Mitigation of Fouling in Refinery Heat Exchanger Networks by Optimal

1038

Energy & Fuels 2001, 15, 1038-1056

Mitigation of Fouling in Refinery Heat Exchanger Networks by Optimal Management of Cleaning F. Smaı¨li, V. S. Vassiliadis, and D. I. Wilson* Department of Chemical Engineering, Pembroke Street, Cambridge, CB2 3RA, U.K. Received March 2, 2001. Revised Manuscript Received June 8, 2001

A method for optimizing the cleaning schedule in large continuously operating heat exchanger networks (HENs) such as refinery crude preheat trains is presented. The method is based on two heuristics: (i) discretisation of the operating horizon into a number of equally long periods in which cleaning decisions are allocated and (ii) solution of the resulting mixed integer nonlinear programming (MINLP) problem by a multiple starting point strategy and selection of the best locally optimal solution. The network performance equations are written in terms of the binary decision variables and are not linearized further, giving rise to a nonconvex objective function. The formulation includes constraints set by pumparound operation and pressure drop. Different deterministic relationships between pressure drop and fouling resistances are discussed. The form of the objective function is discussed, as a fixed horizon approach has a significant effect on the results obtained. Solution of the MINLP model is demonstrated using a commercial MINLP solver (DICOPT++) for two case studies: (I) an idealized network containing 14 exchangers over three years and (II) an operational plant featuring 27 exchangers for a two-year horizon. The fouling models and parameters for study II were obtained by reconciliation of plant data, and the relatively short horizon reported was due to limitations in the NLP solver rather than to the model. The results for study II are compared with those generated from a simpler approach based on a “greedy” algorithm. The case study II results are also interpreted in terms of selection of appropriate fouling mitigation strategies. The effectiveness of the MINLP approach is discussed, particularly with regard to obtaining globally optimal solutions.

1. Introduction Heat exchanger fouling is a long-standing problem in the process industries. Fouling is particularly important in continuous processes as the loss in efficiency over time caused by fouling must be countered by reducing production rates, by using more severe operating conditions (e.g., higher utility temperatures, higher feed pressures) until the system reaches its operability limits, or by a combination of these measures. Other fouling costs include lost production due to downtime for cleaning and extra maintenance. The problem is particularly significant in large networks of heat exchangers, such as those found on oil refinery crude distillation unit (CDU) preheat trains, which are required to operate continuously over several years between shut-downs. CDU furnaces are major energy consumers on any refinery, and the extra energy cost caused by preheat train fouling has become more important following the 1997 Kyoto Agreement, owing to the imposition of energy taxes and CO2 emission levels. Effective methods for mitigating fouling are therefore becoming more important. Fouling mitigation techniques include: (i) Reducing the rate of fouling by interrupting the mechanisms causing fouling. This is achieved in crude oil systems by adding antifoulant chemicals, which adds * To whom correspondence should be addressed. E-mail: ian_wilson@ cheng.cam.ac.uk.

directly to the operating cost of the systems. Other methods include careful control of feedstock composition and operating conditions in order to minimize the processing of unstable blends prone to cause fouling.1 (ii) Using more robust heat transfer equipment, usually at increased capital cost. This strategy can take the form of specifying individual units, such as fluidized bed exchangers,2 for duties subject to severe fouling, or a more flexible network configuration featuring oversized or duplicate units for such duties. However, such hardware-based solutions may not be viable, or capable, for all installations. Concepts of network resilience and robustness with regards to fouling have also been discussed (e.g., ref 3). (iii) Regular cleaning of fouled units during the operating cycle, to restore thermal and hydraulic performance. Cleaning affects both the operating and capital costs, as there will be penalties incurred while a unit is being cleaned, and the network must be configured so that the units can be taken off-line for cleaning. Refineries frequently use a combination of the above techniques in order to minimize the overall operating cost. Identification of the most favorable combination involves a techno-economic cost calculation which is complicated by the unknown fouling propensity of many (1) Wiehe, I. A.; Kenedy, R. J. U.S. Patent 763652, 1996. (2) Klaren, D. G.; Bailie, R. E. Hydrocarbon Processing 1989, 68 (7), 48-50. (3) Fryer, P. J.; Paterson, W. R.; Slater, N. K. H. Chem. Eng. Res. Des. 1987, 65, 267-271.

10.1021/ef010052p CCC: $20.00 © 2001 American Chemical Society Published on Web 07/26/2001

Mitigation of Fouling

crude feedstocks and the difficulty in generating cleaning schedules for large networks. This paper addresses the latter aspect; the aim of the current work is to develop methods which will allow plant operators to estimate the optimal (lowest) likely operating cost associated with cleaning strategies, for comparison with other mitigation scenarios. The focus of this paper is on oil refinery operations, but the approach can be applied to other sectors featuring networks subject to fouling, e.g., sugar, bauxite, Kraft pulping, or desalination plants. Calculation of the cost of a cleaning strategy requires optimization of the cleaning schedule for the whole network, which has not received much attention until recently. The optimal cleaning schedule for an individual heat exchanger can be readily calculated using deterministic,4 statistical,5 or optimal control6 approaches. Scheduling in networks is complicated by the redistribution of heat duties around the network when a unit is taken off-line for cleaning. Furthermore, the response of the network to a unit being cleaned changes over time because units foul at different rates, often affecting the sensitivity of the system to changes in configuration. The scheduling problem for a continuously operating heat exchanger network (HEN) seeks to determine the best combination of discrete events, namely, the removal or bypassing of exchangers for cleaning, while maintaining its operability. It therefore features a combination of continuous temporal variabless temperatures, pressure drops, etc.swith integer variables, describing cleaning decisions. The problem features an objective function, usually based on cost, constraints related to operability and safety, and a set of nonlinear equations relating the heat duties and temperatures in the network. The HEN scheduling problem therefore requires three elements: (i) a reliable simulation of the network performance; (ii) representative models for fouling behavior; (iii) a robust method for schedule optimization. Item ii is frequently a source of difficulty, as fouling data may not be available or feature large uncertainties. Mu¨ller-Steinhagen7 reported the successful combination of all three elements in considering fouling mitigation, including cleaning, for a bauxite refinery, where the feedstock was relatively consistent. This work is concerned with the development of robust scheduling methods for use in networks with poor quality fouling data. A robust, reasonably fast optimization method is required, as it is expected that several scenarios will need to be tested. In practice, the schedule would be based on fouling models based on plant data and updated in an adaptive mode as more data become available to revise the fouling model parameters. Recent work based on mixed integer nonlinear programming (MINLP) approaches has shown that the HEN scheduling problem can be solved using modern optimization codes. Smaı¨li et al.8 demonstrated the use (4) Epstein, N.; Ma, R. S. T. Can. J. Chem. Eng. 1981, 59, 631633. (5) Zubair, S. M.; Sheikh, A. K.; Budair, M. O.; Badar, M. A. In Understanding Heat Exchanger Fouling and its Mitigation; Panchall, C. B., Bott, T. R., Somerscales, E. F. C., Toyama, S., Eds.; Begell House: New York, 1997; pp 397-406. (6) Ritter, N. L. AIChE Symp. Ser. 1983, 79, 39-47. (7) Mu¨ller-Steinhagen, H. Chem. Eng. Res. Des 1998, 76, 97-107.

Energy & Fuels, Vol. 15, No. 5, 2001 1039

of an MINLP model based on time discretisation for smaller networks and its subsequent application to larger networks. Georgiadis et al.9 developed another MINLP formulation, also based on time discretisation, to solve the scheduling problem for larger networks subject to rapid fouling. Georgiadis and Papageorgiou10 subsequently reported the application of their formulation to the “wraparound” problem where the schedule is to be repeated on a regular basis. Their papers contain useful summaries of related work in the scheduling and MINLP optimization literature. Their formulation differed from that reported here principally in the use of the arithmetic mean temperature difference instead of the logarithmic mean in the heat exchanger performance equation, viz.

Q ) UA∆Tlm

(1)

Linearization of the resulting equations yielded a mixed integer linear programming (MILP) problem that could be solved using standard optimization software to yield a globally optimal solution. This paper describes a similar approach to the network scheduling problem, (formulation of an MINLP model) but maintaining the nonlinear form of eq 1 as the arithmetic form is not appropriate for large networks which feature extensive feedback of hot (and/or cold) streams. The scheduling formulation is similar in concept to the earlier work;8 the major developments are the inclusion of pressure drop and pumparound operation constraints, nonlinear fouling rates, and the extension to larger, integrated networks. Section 2 describes the MINLP formulation, including process constraints of particular importance in CDU HENs, namely, pressure drop and pumparound limitations. Solutions to two case studies obtained using a commercial optimization solver are presented in section 3: an idealized network and a problem based on an operating refinery CDU. The MINLP solutions for the latter are compared with those generated using a simpler optimization approach, based on a greedy algorithm. Section 4 contains a discussion of the advantages and drawbacks to the MINLP approach for this type of scheduling problem. 2. Formulation of the Scheduling Problem 2.1. Objective Function and MINLP Approach. The scheduling problem for a fixed network involves the optimization (minimization) of the total operating cost due to fouling over the operating horizon of the plant, tF. The objective function is often presented in the form

Obj )

∫0

tF

NC

{CEQFurnace(t) + CM(t) + CX(t)} dt +

∑j Cc,j (2)

where CEQ(t) describes the extra energy cost caused by fouling, often based in refineries on the coil inlet (8) Smaı¨li, F.; Angadi, D. K.; Hatch, C. M.; Herbert, O.; Vassiliadis, V. S.; Wilson, D. I. Trans. Inst. Chem. Eng. 1999, 77C, 159-164. (9) Georgiadis, M. C.; Papageorgiou, L. G.; Macchietto, S. Ind. Eng. Chem. Res. 2000, 39, 441-454. (10) Georgiadis, M. C.; Papageorgiou, L. G. Trans. Inst. Chem. Eng. 2000, 78A, 168-179.

1040

Energy & Fuels, Vol. 15, No. 5, 2001

Smaı¨li et al.

The formulation thus involves the following indices:

n ) 1, ..., NE where NE is the number of exchangers to be considered for cleaning p ) 1, ..., NP Figure 1. Time discretisation in MINLP formulation showing labeling of sub-periods.

temperature (CIT) on the CDU furnace downstream of the network. Initial calculations for UK scenarios indicated that the cost of cooling hot streams was significantly smaller than furnace firing, so CE here is based on CIT alone. CM and CX describe the extra maintenance and lost “opportunity” costs incurred by fouling, respectively; Cc,j is the cost of the jth cleaning action, with a total of NC cleaning actions in the interval 0-tF. The first term in eq 2 gives an energy optimization calculation: a detailed description of the cost terms in eq 2 for a single heat exchanger was given by Casado.11 This work considers the scenario where CM ) CX ) 0 and Cc is constant, without any loss of generality in the formulation. The time horizon, tF, to be considered in refinery networks presents a challenge for schedule optimization because the natural choice for tF, namely, the interval between planned shut-downs, is currently being increased where possible in order to maximize productivity. Intervals of 5-7 years for such networks give rise to large problems. In case study II, we compare the results of the MINLP model with an alternative approach to optimizing Obj on the basis of the simple greedy algorithm. It is noteworthy that the objective function presented in eq 2 does not consider the state of the network at time tF or beyond. There is thus little incentive to clean units near this point, as the opportunity to recoup energy lost during the time off-line is limited. It will be seen that this has significant effects on the schedules generated. The objective could be modified to include “future” operations by (i) adding a period of time beyond tF in which energy costs are counted but cleaning is not allowed or (ii) adding a penalty cost for cleaning each exchanger at time tF, (i.e., at shutdown) related to its fouled state (quantified by its fouling resistance, Rf). The principal step in the MINLP formulation is discretising the operating horizon into NP equal periods, of length ∆t ) tF/NP, and only allowing cleaning actions to be performed at the period nodes. The finite length of time required to clean an exchanger is incorporated by further sub-dividing each period into a cleaning subperiod, of length ∆tcl, in which cleaning actions may be performed, and a processing sub-period, of length ∆tpr, in which all units are on-line, (Figure 1). The length of each sub-period can be adjusted as necessary (e.g., weekdays vs weekends, or even ∆tpr ) 0). The cleaning status of each exchanger in each cleaning sub-period p is described by a binary variable, yn,p, where

yn,p ) 1 if unit n is on-line over cleaning sub-period p 0 otherwise (3)

{

where NP is the number of periods

Given the above criteria, the objective function can then be written as N P NE

Obj )

∫0t CEQ(t) dt + ∑ ∑ Cc(1 - yn,p) F

(4)

p)1n)1

where the energy integral is calculated using an appropriate quadrature method. The cleaning cost calculation in eq 4 incorporates a cleaning cost, Cc, which could be tailored to the type and size of each exchanger, although in the following case studies a uniform value is used. The cost of cleaning an exchanger can also depend on how long the unit has been in service, particularly if deposit aging (e.g., “coking”) occurs. A time-dependent cleaning cost is not considered here, as (i) a significant fraction of the labor cost is due to isolation and removal of tube bundles and (ii) there is a scarcity of data available to support any aging model. An alternative approach to the coking problem would be to employ an operational constraint, specifying the maximum length of time between cleaning actions. The MINLP solver requires the model to be written in terms of a continuous decision variable, θn,p

0 e θn,p e 1

(5)

which is subsequently constrained to take the value 0 or 1. The speed of the MINLP solver is directly related to the number of binary variables involved. Several time scales must be considered in the selection of an appropriate discretisation period. The value of ∆t chosen must be small enough to give reasonable accuracy in integrating eq 4 and smaller than any characteristic period associated with fouling in any exchanger. It is important to recognize that the introduction of a regular time discretisation represents the application of a heuristic to the scheduling problem, reducing the choice of timings permitted to cleaning actions and possibly excluding the globally optimum schedule from the solution space. This feature can only be countered by very fine discretisation (large NP) or by progressively testing the sensitivity of the results to the value of NP. Alternatively, the length of time between cleaning actions could be allowed to vary, giving a more complex optimization problem. Related scheduling approaches have been described by Iepatritou and Floudas.12 The task is to find the combination of {yn,p, n ∈ 1, ..., NE; p ∈ 1, ..., NP} which minimizes Obj given a fixed network configuration, process parameters, and constraint sets. 2.2. HEN Simulation. A heat exchanger network consists of a number of hot and cold streams which pass through a set of heat transfer units in a fixed configuration, as shown in Figure 2a. We assume that each unit (11) Casado, E. Hydrocarbon Processing 1990, 69 (8), 71-76. (12) Ierapetitou, M. G. Floudas, C. A. Ind. Eng. Chem. Res. 1998, 37, 4360-4374.



exchanger will not vary demonstrably over time in most exchangers, so this assumption is reasonably valid. In a similar vein, film heat transfer coefficients are assumed to remain constant despite changes in Prandtl and Reynolds numbers caused by different temperatures and fouling. (III) Fouling Models Used. Individual heat exchangers are modeled using a lumped parameter approach, using the first-order models described below. This approach is justified on the basis of accuracy of data available, the computational time required to perform a full thermohydraulic simulation of real units, and the uncertainty attached to the fouling models. Fouling is modeled using the lumped fouling resistance approach, where the change in overall heat transfer coefficient U is related to the fouling resistance, Rf, via

1 1 + Rf(t) ) U(t) Uclean Figure 2. (a) Schematic diagram of a heat exchanger network. (b) Individual heat exchanger unit showing nomenclature and bypasses for isolation during cleaning. Solid lines, cold streams; dotted lines, hot streams; circles, heat exchangers; H, hot utility (e.g., furnace); C, cold utility.

can be taken off-line or bypassed for cleaning without interrupting the process, as indicated in Figure 2b. The relevant hot and cold streams either bypass the unit or are diverted to another exchanger. The HEN simulation therefore consists of a set of nonlinear equations describing the performance of each exchanger and a set of linear equations describing linking of units, splitting and mixing of streams. The simulation model used here is based on the following assumptions: (I) Constant Flow Rates. The mass flow rates of streams entering the network during each period are fixed at values which may vary between periods in a preordained manner to reflect seasonal variations. This assumption excludes corrective control actions in order to meet process constraints, e.g., by partial bypassing of an exchanger or reducing the input crude flow rate in order to avoid a furnace firing limitation. The flow rate through an exchanger only depends on its cleaning status, or the status of other units sharing the same hot stream in the case of flow splitting. The scenario being studied will therefore correspond to maximum throughputsthat of greatest interest to refinery operations. The combined scheduling and optimal control problem presents a significantly more difficult problem. The work of See et al.13 featured the use of a similar MINLP formulation to solve the combined scheduling and control problem in reverse osmosis desalination systems, where network configurations are considerably simpler. (II) Constant Physical Parameters. The heat capacity of each stream is assumed not to vary significantly with temperature. Significant variations in fluid heat capacity, caused by phase changes in hot streams, for example, would require more complex exchanger models than are used here. The temperature range in a given (13) See, H. J.; Vassiliadis, V. S.; Wilson, D. I. In Water Industry Systems: Modelling and Optimization Applications; Savic, D. A., Walters, G. A., Eds.; Research Studies Press: Baldock, 1989; Vol. 2, pp 393-405.

(6)

where Uclean is the overall heat transfer coefficient in the absence of fouling. The rate of fouling will vary with fluid composition, flow rate and temperature (e.g., ref 14), but variations in crude feedstock composition render detailed modeling difficult. Most studies of refinery fouling have reported strong temperature effects;15,16 within an exchanger, the distribution of deposition is very sensitive to the temperature distribution. For network modeling purposes, however, the range of stream temperatures across a HEN unit will be reasonably constant, and process plant data will be, at best, averaged across a given unit. Simple fouling models with constant parameters, based on approximate temperature ranges for individual units, are therefore used here. In practice, plant data would be used to indicate appropriate fouling models, as demonstrated in case study II. The fouling resistance approach is used here as it builds on the existing fouling literature and also provides a deterministic framework for relating heat transfer and pressure drop. The two most common forms of fouling model are considered. Fouling induction times are taken to be negligible. (i) Linear fouling

R˙ f ) c

(7)

where c is constant for a particular exchanger. Systems with well-defined fluid composition, etc., can feature more complex fouling models (e.g., ref 7). (ii) Asymptotic fouling

Rf(t) ) R∞f (1 - exp(-t′/τ))

(8)

where R∞f is the asymptotic fouling resistance, t′ is the time elapsed since the last cleaning action, and τ is a fouling time constant. (14) Wilson, D. I.; Watkinson, A. P. Eng. Therm. Sci. 1997, 14, 361374. (15) Crittenden, B. D.; Kolaczkowski, S. T.; Downey, I. L. Chem. Eng. Res. Des. 1992, 70, 547-557. (16) Panchal, C. B.; Kuru, W. C.; Liao, C. F.; Ebert, W. A.; Palen, J. W. In Understanding Heat Exchanger Fouling and its Mitigation; Bott, T. R., Melo, L. F., Panchal, C. B., Sommerscales, E. F. C., Eds.; Begell House: New York, 1999; pp 273-282.

1042


Smaı¨li et al.

The performance of individual heat exchangers is calculated using the NTU-effectiveness method, whereby eq1 is rearranged in the form of a rating calculation. Exchangers are modeled here as simple countercurrent units, as shown in Figure 2b; the relationships for more complex configurations can be found in standard texts (e.g., ref 17). The parameters R and R are defined by

R)

UA FhChp

(9)

(note that the initial value (p ) 1) of U need not be Uclean) and (b) processing the sub-period pr Un,p )

cl Un,p cl cl 1 + Un,p R˙ fn,p ∆tcl p

yn,p + (1 - yn,p)Uclean,n

(17)

Equations 16 and 17 involve the fouling rate in each sub-period, which is given by (i) Linear fouling pr cl ) R˙ fn,p ) cn R˙ fn,p

(18)

h

R)

F Chp FcCcp

(10)

Most preheat configurations will feature R < 1, so the following equations must be amended if the alternate case, R > 1, arises. Assuming no energy losses, an enthalpy balance gives

Q ) FcCcp(Tc,out - Tc,in) Q)F

h

Chp(Th,in

h,out

-T

(11)

)

(12)

Rearranging (eqs 1, 11, and 12) gives

[

]

R(exp(-R(R - 1)) - 1) h,in T + Tc,out ) exp(-R(R - 1)) - R (1 - R) exp(-R(R - 1)) c,in T (13) exp(-R(R - 1)) - R

[

]

or

Tc,out ) φhTh,in + φcTc,in

(14)

and

Th,out ) Th,in -

1 c,out - Tc,in) (T R

(15)

In the MINLP model, the overall heat transfer coefficient for exchanger n in period p, Un,p, is written in terms of the decision variable θn,p, which is subsequently constrained to take binary [0,1] values. In the following equations, the binary decision variable, yn,p, is used in order to maintain clarity. Equation 14 is then written as c,out h h,in c c,in ) φn,p Tn,p yn,p + (1 - yn,p + yn,pφn,p )Tn,p Tn,p

(14a)

The heat transfer coefficients in each sub-period are calculated at the beginning of each sub-period and are related to the values in the previous sub-period by (a) cleaning the sub-period cl Un,p*1

)

pr Un,p-1 pr pr 1 + Un,p-1 R˙ fn,p-1 ∆tpr p

(16)

(ii) Asymptotic fouling. The time elapsed since the last cleaning action is accounted for by linking the fouling rate to that in the previous period. (a) Cleaning sub-period cl pr ) R˙ fn,p-1 exp(-∆tpr R˙ fn,p p /τ)

(b) Processing sub-period pr cl R˙ fn,p ) R˙ fn,p yn,p exp(-∆tcl p /τ) +

R∞fn (1 - yn,p) τ

(20)

cl The initial rate, R˙ fn,1 , is related to the initial fouling resistance by

cl ) R˙ fn,1

(

)

R∞fn Rfn(0) 1τ R∞ fn

(21)

Derivation of eqs 19 and 21 is given in Appendix A. Pressure Drop. Deposition of fouling layers increases the pressure drop across an exchanger for a given flow rate due to reduction of duct size and/or channel blockage. Increases in pressure drop across a network can be countered by increasing the inlet pressure, incurring extra energy costs, or by reducing throughput. In the case of pumping liquids (e.g., in refinery CDU HENs), energy costs are often less important than the limits set by the maximum pressures permitted in equipment so that pressure drop effectively imposes a constraint on the operation of the network. For the constant flow rate scenario, there will therefore exist a maximum increase in pressure drop due to fouling beyond which the throughput has to be reduced. Pressure drop is therefore considered as part of the constraint set in the following section. The fouling resistance concept allows the change in pressure drop across an exchanger to be linked to the change in thermal efficiency of that unit. The increase in pressure drop due to fouling is estimated here on the basis of reduction of channel duct dimension under turbulent flow conditions, assuming that the roughness and thermal properties of the deposit remain constant. Consider the situation where crude flows through the tubes of a shell and tube heat exchanger. If the fraction of the overall Rf value due to crude side fouling, xf, remains constant, the derivation in Appendix B shows that the change in the pressure drop ∆Pt on the tube side of a shell and tube heat exchanger can be approximated by

∆Pt(t) (17) Incropera, F. P.; De Witt, D. P. Fundamentals of Heat & Mass Transfer; Wiley: New York, 1996.

(19)

∆Pt(clean)

(

)

λ fx f Dt

) 1 - 2Rf(t)

-4.75

(22)



and the change in pressure drop due to fouling across the shell side, ∆Ps, is given by

(

4z2 - πDs2

)

Ds + 2Rfλfxf ) 2 2 Ds ∆Ps(clean) 4z - π(Ds + 2Rfxf) ∆Ps(t)

(

z - Ds Ds + 2Rfλfxf

)

1.19

and/or

(23)

(24)

Tm,p g XL

(25)

Seasonal variations in operating strategies could be incorporated by specifying XU or XL on a period by period basis. (iii) Target Constraints, Where the Target Involves a Combination of Process Variables. Important examples in refinery networks are furnace firing limits set by the capacity of the CDU furnace, pressure drops ,and pumparound duties on distillation columns. These variables all feature a strong dependence on flow rate, which is suspended here by the constant throughput statement (assumption I). These constraints can introduce significant bilinearities into the model, creating problems in converging solutions. With assumption I, the furnace firing capacity gives a lower limit temperature on CIT (eq 25). The pressure drop criterion is represented by a sum of terms; for example, the cold stream passing through exchangers A-C in Figure 2a may be subject to operating limits described by U ∆Pt,A,p + ∆Pt,B,p + ∆Ps,C,p e ∆PA-C

(27)

Equation 14 is then written as

(ii) Target Constraints, Involving Single-Process Variables. Examples are “run down” temperatures for hot streams, operating ranges for desalters or flash drums, etc., which take the following form for exchanger m

XU g Tm,p

QPA ) FcCcp(Tc,out - Tc,in) ) FcCcpDTPA

1.81

Appendix B also features a discussion of a model based on tube blockage, which gives a different result for eq 22. 2.3. Constraints. The scheduling problem in large continuously operating HENs will feature a number of constraints on continuous (process) variables and decision (integer) variables, which are categorized here as (i) Bounds. The continuous decision variable, θn,p, is allowed to vary from 0 to 1

0 e θn,p e 1

around results in a uniform increase in cold stream temperature, DTPA

(26)

Assumption I means that the model described here does not therefore incorporate partial bypassing of exchangers in order to counter excessive pressure drop through a badly fouled unit. Pumparounds on distillation columns require constant heat duties, which are affected both by flow rates and approach temperatures. Pumparound streams usually feature automated bypassing and trim cooling, i.e., local control, which is difficult to incorporate into the larger scheduling problem. A “coarse” approach is presented here and demonstrated in case study II, which features a number of pumparound streams. The approach taken is to modify the heat exchanger performance equations. Under assumptions I and II, a pump-

Tc,out ) Tc,in + DTPA

(28)

Equation 15 does not require alteration. The feasibility of pumparound operation is maintained by comparing DTPA with the maximum possible increase in cold stream temperature for the calculated approach temperatures, DTPA,max, given by (see eq 14)

DTPA,max ) φhTh,in + (φc - 1)Tc,in

(29)

and by the constraint

DTPA,max g DTPA

(30)

The choice of DTPA requires careful consideration, as the pumparound duty may be spread across more than one exchanger, either in parallel or in series so that QPA is given by the sum of several contributions. Both DTPA and DTPA,max may then also need to be modified to accommodate the case where one of the units is off-line for cleaning. (iv) Selection Constraints Imposed on Binary Variables. Processing considerations are likely to result in target criteria which can be more readily incorporated into the optimization model as constraints on the combinations of cleaning actions allowed. Such selection constraints reduce the solution space and the number of network simulations to be performed, usually resulting in faster convergence to a solution. For example, the plant engineer for the network shown in Figure 2a could readily calculate that only one of exchangers A-C can be taken off-line for cleaning at any time in order to avoid a furnace firing limitation, expressed by

yA,p + yB,p + yC,p g2

(31)

Other selection constraints can embody common sense or plant protocols, e.g.

yn,p + yn,p-1 g 1

(32)

this prevents a unit being cleaned in the period after it was last cleanedswhich may not be valid if the interval between cleaning actions were very long. 2.4. Implementation. The equations representing the network and its constraints were written in terms of {θn,p} in the GAMS programming environment (GAMS Development Corp., Washington, DC) and solved using DICOPT++ on a Unix workstation (Sun Sparc 10). DICOPT++ uses the deterministic outer approximation/ extended relaxation (OA/ER) method in decomposing the main MINLP problem into a series of nonlinear programming (NLP) and mixed-integer programming (MIP) sub-problems.18 The formulation presented here involves a nonconvex objective function, so the OA/ER algorithm provides locally optimal solutions. The solution technique em(18) Viswanathan, J.; Grossmann, I. E. Comput. Chem. Eng. 1990, 14, 769-782.

1044


Figure 3. Schematic of temporal formulation of the scheduling approaches on a typical CIT time profile. The dashed line shows the idealized effect of cleaning an exchanger in period j. Solid line, no cleaning (worst case) schedule.

ployed here therefore used a number of starting points (often 10-50), followed by review to identify the best solution obtained. The nonconvexity is a result of the combinatorial nature of the problem, and the aim of this investigation was to ascertain whether a nonlinearized model could be solved successfully. The problems presented here required up to 1000s of CPU time on the machine used; CPU times and numerics are not discussed in detail. The multiple starting point approach represents the application of a second heuristic, after time discretisation, so there is no guarantee that a solution to the MINLP problem obtained by exhaustive search techniques would be the global optimum. A helpful feature of the OA/ER method is that the algorithm calculates the objective function for the relaxed MINLP (RMINLP) problem, and this value can be compared with the MINLP objective. The RMINLP value is nonfeasible but provides an upper bound (maximum saving) for the MINLP solutions, while the no-cleaning scenario represents the corresponding lower bound. Comparison of the MINLP and RMINLP objective function values gives some indication of how good the MINLP value is. Close agreement of these values is likely to be sufficient, given the uncertainty in the model parameters and the effort required to find better solutions. 2.5. Greedy Algorithm. The MINLP model described above is complex to implement and solve, so case study II features a comparison with a simpler scheduling approach based on the “greedy algorithm”. The difference in approach is illustrated in Figure 3. Whereas the MINLP model described above considers all possible cleaning actions over the horizon 0-tF (i.e., yn,p, n ∈1, ..., NE; p ∈ 1, ..., NP), the greedy algorithm only considers cleaning actions in the current period (i.e., period j, yn,j, n ∈ 1, ..., NE) and the effect of those actions over a “sliding” horizon of Ns periods. No cleaning actions in future periods are considered in the scheduling decision. Decisions are based on the return on cleaning exchanger m, which is the difference between the costs when unit m is cleaned and the base case where no cleaning occurs. With CM ) CX ) 0, the decision parameter Gm is given by

Gm|period j ) CE

∫tt j

j+Ns

(Q(t)|no cleaning Q(t)|clean m in period j) dt - Cc,m (33)

where tj and tj+Ns are the times corresponding to periods j and j + Ns respectively. Cleaning decisions are based

Smaı¨li et al.

Figure 4. Configuration of crude oil preheat train in case study I. Solid lines, cold streams; dotted lines, hot streams; H, hot stream input. Clean heat exchanger duties (in MW) are shown in italics beside each unit.

on a threshold heuristic: only cleaning actions which generate an improvement are considered, i.e., those with

Gm > ∆

(34)

where ∆ is a threshold parameter with ∆ > 0. The cleaning action selected is that with the largest Gm value satisfying eq 34 unless cleaning another unit is necessary to avoid a constraint violation in the period under consideration. This algorithm thus involves a significantly smaller combinatorial problem in each period, but features three drawbacks. First, eq 34 indicates that G is very sensitive to Ns and a minimum value of Ns will exist for Gm > 0. Furthermore, since different units foul at different rates, Gm values for units can change ranking if a larger value of Ns is used. Second, the use of the threshold selection criterion reduces the scope for optimization. The ∆ parameter represents the net gain on cleaning, and the algorithm does not consider whether this can be maximized by delaying cleaning by a period or more. Furthermore, the algorithm does not consider the best way to handle future control constraint violations. Third, the simple greedy algorithm presented here requires G to be calculated for each exchanger in each period, which therefore involves many simulation iterations. More refined dynamic programming methods would increase the calculation speed. The greedy algorithm was implemented for case study II using the GAMS model constructed for the MINLP model. The cleaning decision heuristic used featured ∆ ) 1000 £ and only considered one exchanger to be cleaned per period. 3. Case Studies 3.1. Case Study IsModel Heat Exchanger Network. The heat exchanger network shown in Figure 4 is based on an existing network and features seven hot streams. The crude oil flow between the flash unit and the furnace is split, as exchangers 1A/B to 3A/B are subject to severe fouling and are more likely to require cleaning. The hot streams passing through these units are subsequently contacted with the incoming crude earlier in the network so that there is significant feedback in the network. The design and operating parameters of the system are summarized in Table 1. The linear fouling rates in Table are taken from plant studies.19 Inspection of the rates shows an increase in fouling rate with bulk crude temperature, with the


Energy & Fuels, Vol. 15, No. 5, 2001 1045 Table 1. Summary of Design and Fouling Parameters for Case Studya

unit

Th,in/ °C

Tc,in/ °C

Fh/ kg s-1

Fc/ kg s-1

Chp/kJ kg-1 K-1

Ccp/kJ kg-1 K-1

Uclean/kW m-2 K-1

A/ m2

R˙ f linear × 1011/m2 K J-1

R˙ f asymptotic × 1011/m2 K J-1

1A 2A 3A 1B 2B 3B 4 5 6 7 8 9 10 11

334 286 249 334 286 249 254 205 285 237 170 197 296 194

210 191 178 210 191 178 167 161 135 116 101 50 45 26

17.4 22.8 9.5 17.4 22.8 9.5 45.5 55.8 34.8 49.7 49.7 55.8 3.3 19.1

46.0 46.0 46.0 46.0 46.0 46.0 95.0 95.0 95.0 95.0 95.0 95.0 95.0 95.0

2.8 2.9 2.8 2.8 2.9 2.8 2.9 2.6 2.8 2.6 2.6 2.6 2.9 2.8

2.4 2.4 2.4 2.4 2.4 2.4 2.3 2.3 2.3 1.92 1.92 1.92 1.92 1.92

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

55.6 61.3 91.0 55.6 61.3 91.0 67.1 67.2 110.1 121.6 112.9 208.3 8.9 56.6

1.9 1.8 1.6 1.9 1.8 1.6 1.5 1.1 1.5 0.8 0.8 0.6 0.9 0.6

12.3 12.0 9.0 12.3 12.0 9.0 9.7 7.3 10.1 5.0 5.0 3.8 6.1 4.2

a

Temperatures shown are for initial (clean) conditions.

exception of unit 6. The relatively large fouling rate in this unit arises because stream H1 represents vacuum residue, which gives rise to high wall temperatures and shell-side fouling. The asymptotic fouling rate parameters for each exchanger were calculated from eq 8 using a common decay constant τ of 180 days to give the same value of Rf after three years without cleaning as in the linear fouling scenario. The initial fouling rates in the nonlinear fouling scenario are therefore all larger than in the linear scenario so that more cleaning will be required. The difference in initial fouling rates is very sensitive to the value of the decay constant, and that used here is somewhat arbitrary: the main purpose of these scenarios in case study I is to demonstrate that the scheduling formulation can handle these nonlinear forms. Case study II features a mixture of fouling models based on plant data reconciliation. This case study considers the optimal cleaning schedule for the network in Figure 4 over a three-year period, starting from an initially clean state. Monthly intervals were used, with ∆tcl ) ∆tpr ) 15 days. All exchangers were considered in the optimization problem, giving 518 binary decision variables. Fifty different starting points were used, and the solution corresponding to the best local optimum generated is presented. Where a hot streamflow was split to pass through identical exchangers (e.g., H 1 through 1A and 1B), cleaning of either unit resulted in the whole hot streamflow being directed through the other unit. This construction could be modified to incorporate a maximum hot streamflow rate if hot stream pressure drop was a limiting factor. Constraints. The following constraints were implemented in addition to eq 32. (i) Furnace firing limit, based on CIT after three years’ operation without cleaning. Cleaning actions should not cause a furnace limitation and thus throughput reduction

(iii) Selection constraints were used instead of temperature targets for hot streams passing through several exchangers in series, e.g., (∀p)

y1A,p + y1B,p + y6,p g2

(36a)

y2A,p + y2B,p + y4,p g2

(37b)

y3A,p + y3B,p + y11,p g2

(37c)

(iv) The following selection constraints were used to maintain a feasible (or controllable) network (∀p):

ymA,p + ymB,p g 1 y1i,p + y2i,p + y3i,p g 2

m ) 1, 2, 3 i ) A, B

(38) (39)

y4,p + y5,p + y6,p g 2

(39)

y7,p + y8,p + y9,p + y10,p + y11,p g 2

(40)

(v) Cold stream pressure drop across the hot section of the network (units 1A-3A and 1B-3B)

∆PA(t) ) ∆P1,A(t) + ∆P2,A(t) + ∆P3,A(t) e 1.5∆PA(t)0) (41) with an analogous expression for the B units. Fouling was assumed to occur only on the crude (cold side) so that xf ) 1. The crude passes through the tubes (D ) 0.025 m) in all units except exchanger 11, where the hot stream flows through the tubes to facilitate cleaning. The foulant thermal conductivity was taken to be 0.2 W m-1 K-1, with all fouling assumed to be on the crude side. Pumparound constraints are considered in case study II. Results

(ii) Temperature targetssinspection of plant data for desalter operation established the following relationship:

Table 2 summarizes the results from a number of different cleaning scenarios using different values of Cc and constraint sets. A uniform value of Cc is used here, which could be modified to account for different exchanger types or sizes as required. Cleaning on a refinery usually involves isolation of the unit, removal of a tube bundle and treatment at a remote location, so

c,in c,out ) T7,p - 10 T6,p

(19) Wilson, D. I.; Condron, P. I.; Garrett, S. AIChE Symp. Ser. 314 1997, 93, 305-310.

c,out T1A,p

+

c,out T1B,p

L

g2CIT ≈ 400 °C

(∀p) (35)

(36)

1046


Smaı¨li et al.

Figure 5. CIT profiles for case study I, selected scenarios. Bold line, no cleaning, linear fouling (L1); bold, dotted line, no cleaning, asymptotic fouling (NL1); single line, zero cleaning cost, linear fouling (L2); dotted single line, cleaning cost ) £4000, ∆P constraint, linear fouling (L5). Table 2. Summary of Optimal Cleaning Schedule Results for Case Study I

case #

fouling type

L1 L2

linear “

constraints

Cc/£

L3

“

L4

“

∆P

0

L5

“

∆P

4000

L6

“

4000

L7

“

constrained U(t) constrained U(t), ∆P

NL1 NL2

nonlinear (asymptotic)

NL3

“

NL4

“

∆P

NL5 NL6

“ “

NL7

“

∆P constrained U(t) constrained U(t), ∆P

0 4000

4000 0 4000 0 4000 4000 4000

Obj (Obj RMINLP) /k£ 1320 627 (626) 755 (753) 628 (624) 760 (758) 757 (756) 755 (754) 2132 1418 (1411) 1657 (1656) 1414 (1413) 1654 1655 1658 (1657)

NC 42 25 36 28 23 27 75 44 82 48 51 50

Cc is less sensitive to the extent of fouling: the cost of cleaning in cleaning-in-place (CIP) systems is, however, likely to be related to the extent of fouling. Figure 5 shows the CIT-time profiles obtained for a selected set of solutions, while Figure 6 shows some of the cleaning schedules presented as charts. The CIT profiles in Figure 5 shows that fouling has a significant effect on the performance of the network. The larger initial fouling rates for the asymptotic fouling cases result in a rapid decay in CIT, resulting in a notably larger Obj value for the uncleaned (worst) cases 1320 k£ (L1, linear fouling) cf. 2132 k£ (NL1, asymptotic). Table 2 shows that more cleaning actions are performed in all the nonlinear fouling scenarios than the corresponding linear ones. The CIT profiles in Figure 5 for scenarios with cleaning show several features observed in all other cases, namely: (i) All profiles featured an absence of cleaning actions near times t ) 0 and tF due to the proximity of the temporal boundaries. There is little reward for cleaning

a relatively clean unit, while the objective function posed in eq 4 gives little time for the cost of cleaning a unit near tF to be recovered. The length of the no-cleaning zones were noticeably shorter in the nonlinear fouling scenarios, owing to the larger initial fouling rates. This is evident in comparing panels b and c of Figure 6, corresponding to identical scenario sets but different fouling models. (ii) The profiles all exhibit noticeable drops in CIT when a unit is taken off-line for cleaning, without violating the CIT limit (eq 35). Figure 5 also shows the effect of increasing Cc from 0 to 4000 £/operation. The former case is the energy maximization scenario (L2), where 42 cleaning actions result in an Obj value of 627 k£, yielding a saving of 693 k£. The resultant CIT profile is very busy, with cleaning occurring in most intervals. Increasing Cc requires a more selective schedule so that fewer actions are performed (L3-25) and the Obj value increases to 755 k£ (saving of 560 k£). The latter value reflects the optimized cost of a cleaning mitigation strategy, which can subsequently be compared with costs of capital modifications or chemical treatment programs. Increasing Cc would further decrease NC (at increased Obj), and could be used to determine which cleaning actions are most important. Schedules such as that for scenario L3 in Figure 6a show that the units subject to more severe fouling (1A,B; 2A,B) are cleaned more often and in a quasi-periodic manner. Those units with lower fouling rates (e.g., 7-11) are cleaned less often, near the middle of the time horizon. The order of cleaning is not, however, intuitive. Fouling rate is not, however, the only criterion determining the regularity of cleaning. Unit 4 is cleaned more often than units 3A/B in Figure 6a, despite similar fouling rates, owing to its importance in the network, while unit 10 is not cleaned at all. Scenarios L4,5 and NL4,5 correspond to L2,3 and NL2,3, respectively, with the addition of the constraint on pressure drop (eq 42). The increase in pressure drop due to fouling did not exceed 1.5 bar in these scenarios because thermal considerations and a relatively low Cc value proved to dominate cleaning decisions. The pressure drop constraint is likely to become active if Cc was increased and/or a maximum (small) value of NC was specified. Comparison of the results in Table 2 shows that the pressure-drop-limited cases usually resulted in a slightly larger value of Obj and more cleaning operations; the difference in Obj values reported, however, is not considered to be significant given the uncertainty in the data and simulation accuracy. The schedules obtained under each constraint set were noticeably different. The schedule for L5 (with pressure drop limitation) in Figure 6b shows a more regular pattern than that obtained for the free problem, Figure 6a. These differences illustrate the nature of the optimization problem; the optimal solution is not sharply defined, and a number of feasible solutions exist with Obj values near the lower bound. It is noteworthy that the Obj values obtained for feasible solutions are close to the corresponding RMINLP values. Furthermore, the RMINLP values in Table 2 are similar across the range of constraint scenarios for each Cc value. The schedules shown in Figure 6 constitute those with the lowest Obj values; a number of other schedules, with



Figure 6. Cleaning schedules for selected scenarios, case study I. Filled box denotes cleaning at the start of the period: (a) L3 - Cc ) £4000, linear fouling, no ∆P constraint; (b) L5, cleaning cost ) £4000, linear fouling, active ∆P constraint; (c) NL5, cleaning cost ) £4000, asymptotic fouling, active ∆P constraint.

similar Obj values but different distributions, were also obtained. Selection of the most suitable candidate from an operational standpoint from this group has not been considered here; our experience of discussing such results is that operational criteria arise which should have been incorporated into selection constraints. Some of these schedules represent degenerate solutions, which arise in this case because of design degeneracy (i.e., exchangers 1A/2A/3A and 1B/2B/3B are identical). This has been discussed previously8 and is not considered further here since the network in case study II features different fouling rates in some exchangers operating in parallel. The results for the asymptotic fouling scenarios in Table 2 exhibit the same trends as those above. The high

initial rates of fouling lead to larger NC values owing to the early loss of efficiency in the exchangers. Figure 6c shows the asymptotic fouling result corresponding to Figure 6b (NL5:L5), with 49 cleans cf. 28 cleans, respectively. It is noteworthy that the paired units of exchangers, e.g., 1A,B, experience scheduling similar to those in the linear fouling scenarios. Comparison of the results in Figure 6 indicates that the network performance is most sensitive to fouling in units 1-4 and 6. The scheduling methodology could be readily used to explore potential retrofit options, such as replacing units 4 and 6 with pairs of exchangers, by modifying the simulation. An alternative scheduling methodology would be to specify that cleaning is performed when the fouling

1048


Smaı¨li et al.

Table 3. Summary of Model Parameters for Case Study IIa unit

A/ m2

Uclean/kW m-2 K-1

fouling model/ m2 K kW-1day-1

Fh/ kg s-1

Ccp/kJ kg-1 K-1

Th,in/ °C

DTPA/ K

∆P(t)0)/ bar

1-3 4-6 7-9 10-12 13-15

200 400 130 800 150

0.16 0.15 0.33 0.17-0.20 0.33

13.9 38.3 8.0 63.0 10.4

2.6 2.8 2.9 2.9 3.0

250 355

20 32.2 -

0.3 0.9 0.9 1.7† 1.0

ym,p ) 1 ym,p ) 1 ym,p ) 1 ym,p ) 1

16-18 19-21 22-24 25-27

350 108 800 320

0.05-0.08 0.33 0.25 0.33

R˙ f ) 0 R˙ f ) 0.001 R˙ f ) 0 R˙ f ) 0 Rf,13∞ ) 5, τ ) 400 day Rf,14∞ ) 5, τ ) 400 day Rf,15∞ ) 2.8, τ ) 350 day R˙ f ) 0 R˙ f ) 0.010 R˙ f ) 0.004 R˙ f ) 0.020

13.9 4.0 46.9 13.9

2.6 3.0 3.0 2.6

310 327 335

1.6 22.5 -

1.0 0.8 2.0b 1.5b

ym,p ) 1

comment

c,in Fc ) 83.6 kg/s; Ccp ) 2.6 kJ kg-1 K-1; C1-3,p ) 122 °C; xf ) 1.0; Dt ) 0.025 m; λf ) 0.2 W m-1 K-1. b High-pressure drop due to two exchangers in series.

a

resistance in a unit reaches a fraction of its Tubular Exchanger Manufacturers’ Association (TEMA)20 or design value. This fixed approach ignores the changing thermal response of the network and has been shown in previous work8 to give poorer results than the MINLP model described here. An inspection of the fouling resistances in units just before cleaning indicated that the “dirty” Rf values varied from exchanger to exchanger and across the time horizon but were consistently smaller than the design values for each unit. Table 2 includes a set of scenarios which were repeated with a further constraint on Un,p, namely, Un,p > 0.3 kW/m2K (∀p), or fouling Biot number Bf ≡ UcleanRf < 0.65. The results obtained for the linear fouling scenarios (L3, 5-7) show little effect of this constraint; the simulation records indicated that this constraint was not active in most of the units. Similar trends were observed in the corresponding nonlinear scenarios (NL3, 5-7). 3.2. Case Study IIsRefinery Plant Study. This study illustrates the application of the scheduling model to an operating refinery unit and comparison of the results with those obtained using the simpler greedy algorithm. The CDU preheat train considered is part of a major UK oil refinery and processes approximately 1050 m3/h of a mixed crude slate. The train features 67 individual exchangers, but a preliminary audit of fouling behavior across the network, involving a study of plant operating data over the previous six years, indicated that the most serious fouling effects occurred in the hot section of the preheat train downstream of the flash vessel, corresponding to units 1-3 A/B in Figure 4. An operability audit also indicated that the performance of the cold section was not very sensitive to the temperatures of the hot streams leaving the hot section, so the scheduling simulation could be based on the 27 units shown in Figure 7. The Figure shows that the crude was split into three cold streams and exchanged heat with six hot streams, including four pumparounds, through nine sets of identical exchangers. The network features significant feedback via streams H1 and H4, which results in very nonlinear response to configuration changes imposed by cleaning actions. The Figure also shows that some bypasses isolated two exchangers at a time. Exchanger model parameters and stream properties are summarized in Table 3. The initial parameter values given in Table 3 correspond to those of the (20) Standards of the Tubular Exchanger Manufacturers’ Association; TEMA, New York; 1978.

Figure 7. Case study IIsschematic layout of crude preheat train downstream of flash units. Total duty when clean ≈ 106 MW. Solid lines denote cold (crude) flows; dotted lines, hot streams; dashed lines, pumparound hot streams. Note that the flow rate in H4* is smaller than that in H4. Hot stream splitting and bypasses are omitted for clarity. The hot streamflows are split evenly between the available exchangers. Cold stream bypasses indicate that units 16/19, 17/20, and 18/21 are isolated together.

network following a major shut-down where most units were cleaned. Data Reconciliation. The fouling model parameters in Table 3 were obtained by reconciliation of plant operating data over the six-year period between shutdowns prior to 1999. Data were filtered, and the overall heat transfer coefficient U was calculated from eq 1. Fouling resistances were obtained using eq 6 and the Rf values compared with three fouling models: (i) negligible fouling, (ii) linear fouling, and (iii) asymptotic fouling. Table 3 shows that models i and ii were most common, but asymptotic fouling was clearly evident in units 13-15. Figure 8 shows examples of each fouling model and illustrates the degree of scatter present in the data. There was insufficient information on bypass settings, etc., available to apply more rigorous reconciliation techniques such as those described by Takemoto et al.21 It is noteworthy that most of the fouling rates in Table 3 are lower than those used for similar units in case study I. Pumparounds are a dominant feature of this network and have significant effects on the impact of fouling. For instance, under steady operation, the temperature of the cold stream entering unit 25 is directly linked to the temperature at which it leaves units 16, with little damping from units 19 or 22. The DTPA values in Table 3 were obtained from long-term trending of plant data. (21) Takemoto, T.; Crittenden, B. D.; Kolaczkowski, S. T. Chem. Eng. Res. Des. 1999, 77, 769778.



Figure 8. Examples of preheat train heat transfer data plotted as overall thermal resistances for case study II. (+) Unit 13, asymptotic fouling behavior; solid line, fouling model for Unit 13 obtained by regression; (∆) unit 26, showing pseudolinear behavior, cleaning actions, and scatter.

The network simulation model was used to establish the sensitivity of the system to fouling in different units. The sensitivity parameter chosen was the cost of extra energy due to fouling over the first 12 months of operation, and the base case was the data set given in Table 3. This analysis indicated that the effect of fouling was most noticeable in exchangers 13-15 and 25-27; Figure 9 shows the effect of fouling model parameter variation in these units on annualized cost. The Figure also shows that sensitivities based on a six-month calculation period were not noticeably different. Optimization Model. The energy term in the objective function (eq 4) was based on furnace CIT, calculated from

CITp )

c,out T25,p

+

c,out T26,p

+

c,out T27,p

(42)

and parameters CM ) CX ) 0; Cc ) 5 k£/clean and CE equivalent to 2100 £ (K drop in CIT)-1 month-1. Time was initially discretized into 30-day monthly periods, with ∆tcl ) ∆tpr ) 15 days. Schedules are presented here for tF ) 24, which is a relatively short horizon given the likely interval between shut-downs, of o(60 months). This reduced length was due to the limitations of the NLP solver used in these calculations. Although the number of binary variables was not large, the number of continuous variables was large, and the complexity of the network caused the NLP solver difficulty in converging. A tailored NLP solver which exploits the sequential structure of the simulation problem is under development in conjunction with a stochastic solution method.22 An extended schedule, achieved by extending the monthly period length rather than the number of periods, is presented. The results presented here nevertheless demonstrate the potential for scheduling optimization, and the differences between the two approaches. Selection Constraints. The fouling rates in units 1-3, 4-6, 7-9, 10-12, and 16-18 were either negligible or too small to give significant effects, so these units were excluded from the scheduling calculation

yj,p ) 1

∀p, j ) 1-12, 16-18

(44)

Figure 9. Sensitivity analysis on fouling model parameters in key exchangers in case study II. Solid lines, annualized cost calculated on a 12 month period; dashed lines, annualized cost calculated on a 6 month period. (i) Black lines, units 25-27 (linear fouling); (ii) dark gray lines, unit 13 (asymptotic fouling, R∞f parameter); light gray lines, unit 13 (asymptotic fouling, τ parameter).

This gave 12 decision variables per period, or 288 binary variables for the 24 month horizon. Equation 32 prohibited cleaning in successive months. The following constraint permits only one unit to be cleaned from each pumparound stream pass:

yi,p + yi+1,p + yi+2,p g 2

∀p, i ) 4, 19, 22

(45)

A similar constraint is applied to units 13-15 and 2527, on hot streams H1 and H4, respectively. These hot streams pass through other units downstream, but those units are not subject to significant fouling, so the constraint is reduced to the form of eq 45. The bypass connections mean that units 16 and 19 on cold stream A are isolated together, viz.

y16,p ) y19,p

(46)

and similarly for the analogous units on streams B and C. This type of selection constraint can give rise to problems, as only one unit may actually be cleaned while the exchangers are isolated. This situation can be modeled by replacing y by two binary variables: ζ, denoting isolation and replacing y in the network performance equations, and ξ, denoting cleaning and replacing y in the heat transfer equations and the objective function. The two variables are related by eq 47, which only allows cleaning when the unit is isolated

ζm,p g ξm,p

(47)

The linked bypasses do not cause difficulties in this case study, as units 16-18 do not foul appreciably so that they can be described by a single decision variable, with Cc,16-18 ) 0. No more than two exchangers on each cold (22) Smaı¨li, F.; Vassiliadis, V. S.; Wilson, D. I. Comparison of MINLP with a Backtracking Threshold Acceptance Algorithm. Ind. Eng. Chem. Res., submitted for publication.

1050


Smaı¨li et al.

Table 4. Summary of Scheduling Results for Case Study IIs24-Month Perioda I: base case

II: 50% fouling (use of antifouling chemical additives) a

solution method

obj/k£

NC

saving/k£

CPUsb

no cleaning MINLP, Cc ) 0 MINLP, Cc ) 5 k£ greedy greedy (truncated) no cleaning

173 99.4 151 176 158 104

60 5 14 9 -

N/A 22 -3 15 69

708 s 1808 s 135 s o(500 s) 751 s

MINLP, Cc ) 5 k£

99

3

74

15.8 s

Cc ) 5 k£/clean. CPU s on a Sun Sparc 10 workstation b

stream may be cleaned in a given period, e.g., for stream A

y1,p + y4,p + y7,p + y10,p + y13,p + y16,p + y19,p + y22,p + y25,p g 7 (48) and similarly for streams B and C. The greedy algorithm was implemented using Cc ) 5000 £/clean and ∆ ) £1000 with a one-year horizon for the calculation of energy recovery. The uncertainty in the degree of fouling on the crude side, xf, and the absence of information to indicate the appropriate pressure drop-Rf relationship meant that the pressure drop constraints were not activated in case study II. Results. The results of the 24-month scheduling studies are summarized in Table 4. Two scenarios are considered, namely, (i) forecasting using the fouling models based on the historical data in Table 3 and (ii) forecasting using these fouling rates reduced by 50%. The fouling rates in the units subject to asymptotic fouling were reduced by halving the R∞f parameter. The latter scenario can be viewed as either a sensitivity analysis of the scheduling approach, demonstrating the effect of uncertainty in fouling model parameters on the cleaning schedules, or as a comparison of mitigation methods, whereby the application of chemical additives causes a decrease (or elimination) of fouling. Figure 9 demonstrates that network interactions strongly dampen the impact of changes in fouling model parameters on the objective function. These data were obtained from simulations of the network over 6 or 12 months, starting from the clean condition. The objective function (cost) is presented on an annualized basis to compare the effect of different rates over the two time periods. The Figure shows that changes in rate in a given unit do not result in proportional changes in cost, so any the effect of different rates on scheduling calculations will be similarly damped and difficult to establish without detailed simulation. The results from scenario ii are therefore reported from the perspective of chemical mitigation, e.g., as a potential alternative to cleaning. The base case CIT profile in Figure 10 features a noticeably smaller decay than that observed with case study I (Figure 5), owing to the smaller fouling rates in this case study and the buffering action of the pumparound exchangers. The duties specified for the latter units were comfortably below their maximum values, indicating that the network is not configured for maximum energy recovery. For the two-year period considered here, no pumparound units were scheduled for cleaning except in the energy minimization case (MINLP Cc ) 0).

Figure 10. Selected CIT profiles for case study II, 24-month runs. Dashed line, no-cleaning scenario; dotted line, MINLP schedule I, Figure 11a; solid line, greedy algorithm schedule, Figure 11b; crossed line, MINLP schedule III, Figure 11c. greedy algorithm value is averaged over cleaning and processing sub-periods. MINLP model values are averages for each sub-period and hence show impact of cleaning.

Table 4 shows that the cost of the no-cleaning base case was 173 k£ over 2 years, and the scope for cleaning, represented by the energy minimization case (Cc ) 0), was 73.6 k£. The latter schedule featured an (impracticably) high number of cleaning actions, at 60, including cleaning pumparound units in order to maintain pumparound duties while many other units were being taken off-line. The results obtained for Cc ) 5 k£ show more realistic values of NC and indicate that relatively small savings can be obtained by cleaning during the initial two-year period. Comparison of the MINLP and greedy algorithm shows that the former method performs better, but inspection of the corresponding schedules in Figure 11 and CIT profiles in Figure 10 demonstrate the fundamental differences in approach between the two methods. As mentioned in section 2.4, the MINLP model ignores network behavior after tF, so the cleaning actions are grouped toward the middle of the horizon, and CIT is 247.2 °C after 24 months. The greedy algorithm does not optimize cleaning decisions but does look forward so that CIT is noticeably larger at tF, at 249.5 °C. Furthermore, the schedule in Figure 11b features recognizable patterns in cleaning sets of units, albeit not at regular intervals. The breakdown of cleaning decisions for units 13-15 and 25-27 for the greedy algorithm are summarized in Table 5. No firm conclusions could be drawn concerning the different patterns in the schedules owing to the effect of the



Figure 11. Optimized schedules for case study II. Shaded periods denote cleaning at start of that period. (a) MINLP model, scenario (i); (b) greedy algorithm, scenario (i); (c) MINLP model, scenario (ii). Table 5. Record of Greedy Algorithm Gm Values and Decisions for Case Study IIa unit month

13

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

-5205 -5196 -3665 -2623 -1685 -925 -228 401 871 1448 3365 N/A -5121 -3914 N/A -2085

a

-229 22 629 N/A -3083

14

15

25

-5523 -5919 -4359 -5670 -4079 -2888 -1795 -765 166 -228 -2343 1042 370 -1925 1991 988 -1550 N/A 1574 -1215 -4079 N/A -867 -572 -1817 -4173 -288 -793 -3201 25 134 N/A 269 1097 -1618 541 N/A -909 582 34 N/A 266 -5648 -1784 N/A -5000 131 -3597 1097 N/A -2173 -4079

26

27

decision

-4364 -4380 -4093 -1810 -736 -5000 148 1087 1022 clean 26 N/A 1959 clean 25 -4078 2874 clean 27 -2858 N/A clean 14 -1796 clean 13 -734 204 -1798 1087 -764 clean 26 N/A 237 clean 25 -4078 1207 clean 27 -2865 N/A clean 15 -697 -2920 clean 14 177 clean 13 1094 -783 clean 26 N/A 212 clean 25 1000 clean 27 -2863 N/A

NC ) 14. Obj ) 176 k£.

relatively short horizon on the MINLP model. Regular groupings are evident in the MINLP schedule for case study I in Figure 6b and would therefore be expected in this case if an extended horizon were achieved. Table 4 also reports the results for a truncated version of the greedy algorithm schedule, where the last five cleaning actions in Figure 11b, scheduled for month 18 onward, were not implemented. This schedule features similar no-cleaning zones to Figure 11a, and yields an Obj value within 10% of the optimized value (158 vs 151 k£). The greedy algorithm is not expected to yield an optimal result, but this case study suggests that this

simple threshold technique represents a robust, readily programmable and reasonably effective alternative to the MINLP method. The CPU times indicate that the greedy algorithm is computationally expensive, partly because the model used here was developed for comparison without much attention given to improve solution speed. The noticeable differences observed in CPU times between the no-cleaning and Cc ) 5 k£ cases arise because the former were run first to generate the initial estimates for subsequent MINLP studies. The large CPU times for solving the no-cleaning casessessentially a simulationsillustrates the complexity of these networks. The antifouling chemical scenario results in Table 4 indicate a noticeable reduction in the scope for reducing costs by cleaning when the fouling rates are reduced. Figure 11c shows that the optimal cleaning schedule features only three cleaning actions involving the most heavily fouled units (25-27), grouped again toward the middle of the horizon. A more equitable comparison of the greedy algorithm and the MINLP approach was conducted by considering the performance of the unit over a 36-month horizon. The greedy algorithm results for 24 months were extended over a further 12 months, but in the latter periods, the time for recovering cleaning costs, Ns, was adjusted so that integral in eq 4 terminated at tF. No further cleaning actions were added, as can be seen in Figure 12b. An MINLP solution for the 36 month case was achieved using ∆tpr ) 30 days, i.e., a coarser scheduling framework. The corresponding schedule in Figure 12a shows a more regular series of cleaning actions, spread over the horizon. The cost of fouling over the three-year period without cleaning, i.e., the (worst) base case, was 304 k£, indicating that both calculation approaches yielded appreciable savings of (a) 66 k£ and (b) 59 k£. The superiority of the total horizon approach was principally due to its review of all possible combinations of cleaning actions. The nonconvexity of the MINLP problem renders it difficult to guarantee a global optimum, so the total horizon calculation generates a number of locally optimal solutions from a range of randomly selected starting points (here, 50). Figure 12a represents the best MINLP solution obtained, while Figure 13 indicates that a number of schedules with similar performance were obtained in the same calculation. Many of these outperform the greedy algorithm, but not substantially so. The difference, ca. 6 k£, must be compared with the degree of uncertainty in the model, particularly the uncertainties associated with the fouling model parameters and their propagation over long-term integration of costs in the objective function. Figure 13 is included to demonstrate some of the complexities in this type of scheduling calculation. The greedy algorithm may be less efficient, but it is more robust and can be used to solve extended problems owing to the sequential nature of the solution algorithm. Nevertheless, it is expected that the total horizon approach would yield better performance if the solver could have handled more periods of this complex network. The MINLP model is here used briefly to explore scenarios representing different mitigation strategies for the network operating over a 3.5-year horizon (∆tcl )

1052


Smaı¨li et al.

Figure 12. Optimal schedules for case study II, 36-month horizon. C’s denote cleaning during that period: (a) MINLP model; (b) greedy algorithm.

which this strategy became less attractive than cleaning alone. Note that all these values do not include any costs for cleaning at, or after, tF. These calculations illustrate the potential for an effective cleaning strategy tool. The above results also highlight how the calculations are very sensitive to the fouling model parameters. 4. Discussion

Figure 13. Distribution of solutions generated by the MINLP approach for case study II for a 36-month horizon. Solid line indicates greedy algorithm Obj value.

15 days; ∆tpr ) 35 days). The GAMS/DICOPT model did not perform satisfactorily for longer periods. The results are summarized in Table 6. The different cleaning cost scenarios (5 and 10 k£/clean) illustrate how the optimal schedule is very sensitive to the parameters in the objective function; as cleaning becomes more expensive, only critical cleaning actions are performed. The use of antifoulant chemicals is represented by reducing fouling rates. The maximum benefit from use of such chemicals is remediation of all fouling, at 350 k£, but this may not be practicable (or affordable). The table shows that the return on reducing the fouling rate by 50% in the most severely affected units (25-27) is 59 k£ over three years. This value does not include the cost of chemicals, and is less attractive than a cleaning-only policy if Cc were 5 k£/unit, where the saving ) 82 k£. The final scenario represents a combined cleaning/chemical strategy, the difference of 39 k£ setting the chemical cost at

The MINLP model for the network scheduling problem has been implemented in two case studies and has successfully generated a range of possible cleaning schedules using a commercial MINLP solver. The results demonstrate several expected features, e.g., the regular cleaning of exchangers with significant impact on overall network performance, and nonintuitive ones, such as the ordering of cleaning actions. In their related work on MINLP scheduling approaches, Georgiadis et al. stated that their MINLP model could not be used for the solution of problems with more than 10-12 exchangers over 24 operating periods (ref 9, p 453). Our experience with networks such as those described here is that larger problems are soluble by these routes, as long as the NLP problems are tractable. The limitations of the NLP solver used in this work prevented us from generating more satisfactory results for case study II and have prompted us to consider the development of more effective solver algorithms. Similar considerations have prompted us to consider the use of stochastic solution techniques based on combinatorial optimization; these have proved successful in a parallel study on scheduling in reverse osmosis networks.23 The differences between the MINLP and greedy algorithm schedules demonstrated the importance of the form of the objective function in these calculations. The MINLP function used here (eq 2) is very sensitive to the approach to tF, and a number of alternatives have been proposed to counter this sensitivity in order to (23) See, H. J.; Vassiliadis, V. S.; Wilson, D. I. Chemeca 2000; Institution of Engineers Australia: Perth, Australia, 2000; pp 199204.


Energy & Fuels, Vol. 15, No. 5, 2001 1053 Table 6. Summary of Scheduling Results for Case Study IIs3.5-Year Horizon

I: base case II: 50% fouling (use of antifouling chemical additives)

solution method

obj/k£

NC

saving/k£

no cleaning MINLP, Cc ) 5 k£ MINLP, Cc ) 10 k£ no cleaning

350 268 318 291

14 or 15 6 -

82 32 59

MINLP, Cc ) 5 k£

229

12

generate “acceptable” results for industrial applications. The ultimate aim must be to achieve convergence over long horizons such that “end effects” are marginalised. Ultimately one could consider operation beyond a shutdown by incorporating different selection constraints and objective function parameter values for the period(s) representing the shutdown, such as CE ) 0 and a reduced value of Cc reflecting the ease of cleaning a unit when the plant is not operating. The value of long-term scheduling, however, must be considered in the context of the uncertainty attached to the fouling models and likelihood of changes in plant operating policies over such extended time frames. Neither the MINLP model nor the greedy algorithm can guarantee global optimality for these systems. The MINLP model is more efficient when it can be converged, as it considers all the options available to it over the whole horizon. One aspect which was not encountered in case study II, owing to the short time horizon, was the ability of the greedy algorithm to handle operating constraints. In the formulation described here, the algorithm will proceed until a constraint becomes active, whereas the MINLP can “anticipate” such developments and determine the most profitable path around or over them. Furthermore, all efforts to achieve global optimality must consider the degree of uncertainty associated with the fouling model data. It is our opinion that the schedules generated by these methods are reasonably good, and further effort may be better channelled into determining the sensitivity of the results toward variations in the model parameters than seeking a global optimum. A feature of both case studies was the identification of “key” exchangers, namely, units whose loss in efficiency due to fouling had significant impact on the performance of the whole system. Reducing this sensitivity by modifying the network, namely, mitigation option iisretrofittingswill involve a techno-economic evaluation of alternative designs. The scheduling formulations presented here could be used to determine the optimal operating costs of the design. Alternately, these techniques could be incorporated as part of the operating cost calculation in a HEN design strategy. The latter represents a logical extension of the work by Fryer et al.3 on HEN network design incorporating fouling, where an experimentally based fouling model was used to calculate heat exchanger areas for different designs. They did not, however, consider duplicating or splitting exchangers and cleaning as a possible operating strategy. 5. Conclusions The important industrial problem of scheduling heat exchanger cleaning actions in large-scale continuously operating networks such as oil refinery crude preheat trains has been considered. An MINLP model for the

121

scheduling problem, based on a regular time discretisation heuristic, has been formulated. This approach is able to incorporate many of the important features of industrial networks and can be solved using a commercial solver. The formulation is not currently able to incorporate local control actions into the model. The problem is nonconvex so solutions were selected from local optima generated by starting from a number of starting points. The schedules generated exhibited several nonintuitive features and were obtained in reasonable CPU times when the problem did converge. Comparisons of the MINLP model with a simple greedy algorithm showed the former to converge faster, while the latter returned a sub-optimal result but proved to be robust. Nomenclature A, Aclean Bif CIT Cc CE Cf CM Cp CX c Dt, Ds DTPA F Gm MINLP m ˘ N, No Ns NC NE NP Obj RMINL ∆P Q R Re Rf R˙ f R∞f t t′ tF ∆t T ∆Tlm U, Uclean um X xf

heat transfer area and clean condition, m2 fouling Biot number coil inlet temperature, K cost of cleaning heat exchanger, £/clean energy cost due to fouling, £ kW-1 day-1 Fanning friction factor maintenance cost factor, £/day specific heat capacity, kJ kg-1 K-1 opportunity cost factor, £/day constant diameters of heat exchanger tube and shell, m cold stream temperature rise across a pumparound unit, K mass flow rate, kg/s return on cleaning unit m, greedy algorithm, £ mixed integer nonlinear programming mass flow rate, kg/s number of tubes per pass and unblocked number number of periods in greedy algorithm horizon number of cleaning actions number of exchangers number of periods objective function in optimization problem, £ prelaxed mixed integer nonlinear programming pressure drop across heat exchanger, bar heat exchanger duty, kW ratio of capacity flow rates Reynolds number fouling resistance, m2 K kW-1 fouling rate, m2 K kJ-1 fouling resistance, asymptotic value, m2 K kW-1 times or day time elapsed since last cleaning action, day time at end of cleaning horizon; horizon length, day length of period or sub-period, day temperature, K log mean temperature difference, K overall heat transfer coefficient and clean condition, kW m-2 K-1 mean flow velocity, m/s bound value fraction of fouling resistance due to tube-side fouling

1054


y z

Smaı¨li et al.

binary decision variable tube pitch, m

Greek R δf ∆ φ λf θ F τ

effectiveness term, eq 9 fouling layer thickness, m threshold, greedy algorithm decision parameter, £ temperature rise term, eq 14 thermal conductivity of fouling deposit, W m-1 K-1 continuous decision variable fluid density, kg/m3 decay time, asymptotic fouling model, day

Subscripts n p

Figure 14. Schematic of a fouled heat exchanger tube.

heat exchanger label period label

and the fouling rate in the first cleaning sub-period can be calculated from (A1), viz.

Superscripts h c in L out cl pr PA U

hot stream cold stream input to exchanger lower bound output from exchanger cleaning sub-period processing sub-period pumparound upper bound

cl R˙ fn,1

Asymptotic fouling, described by eq 8, requires that the time elapsed since the unit was cleaned, t′, be known:

Rf(t) ) R∞f (1 - exp(-t′/τ))

(8)

The elapsed time construction can be incorporated in the discretized time framework by considering rates. The fouling rate at time t′ is given by

R∞f dRf |t′ ) (exp(-t′/τ)) dt τ

(A1)

and at time t′ + ∆t

|

t′+∆t

)

fn

)

(21)

Appendix B: Pressure Drop Formulation

Appendix A: Approximation for Asymptotic Fouling Rate

dRf dt

(

R∞f Rfn(0) ) 1τ R∞

R∞f (exp(-t′/τ))(exp(-∆t/τ)) ) τ dRf exp(-∆t/τ) (A2) dt t′

|

In the discretized time framework shown in Figure 1, the asymptotic fouling rate in a cleaning sub-period, calculated at the start of the sub-period, is then given by cl pr ) R˙ fn,p-1 exp(-∆tpr R˙ fn,p p /τ)

(19)

where the terms in yn,p reset the fouling rate whenever the unit is cleaned. The asymptotic fouling rate in a processing sub-period is given by

R∞fn pr cl (1 - yn,p) ) R˙ fn,p yn,p exp(-∆tcl R˙ fn,p p /τ) + τ

(20)

The growth of a fouling film can result in the narrowing of the flow duct dimension, as shown in Figure 14, and may ultimately result in the blockage of the duct. The latter case is often caused by the presence of foreign objects or by regions of deposit “spalling off” and becoming stuck in narrowed ducts or covering flow manifolds. This treatment considers the relationship between fouling resistance and pressure drop in a typical shell and tube heat exchanger. Analogous relationships can readily be obtained for other exchanger configurations. B1. Duct Reduction. The following assumptions are made (i) Constant fluid properties. (ii) Constant foulant layer properties (e.g., negligible aging of deposit). (iii) Constant mass flow rate of fluid. (iv) Bare wall and deposit surface have similar roughness characteristics (e.g., smooth). (v) Pressure drop is dominated by losses along exchanger tubes, not in headers, etc. (vi) Fouling layers are evenly deposited across a tube surface. Consider the case where crude passes through the tubes and causes fouling. The fouling resistance defined by eq 6 includes the thermal resistance of the inner (i) and outer (o) fouling layers

Rf ) Rf,i + Rf,o

(B1)

If fouling occurs on both surfaces at constant rates so that the fraction of the overall resistance due to the inner layer remains constant and is given by xf, then

Rf,i ) Rfxf

(B2)

The thermal resistance of a fouling layer of thickness



δf inside a tube and uniform thermal conductivity λf is given by

Rf,i ) )

[

Dt Dt ln 2λf D - 2δf

eq 6 implicitly assumes constant surface area in calculating the overall transfer coefficient

]

Rf )

- Dt δf ln[1 - 2δf/Dt] ≈ 2λf λf

(B3)

We note that the linear approximation in B3 gives errors of 11% when δf/Dt ≈ 0.10; the uncertainty in λf is likely to be of this magnitude. For turbulent flow through a tube, the pressure drop is given by 2

4L Fum ∆Pt ) Cf Dt 2

1 1 U Uclean

(6)

When both U and A are affected by fouling, the lumped fouling resistance, based on Aclean, can be calculated thus

Rf 1 1 ) Aclean UA UcleanAclean

(B7)

where UA is estimated from exchanger performance data as ∼Q/∆Tlm. This can be rearranged to give

UcleanRf ≡ Bif )

UcleanAclean -1 UA

(B8)

2

1 m ˘ ∝ Cf Dt D 4

(B4)

t

where Cf is a weak function of Re (e.g., Cf ) 0.079Re-0.25). Combining B2-B4 gives

∆Pt(t) ∆Pt(clean)

)

[

Dt

] ( 4.75

(Dt - 2δf)

) 1 - 2Rf

)

xfλf Dt

-4.75

(B5)

Alternatively, one could ignore the dependency of Cf on Re and obtain the approximation that ∆P ∝ D-5, giving


)

[

Dt

]

(Dt - 2δf)

5

()

δf δf ≈ 1 + 10 + 40 Dt Dt

Let the area for heat transfer A be proportional to the number of unblocked tubes, N, where N ) No under clean conditions. Under the constant mass flow rate assumption, the flow rate through unblocked tubes will increase as fouling continues, and the overall heat transfer coefficient will therefore increase. Assuming that Uclean increases uniformly with deposit growth and is proportional to (tube velocity)0.4sthe actual power depending on the ratio of tube-side to shell side film resistancessthen

Bf )

2

Uclean Aclean -1) U A

(B6)

It is useful to compare the size of the effect predicted by this analysis. At δf/Dt ) 0.10, the ratio of pressure drops is 2.89 (equation B5), 2.0 (equation B6, linear terms), and 2.4 (equation B6, quadratic terms). These ratios are likely to be unacceptable in practice, so the thermal resistance approximation (B3) is therefore unlikely to be a significant source of error. Moreover, substituting typical values for petroleum deposits (λf ≈ 0.2 W/mK, tubeside fouling only - xf ) 1, Dt ≈ 0.025 m) gives the corresponding Rf value to be 0.0125 m2 K W-1. For a typical Uclean value of 300 W m-2 K-1, this gives a fouling Biot number (Bf ≡ UcleanRf) of ≈ 6.25, or U ) 41 W m-2 K-1, which is unlikely to be acceptable from a thermal efficiency standpoint. The relationship for shell side fouling, eq 23, can be derived in an analogous manner, using shell-side pressure drop relationships such as those described in Hewitt et al.24 B2. Tube Blockage. Fouling resistances are usually estimated from thermal performance data by assuming that the deposit is evenly distributed, as in section B1. An alternative relationship between Rf and pressure drop is obtained if the loss in heat transfer is dominated by tube blockage. Consider the case where fouling only causes tube blockage, i.e., no shell side fouling or tubeside fouling layer build-up. Tube blockage will result in the area for heat transfer decreasing from Aclean to A(t), whereas the definition of fouling resistance in (24) Hewitt, G. F.; Shires, G. L.; Bott, T. R. Process Heat Transfer; Begell House: New York, 1995.

[( ) ( )] N No

0.4

No N

-1)

() No N

0.6

- 1 (B9)

Since the pressure drop across a smooth tube ∝ (tube velocity)1.8-2, it follows that


≈ (1 + RfUclean)3-3.3

(B10)

which features a different sensitivity to the result for even deposit growth (B5). Substituting in the aforementioned values over the linear δf/Dt approximation range gives

Film growth ∆Pt(t) ∆Pt(clean)

≈ (1 - 16Rf)-5 ≈ 1 + 80Rf + 160Rf

2

(B11) Tube blockage ∆Pt(t) ∆Pt(clean)

≈ (1 + 330Rf)3 ≈ 1 + 990Rf + 2

326700Rf (B12) 0 e Rf < 0.0125 m2 K W-1 It can be seen that any contribution from tube blockage will have a significant effect on the observed pressure drop behavior. The pressure drop/fouling behavior in a real unit will lie between these two limiting cases, and

1056


will most likely be closer to the lower bound provided by the film growth model. Acknowledgment. The authors wish to acknowledge the support of Esso Petroleum (UK) and NalcoExxon Energy Chemicals, and funding from the EPSRC under Grant GR/L 30658. The assistance of Ms. Andre´e Cracknell and Messrs. Jim McGillviray, Dom Marnell and Ed Lapham (Esso) is gratefully acknowledged, as are discussions with David Dankworth and Himanshu

Smaı¨li et al.

Goshi from ExxonMobil and Dr. Andy Philpott at the University of Auckland. Permission from Esso Petroleum (UK) to publish the plant data in case study II is particularly appreciated. Different parts of the case study II material were presented at the Chemeca ‘2000 Conference in Perth, WA, and at the 2nd International Conference on Petroleum and Gas Phase Behavior and Fouling, Copenhagen, Denmark, 2000. EF010052P

Mitigation of Fouling in Refinery Heat Exchanger Networks by Optimal

Recommend Documents