Numerical Analysis of a Periodically Forced Dyeing Process

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Numerical Analysis of a Periodically Forced Dyeing Process Erasmo Mancusi,*,† Selene Guelli Ulson de Souza,‡ and Antoˆnio Augusto Ulson de Souza‡ Facolta` di Ingegneria, UniVersita` del Sannio, Piazza Roma, 82100, BeneVento, Italy, and Departamento de Engenharia Quı´mica e Engenharia de Alimentos, UniVersidade Federal de Santa Catarina, Laborato´rio de Simulaçaõ Nume´rica de Sistemas Quı´micos, LABSIN, Campus UniVersita´rio Cx. P. 476, 88.040-900, Floriano´polis (SC), Brazil

In this work, a forcing strategy based on a cyclic flow reversal of a dyeing process of bobbin threads is investigated through numerical simulation. Convection, dispersion, and adsorption of dye on the threads are modeled considering that the system is operated by a cyclic reversal of the flow direction through the bobbins. The periodic forcing is modeled by an ad hoc discontinuous periodic function, and a mathematical model that takes it into account is developed. The mathematical model is a set of partial differential equations that is reduced to a system of ordinary differential equations by an orthogonal collocation on finite elements technique. The comparison between forced and unforced processes has been carried out analyzing the dye distribution factor and the total amount of adsorbed dye during the transient regime for both of the processes. The main effect of periodic forcing is to enhance a more even dye distribution. 1. Introduction As result of the development of new dyes and textile fibers, a large number of studies have been carried out focusing on the synthesis of new effective dyeing technologies. The main objectives of these studies have been to increase the productivity of the process, to improve the quality of the dyed product, and to reduce environmental pollution (see, for example, refs 1–5). These studies are largely empirical in nature and provide a thorough analysis of the physical mechanisms responsible for the characteristics of the dyed product. The analysis of the dyeing process of threads bobbins is, in general, complex (e.g., see refs 6 and 7). The dyeing process is intrinsically heterogeneous: Dye is transferred to the liquid phase by convection and dispersion, and at the same time, the liquid phase exchanges dye with the solid phase (threads). Various models have been proposed to describe the dyeing process and/ or relate the operating conditions during a dyeing process to the quality of the resulting dyed product. Some studies have focused on the online control of dyeing processes.8–11 Others have dealt with the physical chemistry of dye adsorption and transport of dye into the threads, providing a comprehensive investigation of the effect of various process parameters on dye distribution and dye uptake.6,7,12–17 In the standard dyeing processes, the bobbins of threads are fixed to perforated supports and receive dye from the liquid crossing the bobbins and recirculating to a mixing tank (e.g., see ref 7). Under these conditions, the dye may not penetrate completely into the fibers, and some areas may be left without dyeing even with large dye recirculation flow rates. As a result, unacceptable dye distributions may be achieved (e.g., see ref 14). To overcome these difficulties, Burley et al.18 proposed to force the dyeing process by adding and/or removing dye during the process. It must be remarked, in this context, that the analysis and design of periodically forced processes have been, over the past decade, the subject of intense research. Many forcing strategies have been proposed in this area, enabling researchers to significantly improve the performance of traditional processes * To whom correspondence should be addressed. Tel: +39 0824305587. Fax: +39 0824325246. E-mail: [email protected]. † Universita` del Sannio. ‡ Universidade Federal de Santa Catarina.

(see refs 19 and 20). Periodically forced conditions are typically realized by temporally varying one or more input to the system as, for instance, inlet concentration and temperature. Other forcing strategies are realized by periodically inverting the flow direction inside the system, leaving unchanged the input variables. This operation mode is commonly referred to as reverse flow operation21 and has been successfully used for the purification of industrial exhaust off-gas streams22 and to improve equilibrium-limited exothermic reactions such as methanol synthesis.23 The capability of the reverse flow operation to generate a great driving force between two phases has also been exploited in the area of adsorption process that is pressure swing adsorption (PSA) (see, for example, refs 24–27) and in vapor liquid tray separation process.28 PSA processes have been suggested as an energy-saving process and as an alternative to traditional separations, distillation, and absorption for bulk gas separations (e.g., air purification,29 propane/ propylene separation,30 and hydrogen purification31). In the present work, the effect of periodically inverting the flow in a traditional dyeing process is investigated. The transport of dye through the bobbins is described by a set of timedependent partial differential equations, accounting for convection, dispersion, and adsorption of dye. The effect of the periodic change of the flow direction is described by a discontinuous periodic function.32 Numerical simulations are performed with the objective of analyzing the transport phenomena and evaluating the feasibility of the proposed forcing strategy with respect to the classical unforced process. The dye distribution throughout the package, the rate of dye uptake, and the total amount of adsorbed dye are analyzed to characterize the quality of the dyed product. Spatial profiles and time series are presented for different values of the operating parameters to elucidate the effect the cyclic reversal. The advantages of forced process compared to unforced one are outlined. The paper is organized as follows. In section 2, the mathematical model of the unforced dyeing process is described, and an ad hoc discontinuous function is introduced to account for the effect of cyclic reversal of the flow direction. In section 3, the performances of the periodically forced process are studied by comparing the spatial profiles, the dye distribution throughout the bobbins, the rate of dye uptake, and the total amount of

10.1021/ie9017012  2010 American Chemical Society Published on Web 08/10/2010

Ind. Eng. Chem. Res., Vol. 49, No. 18, 2010

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adsorbed dye for forced and unforced operations. Final remarks end the paper.

r ) RI ⇒

∂C )0 ∂r

(2)

2. Mathematical Model

r ) RE ⇒ C ) CM

(3)

We present in this section the mathematical model of the forced dyeing process. To this aim, in subsection 2.1, we discuss a mathematical that describes a standard dye equipment (e.g., refs 33 and 34), and then, the forcing function and the model equations of forced process are reported in subsection 2.2. Finally, we briefly discuss in subsection 2.3 the adopted numerical procedure. 2.1. Unforced Dyeing Process. A schematic representation of the dyeing process is reported in Figure 1 where the overall dye equipment and a section of the cylindrical bobbins threads are sketched. Because of the natural symmetry of the problem, only half of the bobbin is here considered (see Figure 1). The mathematical model studied in this work results from mass balances on the solid phase (threads), the liquid phase (dyeing bath), and the mixing tank. The most important assumptions are as follows: 1. The external dye bath is well mixed. Therefore, the liquid concentration inside the bobbins is assumed to be uniform with respect to the height coordinate. 2. The process is isothermal. 3. The bobbin porosity is constant. 4. Dispersion and equilibrium constants are constant. 5. A linear adsorption isotherm is considered to describe the adsorption process.35 6. The time delay inside the pipe is negligible. The mass balance in the radial coordinate for transport in the liquid phase based on a convection-dispersion-adsorption equation can be derived as shown in eq 1 Da ∂ ∂C ∂C (1 - ε) ∂q ∂C ) r + Vr ∂t r ∂r ∂r ∂r ε ∂t

( )

(1)

A flow direction from the east to the west surface is here assumed. The concentration of dye in the circulating liquid will be referred to as the liquid concentration (C), while the amount of dye absorbed by the threads will be called the thread concentration (q). All of the symbols are explained in the Nomenclature section. For eq 1, the following boundary conditions are assumed.

To accurately describe the dye distribution in the solid phase, intrafiber transport phenomena should be, in principle, considered (e.g., ref 7). However, because it can be computationally very time-consuming, we here assume a uniform thread concentration profile. Under this assumption, the mass balance on threads can be described by a space-independent expression for the adsorption rate. A common approach is that given by the linear driving force model (e.g., ref 25): ∂q ) kt(q* - q) ∂t

(4)

Here, kt is the effective mass transfer coefficient, while q* is the equilibrium thread concentration that is related to the liquid concentration phase by the so-called adsorption isotherm. Following the experimental results of Revello et al.,35 the q* can be expressed as follows: q* ) RC

(5)

A mass balance for the mixing tank is formulated under the assumption of vigorous agitation to capture the temporal evolution of CM, as shown in the following equation: dCM Q ) (C| RI - CM) dt V

(6)

Here, Q is the recirculation flow rate, V is the total volume of the mixing tank, and C|RI is the inlet concentration of the mixing tank, namely, the concentration of dye in the liquid stream leaving the bobbins (see Figure 1). The solution of eq 6 provides the time dependence of CM, which is required in the boundary condition (eq 3). The concentration C|RI in eq 6 is evaluated at the exit condition of the bobbins. The system of eqs 1-6 describes the unforced dyeing of the threads bobbins process, and the parameter values and the initial conditions used in the present work are reported in Table 1. 2.2. Periodically Forced Dyeing Process. The valves system sketched in Figure 1 allows the cyclic reversal of the flow direction inside the bobbins. We refer, from now on, to the switch time τ as the time at which the flow direction is reversed.

Figure 1. Simplified scheme of the equipment for the dyeing process and a schematic representation of the bobbins.

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Table 1. Parameter Values and Initial Conditions33,34 RE H Da ε R C(r, 0)

0.0975 m 0.1475 m 10-5 m2min-1 0.57 562.32 0.15 g/L

RI V kt F CM(0) q(r, 0)

0.0385 m 15 L 2.9 × 10-2 min-1 1170 kg m-3 0.15 g/L 0

In particular, during the interval [kτ, (k + 1)τ[, the valves (B) are open, while the other couple (A) is closed and the liquid flows from the east surface to the west surface. In the time interval [(k + 1)τ, (k + 2)τ[, the valves (B) are closed, and the valves (A) are open, and the liquid flows from the west surface to the east one. From a mathematical point of view, the reverse operation means that, according to the flow direction, the inlet concentration to the mixing tank is C|RI when the liquid flows from west to east and is C|RE when the flow direction is reversed (see Figure 1). To take into account the periodic reverse of the flow direction, we here consider a discontinuous periodic function g(t):32 g(t) ) int

( τt ) - 2 int( 2τt )

(7)

where int(x) denotes the integer part of the variable x. g(t) is a discontinuous rectangular wave type function of unit amplitude and is periodic with a minimum period T ) 2τ. According to the introduction of the periodic function g(t), we can reformulate the mathematical model to take into account the cyclic reversal of the flow direction. A change of the flow direction inside the bobbins implies the reverse of the velocity direction and the mirror change of the boundary conditions at each switch time τ. Da ∂ ∂C ∂C (1 - ε) ∂q ∂C ) r + [1 - 2g(t)]Vr ∂t r ∂r ∂r ∂r ε ∂t

( )

(8) According to g(t), the boundary conditions and the mass balance on the mixing tank can be recast as follows: ∂C [1 - g(t)] + (C| RI - CM)g(t) ) 0 ∂r ∂C g(t) + (C| RE - CM)[1 - g(t)] ) 0 r ) RE ⇒ ∂r

r ) RI ⇒

dCM Q ) {C| REg(t) + C| RI[1 - g(t)] - CM} dt V

(9)

(10)

Therefore, the liquid and solid mass balances (eqs 8 and 4), the boundary conditions (eq 9), and the balance on the mixing tank (eq 10) describe the periodically forced dyeing process that we study in the present work. 2.3. Numerical Approach. The development of efficient numerical methods for numerical simulation of the periodic processes is not a simple task (e.g., ref 36). In fact, because the process is periodically forced, the concentration profiles are characterized by step-moving fronts inside the bobbins. To overcome this problem, we have developed a software simulator exploring collocation methods on finite elements. Precisely, the approach is to apply the orthogonal collocation on various fixed subdomains (or finite elements) rather than on the whole domain of the space integration. This choice is also motivated by the need to overcome the intrinsic problems of classical polynomial collocation algorithms to handle a number of collocation points larger than 20 (see refs 37 and 38 and references therein).

To determine the minimum number of collocation points and space elements required for the attainment of precise solutions, preliminary runs employing a different number of collocations points and space elements were performed, and the results were compared. For the sake of simplicity, we only report that four elements equally spaced with eight collocation points are enough to the attainment of precise integration of the mathematical model. Homemade software based on the robust and popular routine VODE libraries39 for the time integration of ODEs has been developed. 3. Results The effect of the cyclic reversal operation on the performances of dyeing process has been studied by comparing the dye distribution factor (DDF) throughout the package, the total amount of adsorbed dye, and the rate of dye uptake. The dimensionless dye distribution factor (DDF) is the ratio between the highest to the lowest thread concentration, and it provides a measure of the level of dyeing of the threads. For the unforced dyeing process, it is reasonable to assume that the DDF represents the ratio between the thread concentration values observed at the inlet and exit flow points from the bobbins.12 On the contrary, for the reverse flow operation, the DDF is here evaluated as the ratio between maxima and minima thread concentrations. Because of the batch nature of the dyeing process, the total amount of adsorbed dye can be approximated by computing at each time the dye concentration in the mixing tank. Finally, the rate of dye uptake can be estimated by time history of thread concentration vs time. More precisely, the angle between the tangent line of the thread concentration curve vs time and the time line coordinate represents the rate of dye uptake.12,40 In the unforced process, the thread concentration is measured at the point where the flow leaves the bobbins.12 Because for the forced process, feed and exit positions are periodically inverted, the rate of dye uptake is estimated by the time history of the thread concentration in the middle of the bobbins. Spatial profiles and time series are presented to elucidate the effect of parameters on performances of the dyeing process. In particular, different values of the switch time (τ) and the recirculation flow rate (Q) have been considered. These two operating parameters are very important and are key parameters. Indeed, the switch time is an intrinsic parameter of the periodic forcing, and it directly affects the efficiency of the periodic action, whereas the recirculation flow rate strongly affects the time needed for the liquid to cross the bobbins. Finally, it is important to stress that all of the results discussed in the present work have been obtained by using the same uniform initial conditions. In particular, the initial dye concentration in the liquid phase has been fixed to the same value of the external bath, while a zero initial concentration has been assumed for the solid phase (see Table 1). 3.1. Thermodynamic Equilibrium between the Phases: Regime Profiles. Before we analyze the differences between the forced and the unforced processes, we wish to stress that for both of the processes, the system is intrinsically batch. Therefore, we can observe a transient behavior as far as the two phases exchange dye. Then, when the equilibrium between the two phases is reached, flat concentration profiles are observed, and no further change occurs. Such an equilibrium regime only depends on the physics of the problem, and the way the process is forced does not change the concentrations profile reached after a long time (regime). Therefore, the regime of a periodically forced dyeing process will be the same as the


Figure 2. Time series of mixing tank concentration for different values of the switch time. The recirculation flow rate used is Q ) 1 L s-1, while the other parameters are fixed to the values reported in Table 1.

unforced process. On the other hand, we show in this section that the reverse operation enables us to rapidly achieve a more uniform dye distribution. This is of great practical relevance, since the process is stopped before reaching thermodynamic equilibrium conditions. Figure 2 shows the mixing tank concentration (CM) history for different values of the switch time. The dye concentration CM decreases, showing damped oscillations during the transient, and asymptotically tends to the unforced behavior represented with a dotted line in Figure 2. The oscillating transient behavior of CM is due to the reverse of the flow direction inside the bobbins, and the period of damped oscillations is equal to the switch time τ. Independently on the switch time values, we observe in Figure 2 that the CM rapidly decreases in the first time interval (∼[0, 15 min]) due to the adsorption of dye by threads. After (approximately) 3 h, no more changes in the mixing tank can be observed (CM ≈ 10-3 g/L). On the basis of the results reported in Figure 2, it is possible to argue that the total amount of dye adsorbed by the threads is invariably the same for the forced and the unforced processes. Therefore, the reverse operation does not affect the total amount of dye adsorbed by the threads as compared to the classical unforced operation. In all cases here discussed, the great part of the dye is adsorbed in the first 15 min, but at the same time, no information is given about the dye distribution. This information is reported in Figure 3 where the DDF time history for three different switch times and for the unforced process is reported. During the first time interval [0, τ], the DDF quickly increases for the unforced process (dotted line). Indeed, at the beginning, the thread concentration near the inlet flow surface rapidly increases, since this volume of threads receives dye directly from the mixing tank. On the contrary, the most internal threads receive dye transferred by convection/dispersion mass transport. The great difference between the dye concentration close to the inlet and those close to the outlet gives rise to a rapidly increasing DDF. As the process runs, the adsorbed dye is spread along all of the bobbin threads, and the DDF decreases, asymptotically reaching the unit value that corresponds to a perfect dye distribution. For the forced process, the DDF starts to decrease after the first switch and approaches the asymptotic value of uniform distribution faster than the unforced process. In fact, after each switch, the bobbin surface that is fed by “fresh” dye changes. In other words, the internal threads do not have to wait for the convective and dispersive mass transport to receive liquid rich of dye. After

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Figure 3. DDF vs time for different values of the switch time. The recirculation flow rate used is Q ) 1 L s-1, while the other parameters are fixed to the values reported in Table 1.

30 min, the percent difference between the DDF for the unforced process and the forced one is of 52% for τ ) 1 min, 34% for τ ) 5 min, and 25% for τ ) 10 min, respectively. After 1 h, the percent difference becomes 35% for τ ) 1 min, 23% for τ ) 5 min, and 13% for τ ) 10 min, respectively. We have to wait 90 min to have percent differences between forced and unforced processes less than 5%. Moreover, the effect of reverse flow operation is enhanced as the switch time approaches the recirculation characteristic time (approximately 1 min for the recirculation flow rate chosen). For large switch time values, the benefit of the cyclic reversal of the flow direction is lost in spite of the natural recirculation imposed by the process. A better insight into the dye distribution can be obtained by analyzing the spatial profiles of liquid and thread concentrations reported in Figure 4. While the liquid concentration rapidly decreases for the unforced process (Figure 4a), the cyclic reversal of the feed position means that both the internal and the external surfaces receive periodically large dye concentrations. Therefore, a larger liquid concentration close to the internal surface of the bobbins is obtained. As a result, the unadsorbed dye is better distributed across the bobbins as shown by the wavy shape of the spatial profile reported in Figure 4b. For the unforced process, the spatial profiles of the thread concentration exhibit a maximum during the first time instance close to the feed surface of the bobbins (see Figure 4c). As the process runs, this maximum value decreases and moves inside the bobbins. The dye is transferred to the internal threads only by convection/dispersion mass transport. Therefore, the thread concentration in the most internal threads slowly increases as shown in Figure 4c. The external thread volumes work as lungs for the dye. In the first time interval, this volume captures the dye initially present in the liquid, and then, like a lung, such volume releases the dye that is transferred by convection and dispersion to the more internal threads. When the process is forced, the internal threads receive periodically fresh dye, and the thread concentration “symmetrically” increases as represented in Figure 4d. From the analysis of the results reported in Figure 4, it is possible to state that the reverse flow operation allows a greater mass flux between the liquid and the solid phases. In fact, while for the unforced process (Figure 5a) the mass flux rapidly decreases, reaching an almost flat profile, the wavy shape (Figure 5b) for the forced process indicates a mass flux well distributed

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Figure 4. Spatial profiles of liquid and thread concentrations in the unforced (a and c) and forced (b and d) dyeing process. For τ ) 1 min and Q ) 1 L s-1, while the other parameters are fixed to the values reported in Table 1.

Figure 5. Spatial profiles of the driving force between the two phases in the unforced (a) and forced (b) dyeing process. For τ ) 1 min and Q ) 1 L s-1, while the other parameters are fixed to the values reported in Table 1.


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for τ ) 1 min, respectively. After 1 h, the percent difference between the unforced process and the forced one is 10% for τ ) 30 s and 7% for τ ) 1 min, respectively. Practically, it is necessary to wait more than 1 h, to have percent differences between forced and unforced processes less than 5%. 4. Conclusions

Figure 6. Dye uptake vs time for different values of the switch time and for Q ) 1 L s-1, while the other parameters are fixed to the values reported in Table 1.

In this work, we have presented and studied a dyeing strategy based on cyclic reversal of the flow direction inside the bobbins. By means of mathematical modeling, we show how a periodically forced flow in a dyeing process enhances a more even distribution of the dye relative to the unforced process. In the classical dyeing process, just one side of the thread bobbin is fed by fresh dye, which has to be transported by convection and dispersion to the internal core of the threads. The cyclic reversal of the flow direction reduces this problem. In fact, when the flow direction is reversed, the internal core of the bobbins immediately receives fresh dye. The main effect of the flow reversal operation is to guarantee a driving force greater and better distributed along the threads volume and thus a better dye distribution. This effect is well explained by comparing the dye distribution factor and spatial profile for classical and periodically forced dyeing strategy. By an appropriate control of the switch time, the periodically forced process could allow a dye distribution factor 20-50% lower than the DDF of the unforced process. Finally, we have shown that the best performance of the reverse flow operation is obtained when the switch time magnitude is comparable to the time scale of transport phenomena mainly with the convection time. Nomenclature

Figure 7. DDF vs time for different values of the switch time and for Q ) 10 L s-1, while the other parameters are fixed to the values reported in Table 1.

over all volumes of the threads. Therefore, the reverse flow operation allows for a better distribution of the driving force between the two phases, leading to a more uniform thread concentration profile. In Figure 6, the thread concentration in the center of the bobbins for the periodically forced process and at r ) RI for unforced are reported. It is apparent that the rate of dye uptake of the periodically forced process is slightly greater than the dye uptake of the unforced one. Moreover, it is important to stress that the rate of dye uptake is not affected by the switch time. In fact, after a quick increase, the thread concentration increases with a slope (that is the rate of dye uptake) that can be reasonably considered independent by the switch time values. In Figure 7, the dye distribution factor for a greater value of the recirculation flow rate is reported. Increasing the recirculation flow rate value, the recirculation time decreases. Thus, it is necessary to increase the forcing frequency to benefit the forcing operation. Indeed, the DDF factor for a forced process when the switch time is 5 min or greater is very similar to those of an unforced process. On the other side, with a reasonable switch time of magnitude of 1 min, the favorable effects of the cyclic reversal of the flow direction can be noticeable, as is apparent in Figure 7. After 30 min, the percent difference between the unforced process and the forced one is 30% for τ ) 30 s and 10%

C ) liquid concentration of dye CM ) mixing tank concentration DA ) dispersion coefficient DDF ) dye distribution factor g(t) ) forcing function defined in eq 7 H ) height of the bobbins k ) integer number kt ) global mass transfer coefficient q ) thread concentration q* ) equilibrium thread concentration Q ) flow rate t ) time V ) volume of dyeing bath in the mixing Greek Letters R ) equilibrium constant ε ) bobbins void fraction F ) fiber density Vr ) interstitial fluid velocity in radial direction τ ) switch time Subscripts and Superscripts M ) mixing tank RE ) external bobbins radius RI ) internal bobbins radius

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ReceiVed for reView October 29, 2009 ReVised manuscript receiVed July 5, 2010 Accepted July 24, 2010 IE9017012

Numerical Analysis of a Periodically Forced Dyeing Process

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