16 - 17 October 2010, Perlis, Malaysia
Abstract - Distillation column processing is the most vital separation technology for today’s industry. An inclination to controlling distillation columns in a manner that is economically efficient depends much on the selection of reliable systems. This paper presents a detailed methodology to derive a mathematic modeling and simulation analysis for a distillation column based on continuous and non-linear model of the mass and energy balance. The mathematical modeling is then used as the reference model system to examine the ability of the developed distillate column tower dealing with the model-plant mismatch and feed disturbances inputs.
Index Terms— Mathematical modeling; Distillate column;
Dynamic simulation
I. INTRODUCTION
Distillation column processing is used to separate a mixture of different components, from crude oil or natural gas, as each component has different volatilities. With different boiling behaviors of each component, the lighter components normally carry a higher vapour concentration.
By stacking a number of trays with the boiling mixture on top of each other to a column, and allowing a certain amount of liquid and vapour to be exchanged between the trays, higher purities of the light and heavy components can be achieved towards the top and the bottom of the column. At the top of the column, the vapour is condensed into liquid and it partially flows out from the column as distillate, and eventually flowing back into the column [1][2].
At the bottom of the column, a partial portion of the accumulating liquid will flow out as bottom product and the remainder will be evaporated by re-boiling the liquid. The mixture feeds into the middle of the column. In column operations, a stable operation has to be ensured by controlling certain physical quantities inside the column, typically the temperature and pressure, and the levels in the condenser and reboiler. The L-V energy balance structure of the distillation system is the common energy control structure for the binary column system. In the energy structure the pressure to the column will remain at constant value as an assumption. The system control structure input is respectively classified dually as liquid flow rate, L and vapour flow rate, V [3].The Column system controller will uphold the product concentrations of column bottom
product, XB and distillate/overhead product, XD correspondingly regardless of the disturbance input feed flow, F and the feed concentration, ZF. (Figure 1)
Figure 1 : Distillation Flowsheet II. SCOPE
The scope of study is to identify an efficient optimization algorithm for the distillation column. In order to represent the realistic operation of an actual distillation column, a rigorous nonlinear mathematical model needs to be materialized. The mathematic model is, in principle, a collection of mathematical relationships among process variables which purport to describe the behavior of a physical system. The mathematical model therefore functions mainly as a convenient surrogate for the physical systems, making it possible to investigate system response under various input conditions both rapidly and inexpensively without necessarily tampering with the actual physical entity.
The distillation model is derived both from first principles involving dynamic material and component and algebraic energy equations supported by vapour–liquid equilibrium
Mathematical modeling and simulation of a distillation column
Khoo Boo Kean and Vu Trieu Minh
Mechanical Engineering Department, Universiti Teknologi Petronas, Bandar Seri Iskandar, 31750 Tronoh, Perak Darul Ridzuan, Malaysia.
Email: [email protected], [email protected]
Stripping Section Rectifying Section
Reboiler
Bottom product
X
BB ,
Boilup B B
Y V ,
Enriched
Condenser
Accumulator Feed
Condensate
ZF
F,
Heat flow Q Reflux
D
n X
L , Overhead product
XD
D,
B
B X
L ,
n n Y V ,
16 - 17 October 2010, Perlis, Malaysia and physical properties. The simplified case with
mathematical modeling will reduce the problem to its most elementary form so that the basic structure of the equations can be clearly evident. The design of an effective model is facilitated significantly by a proper understanding of the dynamic distillation process. Since process models are most effectively couched in the language of mathematics, it is fairly accurate to suggest therefore that the first step in the analysis and/or design of a system is the development of an appropriate mathematical model.
III. DISTILLATION COLUMN CONTROL
The L-V energy balance structure is the genuine control structure for a typical binary distillation column control. The reflux flow, L and the boil-up flow, V are the main output control variable to determine product output requirement.
However, the inner drum liquid holdups distillate flow rate, D and inner column bottom case flow rate, B are the associate variable that have momentous role towards product output requirement.
These are the material structures balance control, in abbreviation is called D-V and L-B structure. The differences between D-V versus the L-B structures are the roles of L and D are substitute among each other. The L-B structure is very responsive to feeding disturbances. The possibility of “inverse response” occurrences between the reboiler liquid level and the boil-up flow, will caused difficulties of controlling bottom section stock [7].
IV. MATHEMATICAL PROCESS MODEL
The requirement of the condensate output products is measure by purity of the distillate overhead product, XD, higher/equal than 98% and the impurity of the bottom product, XB, less/equal than 2%. A distillate tower which consists of 14 trays can be calculating mathematically to 16 non-linear differential equations. The feed stock are lumped some components together which will be used for analysis.
The feedstock is considered as a binary component as is the combination of liquefied petroleum gas (iso-butane, n- butane and propane) together with Naphthas (iso-pentane, n- pentane, and heavier components). The characteristic of the binary components will be fluctuate within below range depends on the distillate tower feeding composition. Under several assumptions [4], rigorous model equations are developed for 16 trays including reboiler and condenser section:
A. Model equation
(i) Condenser (N+2) :
D D N
f N N N
cX V V V Y L X D X
M =( + 1+ ) 1− , − ,
+
+ (1)
(ii) Feed Tray :
) , ,
(
) )(
(
1 1
1 1
+ +
+ +
+
−
− +
+
=
N N N N
n N f N N N N
X L X L
Y Y V V V X
M (2)
(iii) Upper feed :
f f N N N N
n N f N N N N
Y V X L X L
Y Y V V V X M
, ) , ,
(
) )(
(
1 1
1 1
+ +
−
− +
+
=
+ +
+
+ (3)
(iv) Bottom feed :
f f N N N N
n N f N N N N
Y L X L X L
Y Y V V V X M
, ) , ,
(
) )(
(
1 1
1 1
+ +
−
− +
+
=
+ +
+
+ (4)
(v) Reboiler :
B B
N f N N N B
Q V X B
X L L L X M
, ,
)
( 1
−
−
+ +
= +
(5)
B. Calculating liquid hold up (i) Feed rate calculation
The refinery is capable of processing 156,000 bpd running manufacturing for 24 hours non stop throughout the whole year. Therefore, the feed rate of the feedstock is as follow:
hour per barrel C
days X
hr C bpd
B f
B f
8082 . 17
365 24
000 , 156
=
=
Tons (metric) per hour:
hour per metric tons C
days X
hr
metric C tons
T f
T f
3745 . 2
365 24
800 , 20
=
=
* Assuming 7.5 barrel = 1 tons metric (ii) Liquid holdup calculation Liquid holdup on each tray
) ( 4
2
Molar d
M πhT h
=
kmole M
76 . 5
58 691 4
) 57 . 1 )(
25 . 0 )(
14 . 3
( 2
=
=
Liquid holdup in column base
) ( 4
2
B B B
B Molar
d T M πh
=
kmole MB
26 . 25
76 710 4
) 35 . 1 )(
89 . 1 )(
14 . 3
( 2
=
=
Liquid holdup in reflux drum
ond D MD L
sec 60
) (
5 +
=
kmole MD ond
12 . 13
sec 60
) 15 . 82 30 . 75 ( 5
=
+
=
In summary, liquid holdup based on calculation generated above are :
• Holdup on tray, M = 5.76 kmole
16 - 17 October 2010, Perlis, Malaysia
• Holdup in column base, MB = 25.26 kmole
• Holdup in reflux drum, MD = 13.12 kmole (iii) Flow rates calculation.
Tray at above feeding
L9=...=L15 =L=75.64kmole per hour
hour per kmole
hour per kmole V
V V
V F
62 . 164
51 . 98 11 . 66 ... 15
9
= +
= +
=
=
=
Tray at below feeding
hour per kmole
hour per kmole L
L L
L F
89 . 179
25 . 104 64 . 75 ... 8
1
= +
= +
=
=
=
hour per kmole V
V
V1=...= 8 = =66.11
Distillate flow : D = 92.76 kmole per hour Bottoms flow : B = 110.92 kmole per hour (iv) Flash equation
Flash equation solving via MATLAB are XF = 0.231 and YF = 0.629
C. Calculating vapour liquid equilibrium
Assume constant relative volatility throughout the column and perfect mixing equilibrium on all stages. The formula used,
j i ij K
K
α
= and looking up data in handbook for the operating range of temperature and pressure, the relative volatility is calculated as, αα= 5.68 [5]. Therefore, the vapour liquid equilibrium on each tray is:n n
n X
Y X
68 . 4 1+
=
α (6)
Tray 1:
1 1 1 1 4.68X Y X
+
=
α (7)
Tray 2:
2 2 2 1 4.68X Y X
+
=
α (8)
to…..
Tray 15 ;
15 15 15 1 4.68X Y X
+
=
α (9)
D. Nonlinear differential and empirical model
From the figure calculated the rigorous model is represented by a set of 31 nonlinear differential and empirical equations:
16 16
15
16 164.62 75.63 92.76
12 .
13 X = Y − X − X (10)
) (
63 . 75 ) (
62 . 164 76
.
5 X15 = Y14−Y15 + X16−X15 (11) )
( 63 . 75 ) (
62 . 164 76
.
5 X14 = Y13−Y14 + X15−X14 (12) )
( 63 . 75 ) (
62 . 164 76
.
5 X13 = Y12−Y13 + X14−X13 (13) )
( 63 . 75 ) (
62 . 164 76
.
5 X12 = Y11−Y12 + X13−X12 (14) ) (
63 . 75 ) (
62 . 164 76
.
5 X11 = Y10−Y11 + X12 −X11 (15) )
( 63 . 75 ) (
62 . 164 76
.
5 X10 = Y9−Y10 + X11−X10 (16) 969 . 61 ) (
63 . 75 62 . 164 11 . 66 76 .
5 X9= Y8− Y9+ X10−X9 + (17) 081 . 24 59 . 188 63 . 75 ) ( 11 . 66 76 .
5 X8= Y7−Y8 + X9− X8+ (18) )
( 89 . 179 ) ( 11 . 66 76
.
5 X7 = Y6−Y7 + X8−X7 (19) )
( 89 . 179 ) ( 11 . 66 76
.
5 X6 = Y5−Y6 + X7−X6 (20) )
( 89 . 179 ) ( 11 . 66 76
.
5 X5 = Y4−Y5 + X6−X5 (21) )
( 89 . 179 ) ( 11 . 66 76
.
5 X4= Y3−Y4 + X5−X4 (22) )
( 89 . 179 ) ( 11 . 66 76
.
5 X3= Y2−Y3 + X4−X3 (23) )
( 89 . 179 ) ( 11 . 66 76
.
5 X2= Y1−Y2 + X3−X2 (24)
1 1
2
1 179.89 66.11 110.92
26 .
25 X = X − Y − X (25)
V. SIMULATION RESULTS
Figure 2 is the MATHLAB model represented with 16 sets of nonlinear differential equations and Figure 3 is the general modular of the distillate tower.
16 - 17 October 2010, Perlis, Malaysia
Figure 2 Distillate tower module
Figure 3 General modular tray in the distillate tower
Figure 4 is the simulation results outcome of the values concentration for each liquefied petroleum gasoline tray based on the developed mathematical modeling. (Tray 1 to tray 14).
Figure 4 Steady state of concentration on each tray The feed rate is significant towards the upstream processes.
The changes of the feed stream, which the feed flow rate and feed composition changes is known as disturbance.
Table 1 shows the steady state values of concentration of XN and YN on each tray.
Stage Bottom Tray 1 Tray 2 Tray 3 Tray 4 Tray 5 Tray 6 Tray 7 Xn (Liquid flow) 0.00364 0.00970 0.02163 0.04322 0.07729 0.12070 0.16330 0.19540 yn (vapor flow) 0.02031 0.05271 0.11500 0.20420 0.32240 0.43810 0.52570 0.57980 Stage Tray 8 Tray 9 Tray 10 Tray 11 Tray 12 Tray 13 Tray 14 Distillate Xn (Liquid flow) 0.21620 0.21870 0.22620 0.24840 0.31020 0.45440 0.68680 0.98500 yn (vapor flow) 0.61040 0.61380 0.62410 0.65250 0.71870 0.82550 0.92570 0.99650
Table 1 Output concentration of XN and YN on each tray.
The output concentration meets the desire output purity with more than 98%. Feed disturbance feasibilities studies will perform on the design model under several conditions.
C. Feasibility study on feed disturbance influences
(i) Feed rate randomly increased to 2%
Input feed rate randomly raised 2% via uniform random number
Figure 5 Purity of the LPG versus time after feed rate raised to 2%.
Figure 6 Purity of LPG when noise increased 2%
The process operational goals is achieved as the purity of the LPG,
X
D is >98% after incurred 2 % random noise.(ii) Feed rate randomly decreased to 2%
Input feed rate randomly reduced 2% via uniform random number
Figure 7 Purity of the LPG versus time after feed rate reduced to 2%.
16 - 17 October 2010, Perlis, Malaysia
Figure 8 Purity of LPG when noise decreased 2%
The process operational goals is achieved as the purity of the LPG, XD is >98% after incurred 2 % random noise.
(iii) Feed rate randomly increased to 5%
Input feed rate randomly raised 5% via uniform random number
Figure 9 Purity of the LPG versus time after feed rate raised to 5%.
Figure 10 Purity of LPG when noise increased 5%
The process operational goals is not achieved as the purity of the LPG,
X
D is <98% after incurred 5% random noise.(iv) Feed rate randomly decreased to 5%
Input feed rate randomly reduced 5% via uniform random number
Figure 11 Purity of the LPG versus time after feed rate reduced to 5%.
Figure 12 Purity of LPG when noise decreased 5%
The process operational goals is not achieved as the purity of the LPG, XD is <98% after incurred 5% random noise.
Therefore, the simulation with disturbance exhibit the process operational stabilities will not satisfied if disturbances exceed its process limitation. The external disturbances compromise on feed flow rate and also feed composition fluctuation. Table 2 shows the summary when various feed rate randomly incurred.
Feed rate
Feed rate Increased 2%
Feed rate Reduced 2%
Feed rate Increased 5%
Feed rate Reduced 5%
98.54
97.78 89.92
0.36
0.84 0.22 98.26
98.27 0.39
0.39 Product XD (%)
Purity of the Distillate Impurity of the Bottoms Product XB (%)
Table 2 Product qualities on the changes of feed rates
16 - 17 October 2010, Perlis, Malaysia VI. CONCLUSION
Above paper described a procedure to set up a mathematic model of a condensate distillation column based on the energy balance (L-V) structure for analysis and simulation.
The mathematic modeling and simulation outcome achieved the desire operational objective of the product quality of XD≥98% when disturbance increased or decreased at 2%, but however when the random disturbances increased or decreased to 5% the purity of the output deteriorate. The design system is proficient to endure disturbance during feeding with limitation.
However, the calculation of mathematical model design and analysis is merely based on the physical laws of the process.
The actual process variables include the production factors;
process parameters tolerance and the real system validation are not discussed.
An appropriate modify MRAC system can be an option to enhance the distillate tower controlling to produced comparable output purification when dealing with greater plant divergence.
ACKNOWLEDGMENT
The authors would like to thank the comments provided by the anonymous reviewers and editor, which help the authors improve this paper significantly. The authors have taken into consideration all comments of the reviewers in the final version of the paper. This work was supported by Universiti Teknologi PETRONAS (UTP).
REFERENCES
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[2] James G Speight and Baki Ozum, “Petroleum Refining Process”, Marcel Dekker, Inc, 2002.
[3] Surinder Parkash, “Refining Processes Handbook”, Gulf Professional Publishing, New York, 2003.
[4] G. Speight, “Chemistry and Technology of Petroleum”, Marcel Dekker, New York, 2000.
[5] G. Fernholz and A. Gorak, “Dynamics and Control of Process Systems”, Gordon and Breach, Philadelphia, 2001.
[6] Johan G. Stichlmair and James R. Fair, “Distillation Principles and Practice”, Wiley-VCH, Santiago,1998.
[7] Micheal J. Grinble, “Robust Industrial Control System”, John Wiley and Sons Limited, San Francisco, 2006.
