⎜ ⎟ ⎝ ṁ 0 ⎠
4/5
⎛ ṁ ⎞ ⇒ U = U0⎜ ⎟ ⎝ ṁ 0 ⎠
4/5
The actual variation of pressure with the manipulated variable (dP/dṁ ) is a complex function that depends on the energy holdup of the top of the column. From this perspective, the control of pressure can be identified as a control of the energy inventory of the column. The relation can be decomposed into a term (dP/dQcnd) that depends on the energy balance to the top of the column and the term dQcnd/ dṁ . Therefore, the control action can be assessed by studying how the cooling duty in the condenser varies with respect to the manipulated variable to compare different controllers and providing an indication of the process gain. Combining eqs 11 and 16, the following derivative is obtained dṁ
= ΔTCp[0.2NTU(ε − 1) + ε]
lim
NTU →∞
The main difference in this mass balance is the inclusion of the term Veq, where the superscripts R and cnd indicate equalizing vapor flows from the reflux drum and from the condenser, respectively. The equalizing vapor flow is determined as
(17)
(18)
f (NTU) = 1
(23)
⎧ ⎪K eq Pcnd − PR if Pcnd − PR > 0 cnd =⎨ V eq ⎪ ⎩0 otherwise
(24)
Lc = uV KV (Pcnd + ρgh) − PR
if (Pcnd + ρgh) − PR > 0 (25)
where uV is the valve opening (the manipulated variable); KV is the valve constant; and h is the height of liquid in the condenser, which is responsible for the hydrostatic head. As with the equalizing line, the liquid can also ascend to the condenser if the pressure difference is reversed. However, in such a case, the liquid needs to overcome the head difference due to the liquid holdup in the condenser and the level difference between the two vessels
d f (NTU) = 0.2NTU exp( −NTU) dNTU ∀ NTU ≥ 0
⎧ ⎪K eq PR − Pcnd if PR − Pcnd > 0 R =⎨ Veq ⎪ ⎩0 otherwise
where Keq is a constant representing the resistance to vapor flow of the equalizing line. Because the goal of the equalizing line is to allow vapor flow to ensure that the pressure is equal in the two compartments, Keq was estimated and fitted so that the maximum pressure difference was lower than 1 Pa during the dynamic simulations. The liquid flow to the reflux drum, Lc, was modeled using a valve equation as
On the other hand, the first derivative of f is a continuous function that is always positive for any positive NTU
+ 0.8 exp(− NTU) > 0
R cnd yR, j − V eq ycnd, j + LcR x R, j − Lcxcnd, j = V1y1, j + Veq
(22)
If we restrict the function to positive values of NTU, the results for the extremes of the interval are and
(21)
dt
f (NTU) = 0.2NTU(ε − 1) + ε
f (0) = 0
d ⎛ dQ cnd ⎞ 2 ⎜ ⎟ = ΔTε(NTU + NTU + 1) dCp ⎝ dṁ ⎠
dMcnd, j
Although the two terms in eq 17 have different signs, it can be proven that the result of the equation remains greater than zero for positive values of ṁ , and the cooling duty increases with an increase in water flow. Let f be the term that depends on NTU in eq 17 (between square brackets). As can be seen, f is a continuous function for all values of NTU
= 1 − (0.2NTU + 1) exp( −NTU)
(20)
2.4. Model of Control Structure B. Control structure B (Figure 3), in contrast to the previous control structure, is modeled considering two control volumes: one for the condenser and one for the reflux drum. As a result, the condenser has its own holdup and mass and energy balances related to the mass streams. The mass balance reads
(16)
dQ cnd
(1 − ε)(0.2NTU + 0.8) d ⎛ dQ cnd ⎞ ⎜ ⎟ = ΔT dU0A ⎝ dṁ ⎠ ṁ 0.2ṁ 0 0.8
(19)
Hence, f grows monotonically from 0 to 1, and eq 18 always has a positive sign. From the previous analysis it can be concluded that the “gain” of the process always has the same sign for all the values of the manipulated variable. Besides, it is directly proportional to the temperature difference between fluids, increases with UA, and decreases asymptotically to zero when increasing ṁ . This nonlinear relation between the manipulated variable and the process gain may require a nonlinear controller or at least any type of gain scheduling if the range of operation is large. With respect to the rest of the variables, the gain of the process is directly proportional to the temperature difference between the two fluids and increases with UA and Cp. In effect, differentiating again with respect to UA and Cp, it can be shown that the gain always increases with UA and Cp, giving easier control action
LcR = uV KV (PR − [Pcnd + ρg (h + h0)] if [PR − (Pcnd + ρg (h + h0)] > 0
(26)
Given the previous definitions of the molar flows, the energy balance to the condenser becomes L d(M Thcnd ) L R V cnd V = V1h1V + Veq hR − V eq hcnd + LcR hRL − Lchcnd dt
+ Q cnd
(27)
Likewise, the mass and energy balances for the reflux drum are 14779
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and each condenser has an effective exchange area of 200 m2. A diagram of the column is presented in Figure 4.
cnd R ycnd, j − Veq yR, j + Lcxcnd, j − LcR x R, j = V eq
dt
− (R + D)x R, j
(28)
d(MR hRL) cnd V R V L = V eq hcnd − Veq hR + Lchcnd − LcR hRL − RhRL dt − DhRL
(29)
The cooling duty in the condenser now depends on the heat transfer in the vapor area and the flooded area. If one assumes that the variation of temperature due to subcooling is negligible compared to the temperature difference between the two fluids, then eqs 11−13 still hold, but the number of transfer units has to be redefined as NTU =
UVAV + ULAL mC ̇ p
(30)
where the subscripts V and L indicate the vapor area and the flooded area of the condenser, respectively. Following the same reasoning as previously, the control action depends on the effect of the manipulated variable uV on the cooling duty dQ cnd du V
⎡ d(VL /VT) ⎤ = ΔTA T(UV − UL)(1 − ε)⎢ − ⎥ du V ⎦ ⎣
(31)
An examination of eq 31 shows that the gain of the process depends on the temperature difference, the exchange area, the difference in heat coefficients between the vapor area, and the flooded area and the capacity of the valve to modify the flooded volume. Hence, the sizing of the valve is essential for the success of this control strategy. Finally, it can be noted that the right-hand side of eq 31 is, in most cases, positive, that is, the cooling duty increases when the valve opening increases because more area is exposed to the vapor. However, if the pressure difference between the reflux drum and the condenser is reversed, opening the valve increases the reverse flow and the flooded volume. As a consequence, the sign of the derivative dVL/duV changes, and the cooling duty decreases. This is not expected to happen in normal operation because the vapor equalizing line would reduce the pressure difference until the liquid flowed in the normal direction.
Figure 4. Diagram of the distillation column and two condensers in series for the case study.
The purpose of the column is to recover the 2-propanol at a concentration of at least 80% (w/w) (0.565 mole fraction) in the distillate while removing all of it from the bottoms stream. With this aim, the temperature of the 15th tray is automatically controlled close to 101 °C (pure water at the corresponding pressure). The distillate concentration is monitored online by density measurements and manually controlled by manipulating the reflux flow. A model of the column was developed based on the previous equations (eqs 1−10) and was calibrated using the following decision variables: CVi , which is a constant relating the vapor loading and the pressure drop in each tray; the weir heights hW; and the holdups in the reflux drum and reboiler. Because the concentration of 2-propanol in the feed was not known, a feed profile was estimated based on mass balance closure for different horizon windows. Vapor−liquid equilibrium was described by the NRTL model using the parameters given by Marzal et al.11 The goal of this case study was to investigate which of the presented pressure control strategies is most suitable in terms of control and operation performance. Furthermore, because the distillate concentration was not automatically controlled, the synergic benefits between pressure and distillate control were also explored. 3.1. Definition of Simulation Tests. A number of control structures were considered for benchmarking through simu-
3. INDUSTRIAL DISTILLATION CASE STUDY The benefits and disadvantages of using either of the specified structures for pressure control were analyzed by simulation tests on a real case of recovery of 2-propanol from an aqueous solution. The distillation column is fed with 846 kmol/h of an aqueous stream with a mole fraction of 0.176 of 2-propanol. Other impurities are present in small amounts (
