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Dynamic Analysis of Fouling Build Up in Heat Exchangers Designed According to TEMA Standards Carolina Borges de Carvalho, Esdras P. Carvalho, and Mauro Ravagnani Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b05306 • Publication Date (Web): 26 Feb 2018 Downloaded from http://pubs.acs.org on February 26, 2018

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Industrial & Engineering Chemistry Research

Dynamic Analysis of Fouling Build Up in Heat Exchangers Designed According to TEMA Standards Carolina Borges de Carvalho, † Esdras P. Carvalho,‡ and Mauro A. S. S. Ravagnani∗,† †Department of Chemical Engineering, State University of Maringá, Maringá - Brazil ‡Department of Mathematics, State University of Maringá, Maringá - Brazil E-mail: [email protected] Abstract The prediction of important control properties is a challenging task for shell and tube heat exchangers design. Dynamic models, which include fouling effects, are still poorly investigated in the recent literature. In order to study the behavior of variable conditions over time and its influence on the dynamic aspect of the system, it has been proposed a design approach based on TEMA standards, aiming to analyze the process in a more realistic equipment. To study the behavior of time-varying conditions and its influence in the system dynamics, the model considered in this work is based on the idea of heat exchanger cells as the basic modeling element. This kind of approach has some advantages over the distributed model, such as continuous variables in time and discrete in space, leading to ordinary differential equations (ODE) and also providing the possibility of controlling the model complexity by adjusting the number of modeling cells. The ratio between output and input signals in each cell generates eight operator transmittances that describe the dynamic of the system. The

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influence of fouling in the dynamic behavior is evaluated by considering its resistance (Rf ) as a function of time. The model was implemented in MATLAB/Simulink and simulations have been carried out for different Rf values with a step change in three input variables. Open loop responses showed that as Rf increases, the quality of response is deteriorated and the system is affected mainly by the inertia and thermal exchange inefficiency.

Introduction Undesired deposits in heat exchangers (HE) surfaces affect nearly every chemical plant, introducing additional costs which are mainly related to the conservation of energy (burn extra fuel to overcome the effects of fouling), operation (production losses due to planned and unplanned cleaning interventions) and capital investment (oversized equipment). Besides increasing heat transfer resistance and hence leading to the reduction of the heat duty of an existing heat exchanger, fouling deposition also affects the restriction to flow, increasing fluid velocity and, consequently, increases pressure drop as well. Operational evidence of fouling resistance (R f ) values are provided since 1950s by a compilation published in Tubular Exchangers Manufacturers Association (TEMA)1 and those values are still the basis for the design of most heat exchangers worldwide.2 However, the uncritical use of the fixed TEMA fouling resistances leads to several problems that are mostly related to dynamic operation. Depending on the mechanism of fouling deposition, some strategies can be adopted to mitigate its effects on heat recovery, keeping it at acceptable levels. In recent years, several methods to monitor and mitigate fouling effects have been reported in literature. The general preference is to mitigate HE fouling by proper equipment design and then by on-line techniques.3 Regarding mechanical aspects, the formation of deposits on finned tubes in CaSO4 solutions was investigated,4 showing that increasing fin density, fouling resistance was 2


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significantly decreased. Computational Fluid Dynamics (CFD) simulation was applied to develop a fouling model for fluid flow in a bare tube and in a tube with inserts, allowing to predict fouling threshold conditions for both cases.5 On the basis of cleaning interventions, energy scheduling and periodic cleaning problem of complex heat exchanger networks (HEN) under fouling were optimized simultaneously to minimize utility requirements.6 Also, aspects regarding aging7 and models for chemical and mechanical cleaning interventions8 are frequently investigated. Another approach is to consider fouling mitigation during the equipment operation by optimizing flow rate itself9 or combining it with optimal cleaning schedules.10 With a different approach, HE optimization is proposed jointing both equipment design and cleaning policy optimization.11 The method sets a threshold level for the maximum tolerated fouling resistance (R f ), implying a trade-off decision between the choice of low R f and maintenance stops. Some important features, like pressure drop and fouling effects, were considered in heat exchangers network synthesis with detailed equipment design.12 Pinch analysis was used to obtain the HEN with maximum energy recovery combined with Bell-Delaware method for shell side was applied for the heat exchanger design. Pressure drop and fouling effects are frequently neglected during HEN design, and this methodology provides more realistic values from the industrial point of view. Detailed equipment design, following rigorously the TEMA standards, was later proposed13 and then extended for a HEN design14 considering shell and tube fouling resistances using a mixed integer non-linear programming (MINLP). In a different approach, the HE design was formulated as an optimization problem solved with particle swarm optimization (PSO), also considering pressure drops and fouling limits, rigorously following TEMA standards and Bell-Delaware method for shell side calculations.15 Although metaheuristic methods, like PSO, tend to produce near global optimal solutions, depending on the complexity of the problem, computer efforts can be

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significantly higher than a MINLP approach. However, fouling is a time dependent phenomenon and transient states of heat exchangers may also have detrimental effects on the overall process.16 All the literature cited above were applied to HE on stationary behavior, focusing on steady-state heat recovery. Only few papers focus on dynamic models for heat exchangers which incorporate fouling resistance as a time dependent variable.17 As a result, oversized equipment may be designed causing huge economic losses due to several factors, e.g., higher operating costs, different mitigation measures, more frequently cleaning interventions and increased demand for utilities to meet the process requirements. Moreover, dynamic models are essential to assess controllability and stability of the system. As industrial plants are frequently affected by internal and external disturbances, timevarying models are required to analyze and develop control strategies in order to minimize disturbances effects. The first significant approach regarding Rf dynamic behavior was developed almost six decades ago.18 The authors proposed the modeling of fouling process based on the difference between deposition (φD) and removal rates (φR) of deposits on the thermal exchange surface: dRf dt

= φD − φR.

(1)

This model became the basis for future researches, since fouling build-up depends on several factors for deposition or removal, as operational conditions, chemical and biological reactions, material adherence of the deposits and the specific nature of the streams, which are complex to consider in this first model. Three main approaches for fouling rate models can be enumerated: 19 deterministic, semiempirical and artificial neural network. In deterministic models, the fouling rate is based on phenomenological analysis and have become one of the most complex to study, losing importance in recent researches. 4


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Semiempirical models are based on operational variables, requiring reliable process data and parameter estimation procedures to determine model parameters. The last approach includes statistical learning algorithms which create relationships between different variables related to fouling deposition and removal. 20 A procedure for the design of HE where fouling occurs in the tube side was recently proposed based on a semiempirical approach.20 The authors presented a design procedure by calculating the fouling resistance from the thermofluidynamic conditions of the design itself, incorporating fouling modeling into HE design. In regard to artificial neural networks, some tools are used to detect and estimate the fouling resistance, such as Kalman filters,21 statistical tests on estimated values of fouling factors obtained by the neural network22 and also a comparison of neural networks and Kalman filters23 and other fuzzy approaches24 performances can be found in recent literature. Based on those considerations, the authors noticed a gap in the field of process dynamic related to fouling build up effects in the design and operation of heat exchangers. Therefore, this work aims to assess the influence of fouling build up in the dynamic behavior of heat exchangers designed following TEMA standards, evaluating its impact on the process.

Shell and Tube Heat Exchanger Design Shell and tube heat exchangers design is a very known subject in the literature and some classical methods, as Kern25 and Bell-Delaware, are used in practice. The present work intends to evaluate the influence of fouling deposition in heat exchangers designed according to TEMA standards, introducing the study on HE dynamic behavior in a more realistic problem. For this purpose, the model implemented is based on a MINLP model13 using a generalized disjunctive programming (GDP) problem to formulate the model and a mixed integer nonlinear programming (MINLP) reformulation

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to determine the optimal heat exchanger design that minimizes area costs or the total annualized costs, depending on data availability. The objective function is also constrained to allowable pressure drops and fouling factor limits. For the formulation, inlet data for both fluids are: Tsi and Tso (inlet and outlet temperatures for shell-side), Tti and Tto (inlet and outlet temperatures for tube-side), Ms and Mt (mass flowrate for shell and tubes), ρs and ρt (density for shell and tube sides), Cps and Cpt (heat capacity for shell and tube sides), µs and µt (viscosity for shell and tube sides), k (thermal conductivity), ∆Ps and ∆Pt (allowable pressure drops for shell and tube sides) and R f s and R f t (fouling factors for shell and tube sides). The mechanical variables to be optimized regarding the tube-side are tube inside diameter (D1), tube outside diameter (D2), tube arrangement (arr, triangular or square), tube pitch (pt), tube length (L), number of tube passes (Ntp ) and number of tubes (nb) per exchanger pass. To the shell-side, the desired variables are the external diameter (Ds), the tube bundle diameter (D3), baffles number (Nb f ) and spacing (b fs). Finally, thermalhydraulic variables to be calculated are heat duty (Q), heat exchange area (A), tube-side and shell-side film coefficients (ht and hs), dirty and clean overall heat transfer coefficient (Ud and Uc), log mean temperature difference (LMTD) and the correction factor of LMTD (Ft).13 Some parameters must be previously defined by engineer: hot fluid allocation (shell or tubes), construction materials and fouling resistance limits. The physical properties (ρ, µ, C p , and k) are assumed to be temperature invariant along HE length for the design. Detailed information about the formulation and optimization for the mechanical variables is available on the Supporting Information provided free of charge with this material.

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Dynamic Modeling Processes under time varying conditions, such as fouling resistance in HE, require appropriate models to evaluate system dynamics. Distributed and lumped parameters models are the most investigated approaches, the range of application of each one is different and both approaches have their strong sides and associated problems. Distributed models use infinitely small differential units as the basic modeling elements, resulting in a set of partial differential equations (PDE) with regard to both space and time. This type of equations may require advanced computational efforts, especially for multi-pass HE or different flow configurations.26 Otherwise, lumped parameters models introduce the concept of modeling cell, defined as a perfectly stirred tank, exchanging heat only with each other through a dividing wall.16 A double-pass heat exchanger divided into cells is illustrated in Figure 1.

Figure 1: Scheme of the division and cells interdependency in a two-pass heat exchanger In Figure 1, a two-pass heat exchanger (1 − 2) is divided into n modeling cells for each tube pass. The red continuous line represents shell-side with a hot fluid. The dashed blue 7



line depicts tube-side with cold fluid. In the cell (1, 2), a detailed information about the interdependency between input and output variables for this arrangement is provided. Therefore, the dynamic model based on heat exchanger cells leads to a set of ordinary differential equations (ODE) for each cell with regard to time only. Other qualitative differences for both distributed and lumped parameters models can be listed. It is assumed a perfect mixing in each modeling cell for lumped models as a simplifying option for multi-pass heat exchangers, since most distributed models consider only single-pass heat exchangers due to the complexity of the resulting equations, leading to differential with regard to both space and time.26 Moreover, the solution methods for lumped models are simpler, approximating the time derivatives by numerical integration. The equivalent problem in a distributed model would require a more advanced technique for the solution. Lumped models are preferred to empirical models to get a rational transfer function and to be able to simulate different flow configurations. 27 In addition, distributed model behavior can be achieved by using an adequate number of cells. Based on those topics, a lumped model was chosen to describe the dynamic processes studied by this work. This approach was investigated by several authors28 and simulation results were satisfactory to predict heat exchanger dynamic behavior 29 allied to computational simplicity.26 Although a lumped parameter model described by cells results in a considerable number of equations, they are simple and a satisfactory trade-off between model accuracy and complex process can be established. In this paper, three examples of two-pass shell and tube heat exchangers are considered. Based on similar models, the following simplifications are usually assumed for fully developed turbulent flows when convective heat transfer is greater than heat conduction between cells:16 • Assumption of perfect mixing in all cells. Therefore, fluid and tube-wall thermophys8


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ical parameters are constant regarding to space inside each cell. • Wall temperature is considered uniform within the cell volume, that is, wall resistance to heat transfer is negligible in comparison with the convective resistance. • Wall heat capacity is taken into account due to its influence in temperature distribution. • Heat losses to the environment or heat conduction between cells are negligibly small.

Lumped-Parameter Energy Balances For each modeling cell (i, j) defined before, there will be three equations derived from the energy balances of a control volume for the cold fluid, hot fluid and wall. The minimum number of cells (NC) that accurately describe the system should be one above the number of baffles (Nb f ) times the number of tube passes (N tp ),27 or: NC = (1 + Nb f )N tp.

(2)

Thereby, the number of model equations depends directly on the cell number and vary for each heat exchanger. Due to the complexity of the entire arrangement, a splitting scheme into cells is detailed in Figure 2 for cells (1, 2) and (1, 3). From thermodynamics, assuming kinetic and potential energies negligibly small and no work generation, the general energy balance for each cell is: dH = − ∆H ± Q˙ C , dt

(3)

where H represents the enthalpy and Q˙ C represents the rate of heat transfered and is positive for energy entering the cell and negative for energy exiting the cell. As a result, Eq. (3) can be applied for tube-side fluid, tube walls and shell-side fluid as follows: 9



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Figure 2: Detailed scheme for cells (1, 2) and (1, 3) with flows and temperatures inlets and outlets

dTto

=

dt dTw = dt dTso = dt

Mt .(T − T ) + h f t.nb.π.D1.l .(T − T ) to tw to

(4)

ti

ρt.Vt.Cpt h .n .π.D2.l .(Tto − Ttw) + f s b .(Tso − Tsw) ρw.Vw.Cpw ρw.Vw.Cpw Ms .(T − T ) + h f s.nb.π.D2.l .(T − T ) so sw so

ρt.Vt. h f t.nb.π.D1.l

ρs.Vs

si

(5) (6)

ρs.Vs.Cps

Adopting geometric constants a1 - a6, Eqs. (4) - (6) can be transformed to the following form:

dTto dt dTw dt dTso dt

= a1 Mt(Tti − Tto ) + a2(Ttw − Tto )

(7)

= a3(Tto − Tw) + a4(Tso − Tw)

(8)

= a5 Ms(Tsi − Tso ) + a6(Tsw − Tso ),

(9)

10


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and the constants a1 - a6 are defined as:

1 a1 = ρ .V t t

(10)

a2 =

nb .π.D1 .l.h f t ρt.Cpt.Vt

(11)

a3 =

nb .π.D1 .l.h f t ρw.Cpw.Vw

(12)

a4 =

nb .π.D2 .l.h f s ρw.Cpw.Vw

(13)

1 a5 = ρ .V s s a6 =

(14)

nb .π.D2 .l.h f s ρs.Cps.Vs .

(15)

In a two-pass heat exchanger, each cell assumes the shape of a half-cylinder. Hence shell, tube and tube-wall material volumes can be calculated as follows, respectively:

/ Vs = Vt = V w

=

π.D2 π.D22.nb 3 8 − 4

π.D12.l.nb 4 π.(D2 − D2).l.nb 2

4

.l

(16) (17)

.

(18)

1

As fluid temperature changes along HE length, all thermophysical parameters, such as density, heat capacity, viscosity and thermal conductivity, assume different values in each cell. In order to predict the temperature distribution along the heat exchanger and, consequently, the thermophysical parameters distribution as well, the HE was simulated in Aspen Exchanger Design & Rating V8.0 (EDR) from Aspen PLUS. Although Aspen EDR does not provide the detailed calculation, it has a reliable database that accurately calculates the local temperature distribution.30 11



In addition, since this work only considers heat exchangers constrained to TEMA standards, Aspen EDR designs, checks and simulates all TEMA types. 31 Therefore, for each heat exchanger designed, Aspen EDR was used to predict the temperature distribution along the equipment length, assembling a set of variables for each HE cell (i, j). At this point, the thermal resistance of fouling was considered constant, corresponding to the maximum value that the equipment is capable to attend the heat duty. For each heat exchanger simulated, the database for cell (i, j) is composed by the following parameters: • Temperature: Tt(i, j) and Ts(i). • Heat capacity: Cpt (i, j), Cps (i, j) and Cpw(i, j). • Density: ρs(i, j), ρt(i, j) and ρw(i, j). • Mass flow: Mt and Ms. • Film coefficient: ht(i, j) and ht(i, j). Aiming to distinguish the steady-state properties obtained by Aspen EDR, the subscript e was used. The dynamic analysis was made regarding some defined variables. Flow rate and inlet temperatures for shell and tube (Ms, Mt, Tsi (i, j), Tti (i, j)) are considered as disturbances or process variables (inputs), and the outlet temperatures for both shell and tube (Tso (i, j), Tto (i, j)) are the desired variables or outputs. A relationship between output and input variables is desirable to describe a dynamic system and is usually provided by transfer functions. Since Eqs. (7) and (9) produce a nonlinearity by the product between an input and output variables, it is necessary to linearize them by Taylor series expansions in the neighborhood of steady-state point value. After that, it is applied the Laplace transform in the three energy balance equations resulting in 12


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an algebraic linear system in the frequency domain. This procedure is similar to the one adopted by previous authors,16 although resulting in different transfer functions. Expressed in terms of deviation variables (denoted with a bar), the system solution provides eight operator transmittances, G(s), for each cell (i, j), as follows:

a1 a3 a6(Ttie − Ttoe) T¯ so G1(s) = ¯ = Mt (s + a1 Mte + a2)[(s + a3 + a4)(s + a5 Mse + a6) − a4 a6] − a2 a3(s + a5 Mse + a6) (19)

G2(s) =

G (s ) = 3

G (s ) = 4

G5(s) =

G6(s) =

a5(Tsoe − Tsie) T¯ so = ¯s M s + a5 Mse + a6 − a4 a6(s + a1 Mte + a2)[(s + a3 + a4)(s + a1 Mte + a2) − a2a3]− 1 (20)

T¯ so

(s + a5 Mse + a6) − a4 a6(s + a1 Mte + a2)[(s + a3 + a4)(s + a1 Mte + a2) − a2 a3]−1 (21)

T¯ si

T¯ so T¯ ti

a5 Mse

=

a1 a3 a6 Mte

=

(s + a1 Mte + a2)[(s + a3 + a4)(s + a5 Mse + a6) − a4 a6] − a2 a3(s + a5 Mse + a6) (22)

a1(Ttie − Ttoe) T¯ to = ¯t M s + a1 Mte + a2 − a2 a3(s + a5 Mse + a6)[(s + a3 + a4)(s + a5 Mse + a6) − a4a6]− 1 (23)

a2 a4 a5(Tsie − Tsoe) T¯ to = ¯s M (s + a1 Mte + a2)[(s + a3 + a4)(s + a5 Mse + a6) − a4 a6] − a2 a3(s + a5 Mse + a6) (24) 13



G ( s) = 7

T¯ to T¯ ti

a2 a4 a5 Mse

=

(s + a1 Mte + a2)[(s + a3 + a4)(s + a5 Mse + a6) − a4 a6] − a2 a3(s + a5 Mse + a6) (25)

T¯ si

G (s ) = 8

T¯ to

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a1 Mte

=

s + a1 Mte + a2 − a2 a3(s + a5 Mse + a6)[(s + a3 + a4)(s + a5 Mse + a6) − a4 a6]−1 (26)

Defining the subscript f for film coefficients (h) under fouling conditions, the following relationships are used to calculate their values after certain periods of operation:

hfs =

h

ft

=

hs hs.Rf s + 1 ht ht.Rf t + 1

.

(27)

(28)

For the purposes of this work, it is considered five periods of operation, varying R f since the clean condition until the maximum fouled condition used to design the equipment for each fluid, simulating the dynamic behavior in different stages of uninterrupted operation.

Block Diagram Model In industrial applications, processes are usually subjected to sudden and sustained input changes that can be approximated by a step disturbance of magnitude N in a process variable at the time t = 0: us ( t) =

0,

t

Dynamic Analysis of Fouling Buildup in Heat ... - ACS Publications

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