7.1 INTRODUCTION
Distillation columns fractionate a feed mixture to produce product streams of desired purity, either for direct sale or for use in other processes. As mentioned in Chapter 6, when a distillation column separates a feed mixture of two components, that process is called as binary distillation column. If an ideal mixture having two components is dealt in a fractionating column, the column is usually referred to as ideal binary distillation column. In another sense, if all the trays of a binary distillation are ideal (100%
efficient), then the column may be called ideal binary distillation column.
The previous chapter develops a mathematical model for a simple staged distillation distillation column. The column has been considered as a compartment system, in which, a number of stages are lumped. In this chapter, we will develop a dynamic model for a little more complex distillation process.
Few assumptions, which are used to simplify a distillation model to a compartmental distillation model, will be avoided in the derivation of the ideal binary distillation column model. The present study also includes the dynamic simulation of the process model.
7.2 THE PROCESS AND THE MODEL 7.2.1 Process Description
The schematic diagram of a distillation column, which is dealt here, is shown in Figure 7.1. It consists of 15 trays, a reboiler and a partial condenser. The trays are numbered from the bottom to the top; the bottom tray has number 1 and the top tray has number 15. The feed is a mixture of components 1-propanol and ethanol. On the fifth tray of the column, a partially vaporized feed (liquid + vapour) is introduced. The overhead vapour is partly condensed in the partial condenser and the noncondensed vapour (not superheated) is withdrawn from the reflux drum as vapour distillate. A fraction of the condensed liquid (not subcooled) is recycled back in the column as reflux stream and some of the liquid is removed as liquid distillate product. At the base of the column, the bottom product is collected as a liquid stream. The boil-up vapour is generated using steam as a heating medium in the reboiler and the produced vapour returns to the bottom plate. The vapour and liquid distillates in the reflux drum, and the boil-up vapour and bottom product streams in the reboiler are in equilibrium.
FIGURE 7.1 Schematic representation of the example of distillation column.
7.2.2 Mathematical Model
Assumptions
The following assumptions have been adopted to develop the mathematical process model.
Negligible vapour holdup is assumed.
The molar heats of vaporization of both components are about the same.
The liquid holdup varies on each tray (excluding reflux drum and column base), and the liquid hydraulics are calculated from the Francis weir formula. In practice, the liquid holdups in reflux drum and column base are generally tightly controlled implementing level controllers with the manipulation of distillate and bottom product flow rates respectively.
The liquid is perfectly mixed on each stage. For the nth stage, it reveals that liquid composition anywhere on the stage = xn...(7.1)
The total amount of liquid accumulated in the reboiler as well as in the base of the column is
considered as the column base holdup.
The heat losses from the column to the surroundings are assumed to be negligible.
The relative volatility is invariant with time and with column length.
Each tray is assumed to be ideal (i.e., 100% efficient).
Coolant and steam dynamics are negligible in the condenser and reboiler respectively. The book by Franks (1972) can be consulted for details about the condenser and reboiler dynamics.
Model development
As was mentioned above, in the present study, the distillation process considers varying liquid holdups on each tray excluding reflux drum and column base. The internal liquid flow rate leaving a stage n(Ln) is determined by means of the linearized version of the Francis weir formula:
...(7.2)
where Ln0 is the reference value of the internal liquid flow rate, and mn and mn0 are the actual and reference molar holdups respectively on tray n. The hydraulic time constant is typically 3 to 6 seconds per tray (Luyben, 1990).
If VB be the vapour boil-up rate, VS the vapour flow rate throughout the stripping section (tray 1 to tray 4) and the vapour flow rate leaving the feed plate (nF), the stated assumptions imply that:
= VS = VB...(7.3) and
VR = + FV = VB + FV...(7.4)
In the above equation, FV is the flow rate of vapour feed and VR the vapour flow rate along the rectifying section (tray 6 to tray 15). Note that these V terms are not necessarily constant with time. In the industrial column, usually the bottom product composition (xB) is controlled by manipulating the flow rate of boil-up vapour (or steam input to the reboiler). Therefore, the other vapour flow rates are also changed accordingly.
Here, the vapour–liquid equilibrium (VLE) describes the relation between the vapour-phase composition of more volatile component (here ethanol) y and the liquid-phase composition of ethanol x for any stage n of the column. The relationship is given below as a nonlinear expression (details in Chapter 6):
...(7.5)
where (or 12) is the relative volatility of component 1 (here ethanol) with respect to component 2 (here 1-propanol).
In the following, the tray-wise modelling equations for the complete distillation column will be derived based on the assumptions made.
Condenser–Reflux Drum System (subscript ‘D’)
The partial condenser and the reflux drum are shown as a combined system in Figure 7.2. The total and
component continuity equations, which represent the mathematical model of the combined system, can be obtained by making balance based on the following form:
Rate of input – Rate of output = Rate of accumulation...(7.6)
Here, this relationship will be used with the unit of pound-moles per hour (lbmol/h) and to derive the continuity model of all the distillation trays.
FIGURE 7.2 Condenser–Reflux drum system.
Total continuity equation
...(7.7)
In this equation, is the flow rate of a vapour stream with composition leaving top tray (nT) , R the reflux flow rate having composition xD, DL the flow rate of the liquid distillate having composition xD and DV the flow rate of the vapour distillate with composition yD. Since the liquid holdup in the reflux drum (mD) is assumed constant and = VR, therefore
DL = VR – R – DV...(7.8) Component continuity equation
...(7.9)
Rearranging
...(7.10)
Top Tray (subscript ‘nT’)
The top tray along with the associated flow schemes is shown in Figure 7.3. To represent this tray mathematically, the balance equations have been derived as follows:
FIGURE 7.3 Top tray.
Total continuity equation
...(7.11)
Since , so
...(7.12)
Component continuity equation
...(7.13) This equation gives
...(7.14) or
...(7.15)
Simplifying,
...(7.16)
nth Tray (subscript ‘n’, where n = 7, ..., 14)
The sketch of the incoming and outgoing flow streams in addition to the liquid holdup is shown for nth tray in Figure 7.4. The following continuity equations can represent the nth tray model.
Total continuity equation
...(7.17) Simplifying
...(7.18)
FIGURE 7.4 nth tray.
Component continuity equation
...(7.19)
Considering dynamic tray holdup and rearranging, we obtain:
...(7.20)
Above Feed Tray (subscript ‘nF+1’)
The feed stream of the sample distillation column is a mixture of both liquid and vapour streams. The feeding of this mixture to the column leads to the down flow of the liquid feed (FL) through tray nF and up flow of the vapour feed (FV) through tray nF + 1. The material balance equations for tray nF + 1 have been derived as per the description given in Figure 7.5.
Total continuity equation
...(7.21)
Here, and , and
...(7.22) Then, Equation (7.21) yields
(7.23) Component continuity equation
...(7.24)
where ZV is the composition of vapour feed. Substituting Equation (7.23), the above equation simplifies to
...(7.25)
FIGURE 7.5 Above feed tray.
Feed Tray (subscript ‘nF’)
Figure 7.6 illustrates the feed tray. Here, the liquid feed is considered as an incoming stream to the feed tray. Notice that the input and output vapour flow rates are equal. The mathematical representation of this feed tray is provided by the following balance equations.
Total continuity equation
...(7.26)
Substituting , we get
...(7.27)
FIGURE 7.6 Feed tray.
Component continuity equation
...(7.28)
where ZL is the composition of liquid feed. Simplifying the above equation, we have
...(7.29) nth Tray (subscript ‘n’, where n = 2, ..., 4)
This nth tray (n = 2, ..., 4) is not same with the previous nth tray (n = 7, ..., 14). The input and output flows of this tray are demonstrated in Figure 7.7. Now the modelling equations may be derived by the following way.
Total continuity equation
...[from Equation (7.17)]
We know Vn = Vn–1 = VS and therefore, the above equation yields ...[from Equation (7.18)]
FIGURE 7.7 nth tray.
Component continuity equation
...[from Equation (7.19)]
Considering dynamic holdup and rearranging, we obtain
...(7.30)
Bottom Tray (subscript ‘1’)
Figure 7.8 represents the bottom (1st) tray of the distillation column. The following total and component mass balance equations describe this tray.
Total continuity equation
...(7.31)
It is assumed that V1 = VS = VB and hence, the above equation reduces to ...(7.32)
FIGURE 7.8 Bottom tray.
Component continuity equation
...(7.33)
Inserting Equation (7.32) into Equation (7.33), one finally obtains
...(7.34)
Reboiler–Column Base System (subscript ‘B’)
Figure 7.9 describes an equilibrium stage that comprises of a reboiler and the base of the distillation column. The total liquid holdup in this combined system is mB and the flow rates are shown in the figure.
The model can be developed as follows.
Total continuity equation
...(7.35) Since the liquid holdup is constant,
B = L1 – VB (7.36)
FIGURE 7.9 Reboiler–Column base system.
Component continuity equation
...(7.37) Equation (7.37) yields at constant liquid holdup,
...(7.38)
7.3 DYNAMIC SIMULATION
In order to predict the ideal binary distillation column dynamics, we need to simulate the developed mathematical model. The process model comprises of a set of ordinary differential equations (ODEs) in addition to the algebraic equations (AEs). Among the ODEs, the total continuity equations are used to compute the holdups of tray liquids and the component continuity equations are employed to determine the liquid-phase compositions on all trays. Among the algebraic form of equations, the VLE relationship [Equation (7.5)] provides the vapour-phase compositions and the Francis weir formula [Equation (7.2)]
calculates the internal liquid flow rates. In addition, the vapour flow rates are calculated using Equations (7.3) and (7.4). Therefore, apart from the vapour rate calculations, there are two ODEs (a total continuity equation and a light component continuity equation) and two AEs (a VLE relationship and a liquid-hydraulic relationship) per tray. The operating and steady state conditions of the example distillation column are reported in Table 7.1.
Table 7.1 Operating and steady state conditions Binary system: 1-propanol/ethanol
Flow rate of liquid distillate (lbmol/h), DL Composition of liquid distillate (mol fraction), xD Flow rate of vapour distillate (lbmol/h), DV Composition of vapour distillate (mol fraction), yD Liquid holdup in reflux drum (lbmol), mD
Reflux flow rate (lbmol/h), R
Flow rate of liquid feed (lbmol/h), FL Composition of liquid feed (mol fraction), ZL Flow rate of vapour feed (lbmol/h), FV Composition of vapour feed (mol fraction), ZV Composition of bottom product (mol fraction), xB Liquid holdup in column base (lbmol), mB Vapour boil-up rate (lbmol/h), VB Hydraulic time constant (s), Relative volatility,
Integration time interval (h), dt
27.8
The dynamic process simulator that solves the differential-algebraic system can be developed according to the following logic.
Step 1: Input data for variables: liquid-phase compositions x for all trays (including reflux drum and column base) and liquid holdups m for all trays (excluding reflux drum and column base).
Step 2: Input data for constant inputs/outputs/parameters: liquid feed rate FL with composition ZL, vapour feed rate FV with composition ZV, vapour distillate rate DV, hydraulic time constant , relative volatility , and liquid holdups in reflux drum mD and in column base mB. Note that the values of input disturbances, outputs and uncertain parameters may change during the plant operation.
Step 3: Either input the values of reflux flow rate R and vapour boil-up rate VB or manipulate these flow rates employing the controllers. In our case, no controllers have been implemented and the values of these manipulated inputs are supplied at each time step.
Step 4: Calculate the internal liquid flow rate L using the Francis weir formula [Equation (7.2)] for all trays. Also compute the vapour flow rate V employing Equations (7.3) and (7.4) for all trays.
Step 5: Calculate the vapour-phase composition y for all trays (including reflux drum and column base) from Equation (7.5).
Step 6: Compute the liquid distillate flow rate DL using Equation (7.8) and bottoms flow rate B from Equation (7.36).
Step 7: Update the liquid-phase compositions on all trays (including reflux drum and column base) and liquid holdups on all trays (excluding reflux drum and column base) for the next time step. The Euler method (details in Chapter 2) is used to integrate the ODEs.
Step 8: To predict the column dynamics for the future time step, go back to Step 3.
The complete computer program (Fortran 90 code) for the prescribed ideal binary distillation column is given in Program 7.1. Note that if you run Program 7.1, you shall get the dynamic process response for a step change in liquid feed composition from 0.60 (steady state value) to 0.55 at time equal to zero.
PROGRAM 7.1 Dynamic Distillation Simulator
! B = Bottoms flow rate
! DL = Liquid distillate flow rate
! DV = Vapour distillate flow rate
! FL = Liquid feed flow rate
! FV = Vapour feed flow rate
! L = Internal liquid flow rate
! M = Liquid holdup on the tray
! MB = Liquid holdup in the column base
! MD = Liquid holdup in the reflux drum
! NF = Feed tray
! NT = Total number of trays
! R = Reflux flow rate
! VB = Vapour boil-up rate
! VR = Vapour flow rate throughout the rectifying section
! VS = Vapour flow rate throughout the stripping section
! X = Liquid-phase composition
! XB = Composition of bottom product
! XD = Composition of liquid distillate
! XF = Composition of liquid feed
! Y = Vapour-phase composition
! YB = Composition of boil-up vapour
! YD = Composition of vapour distillate
! YF = Composition of vapour feed PROGRAM DYNAMIC_IBDC
REAL*8,PARAMETER:: BETA=0.001 D0,ALPHA=2.0 D0,dt=0.0001 D0 REAL*8,PARAMETER::R0=400.8636170476530082 D0
OPEN(UNIT=2,File=”IBDC_DS.dat”)
! Initial data
XD(1)=0.8993283514871822 D0
XB(1)=0.4812223878557398 D0 MD=8.0557290338576260 D0 MB=4.8969852099258300 D0 FL=800.0 D0
FV=200.0 D0 DV=200.0 D0 XF=0.55 D0 YF=0.53 D0 Time(1)=0.00 D0
! Tray-wise holdups and compositions M(1,1)=2.3838602106839941 D0
X(1,1)=0.5413852549176267 D0 M(2,1)=2.5699186724215152 D0 X(2,1)=0.5601978537566697 D0 M(3,1)=2.7363668003742906 D0 X(3,1)=0.5657826798790962 D0 M(4,1)=2.8625585439519976 D0 X(4,1)=0.5674147929146519 D0 M(5,1)=2.9135632410532013 D0 X(5,1)=0.5678895664968068 D0 M(6,1)=3.0847017035199670 D0 X(6,1)=0.5042202331519813 D0 M(7,1)=3.3039534593441741 D0 X(7,1)=0.5165359860620628 D0 M(8,1)=3.4025198348688281 D0 X(8,1)=0.5334695233306152 D0 M(9,1)=3.4534614354834354 D0 X(9,1)=0.5563081868270025 D0 M(10,1)=3.4804263798593035 D0 X(10,1)=0.5863240475565120 D0 M(11,1)=3.4947818851582641 D0 X(11,1)=0.6244582053050142 D0 M(12,1)=3.5025169721393036 D0 X(12,1)=0.6708738701789345 D0 M(13,1)=3.5067720060071967 D0 X(13,1)=0.7245107163243557 D0 M(14,1)=3.5091445478460969 D0 X(14,1)=0.7828961533779115 D0 M(15,1)=3.5083822933519770 D0 X(15,1)=0.8424573558588187 D0
DO K=1,i !——Starting of main loop——!
R(K) =400.8636170476530082 D0 VB(K)=428.6624377313841023 D0
! Liquid flow rates DO n=1,NF
L0(n,k)=R0+FL
! Liquid-phase compositions and liquid holdups
X(6,k+1)=X(6,k)+(dt/M(6,k))*((L(7,k)*(X(7,k)-& X(6,k)))+(VS(5,k)*Y(5,k))-(VR(6,k)*Y(6,k))+(FV*YF)) IF(X(6,k+1)>1.0 D0)X(6,k+1)=1.0 D0
IF(X(6,k+1)<0.0 D0)X(6,k+1)=0.0 D0 DO n=7,14
M(n,k+1)=M(n,k)+dt*(L(n+1,k)-L(n,k)) X(n,k+1)=X(n,k)+(dt/M(n,k))*((L(n+1,k)*
& (X(n+1,k)-X(n,k)))+(VR(n-1,k)*Y(n-1,k))-(VR(n,k)*Y(n,k))) IF(X(n,k+1)>1.0 D0)X(n,k+1)=1.0 D0
IF(X(n,k+1)<0.0 D0)X(n,k+1)=0.0 D0 END DO
M(15,k+1)=M(15,k)+dt*(R(k)-L(15,k))
X(15,k+1)=X(15,k)+(dt/M(15,k))*((R(k)*(XD(k)-& X(15,k)))+(VR(14,k)*Y(14,k))-(VR(15,K)*Y(15,k))) IF(X(15,k+1)>1.0 D0)X(15,k+1)=1.0 D0
IF(X(15,k+1)<0.0 D0)X(15,k+1)=0.0 D0
XD(k+1)=XD(k)+(dt/MD)*((VR(15,k)*Y(15,k))-& (R(k)*XD(k))-(DL(k)*XD(k))-(DV*YD(K))) IF(XD(k+1)>1.0 D0)XD(k+1)=1.0 D0
IF(XD(k+1)<0.0 D0)XD(k+1)=0.0 D0 Time(k+1)=Time(k)+dt
PRINT*,Time(k),XB(k),X(1:15,K),M(1:15,K),XD(k)
WRITE(2,FMT=100)Time(k),XB(k),X(1:15,K),M(1:15,K),XD(k) 100 FORMAT (1X,33(2X,F21.16))
END DO !——End of main loop——!
END FILE 2 REWIND 2 CLOSE(2)
END PROGRAM DYNAMIC_IBDC
7.4 SUMMARY AND CONCLUSIONS
This chapter presents the detailed modelling and simulation of an ideal distillation column. A binary 1-propanol/ethanol mixture has been fractionated in the prescribed operation. In the process simulator, a relative-volatility-based VLE relationship has been used to predict the equilibrium properties. In addition, the internal liquid flow rates have been calculated using a simple Francis weir formula. Energy balance equations, phase nonidealities, rigorous tray hydraulics and some other vital considerations have been omitted from the process model for simplicity. In the subsequent chapters, of course, several realistic distillation simulators will be developed considering all these issues.
EXERCISES
7.1 Why is the distillation column that is considered in this chapter called ‘ideal distillation column’?
Explain it.
7.2 What is the physical significance of the term ‘hydraulic time constant’ in the Francis weir formula?
7.3 Why are the energy balance equations not included in the developed model of the ideal binary distillation column?
7.4 Why is the pressure in the base usually higher than that at the top of the distillation column?
7.5 Simulate the model structure, which is developed in this chapter for an ideal binary distillation column, using the fourth-order Runge–Kutta algorithm.
7.6 Develop a mathematical model for an ideal binary distillation column considering two partially vaporized feed streams. One is fed on 5th tray and other one is introduced on 10th tray. The same assumptions and equipment configurations as we considered in this chapter can be used.
REFERENCES
Franks, R.G.E. (1972). Modeling and Simulation in Chemical Engineering, 1st ed., John Wiley & Sons, New York.
Luyben, W.L. (1990). Process Modeling, Simulation, and Control for Chemical Engineers , 2nd ed., McGraw-Hill Book Company, Singapore.