Numerical Analysis of a Periodically Forced Dyeing Process

Numerical Analysis of a Periodically Forced Dyeing Process

Erasmo Mancusi,*,† Selene Guelli Ulson de Souza,‡ and Antonio Augusto Ulson de Souza‡

Facolta di Ingegneria, UniVersita del Sannio, Piazza Roma, 82100, BeneVento, Italy, and Departamento deEngenharia Quımica e Engenharia de Alimentos, UniVersidade Federal de Santa Catarina, Laboratorio deSimulacao Numerica de Sistemas Quımicos, LABSIN, Campus UniVersitario Cx. P. 476,88.040-900, Florianopolis (SC), Brazil

In this work, a forcing strategy based on a cyclic flow reversal of a dyeing process of bobbin threads isinvestigated through numerical simulation. Convection, dispersion, and adsorption of dye on the threads aremodeled 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 modelthat takes it into account is developed. The mathematical model is a set of partial differential equations thatis reduced to a system of ordinary differential equations by an orthogonal collocation on finite elementstechnique. The comparison between forced and unforced processes has been carried out analyzing the dyedistribution 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 onthe synthesis of new effective dyeing technologies. The mainobjectives of these studies have been to increase the productivityof the process, to improve the quality of the dyed product, andto reduce environmental pollution (see, for example, refs 1–5).These studies are largely empirical in nature and provide athorough analysis of the physical mechanisms responsible forthe characteristics of the dyed product.

The analysis of the dyeing process of threads bobbins is, ingeneral, complex (e.g., see refs 6 and 7). The dyeing process isintrinsically heterogeneous: Dye is transferred to the liquid phaseby convection and dispersion, and at the same time, the liquidphase exchanges dye with the solid phase (threads). Variousmodels have been proposed to describe the dyeing process and/or relate the operating conditions during a dyeing process tothe quality of the resulting dyed product. Some studies havefocused on the online control of dyeing processes.8–11 Othershave dealt with the physical chemistry of dye adsorption andtransport of dye into the threads, providing a comprehensiveinvestigation of the effect of various process parameters on dyedistribution and dye uptake.6,7,12–17

In the standard dyeing processes, the bobbins of threads arefixed to perforated supports and receive dye from the liquidcrossing the bobbins and recirculating to a mixing tank (e.g.,see ref 7). Under these conditions, the dye may not penetratecompletely into the fibers, and some areas may be left withoutdyeing even with large dye recirculation flow rates. As a result,unacceptable dye distributions may be achieved (e.g., see ref14). To overcome these difficulties, Burley et al.18 proposed toforce the dyeing process by adding and/or removing dye duringthe process. It must be remarked, in this context, that the analysisand design of periodically forced processes have been, over thepast decade, the subject of intense research. Many forcingstrategies have been proposed in this area, enabling researchersto significantly improve the performance of traditional processes

(see refs 19 and 20). Periodically forced conditions are typicallyrealized by temporally varying one or more input to the systemas, for instance, inlet concentration and temperature. Otherforcing strategies are realized by periodically inverting the flowdirection inside the system, leaving unchanged the inputvariables. This operation mode is commonly referred to asreverse flow operation21 and has been successfully used for thepurification of industrial exhaust off-gas streams22 and toimprove equilibrium-limited exothermic reactions such asmethanol synthesis.23 The capability of the reverse flow opera-tion to generate a great driving force between two phases hasalso been exploited in the area of adsorption process that ispressure swing adsorption (PSA) (see, for example, refs 24–27)and in vapor liquid tray separation process.28 PSA processeshave been suggested as an energy-saving process and as analternative to traditional separations, distillation, and absorptionfor bulk gas separations (e.g., air purification,29 propane/propylene separation,30 and hydrogen purification31).

In the present work, the effect of periodically inverting theflow in a traditional dyeing process is investigated. The transportof dye through the bobbins is described by a set of time-dependent partial differential equations, accounting for convec-tion, dispersion, and adsorption of dye. The effect of the periodicchange of the flow direction is described by a discontinuousperiodic function.32 Numerical simulations are performed withthe objective of analyzing the transport phenomena and evaluat-ing the feasibility of the proposed forcing strategy with respectto the classical unforced process. The dye distribution throughoutthe package, the rate of dye uptake, and the total amount ofadsorbed dye are analyzed to characterize the quality of the dyedproduct. Spatial profiles and time series are presented fordifferent values of the operating parameters to elucidate theeffect the cyclic reversal. The advantages of forced processcompared to unforced one are outlined.

The paper is organized as follows. In section 2, the math-ematical model of the unforced dyeing process is described,and an ad hoc discontinuous function is introduced to accountfor the effect of cyclic reversal of the flow direction. In section3, the performances of the periodically forced process are studiedby comparing the spatial profiles, the dye distribution throughoutthe bobbins, the rate of dye uptake, and the total amount of

* To whom correspondence should be addressed. Tel: +390824305587. Fax: +39 0824325246. E-mail: [email protected].

† Universita del Sannio.‡ Universidade Federal de Santa Catarina.

Ind. Eng. Chem. Res. 2010, 49, 8568–85748568

10.1021/ie9017012 2010 American Chemical SocietyPublished on Web 08/10/2010

adsorbed dye for forced and unforced operations. Final remarksend the paper.

2. Mathematical Model

We present in this section the mathematical model of theforced dyeing process. To this aim, in subsection 2.1, we discussa mathematical that describes a standard dye equipment (e.g.,refs 33 and 34), and then, the forcing function and the modelequations of forced process are reported in subsection 2.2.Finally, we briefly discuss in subsection 2.3 the adoptednumerical procedure.

2.1. Unforced Dyeing Process. A schematic representationof the dyeing process is reported in Figure 1 where the overalldye equipment and a section of the cylindrical bobbins threadsare 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 frommass balances on the solid phase (threads), the liquid phase(dyeing bath), and the mixing tank. The most importantassumptions are as follows:

1. The external dye bath is well mixed. Therefore, the liquidconcentration inside the bobbins is assumed to be uniformwith 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-adsorptionequation can be derived as shown in eq 1

A flow direction from the east to the west surface is hereassumed. The concentration of dye in the circulating liquid willbe referred to as the liquid concentration (C), while the amountof dye absorbed by the threads will be called the threadconcentration (q). All of the symbols are explained in theNomenclature section. For eq 1, the following boundaryconditions are assumed.

To accurately describe the dye distribution in the solid phase,intrafiber transport phenomena should be, in principle, consid-ered (e.g., ref 7). However, because it can be computationallyvery time-consuming, we here assume a uniform threadconcentration profile. Under this assumption, the mass balanceon threads can be described by a space-independent expressionfor the adsorption rate. A common approach is that given bythe linear driving force model (e.g., ref 25):

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

A mass balance for the mixing tank is formulated under theassumption of vigorous agitation to capture the temporalevolution of CM, as shown in the following equation:

Here, Q is the recirculation flow rate, V is the total volume ofthe mixing tank, and C|RI

is the inlet concentration of the mixingtank, namely, the concentration of dye in the liquid streamleaving 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). Theconcentration C|RI

in eq 6 is evaluated at the exit condition ofthe bobbins. The system of eqs 1-6 describes the unforceddyeing of the threads bobbins process, and the parameter valuesand the initial conditions used in the present work are reportedin Table 1.

2.2. Periodically Forced Dyeing Process. The valves systemsketched in Figure 1 allows the cyclic reversal of the flowdirection inside the bobbins. We refer, from now on, to theswitch 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.

∂C∂t

)Da

r∂

∂r(r∂C∂r ) + Vr

∂C∂r

- (1 - ε)ε

∂q∂t

(1)

r ) RI ⇒ ∂C∂r

) 0 (2)

r ) RE ⇒ C ) CM (3)

∂q∂t

) kt(q* - q) (4)

q* ) RC (5)

dCM

dt) Q

V (C|RI- CM) (6)

Ind. Eng. Chem. Res., Vol. 49, No. 18, 2010 8569

In particular, during the interval [kτ, (k + 1)τ[, the valves (B)are open, while the other couple (A) is closed and the liquidflows from the east surface to the west surface. In the timeinterval [(k + 1)τ, (k + 2)τ[, the valves (B) are closed, and thevalves (A) are open, and the liquid flows from the west surfaceto the east one.

From a mathematical point of view, the reverse operationmeans that, according to the flow direction, the inlet concentra-tion to the mixing tank is C|RI

when the liquid flows from westto east and is C|RE

when the flow direction is reversed (see Figure1). To take into account the periodic reverse of the flowdirection, we here consider a discontinuous periodic functiong(t):32

where int(x) denotes the integer part of the variable x. g(t) is adiscontinuous rectangular wave type function of unit amplitudeand 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 accountthe cyclic reversal of the flow direction. A change of the flowdirection inside the bobbins implies the reverse of the velocitydirection and the mirror change of the boundary conditions ateach switch time τ.

According to g(t), the boundary conditions and the massbalance on the mixing tank can be recast as follows:

Therefore, the liquid and solid mass balances (eqs 8 and 4), theboundary conditions (eq 9), and the balance on the mixing tank(eq 10) describe the periodically forced dyeing process that westudy in the present work.

2.3. Numerical Approach. The development of efficientnumerical methods for numerical simulation of the periodicprocesses is not a simple task (e.g., ref 36). In fact, because theprocess is periodically forced, the concentration profiles arecharacterized by step-moving fronts inside the bobbins. Toovercome this problem, we have developed a software simulatorexploring collocation methods on finite elements. Precisely, theapproach is to apply the orthogonal collocation on various fixedsubdomains (or finite elements) rather than on the whole domainof the space integration. This choice is also motivated by theneed to overcome the intrinsic problems of classical polynomialcollocation algorithms to handle a number of collocation pointslarger than 20 (see refs 37 and 38 and references therein).

To determine the minimum number of collocation points andspace elements required for the attainment of precise solutions,preliminary runs employing a different number of collocationspoints and space elements were performed, and the results werecompared. For the sake of simplicity, we only report that fourelements equally spaced with eight collocation points are enoughto the attainment of precise integration of the mathematicalmodel. Homemade software based on the robust and popularroutine VODE libraries39 for the time integration of ODEs hasbeen developed.

3. Results

The effect of the cyclic reversal operation on the performancesof dyeing process has been studied by comparing the dyedistribution factor (DDF) throughout the package, the totalamount of adsorbed dye, and the rate of dye uptake. Thedimensionless dye distribution factor (DDF) is the ratio betweenthe highest to the lowest thread concentration, and it providesa measure of the level of dyeing of the threads. For the unforceddyeing process, it is reasonable to assume that the DDFrepresents the ratio between the thread concentration valuesobserved at the inlet and exit flow points from the bobbins.12

On the contrary, for the reverse flow operation, the DDF is hereevaluated as the ratio between maxima and minima threadconcentrations. Because of the batch nature of the dyeingprocess, the total amount of adsorbed dye can be approximatedby computing at each time the dye concentration in the mixingtank. Finally, the rate of dye uptake can be estimated by timehistory of thread concentration vs time. More precisely, the anglebetween the tangent line of the thread concentration curve vstime and the time line coordinate represents the rate of dyeuptake.12,40 In the unforced process, the thread concentration ismeasured at the point where the flow leaves the bobbins.12

Because for the forced process, feed and exit positions areperiodically inverted, the rate of dye uptake is estimated by thetime history of the thread concentration in the middle of thebobbins.

Spatial profiles and time series are presented to elucidate theeffect of parameters on performances of the dyeing process. Inparticular, different values of the switch time (τ) and therecirculation flow rate (Q) have been considered. These twooperating parameters are very important and are key parameters.Indeed, the switch time is an intrinsic parameter of the periodicforcing, and it directly affects the efficiency of the periodicaction, whereas the recirculation flow rate strongly affects thetime needed for the liquid to cross the bobbins.

Finally, it is important to stress that all of the results discussedin the present work have been obtained by using the sameuniform initial conditions. In particular, the initial dye concen-tration in the liquid phase has been fixed to the same value ofthe external bath, while a zero initial concentration has beenassumed for the solid phase (see Table 1).

3.1. Thermodynamic Equilibrium between the Phases:Regime Profiles. Before we analyze the differences betweenthe forced and the unforced processes, we wish to stress thatfor both of the processes, the system is intrinsically batch.Therefore, we can observe a transient behavior as far as thetwo phases exchange dye. Then, when the equilibrium betweenthe two phases is reached, flat concentration profiles areobserved, and no further change occurs. Such an equilibriumregime only depends on the physics of the problem, and theway the process is forced does not change the concentrationsprofile reached after a long time (regime). Therefore, the regimeof a periodically forced dyeing process will be the same as the

Table 1. Parameter Values and Initial Conditions33,34

RE 0.0975 m RI 0.0385 mH 0.1475 m V 15 LDa 10-5 m2min-1 kt 2.9 × 10-2 min-1

ε 0.57 F 1170 kg m-3

R 562.32 CM(0) 0.15 g/LC(r, 0) 0.15 g/L q(r, 0) 0

g(t) ) int( tτ) - 2 int( t

2τ) (7)

∂C∂t

)Da

r∂

∂r(r∂C∂r ) + [1 - 2g(t)]Vr

∂C∂r

- (1 - ε)ε

∂q∂t

(8)

r ) RI ⇒ ∂C∂r

[1 - g(t)] + (C|RI- CM)g(t) ) 0

r ) RE ⇒ ∂C∂r

g(t) + (C|RE- CM)[1 - g(t)] ) 0

(9)

dCM

dt) Q

V {C|REg(t) + C|RI

[1 - g(t)] - CM} (10)

8570 Ind. Eng. Chem. Res., Vol. 49, No. 18, 2010

unforced process. On the other hand, we show in this sectionthat the reverse operation enables us to rapidly achieve a moreuniform dye distribution. This is of great practical relevance,since the process is stopped before reaching thermodynamicequilibrium conditions.

Figure 2 shows the mixing tank concentration (CM) historyfor different values of the switch time. The dye concentrationCM decreases, showing damped oscillations during the transient,and asymptotically tends to the unforced behavior representedwith a dotted line in Figure 2. The oscillating transient behaviorof CM is due to the reverse of the flow direction inside thebobbins, and the period of damped oscillations is equal to theswitch time τ. Independently on the switch time values, weobserve in Figure 2 that the CM rapidly decreases in the firsttime interval (∼[0, 15 min]) due to the adsorption of dye bythreads. After (approximately) 3 h, no more changes in themixing tank can be observed (CM ≈ 10-3 g/L).

On the basis of the results reported in Figure 2, it is possibleto argue that the total amount of dye adsorbed by the threads isinvariably the same for the forced and the unforced processes.Therefore, the reverse operation does not affect the total amountof dye adsorbed by the threads as compared to the classicalunforced operation.

In all cases here discussed, the great part of the dye isadsorbed in the first 15 min, but at the same time, no informationis given about the dye distribution. This information is reportedin Figure 3 where the DDF time history for three different switchtimes and for the unforced process is reported. During the firsttime interval [0, τ], the DDF quickly increases for the unforcedprocess (dotted line). Indeed, at the beginning, the threadconcentration near the inlet flow surface rapidly increases, sincethis volume of threads receives dye directly from the mixingtank. On the contrary, the most internal threads receive dyetransferred by convection/dispersion mass transport. The greatdifference between the dye concentration close to the inlet andthose close to the outlet gives rise to a rapidly increasing DDF.As the process runs, the adsorbed dye is spread along all of thebobbin threads, and the DDF decreases, asymptotically reachingthe unit value that corresponds to a perfect dye distribution.For the forced process, the DDF starts to decrease after the firstswitch and approaches the asymptotic value of uniform distribu-tion faster than the unforced process. In fact, after each switch,the bobbin surface that is fed by “fresh” dye changes. In otherwords, the internal threads do not have to wait for the convectiveand dispersive mass transport to receive liquid rich of dye. After

30 min, the percent difference between the DDF for the unforcedprocess and the forced one is of 52% for τ ) 1 min, 34% forτ ) 5 min, and 25% for τ ) 10 min, respectively. After 1 h,the percent difference becomes 35% for τ ) 1 min, 23% forτ ) 5 min, and 13% for τ ) 10 min, respectively. We have towait 90 min to have percent differences between forced andunforced processes less than 5%.

Moreover, the effect of reverse flow operation is enhancedas the switch time approaches the recirculation characteristictime (approximately 1 min for the recirculation flow ratechosen). For large switch time values, the benefit of the cyclicreversal of the flow direction is lost in spite of the naturalrecirculation imposed by the process.

A better insight into the dye distribution can be obtained byanalyzing the spatial profiles of liquid and thread concentrationsreported in Figure 4. While the liquid concentration rapidlydecreases for the unforced process (Figure 4a), the cyclicreversal of the feed position means that both the internal andthe external surfaces receive periodically large dye concentra-tions. Therefore, a larger liquid concentration close to theinternal surface of the bobbins is obtained. As a result, theunadsorbed dye is better distributed across the bobbins as shownby the wavy shape of the spatial profile reported in Figure 4b.

For the unforced process, the spatial profiles of the threadconcentration exhibit a maximum during the first time instanceclose to the feed surface of the bobbins (see Figure 4c). As theprocess runs, this maximum value decreases and moves insidethe bobbins. The dye is transferred to the internal threads onlyby convection/dispersion mass transport. Therefore, the threadconcentration in the most internal threads slowly increases asshown in Figure 4c. The external thread volumes work as lungsfor the dye. In the first time interval, this volume captures thedye initially present in the liquid, and then, like a lung, suchvolume releases the dye that is transferred by convection anddispersion to the more internal threads. When the process isforced, the internal threads receive periodically fresh dye, andthe thread concentration “symmetrically” increases as repre-sented in Figure 4d.

From the analysis of the results reported in Figure 4, it ispossible to state that the reverse flow operation allows a greatermass flux between the liquid and the solid phases. In fact, whilefor the unforced process (Figure 5a) the mass flux rapidlydecreases, reaching an almost flat profile, the wavy shape (Figure5b) for the forced process indicates a mass flux well distributed

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

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

Ind. Eng. Chem. Res., Vol. 49, No. 18, 2010 8571

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.

8572 Ind. Eng. Chem. Res., Vol. 49, No. 18, 2010

over all volumes of the threads. Therefore, the reverse flowoperation allows for a better distribution of the driving forcebetween the two phases, leading to a more uniform threadconcentration profile.

In Figure 6, the thread concentration in the center of thebobbins for the periodically forced process and at r ) RI forunforced are reported. It is apparent that the rate of dye uptakeof the periodically forced process is slightly greater than thedye uptake of the unforced one. Moreover, it is important tostress that the rate of dye uptake is not affected by the switchtime. In fact, after a quick increase, the thread concentrationincreases with a slope (that is the rate of dye uptake) that canbe reasonably considered independent by the switch time values.

In Figure 7, the dye distribution factor for a greater value ofthe recirculation flow rate is reported. Increasing the recirculationflow rate value, the recirculation time decreases. Thus, it isnecessary to increase the forcing frequency to benefit the forcingoperation. Indeed, the DDF factor for a forced process whenthe switch time is 5 min or greater is very similar to those ofan unforced process. On the other side, with a reasonable switchtime of magnitude of 1 min, the favorable effects of the cyclicreversal of the flow direction can be noticeable, as is apparentin Figure 7. After 30 min, the percent difference between theunforced process and the forced one is 30% for τ ) 30 s and 10%

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

4. Conclusions

In this work, we have presented and studied a dyeing strategybased on cyclic reversal of the flow direction inside the bobbins.By means of mathematical modeling, we show how a periodi-cally forced flow in a dyeing process enhances a more evendistribution of the dye relative to the unforced process. In theclassical dyeing process, just one side of the thread bobbin isfed by fresh dye, which has to be transported by convectionand dispersion to the internal core of the threads. The cyclicreversal of the flow direction reduces this problem. In fact, whenthe flow direction is reversed, the internal core of the bobbinsimmediately receives fresh dye. The main effect of the flowreversal operation is to guarantee a driving force greater andbetter distributed along the threads volume and thus a betterdye distribution. This effect is well explained by comparing thedye distribution factor and spatial profile for classical andperiodically forced dyeing strategy. By an appropriate controlof the switch time, the periodically forced process could allowa dye distribution factor 20-50% lower than the DDF of theunforced process. Finally, we have shown that the bestperformance of the reverse flow operation is obtained when theswitch time magnitude is comparable to the time scale oftransport phenomena mainly with the convection time.

Nomenclature

C ) liquid concentration of dyeCM ) mixing tank concentrationDA ) dispersion coefficientDDF ) dye distribution factorg(t) ) forcing function defined in eq 7H ) height of the bobbinsk ) integer numberkt ) global mass transfer coefficientq ) thread concentrationq* ) equilibrium thread concentrationQ ) flow ratet ) timeV ) volume of dyeing bath in the mixing

Greek Letters

R ) equilibrium constantε ) bobbins void fractionF ) fiber densityVr ) interstitial fluid velocity in radial directionτ ) switch time

Subscripts and Superscripts

M ) mixing tankRE ) external bobbins radiusRI ) internal bobbins radius

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Ind. Eng. Chem. Res., Vol. 49, No. 18, 2010 8573

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

Accepted July 24, 2010

IE9017012

8574 Ind. Eng. Chem. Res., Vol. 49, No. 18, 2010