20th European Symposium on Computer Aided Process Engineering – ESCAPE20 S. Pierucci and G. Buzzi Ferraris (Editors) © 2010 Elsevier B.V. All rights reserved.
The Steady State Modeling and Simulation of the Chemical Multivariable Systems - Case Studies Cristian Patrascioiu,a Daniel Mihaescu b a
Petroleum – Gas University of Ploiesti, Ploiesti, 39 39 Bd. Bucuresti, Bucuresti, 100680, Romania, Romania,
[email protected] b Oltchim S.A., Ramnicu Valcea, 1 Uzinei Street, 240050, Romania,
[email protected]
Abstract The steady state mathematical modeling of the chemical processes represents an important problem, for both: design stage and chemical plants operation. A class of the chemical processes is the multivariable process systems, characterized by two manipulated variables and two output variables. For these chemical processes, the steady state model may be defined by the general non-linear equations system. The author has studied two particular chemical processes: the radiation section of the tubular heater and the shell and tube heat exchanger. The mathematical model of the radiation section of the tubular heater is developed by using the heat balance equations and the Lobo-Evans model of the heat transfer process. This mathematical model has been used to evaluate the statically characteristics and those that have represented the startup of the development of a new control system. The second case study is represented by the shell and tube heat exchanger. The solutions of this model have been used to determine the characteristics of the chemical process too. Keywords: Keywords: mathematical model, non-linear equations, Jacobian matrix, simulation, steady state characteristics
1. Introduction The chemical systems are characterized by a great complexity, being also classified into many classes: systems with concerted or distributed parameters, monovariable or multivariable systems [1]. The design and the control of the multivariable chemical processes are more difficult as compared in relation to the design and the control of the monovariable chemical systems [2, 3]. To increase the efficiency of the design activity, the chemical systems must be mathematically modeled and numerically simulated into statical and dynamical state too [4, 5]. The authors have studied the steady state modeling and the numerically simulation for two multivariable chemical systems.
2. The structure and the general model of the multivariable chemical systems A class of the standard multivariable chemical processes is characterized by two input variables and two output variables, figure 1. For theses chemical processes, the steady state model may be defined by the general non-linear equations system
⎧⎪ f 1( y1 , y 2 ) = 0 ⎨ f ( y , y ) = 0 . ⎩⎪ 2 1 2
(1)
C. Patrascioiu et al.
Fig. 1 The two inputs – two outputs system
The solution of this non-linear equations system is obtained by using the Newton Raphson or Broyden numerical algorithm. The Jacobian matrix, associated to the nonlinear equations system (1), has the form
⎡ ∂ f 1 ⎢ ∂ y J = ⎢ 1 ⎢ ∂ f 2 ⎢⎣ ∂ y1
∂ f 1 ⎤ ∂ y 2 ⎥⎥ . ∂ f 2 ⎥ ∂ y1 ⎥⎦
(2)
The authors have studied two multivariable chemical systems characterized by two inputs – two outputs: the radiation section of the tubular heater and the shell and tube heat exchanger.
3. Case studies 1. The radiation section of the tubular heater The most important type of the refinery’s heaters is the parallelepiped heater, with one radiation section and one convection section. The radiation section block scheme is presented in figure 2a.
Fig. 2 The structure of the radiation section of the tubular heater: a) the two inputs – two outputs multivariable system; b) internal structure of the radiation section
This chemical system has two input variables: α – the air coefficient and Q fuel – the fuel flow rate. The output variables of the radiation section of the tubular heater are: T g - the specifical burn gases temperature into radiation section, T out – the feed output temperature. The internal structure of the radiation section of the tubular heater is more complicate. This structure has been built by using two subsystems: the combustion subsystem and the heat transfer subsystem, figure 2b. The combustion subsystem is a monovariabile system but the heat transfer subsystem is a multivariable system. The mathematical model of the heat transfer is based on the heat balance relations and the Lobo – Evans model [6, 7]
⎧⎪ f 1 (T g , T out ) = 0 ⎨ f (T , T ) = 0 . ⎪⎩ 2 g out
(3)
The internal variables used in the model are: F – the radiation coefficient of the LoboEvans model, e g – the emission burn gases coefficient, ψ - the screening factor, T e – the
The Steady State Modeling and Simulation of the C hemical Multivariable Systems Case Studies medium temperature of the fluid, Q feed – the feed flow rate. In order to simplify the mathematical model, the authors have introduced the constant coefficients z 1 K z 8 as the functions f 1 and f 2 [6]. The concrete form of the functions f 1 and f 2 are: 4
n
k =1
i =1
(
)
i i =0; f 1 = z 5 − z 6 + Q feed k 2 T out + Q fuel ∑ n k ∑ a ki T g
(4)
) )
4 n z 2 − 17 T g + δ i i − 3,6σ A × f 2 = z 6 − Q fuel ∑ n k ∑ a ki T g ee z 3 + z 4 T g + δ k =1 i =1
(
)
(
× (T g + δ + 273)4 − 625 × 10 −4 (T out + T in + 546)4 − z 7 (T g + δ ) + 0,5 z 7 T out + z 8 = 0
(5)
The radiation coefficient of the Lobo-Evans model is considered a non-linear function of the burn gases temperature and the outlet enthalpy of the feed is considered a linear function of the outlet feed temperature [6]. The Jacobian of the non – linear equation system (3) contains the next elements: 4
n
(
)
i −1 ; ∂ f 1 ∂ T g = Q fuel ∑ nk ∑ aki T g = 1 i k =1
(6)
∂ f 1 ∂ T out = Q feed k 2 ;
(7)
4 n z 2 z 4 + 17 z 3 i −1 ∂ f 2 ∂ T g = −Q fuel ∑ n k ∑ aki T g + 14 ,4σ Aee × [ z 3 + z 4 (T g + δ )]2 k =1 i =1
(
)
× (T g + δ + 273)4 − 625 × 10 − 4 (T out + T in + 546)4 − z 2 − 17 T g + δ
− 14 ,4σ Aee
(
z 3 + z 4 T g + δ
∂ f 2 ∂ T out = 0 ,9 σ Aee
)
× (T g + δ + 273)3 − z 7 .
z 2 − 17 T g + δ
(
z 3 + z 4 T g + δ
)
(T out + T in + 546)3 + 0.5 z 7
(8)
(9)
The authors have elaborated a special simulation program for the radiation section of the crude unit tubular heater. Using the numerical simulations, the authors have studied the statically characteristic of the tubular heater [7]. Figure 3 shows two statical characteristics which depend of the air coefficient variable: the outlet temperature of the feed and the carbon monoxide concentration into burn gases. The outlet temperature of the feed characteristic has a maximum point corresponding to the optimal value of the air coefficient, figure 3a. This characteristic has been used to design the optimal combustion controller for the tubular heater [7]. The carbon monoxide characteristic, figure 3b, represents an important element of the combustion control system because the small values [1K 4] ppm of the carbon monoxide into burn gases determines the real optimal operating point of the tubular heater.
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Fig. 3 The statically characteristic of the tubular heater: a) Outlet fluid feed temperature; b) Carbon monoxide concentration into burn gases
4. Case study 2. The shell and tube heat exchanger The authors have studied, modeled and simulated the shell and tube heat exchanger having fluxes in counter flow [8]. This heat exchanger is characterized by four inlet and two outlet variables, figure 4. The inlet variables are: T h1, Qhot – the input hot temperature and the hot flow rate, T cl , Qcold – the input cold temperature and the flow rate of cold fluid. The outlet variables are: T h2 - the outlet temperature of the hot fluid, T c2 – the outlet temperature of the cold fluid.
Fig. 4 The shell and tube heat exchanger: a) exchanger section; b) structure of the system
Mathematical model of heat exchanger is obtained by using heat balance equations associated with the hot flow and the cold flow, and the Delaware model of the heat exchanger [9]. The authors have obtained a compact form of the mathematical model of the exchanger, similarly to equations system (1) ⎧ f 1 (T h 2 , T c 2 ) = 0 . ⎨ ⎩ f 2 (T h 2 , T c 2 ) = 0
(10)
The more important and complex internal variable is the thermal transfer global factor k which includes the effect of the convection heat transfer into tube-side and shell-side. The non – linear functions of the equations system (10) are in the next form: f 1 = Qhot c p, hot (T h1 − T h 2 ) − Qcold c p, cold (T c 2 − T c1 ) ;
(11)
The Steady State Modeling and Simulation of the Chemical Multivariable Systems Case Studies f 2 = Qhot c p, hot (T h1 − T h 2 ) − kA
(T h1 − T c2 ) − (T h2 − T c1 ) ln
T h1 − T c 2
.
(12)
T h 2 − T c1
The Jacobian matrix associated to the equations system (10) has four components: ∂ f 1 = −Qhot c p, hot ; ∂T h 2
(13)
∂ f 1 = −Qcold c p, cold ; ∂T c 2
(14)
∂ f 2 = −Qhot c p, hot − k A ∂T h 2
∂ f 2 = −k A ∂T c 2
ln
T h1 − T c 2 T h 2 − T c1
− ln
T h1 − T c 2 T h 2 − T c1
− (T h1 − T h 2 + T c1 − T c 2 ) ⎛ T h1 − T c 2 ⎞ ⎜ ln ⎟ ⎜ T − T ⎟ ⎝ h 2 c1 ⎠
− (T h1 − T h 2 + T c1 − T c 2 ) ⎛ T h1 − T c 2 ⎞ ⎜ ln ⎟ ⎜ T − T ⎟ ⎝ h 2 c1 ⎠
2
2
1 T h 2 − T c1
;
(15)
1 T h1 − T c 2
.
(16)
Recently, the authors have elaborated the simulation program for a heat exchanger with the shell and tube used into crude unit process. The hot fluid is the kerosene reflux flow of the crude unit column. The normal operating point of the heat exchanger is defined by the next values: the hot input temperature T h1 = 180°C , the hot flow rate Qh = 163,000 kg / h ,
the
input
cold
temperature T c1 = 103°C ,
the
cold
flow
rate Qc = 500,000 kg / h . The constructive data of the heat exchanger have been defined in [10] and the physically properties of the hot and cold fluid have been calculated using the estimated medium temperature. For the industrial problem formulated in [10] and using the simulation program developed by authors have been calculated the numerical output values: the hot output temperature T h2 = 138.6°C (140°C in the reference) and the cold output temperature T c 2 = 118.1°C (118°C). For theses reasons, the authors have considered that the proposed model and the simulation program have been validated. Using the simulation program, the authors have studied the statically characteristics of this shell and tube heat exchanger. Figure 5-a shows the dependence of the hot fluid flow in function to the output temperature of the heat exchanger. The increased of the hot flow rate in the [100K 200] % domain, in rapport to the normal operating point, do not increase into same percent the cold fluid temperature (max. 4 Celsius degrees). In theses conditions and for this heat exchanger, the hot flow rate is not recommendable for the manipulated variable of the hypothetically temperature control system. The disturbances of the process have different effects: when the input hot fluid temperature increase with 18%, the output cold fluid temperature increases with 6%, figure 5-b. In conclusion, the output variables of the heat exchanger with the shell and tube have an important sensibility to disturbance input temperatures and the output variables are not sensible to flow rate of the input flows.
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Fig. 5 The statically characteristics of the shell and tube heat exchanger: a) the outlet temperatures versus the hot fluid flow rate; b) the outlet temperatures versus the input hot temperature
5. Acknowledgements The authors have studied the steady-state modeling of the two multivariable chemical processes. For each process, the mathematical model is represented by a non-linear equations system. The solutions of the mathematical models have obtained with the same numerical method, respectively the Newthon-Raphson method. The authors have elaborated the numerical simulation programs for each process. The mathematical models and the simulation programs have been validated using the comparison of the numerical results with the industrial or literature data. The numerically results have been utilized to develop the next statically characteristics: the output variable versus the manipulated variable or the output variable versus the disturbance variable. Theses characteristics will be used in the future for design of the control systems associated of the two chemical processes.
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