Dynamics and Control of Process Networks with Large Energy Recycle

Jan 21, 2009 - This paper focuses on the dynamics and control aspects of a class of energy integrated networks with large recycle of energy compared t...
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Ind. Eng. Chem. Res. 2009, 48, 6087–6097

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Dynamics and Control of Process Networks with Large Energy Recycle Sujit S. Jogwar Department of Chemical Engineering and Materials Science, UniVersity of Minnesota, Minneapolis, Minnesota 55455

Michael Baldea Praxair Technology Center, 175 East Park DriVe, Tonawanda, New York 14150

Prodromos Daoutidis* Department of Chemical Engineering and Materials Science, UniVersity of Minnesota, Minneapolis, Minnesota 55455

This paper focuses on the dynamics and control aspects of a class of energy integrated networks with large recycle of energy compared to the input/output energy flows. A prototype network is considered to identify the underlying model structure. A time scale separation in the energy dynamics of such networks is documented. Using singular perturbation techniques, a model reduction procedure is outlined, resulting in nonstiff reduced order models for the dynamics in each time scale. The theoretical results are illustrated with the help of examples and a simulation case study on a reactor-feed effluent heat exchanger network. 1. Introduction Energy integration is ubiquitous in the process and energy industries. It is motivated by the high cost of energy and the corresponding need to minimize utility usage by pairing energy generation and consumption within the same plant. Numerous design modifications are possible to this end, e.g. heat exchangers integrated with reactors, heat exchanger networks, heat integrated and thermally coupled distillation columns, etc. Integrated designs invariably introduce a dynamic coupling between the process units. Several papers have documented positive feedback due to recycle of energy, which can lead to complex dynamic behavior, such as inverse response, openloop instability, and chaotic behavior.1-7 The control of such integrated systems is a challenging problem,7,8 especially in the context of transitions between steady states motivated by the current economic environment which dictates frequent changes in operating targets. The transient operation of integrated networks is even more challenging in view of their nonlinear behavior. However, most of the control studies on such networks are within a multiloop linear control framework.1,7,8 In our previous work,9,11 we focused on networks with large material recycle compared to network material throughput. We showed that a large recycle flowrate induces a time scale separation, with the dynamics of individual units evolving in a fast time scale, with weak interactions, and the dynamics of the overall system evolving in a slow time scale, where these interactions become significant. A systematic framework was developed for model reduction and subsequent control system design using singular perturbation arguments. We further focused on the energetic aspects of such networks, in the presence of large energy sources and sinks, in which case, the large material streams act as energy carriers (examples include high purity distillation columns and reactors with external heat exchangers).12,13 Using singular perturbation arguments, it was shown that the dynamics of these networks evolve over two time scales, with the variables in the energy balance evolving * To whom correspondence should be addressed. Telephone: (612) 625 8818. E-mail: [email protected].

over a short time horizon while the variables in the material balance dynamics exhibiting both fast and slow transients. In the present paper, we focus on networks with large recycle of energy compared to the energy sources and/or sinks as well as the energy input through the feed stream. A prototype network is considered to identify and characterize the underlying dynamic structure. It is shown that the simultaneous presence of energy flows of different magnitudes causes stiffness and is at the origin of a time scale separation in the energy balance dynamics. A model reduction framework is outlined to derive reduced order models valid in each time scale. We then propose guidelines for controller design that account for the two time scale dynamics. The theoretical results derived are illustrated via example networks. Lastly, a simulation case study is performed on a reactor-feed effluent heat exchanger (FEHE) network, demonstrating the dynamic and control aspects of these networks. 2. Dynamics of Networks with Large Energy Recycle Let us consider the network in Figure 1 which represents a prototype network of the class considered. It consists of N units with a single energy recycle loop. We assume that the kinetic and potential energy contributions to the total energy are negligible compared to enthalpy, and let H1, H2,..., HN denote the enthalpies of each unit. The terms h1, h2,..., hN-1 represent the enthalpy flows exiting each unit. These enthalpy flows can be due to material streams acting as energy-carriers connecting the process units or due to heat transfer across heat exchangers. The terms hin and hout are the enthalpy flows associated with the feed and effluent stream and hr represents the rate at which

Figure 1. Prototype network with energy recycle.

10.1021/ie801050b CCC: $40.75  2009 American Chemical Society Published on Web 01/21/2009

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the energy is recycled. The term qi represents a possible additional energy flow for the ith unit, due to external energy inputs (e.g., work done on/by the system) or energy generation/ consumption (e.g., via chemical reaction). The energy balance equations for the units in this network are dH1 ) hin + hr - h1 + q1 dt dHi (1) ) hi-1 - hi + qi for i ) 2, 3, ... , N - 1 dt dHN ) hN-1 - hr + qN - hout dt We consider the case where the internal energy flows (i.e., the energy flows internal to the network, hi) are large compared to the external energy flows (hin, qi, and hout) and are of comparable magnitude to the rate of energy recycle (hr). This case represents tight energy integration through the recovery and recycle of a large fraction of the energy flow in the network and is therefore of high economic interest. An overall steady state energy balance indicates that the external energy inputs (qi) should be, at most, of the order of the energy flows hin and hout. For simplicity, we assume that the external energy flows (qi) are of comparable magnitude to the energy input (hin). Let us now formalize these assumptions mathematically. At steady state (denoted by subscript s), we define a ratio of the rate of energy recycle to the energy input through the feed stream as hr,s/hin,s ) 1/ε . 1. We also define the O(1) steady state ratios (where O(.) is the standard order of magnitude notation) as ki )

hi,s hout,s , kout ) hr,s hin,s

and kqi )

qi,s hin,s

and uqi )

qi qi,s

and the scaled energy flows ui )

hi hout , uout ) hi,s hout,s uin )

hin hr , ur ) hin,s hr,s

Using these definitions, eq 1 becomes

[ [ [

]

ur k1u1 dH1 ) hin,s uin + + kq1uq1 dt ε ε dHi ki-1ui-1 kiui ) hin,s + kqiuqi for i ) 2, 3, ... , N - 1 dt ε ε dHN kN-1uN-1 ur ) hin,s - + kqNuqN - koutuout dt ε ε (2)

]

]

Equation 2 can be written in a vector form as follows: 1 dH ) f(H, us) + g(H, ul) dt ε

[ ]

g ) hin,s

ur - k1u1 k1u1 - k2u2 l kNuN - ur

Due to the presence of the small parameter ε, the ordinary differential equation (ODE) system (3) is stiff and, thus, presents a possibility of dynamic behavior evolving over two time scales. Incorporating such stiff energy balance models in controller design results in ill-conditioned controllers, with high sensitivity toward modeling errors.9 A rational approach, in this case, is to address control objectives in each time scale using reduced order nonstiff representations of the fast and slow dynamics. To derive such reduced order models in each time scale, we use singular perturbation techniques. 3. Model Reduction In what follows, we proceed with a model decomposition of the system (3) and establish that it is a singularly perturbed system albeit in a nonstandard form (i.e., with a nonexplicit separation of fast and slow states).10 In order to obtain a description of dynamics in the fast time scale, we define a fast, “stretched” time scale τ ) t/ε. Using the definition of τ, the process model (3) becomes the following: dH ) εf(H, us) + g(H, ul) (4) dτ In the limit ε f 0 (corresponding to the case of infinitely large recycle of energy), we obtain the description of the dynamics of the system in the fast time scale dH ) g(H, ul) (5) dτ describing the dynamics of individual units. We note that only the scaled energy flows corresponding to the large recycle and the large internal energy flows appear in eq 5. The large internal flows are the manipulated inputs available in this fast time scale to address regulatory control objectives at the unit level and can be manipulated by varying the corresponding material flows, e.g. using bypass streams or by varying the heat transfer rate across heat exchangers, e.g. by varying the heat transfer area in the case of flooded condensers. However, these large internal energy flows do not affect the total enthalpy content of the network, which is governed by the small external energy flows, indicating that the differential equations in eq 5 are not linearly independent. From eq 5, N ∑N i)1 dHi/dτ ) 0, and therefore, dHj/dτ ) -∑i)1,i*j dHi/dτ. Thus only (N - 1) equations are linearly independent. The quasisteady state constraints for the fast dynamics, obtained from eq 5, can thus be written as follows: g(H, ul) ) Bg˜(H, ul) ) 0

(3)

(6)

us ) [uin uout uq1 ... uqN ]T

where, B ∈ B ⊂ R has full column rank. The terms in g(H, ul) correspond to the difference between scaled internal energy flows. The quasi-steady state condition (6) thus states that, in the limit of infinite energy recycle rate, all the scaled internal energy flows are equal. The linearly independent constraints are

ul ) [ur u1 ... uN-1 ]T

g˜(H, ul) ) 0

where H ) [H1 H2 ... HN ]T

[ ]

f ) hin,s

uin + kq1uq1 kq2uq2

l kqNuqN - koutuout

N×(N-1)

(7)

which suggests that the equilibrium manifold for the fast dynamics is at most one-dimensional. Proceeding now to the slow dynamics, we take the limit ε f 0 in the original time scale, under the constraints (7) to obtain the following:

Ind. Eng. Chem. Res., Vol. 48, No. 13, 2009

g˜(H, ul) dH ) f(H, us) + B lim εf0 dt ε 0 ) g˜(H, ul)

(8)

We note that the terms limεf0 g˜(H, ul)/ε are indefinite, but finite. These terms correspond to the difference between the actual large internal energy flows, which now becomes indeterminate in the slow time scale. We denote these as a set of “algebraic” variables z ) limεf0 g˜(H, ul)/ε and the model (8) becomes dH ) f(H, us) + Bz dt 0 ) g˜(H, ul)

(9)

which represents the energy balance model in the slow time scale, in the form of a differential algebraic equation (DAE) system. The scaled energy flows corresponding to the small external energy flows (us), most likely the small external energy inputs are the potential manipulated inputs available in the slow time scale. The algebraic constraints of eq 9 can be differentiated, and an ODE representation (state space realization) of the DAE system (9) can be obtained once the scaled energy flows ul are specified via appropriate energy transfer and energy flow correlations in terms of temperature gradients, heat transfer parameters, etc. The order of this state space realization would be one. Remark 3.1. One way to capture the slow dynamics is through a coordinate change of the form10

[

[]

φ(H) ζ ) T(H) ) g˜(H, u ) η l

]

(10)

In these new coordinates, the model (9) of the slow dynamics becomes dζ ∂φ ∂φ ) f(H, us)H)T-1(ζ) + Bz dt ∂H ∂H H)T-1(ζ) (11) η)0 The function φ(H) can be arbitrarily chosen and can allow a description of the slow dynamics in terms of the enthalpy of any one of the units. Furthermore, φ(H) may be chosen in such a way that (∂φ/∂H)B ) 0. In this case, the model (11) will be independent of z and the corresponding ζ represents a true “slow” variable in the system (whereas the original state variables evolve in both fast and slow time scales). If we choose φ(H) as the total network enthalpy defined as N

Htotal )

∑H

i

i)1

it can be shown that indeed (∂φ/∂H)B ) 0. Thus the total network enthalpy evolves only over a slow time scale. This is also apparent if we consider the energy balance over the entire network: N dHtotal ) hin,s[uin + koutuout + kqiuqi] dt i)1

∑

(12)

which involves only small external energy flows. Remark 3.2. We can note an analogy of this analysis with the case of material recycle, analyzed in our previous work.9 There, it was shown that a large recycle flowrate compared to throughput introduces stiffness in the material balance dynamics and induces a time scale separation. The dynamics of individual units evolve in a fast time scale, and the dynamics of the overall system evolves in a slow time scale. It was further shown that the slow model is of order c, where c is the total number of components present in the network.

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Remark 3.3. The above analysis indicates that the control objectives related to the energy balance of the individual units of the network should be addressed in the fast time scale using the large internal energy flows, while the control and, more importantly, optimization of the energy utilization at the level of the entire network should be undertaken in the slow time scale using the small energy flows of the network as manipulated inputs. This is contrary to the case of networks with high energy throughput, containing large energy sources and sinks (qi), wherein the entire energy balance dynamics of the network evolve in the fast time scale and the energy related control objectives are addressed in the fast time scale.12,13 Remark 3.4. The reduced order fast (5) and slow (9) time scale models can be used to derive controllers in the respective time scales. Closed-loop stability of the overall system (3) can be guaranteed by the results available for the composite control of singularly perturbed systems.14,15 Remark 3.5. This analysis has focused only on the energetic aspects of the network, and the corresponding slow model is one-dimensional. Note that material flowrates enter into the energy balance equations (through the enthalpy flows), implying a coupling with the material balance equations. Considering the network material balance dynamics as well, there can be slow modes corresponding to the material dynamics and the net dimension of the slow model of the network may be greater than one. Remark 3.6. The arguments concerning the magnitude of the energy flows q that were used in the derivations above imply that q are small at steady state. This is not necessarily true during process startup, when a significant amount of energy needs to be accumulated in the process, and in which case the energy input associated with qi may be very large. 4. Illustrative Examples In what follows, we consider representative networks of chemical processes that belong to the class of large energy recycle networks analyzed in section 3. An alternate “energy” representation is considered for these networks, explicitly showing the energy flows in the network. This representation allows us to represent these networks in the form of the prototype framework analyzed previously. 4.1. Heating Tanks in Series. A series of heating tanks, interconnected through energy carrying material recycle streams, represents a simple example for networks with energy recycle. We consider a case of two heating tanks with volumes V1 and V2, operating at temperatures T1 and T2, respectively, as shown in Figure 2. F is the volumetric flowrate of the input stream to tank 1, and R is the recycle flowrate from tank 2 to tank 1. Q1 and Q2 are heat sources/sinks, and Tf is the feed temperature. We assume constant volumes of tanks, constant density (F), and heat capacity (cp), and we further assume that there is no phase change in the heaters. Let us consider an energy representation for this system, shown in Figure 3. H1 and H2 are the enthalpies of the tanks. hin and hout are the enthalpy flows entering and exiting the network, and hr is the enthalpy flow associated with the recycle stream. The energy balance dynamics for the tanks are given as follows: dH1 ) hin + hr - h1 + Q1 dt (13) dH2 ) h1 - hr - hout + Q2 dt Let us consider a case where the recycle flowrate R is much larger than the feed F (i.e., at steady state, Fs/Rs ) ε , 1). We

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Figure 2. Two heating tanks in series with recycle. Figure 4. Reactor-FEHE network with a furnace and a continuously stirred tank reactor (CSTR).

Figure 3. Energy representation for heating tanks in series.

further assume that the energy sources/sinks Q1 and Q2 are comparable in magnitude to hin. In this case, the temperatures (Tf, T1, T2) are of comparable magnitude, and thus, the relative magnitudes of energy flows depend on the corresponding material flows. We define the O(1) steady state ratios kq1 ) Q1,s/[FFcp(Tf Tref)]s and kq2 ) Q2,s/[FFcp(Tf - Tref)]s and the scaled flows uf ) F/Fs, ur ) R/Rs, uq1 ) Q1/Q1,s and uq2 ) Q2/Q2,s, where Tref is a reference temperature. The energy balance dynamics in eq 13 can now be put in the vector form (3) with the following: H ) [H1 H2 ]T us ) [uf uq1 uq2 ]T

f ) FsFcp

[

ul ) [ur]T

(Tf - T1)uf + (Tf - Tref)skq1uq1 (T1 - T2)uf + (Tf - Tref)skq2uq2

[

g ) FsFcp

(T2 - T1)ur (T1 - T2)ur

]

]

Thus this network belongs to the class considered in section 2. The enthalpy of the individual tanks (H1, H2) evolves in both a fast and a slow time scale, and the total network enthalpy (H1 + H2) evolves only in a slow time scale. This was also pointed out earlier17 using an eigenvalue analysis. In practice, the individual unit temperatures, instead of the total enthalpy content, are used as controlled variables. The model reduction framework presented in this paper allows for a representation of the slow dynamics in terms of any of the temperatures (see remark 3.1). The corresponding nonstiff slow model can be used to address the control of that temperature in the slow time scale (see section 5 for an example of the derivation of such a controller). Remark 4.1. Note that the large recycle of energy, in this case, is achieved through the large material recycle stream acting as the energy carrier. These tanks can also serve as chemical reactors, in which case, the presence of large material recycle could be justified. However, the large material recycle stream can also introduce a time scale separation in the material balance dynamics,9 leading to a (c + 1) dimensional overall slow dynamics (with c being the number of components present in the network). 4.2. Reactor-Feed Effluent Heat Exchangers Network. Feed effluent heat exchangers (FEHEs) are key components in the design of heat integrated processes. An FEHE transfers the

Figure 5. Energy representation for the reactor-FEHE network.

heat available in the hot effluent stream from a reactor to the cold reactor inlet stream. A typical setup for energy integration with an FEHE is shown in Figure 4. To improve dynamic operability and allow for startup strategies, the FEHE is usually accompanied by a furnace and a bypass stream. It is required to bring the reactor inlet temperature to the required value in the furnace. Let us consider an energy representation of this reactor-FEHE network, as shown in Figure 5. The cold and hot passes of the FEHE (of duty htr) are represented separately. Let hin, hc, hf, hr, and hout be the enthalpy flows associated, respectively, with the inlet, cold leg outlet, furnace outlet, reactor outlet, and the network outlet streams, and allow Hc, Hh, Hf, and Hr to be the enthalpies of the cold and hot leg of the FEHE, the furnace, and the reactor, respectively. qh is the heat input through the furnace and qgen represents the rate of heat generation in the reactor. The energy balance dynamics for the reactor-FEHE network based on this representation can be represented as follows: dHc ) hin + htr - hc dt dHf ) h c + qh - hf dt (14) dHr ) hf + qgen - hr dt dHh ) hr - htr - hout dt We consider a case where the reactor operates at an elevated temperature and the feed is available at a lower temperature. Tight energy integration via the FEHE requires a high rate of recovery and recycle of energy, which is generally achieved by allowing for sufficiently large heat transfer areas. On the basis of the above, we make the following assumptions regarding the steady state energy flows in the network: 1. The energy recycle term is of higher order of magnitude than the energy inlet through the feed stream, i.e. htr,s/hin,s ) 1/ε. 2. The feed and the effluent stream energy flows are of comparable magnitude, i.e. kout ) hout,s/hin,s ) O(1). 3. The overall heat effect of the reactions is moderately exothermic, i.e. kqgen ) qgen,s/hin,s ) O(1). (Note that this is

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consistent with an adiabatic operation of the reactor; in the case of high heat of reaction, direct cooling would be more appropriate.) 4. As a large amount of energy is recycled, the energy input from the furnace is small and it is assumed to be comparable in magnitude to the feed and effluent streams, i.e. kqh ) qh,s/hin,s ) O(1). 5. The internal energy flows (hc, hf, hr) are of comparable magnitude to the recycle flow. We thus define O(1) ratios kc ) hc,s/htr,s, kf ) hf,s/htr,s, and kr ) hr,s/htr,s. Defining the scaled energy flows uin ) hin/hin,s, utr ) htr/htr,s, uc ) hc/hc,s, uqh ) qh/qh,s, uf ) hf/hf,s, uqgen ) qgen/qgen,s, ur ) hr/hr,s, and uout ) hout/hout,s, eq 14 becomes

[ [ [ [

]

dHc utr kcuc ) hin,s uin + dt ε ε dHf kcuc kfuf ) hin,s + kqhuqh dt ε ε krur dHr kfuf ) hin,s + kqgenuqgen dt ε ε dHh krur utr ) hin,s - - koutuout dt ε ε which is in the vector form (3) with

]

]

]

(15)

H ) [Hc Hf Hr Hh ]T us ) [uin uout uqh uqgen ]T ul ) [utr uc uf ur ]T

f ) hin,s

[ ] uin kqhuqh kqgenuqgen -koutuout

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integration is achieved by exchanging the heat between the top vapor and the bottom streams. Figure 6 shows a typical configuration for a direct vapor recompression distillation column. The vapor coming out from the top of the distillation column is compressed in the compressor so as to make the heat transfer to the bottoms stream possible. The compressed vapor condenses in the reboiler-condenser, and this in turn boils the bottoms stream, generating the vapor entering the stripping section of the column. A trim condenser is used to bring the temperature of the reflux back to the required value as well as to condense the residual vapor. Figure 7 shows an energy representation for this system, wherein the condenser section and the reboiler section of the reboiler-condenser are shown separately, connected via the energy transfer stream htr. The term ht represents energy recycle associated with the reflux stream (material recycle) and hl represents the energy flow associated with the liquid entering the reboiler. Let HR, HD, HC, HF, and HT represent the enthalpy holdups for the reboiler section, the distillation column, the compressor, the condenser section, and the trim condenser, respectively. The terms hf, hr, hv, hc, hrf, hD, and hB represent the enthalpy flows associated with the feed, the vapor leaving the reboiler-condenser, the vapor leaving the column, the vapor leaving the compressor, the condensate leaving the reboilercondenser, the bottoms stream, and the distillate stream, respectively. W is the compressor power input, and Qc is the trim conderser duty. On the basis of the representation in Figure 7, we can write the energy balance equations for the network as follows:

[ ]

g ) hin,s

utr - kcuc kcuc - kfuf kfuf - krur krur - utr

Remark 3.3 helps to classify the control objectives and the manipulated inputs available in each time scale. For example, the bypass stream on the FEHE can be used to manipulate ur (the scaled internal energy flow) by adjusting the material flow entering the hot leg of the FEHE, to address control objectives at the unit level in the fast time scale. On the other hand, the furnace duty can be used to manipulate uqh (the scaled external energy flow) to address control objectives in the slow time scale. Remark 4.2. In the reactor-FEHE network, the large recycle of energy is achieved through a high duty heat exchanger, rather than energy-carrying large material flows. Thus, large energy recycle does not introduce stiffness/time scale separation in the corresponding material balance dynamics. Remark 4.3. The energy balance dynamics of this reactorFEHE network evolves over two time scales. This is contrary to the case of reactor-external heat exchanger networks, where a large material stream from the adiabatic reactor is used reject the heat of exothermic reaction through an external heat exchanger. In those networks, the entire energy balance dynamics evolves over a fast time scale.12 4.3. Vapor Recompression Distillation. Vapor recompression distillation (VRD) is a heat integrated distillation scheme which works on the principle of a heat pump.18 The heat

Figure 6. Vapor recompression distillation configuration: (1) distillation column, (2) compressor, (3) reboiler-condenser, (4) trim condenser, and (5) reflux drum.

Figure 7. Energy representation for the vapor recompression distillation network.

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dHR ) hl + htr - hr - hB dt dHD ) h f + hr + ht - hv - hl dt dHC ) hv - hc + W dt dHF ) hc - htr - hrf dt dHT ) hrf - hD - ht - Qc dt

(16)

g ) hf,s

The economics of VRD favors separations involving components with close boiling points (difficult separations) so that the temperatures of the top and the bottom streams of the distillation column are close. This leads to less compressor power and small trim condenser duty. On the basis of the above, we can make the following assumptions concerning the various steady state energy flows in the network. 1. The energy flows exiting the network (hB,s and hD,s) have the same order of magnitude as that of the feed (hf,s). 2. The compressor energy input and the trim condenser duty are small and we assume them to have the same order of magnitude as that of the feed, i.e. Ws/hf,s ) O(1) and Qc,s/hf,s ) O(1). 3. The latent heat contribution to the enthalpy generally has a higher order of magnitude compared to the sensible heat contribution. The heat transfer across the reboiler-condenser is dominated by the latent heats and so, the energy recycle term htr has a higher order of magnitude compared to the energy entering into the network through the feed, i.e. htr,s/ hin,s ) 1/ε . 1. This also implies that the internal flows (hf,s, hv,s and hc,s) are comparable in magnitude to htr,s. 4. The energy flow associated with the material recycle stream (ht,s) can be of higher order of magnitude than the feed depending on the magnitude of the reflux ratio. However, it will always be of lower order in magnitude compared to htr,s (the latter being the latent heat contribution). For initial analysis, we consider it to be of comparable magnitude with the feed. Similar arguments hold for hl,s. We now define the O(1) steady state ratios kr ) hr,s/htr,s, kc ) hc,s/htr,s, and kv ) hv,s/htr,s and the scaled energy flows utr ) htr/htr,s, ur ) hr/hr,s, uc ) hc/hc,s, and uv ) hv/hv,s, and eq 16 becomes the following:

[ [ [ [

]

dHR utr krur ) hf,s + hl - hB dt ε ε dHD krur kvuv ) hf,s + h f + ht - h l dt ε ε dHC kvuv kcuc ) hf,s +W dt ε ε kcuc utr dHF ) hf,s - hrf dt ε ε dHT ) hrf - hD - ht - QC dt Equation 17 is in the vector form (3) with

]

] ]

H ) [HR HD HC HF HT ]T us ) [hf hD hB W Qc hl ht hrf ]T ul ) [utr ur uv uc ]T

[ ] [ ]

hl - hB hf + ht - hl f) W -hrf hrf - hD - ht - QC

(17)

utr - krur krur - kvuv kvuv - kcuc kcuc - utr 0

Note that for the separation of close-boiling components (for which VRD is typically used), a high reflux ratio and, hence, large internal flows are common. These large material flows are the carriers of a large amount of energy, and they also introduce stiffness in the material balance equations. A more detailed analysis of such a system will be addressed in a future paper. 5. Control of Reactor-FEHE Networks In this section, we illustrate the model reduction and control concepts presented so far on a reactor-FEHE network. The typical control objectives in such networks are controlling the outlet stream composition (a product quality specification), the final effluent temperature (a separation system specification), and the holdup of the network. We assume that m reactions involving n species are being carried out in a CSTR which is operated adiabatically. C represents the vector of concentrations of all species and Co represents the vector of initial concentrations. ∆H ) [∆H1 ∆H2... ∆Hm]T represents the vector of heats of reaction. r ) [r1 r2... rm]T represents the vector of reaction rates and Sn×m represents the matrix of stoichiometric coefficients. Fin is the volumetric flow rate of the input stream to the system, and F is the reactor outlet flow rate; V, VH, VC, and Vf are the holdups of the reactor, the FEHE hot and cold streams, and the furnace, respectively. Tin is the cold leg inlet temperature to the FEHE. TH and TC are the hot and cold leg temperatures for the FEHE. Ti and TR are the reactor inlet and outlet temperatures, and Texit is the network exit temperature. QH is the furnace heat duty. For simplicity, we assume constant heat capacities and densities (i.e., independent of temperature and concentration) and constant heats of reaction. We also assume that there is no phase change in the FEHE and constant holdups for the FEHE and the furnace. The dynamic model of the system then takes the following form: dV ) Fin - F dt dC Fin(Co - C) ) - Sr dt V dTR Fin(Ti - TR) ∆HTr ) dt V Fcp ∂TH ∂TH UA TH - TC ) -VH ∂t ∂z Fcp VH ∂TC ∂TC T TC UA H ) VC + ∂t ∂z Fcp VC QH dTi Fin(TCz)0 - Ti) ) + dt Vf FcpVf

( )( ) ( )( )

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5.1. Model Reduction. As defined earlier, we have

Table 1. Nominal Values of Process Parameters for the Reactor-FEHE Network parameter

value

parameter

value

ko E ∆H Fcp UA R V VH VC Vf L

1.2667 × 107 1/s 142870 J/mol -4.828 kJ/mol 4.184 × 106 J/(m3 K) 83680 W/K 0.1 0.1 m3 0.1 m3 0.09 m3 0.01 m3 1m

cA TR Ti Texit TC,z)0 QH Tin Fin F cAo ε

54.79 mol/m3 922.39 K 910 K 364.86 K 873.48 K 8.82 × 104 W 300 K 5.7667 × 10-4 m3/s 5.7667 × 10-4 m3/s 1000 mol/m3 0.0471

ε)

[FinFcp(Tin - Tref)]s hin,s ) htr,s [FinFcp(TCz)0 - Tin)]s

where Tref is a reference temperature. In order to facilitate such a large recovery of heat, the FEHE must provide a sufficiently large heat transfer area. Specifically, from the energy balance on the cold stream of the FEHE, at steady state, we have [UA(TH′-TC′)]s ) [FinFcp(TC,z)0 - Tin)]s which leads to UA/ Fin,sFcp ) k(O(1))/ε. TH′ and TC′ are the average temperatures of the hot and the cold leg of the FEHE respectively. We define the O(1) ratios: kr ) [FinFcp(TR - Tref)]s/[FinFcp(TCz)0 - Tin)]s

with Texit ) RTR + (1 - R)THz)L

kc ) [FinFcp(TCz)0 - Tref)]s/[FinFcp(TCz)0 - Tin)]s

THz)0 ) TR

kf ) [FinFcp(Ti - Tref)]s/[FinFcp(TCz)0 - Tin)]s

TCz)L ) Tin

ur ) [FinFcp(TR - Tref)]/[FinFcp(TR - Tref)]s

where VH and VC are the velocities of the fluid in the hot and cold compartments of FEHE. L is the length of the exchanger, z is the spatial coordinate, and R is a bypass ratio defined as follows:

uc ) [FinFcp(TCz)0 - Tref)]/[FinFcp(TCz)0 - Tref)]s

R)

flow rate of the bypass stream flow rate of the original stream(F)

uf ) [FinFcp(Ti - Tref)]/[FinFcp(Ti - Tref)]s The energy dynamics equations in (18) thus become:

[ [ (

Figure 8. Steady state energy flows in the reactor-FEHE network.

Figure 9. Evolution of the network variables for the perturbed system.

] ] )

dTR [Fin(Tin - Tref)]s kfuf - krur ∆HTr ) dt V ε Fcp QH dTi [Fin(Tin - Tref)]s kcuc - kfuf ) + dt Vf ε FcpVf ∂TH ∂TH kFin,s TH - TC ) -VH ∂t ∂z VH ε ∂TC ∂TC kFin,s TH - TC ) VC + ∂t ∂z VC ε

(

)

(19)

We define the stretched time scale τ ) t/ε to obtain the description of dynamics in the fast time scale as follows:

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dTR [Fin(Tin - Tref)]s ) (kfuf - krur) dτ V dTi [Fin(Tin - Tref)]s ) (kcuc - kfuf) dτ Vf kFin,s ∂TH )(T - TC) ∂τ VH H ∂TC kFin,s ) (T - TC) ∂τ VC H

(20)

The network exit temperature Texit is related to the temperature of the hot leg exit of FEHE and the reactor temperature through the bypass ratio. We address the control of Texit in the fast time scale using R as the manipulated input to reject the disturbances (e.g. Tin) affecting the dynamics of FEHE. This is consistent with the observation in remark 3.3 regarding manipulating the large internal flows to address the control objectives related to individual units in the fast time scale. Multiplying (19) by ε and taking the limit ε f 0, we obtain the three linearly independent quasi-steady state conditions: 0 ) kfuf - krur i . e . Ti ) TR 0 ) kcuc - kfuf i . e . TCz)0 ) Ti

0 ) T H - TC We now take the limit ε f 0 in the original time scale to obtain the description of dynamics in the slow time scale given by a DAE system of the form (9) as follows: dTR [Fin(Tin - Tref)]s ∆HTr ) z1 dt V Fcp QH dTi [Fin(Tin - Tref)]s ) z2 + dt Vf FcpVf ∂TH ∂TH kFin,s ) -VH z ∂t ∂z VH 3 (21) ∂TC ∂TC kFin,s ) VC + z ∂t ∂z VC 3 0 ) T i - TR 0 ) TCz)0 - Ti 0 ) T H - TC where z1, z2 and z3 are algebraic variables given by lim(kf uf - krur)/ε z1 εf0 (kcuc - kf uf )/ε z ) z2 ) lim εf0 z3 lim(TH - TC)/ε

[][

εf0

]

Figure 10. Closed loop response for +5% change in temperature set point, in the presence of 10% modeling error in ∆H and unmeasured disturbances.

Ind. Eng. Chem. Res., Vol. 48, No. 13, 2009

We address the control of reactor conversion and reactor holdup in the slow time scale. The control of conversion, in this case, is equivalent to the control of reactor temperature, and so, we control TR. The derivation of the model based controller, using the reduced order slow model, is addressed in the following simulation case study. 5.2. Simulation Case Study. Let us consider a reactor-FEHE network where a first-order irreversible exothermic reaction A f B is carried out in the CSTR. The reactant A is fed to the network at concentration cAo and at temperature Tin. The control objectives are to control the reactor temperature (TR), the network exit temperature (Texit), and the reactor holdup (V). The nominal values of the state variables and the process parameters are given in Table 1. Figure 8 shows the steady state values for various energy flows in the network. We note that the internal energy flows have a higher order of magnitude compared to the external energy flows. The furnace duty and the heat generation term are of the same order of magnitude as the network inlet and effluent streams. We initially carry out open-loop studies to analyze the evolution of various network variables in the absence of controllers (simple proportional (P) controllers are used to stabilize holdups). Figure 9 shows the evolution of system, for a 1% perturbation from the steady state. The hot and cold leg exit temperatures for the FEHE show a fast transient dynamics followed by a slow evolution to the steady state. The total network enthalpy evolves, as expected, only over a slow time scale. Turning to the controller design, we address the control of Texit in the fast time scale using R. We use a simple proportional and integral (PI) control law (22) to achieve this objective.

[

R ) Ro + Kc (Texit,set - Texit) +

1 τI

∫ (T t

0

exit,set - Texit)

]

dτ

(22)

where, Ro ) 0.1, Kc ) 0.0018 1/K, and τI ) 10 s. Subsequently, we express the slow model in terms of TR by differentiating the algebraic constraints to obtain explicit expressions for the algebraic variables z. The ODE description of the slow model (along with the material balance dynamics) then becomes the following:

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dV ) Fin - F dt dcA Fin(cAo - cA) ) - koe(E/RTR)cA dt V dTR [Fin(Tin - Tref)]s ∆H (E/RTR) ) z1 ke cA dt V Fcp o

(23)

∆Hkoe(E/RTR)cA QH/Vf z1 z 2 ) + + V Vf [FinFcp(Tin - Tref)]s [FinFcp(Tin - Tref)]s Q H ⁄ Vf z2 z1 + z 2 ) -k Vf VC [FinFcp(Tin - Tref)]s z3 ) 0

( )

As indicated above, the available manipulated inputs are QH and F. These inputs are used to design an input/output linearizing controller for this model to induce first order responses in TR and V: β1

dTR + T R ) V1 dt

β2

dV + V ) V2 dt

where, β1 ) 166.7 min and β2 ) 83.3 min. In order to get an offset free response, we use an external PI controller.

[

V1 ) TR,set + Kc,1 (TR,set - TR) +

[

V2 ) Vset + Kc,2 (Vset - V) +

1 τI,1

1 τI,2

∫ (T t

0

∫ (V t

0

R,set - TR)

set - V)

dτ

dτ

]

]

where, KC,1 ) 8, τI,1 ) 15.8 min, KC,2 ) 5, and τI,2 ) 18.33 min. In the first simulation run, the output tracking performance of the controller is studied for a +5% change in the set point of TR, in the presence of +10% modeling error in ∆H, and unmeasured disturbance in Tin. The corresponding responses are shown in Figure 10. We see that the controller derived based on the reduced order model shows excellent performance in tracking the desired trajectory for the full order model, even in the presence of large modeling errors. We also note the time scales on which the controllers operate, i.e. the Texit is controlled in the fast time scale while TR is controlled over a slower time horizon. Note also that there is a sharp increase in the required furnace duty in the initial transient period. This is due to the mismatch between the actual model and the reduced order slow model in the fast time scale.

Figure 11. Closed loop response for +5% change in temperature set point, in the presence of 10% modeling error in ∆H and unmeasured disturbances for a linear and a nonlinear controller.

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Figure 12. Closed-loop evolution in the presence of +5% increase in production rate.

In the second simulation run, we compare the performance of this nonlinear controller with a linear PI controller tuned using the Ziegler-Nichols technique (Kc ) 595.64 K/W and τI ) 300 s). Figure 11 shows the corresponding responses for the same set point change and modeling error. In this case, the linear controller yields an oscillatory response. Furthrmore, the PI controller imposes sharp variations in the furnace duty (compared to the variations imposed by the proposed nonlinear controller). Finally, we consider the response of the system for a +5% increase in the production rate (by increasing the network throughput). In order to get the same conversion in the reactor, the reactor temperature set point was increased to 924.9 K. From Figure 12, we see that the reduced order model based controller works very well in this case too. 6. Conclusions In this paper, we analyzed the dynamic aspects of a class of energy integrated networks with large recycle of energy compared to input/output energy flows. Using a prototype network representing this class, we showed that the energy balance dynamics are modeled by stiff differential equations,

featuring a transient behavior with two time scales. We proposed a model reduction framework using singular perturbation techniques to obtain separate nonstiff models in each time scale. Our analysis illustrated that the enthalpies of the individual units evolve in both a fast and a slow time scale, while the overall network enthalpy evolves only over a slow time scale. As a result, control objectives related to the energy dynamics of the individual units should be addressed in the fast time scale using the large internal energy flows, while control objectives related to the network energy dynamics should be addressed in the slow time scale using the small external energy flows. The results obtained for the prototype network were then applied to a reactor-FEHE network to propose a control strategy, exploiting this time scale separation. The controllers derived using the nonstiff reduced order models were well-conditioned and showed excellent disturbance rejection and set point tracking performance in the presence of large modeling errors. Acknowledgment Partial financial support for this work by the National Science Foundation, grant CBET-0756363, is gratefully acknowledged.

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(11) Baldea, M.; Daoutidis, P.; Kumar, A. Dynamics and control of integrated networks with purge streams. AIChE J. 2006, 52, 1460. (12) Baldea, M.; Daoutidis, P. Model reduction and control of reactorheat exchanger networks. J. Process Control 2006, 16, 265. (13) Baldea, M.; Daoutidis, P. Modeling, dynamics and control of process networks with high energy throughput. Comput. Chem. Eng. 2008, 32, 1964. (14) Kokotovic, P. V.; Khalil, H. K.; O’Reilly, J. Singular perturbation methods in control: Analysis and design; Academic press: New York, 1986. (15) Kumar, A.; Daoutidis, P. Control of nonlinear differential algebraic equation systems; Chapman & Hall/CRC: New York, 1999. (16) Kumar, A.; Daoutidis, P. Nonlinear model reduction and control of high-purity distillation columns. Ind. Eng. Chem. Res. 2003, 42, 4495. (17) Georgakis, C. On the use of extensive variables in process dynamics and control. Chem. Eng. Sci. 1986, 41, 1471. (18) King, C. J. Separation processes, 2nd ed.; McGraw-Hill: New York, 1980.

ReceiVed for reView July 8, 2008 ReVised manuscript receiVed September 22, 2008 Accepted September 24, 2008 IE801050B