Modeling of Multistage Air-Compression Systems in Chemical

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Ind. Eng. Chem. Res. 2003, 42, 2209-2218

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Modeling of Multistage Air-Compression Systems in Chemical Processes In-Su Han and Chonghun Han* Department of Chemical Engineering, Pohang University of Science and Technology, Pohang, Kyungbuk 790-784, South Korea

A systematic modeling method is proposed to predict the overall efficiency and power requirement of a multistage air-compression system consisting of a multistage compressor, a multistage expander, and an electric motor/generator. First, under a lack of measurements at the interstages of the multistage compression system, a calculation method of the overall efficiency is developed based on thermodynamic compression and expansion equations. Then, the prediction structure of the overall efficiency and actual power consumption/generation rate of the compression system is presented, where either partial least-squares models or artificial neural network models can be employed to predict unknown state variables and the overall efficiency, which are, in turn, used to predict the actual power consumption/generation rate based on compression and expansion equations. Finally, to verify the proposed modeling method, it is applied to two industrial compression systems of a terephthalic acid manufacturing process, and the comparisons with measurements show excellent prediction performances both in the overall efficiency and in the actual power consumption rate. The proposed modeling method can be applied to find better operating conditions of compression systems in various chemical processes. 1. Introduction Air, along with steam and electricity, is the essential utility to operate chemical processes. Hence, most of chemical plants are equipped with compression systems of various types and capacities for pressurizing air. These compression systems can be typically categorized into centrifugal compressors or blowers, reciprocating compressors, and rotary compressors and are selected for use depending on the given process conditions.1 Among these compression systems, the centrifugal compressors or blowers, typically consisting of several stages, are widely used for supplying a large amount of compressed air or gases for various chemical processes such as fluidized catalytic cracking units, terephthalic acid (TPA) plants, polyethylene processes, sulfurrecovery plants, and so forth. In some cases, a chemical process consumes a great portion of the total energy usage only to drive the compression systems, depending on the size of the process. For example, it has been reported that a TPA manufacturing process2 uses about 75-85% of the total electric power consumed by the whole plant to drive the air compression systems only. This situation gives an opportunity to us for saving a considerable amount of energy by optimally operating compression systems. As the first step to achieve this goal, it is important to develop accurate models for compression systems; thus, we can consistently monitor and predict the performances and power requirements of the compression systems as functions of various operating variables. In chemical plants, the major utility systems are boilers, turbines, steam headers, compressors, and expanders, each of which usually uses a large portion * To whom correspondence should be addressed. Tel.: +82-54-279-2279. Fax: +82-54-279-3349. E-mail: chan@ postech.ac.kr.

of the total energy usage. Therefore, a lot of work on these utility systems has been carried out in order to reduce the energy consumption through modeling and optimization of the utility systems.3-6 However, most of this work has mainly been focused on steam production networks rather than on air-supplying systems, and they are usually based on macroscopic mass and energy balances where the performances (efficiencies) of boilers and turbines are assumed to be constant or are expressed as relatively simple correlation equations such as a second-order polynomial function of a single variable. For industrial compression systems, modeling for measuring and predicting performances is not straightforward because of the systems being operated usually at the conditions far from their original design, of aging components that constitute the system, and of a lack of the essential sensors that should be needed for measuring key process variables. A few of the previous studies on modeling of compression systems have appeared in the open literature, and they are usually based on simple correlation equations. In industry, the efficiencies and power requirements of compression systems are usually calculated on the basis of design diagrams or other shortcut methods.7,8,24 However, these methods cannot provide accurate results because some approximations should be involved in these calculations. Shain9 predicted the efficiency and polytropic head of a multistage compressor using a set of thermodynamic and correlation equations in which the efficiency and head are represented as functions of a single variable (the volumetric flow coefficient). For a given set of compressor design data, the errors in predicting the actual horsepower delivered to a three-stage compressor were within 4.6-5.4% using the prediction model. Recently, Gresh10 provided a calculation procedure of the overall efficiency and power of a multisection (multistage) compressor based on field measurement data. Rod-

10.1021/ie020270l CCC: $25.00 © 2003 American Chemical Society Published on Web 04/11/2003

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Figure 1. Schematic of the multistage air-compression system.

riguez-Toral et al.11 presented a model for the air stream based on the Beattie-Bridgman equation of state, and they used the model along with other utility models for optimization of a utility system. The purpose of this study is to develop a systematic modeling method for accurately calculating and predicting the two major performance variables of multistage compression systems in chemical plants: the overall efficiency and the the actual power consumption/ generation rate. The overall efficiency of a multistage compression system, which consists of multistage compressors and expanders and an electric motor/generator, is defined first and then is predicted using two empirical modeling methods: a partial least-squares (PLS) method as a linear modeler and a artificial neural network (ANN) as a nonlinear modeler. Then, the actual power delivered to the electric motor or that generated by the electric generator of the compression system can be predicted by combining the prediction models of the overall efficiency into thermodynamic compression and expansion equations. The proposed modeling method forms a hybrid model structure combining the thermodynamic equations for the power consumption/generation rate with the empirical PLS/ANN models for unknown state variables and the overall efficiency. Thus, we can expect to get more accurate and extrapolative results in predicting the overall efficiency and power consumption/generation rate with this new and systematic modeling method for compression systems. In addition, the proposed modeling method can be applied to find better operating conditions and to use as a core model for optimization of compression systems as well as similar utility systems such as turbines in various chemical processes. 2. Air-Compression Systems Figure 1 shows a schematic diagram of the aircompression system consisting of a multistage compressor to increase the pressure of air and an expander or electric motor to drive the compressor. The compressor consists of one or more compression stages in series with or without the intercooling systems between the stages. The expender (or steam turbine) is comprised of one or more expansion stages in series with or without the reheating systems between the stages and converts the internal energy of gas or steam to the kinetic energy to drive the shaft of the compressor. The electric motor makes up for the deficient amount of power required for driving the compressor, along with the expander or alone. Some compression systems (e.g., air blower of a fluidized catalytic cracking unit) are equipped with an electric generator instead of the electric motor because

the total power generated by the expander or steam turbine is more than that required for the compressor.12,13 There are several auxiliary components including speed-reducing gears, couplings, and bearings that serve as the medium of transferring power among the compressor and its drivers. Typically, these auxiliary components can be the causes of power losses in a compression system. Dry or wet air flows into the suction of the first compression stage and is pressurized up to a desired level through several compression stages. In most of multistage compressors, because an isothermal compression process requires its minimum power, compressed air is cooled with the air-water heat exchangers (also called the intercoolers) installed between the compression stages. In many chemical processes, the gases or steams with considerable amounts of energies are produced as byproducts and they are typically reused to recover the energies using expanders, turbines, or heat exchangers. Sometimes, the temperatures of the byproduct gases entering the expanders are elevated to desired levels to maximize energy recovery using the reheaters installed between the expansion stages. The performances of the compression systems always vary as a result of changing operation and ambient conditions, such as ambient temperature, relative humidity, molecular weight of the gases into the expanders, intercooling and reheating temperatures, and aging of the components including impellers, bearings, and casings. 3. Modeling In the operation of compression systems, it is important to identify the overall efficiency in terms of process variables, which might be associated with the operation of the compression systems, either to predict the actual power consumed by the electric motor or to predict the actual power generated by the electric generator. Therefore, the major modeling aspect of the compression system shown in Figure 1 is to accurately predict the overall efficiency; thus, the actual power consumption/ generation rate can be computed from the overall efficiency predicted. 3.1. Definition of the Overall Efficiency. The major performance indices of compression systems are the efficiencies such as the stage efficiencies of a compressor or expander and the overall efficiency of an entire compression system. The stage efficiencies, such as adiabatic or polytropic efficiencies, represent the performance of a single compression or expansion stage and can be directly calculated after measuring the temperatures and pressures at the suction and discharge of a single stage.1 On the other hand, the overall efficiency colligates the performances of the entire system components including the compressors, expanders, electric motor, generator, bearings, gears, and cooling systems. Therefore, the overall efficiency is recognized as the representative performance index of multistage compression systems and will be concerned with modeling of the compression systems rather than the stage efficiencies in this study. The overall efficiency ηO can be defined for the compression system with an electric motor (eq 1) and for that with an electric generator (eq 2) as follows:

Ind. Eng. Chem. Res., Vol. 42, No. 10, 2003 2211

ηO )

W0C + W0E W0C + W0E ) WC + WE + Wloss WM for

ηO )

WC + WE + Wloss W0C + W0E

)

NE

W0E

W0C

+

W0E

< WM (1)

WG W0C + W0E for WG < W0C + W0E (2)

where WC stands for the actual power required for a multistage compressor and WE denotes the actual power generated by a multistage expander. Wloss means all energy losses in the bearings, couplings, electric motors or generators, and other auxiliary components of a compression system, except in the compressor and expander. From the power balances for the whole compression system, the power delivered to the electric motor (WM) and that generated by the electric generator (WG) are obtained. Both WM and WG can also be directly measured using a wattmeter. W0C denotes the minimum power required for all of the compression stages NC under an adiabatic and reversible compression process and is given by the following equation: NC

W0C

)

0 wC(i) ) ∑ i)1 NC k

[( )

s a(i)Fa(i)Za(i)RTa(i)

∑ i)1 (k

a(i)

- 1)Mwa(i)

]

(ka(i)-1)/ka(i)

d Pa(i)

- 1 (3)

s Pa(i)

W0E is the maximum power generated by all of the expansion stages NE, which can be theoretically obtainable under an adiabatic and reversible expansion process and can be calculated using the following expression: NE

W0E )

0 wE(i) ) ∑ i)1 NE k

[ () ]

s g(i)Fg(i)Zg(i)RTg(i)

∑ i)1 (k

g(i)

- 1)Mwg(i)

d Pg(i)

1-

(kg(i)-1)/kg(i)

s Pg(i)

(4)

To calculate the overall efficiency with eqs 1-4, we need to measure the flow rates, temperatures, and pressures at the discharges and suctions of all of the compression and expansion stages. In addition, we should determine the molecular weights, compressibility factors, and adiabatic exponents of the wet air and gases passing through each compression and expansion stage. For most of compression systems, however, the temperatures and pressures between the compression/expansion stages are not available online or are not even measured. In this case, eqs 3 and 4 can be simplified to reduce the measurement points that we have to provide in order to calculate the overall efficiency: NC

W0C

)

∑ i)1

0 wC(i)

)

[( )

d ˜ aRTe Pa(NC) NCk˜ aF

(k˜ a - 1)M ˜ wa

Pe

(k˜ a-1)/(NCk˜ a)

]

-1

(5)

)

∑ i)1

0 wE(i)

)

[ ( )

NEk˜ gF ˜ gRTsg(1) (k˜ g - 1)M ˜ wg

1-

Pe

Psg(1)

(k˜ g-1)/(NEk˜ g)

]

(6)

In the previous equations, the measurement points only at the first suction and last discharge of the compressor and expander of a compression system are needed to calculate the overall efficiency. The mean flow rates, mean adiabatic exponents, and mean molecular weights are used, which are evaluated at the average of the first suction and last discharge measurements of a compressor or an expander, instead of those evaluated at all of the compression and expansion stages. That is, the measurement values are not required between the stages but only at the first suction and last discharge of the compressor and expander when using eqs 5 and 6 instead of eqs 3 and 4 to calculate the overall efficiency. In this study, the ideal compression system, the overall efficiency of which is equal to 1, follows from the assumptions that there are no energy losses due to frictions or leakages in the compressor, expander, bearings, gears, couplings, and other auxiliary components and that the efficiencies of the electric motor and generator are always 1, respectively. When the overall efficiency must be calculated using eqs 5 and 6 because of a lack of observations measured between the stages, the ideal compression system should satisfy the following conditions in addition to the assumptions mentioned above: (1) The temperature of the compressed air entering each compression stage is equal to the ambient temperature by perfect cooling. (2) The temperature of the gas or steam into each expansion stage is equal to that of the first suction of the expander after reheating. (3) The compression ratios of all of the compression stages of a multistage compressor are equal to each other. (4) The expansion ratios of all of the expansion stages of a multistage expander are equal to each other. (5) There are no pressure drops and temperature increases through the suctions of the first compression and expansion stage. (6) The discharge pressure of the last expansion stage is equal to atmospheric pressure. (7) The compressibility factors of air, steam, and gas are always 1. In general, actual compression systems require more power than the ideal compression system, and thus the overall efficiency is quite less than 1, typically ranging from 0.2 to 0.6. 3.2. Prediction Structure of the Overall Efficiency and the Power Consumption/Generation Rate. In this section, a prediction structure of the overall efficiency defined in the preceding section and the actual power consumption/generation rate of a compression system is presented. As can be seen in eqs 1 and 2, if either the overall efficiency or the actual power consumption/generation rate is predicted, the rest of them can be calculated, in turn, from eqs 1-6. It is convenient to predict the overall efficiency first rather than the actual power consumption/generation rate because we already know the overall efficiency to be obtained between 0 and 1. For predicting the overall efficiency, we have to represent the overall efficiency as a function of various process variables as well as ambient conditions that affect the overall efficiency, some of which may not be known for the prediction. Figure 2 shows the overall prediction structure of the overall efficiency and actual powers of a compression

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Figure 2. Prediction structure of the overall efficiency and the actual power consumption/generation rate. Either PLS or ANN models can be used both in the state variable prediction and in the efficiency prediction, and thermodynamic equations are used in the prediction of actual power consumption/generation rate.

system. In the state variable prediction, some of state (process) variables x3 are predicted with a given set of the manipulated variables x1 and ambient conditions x2. The states variables predicted are the variables that have effects on the overall efficiency and actual powers but cannot be directly known when there is a change in other variables because they are neither manipulated variables nor variables having fixed values. It is assumed that the state variables can be predicted from other variables and that the current values of the ambient conditions, such as the relative humidity, air temperature, and gas compositions, are maintained in the state variable prediction. An empirical modeling tool such as PLS14 or ANN15 can be used in the state variable prediction because it is time-consuming and is not achievable to develop fundamental models exactly representing physical phenomena in a compression system. In the efficiency prediction, the overall efficiency of a compression system is predicted using an empirical modeling tool with a given set of predicted state variables, manipulated variables, and ambient conditions. In the prediction of power consumption/generation, once the overall efficiency and a set of predicted state variables, manipulated variables, and ambient conditions are given, the actual power required to drive the electric motor or that generated by the electric generator can be directly computed using the following relationships, respectively:

WM ) (W0C + W0E)/ηO

for W0C + W0E < WM

(7)

WG ) (W0C + W0E)ηO

for W0C + W0E > WG

(8)

In the previous equations, the thermodynamic compression and expansion equations described in eqs 3-6 nonlinearly relate the process variables to the actual power consumption/generation rate, and adopting them is for the sake of utilizing prior knowledge such as physical properties and thermodynamic laws. Through

this prediction structure, the overall efficiency and the actual power consumption/generation rate of the compression system can be evaluated when the manipulated variables and ambient conditions are changed. 3.3. Empirical Modeling. In this section, empirical modeling methods are presented to predict the state variables and overall efficiency in the prediction structure shown in Figure 2. As a result of the rapid growth in the application of real-time database systems to chemical processes, a great amount of plant operation data have been collected and can be easily utilized for empirical modeling of various processes. The prediction models of the state variables and overall efficiency can be constructed by finding the model parameters, such as regression coefficients or weights, that minimize the errors between the target values from past operations of a compression system and the calculated values from an empirical model. Several types of empirical models can be used. In this study, two powerful empirical modeling tools are employed as alternatives for predicting the state variables and overall efficiency: (1) PLS method as a linear modeler and (2) ANN algorithm as a nonlinear modeler. PLS methods have been widely used as a powerful tool for empirical modeling from laboratory and field measurement data because the resulting models are typically more robust and reliable than those using other modeling tools, such as an ordinary least-squares method, particularly when the data are noisy and highly correlated with each other.16,17 The basic concept of the PLS methods is to project a high-dimensional space of the input and output data obtained from a process onto a low-dimensional feature (latent) space, thus finding the best relation between the feature vectors on the basis of the multivariate statistical projection techniques. It has the advantage of dealing with singular and highly correlated regression problems over the traditional multiple linear regression methods but cannot capture the highly nonlinear relationships between the input and output variables of a process. Hence, several researchers have been developing nonlinear PLS methods to efficiently handle nonlinear characteristics of chemical processes.18-20 In this study, however, only a linear PLS method is considered to model the compression systems. The final form of the PLS model is expressed by the following linear regression equation:

y ) xω + E

(9)

where x denotes the input vector of the process variables that affect compressors, expanders, electric motors/ generators, and other auxiliary components that constitute a compression system. The vector x consists of x1 and x2 when modeling and predicting the state variables using eq 9 in the prediction structure shown in Figure 2 or consists of x1, x2, and x3 when modeling and predicting the efficiency. y is either the output vector of state variables (x3) in the state variable prediction of the prediction structure or the predicted value of the overall efficiency (ηO) in the efficiency prediction. E is the residual vector for the predicted output vector or variable (of state variables or overall efficiency), and ω denotes the matrix or vector of regression coefficients, which is given by

ω ) Φ(GTΦ)-1HT

(10)

In the previous equation, the major model parameters

Ind. Eng. Chem. Res., Vol. 42, No. 10, 2003 2213 Table 1. Variables Used for Modeling of the Compression Systems compression system Comp 1 Comp 2

x1 (manipulated variables)

x2 (ambient conditions)

x3 (predicted state variables)

d s s s Fa(NC), Fg(1), Pa(N , Pg(1) , Tg(1) , Tg(2) C) d Fa(NC), Pa(NC)

Te, Ha, λO2, λCO2, λCO Te, Ha

d Ta(N , φC, φE, ∆PF, SE(1), SE(2) C) d Ta(N , φC, ∆PF C)

are G (loading matrix for measurements of the process variables), H (loading matrix for the measurement of output variables), and Φ (matrix of the weight factors). These parameters are determined from a set of process measurements for the inputs x and outputs y using the nonlinear iterative partial least-squares (NIPALS) algorithm. Details on the general concepts of PLS modeling and the NIPALS algorithm are shown in the literature.14 ANNs are widely used for modeling nonlinear behaviors of chemical processes because they allow more flexibility in determining model structures and typically give good modeling performances if there are a sufficient amount of data.21,22 A basic structure of ANN consists of a number of interconnected computing processors, called neurons or nodes, grouped into input, hidden, and output layers. The strengths of the connections, called weights, among the nodes are adjusted to obtain a desired output behavior using given information and a learning algorithm. Various types of ANNs have been proposed according to their structures and learning algorithms for locating the weights.23 In this study, a feedforward network with one hidden layer is employed to model the compression systems. In predicting the state variables and overall efficiency, the final form of the ANN model is expressed by the following equation:

y ) f(2)[ω(2)f(1)(ω(1)x + b(1)) + b(2)]

(11)

where x stands for the vector of the process variables and y for the vector of the state variables or overall efficiency to be predicted. ω(1) and b(1) denote the weight and bias vectors, respectively, in the connections between the input and hidden layers, and ω(2) and b(2) are those between the hidden and output layers. f(1) is the sigmoid transfer function for the hidden nodes and f(2) the linear transfer function for the output nodes. In eq 11, the weights and biases in the connections are the major model parameters, and they are determined using a back-propagation training algorithm.15 4. Application Case As an example modeling case, the performances of industrial multistage compression systems are measured and predicted using the proposed prediction structure shown in Figure 2. By comparison of the predicted values with measured ones, the proposed modeling method will be verified in this section. 4.1. Process Description. The compression systems are used for supplying the compressed air for oxidation of p-xylene in a TPA manufacturing process2 of Samsung Petrochemical Corp. in Ulsan, Korea. Several compression systems with various capacities and performances are being operated, but only two of them are considered for modeling in this study. The first compression system (Comp 1) consists of a four-stage compressor, two-stage expander, and one electric motor, and the second one (Comp 2) consists of a four-stage compressor and one electric motor without an expander. The compressor of each compression system produces the compressed air flowing into p-xylene oxidizers.

Because only oxygen of the air is consumed for oxidation in the oxidizers, nitrogen (off-gas) along with some byproduct gases is recycled to the expander of each compression system in order to recover their internal energy. 4.2. Measured Variables and Data Collection. Table 1 shows all of the process variables and ambient conditions measured online for the two compression systems. Not all temperatures and pressures are measured online to calculate the overall efficiency and actual power consumption rate using eqs 1, 3, and 4, but we can calculate those using eqs 1, 5, and 6, which require much fewer measurements. In the table, the manipulated variables x1 measured online are the flow rate of air exiting the last discharge, the flow rate of off-gas entering the first suction, the air and off-gas pressures at the first suction and last discharge, and the temperatures of off-gas entering the expansion suctions. The ambient conditions x2 are ambient temperature, relative humidity, and composition of the off-gas from oxidizers, which are not changed with other process variables. The predicted state variables x3 are the discharge temperatures of the last compression stages, the position of inlet guide vanes that are used for adjusting flow rates, the pressure drops through air filters, and the expander rotating speeds, and they are predicted from the manipulated variables (x1) and ambient conditions (x2). The discharge temperature can be used as an indirect measurement of the efficiency, and the position of the inlet guide, which is strongly correlated with the mass flow rate, also affects the efficiency.1,24 Thus, the discharge temperature and the inlet guide vane position are included in the set of predicted state variables. Because an increase in the pressure drop through air filters decreases the overall efficiency and the expander rotating speeds always vary affecting the efficiency, these variables are also selected as the predicted state variables. The real-time database system, running at Samsung Petrochemical Corp., collects measurements of the process operating variables and ambient conditions shown in Table 1, and the data sets for modeling were collected every 10 min for about a 6-month period (total number of observations ) 26 350). The data were collected during a normal operation of the process, and the 6-month period is sufficiently long to cover most of the possible operating regions of the compression systems.24 If there is a significant change in the operating condition due to a startup or shutdown of the plant, the data should be collected again after the condition has changed. Statistical outliers, which may be caused by measurement errors or abnormal operations, were removed on the basis of principal component analysis.25 After the outliers are removed (1845 points), threequarters (modeling data set ) 18 379 points) of the data sets were used to build the models for the state variables and efficiency predictions, and the rest (validation data set ) 6126 points) of the data sets were for validating the resulting models. 4.3. Prediction Models. Two types of models are built using PLS and ANN. First, PLS models are

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Figure 3. Effects of the variables on the overall efficiency of Comp 1.

Figure 5. Comparison of the measured and predicted values of the overall efficiency (a) and the actual power consumption rate (b) using the PLS models for Comp 1.

Figure 4. Effects of the variables on the overall efficiency of Comp 2.

constructed both for the state variable and for overall efficiency predictions shown in Figure 2, which capture only the linear relationship between the process variables and the overall efficiency. Then, ANN models are built using the same data sets as those used in the PLS models for both of the predictions in Figure 2, which may capture the nonlinear relationship between the process variables and the overall efficiency. In the PLS models, the number of latent variables is the major parameter to be determined, and in the ANN models, the number of nodes in the hidden layer is the major paramater. The numbers of latent variables and nodes are determined at the number that makes the prediction error minimum when using the validation data sets. On the basis of modeling data sets and the overall efficiency calculated from these data sets, PLS regression coefficients and ANN weights are obtained using the NIPALS and the back-propagation algorithm, respectively. Then, the actual power consumption rates of electric motors can be calculated from the predicted overall efficiencies and eqs 5 and 6. In the calculation, the mean flow rates, the mean adiabatic exponents, and the mean molecular weights are used, and they are given in the appendix. 4.4. Results and Discussion. Figures 3 and 4 display the regression coefficients ω of the PLS models for the overall efficiencies of Comp 1 and 2, respectively.

The higher a bar in the figures is, the more effect on the overall efficiency the corresponding variable has. In the figures, the sign of a regression coefficient denotes the direction of the effect on the overall efficiency. As shown in Figure 3, the ambient temperature (Te) and the flow rate of the air exiting the last discharge (Fa(NC)) have strong effects on the overall efficiency of Comp 1. That is, a slight change in these variables in the positive direction raises the overall efficiency quite much. Note that the correlations between these variables (Te and Fa(NC)) and the overall efficiency can be changed, depending on the operating region and configuration of a compression system. In the operating region where the compression system has been operated, the overall efficiency mostly rose with an increase in either the ambient temperature or the mass flow rate. However, the power delivered to the electric motor went up with an increase in the ambient temperature and mass flow rate, as can be expected from eq 5, though the overall efficiency was increased in this operating region. It can be also known that the overall efficiency somewhat increases with a decrease in the flow rate of the off-gas entering the first suction (Fa(1)) and the inlet guide vane position of the compressor (φC) and with an increase in the pressure of the off-gas entering the first suction s (Pg(1) ). Other variables, such as the temperature of the s ), expander rooff-gas entering the first suction (Tg(1) tating speeds (SE(1) and SE(2)), and off-gas composition (λO2, λCO2, and λCO), slightly change the overall efficiency. Similarly to the results for Comp 1, the overall efficiency of Comp 2 was strongly affected by the flow rate of the air exiting the last discharge and the ambient temperature but weakly affected by the other variables as shown in Figure 4. Figures 5 and 6 show the prediction performances of the PLS models for Comp 1 and 2, respectively. As shown in the figures, the predicted values of the overall efficiency and actual power consumption rate give excellent agreements with the measured ones. For Comp 1, the root-mean-squared error (RMSE) between the predicted and the measured values of the overall

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Figure 6. Comparison of the measured and predicted values of the overall efficiency (a) and the actual power consumption rate (b) using the PLS models for Comp 2.

Figure 8. Comparison of the measured and predicted values of the overall efficiency (a) and the actual power consumption rate (b) using the ANN models for Comp 2. Table 2. RMSEs in Predicting the Overall Efficiencies and the Actual Power Consumption Rates Using the PLS and ANN Models PLS model

ANN model

compression system RMSE (ηO) RMSE (WM) RMSE (ηO) RMSE (WM) Comp 1 Comp 2

Figure 7. Comparison of the measured and predicted values of the overall efficiency (a) and the actual power consumption rate (b) using the ANN models for Comp 1.

efficiency is 0.0046 (which corresponds to 0.90% of the average overall efficiency measured), and the RMSE of the actual power consumption rate is 77.7 kW (which corresponds to 0.90% of the average power consumption rate measured), as summarized in Table 1. For Comp 2, the RMSE of the overall efficiency is 0.0026 (corresponding to 0.49%), and the RMSE of the actual power consumption rate is 25.3 kW (corresponding to 0.50%). Note that the prediction performance of the PLS model for Comp 2 is somewhat better than that for Comp 1 because Comp 2 has no expander, which makes Comp 2 less complex than Comp 1 from the viewpoint of modeling. Figures 7 and 8 illustrate the prediction results from the ANN models for Comp 1 and 2, respectively. The figures show excellent prediction performances, as good

0.0046 0.0026

77.7 25.3

0.0039 0.0023

66.4 21.6

as can be obtained using the PLS models. In predicting the overall efficiencies with the ANN models, the RMSEs are 0.0039 and 0.0023, which correspond to only 0.76% and 0.44% prediction errors for Comp 1 and 2, respectively. The RMSEs of the actual power consumption rates are 66.4 kW (which correspond to 0.77% of the average value of the measurements) and 21.6 kW (corresponding to 0.42%) for Comp 1 and 2, respectively. For the same reason as was applied to the PLS models, the prediction performance of the ANN model for Comp 2 is somewhat better than that for Comp 1. As shown in Table 2, the ANN models give slightly better prediction performances than the PLS models for all of the compression systems modeled. However, the PLS models also give excellent and acceptable modeling performances although the PLS models can capture only linear behaviors of the compression systems. This means that the overall efficiency defined in this study can be successfully modeled using a linear modeling tool because the overall efficiency is almost linearly related to the process variables in normal operating ranges of the compression systems. Typically, ANN models require lots of learning procedures, while they provide lots of flexibility in representing nonlinear functions. On the other hand, PLS models can be validated with a relatively small effort compared to the ANN models but cannot capture nonlinear behaviors. However, because the ANN models used in the prediction structure shown in Figure 2 have only one hidden layer and a few input (less than 6) and output (less than 6) nodes, it would not require lots of effort to find the optimal model structure for the learning procedure. Hence, the proposed modeling

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method adopting either the ANN or PLS model can be easily applied to similar compression systems without a large amount of effort. 5. Conclusions A systematic modeling method for a multistage compression system, consisting of a multistage compressor, a multistage expander, and an electric motor/ generator, was proposed. First, the overall efficiency was defined based on thermodynamic compression and expansion equations for the multistage compression system. Without measurements at the interstages of a multistage compressor and expander, the overall efficiency can be calculated using the proposed compression and expansion equations. Then, the prediction structure of the overall efficiency and actual power consumption/generation rate was designed, which consists of the state variable prediction based on the PLS or ANN model, the efficiency prediction based on the PLS or ANN model, and the prediction of actual power consumption/generation rate based on compression and expansion equations. The proposed modeling method was applied to two industrial compression systems of a TPA manufacturing process and gave excellent performances in predicting both the overall efficiency and the actual power consumption rate. It was also found that both the PLS and ANN models can be successfully used for predicting the overall efficiency of an industrial compression system because the overall efficiency defined in this study is an almost linear function of the operating variables and ambient conditions of the compression system under normal operating conditions. The proposed modeling method can be applied to find better operating conditions and to be used as a core model for optimization of multistage compression systems in various chemical processes.

Psw )

-78.994 - 37.955Te + 0.145Te2 1 + 14.289Te - 0.043Te2

(13)

Because off-gas is not condensed through an expander, the mean flow rate of the off-gas is equal to the flow rate of the off-gas entering the first section:

F ˜ g ) Fg(1)

(14)

The mean adiabatic exponents of the air and off-gas are simply given by the following equations, respectively:

k˜ a ) k˜ g )

C ˜ pa(T ˜ a,Ca)

(15)

C ˜ pa(T ˜ a,Ca) - R/M ˜ wa C ˜ pg(T ˜ g,Cg)

(16)

C ˜ pg(T ˜ g,Cg) - R/M ˜ wg

where the mean heat capacities of air and off-gas are ˜ g) and functions of temperatures (T ˜ g) and compositions (C the mean molecular weights of humid air and off-gas are calculated from the following expressions, respectively:

(

M ˜ wa ) 0.5Mwa 1 + M ˜ wg )

Mww λaMww + λwMwa

)

1 λCO2 λCO + + + MwO2 MwN2 MwCO MwCO2 λO2

λN2

(17) (18)

where λa and λw are the weight fractions of air and water, respectively, contained in ambient air and the off-gas is composed of oxygen, nitrogen, carbon monoxide, and carbon dioxide.

Acknowledgment This work was supported by the Brain Korea 21 Project, and the authors gratefully acknowledge the support provided by Woochang Lee, Euichul Noh, and Kyunghoon Lee of Samsung Petrochemical Corp., Ulsan, Korea. Appendix: Calculations of Mean Flow Rates and Properties The following are the mean flow rates, the mean adiabatic exponents, and the mean molecular weights used in the modeling of the two compression systems of Samsung Petrochemical Corp. Most of the moisture contained in air is condensed through several intercoolers of a compression system. Assuming that there is no water in the air discharged from the last stage of the compressor, the mean flow rate of the air is given by

[

F ˜ a ) Fa(NC) 1 +

0.5MwwHaPsw Mwa(Pe - Psw)

]

(12)

where Psw is the vapor pressure of saturated steam at the ambient temperature Te and can be calculated as follows [this equation is applicable when the ambient temperature (Te) ranges from 273 to 315 K]:

Nomenclature b ) vector of biases in the ANN model C ) composition of air or gas C ˜ pa ) mean heat capacity of air through a multistage compressor [kJ/kg‚K] C ˜ pg ) mean heat capacity of gas through a multistage expander [kJ/kg‚K] f(1) ) sigmoid transfer function, f(1) ) 1/[1 + exp(-x)] f(2) ) linear transfer function, f(2) ) x Fa(i) ) flow rate of air at the compression stage i [kg/s] F ˜ a ) mean flow rate of air through a multistage compressor [kg/s] Fg(i) ) flow rate of gas at the expansion stage i [kg/s] F ˜ g ) mean flow rate of gas through a multistage expander [kg/s] G ) loading matrix for the measurements of the process (input) variables H ) loading matrix for the measurements of the output variables Ha ) relative humidity of ambient air ka(i) ) adiabatic exponent of the air in the single compression stage i k˜ a ) mean adiabatic exponent of air kg(i) ) adiabatic exponent of the gas in the single expansion stage i k˜ g ) mean adiabatic exponent of gas Mw ) molecular weight [kg/kg‚mol] Mwa ) molecular weight of dry air [28.96 kg/kg‚mol]

Ind. Eng. Chem. Res., Vol. 42, No. 10, 2003 2217 M ˜ wa ) mean molecular weight of air [kg/kg‚mol] M ˜ wg ) mean molecular weight of gas [kg/kg‚mol] Mww ) molecular weight of water [18.02 kg/kg‚mol] NC ) total number of compression stages of a multistage compressor NE ) total number of expansion stages of a multistage expander d ) discharge pressure of the air at the compression Pa(i) stage i [kPa] s Pa(i) ) suction pressure of the air at the compression stage i [kPa] Pe ) atmospheric pressure [kPa] d Pg(i) ) discharge pressure of the gas at the expansion stage i [kPa] s Pg(i) ) suction pressure of the gas at the expansion stage i [kPa] Psw ) vapor pressure of saturated steam [kPa] R ) universal gas constant [8.314 kJ/kg‚mol‚K] SE(i) ) impeller rotating speed of the single expansion stage i [rpm] Te ) ambient temperature [K] T ˜ a ) average air temperature through a multistage compressor [K] s Ta(i) ) suction temperature of the air at the compression stage i [K] T ˜ g ) average gas temperature through a multistage expander [K] s ) suction temperature of the gas at the expansion Tg(i) stage i [K] WC ) actual power required for a multistage compressor [kW] W0C ) minimum power required for a multistage compressor under an adiabatic and reversible process [kW] 0 ) minimum power required for the single compreswC(i) sion stage i under an adiabatic and reversible process [kW] WE ) actual power generated by a multistage expander [kW] W0E ) maximum power generated by a multistage expander under an adiabatic and reversible process [kW] 0 ) minimum power required for the single expansion wE(i) stage i under an adiabatic and reversible process [kW] WG ) actual power generated by an electric generator [kW] Wloss ) energy losses in auxiliary components of a compression system [kW] WM ) actual power delivered to an electric motor [kW] x ) vector of all of the process variables, [x1x2x3] x1 ) vector of manipulated variables x2 ) vector of ambient conditions x3 ) vector of state variables predicted y ) vector of output variables predicted from an empirical model Z ) compressibility factor Greek Symbols ∆PF ) pressure drop through a compressor filter [kPa] E ) residual vector in the PLS model ηO ) overall efficiency of a compression system [0-1] λ ) weight fraction [0-1] Φ ) matrix of the weight factors in the PLS model φC ) position of the inlet guide vane of a compressor [deg] φE ) position of the inlet guide vane of an expander [deg] ω ) vector of PLS regression coefficients or vector of ANN weights Subscripts a ) air

C ) compressor or compression stage CO ) carbon monoxide CO2 ) carbon dioxide E ) expander or expansion stage e ) ambient condition G ) electric generator g ) gas M ) electric motor N2 ) nitrogen O2 ) oxygen w ) water (i) ) compression or expansion stage i of a compression system Superscripts 0 ) theoretically minimum or maximum state (1) ) hidden layer in a neural network structure (2) ) output layer in a neural network structure s ) suction of a single stage d ) discharge of a single stage ∼ ) mean (average) value

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Received for review April 12, 2002 Revised manuscript received November 26, 2002 Accepted February 7, 2003 IE020270L