Dynamic optimization of multiple-zone air impingement drying processes

Computers and Chemical Engineering 30 (2006) 467–489

Dynamic optimization of multiple-zone air impingement drying processes Mariana Barttfeld, Norbert Alleborn, Franz Durst ∗ Lehrstuhl f¨ur Str¨omungsmechanik, Universit¨at Erlangen-N¨urnberg – Cauerstr. 4, D-91058 Erlangen, Germany Received 10 November 2004; received in revised form 29 September 2005; accepted 4 October 2005

Abstract This article introduces a mathematical programming model for the rigorous optimization of multiple-zone air impingement dryers for drying thin liquid films on continuous substrates. The formulation is based on a modular representation of the drying process and includes a non-linear Partial Differential Equation (PDE) governing the mass transport in the film. A simultaneous approach is selected to solve the PDE dynamic optimization problem. By the full discretization of all variables, a Non-Linear Programming (NLP) model is obtained and the system of differential equations is then solved simultaneously with the optimization, resulting in a large-scale optimization problem. This approach allows an easy implementation of the process operation limits as inequality constraints, e.g. constraints to avoid bubble formation in the film, risk of solvent vapor explosion in the exhaust air. Various drying scenarios are investigated by considering different objective functions in the formulation, e.g. minimization of heat consumption, maximization of the production rate. Numerical computations illustrate the proposed optimization method for different drying scenarios for which optimal operation conditions with respect to the objective function and constraints are determined. Trends for the optimal operation of air impingement drying processes are extracted from the optimization results obtained for the drying scenarios. © 2005 Elsevier Ltd. All rights reserved. Keywords: State variables; Decision variables; Optimal solution; Drying; Impingement drying

1. Introduction The drying process is the most important step following the application of a uniform thin liquid film onto a moving substrate in the production of many coated products, such as coated paper, photographic films or medical products. In a thermal drying process heat is supplied to the liquid film to evaporate the solvent and to immobilize the coating layer. The drying section in an industrial coating plant usually consists of a number of drying modules (up to 12–15), making up by far the largest part of the plant; see Fig. 1(a). Among the broad range of drying concepts, air impingement dryers are commonly used in many industrial applications due to the high heat and mass transfer rates they provide. A massive amount of energy is required to heat the drying air, so that the drying process dominates not only the capital costs but also the operation costs. Finding the best choice for dryer design and operation is therefore essential for a profitable production.

∗

Corresponding author. E-mail address: [email protected] (F. Durst).

0098-1354/$ – see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2005.10.016

A typical task in film drying, for instance, is to choose the operation conditions (control variables) of each module, such as drying air temperature and velocity, in a way to minimize at a certain production rate the energy consumption of the plant (objective function) while reaching the desired residual solvent content and preserving the function and quality of the film, i.e. avoiding defects, such as bubble formation (process constraints). An empirical determination of the optimal operation profile of a modular drying section, as still practiced in some industrial applications, involves high operation and material costs and becomes prohibitively sophisticated when a large number of modules and operation parameters are present. Consequently, there is a clear need of developing rigorous optimization techniques to systematically determine the optimal operation conditions for a modular drying plant. Numerical models have been developed in the past to describe the drying process of mixtures of volatile solvents and a non-volatile solute, see, e.g. Alsoy (1999), Aust, Durst, and Raszillier (1997), Cairncross, Francis, and Scriven (1996), Guerrier, Bouchard, Allain, and Benard (1998), Price, Wang, and Romdhane (1997), Saure (1995) and Wagner, Schabel, and Schl¨under (1999). The core of these models is constituted by the solvent mass transport in the liquid film, described by a diffusion

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Nomenclature a1 b d D Cp1 Cp2 Cpweb h hfilm l M1 Mw Mair m1 m2 m ˙ mv mweb n Nk Nt Nz P p1 Q qb q qt R s T t V2∗ X x1 xw wweb y1 z Nu Le

solvent activity coefficient slot width (m) nozzle-coating distance (m) diffusion coefficient (m2 /s) solvent heat capacity (kJ/kg K) polymer heat capacity (kJ/kg K) web heat capacity (kJ/kg K) enthalpy (kJ/s) thickness (m) module length (m) solvent molar mass (kg/kmol) water molar mass (kg/kmol) air molar mass (kg/kmol) solvent specific mass (kg/m2 ) polymer specific mass (kg/m2 ) mass flow (kg/s) solvent evaporation rate (kg/m2 s) substrate specific mass (kg/m2 ) number number of drying modules number of time discretization points number of space discretization points total pressure of the dryer (Pa) partial pressure of solvent (Pa) total heat exchanged (kW) convective heat input at the film bottom (kW/m2 ) heat exchanged (kW) convective heat input at the film surface (kW/m2 ) ideal gas constant (J/kmol K) distance between nozzles (m) temperature (K) time (s) polymer specific volume (m3 /kg) solvent mass fraction (dry basis) (kg/kg) solvent mass fraction in air (dry basis) (kg/kg) water mass fraction in air (dry basis) (kg/kg) web width (m) solvent mole fraction (kmol/kmol) space (m) Nusselt number Lewis number

Subindices 0 initial conditions i space discretization point j time discretization point k drying modules L lower bound U upper bound Superindices bot bottom of the film ex exhausting air film film

final in ins L ma nozzle o out slot specif su surface top V vap

final time inlet infiltrating liquid make-up (fresh) air nozzle reference outlet slot specified suction air film surface top of the film vapor vapor/vaporization

Greek symbols α heat transfer coefficient (kW/m2 K) β mass transfer coefficient (m/s)  variation (kJ/kg) ε density ratio λ thermal conductivity (W/m K) ρ density (kg/m3 ) ρ1 solvent density (kg/m3 ) polymer density (kg/m3 ) ρ2 δ solvent air diffusion coefficient (m2 /s) υ velocity (m/s)

equation. Flow, heat and mass transfer in the dryer is characterized by (averaged) heat and mass transfer coefficients that enter the boundary conditions, giving flexibility towards a change of the dryer type and allowing, e.g. to compare new dryer concepts with conventional ones, cf. Alleborn (2001a, 2001b). In Aust (1996) and Aust et al. (1997) a detailed model for the numerical simulation of multiple-zone air impingement dryers has been developed, that takes into account also process engineering issues like air handling, energy consumption and explosive levels in the drying air, in addition to the transport processes in the wet film. Systematic parameter studies, as performed with these models to obtain insight into the drying mechanisms, basically compare a finite number of drying strategies, i.e. particular choices of operation parameters, from which an optimum is chosen with respect to suitable objective functions and constraints. However, it cannot be decided generally whether operation conditions obtained in this way are optimal on the continuous set of control variables. The rigorous optimization for liquid film drying in multiplezone dryers has not yet received considerable attention in the literature. Recently, Price and Cairncross (2000) have addressed the optimization of drying of coatings in a single-zone convection dryer using mathematical programming. The optimization criterion considered by these authors was the minimization of the residual solvent mass in a polymer film with a specified

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Fig. 1. Industrial drying process. (a) Multiple-zone air impingement dryer (Benzi, 1991) and (b) modular representation.

web speed in order to produce a film free of bubbles. Promising optimization results for a modular dryer consisting of six air impingement zones were presented by Alleborn (2001a) using a sequential approach, integrating the differential equation system to compute the objective function and carrying out the optimization using non-gradient methods. A major challenge in solving the optimization problem of drying processes by mathematical optimization algorithms relies on the fact that the formulations involve Partial Differential Equations (PDE) to describe the mass transport in the film, giving rise to a large-scale PDE constrained optimization problem with a strong non-linear and non-convex nature. This article presents a method for the rigorous optimization of drying processes for thin liquid films in multiple-zone air impingement dryers. In contrast with the optimization methods proposed in the literature (Price & Cairncross, 2000; Alleborn, 2001a), in this work a simultaneous approach (full discretization method) is considered to solve the mathematical formulation (see Biegler, Ghattas, Heinkenschloss, & van Bloemen Waanders, 2003; Biegler & Grossmann, 2004). Control and state variables are discretized and the problem is converted into a Non-Linear Programming (NLP) problem by the full discretization of decision and state variables. In this approach, the solution of the differential equations system is then directly coupled with the optimization problem. This approach appears to be suitable and very flexible to optimize drying processes. Many different drying scenarios can be explored simply by exchanging the objective functions in the algorithmic formulation. Furthermore, inequality (path) constraints which model the process operation limits (e.g. bounds on the drying air velocity and temperature or risk of bubble formation in the film) can be easily implemented in the formulation. However, the major advantage of the simultaneous approach is the fact, that the system of governing equations is solved just once, at the optimal point, therefore avoiding repetitive computations of intermediate solutions that may require excessive computational effort, as in the sequential approach (Biegler & Grossmann, 2004). This paper is organized as follows: in the next section, the process engineering framework of the optimization problem is briefly sketched. In Section 3, four drying scenarios for the optimization are defined and the respective optimization problem is stated in Section 4. In Section 5, the fully discretized NLP opti-

mization problem is formulated in detail. In Section 6, numerical computations are presented to illustrate the scenarios and to discuss details of the optimization method.

2. Drying process representation A sketch of a coating plant with a multiple-zone air impingement drying section is shown in Fig. 1. The substrate (web) moves at a certain speed υweb , which determines the production rate of the coated product. The coating unit applies a thin liquid film of uniform thickness onto the substrate, which afterwards moves then through a drying section with several modules. A module, as sketched in Fig. 1(b), is defined as an enclosed region of the drying section where uniform operation conditions, such as drying temperature and drying air velocity, are set. Different modules, labeled k = 1, . . ., Nk (with Nk denoting the total number of modules), may operate at different conditions. The present paper focuses on the investigation of modular dryers consisting of air impingement modules with arrays of slot nozzles in the configuration sketched in Fig. 2 for a single module. The specification of the nozzle geometry and the air handling system of the impingement dryer module shown in Fig. 2 follows Aust et al. (1997). Each drying module k is fed by fresh air (makema up air in Fig. 2, with mass flux m ˙ ma k , moisture content xk and ma temperature Tk ) which along with the recycled air (superscript re in Fig. 2) is heated (heat input q˙ k ) to achieve the required drying air temperature Tknozzle at the nozzle exit. The drying air leaves the nozzles with nozzle exit velocity υknozzle and impinges onto the top and bottom surface of the coated web. The shifted arrangement of the slot nozzle arrays on the top and bottom part of the module, as indicated in Fig. 2, ensures a stable, slightly sinusoidal motion of the web. When the module is operated at slightly sub-ambient pressure, as to avoid leakage of organic solvent vapor into the environment, the drying air mixes with infiltrating ambient air (superscript ins). The solvent rich drying air is collected in the exhaust hoods to be removed from the process (suction air, superscript su). A fraction of the suction airflow can be fed back into the make-up airflow to exploit its heat content and reduce the energy consumption. The maximum amount of exhaust air that may be recycled will depend on process constraints, such as, e.g. lower explosive limits and

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Fig. 2. Single drying module representation.

the required residual solvent content in the film and will therefore be a result of the optimization. The representation of the air handling system in a dryer module is given by (global) mass and enthalpy balances for the material streams in control volume I and II, respectively, as sketched in Fig. 2, resulting in a set of algebraic equations. The representation of the heat and mass transport in the thin liquid film and substrate is considered separately, coupled by heat and mass transfer boundary conditions with the gaseous phase of the dryer module. Fig. 3 shows a sketch of the configuration. In typical coating applications the thickness of the film and substrate is much smaller than the lateral dimensions (substrate width, dryer length), introducing a slenderness parameter that allows a simplification of the governing equations, cf. Alleborn (2001a) and Aust (1996). The mass transport in the liquid film is governed by a partial differential equation (diffusion equation) for the solvent content. For sufficiently thin films and substrates, a constant average temperature across the film and substrate may be assumed, resulting in an ordinary differential equation for the energy transport. The dryer module is character-

ized by average heat and mass transfer coefficients that depend on its geometry and operation conditions and determine the heat and mass transfer through the film surface and the substrate bottom, cf. Fig. 3. The physical and process engineering background of the representation as well as the mathematical model for the heat and mass transport is described in detail in the work of Aust (1996) and Aust et al. (1997), which will be followed closely in the present paper. The governing equations and boundary conditions will be stated in Section 5 and embedded into the framework of the mathematical optimization algorithm for the large-scale NLP-problem. The control variables for the optimization problem are the velocity and temperature of the drying air at nozzle exit in each module. In this paper a simultaneous approach is considered for the optimization and both parts of the process representation, air handling in the dryer and transport in the wet film and substrate, are modeled and solved simultaneously. As it was previously mentioned, the simultaneous approach has the advantage that the operation limits of the drying process can be easily implemented

Fig. 3. Representation of the heat and mass transfer into the wet film and web.

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in the formulation as inequality constraints, providing a high flexibility in investigating different process scenarios. 3. Drying scenarios for optimization The optimization of a process involves the minimization (or maximization) of a certain criterion (objective function) while a set of process requirements have to be satisfied (constraints). In the optimization of drying processes, an essential point is the consideration of the constraints under which the drying operation has to be performed, e.g. a maximum permissible oven temperature, protecting the wet film from disturbances, assuring a certain coating production rate (analogously, a certain web speed), demands on operational security. Many different objective functions can be defined according to the criteria, which have the highest priority in the process. As in many other applications, the optimization of a drying process requires a trade-off between the different criteria of interest, e.g. fast drying (high production rate) involves high heat consumption; the removal of high amounts of solvent is not energy efficient and fast at the same time. Therefore, the definition of a high priority optimization criterion is essential and defines a drying scenario. Bounds can be defined for secondary criteria, e.g. minimize the heat consumption (objective function, highest priority criterion) to generate a minimum required production rate of a film with a certain solvent specification (constraint, secondary criterion). In this article, the following drying scenarios are considered: • Minimization of the total heat consumption in the process to obtain a fixed production rate of a film with a certain final solvent content (scenario 1). • Maximization of the coating production rate to obtain a film with a certain final solvent content (scenario 2). • Minimization of the final solvent mass in the film (scenario 3). • Maximization of the total flow of evaporated solvent (scenario 4). 4. Optimization problem statements The general optimization problem can be stated as follows. Given is an existing multiple-zone dryer with Nk modules. Given are the composition, humidity and temperature of the make-up ma , xma and T ma , respectively. Given is the initial solvent air, x1k wk k specific mass in the film, m10 , a uniform initial temperature, T0 and a uniform initial solvent distribution along the film thickness, ins X0 = m10 /m2 . Given are the flow rate, m ˙ ins k and temperature, Tk of the infiltrating air. Given is an upper bound for the solvent nozzle . Concerning the quality of the content in the drying air, x1U coated product, an upper limit on the film temperature, TU , is given. The following optimization problem can be stated for the different drying scenarios: • Scenario 1. The optimization problem consists in optimiznozzle ), temperature (T nozzle ) ing the drying air composition (x1k k

471

and velocity (υknozzle ) for each drying module to produce a specif film with a solvent mass of at most the specification, m1 , without generating bubbles involving minimum total heat consumption. • Scenario 2. The problem determines the optimal drying opernozzle , T nozzle and υ nozzle ) for each module ation conditions (x1k k k in order to achieve a film at maximum production rate with at specif most the desired film specification, m1 and without generating bubbles. • Scenario 3. The problem determines the optimal drying opernozzle , T nozzle and υ nozzle ) for each module ation conditions (x1k k k in order to achieve a coating product (free of bubbles) with a minimum final solvent mass. • Scenario 4. The problem determines the optimal drying opernozzle , T nozzle and υ nozzle ) for each module ation conditions (x1k k k in order to produce maximum solvent evaporation in the drying process. It should be noted that for all drying scenarios, the solution of the optimization problem also yields optimum values for the film temperature and solvent content and for the rest of the state variables associated with the process. 5. Optimization problem formulation The fully discretized dynamic optimization model formulation is presented next. In Section 5.1, the model discretization, which converts the PDE model into an NLP formulation is described. In Section 5.2, the model constraints are presented and in Section 5.3 different alternatives for the model objective function are proposed for each drying scenario. 5.1. Model discretization The NLP formulation is obtained by discretizing the film residence time t, the film thickness coordinate z as well as all the variables involved in the differential and algebraic equations. In Fig. 4, the discretization scheme is shown. For each drying module, the time is discretized in Nt points such that the point j = 1 corresponds to the initial time (t = t0 ) and j = Nt corresponds to the residence time of the film in the dryer. The film thickness is discretized with Nz points. The film bottom corresponds to i = 1 (z = z0 ) while the surface is defined with i = Nz (see Fig. 4(a)). The drying modules are then connected so that the last time point of module k is linked with the first time point of module k + 1 (see Fig. 4(b)). For the complete discretization of the PDE optimization formulation consider the following sets definition. Let K be the set of drying modules: K = {k|k = 1, . . ., Nk }. Let Jk be the set of time points which discretize the drying module k: Jk = {j|j = 1, . . ., Ntk }. Let I be the set of space points which discretize the thickness of the film: I = {i|i = 1, . . ., Nz }. Then, the residence time of the film in module k, tk , is discretized over Ntk time points and the film thickness is discretized over Nz points according to (1):

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Fig. 4. Discretization scheme.



ˆtk,j

t11  . =  .. tNk ,1

 t1,Ntk ..  ..  . .  · · · tNk ,Ntk ···

zˆ = (z1 , z2 , . . . , zi , . . . , zNz )t t1,1 = t0

Note that not necessarily all the drying modules are discretized with the same amount of time points. The discretized residence times for the complete time horizon are computed in (5). Note that in (5), the index j is redefined as jj. (1)

tk,Ntk = tkfinal z1 = z0

zNz = hfilm dry Note that t0 is the initial time, z0 the initial space (generally both considered as zero) and tfinal is the film residence time in the dryer. The drying process is considered in a reference frame moving at the web velocity, υweb through the dryer, so that the residence time tkfinal is related to the module length, lk = υweb tkfinal . The film thickness, hfilm dry , refers to the dry film thickness. The actual thickness of the wet film diminishes as the film moves through the dryer due to the solvent evaporation constituting essentially a moving boundary problem, which can be conveniently solved by using a solid frame of reference based on the dry film thickness, as widely used in the literature, e.g. see Aust et al. (1997). Note that the dry film thickness, hfilm dry , is computed as a function of the dry coating mass, m2 , and its density, ρ2 : hfilm dry = m2 /ρ2 . The film thickness hfilm dry is discretized using Nz space points and a vector of components zi is generated (see (2)) according to the space step z defined in (3). zi = z0 + (i − 1)z z =

hfilm dry Nz − 1

(2) (3)

The residence time for one drying module (or equivalently, the length of one drying module) is discretized over Ntk time points (see first equation in (4)) as a function of the module k length, lk , and the web speed, υweb . The time steps for each module k, tk can be then defined.  lk   tkfinal = υweb  , k∈K (4) tkfinal    tk = Ntk − 1

k = 1, j = 1 tk,j = t0 , tk,j = tk,jj + (j − 1)tk , ∀k ∈ K, j = 2, . . . , Ntk , jj = 1 tk,j = tk−1,jj , ∀k = 2, . . . , Nk , j = 1, jj = Ntk (5) 5.2. Model constraints The constraints of the optimization model are classified in the following groups: • • • •

differential equations; algebraic equations modeling the energy transport; algebraic equations modeling the mass transport; interconnection (algebraic) equations linking the drying modules; • algebraic equations modeling the air handling at each module; • inequality constraints for the process operation limits formulation. The system of equations governing the heat and mass transfer is adopted from Aust (1996) and Aust et al. (1997) where a more detailed discussion of the physical background can be found. 5.2.1. Differential equations: finite difference approximation The differential equations are fully discretized using finite difference approximation methods. For the energy transfer, the ordinary differential equation: vap

dT (t) qt (t) + qb (t) − mv (t)h1 (t) = dt (mweb Cpweb + m2 Cp2 + m1 (t)CpL (t)) 1

T (t) = T0 ,

(6)

t = t0

models the temperature distribution in the film, T, over the time. Before the film enters the dryer (initial condition) it has a constant and uniform temperature, T0 . In Eq. (6), qt and qb are the heat fluxes at the film top and substrate bottom, respectively, mv is the solvent evaporation rate,

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hvap = hvap (T(t)) is the heat of vaporization as a function of the film temperature T(t) (see Appendix A), m1 is the solvent mass and CpL = CpL (T (t)) is the solvent heat capacity formulated as a function of the film temperature (see Appendix A). Note that this 1 1 equation does not account for the temperature distribution along the film thickness since the temperature distribution in thin layers of materials is almost uniform (Aust et al., 1997; Cairncross, 2002). Eq. (6) are fully discretized in time with a Forward Difference approximation to evaluate the differential term dT dt at the point j for each module k: vap

qtk,j + qbk,j − mvk,j h1k,j Tk,j+1 − Tk,j = , tk,j+1 − tk,j mweb Cpweb + m2 Cp2 + m1k,j CpL

∀k ∈ K, j = 1, . . . , Ntk − 1

1k,j

Tk,j = T0 ,

(7)

k = 1, j = 1

In the discretized space, each variable is defined for each module k at each time j, e.g. Tk,j denotes the film temperature in module k at the residence time j. The mass transport that takes place in the film is modeled with a partial differential equation, the diffusion equation in a solid frame of reference given in (8) with boundary conditions at the film surface and bottom and the initial condition (t = t0 ) of constant and uniform solvent content over the film thickness, X0 :  ∂X(t, z) ∂ D(X, T ) ∂X(t, z) = ∂t ∂z (1 + ε(t)X(t, z))2 ∂z ∂X(t, z) V ∗ [1 + ε(t) X(t, z)]2 ∀t, z = hfilm =− 2 mv (t), dry (8) ∂z D(X, T ) ∂X(t, z) = 0, ∀t, z = z0 ∂z X(t, z) = X0 , ∀z, t = t0 where X is the solvent content (dry basis), D = D(X, T) is the diffusion coefficient which may depend on the temperature and composition and ε is the ratio of densities of the non-volatile component and the solvent, ε(t) = ρ2 /ρ1L (t) with ρ1L = ρ1L (T (t)). At the film surface, z = hfilm dry , the change in the solvent composition is a function of the solvent evaporation rate mv (t) and the dry coating specific volume V2∗ . At the film bottom (z = z0 ), no flux of solvent takes place due to the contact with the impermeable substrate. The PDE equation in (8) can be reformulated as an expanded expression in Eq. (9) where all variables are discretized at each drying module k in time j and film thickness i:  

2

∂X

1 ∂D

∂X

Dk,j,i ∂2 X

2εk,j Dk,j,i = + − , ∂t k,j,i ∂z k,j,i (1 + εk,j Xk,j,i )2 ∂X k,j,i (1 + εk,j Xk,j,i )3 (1 + εk,j Xk,j,i )2 ∂z2 k,j,i ∀k ∈ K, j = 2, . . . , Ntk , i ∈ I

(9)

In Eq. (10), a Forward Difference approximation is used to evaluate the first order differential terms of Eq. (9) while the second order spacial derivative is approximated with a Crank–Nicholson Implicit (C–N) method, using the average of the central difference expressions at (j, i) and (j + 1, i):   Xk,j+1,i − Xk,j,i 1 Xk,j,i+1 − Xk,j,i 2 (Dk,j,i+1 − Dk,j,i )(Xk,j,i+1 − Xk,j,i ) 2εk,j Dk,j,i = − tk,j+1 − tk,j zi+1 − zi (1 + εk,j Xk,j,i )2 (zi+1 − zi )2 (1 + εk,j Xk,j,i )3  (Xk,j,i+1 − 2Xk,j,i + Xk,j,i−1 ) + (Xk,j+1,i+1 − 2Xk,j+1,i + Xk,j+1,i−1 ) 1 Dk,j,i , + 2 (1 + εk,j Xk,j,i )2 (zi+1 − zi )2 ∀k ∈ K, j = 2, . . . , (Ntk − 1), i = 2, . . . , (Nz − 1)

(10)

The boundary conditions are discretized using a Forward Difference approximation at the grid point (j, i). Also, the initial condition is discretized in (11):  Xk,j,i+1 − Xk,j,i V ∗ (1 + εk,j Xk,j,i )2   =− 2 mvk,j , i = Nz − 1  zi+1 − zi Dk,j,i , ∀k ∈ K, j = 2, . . . , Ntk Xk,j,i+1 − Xk,j,i (11)    = 0, i=1 zi+1 − zi Xk,j,i = X0 , ∀k = 1, j = 1, i ∈ I

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5.2.2. Energy transfer constraints The convective heat input at the film bottom, qbk,j , is modeled in Eq. (12): qbk,j =

αk,j (Tknozzle

− Tk,j ),

∀k ∈ K, j ∈ Jk

(12)

1k,j

∀k ∈ K, j ∈ Jk 1k,j

 m1k,j = m2

= f (Tk,j ) is the heat capacity of the solvent vapor

mvk,j = βk,j

1  Xk,j,i m2 , Nz Nz

m1k,j =

∀k ∈ K, j = 1, . . . , Ntk

(18)

i=1

The solvent evaporation rate for each module k at each time j, mvk,j , is defined in Eq. (19):

PM1 ln + Tk,j )R

0.5(Tknozzle

distribution. The heat transfer coefficient αk,j is computed with Eq. (14): Nuk,j λsurface k,j

,

∀k ∈ K, j ∈ Jk

(14)

as a function of the Nusselt number, Nuk,j , the thermal conductivity of the mixture solvent vapor-drying air at the film surface, λsurface (see Appendix A for detailed computation of properties k,j at the film surface) and the nozzle slot width b. In Eq. (15), the Reynolds and the Prandtl numbers are defined to compute the Nusselt number using the formulas of Martin (1977) for arrays of slot nozzles, see also Aust (1996).  surface (nslot b)  υknozzle ρk,j  k  Rek,j =   surface  µk,j     surface   µk,j Cpsurface   k,j  Prk,j = surface λk,j , ∀k ∈ K, j ∈ Jk  2/3          2Re 2 0.75 0.42  k,j    Nuk,j = fo Pr k,j      fk fo 3   +  fo fk (15) where fk and fo are geometric factors (see definition in Appendix A). 5.2.3. Mass transfer constraints The initial condition for Eq. (9) is given in Eq. (16). At initial time, a constant solvent content X0 over the film thickness is assumed: Xk,j,i = X0 ,

(17)

where m2 is the polymer mass (dry coating mass) and dz is the differential of space. In case an equally spaced grid distribution is considered for the film thickness discretization, Eq. (17) can be approximated with the algebraic equation in (18):

mvk,j = 0,

b

∀k ∈ K, j ∈ Jk

Xk,j dz,

(13)

at the time point j defined as a function of the film temperature

αk,j =

hfilm

0

where αk,j is the heat transfer coefficient and Tknozzle is the temperature of the nozzle (drying) air in module k. In the presence of evaporation, the heat input at the film surface, qtk,j , is modeled by Eq. (13), see Aust (1996), Krischer and Kast (1978).   vap CpV αk,j h1k,j 1k,j nozzle qtk,j = ln 1 + − Tk,j ) , vap (Tk CpV h1k,j

where CpV

The specific average mass of solvent can be expressed as a function of the solvent content distribution in the film:

∀k = 1, j = 1, i ∈ I

(16)



P − pnozzle 1k top P − p1k,j

k = 1, j = 1



(19)

, ∀k ∈ K, j = 2, . . . , Ntk

where βk,j is the mass transfer coefficient, M1 the solvent molar the solvent partial mass, P the total pressure in the dryer, pnozzle 1k top pressure in the nozzles (drying air) and p1k,j is the solvent partial pressure at the film surface, defined in Eqs. (20) and (21), respectively.    

nozzle P = y1k pnozzle 1k nozzle = y1k

top

nozzle x1k ,  nozzle + xnozzle M1 + M1   x1k wk Mw Mair

top

vap

p1k,j = a1k,j pk,j ,

∀k ∈ K

∀k ∈ K, j ∈ Jk

(20)

(21)

top

vap

In Eq. (21), a1k,j is the solvent activity coefficient and p1k,j = f (Tk,j ) is the vapor pressure of the pure solvent computed as a function of the film temperature. The mass transfer coefficient βk,j is formulated in Eq. (22) from the Chilton–Colburn equation for turbulent flow (Chilton & Colburn, 1934; Krischer & Kast, 1978) as a function of the heat transfer coefficient, αk,j and the Lewis number, Lek,j (see Appendix A for definition of properties at the film–air interface). βk,j =

αk,j surface Le0.58 Cpsurface ρk,j k,j k,j

Lek,j =

λsurface k,j surface δsurface Cpsurface ρk,j k,j

          

,

∀k ∈ K, j ∈ Jk

(22)

k,j

5.2.4. Modules interconnecting constraints The modules are linked by connection balances formulated in Eq. (23) for the variables related to the energy transfer in the

M. Barttfeld et al. / Computers and Chemical Engineering 30 (2006) 467–489

film and in Eq. (24) for the variables related to the mass transfer.  Tk,j = Tk+1,jj      αk,j = αk+1,jj      qtk,j = qtk+1,jj      qbk,j = qbk+1,jj     Nuk,j = Nuk+1,jj      vap vap  hk,j = hk+1,jj       CpV = CpV   1k,j 1k+1,jj , ∀k ∈ K, j = Ntk , jj = 1 (23) CpL = CpL  1k,j 1k+1,jj    Cpsurface = Cpsurface    k,j k+1,j     surface = ρsurface  ρk,j  k+1,j    surface surface  δk,j = δk+1,j     surface surface  Lek,j = Lek+1,j      surface surface  Prk,j = Prk+1,j     surface surface Rek,j = Rek+1,j     Xk,j,i = Xk+1,jj,i  , ∀i ∈ I     Dk,j,i = Dk+1,jj,i      βk,j = βk+1,jj   , ∀k ∈ K, j = Ntk , jj = 1 m1k,j = m1k+1,jj    mvk,j = mvk+1,jj      surface surface  p1k,j = p1k+1,jj     a1k,j = a1k+1,jj (24) Note that for these equations, the index j was renamed as jj to link the variables defined for the last time discretization point of a module k with the first time point of module k + 1. 5.2.5. Air handling equations The overall mass and energy balances involve time and space invariant parameters, which model the drying air handling at each drying module. Eqs. (25) and (26) model the control volume I (see Fig. 2): m ˙ nozzle +m ˙ ins ˙ su k k =m k , =

∀k = 1, K

∀k = 2, . . . , K − 1  vap nozzle + m su m ˙ nozzle x1k ˙k =m ˙ su k k x1k   nozzle = m su , ∀k ∈ K m ˙ nozzle xwk ˙ su k k xwk   web nozzle su hk + hk = hk

m ˙ nozzle k

m ˙ su k ,

(25)

(26)

The first equation in (25) models the mass balances for the air handling considering that air is infiltrating at the inlet (first module) and outlet (last module) of the dryer. The enthalpy of the coating is computed in Eq. (27): in o hweb = mweb Cpweb + m2 Cp2 + min 1k Cpin (Tk − T )wweb υweb k 1k

−mweb Cpweb +m2 Cp2 +mout (Tkout 1k Cpout 1k

− T o )wweb υweb (27)

475

out The solvent specific mass (min 1k and m1k ) and temperature of the film (Tkin and Tkout ) entering and leaving a module k are defined in Eqs. (28) and (29), respectively.  min 1k = m1k,j , ∀k ∈ K, j = 1 (28) Tkin = Tk,j  mout 1k = m1k,j , ∀k ∈ K, j = Ntk (29) Tkout = Tk,j vap

The total evaporated solvent flow in module k at time j, m ˙k , is defined in Eq. (30): vap

m ˙k

out = (min 1k − m1k )wweb υweb

(30)

where wweb is the film width and υweb is the web speed. The mass flow of drying air at the nozzle outlet in module k is defined by the first equation in (31) while its density is modeled assuming ideal gas behavior:  nozzle = υknozzle bwnozzle nnozzle ρair m ˙ nozzle  k k k k    nozzle M + ynozzle M P[y1k 1 w wk , ∀k ∈ K (31) nozzle − ynozzle )M ]  + (1 − y1k air  wk nozzle   ρair k = RTknozzle where wnozzle is the nozzle width in module k, nnozzle the number k k nozzle the density of the drying air in of nozzles in module k, ρairk module k, Mw and Mair the mole mass of water and air, respecnozzle the solvent mole fraction of solvent in the drying tively, y1k nozzle is the water mole fraction in the drying air in air and ywk module k. Eq. (32) model the control volume II defined for each drying module (see Fig. 2):  ˙ ma ˙ ex ˙ nozzle m ˙ su  k +m k =m k +m k    su su ma ma ex ex nozzle nozzle  m ˙ k x1k + m ˙ k x1k = m ˙ k x1k + m ˙k x1k     su su ma ma ex ex nozzle nozzle m ˙ k xwk + m ˙ k xwk = m ˙ k xwk + m ˙k xwk    su ma ex nozzle , ∀k ∈ K qk + hk + hk = hk + hk   ex su   x1k = x1k    ex su   xwk = xwk    ex su Tk = Tk (32)

5.2.6. Process operation limits constraints The operation limits for the drying process are inequality constraints, which are included in the formulation to model the limitations concerning the dryer operating conditions, the quality of the final coated product as well as safety conditions. Regarding the drying air operation, commonly there exist lower and upper bounds for the temperature ranges at which the drying air can be heated. Similarly, the velocity of the drying air can be restricted between maximum and minimum values according with the plant operation policies. These restrictions constraints are added to the optimization formulation by the

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476

following set of inequality constraints:  Tknozzle ≤ TUnozzle      nozzle nozzle Tk ≥ TL , ∀k ∈ K nozzle  υknozzle ≤ υU     υknozzle ≥ υLnozzle

Another possibility is to maximize the coating production, i.e. maximize the web speed (drying scenario 2). In (40), the dryer web speed, υweb , is maximized: (33)

Regarding the quality of the coated product, the formation of bubbles is an undesired effect. Therefore, in order to prevent bubble formation in the film, the maximum solvent partial pressure at the film bottom, pbot 1k,j (where there is the maximum solvent content) must be below the total pressure in the dryer, P (for definition of activity coefficient, see Appendix A): vap

bot pbot 1k,j = a1k,j pk,j ≤ P,

∀k ∈ K, j ∈ Jk

(34)

Formation of bubbles (or blisters) is one among numerous drying defects, which are in general difficult to quantify, cf., e.g. Cohen and Gutoff (1992). The criterion for bubble formation stated in Eq. (34) has been adopted from literature (Aust, 1996; Price & Cairncross, 2000) in order to illustrate the inclusion of a constraint regarding coating quality into the optimization model. This criterion is widely used in numerical simulations of film drying processes, however, it neglects details of the bubble formation process, such as nucleation and transport phenomena, or the influence of dissolved gases. Another operation limit related to the coating product quality imposes the condition that the temperature of the film must not exceed a certain value (upper bound): Tk,j ≤ TU ,

∀k ∈ K, j ∈ Jk

(35)

The solvent content in the drying air can be restricted to be nozzle ) by including the inequality (36) below an upper bound (x1U nozzle nozzle x1k ≤ x1U

(36)

Safety conditions are related with the risk of solvent explosion in the dryer. This condition is modeled in the inequality (37) where a maximum concentration of solvent (depending on su is allowed in the suction air: the solvent nature) x1U su su x1k ≤ x1U ,

∀k ∈ K

(37)

For several applications, the solvent mass at the end of the process should not exceed a certain specification. The film specspecif ification is formulated in Eq. (38), where m1 is the specified final solvent mass: specif

m1k,j ≤ m1

,

∀k = Nk , j = NtNk

(38)

5.3. Objective functions For the minimization of the total heat consumed in the process, Q (drying scenario 1) expression (39) is included in the formulation as the objective function: min Q =

Nk  k=1

qk

(39)

(40)

max υweb

In some industrial applications, e.g. paper coating, the minimization of the solvent mass at final time, mout 1Nk , is desired (scenario 3). Then, in (41) is formulated as the objective function for the optimization problem: min mout 1Nk

(41)

The minimization of the final solvent mass can be alternatively optimized by maximizing the total flow of evaporated solvent in the film (scenario 4). Then, Eq. (42) is formulated as the objective function: max

Nk 

vap

m ˙k

(42)

k=1

6. Numerical results In this section numerical results are presented to illustrate the proposed optimization formulation. All the examples involve the dryer geometry and the drying air operation limits used by Aust et al. (1997), which are presented in Table 1. The operation limits for the drying air temperature are 60–120 ◦ C and for the velocity 10–50 m/s. The solvent concentration in the drying air is assumed to be 2 vol%. A constant pressure of 1.013 × 105 Pa is considered in the dryer. In order to illustrate the optimization of drying processes, a constant diffusion coefficient is assumed in this paper. A value of 1 × 10−9 m2 /s is adopted which is a typical value, e.g. for aqueous solutions of PVOH, starch or gelatin (Aust, 1996; Uzman & Sahbaz, 2000; Gehrmann, 1979). Although detailed models for the concentration and temperature dependence of the diffusion coefficient exist for some solvent/solute systems, see, e.g. Aust (1996), Neogi (1986), for many coating recipes used in industry, this transport property is not available and has to be estimated, see, e.g. Price et al. (1997). In Section 6.1, results obtained with the proposed discretization scheme (see Section 5.2) are verified using the simulation software RASTA (Aust, 1996). Afterwards, four numerical drying scenarios are investigated. Drying scenarios 1–3 are illustrated with a film of water in polyvinyl alcohol (PVOH), as processed Table 1 Geometric data and drying air operation limits b (mm) d (cm) nnozzle k nslot k s (cm) wweb (m)

Slot width Nozzle-coating distance Number of nozzles in each module k Number of slots per nozzles in module k Distance between nozzles Coating width

TUnozzle (◦ C) TLnozzle (◦ C) nozzle (m/s) υU υLnozzle (m/s)

Drying air temperature upper bound Drying air temperature lower bound Drying air velocity upper bound Drying air velocity lower bound

2 3 40 1 15 1 120 60 50 10

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for instance in the paper coating industry. For the illustration of scenario 4, a different system comprising methanol as solvent in polyvinyl acetate (PVAc) is considered. The optimization problem solved in scenario 1 involves the minimization of the total heat consumption of the process. Scenario 2 shows optimization results corresponding to the maximization of the process production rate. In scenario 3, the final solvent mass in the film is minimized while scenario 4 considers the maximization of the total evaporated solvent flow in the process. The property data for the polymers is taken from Brandrup and Immergut (1975). The temperature dependence of all properties for pure methanol and water are calculated with temperature-dependent correlations, from Daubert (1989) and Daubert and Danner (1985). Different modeling systems were tested for the implementation and solution of the drying formulation presented in Section 5. However, according to our experience, the software package “A Modeling Language for Mathematical Programming” (AMPL, Fourer, Gay, & Kernighan, 2003) in combination with the NLP solver IPOPT (W¨achter & Biegler, 2004) proved to be the best choice to tackle this problem. The numerical results for all the drying scenarios presented here were obtained on a Pentium IV computer with 2.26 GHz clock speed and 512 KB RAM. The CPU seconds stated in the following refer to this machine. Regarding the discretization of the model, for all scenarios, 10 space points are used to discretize the film thickness and 20 points discretize the time horizon corresponding to each drying module, i.e. Ntk = 10 for all k in K. This discretization scheme appears to be appropriate to solve the differential equations with sufficient accuracy (see Section 6.1). 6.1. Verification of the optimization algorithm In this section, the discretization scheme proposed in Section 5.2 is verified with an example problem, which considers the drying of a water–PVOH film with a total residence time of 30 s in a ten module dryer. A constant initial temperature of 20 ◦ C is assumed as the initial condition of Eq. (7) and an initial solvent content of 4 (kg/kg) is given as the initial condition of Eq.

477

Table 2 General properties: scenarios 1, 2 and 3 Cp2 (kJ/kg K) Cpweb (kJ/kg K) D (m2 /s) Mair (kg/kmol) Mw (kg/kmol) m10 (g/m2 ) mweb (g/m2 ) m2 (g/m2 ) P (Pa) T0 (◦ C) Tkma (◦ C) V2∗ (m3 /kg) x0 (kg/kg) ma xwk (kg/kg) ρ2 (kg/m3 )

Dry coating heat capacity Substrate heat capacity Diffusion coefficient Air molar mass Water molar mass Initial solvent specific mass Substrate specific mass Dry coating specific mass Total pressure of the dryer Initial temperature of the film Make-up air temperature Dry coating specific partial volume Initial solvent content (dry basis) Water content in fresh air (dry basis) Dry coating density

1.5 1.6 1 × 10−9 28.85 18 80 60 20 1.013 × 105 20 20 8.33 × 10−4 4 0 1200

(10). For this example, the dryer geometry given in Table 1 and the general properties presented in Table 2 are considered. The drying air velocities are set to 50 m/s in all the modules while the drying air temperature is 60 ◦ C in the first nine modules and 80 ◦ C in the last one. In Fig. 5, the solution of Eqs. (7) and (10) with the boundary conditions (11) implemented in AMPL and solved with IPOPT is shown. In Fig. 5(a), the film temperature profile is shown in contrast with the numerical solution obtained with RASTA. As it can be seen from Fig. 5, a good approximation is obtained with the proposed discretization scheme using 20 time points to discretize the residence time of each drying module and 10 points to discretize the film thickness. Note that even if a smaller amount of points are used (e.g. five space and five time points per module), the solution obtained with IPOPT is still fairly good (see Fig. 5(a)). In Fig. 5(b), the solvent composition profile at the film surface is also in good agreement with the simulation results yielded by RASTA. 6.2. Scenario 1: minimization of total heat consumption In this example, a dryer with 10 modules is considered (i.e. Nk = 10). Each drying module has a length of 3 m and a web

Fig. 5. Implicit discretization scheme in AMPL (IPOPT) vs. RASTA.

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Table 3 Scenario 1: optimization model statistics Variables number Equations number

10048 10028

Objective function value (kW) Time (CPU seconds)

406.92 13.13

speed of 1 m/s is considered. The web speed together with the module length define the discretization grid formulated in (1)–(5). The complete time horizon is then 30 s discretized with time steps of 158 ms. Note that considering a variable web speed with lower and upper limits would lead to a trivial optimal value for this variable. In order to minimize the heat consumption, at the optimal solution, the web speed would match its lower bound in order to maximize the film residence time. A film of water in polyvinyl alcohol with an initial solvent specific mass of 80 g/m2 (80% solvent mass fraction) and an initial temperature of 20 ◦ C is considered. The minimum required final solvent content is specified 0.3 g/m2 (1.5% solvent mass fraction). An upper bound on the film temperature of 90 ◦ C is assumed in order to safely avoid the melting and degradation of the product. The general properties and data for this example are presented in Table 2. The optimization model is solved minimizing Eq. (39) subject to the constraints (1)–(5), (7), (10)–(35), (38) and Eqs. (43)–(57). (see Appendix A for definition of Equations (43)–(57)). The model involves 10,048 variables and 10,028 equations (see Table 3). Note that the degree of freedom of the optimization problem is 20: each drying module has two control variables, the drying air temperature and velocity. The model is solved in 13.13 CPU seconds and the optimal objective function value (total heat consumption) is 406.92 kW (see Table 3). The optimal results for the drying air temperatures and velocities are shown in Fig. 6(a). For the optimal dryer operation, the drying air velocity is selected to 50 m/s (upper bound) in all modules and its temperature to 60 ◦ C (lower bound) for the first nine modules and 76.83 ◦ C for the last drying module. In Fig. 6(b), the results of the heat consumption, the fraction of recycled air (m ˙ su ˙ ex ˙ su k −m k )/m k and the fraction of evaporated solvap Nk vap vent m ˙ k / k=1 m ˙ k at each drying module are shown. Since the total heat consumption is minimized, the fraction of recycled drying air in all the modules is high (approximately 0.9, see Fig. 6(b)). The fraction of evaporated solvent is higher in the first modules (around 0.1) and toward the last modules decreases to

approximately 0.06. In Fig. 7(a), the film temperature profile is shown along the dryer. Note that the values for the optimal solution are below the upper bound imposed for this variable (90 ◦ C). The three distinct regions of drying behavior can be observed in the film temperature profile. The warm-up region in the first 3 s where the film enters the dryer and its temperature increases rapidly until the rate of energy supplied by the air matches the rate of energy consumption by evaporation (due to the evaporative cooling, see Eq. (6)). After this period, a constant rate region of drying takes place starting after 3 m of the dryer until 22 m approximately. Finally, the falling rate period starts, in this case because the film runs out of solvent and the activity coefficient drops sharply (see Fig. 7(d)). The solvent mass can be observed in Fig. 7(b) (dotted line) which just meets the specification at the 2 end of the dryer (mout 1k = 0.3 g/m , k = 10). The solvent content profiles (X) along the dryer length are also shown in Fig. 7(b). Note that the profiles are plotted every 1.90 ␮m of the (dry) film thickness (17.1 ␮m), which corresponds to the value of the step dz computed in the discretization (see Eq. (3)). Note that towards the end of the dryer the solvent content profiles are close to each other since the gradients of the solvent content with respect to the film thickness decrease as the film gets dry. The solvent evaporation rate is shown in Fig. 7(c). Note that in the warm-up region the evaporation rate increases rapidly, holds approximately constant in the constant rate period (2.56 × 10−5 kg/m2 s) and decreases towards zero in the falling rate. It should be pointed out that the dryer operation takes place without the formation of bubbles in the film since the solvent partial pressure in the film is considerably smaller than the total (atmospheric) pressure in the dryer (see Fig. 7(d)). The optimal solution was compared with simulations of different drying strategies, which were performed with the simulation software RASTA (Aust et al., 1997). In the optimal solution, for all the drying modules the drying air velocities are in the upper bound (50 m/s) and the temperature in the lower bound (60 ◦ C) except for the last module (76.83 ◦ C). On a first sight, the high airflow rate of the optimal solution seems to imply higher heat consumption, and therefore a strategy with lower flow rate and high temperatures would appear to yield at least comparable if not even lower heat consumption, under the same constraints. This operation strategy is then considered by setting the drying air temperature in 120 ◦ C and a velocity of 10 m/s for all modules. In Fig. 8, the results of the simulated solution are shown (black dotted line, simulation case 1). Note that with this

Fig. 6. Scenario 1: (a) optimal drying air operating conditions and (b) recycled air fraction, evaporated solvent fraction and heat consumption at each drying module.

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479

Fig. 7. Scenario 1: optimal solution.

operation conditions the heat consumption is 454.5 kW, higher than the optimal solution (406.92 kW), and the desired final film specification is not even achieved (see Fig. 8(a)), so that simulation case 1 is an infeasible scenario of the optimization problem. If the drying air velocity is increased to 20 m/s only in the first module of the simulated case 1, the desired film specification is obtained with a heat consumption of 503.7 kW (see cross dark grey line in Fig. 8, simulation case 2). However, note that in simulation 2, the upper bound on the film temperature is violated in the last module. This fact gives an indication that the drying air temperature has to be reduced in order to generate a feasible solution of the original problem and at the same time,

the drying air velocity increased in order to achieve the final film specifications. Simulation case 3 involves a drying air temperature of 90 ◦ C in all modules and the air velocity is set to 30 m/s in the first module and in 20 m/s in the rest of the modules. This drying scenario generates a feasible solution for the problem with no bubbles generation in the film (see Fig. 8, black circles line). However, this solution involves a heat consumption of 478.5 kW, again considerably higher than the optimal solution (406.92 kW). If the drying air temperatures are decreased towards the lower bound (60 ◦ C) and the air velocities increased towards the upper bounds (50 m/s), drying scenarios involving smaller heat consumption are obtained. The comparison with

Fig. 8. Scenario 1: optimal solution compared with simulated strategies in RASTA.

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Fig. 9. Scenario 1: sensitivity analysis with respect to changes in the diffusion coefficient. (a) Optimal objective function and (b) optimal drying air temperatures.

this parametric study confirms that the solution obtained in AMPL (IPOPT) is indeed an optimal solution for scenario 1. It should be noted that the energy balance considered in this paper focuses on the heat consumption. If in addition the energy for the fans is taken into account, a low air flow strategy may become competitive again when the pressure losses in the air handling system require a high fan power. When pressure losses for the specific dryer configuration are known, they could be easily included in the formulation of the objective function. The impact of the diffusion coefficient value on the optimal solution was also analyzed. In Fig. 9(a), the optimal total heat consumption (objective function) is shown for a range of diffusion coefficient values. For values of the diffusion coefficient larger than 1 × 10−9 m2 /s, the optimal total heat consumption remains almost constant in 405.45 kW. For values smaller than 1 × 10−9 m2 /s, the optimal heat consumption increases drastically as the diffusion coefficient decreases (see Fig. 9(a)). The optimal drying temperatures for four selected values of the diffusion coefficient can be observed in Fig. 9(b). The optimal drying air velocities were 50 m/s in all four cases. Summarizing the sensitivity results for this scenario, as the diffusion coefficient decreases, higher drying temperatures are needed in order to achieve the specified final solvent mass. As a consequence, more energy is consumed in the drying process. On the contrary, an increment in the diffusion coefficient values yields lower drying temperatures and lower energy consumption. However, when the drying temperatures are almost at their lower bounds (see Fig. 9(b), D = 1 × 10−6 m2 /s, Q = 405.45 kW), a further increment on the diffusion coefficient does not influence the value of the objective function, which remains practically constant. In all cases analyzed in this example the drying air velocity remained in the optimal value of 50 m/s. 6.3. Scenario 2: maximization of the coating production rate As in the previous example, a 10-module dryer of 3 m each and a film of water in PVOH with an initial solvent specific mass of 80 g/m2 (80% solvent mass fraction) and a temperature of 20 ◦ C are considered. The minimum required solvent content is 0.3 g/m2 (1.5% solvent mass fraction) and an upper bound on the film temperature of 90 ◦ C is also assumed in this example.

The general properties and data for this example are presented in Table 2. In the optimization problem, the web speed is maximized under the conditions that the minimum desired final solvent mass is achieved. The decision variables are the drying air temperatures and velocities of each drying model. The optimization model is solved minimizing Eq. (40) subject to the constraints (1)–(5), (7), (10)–(35), (38) and Eqs. (43)–(57). The model involves 10,048 variables and 10,028 equations (see Table 4) and like the problem presented in Section 6.2, the optimization model has 20 degrees of freedom. The model is solved in 14.72 CPU seconds and the optimal objective function value (web speed) is 2.53 m/s (see Table 4) yielding a residence time for the film in the dryer of 11.86 s. The optimal results for the drying air temperature and velocity coincide with the upper bounds in all the modules (120 ◦ C and 50 m/s, respectively). In Fig. 10(a), the film temperature profile is shown. Note that in the optimal solution, the temperature of the film is below the imposed upper bound (90 ◦ C). The solvent mass profile can be observed in Fig. 10(b) plotted every 1.9 ␮m of the film thickness. In the optimal solution the final solvent mass just 2 meets the specification at the end of the dryer (mout 1k = 0.3 g/m , k = 10). The solvent evaporation rate and the solvent pressure at the film surface are presented in Fig. 10(c). Since the solvent pressure is below the total pressure in the dryer (1.013 × 105 Pa) no risk of bubble formation in the film takes place. In Fig. 10(d), the solvent content (dry basis) is shown along the wet film thickness (every approximately 1.2 s). Note that as the time increases the film thickness decreases (Fig. 10(d)). In Fig. 11, the results of the heat consumption (1516.3 kW, total heat consumed), the fraction of recycled air (m ˙ su k − ex su m ˙ k )/m ˙ k and the solvent content in the suction air at each drying module are shown. The fraction of recycled air in all the modules is close to 0.7 in most of the drying modules (see Fig. 11). However, since the solvent content in the suction air is lower in the Table 4 Scenario 2: optimization model statistics Variables number Equations number

10048 10028

Objective function value (m/s) Time (CPU seconds)

2.53 14.72

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481

Fig. 10. Scenario 2: optimal solution.

last modules of the dryer, the fraction of recycled air increases to approximately 0.8 and as a consequence the heat consumption in these modules decreases (see Fig. 11). There exist heuristic rules in order to select the optimal conditions for the dryer in which the maximization of the production rate is desired. Aust et al. (1997) have investigated three different operating strategies for organic solvents:

Fig. 11. Scenario 2: recycled air fraction, heat consumption at each drying module and solvent content in the suction air.

1. Operate all modules at maximum air velocity and possible maximum air temperature without generating bubbles. 2. Operate all modules at maximum air temperature as far as possible (without violating upper bound on the film temperature) and possible maximum air velocity without generating bubbles. 3. Operate all modules at maximum air temperature as far as possible (without violating upper bound on the film temperature) and minimum air velocity. Aust et al. (1997) have concluded that strategies 1 and 2 do not differ much regarding the dryer performance; moreover, the inputs in thermal energy are also similar. Note that the optimal solution just presented coincides with operations 1 and 2 since the drying air temperature and velocity can achieve the respective upper bounds without generating bubbles (see Fig. 10(c)). In Fig. 12, simulation results for the film water–PVOH are presented for the different strategies in contrast with the optimal solution (dotted line). For this example, strategies 1 and 2 coincide with the optimal solution yielding a web speed of 2.53 m/s. If a further increment of the web speed is considered, e.g. 2.75 m/s, the final coated product will not achieve the desired 2 specification (mout 1k = 0.3 g/m , k = 10), see Fig. 12(a). As it was

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Fig. 12. Scenario 2: optimal solution compared with simulated scenarios in RASTA. Table 5 Minimization of heat vs. maximization of production Scenario 1: min Q

Scenario 2: max υweb

Coating production (g/s) Heat consumption (kW)

0.68 406.92

1.72 1516.3

Heat per mass unit (kJ/g)

598

882

also concluded by Aust et al. (1997), strategy 3 does not yield an optimal solution since the web speed can only be increased to 1.42 m/s (see Fig. 12, line with circles). If the web speed is increased above this value, the desired final solvent specification will not be achieved. After considering the examples where the total heat consumed in the process is minimized (see scenario 1) and the coating production is maximized (scenario 2), the question arises about which of these two scenarios is preferred to achieve a more profitable drying production. The heat consumption in scenario 1 is 406.92 kW while in scenario 2 is 1516.3 kW (see Table 5). These amounts of heat are used to produce 0.68 and 1.72 g/s of coated product in scenarios 1 and 2, respectively. Therefore, in scenario 2, the heat required to produce 1 kg of coated product is higher than the heat required in scenario 1, 882 kJ/g versus 598 kJ/g (see Table 5). According to this result, a process which consumes less energy per unit mass of coated product is obtained if the total heat consumption is minimized. Therefore, the scenario where the coating production rate is maximized should be

only considered for those cases in which there is a clear need of increasing the actual coating production rate, taking into account additional factors, such as costs of labor. The impact on the drying air conditions for different values of the diffusion coefficient was explored. In Fig. 13(a), the optimal values for the web speed (objective function) are shown for a range of diffusion coefficient values. Note that as the diffusion coefficient decreases, the web speed also decreases in order to achieve the specified final solvent mass since for this scenario the optimal drying air temperatures are already in their upper bounds in almost all the drying modules (see Fig. 13(b)) as well as the air velocity (50 m/s). For values of the diffusion coefficient larger than 1 × 10−9 m2 /s, the optimal web speed remains almost constant (Vw = 2.6 m/s, Q = 1545 kW, see Fig. 13(b)). 6.4. Scenario 3: minimization of the final solvent mass In this scenario, the final water specific mass is minimized in the 10 modules dryer which coats a film of water in PVOH. Again, the initial water specific mass is 80 g/m2 (80% solvent mass fraction) and the initial temperature 20 ◦ C. The general properties and data for this example are presented in Table 2. The same dryer geometry, discretization scheme and upper limit for the film temperature as in the previous examples are assumed. The dryer web speed is set to 1 m/s and the residence time of the film in the dryer is then 30 s. The optimization problem selects the optimal drying air operation conditions (decision variables) in order to achieve a film

Fig. 13. Scenario 2: sensitivity analysis with respect to changes in the diffusion coefficient. (a) Optimal objective function values and (b) optimal drying air temperatures.

M. Barttfeld et al. / Computers and Chemical Engineering 30 (2006) 467–489 Table 6 Scenario 3: optimization model statistics Variables number Equations number

9849 9829

Objective function value (g/m2 ) Time (CPU seconds)

0.127 35.6

with minimum water specific mass at the end of the process. The formulation involves the minimization of the objective function (41), subject to the constraints (1)–(5), (7), (10)–(35), (38) and Eqs. (43)–(57). The formulation involves 9849 variables and 9829 equations and is solved in 35.6 CPU seconds yielding an optimal objective function value (final solvent specific mass) of 0.127 g/m2 (see Table 6). In Fig. 14(a), the optimal film temperature and solvent mass profiles are presented. In Fig. 14(c), the solvent content (dry basis) is shown along the dryer length. Note that in this plot the film thickness increases as the time decreases. In Fig. 14(d), the optimal values for the drying air temperature and velocity as well as for the heat consumed at each module are shown. The optimal dryer operation involves drying air temperatures decreasing from 110 ◦ C towards the middle of the dryer to approximately 60 ◦ C (lower bound) and increasing then towards the end of it. A similar trend is observed for the drying air velocity, which reaches its upper bound at the beginning of the process and towards the end of the operation. As a consequence of this operation strategy, the heat consumed in the process is higher at the beginning and at the end of the dryer, despite the last module

483

where the heat consumption is only 0.058% of the total heat consumption (480.7 kW). It should be noted that in the last modules of the dryer, the film temperature hits its upper bound of 90 ◦ C and the change in the water specific mass is very small (in the order of 10−5 g/m2 ). This fact can also be observed in Fig. 14(b), where the solvent evaporation rate is in the order of 10−5 –10−6 kg/m2 s towards the end of the dryer. For practical purposes, it can be considered that after 28 m of the dryer length practically no change in the specific solvent mass takes place, i.e. the last 2 m of the equipment (one-third of the last module) essentially do not participate in the process. This flat region of the profiles obtained by minimizing the final solvent mass could be eliminated by considering a two steps optimization scheme. In the first step, the final solvent mass is minimized. In case that flat regions are developed in the profiles, a second optimization step can follow afterwards, e.g. by minimizing the heat consumption or by maximizing the coating production. After minimizing the final solvent mass, an upper bound on the solvent mass is defined according to the optimal value obtained for this problem (0.127 g/m2 ). For this example, 0.17 g/m2 can be considered as the limit in the specific mass profile for which no further significant decrement on the solvent mass occurs. Then, this value is considered as the upper bound for the residual water content in the film at the end of the process (see Eq. (38)). The second optimization step can involve the heat consumption minimization (see scenario 1) or the coating production maximization (see scenario 2). In this example, the second optimization step will be illustrated with

Fig. 14. Scenario 3: optimal solution.

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M. Barttfeld et al. / Computers and Chemical Engineering 30 (2006) 467–489

Fig. 15. Scenario 3: two step optimization.

the minimization of the total heat consumption. In Fig. 15(a), the film temperature and solvent mass profiles for the second step optimization are shown (empty dot line) in contrast with the first step optimal profiles (full dot line). Note that the flat regions in the profiles were eliminated, the final film temperature is below the upper bound (90 ◦ C) and the final solvent meets the specification at the end of the dryer. The optimal operation for the drying air can be observed in Fig. 15(b). In all modules, the optimal drying air temperature is close to 80 ◦ C while the optimal velocity is 50 m/s (upper bound). This operation strategy yields a heat consumption of close to 40 kW at each drying module. The total heat consumption (objective function value) is 412.79 kW. Therefore, by applying a two steps optimization procedure, an optimal drying operation is obtained where the final solvent mass is very close to the minimum value (obtained in the first step) and the heat consumption is considerably decreased with respect to the first step solution (412.79 kW versus 480.7 kW). The sensitivity of the objective function values with respect to changes in the diffusion coefficient was also investigated in this scenario. We have observed that the values of the optimal objective function (final solvent mass) remained practically constant in 0.127 g/m2 for a broad range of diffusion coefficient values ([1 × 10−10 to 1 × 10−6 ] m2 /s). However, the optimal drying air temperatures and velocities in the drying modules are influenced (see Fig. 16(a and b) for three selected cases). As the diffusion coefficient decreases, the drying air temperatures and velocities increase. For larger values of diffusion coefficient the drying conditions are milder (see Fig. 16) as expected.

Table 7 Scenario 4: material and operation parameters Cp2 (kJ/kg K) Cpweb (kJ/kg K) D (m2 /s) M1 (kg/kmol) m10 (kg/m2 ) mweb (kg/m2 ) m2 (kg/m2 ) P (Pa) T0 (◦ C) Tkma (◦ C) x0 (kg/kg) xkma (kg/kg) ρ2 (kg/m3 )

Dry coating heat capacity Substrate heat capacity Diffusion coefficient Methanol molar mass Initial solvent specific mass Substrate specific mass Dry coating specific mass Total pressure of the dryer Initial temperature of the film Make-up air temperature Initial solvent content (dry basis) Solvent content in fresh air (dry basis) Dry coating density

1.465 1.6 1 × 10−9 32.042 0.1 6 × 10−2 2 × 10−2 1.013 × 105 20 20 5 0 1171

6.5. Scenario 4: maximization of the total flow of evaporated solvent This example considers a dryer with 6 modules (i.e. Nk = 6) of 4 m each moving with a web speed of 1 m/s with the same geometry and drying air operation limits as in the previous examples (see Table 1). In order to illustrate the proposed formulation with a different system, a film of methanol in polyvinyl acetate is considered to illustrate this drying scenario. The film enters the dryer with an initial solvent specific mass of 100 g/m2 (83% initial solvent mass fraction). The process data are presented in Table 7. In this case, note that no upper limit is imposed to the film temperature. The general properties and data for this example are presented in Table 7.

Fig. 16. Scenario 3: sensitivity analysis with respect to changes in the diffusion coefficient. (a) Optimal drying air temperatures and (b) optimal drying air velocities.

M. Barttfeld et al. / Computers and Chemical Engineering 30 (2006) 467–489

485

Fig. 17. Scenario 4: optimal solution.

This example considers the maximization of the total evaporated solvent mass flow. Note that this optimization scenario is closely related to the minimization of the final solvent mass (see scenario 3). However, the optimization formulation (objective function) is different, for this reason is illustrated separately from the previous numeric example. The optimization model is then solved minimizing Eq. (42) subject to the constraints (1)–(5), (7), (10)–(35) and Eqs. (43)–(57). The model involves 5883 variables and 5871 equations (see Table 8) and is solved in 23.63 CPU seconds. In Fig. 17(a), the film temperature profile is shown. Since no upper bound for the film temperature is imposed in the formulation, the film leaves the dryer with a temperature of 120 ◦ C which coincides with the upper bound of the drying air temperature. The optimal profiles for the solvent specific mass and solvent are shown in Fig. 17(b). Over the entire drying section 0.4912 kg/s of solvent are evaporated (optimal objective function value) and the film leaves the dryer with a final solvent mass 2 of mout 1k = 0.0129 g/m (k = 6) (see Table 8). Note that a logarithTable 8 Scenario 4: optimization model statistics Variables number Equations number

5883 5871

Objective function value (kg/m2 s) Time (CPU seconds)

0.4912 23.63

mic scale is considered in Fig. 17(b) as well as for the solvent evaporation rate which is shown in Fig. 17(c). In Fig. 17(d), the optimal operating conditions for the drying air are presented as well as the evaporated solvent at each module. In modules 1 and 2, the solvent evaporation is high compared with the last modules of the dryer, where the solvent evaporation rate is in the order of 0.1 kg/h. As a consequence of this fact, the heat consumption in the final modules is lower compared with the amount consumed at the beginning of the process. 6.6. Discussion of the drying scenarios In the framework of the physical model and the parameters used in the present paper for describing the drying process in modular air impingement dryers some trends can be extracted from the optimization results. The minimization of the heat consumption (defined as scenario 1) in some cases suggests a drying strategy which involves low drying air velocities and high temperatures. However, the illustration of scenario 1 (see Section 6.2), these conditions may not be sufficient to achieve the desired final solvent content in the film. For this reason, the drying air velocities have to be increased together with a decrement of the air temperature in order that the film temperature does not increase above its upper bound (if imposed). The results confirm the well-known fact that

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higher residence times (longer drying section) allow for milder drying air operating conditions. In the case of the production rate maximization, i.e. web speed maximization (scenario 2), the optimization results suggest to operate the dryer with the highest drying air velocity and temperatures. If the film temperature exceeds its upper limit, a decrease of the drying air temperature is chosen by the optimization algorithm to meet this constraint. A comparison of scenario 1 and scenario 2 revealed that the minimization of heat consumption (scenario 1) requires per unit mass of product only about 67% of the thermal energy of scenario 2. On the other hand, scenario 2 yields the same mass of coated product about 2.5 times faster than in scenario 1, bringing economical issues such as operating time or cost of labor into play, which would broaden the scope of the optimization task towards multi-objective optimization. As it was concluded by Price and Cairncross (2000), the minimization of the final solvent mass (scenario 3) does not necessarily take place at the highest drying air velocity and temperature. The optimization results show that the minimization of the final solvent mass is achieved at high temperature and velocities but not in the upper limits of these variables in all the drying modules. As expected, a similar trend is observed for the closely related scenario 4, the maximization of the total evaporation rate. 7. Conclusions This paper presented a formulation for the rigorous optimization of multiple-zone air impingement dryers based on a modular representation of the process. The mathematical representation of the problem allows a flexible implementation of the process operation limits, film quality-related constraints (e.g. risk of bubble formation) as well as different objective functions which give rise to the different drying scenarios. The drying air operating conditions are the degree of freedom of the optimization formulation, i.e. the decision variables. A simultaneous optimization approach (full discretization) is proposed to solve this problem. Then, all the variables involved in differential and algebraic equations are fully discretized and the PDE optimization model is converted into an NLP problem which was efficiently implemented in AMPL and solved with IPOPT. Numerical computations were presented to illustrate the performance and robustness of the proposed optimization model which was successfully solved for different drying scenarios with the systems water–polyvinyl alcohol and methanol–polyvinyl acetate. Moderate/low solution times were involved despite the large size of the formulations. Future developments of this optimization method will address more detailed formulations, e.g. taking into account a concentration and temperature-dependent diffusion coefficient. First results obtained for a diffusion model for water–PVA, however, indicate that the convergence behavior and the robustness of the algorithm will need improvement. The particular challenge in using concentration and temperature-dependent diffusion models for polymer solutions relies on the fact that the diffusion coefficient may vary over several orders of magnitude with concentration. For the Free-Volume-Theory model, cf., e.g. Neogi

(1986), the diffusion equation becomes a degenerate parabolic differential equation, since the diffusion coefficient vanishes as the concentration tends to zero. Steep concentration gradients occur near the film surface for such systems, causing a decrease of the drying rate (“skinning”, cf. Price & Cairncross, 2000). However, an extension of the optimization algorithm towards such diffusion models, to solve the steep concentration profiles and to handle the singularities, will be a formidable task and is beyond the scope of the present work. Initialization (preprocessing) strategies may be then required to achieve the convergence of the formulations. In this context, the solutions involving constant diffusion coefficient could be used as initial guesses of more complex formulations. In a further step, the optimal design of drying processes, i.e. to determine the number of nozzles, slot length of a nozzle and geometry of nozzles among others, can also be considered for a future outlook.

Acknowledgments The authors want to thank Prof. Lorentz T. Biegler (Carnegie Mellon University) who kindly provided to this research project the software AMPL and Dr. Andreas W¨achter who shared his open source NLP solver IPOPT. The authors are also very grateful to Dr. David Gay (developer of AMPL) for the support in this project and to Prof. H. Raszillier (University of ErlangenN¨urnberg) for the interesting discussions. The financial support from Bayerische Forschungsstiftung (BFS) is gratefully acknowledged.

Appendix A A.1. Geometric factors (Martin, 1977): fo = 

1 

d −2 60 + 4 2b fk =

2 (43)

nslot k b s

where d is the nozzle-coating distance (see Table 1).

A.2. Thermodynamic properties at the film–air interface The properties at the film interface are evaluated at temperasurface (average of the film temperature, T , and the drying ture Tk,j k,j air temperature, Tknozzle ) and at solvent partial pressure psurface 1k,j (average of the drying air solvent partial pressure, pnozzle , and 1k the solvent partial pressure in the film surface, p1k,j ): surface = 0.5(T nozzle ) Tk,j k,j + Tk,j

= 0.5(pnozzle + p1k,j ) psurface 1k,j k

 ,

∀k ∈ K, j ∈ Jk

(44)

M. Barttfeld et al. / Computers and Chemical Engineering 30 (2006) 467–489

The solvent, water and air molar fractions at the film–air interface are computed in Eq. (45):

surface = y1k,j

psurface 1k,j P

nozzle xwk nozzle + xnozzle M1 + M1 x1k wk Mw Mair surface surface ) = 1 − (y1k,j + ywk,j

surface = ywk,j

surface yairk,j

                

1995): top

∀k ∈ K, j ∈ Jk

P

top

2

top

Φ1k,j = 1 − Φ2k,j 1 top Φ2k,j = top 1 + εXk,j top

Xk,j = Xk,j,i ,

          

i = Nz

bot Φbot 1k,j = 1 − Φ2k,j 1 Φbot 2k,j = bot 1 + εXk,j bot Xk,j = Xk,j,i ,

,

∀k ∈ K, j = Jk

(48)

,

∀k ∈ K, j = Jk

(49)

    

Note that υair = 20.1, υ1 = 12.7 for water and υ1 = 29.9 for methanol. Interconnecting equations are included in the formulation to link the properties through adjacent drying modules:

i=1

     

For the system methanol–PVAc, the interaction parameter has been experimentally determined: χ12k,j = 1.76Φ2k,j − 0.42

surface surface = (ysurface M1 + ywk,j Mw + yairk,j Mair ) surface 1k,j RTk,j surface √M µ surface √M µ surface √M µ y1k,j 1 1k,j + ywk,j w wk,j + yairk,j air airk,j surface = µk,j √ √ √ surface surface surface y1k,j M1 + ywk,j Mw + yairk,j Mair surface surface surface surface λk,j = y1k,j λ1k,j + ywk,j λwk,j + yairk,j λairk,j surface M C surface surface y1k,j 1 p1k,j + ywk,j Mw Cpwk,j + yairk,j Mair Cpairk,j Cpsurface = surface M + ysurface M + ysurface M k,j y1k,j 1 w air wk,j airk,j  1.75  surface 10−7 Tk,j M1 + Mair δsurface =  2 k,j √ √ 3 3 υ M1 Mair air + υ1

surface = ρsurface  ρk,j k+1,jj     surface  µk,j = µsurface k+1,jj    surface    λsurface = λ k,j k+1,jj     Cpsurface = Cpsurface   k,j k+1,jj    surface = σ surface  σk,j k+1,jj

top

2

and the properties for the mixture vapor solvent and air are defined in (46) (Aust, 1996). The binary diffusion coefficient correlation for the vapor solvent in air, δsurface , was taken from k,j Fuller, Schettler, and Giddings (1966).

surface = T surface  Tk,j k+1,jj     surface  p1k,j = psurface 1k+1,jj     surface = ysurface   yk,j 1k+1,jj     surface surface  ywk,j = ywk+1,jj     surface = ysurface   yairk,j airk+1,jj

top

bot bot bot bot ln ak,j = ln Φbot 1k,j + Φ2k,j + χ12k,j (Φ2k,j )  

(45)

surface ρk,j

top

ln ak,j = ln Φ1k,j + Φ2k,j + χ12k,j (Φ2k,j )   top

,

487

                                        

,

∀k ∈ K, j ∈ Jk

(46)

(Saure & Schl¨under, 1995). For the system water–PVOH we have adopted the value determined by Zielinski and Duda (1991): χ12k,j = 0.393. Interconnecting Eq. (50) for the film top and bottom are included in the formulation to link the properties through the drying modules:  ak,j = ak+1,jj   Φ1k,j = Φ1k+1,jj , ∀k ∈ K, j = Ntk , jj = 1 (50)   Φ2k,j = Φ2k+1,jj A.4. General properties

,

∀k ∈ K, j = Ntk , jj = 1

(47)

A.3. Activity coefficient model The activity coefficient for the solvent is predicted with a simplified form of the Flory Huggins equation (Saure & Schl¨under,

The general temperature-dependent correlations for the pure components (water, methanol, water and air) are taken form Daubert and Danner (1985) and from Daubert (1989): A     T D

ρ(T ) = B

(51)

1+ 1− C

  B E Pv (T ) = exp A + + C ln(T ) + DT T   B+C T +D T 2 +E T 3  Tc Tc Tc T hvap (T ) = A 1 − Tc

(52)

(53)

M. Barttfeld et al. / Computers and Chemical Engineering 30 (2006) 467–489

488

A

B

C

D

E

2.3072 × 10−1 4.046 × 10−6 4.7678 × 10−1 1 × 10−3 7.724 × 10−1 4.4782 × 106 −4.670 × 103

– 2 0 0 – – –

1.7272 × 10−1 1.5281 × 10−2 8.4 × 10−2 0 1.1628 6.77 × 109 0

– 1

Water ρ(T) (kmol/m3 ) Pv (T) (Pa) hvap (T) (J/kmol) CpL (T ) (J/kmol K) CpV (T ) (J/kmol K) λ(T) (W/m K) µ(T) (Pa s)

4.6137 7.255 × 101 5.761 × 107 5.2634 × 104 3.3252 × 104 1.062 × 102 7.619 × 10−8

2.6214 × 10−1 −7.2067 × 103 6.964 × 10−1 2.4119 × 102 6.0104 × 104 −8.681 × 10−1 9.2758 × 10−1

6.4729 × 102 −7.1385 −7.797 × 10−1 −8.5085 × 10−1 4.1899 × 102 −2.7775 × 103 2.116 × 102

Methanol ρ(T) (kmol/m3 ) Pv (T) (Pa) hvap (T) (J/mol) CpL (T ) (J/mol K) CpV (T ) (J/mol K) λ(T) (W/m K) µ(T) (Pa s)

1.2057 1.0993 × 102 3.97 × 107 1.076 × 105 3.8188 × 104 −7.763 3.0663 × 10−7

1.9779 × 10−1 −7.471 × 103 −2.79 × 10−1 −3.806 × 102 1.0424 × 105 1.0279 6.9655 × 10−1

5.1263 × 102 −1.3988 × 101 5.08 × 10−1 9.79 × 10−1 2.1867 × 103 −7.436 × 107 2.050 × 102

1.026 × 104 4.883 × 10−1 5.023 × 10−1

2.19 × 104 2.565 × 102 1.080 × 102

Air CpV (T ) (J/kmol K) λ(T) (W/m K) µ(T) (Pa s)

2.8985 × 104 2.872 × 10−3 1.4373 × 10−6

CpL (T ) = A + BT + CT 2 + DT 3 + ET 4  CpV (T ) = A + B exp λ(T ) =

µ(T ) =

AT B 1+

C T

+

D T2

AT B 1+

C T

+

D T2

−C TD

(54)

(55)

(56)

(57)

References Alleborn, N. (2001a). Analyse der Str¨omung und des W¨arme - und Stofftransports in einer Trocknerkammer. Universit¨at Erlangen-N¨urnberg, Dissertation (Doctoral Thesis). Alleborn, N. (2001b). Progress in coating theory. Thermal Science, 5, 131. Alsoy, S. (1999). Modeling of multicomponent polymer drying processes in multiple zone industrial dryers. In F. Durst & H. Raszillier (Eds.), Advances in coating and drying of thin films: 3rd European coating symposium (pp. 393–398). Aachen: Shaker-Verlag. Aust, R. (1996). Verfahrenstechnische Aspekte der Trocknung beschichteter Materialbahnen. University Erlangen-N¨urnberg, Doctoral Thesis. Aust, R., Durst, F., & Raszillier, H. (1997). Modeling a multiple-zone air impingement dryer. Chemical Engineering and Processing, 36, 469. Benzi, M. (1991). Anlagen zur Herstellung selbstklebender B¨ander mit w¨assrigen Systemen. Coating, 24(4), 120. Biegler, L. T., & Grossmann, I. E. (2004). Retrospective on optimization. Computers & Chemical Engineering, 28(8), 1169. Biegler, L. T., Ghattas, O., Heinkenschloss, M., & van Bloemen Waanders, B. (2003). Large-scale PDE-constrained optimization. Berlin, Heidelberg: Springer-Verlag. Brandrup, J., & Immergut, E. H. (1975). Polymer handbook. (2nd ed.). New York.

1.4543 −6.49 × 103 0

0 – – – – – –

Cairncross, R. A. (2002). Residual solvent in drying coatings. Adhesives Age, 45(7), 37. Cairncross, R. A., Francis, L. F., & Scriven, L. E. (1996). Predicting drying in coatings that react and gel: Drying regime maps. AIChE Journal, 42, 55. Chilton, T. H., & Colburn, A. P. (1934). Mass transfer (absorption) coefficients. Industrial and Engineering Chemistry, 26(11), 1183. Cohen, E. D., & Gutoff, E. B. (1992). Modern coating and drying technology. New York. Daubert, T. E., & Danner, R. P. (1985). Data compilation tables of properties of pure compounds. New York. Daubert, T. E. (1989). Physical and thermodynamic properties of pure chemicals. New York. Fourer, R., Gay, D. M., & Kernighan, B. W. (2003). AMPL. A modeling language for mathematical programming. United Kingdom. Fuller, E., Schettler, P., & Giddings, J. (1966). A new method for the prediction of binary gas-phase diffusion coefficients. Industrial and Engineering Chemistry, 58, 19. Gehrmann, D. (1979). Theoretische und experimentelle Untersuchuungen zur konvektiven Trocknung d¨unner, ebener Gelatineschichten. TH Darmstadt, Dissertation (Doctoral Thesis). Guerrier, B., Bouchard, C., Allain, C., & Benard, C. (1998). Drying kinetics of polymer films. AIChE Journal, 44, 791. Krischer, O., Kast, W. (1978). Die wissenschaftlichen Grundlagen der Trocknungstechnik. Berlin. Martin, H. (1977). Heat and mass transfer rates between impingement gas jets and solid surfaces. Advances in Heat Transfer, 13, 1. Neogi, P. (1986). Diffusion in polymers. New York: Marcel Dekker Inc. Price, P. E., & Cairncross, R. A. (2000). Optimization of a single-zone drying of polymer solution coatings using mathematical modeling. Journal of Applied Polymer Science, 78, 149. Price, P. E., Wang, S., & Romdhane, I. H. (1997). Extracting effective diffusion parameters from drying experiments. AIChE Journal, 43, 1925. Saure, R., & Schl¨under, E. U. (1995). Sorption isotherms for methanol, benzene and ethanol on polyvinyl acetate (PVAc). Chemical Engineering and Processing, 34, 305. Saure, R. (1995). Zur Trocknung von l¨osungsmittelfeuchten Polymerfilmen. VDI Reihe 3, No. 407. D¨usseldorf. VDI-Verlag - Fortschr.-Ber.

M. Barttfeld et al. / Computers and Chemical Engineering 30 (2006) 467–489 Uzman, D., & Sahbaz, F. (2000). Drying kinetics of hydrated and gelatinited corn starches in the presence of sucrose and sodium chloride. Journal of Food Science, 65(1), 115. Wagner, G. R., Schabel, W., & Schl¨under, E.-U. (1999). Enhancing the drying rate of polymeric solvent coatings using a pre-loaded gas phase. In F. Durst & H. Raszillier (Eds.), Advances in coating and drying of thin films: 3rd European coating symposium (pp. 417–422). Shaker-Verlag: Aachen.

489

W¨achter, A., & Biegler, L. T. (2004). On the implementation of an interiorpoint filter line-search algorithm for large-scale nonlinear programming research report RC 23149. Yorktown, USA: IBM T.J. Watson Research Center. Zielinski, J., & Duda, J. (1991). Predicting polymer/solvent diffusion coefficients using free-volume theory. AIChE Journal, 38(3), 405.