This article was downloaded by: [University of Connecticut]On: 11 October 2014, At: 11:28Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

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TWO-SIDED DRYING OF COATINGS ON POROUS WEBSEDGAR B. GUTOFF aa Consulting Chemical Engineer , 194 Clark Rd, Brookline, MA, 02445-5848, U.S.A.Published online: 27 Apr 2007.

To cite this article: EDGAR B. GUTOFF (1999) TWO-SIDED DRYING OF COATINGS ON POROUS WEBS, Drying Technology: AnInternational Journal, 17:4-5, 691-709, DOI: 10.1080/07373939908917564

To link to this article: http://dx.doi.org/10.1080/07373939908917564

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DRYING TECHNOLOGY, 17(4&5), 691-709 (1999)

TWO-SIDED DRYING OF COATINGS ON POROUS WEBS

Edgar B. Gutoff Consulting Chemical Engineer

194 Clark Rd., Brookline, MA 02445-5848, U.S.A.

Key Words and Phrases: continuous webs, constant rate, falling rate, porous webs

ABSTRACT

A relatively simple spreadsheet-based model of the drying of coated webs has been extended to cover coatings on porous webs. Two cases have been modeled. In one the coating is assumed to fully saturate the porous web, with the drying being completely symmetrical about the center plane except for differences in the air velocities and temperatures. The drying rate in the constant rate period is then increased, but because of the greater thickness that the coating now occupies, diffusion is much slower and the falling rate drying can be slower, even though it takes place on both sides. In the other case the coating is assumed to lie on top of the porous web, with solvent vapor difhing through the web. Here the constant rate drying is only slightly faster than in the base case, but in the falling rate period the diymg can be greatly accelerated Both models predict the temperatures and solvent levels throughout the length of the dryer.

Copyright 0 1999 by Marecl Dckker. Inc.

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INTRODUCTION

A relatively simplified model ofthe drylng of solvent coatings on impervious webs has been described in an earlier publication (1) and implemented as the spreadsheet program DRYWEB. However, not all coatings are made on impervious webs. Paper is one of the most common webs, and it is highly porous. Its porosity depends in part on previous coatings or treatments. A coating on a porous web should certainly dry faster than one on an impervious web, assuming that drying air is also circulated on the backside. Modeling the drying through the back side of the web is, however, more involved than when drying is only from the top of the coating.

In the earlier publication it was demonstmted that with most coatings the temperature can be assumed uniform across any section, from the top of the coating to the bottom of the web. Finite difference methods are used to follow the diffusion of the solvent to the surface in the falling rate period Mixed solvents may be used, and the vapor pressure of the first vapor evaporating is calculated using the Wilson method to find the activity coefficients. The vapor pressure is reduced by an adjustable vapor pressure factor to allow for the reduction in vapor pressure due to dissolved species. Theoretically one should calculate activity coefficients here, but in most cases the data is not available. For mixed solvents the solvent composition is assumed not to change throughout the course of drying. This obviously is not true because the more volatile solvent evaporates first, but it appears that it is not a bad assumption The temperature of the coating in any one constant rate zone does not change appreciably, as it would for mixed solvents if the solvent activity changed significantly. In addition, the model does do a good job in predicting both the coating temperatures throughout the length of the dryer and also the solvent level in the dried coating. The model also predicts the solvent levels in the coating throughout the dryer.

In the base model there are several parameters that need to be chosen by matching the predicted temperatures and the final solvent level to one or two experimental runs. One is the vapor pressure factor, mentioned above, which varies between zero and one. This factor is adjusted to gwe the correct coating temperature in the constant rate pied. If there is no constant rate period then it is not used. There is also the heat transfer coefficient at a reference air velocity or reference plenum air pressure. This is rarely known. It, and the two of the three constants in the diffusivity equation, are adjusted so as to have the location of the end of the

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TWO-SIDED DRYING OF COATINGS 693

constant rate period, the predicted coating temperatures, and the final solvent level match the experimental runs. If the air temperahues do not vary from run to run then the activation energy for diffusion does not need to be adjusted but can be left at a typical value, such as 15,000 Jlmol (or 3600 calimol), and only one (rather than two) experimental run is needed to find the parameters. One of the other diffusivity constants can be chosen arbitrarily when, as is almost always the case, the solvent levels in the coating are not measured at different locations within the dryer. Then just the one remaining diffusivity constant needs fitting. The utility of the model depends on these semiempirical parameters, which can also be considered as "fudge factors" If the geometry of the dryers changes from one zone to the next it becomes difficult to pick the reference heat transfer coefficient for each zone. The reference heat transfer coefficients should then be measured, by back calculation from the temperature rise in each d y n g m e of a heavy uncoated web, when heat is supplied to one zone at a time.

CASE 1. THE COATING COMPLETELY IMPREGNATES THE WEB

The Constant Rate Period

In the constant rate period the rate of evaporation is determined by the processes that occur in the d y n g air and not in the material being dried; that is, the rate of mass transfer of solvent vapors in the air from the surface to the bulk is slower than the rate of diffusion of solvent to the surface. This occurs when there is a "pool" of solvent at the surface. At equilibrium the evaporation rate is constant and remains constant as long as the surface remains wet with solvent. However, while the system is approaching equilibrium both the evaporation rate and the temperature of the coating vary. In the falling rate period the surface appears dry, the evaporation rate is determined by the rate of diffusion of the solvent to the surface, and the d y n g air has little effect on the rate, except as it affects the temperature of the system. The diffusion rate is strongly affected by the temperature of the coating and by the solvent level in the coating.

This treatment here follows closely the earlier work (4). In the heat balance used to calculate the coating temperature there is now another term, that for the heat

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consumed in vaporizing solvent from the back side. Thus we equate the rate of heat transfer into the system by convection from both sides and by radiation to the rate that heat is lost in vaporizing the solvent from both sides plus the rate of accumulation of heat in the coated web due to the rise in temperature, giving

h<(T".< - TJ+h,(Ta.* - T J +9,, =

where emtiand c, are the solvent concenhations in the air at the surface and in the bulk air on the coating (c) side or the base (6) side, k. is a mass transfer coefficient, W is the coverage of the components in the coated web, A is the latent heat of vaporization, and 0 is time.

The concentration of solvent in the bulk air is related to the partial pressure of solvent in the air and to the air temperature, either on the coating or the base sides, by means of the perfect gas laws. In SI units, for both the coating and back sides, this results in

where M, is the molecular weight of the solvent and, P, is the partial pressure of the solvent in the air, and the 1000 in the denominator converts the molecular weight to kglmol.

If solvent-free air is used in drying then the solvent partial pressure, P,, is zero. If recirculated air is used then the solvent partial pressure can be found from a material balance.

Adjacent to the surface of the web, the vapor pressure of the solvent will be in equilibrium with the solvent in the coating at the temperature of the coating, T, The concentration of solvent at the vapor pressure P, will then be

The actual partial pressure of the solvent in the coating may be less than the vapor pressure of pure solvent (or solvent mixture) due to the vapor pressure

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TWO-SIDED DRYING OF COATINGS 695

lowering by dissolved species. This is taken into account by the vapor pressure factor.

The mass transfer coefficient is related to the heat transfer coefficient by the Chilton-Colburn (3) relationship

where N,, is the dimensionless Lewis number, equal to the thermal d i h i v i t y divided by the mass diffusivity.

The density of the air next to the coating is found using the perfect gas laws at the temperature of the coating, or

where n is atmospheric pressure. The heat capacity of the solventantaining air on tbe coating side is

where S is the absolute saturation, kg solvent vaporskg of air. S is equal to

Combining equations >I 1 and assuming that the partial pressure of solvent in the mylng air is the same on both the coating and back sides gives for the rate of temperature rise

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696 GUTOFF

which hffers from equation 12 in Gutoff (4) by the addtional term on the far right

in the numerator. K is a grouping of cornitants that is unchanged &om the earlier publication

As before, the evaporation rate is the rate that heat flows into the system minus the energy being used to heat up the coated -web, all divided by the heat of vaporization, or

alws evap rate = - -- = 0'0

Equations 8 and 10 form a couplled initial value problem which can be integrated by any of EL number of standard methods. The Runga-Kutta method is very efficient and is used here to solve for the coating temperature and the solvent level in the coating.

The time interval, A@, should be chosen such that increasing it by a factor of two does not significiantly chmge the final values of temperature or of solvent. If the time interval chosein is too large, a mathematical instability may appear in the response. Thrs will be obviois in the graplhs which the program prints out. On the other hand, if the tirne interval is too smaltl, then unnecessary computation time is consumed. The maximum tirne interval alllowed. is the time to cover one-tenth of a zone.

The Falling Rate Period

In the falling rate period the coatirig is divided into a number of dices and finite difference methods are used to follow the cliffilsion of solvent between slices. As there is no accumulation at the surface, the rate cf evaporation equals the rate of

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TWO-SIDED DRYING OF COATINGS 697

dliffusian to the surface. With the symrnetris,al Case I system, the rate of evaporation, equal to the rate of diffusion to both surfaces, is twice that of the base

case of a coating on an impervious web, or

vvhere C is the concentration (of the solvent in the surface or subsurface layers in gjcrn3, D, is the difhivity of the solvent in the surface: layer in cmz/s, and h, is the thickness of the surface layer in cm. With these units, the evaporation rate is in units of g/cm2-s.

As the coating; comple:t;ely permeates the web, the thickness of any layer is equal to the thickness contributed by the base, the solids in the coating, and the solvent. Thus, the web matrix is assumed to shrink as drying proceeds.

For the slices between the central dice and the surface the finite difference equations remain mcjhanged from before (4), as

where Cm, is the concentration of solvent in the mth slice at time n, A 8 is the time interval between n and n+1, and Dm is evaluated at the mean concentration in the slice, (C,-, +CJ/'2. For a mmidldle slice, just above: or below the plane of symmetry, the material balance remains unchanged, resulting in

For mathematical stablility the concentration of solvent in any slice at any time must not have a negative affect on the solvent concentration at the next time interval. This means, for equations 12 and 13, and neglecting subscripts, that the

time interval must be sufficiently small that

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For the diffusivitythe free-volume model developed by Beels and Claes (I), with the addition of an activation energy to allow for temperature effects, is still used. The diffusivity is:

where A is the pre-exponential factor, R accounts for the concentration effect, E is the activation energy for diffusion, and /I,, , and h,,,, are the thicknesses in a slice contributed by the solvent and the solids plus base.

he concentration of the solvent at the surface bf the coating is in equilibrium with the solvent concentration of the drying air. If the air contains no solvent then the concentration of solvent at the surface should be zero. If air recirculation is used, then there will be solvent in the air which could be found by a material balance. Often one does not have enough data to do calculate this.

Also, one rarely has dataon the equilibrium relationship between the solvent in the air and the solvent concentration at the surface. In this case, one should assume that the surface concentration is zero at all times. One can adjust the diffusivity constants to give the correct final solvent concentration at the end of the dryer, and as long as the solvent levels in the drying air do not change greatly the model predictions should still be good.

In the falling rate period the web temperature at the end of any time interval is found from a heat balance. It equals the temperature at the start of the time interval plus the net accumulation of heat (heat in minus the heat used to vaporize solvent) divided by the heat capacity of the system. Including the possibility of infra-red heat, this remains as

- TWM - Tw.& +

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TWO-SIDED DRYING OF COATINGS

The End of the Constant Rate Period

The constant rate period ends and the falling rate period begins when the constant rate evaporation rate becomes geater than the falling rate evaporation rate. To calculate what the falling rate evaporation rate would be in the constant rate period, the solvent concentration is assumed uniform in all the slices except the surface slice, where it is in equilibrium with the drying air. The surface concentration is usually assumed to be zero.

CASE U. THE COATING IS ON TOP OF THE WEB

The Constant Rate Period

On the back side the solvent vapors diffise through the base. The vapor diffusivity is taken to be constant at its value in air, multiplied by a constant less than unity, E, which may be considered a porosity. During the constant rate period the solvent vapors that flow through the base by diffusion will be at a quasi-steady state. The concentration gradient will then be linear, gving for the rate of evaporation from the back

Using the gas laws to convert concentrations to solvent partial pressures gives

and equation 8, the rate of temperature rise, is now

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GUTOFF

The evaporation rate remains as in Equation 10.

The Falling Rate Period

In the falling rate period the equation for diffusion between the slices remains the same. The evaporation rate is now the rate of diffusion to the surface on the top, or equation 11 without the "2". plus the rate of diffusion from the back. For the material balance on the bottom slice one must also take into account the rate of diffusion from the back. The diffusion through the back is the same as it is in the constant rate period, equation 17, except that the partial pressure of the solvent is now lower because of the reduced concentration of solvent in the bottom slice. As

the lowest solvent conceneation which still gives the same vapor pressure is at the end of the constant rate period, we approximate the solvent partial pressure by assuming it is the vapor pressure of the solvent times the vapor pressure factor, reduced by the ratio of the actual concentration of solvent in the bottom slice divided by the solvent concentration in the bottom slice at the end of the constant rate period.

For the concentration of the bottom slice, there is now another negative term on the right-hand side of equation 13, equal to the back evaporation rate times twice delta time and divided by the thickness of the bottom slice, or

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TWO-SIDED DRYING OF COATINGS

Test A

0 I 0 20 30 40

Length in Dlyer, rn

Figure 1. The graph for Test A in DRYWEB, showing that the air temperatures are constant in each zone, the web temperature rises to approach the air temperatures, and the solvent level falls throughout the dryer.

THE MODELS

These models have been implemented in two Excel spreadsheets, DRYWEBPS for the symmetrical case where the coating klly impregnates the web, and DRYWEBPU for the unsymmetrical case where the coating lies on top of the porous web. To compare the models, Figure 1- 3 are graphs printed out by DRYWEB (the base case), DRYWEBPS, and DRYWEBPU, all for the same system, the details of which are given as Figure 4, the printout for the base case. The data are summarized in Table I.

Table I shows that when the coating completely penetrates the web and the drylng takes place equally from both sides, then the constant rate period is greatly reduced. Here the location of the end of the constant rate period has dropped from 15.4 m into the dryer to just 1.0 m. This is not unexpected However, it is surprising

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Test B

0 10 20 30

Length in Dryer, m

Figure 2. The graph for 7'esr R in DRYWEBPS, showing that the solvent level falls very slowly, as compared to Test A in Figure 1

to see that the drying in the falling rate period was greatly reduced, with the coating still having appreciable solvent at the end of the dryer. This is because the coating is much thicker due to the volume occupied by the web, which has been completely penetrated by the coating. Solvent may have to diffuse further, depending on the thickness of the web, even though solvent loss occurs from both sides. This is especially likely toward the end of the dryer when the almost-dry coating is thin and

may be much thinner than the web. Also, the solvent diffusivity itself is reduced by the presence of the web. In case considered here, the thickness of the web, 100 pm, is about the same as the thickness ofthe wet coating and much, much more than the thickness of the almost dry coating.

When the coating does not penetrate the porous web the constant rate period

does not change much from the base case, since diffusion through the web is relatively slow compared to evaporation from the very wet surface. For

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Test C

Length in Dryer, m

Figure 3. The graph for Test C in DRYWEBPU, showing that the residual solvent level is less than for Test A in Figure 1 .

Wlf mr.2 s b M U sole,

. . 1 w dmr3

-1wo , T ~ A I rmw

0 40 DIFNSMlY WNSTINTS lmprCnr3 k m 2mXIE5 2000 ",% 8 2w n c E. M IYXD

025 - a w e pn"d 1s 16 m mdwr

45 nm

Figure 4. The conditions for Tes1.s A. H, and C. This is the printout from DRYWEB for Test A. The several different conditions for Tests B and Care pven in Table I .

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TABLE I COMPARISON OF DRYING MODELS IN ONE TEST

I Model

Test

End of constant rate period,

Solvent at end of zone I

DRYWEB

Impervious

DRYWEBPS DRYWEBPU

100 (when comuressed)

varies 1 0.3

DRYWEBPU the location of the end of the constant rate period has decreased slightly from 15.4 m into the dryer for the base case to 15.1 m. However, in the falling rate the drying rate has greatly increased, as shown by the lower solvent levels at the end of each zone in Table I.

DETERMINATION OF T E PARAMETERS IN THE MODEL

The model is not complete until all the constants needed for the solution of the equations have been determined

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The Heat Transfer Coefticient a t Reference Conditions

Cary and Gutoff (2) pointed out that the heat transfer coefficient may be taken as proportional to the air velocity to the 0.78 power or to the plenum pressure to the 0.39 power. Thus, the heat transfer coefficient is equal to

where h, is the heat transfer coefficient at the reference velocity V, or at the reference pressure P.. When the geomehy in the top and bottom sections of all zones are the same then the heat transfer coefficient at reference conditions will be same for all sections in all zones. The reference conditions might be, for example, an air velocity of 10 m/s or a plenum pressure of 250 Pa (approximately 1 inch of water). This reference heat transfer coefftcient can be estimated, as Martin (5) has done. However, this estimation is not accurate enough for our use. It also can be determined from measurements of the rate of temperature rise in a thick uncoated web. However, it, along with the other parameters, is usually found by choosing values which gives a good match behueen the predicted and measured coating temptures, including the location of the end of the constant rate period, and also the solvent level in the coating at the end of the dryer. To measure the web temperatures along the length of the dryer is not always easy.

For dryers with different 'in different zones or on the top and bottom, it becomes too difficult to determine the several reference heat transfer coefficients by matching predicted to measured values of temperature and solvent level. Also, when Case ll is used with the additional parameter of the porosity, the reference heat transfer coefticient cannot be estimated In these cases it must be measured, by back calculation from the rate of temperature rise in a thick uncoated web. For the case with several different geometries, it must be done with only one zone or only the top or the bottom section in a zone actively heated at a time.

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706 GUTOFF

The end of the constant rate period, the falling rate temperatures, and the residual solvent level are all affected by the heat transfer coefficient and by rate of diffusion ofthe solvent. Thus, both the diffusivity constants and the reference heat transfer coefficient are usually determined at the same time by comparison with one or two test runs.

The Vapor Pressure Factor

The vapor pressure factor multiplies the pure solvent vapor pressure in the constant rate period, and is the main determinant of the equilibrium constant rate temperature. The heat transfer coefficient does not affect this temperature; however, changes in the ratio of the front-side coefficient to the back-side coefficient will change it. The vapor pressure factor should be adjusted so that the model predicts the correct constant rate temperature. If the constant rate period is very short, or if the equilibrium constant rate temperature is never reached and the transient temperature is determined by the heat capacity of a heavy web, then the vapor pressure factor is dificult to determine and may be chosen arbitrarily. In aqueous systems it is usually one; in solvent systems it often is on the order of 0.25.

The Three Diffusivity Constants

Although it may be best to determine these three constants in the laboratory, this is very difficult and rarely done. The constants A and B determine the solvent level in the coating throughout the falling rate period As this is almost never known except at the end of the dryer, one constant can be chosen arbitrarily. Usually B is chosen between 2 and 5, and then A is adjusted (along with the heat transfer coefficient at reference conditions) to give a correct match with the residual solvent level and the location of the end of the constant rate period. Higher values of A and lower values of B give higher diffusion rates and lower residual solvent levels.

If the dispersed solids are below the critical solids concentration, such that when the coating dries the binder cannot fill all the space between the particles, then air is drawn into the system and solvent is drawn to the surface by capillary action and not by diffusion. However, the same diffusion equations can be used as strictly empirical equations to represent the drying process. Capillary action is much faster

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TWO-SIDED DRYING OF COATINGS 707

than diffusion, and the value of A is much higher than in the case of pure liquid diffusion to the upper surface.

The activation energy for diffusion, E, affects the system when the air temperatures differ from run to run. Two runs are needed to match all the parameters. In one case the activation energy for diffusion was found to be 25,000 Jlmol(6000 cal/mol). This is higher than the typical values of 3000 - 5000 callmol. If the air temperatures do not change, and this occurs in some dryers, then the activation energy can be left at a typical value for diffusion, say 15,000 JImol(3600 cal/mol), and then only one experimental run is needed to pick the parameters.

The Porosity

For Case n the heat transfer coefficient at reference conditions has to be

found independently, and the porosity is found, along with the vapor pressure factor and the diffusivity constants, by adjusting them to match one or two experimental ms.

CONCLUSIONS

One or the other of the two models developed should adequately model the drying of a given coating on a porous web.

When the coating fully saturates the web, then d y n g rate is greatly increased in the constant rate period, but in the falling rate period the drying rate can be

slower. When the coating sits on top of the web, the drying rate may be only slightly

faster in the constant rate period, but can be significantly higher in the falling rate period, and the residual solvent level can be significantly reduced.

NOMENCLATURE

Typical SI units A pre-exponential factor in diffusivity equation m2/s B constant in diffusivity equation accounting for concentration effect

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concentration of solvent in the air heat capacity concentration of the solvent in the coating diffusivity activation energy for diffision heat transfer coefficient thickness of a slice in the coating evaporation rate thermal conductivity mass transfer coefficient grouping of constants, defined by eq. 9 molecular weight Lewis number, alD parhal pressure of the solvent in air rate of radiant energy absorption by the coated web gas constant absolute saturation of air with respect to solvent temperature coverage

Greek E porosity 0 time A heat of vaporization a atmospheric pressure p density of air

Subscrijm a air b base side of the web c coated side of the web I ith component m slice rn n time n s solvent sur surface

surf air next to the surface w web or coating 0.1.2 slice number or location

kg/m3 Jkz

kg/m3 m2/s

Jlmol W/m2-K

m kg/mz/s

W/m2-(Kh) kg/s-m2-(kg/m3)

PaK daltons

Pa W/m2

8.3 14 Pa-m3/mol-K kg solventkg air

K or "C kg/mz

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TWO-SIDED DRYING OF COATINGS

REFERENCES

1. R. Beels and F. H. Claes, 1977, Diffusion Phenomena in Gelatin Sheets, Photogr. Sci. Eng. 21,336-42.

2. J. C. Cay and E. B. Gutoff. Feb. 1991, Analyze the Drying of Aqueous Coatings, Chem. Eng. Prog. 87(2), 73-79.

3.T. K Chilton and A. P. Colburn, 1934, Mass Transfer (Absorption) Coefficients, Ind Eng. Chem 26, 1183 ; see also, for example, R H. Perry and C. H. Chilton, eds., 1973, Chemical Engineers' Handbook, 5th Ed., pp. 12:2, McGmw-Hill, New York.

4. E. B. Gutoff, 1996, Modeling Solvent Drying of Coated Webs Including the Initial Transient. Drying Technology 14, 1673-1693.

5. H. Martin, 1977, Heat and Mass Transfer Rates Between Impinging Gas Jets and Solid Surfaces, in Advances in Heat Transfer, v. 13, pp. 160, J. P. Hartnett and T. F. I ~ n e , Jr., eds., Academic Press, New York.

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