Modeling and Control of Distillation Column in a Petroleum Process

Abstract

This paper introduces a calculation procedure for modeling and control simulation of a condensate distillation column based on the energy balance (𝐿-𝑉) structure. In this control, the reflux rate 𝐿 and the boilup rate 𝑉 are used as the inputs to control the outputs of the purity of the distillate overhead and the impurity of the bottom products. The modeling simulation is important for process dynamic analysis and the plant initial design. In this paper, the modeling and simulation are accomplished over three phases: the basic nonlinear model of the plant, the full-order linearised model, and the reduced-order linear model. The reduced-order linear model is then used as the reference model for a model-reference adaptive control (MRAC) system to verify the applicable ability of a conventional adaptive controller for a distillation column dealing with the disturbance and the model-plant mismatch as the influence of the plant feed disturbances.

1. Introduction

Distillation is the most popular and important separation method in the petroleum industries for purification of final products. Distillation columns are made up of several components, each of which is used either to transfer heat energy or to enhance mass transfer. A typical distillation column contains a vertical column where trays or plates are used to enhance the component separations, a reboiler to provide heat for the necessary vaporization from the bottom of the column, a condenser to cool and condense the vapor from the top of the column, and a reflux drum to hold the condensed vapor so that liquid reflux can be recycled back from the top of the column.

Calculation of the distillation column in this paper is based on a real petroleum project to build a gas processing plant to raise the utility value of condensate. The nominal capacity of the plant is 130 000 tons of raw condensate per year based on 24 operating hours per day and 350 working days per year. The quality of the output products is the purity of the distillate, 𝑥_𝐷, higher than or equal to 98% and the impurity of the bottoms, 𝑥_𝐵, less/equal than 2%. The basic feed stock data and its actual compositions are based on [1].

Most of distillation control systems, either conventional or advanced, assume that the column operates at a constant pressure. Pressure fluctuations make the control more difficult and reduce the performance. The L-V structure, which is called energy balance structure, can be considered as the standard control structure for a dual composition control distillation. In this control structure the liquid flow rate 𝐿 and the vapor flow rate 𝑉 are the control inputs. The objective of the controller is to maintain the product outputs concentrations 𝑥_𝐵 and 𝑥_𝐷 despite the disturbance in the feed flow 𝐹 and the feed concentration 𝑐_𝐹 (Figure 1).

Figure 1 

Distillation flowsheet.

The goals of this paper are twofold: first, to present a theoretical calculation procedure of a condensate column for simulation and analysis as an initial step of a project feasibility study, and second, for the controller design: a reduced-order linear model is derived such that it best reflects the dynamics of the distillation process and used as the reference model for a model-reference adaptive control (MRAC) system to verify the ability of a conventional adaptive controller for a distillation process dealing with the disturbance and the plant-model mismatch as the influence of the feed disturbances.

In this study, the system identification is not employed since experiments requiring a real distillation column are still not implemented yet. So that a process model based on experimentation on a real process cannot be done. A mathematical modeling based on physical laws is performed instead. Further, the MRAC controller model is not suitable for handling the process constraints on inputs and outputs as shown in [2] for a coordinator model predictive control (MPC). In this paper, the calculations and simulations are implemented by using MATLAB (version 7.0) software package.

2. Process Model and Simulation

The feed can be considered as a pseudobinary mixture of Ligas (iso-butane, 𝑛-butane and propane) and Naphthas (iso-pentane, 𝑛-pentane, and higher components). The column is designed with 𝑁=14 trays. The model is simplified by lumping some components together (pseudocomponents) and modeling of the column dynamics is based on these pseudocomponents only [3].

For the feed section, the operating pressure at the feed section is given at 4.6 atm. The feed temperature for the preheater is the temperature at which the required phase equilibrium is established. Consulting the equilibrium flash vaporization (EFV) curve at 4.6 atm, the required feed temperature is selected at 118^∘C corresponding to the point of 42% of the vapor phase feed rate 𝑉_𝐹.

For the rectifying section, the typical pressure drop per tray is 6.75 kPa. Thus, the pressure at the top section is 4 atm. Also consulting the Cox chart, the top section temperature is determined at 46^∘C. Then, we can calculate the reflux flow rate 𝐿 via the energy balance equation.

For the stripping section, the column base pressure is approximately the pressure of the feed section (4.6 atm) because the pressure drop across this section is neglected. Consulting the EFV curve and the Cox chart, the equilibrium temperature at this section (4.6 atm) is determined at 144^∘C. Then, we can calculate the reboiler duty or the heat input 𝑄_𝐵 to increase the temperature of stripping section from 118^∘C to 144^∘C.

Table 1 summarizes the initial calculated data for the main streams of input feed flow rate (Condensate), output distillate overhead product: (LPG) and output bottom product (Raw gasoline).

Table 1 

The main streams.

The vapor boilup 𝑉 generated by the heat input to the reboiler is calculated as [4]: 𝑉=(𝑄_𝐵−𝐵𝑐_𝐵(𝑡_𝐵−𝑡_𝐹))/𝜆 (kmole/h), where 𝑄_𝐵 is the heat input (kJ/h); 𝐵 is the flow rate of bottom product (kg/h); 𝑐_𝐵 is the specific heat capacity (kJ/kg⋅^∘C); 𝑡_𝐹 is the inlet temperature (^∘C); 𝑡_𝐵 is the outlet temperature (^∘C); 𝜆 is the latent heat or the heat of vaporization (kJ/kg). The latent heat at any temperature is described in terms of the latent heat at the normal boiling point [5] 𝜆=𝛾𝜆_𝐵(𝑇/𝑇_𝐵), where 𝜆 is the latent heat at the absolute temperature T in degrees Rankine (^∘R); 𝜆_𝐵 is the latent heat at the absolute normal boiling point 𝑇_𝐵 in degrees Rankine (^∘R); and 𝛾 is the correction factor obtained from the empirical chart.

Major design parameters to determine the liquid holdup on tray, column base and reflux drum are calculated mainly based on [6–8].

Velocity of vapor phase is arising in the column 𝜔_𝑛=𝐶(𝜌_𝐿−𝜌_𝐺)/𝜌_𝐺(m/s), where 𝜌_𝐿 (kg/m³) is the density of liquid phase; 𝜌_𝐺 (kg/m³) is the density of vapor phase; 𝐶 is the correction factor depending flow rates of two-phase flows.

The actual velocity 𝜔 is normally selected at 𝜔=(0.80−0.85)𝜔_𝑛 for paraffinic vapor.

The diameter of the column is calculated on the formula: 𝐷_𝑘=4𝑉_𝑚/3600𝜋𝜔(𝑚), where 𝑉_𝑚 (kmole/h) is the mean flow of vapor in the column.

The holdup in the column base is 𝑀_𝐵=(𝜋𝐻_𝑁𝐵𝐷²_𝑘/4)(𝜌_𝐵/(𝑀𝑊)_𝐵) (kmole), where 𝐻_𝑁𝐵 (m) is the normal liquid level in the column base; (𝑀𝑊)_𝐵 is the molar weight of the bottom product (kg/kmole); 𝜌_𝐵 is the density of the bottom product (kg/m³).

Similarly, the holdup on each tray is 𝑀=(0.95𝜋ℎ_𝑇𝐷²_𝑘/4)(𝜌_𝑇/(𝑀𝑊)_𝑇) (kmole), where ℎ_𝑇 is the average depth of clear liquid on a tray (m); (𝑀𝑊)_𝑇 is the molar weight of the liquid holdup on a tray (kg/kmole); 𝜌_𝑇 is the mean density of the liquid holdup on a tray (kg/m³). And the holdup in the reflux drum 𝑀_𝐷=5(𝐿_𝑓+𝑉_𝑓)/60 (kmole), where 𝐿_𝑓 is the reflux flow rate (kmole/h); 𝑉_𝑓 is the distillate flow rate (kmole/h).

The rate of accumulation of material in a system is equal to the amount entered and generated, less the amount leaving and consumed within the system. The model is simplified under assumptions in [9].

(i) Constant relative volatility throughout the column and the vapor-liquid equilibrium relation can be expressed by 𝑦_𝑛=𝛼𝑥_𝑛1+(𝛼−1)𝑥_𝑛, (2.1) where 𝑥_𝑛 is the liquid concentration on 𝑛th stage; 𝑦_𝑛 is the vapor concentration on 𝑛th stage; 𝛼 is the relative volatility.(ii) The overhead vapor is totally condensed.(iii) The liquid holdups on each tray, the condenser, and the reboiler are constant and perfectly mixed.(iv) The holdup of vapor is negligible throughout the system(v) The molar flow rates of the vapor and liquid through the stripping and rectifying sections are constant.

Under these assumptions, the dynamic model can be expressed by the following equations:

(i)  condenser (𝑛=𝑁+2):

𝑀_𝐷𝑥̇_𝑛=(𝑉+𝑉_𝐹)𝑦_𝑛−1−𝐿𝑥_𝑛−𝐷𝑥_𝑛, (2.2)

(ii)  tray 𝑛(𝑛=𝑓+2to𝑁+1):

𝑀𝑥̇_𝑛=(𝑉+𝑉_𝐹)(𝑦_𝑛−1−𝑦_𝑛)+𝐿(𝑥_𝑛+1−𝑥_𝑛), (2.3)

(iii)  tray above the feed flow (𝑛=𝑓+1):

𝑀𝑥̇_𝑛=𝑉(𝑦_𝑛−1−𝑦_𝑛)+𝐿(𝑥_𝑛+1−𝑥_𝑛)+𝑉_𝐹(𝑦_𝐹−𝑦_𝑛), (2.4)

(iv)  tray below the feed flow (𝑛=𝑓):

𝑀𝑥̇_𝑛=𝑉(𝑦_𝑛−1−𝑦_𝑛)+𝐿(𝑥_𝑛+1−𝑥_𝑛)+𝐿_𝐹(𝑥_𝐹−𝑥_𝑛), (2.5)

(v)  tray 𝑛(𝑛=2to𝑓−1):

𝑀𝑥̇_𝑛=𝑉(𝑦_𝑛−1−𝑦_𝑛)+(𝐿+𝐿_𝐹)(𝑥_𝑛+1−𝑥_𝑛), (2.6)

(vi)  reboiler (𝑛=1):

𝑀_𝐵𝑥̇₁=(𝐿+𝐿_𝐹)𝑥₂−𝑉𝑦₁−𝐵𝑥₁. (2.7) Although the model is simplified, the representation of the distillation system is still nonlinear due to the vapor-liquid equilibrium relationship between 𝑦_𝑛 and 𝑥_𝑛 in (2.1).

The distillation process simulation is done using Matlab Simulink as shown in Figure 2. The dynamic model is represented by a set of 16 nonlinear differential equations: 𝑥₁=𝑥_𝐵 is the liquid concentration in bottom; 𝑥₂ is the liquid concentration in the 1st tray, 𝑥₃ is the liquid concentration in the 2nd tray;…;𝑥₁₅ is the liquid concentration in the 14th tray; and 𝑥₁₆=𝑥_𝐷 is the liquid concentration in the distillate.

Figure 2 

Model simulation with Matlab Simulink.

If there are no disturbance in the operating conditions as shown in Figure 3, the system is to reach the steady state such that the purity of the distillate product 𝑥_𝐷equals 0.9654 and the impurity of the bottoms product 𝑥_𝐵equals 0.0375.

Figure 3 

The steady-state values of concentrations 𝑥𝑛 on each tray.

Table 2 indicates the steady-state values of concentration of 𝑥_𝑛 and 𝑦_𝑛 on each tray.

Table 2 

The steady state values of concentrations 𝑥_𝑛 and 𝑦_𝑛 on each tray.

Since the feed stream depends on the upstream processes, the changes of the feed stream can be considered as disturbances including the changing in feed flow rates and feed compositions. Simulations with these disturbances indicate that the quality of the output products gets worse if the disturbances exceed some certain ranges as shown in Table 3.

Table 3 

Product quality depending on the change of the feed rates.

The designed system does not achieve the operational objective of the product quality (𝑥_𝐷≥0.98 and 𝑥_𝐵≤0.02) and the product quality will get worse dealing with disturbances. Hence we will use an adaptive controller—MRAC—to take the system from these steady-state outputs of 𝑥_𝐷=0.9654 and 𝑥_𝐵=0.0375 to the desired output targets.

3. Linearization of the Distillation Process

In order to obtain a linear control model for this nonlinear system, we assume that the variables deviate only slightly from some operating conditions [10]. Then the nonlinear equation in (2.1) can be expanded into a Taylor’s series. If the variation 𝑥_𝑛−𝑥_𝑛 is small, we can neglect the higher-order terms in 𝑥_𝑛−𝑥_𝑛. The linearization of the distillation column leads to a 16th-order linear model in the state space form:

𝑧(̇𝑡)=𝐴𝑧(𝑡)+𝐵𝑢(𝑡),𝑦(𝑡)=𝐶𝑧(𝑡), (3.1) where

𝑥𝑧(𝑡)=¹(𝑡)−𝑥_1SteadyState𝑥₂(𝑡)−𝑥_2SteadyState⋮𝑥₁₆(𝑡)−𝑥_{16SteadyState},𝑢(𝑡)=𝐿(𝑡)−𝐿_SteadyState𝑉(𝑡)−𝑉_SteadyState,𝑥𝑦(𝑡)=¹(𝑡)−𝑥_1SteadyState𝑥₁₆(𝑡)−𝑥_{16SteadyState}. (3.2) The matrix A elements (n for each stage) are

(i)  reboiler:

for𝑛=1,𝑎_1,1=−(𝐾₁𝑉+𝐵)𝑀_𝐵,𝑎_1,2=(𝐿+𝐿_𝐹)𝑀_𝐵, (3.3)

(ii)  stripping section, tray 1÷6:

for𝑛=2÷7,𝑎_{𝑛,𝑛−1}=(𝐾_𝑛−1𝑉)𝑀,𝑎_𝑛.𝑛=−(𝐾_𝑛𝑉+𝐿+𝐿_𝐹)𝑀,𝑎_𝑛,𝑛+1=(𝐿+𝐿_𝐹)𝑀, (3.4)

(iii)  feeding section, tray 7÷8: for𝑛=8,𝑎_8,7=(𝐾₇𝑉)𝑀,𝑎_8.8=−(𝐾₈𝑉+𝐿+𝐿_𝐹)𝑀,𝑎_8,9=(𝐿)𝑀,for𝑛=9,𝑎_9,8=(𝐾₈𝑉)𝑀,𝑎_9.9=−(𝐾₉𝑉+𝐿)𝑀,𝑎_9,10=(𝐿)𝑀, (3.5)

(iv)  rectifying section, tray 9÷14:

for𝑛=10÷15,𝑎_{𝑛,𝑛−1}=(𝐾_𝑛−1(𝑉+𝑉_𝐹))𝑀,𝑎_𝑛.𝑛=−(𝐾_𝑛(𝑉+𝑉_𝐹)+𝐿)𝑀,𝑎_𝑛,𝑛+1=(𝐿)𝑀 (3.6)

(v)  condenser:

for𝑛=16,𝑎_16,15=(𝐾₁₅(𝑉+𝑉_𝐹))𝑀_𝐷,𝑎_16,16(=−𝐿+𝐷)𝑀_𝐷, (3.7) where 𝐾_𝑛 is the linearized Vapor-Liquid Equilibria (VLE) constant:

𝐾_𝑛=𝑑𝑦_𝑛𝑑𝑥_𝑛=𝛼(1+(𝛼−1)𝑥_𝑛)²=5.68(1+4.68𝑥_𝑛)². (3.8) The matrix 𝐵 elements are

for𝑛=1,𝑏_1,1=(𝑥₂)𝑀_𝐵𝐿,b_1,2(=−𝑦₁)𝑀_𝐵𝑉,for𝑛=2÷15,𝑏_𝑛,1=(𝑥_𝑛+1−𝑥_𝑛)𝑀𝐿,𝑏_𝑛,2(=−𝑦_𝑛−𝑦_𝑛−1)𝑀𝑉,for𝑛=16,𝑏_16,1(=−𝑥₁₆)𝑀_𝐷𝐿,𝑏_16,2(𝑦₁₅)𝑀_𝐷𝑉. (3.9) The output matrix 𝐶 is

𝐶=10000000000000000000000000000001. (3.10) The full-order linear model which represents a two inputs-two outputs plant in equation in (3.3) can be expressed as a reduced order linear model as in [11, 12]:

𝑥_𝐷𝑥_𝐵=11+𝜏_𝑐𝑠𝐿𝑉𝐺(0), (3.11) where 𝐺(0) is the steady-state gain: 𝐺(0)=−𝐶𝐴⁻¹𝐵, 𝜏_𝑐 is the time constant: 𝜏_𝑐=𝑀_𝐼𝐼_𝑠+𝑀ln𝑆^𝐷(1−𝑥_𝐷)𝑥_𝐷𝐼_𝑠+𝑀_𝐵(1−𝑥_𝐵)𝑥_𝐵𝐼_𝑠, (3.12) where 𝑀_𝐼 (kmole) is the total holdup of liquid inside the column; 𝑀_𝐷 (kmole) is the liquid holdup in the condenser; 𝑀_𝐵 (kmole) is the liquid holdup in the reboiler; 𝐼_𝑠 is the “impurity sum”; 𝑆 is the separation factor.

As the result of calculation, the reduced-order linear model of the plant is a first-order system with a time constant of 𝜏_𝑐=1.9588(ℎ):

𝑥_𝐷𝑥_𝐵=1𝐿1+1.9588𝑠𝑉0.0042−0.0062−0.00520.0072. (3.13) Equation (3.13) is equivalent to the following linear model in state space: 𝑧̇_𝑟(𝑡)=𝑧−0.510500−0.5105^𝑟𝑦(𝑡)+1001𝑢(𝑡),_𝑟(𝑡)=𝑧0.0021−0.0031−0.00260.0037^𝑟(𝑡), (3.14) Where 𝑧_𝑟=[^𝑧𝑟1_𝑧𝑟2] are state variable, 𝑢=[^𝑑𝐿_𝑑𝑉] are two manipulated inputs, and 𝑦_𝑟=[^𝑑𝑥𝐵_𝑑𝑥𝐷] are two outputs of LPG and gasoline product.

Stability test. The system is asymptotically stable since all eigenvalues of the state matrix are in the left half of the complex plane ([−0.5105,−0.5105]).

4. MRAC Building and Simulation

Adaptive control system is the ability of a controller which can adjust its parameters in such a way as to compensate for the variations in the characteristics of the process. Adaptive control is widely applied in petroleum industries because of the two main reasons: firstly, most of processes are nonlinear and the linearized models are used to design the controllers, so that the controller must change and adapt to the model-plant mismatch; secondly, most of the processes are nonstationary or their characteristics are changed with time, and this leads again to adapt the changing control parameters.

Figure 4 

MRAC block diagram.

The general form of an MRAC is based on an inner-loop Linear Model Reference Controller (LMRC) and an outer adaptive loop shown in Figure 4. In order to eliminate errors between the model and the plant and the controller is asymptotically stable, MRAC will calculate online the adjustment parameters in gains 𝐿 and 𝑀 by 𝜃_𝐿(𝑡) and 𝜃_𝑀(𝑡) as detected state error 𝑒(𝑡) when changing 𝐴, 𝐵 in the process plant.

Simulation program is constructed using Maltab Simulink with the following data.

(1)  Process Plant:
𝑧̇=𝐴𝑧+𝐵𝑢+noise,𝑦=𝐶𝑧, (4.1) where 𝐴=[^𝛼10_0𝛼2], 𝐵=[^𝛽10_0𝛽2], 𝐶=[^{0.004−0.007}_{−0.00110.0017}], and 𝛼₁, 𝛼₂, 𝛽₁, 𝛽₂ are changing and dependent on the process dynamics.

(2)  Reference Model:
𝑧̇_𝑚=𝐴_𝑚𝑧_𝑚+𝐵_𝑚𝑢_𝑐,𝑦_𝑚=𝐶_𝑚𝑧_𝑚, (4.2) where 𝐴_𝑚=[^−0.26160_0−0.2616], 𝐵_𝑚=[¹⁰₀₁], 𝐶_𝑚=[^{0.004−0.007}_{−0.00110.0017}]

(3)  State Feedback:
𝑢=𝑀𝑢_𝑐−𝐿𝑧, (4.3) where 𝐿=[^𝜃10_0𝜃2] and 𝑀=[^𝜃30_0𝜃4].

(4)  Closed Loop:
𝑧̇=(𝐴−𝐵𝐿)𝑧+𝐵𝑀𝑢_𝑐=𝐴_𝑐(𝜃)𝑧+𝐵_𝑐(𝜃)𝑢_𝑐 (4.4)

(5)  Error Equation:
𝑒=𝑧−𝑧_𝑚=[^𝑒1_𝑒2] is a vector of state errors, 𝑒̇=𝑧̇−𝑧̇_𝑚=𝐴𝑧+𝐵𝑢−𝐴_𝑚𝑧_𝑚−𝐵_𝑚𝑢_𝑐=𝐴_𝑚𝑒+(𝐴_𝑐(𝜃)−𝐴_𝑚)𝑧+(𝐵_𝑐(𝜃)−𝐵_𝑚)𝑢_𝑐=𝐴_𝑚𝑒+Ψ(𝜃−𝜃⁰), (4.5) where Ψ=[^{−𝛽1𝑧10𝛽1𝑢𝑐10}_{0−𝛽2𝑧20𝛽2𝑢𝑐2}].

(6)  Lyapunov Function:
1𝑉(𝑒,𝜃)=2(𝛾𝑒^𝑇𝑃𝑒+(𝜃−𝜃⁰)^𝑇(𝜃−𝜃⁰)), (4.6) where 𝛾 is an adaptive gain and 𝑃 is a chosen positive matrix.

(7)  Derivative Calculation of Lyapunov Function:
𝑑𝑉𝛾𝑑𝑡=−2𝑒^𝑇𝑄𝑒+(𝜃−𝜃⁰)^𝑇(𝑑𝜃𝑑𝑡+𝛾Ψ^𝑇𝑃𝑒), (4.7) where 𝑄=−𝐴^𝑇_𝑚𝑃−𝑃𝐴_𝑚.
For the stability of the system, 𝑑𝑉/𝑑𝑡<0, we can assign the second item (𝜃−𝜃⁰)^𝑇((𝑑𝜃/𝑑𝑡)+𝛾Ψ^𝑇𝑃𝑒)=0 or 𝑑𝜃/𝑑𝑡=−𝛾Ψ^𝑇𝑃𝑒. Then we always have 𝑑𝑉/𝑑𝑡=−(𝛾/2)𝑒^𝑇𝑄𝑒. If we select a positive matrix 𝑃>0, for instance, 𝑃=[¹⁰₀₂], then we have 𝑄=−𝐴^𝑇_𝑚𝑃−𝑃𝐴_𝑚=[^0.52320_01.0465]. Since matrix 𝑄 is obviously positive definite, then we always have 𝑑𝑉/𝑑𝑡=−(𝛾/2)𝑒^𝑇𝑄𝑒<0 and the system is stable with any plant-model mismatches.

(8)  Parameters Adjustment:
𝑑𝜃𝑑𝑡=−𝛾−𝛽₁𝑧₁00−𝛽₂𝑧₂𝛽𝑐₁𝑢₁00𝛽₂𝑢_2𝑐𝑒[𝑃]¹𝑒₂=𝑑𝜃₁/𝑑𝑡𝑑𝜃₂/𝑑𝑡𝑑𝜃₃/𝑑𝑡𝑑𝜃₄=/𝑑𝑡𝛾𝛽₁𝑧₁𝑒₁2𝛾𝛽₂𝑧₂𝑒₂−𝛾𝛽₁𝑢_𝑐1𝑒₁−2𝛾𝛽₂𝑢_𝑐2𝑒₂. (4.8)

(9)  Simulation Results and Analysis:
We assume that the reduced-order linear model in (3.14) can also maintain the similar steady-state outputs as the basic nonlinear model. Now we use this model as an MRAC to take the process plant from these steady-state outputs (𝑥_𝐷=0.9654 and 𝑥_𝐵=0.0375) to the desired targets (0.98≤𝑥_𝐷≤1 and 0≤𝑥_𝐵≤0.02) amid the disturbances and the plant-model mismatches as the influence of the feed stock disturbances.
The design of a new adaptive controller is shown in Figure 5 where we install an MRAC and a closed-loop PID (Proportional, Integral, Derivative) controller to eliminate the errors between the reference setpoints and the outputs.
We run this controller system with different plant-model mismatches, for instance, a plant with 𝐴=[^−0.500_0−0.75], 𝐵=[^1.50_02.5] and an adaptive gain 𝛾=25. The operating setpoints for the real outputs are 𝑥_𝐷𝑅=0.99 and 𝑥_𝐵𝑅=0.01. Then, the reference setpoints for the PID controller are 𝑟_𝐷=0.0261 and 𝑟_𝐵=−0.0275 since the real steady-state outputs are 𝑥_𝐷=0.9654 and 𝑥_𝐵=0.0375. Simulation in Figure 6 shows that the controlled outputs 𝑥_𝐷 and 𝑥_𝐵 are always stable and tracking to the model outputs and the reference setpoints (the dotted lines, 𝑟_𝐷 and 𝑟_𝐵) amid the disturbances and the plant-model mismatches.

Figure 5 

Adaptive controller with MRAC and PID.

Figure 6 

Correlation of plant outputs, model outputs, and reference setpoints.

5. Conclusion

We have introduced a procedure to build up a mathematical model and simulation for a condensate distillation column based on the energy balance (L-V) structure. The mathematical modeling simulation is accomplished over three phases: the basic nonlinear model, the full-order linearized model and the reduced-order linear model. Results from the simulations and analysis are helpful for initial steps of a petroleum project feasibility study and design.

The reduced-order linear model is used as the reference model for an MRAC controller. The controller of MRAC and PID theoretically allows the plant outputs tracking the reference setpoints to achieve the desired product quality amid the disturbances and the model-plant mismatches as the influence of the feed stock disturbances.

In this paper, the calculation of the mathematical model building and the reduced-order linear adaptive controller is only based on the physical laws from the process. The real system identifications including the experimental production factors, specific designed structures, parameters estimation, and the system validation are not mentioned here. Further, the MRAC controller is not suitable for the on-line handling of the process constraints.