Strategy for Obtaining Bimodal Bubble Size Distribution in Water

Jun 19, 2015 - TATA Consultancy Services Limited, 33 Grosvenor Place, London SW1X 7HY, U.K.. ‡. Department of Chemical Engineering, Indian Institute...
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Strategy for Obtaining Bimodal Bubble Size Distribution in Water Blown Polyurethane Foams Debdarsan Niyogi,† R. Kumar,*,‡ and K. S. Gandhi*,‡ †

TATA Consultancy Services Limited, 33 Grosvenor Place, London SW1X 7HY, U.K. Department of Chemical Engineering, Indian Institute of Science, Bangalore-560 012, India

‡

ABSTRACT: Polyurethane foams with multimodal cell distribution exhibit superior mechanical and thermal properties. A technique for generating bimodal bubble size distribution exists in the literature, but it uses supercritical conditions. In the present work, an alternative based on milder operating conditions is proposed. It is a modification of reaction injection molding (RIM), using reactants already seeded with bubbles. The number density of the seeds determines if two nucleating events can occur. A bimodal bubble size distribution is obtained when this happens. A mathematical model is used to test this hypothesis by simulating water blown free rise polyurethane foams. The effects of initial concentration of bubbles, temperature of the reactants, and the weight fraction of water are studied. The study reveals that for certain concentrations of initial number of bubbles, when initial temperature and weight fraction of water are high, it is possible to obtain a second nucleation event, leading to bimodal bubble size distribution.

propose a mechanism of formation of the bimodal cell distribution but implicates a role for microemulsified CO2. The present study is concerned about discovering process conditions to produce a bimodal cell distribution in foams manufactured by reaction injection molding (RIM), without using supercritical conditions. We consider the case of polyurethane foam formed by mixing water, a chemical blowing agent, with an isocyanate and a polyol. A multimodal distribution can be generated if one can create nucleation events separated by reasonably large time intervals. We refer to the first nucleation event as primary nucleation and the subsequent events as secondary nucleation. In the RIM process, water reacts with isocyanate producing CO2, and the latter nucleates bubbles when the mixture becomes supersaturated. Usually, primary nucleation is sufficiently intense and occurs continuously over a time interval. Hence, bubbles created are large enough in number, and they grow rapidly enough to absorb the CO2 being produced, to prevent secondary nucleation. Hence, a strategy for creating nucleation after primary nucleation must incorporate a step to reduce the latter’s intensity. We propose that this strategy can be realized if bubbles of small size are present initially in the reaction mixture. The idea is that these bubbles will absorb CO2 being produced, and reduce the extent of supersaturation being generated. This in turn will reduce the number of bubbles created by primary nucleation. Hence, their numbers will not be large enough to absorb the fresh CO2 being produced, thus creating significant supersaturation at a later stage, and consequently, secondary nucleation. It should be noted that the bubble size distribution created is bimodal not due to the initially introduced bubbles, but because of the multiple

Professor D. Ramkrishna’s contribution to population balance framework has become the cornerstone for all attempting to describe behavior of discrete dispersed phase systems. Contrary to the fate of many classics, his book on population balances is often quoted and widely read! Two authors of this paper have collaborated with him on small systems in the context of nanoparticle synthesis. Here, we direct our efforts to the other end: a large population of macroscopic bubbles in reaction injection molded polyurethane foams. The authors are very happy to contribute this article to the special issue of Industrial & Engineering Chemistry Research honoring Professor Ramkrishna.

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INTRODUCTION

The mechanical and thermal properties of polymer foams depend on their architecture and the properties of the polymer.1,2 Additionally, they are sensitive to the bubble size distribution in it. Cells can be packed more compactly if their size distribution is multimodal rather than being narrowly dispersed, which does influence properties of foams. It is known that polyurethane foams with bi- or multimodal bubble size distribution have superior thermal and mechanical properties. Hence, a considerable body of patented literature exists on methods of preparation of foams with multimodal size distributions and some of it is cited in the patent by Lindner et al.3 In their patent,3 they proposed the following method for obtaining a bimodal distribution. Polyol, isocyanate, along with surfactants and catalyst are mixed with a blowing agent CO2 or mixtures of it with other blowing agents (like fluoralkanes, N2, O2, water, and argon), and it is ensured that the mixture is in a supercritical state by pressurizing it in the range of 80−260 bar. The mixture is discharged from a mixing head into a mold, whereupon the pressure is reduced to atmospheric pressure. It is reported that, upon polymerization, the mixture produces foam with a bimodal cell distribution. The patent does not © 2015 American Chemical Society

Special Issue: Doraiswami Ramkrishna Festschrift Received: Revised: Accepted: Published: 10520

March 31, 2015 June 19, 2015 June 19, 2015 June 19, 2015 DOI: 10.1021/acs.iecr.5b01198 Ind. Eng. Chem. Res. 2015, 54, 10520−10529

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Industrial & Engineering Chemistry Research nucleation events. The bubbles initially added are ignored in our discussion of bubble size distribution. Two avenues are open to implement this strategy. One possibility is to use gas inducing impellers. This is unlikely to produce very small bubbles. The second method is to pressurize the reactants separately under a gaseous atmosphere, saturate the liquids with the gas, and to depressurize them to the mold pressure. Nucleation and growth will occur, and the reactants will be seeded with bubbles of small size if the gas is sparingly soluble. Here it is assumed that the bubbles are of gases like nitrogen, argon, etc., but it is entirely possible CO2 can be also be used with the same effect. Preliminary calculations indicate that the pressures employed in this strategy are of the order of 10 bar and are very small compared to those employed by Lindner et al.3 This method is therefore different from that proposed by them. Returning to our discussion of the process, the bubble laden liquids can then be pumped into the mold. As the residence time in the pump is small, it is unlikely the bubbles will dissolve during this operation. It is also possible that the bubbles will rise through the liquids when they are stored. But such a loss may be minimized by manipulating their size. However, we do not focus on the avenues of creating bubble laden liquids in this paper, and proceed to test the feasibility of the hypothesis itself through a mathematical model. Very robust mathematical models of the foaming process have been developed4−7 which have been verified against experimental data. However, these models4−7 predict only an average bubble size, and a model has been developed by Niyogi et al.8 for predicting bubble size distribution in water blown free rise polyurethane foam. It has recently been extended9 to the case where both physical and chemical blowing agents are used. The model of Niyogi et al.8 is used here. It considers energy balance, kinetics of polymerization, and blowing reaction along with nucleation and bubble growth. Bi- or multimodal distributions will not be realized if the primary and secondary nucleation events are not separated by a sufficiently large time interval. It is clear that the number density of seed bubbles is the critical parameter that ensures this. Using the model, we determine the suitable range of the number density of seed bubbles for different temperatures and input water contents to produce bimodal distribution.

∼NCO + H 2O → ∼NHCOOH → ∼NH 2 + CO2 (g)

∼NH 2 + ∼NCO → ∼NHCONH∼+ΔHW

CO2 being generated by this reaction diffuses into the bubbles already present, resulting in their growth. If the area of the seed bubbles is not too large, concentration of CO2 in the reaction mixture grows, and eventually, the liquid becomes supersaturated. As the existing bubbles are unable to consume all the CO2 being generated, primary nucleation begins. Nucleation, and growth of seed as well as as the newly created bubbles, will continue for some time. If and when bubble growth lowers the supersaturation to a sufficiently low level, primary nucleation will cease and only bubble growth occurs from then on. If, however, rise in the temperature, and CO2 still being generated raise the supersaturation to a sufficiently high level, secondary nucleation may occur. Finally, the microstructure freezes when the gel point is reached. To investigate if secondary nucleation can occur, effects of three main process parameters that control nucleation and bubble growth are investigated: (a) Initial concentration of bubbles, nb,0 (contributes to consumption of supersaturation) (b) Initial concentration of water, WW,0 (responsible for rate and amount of production of CO2) (c) Initial temperature of the reaction mixture, T0 (controls the rate of blowing as well as gelling reaction kinetics) Mathematical Formulation. The assumptions and the mathematical modeling of the foaming process are same as that described elsewhere,8 and are not repeated here. Thus, the energy balances, kinetics, and rate of depletion of CO2 are solved along with nucleation and bubble growth. Here, at the start of the process itself, the foaming mixture contains bubbles of size lN. The governing equations are given below for ready reference. Energy Balance. Under adiabatic conditions, the unsteady heat balance for a batch reactor for both gas (CO2) and liquid phase (polymer) is given as Cp

p

d(mpT ) dt

+ Cp

CO2,g

= ( −ΔHOH)COH0

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DESCRIPTION OF THE MODEL As explained in the previous section, the proposed RIM process starts with two bubble laden liquids. One is a polyol containing some amount of water. The second is an isocyanate along with a catalyst, and surfactant. Desired amounts of the two liquids are pumped in stoichiometric proportion into a cup (mold), which is at the molding pressure. Hence, in contrast with the previous model,8 here the foaming mixture initially itself contains bubbles. These bubbles are assumed to be of uniform size equal to that of critical nuclei size, lN, having zero concentration of CO 2 . This assumption is made for convenience and can be easily altered to take into account any initially present bubble size distribution. An exothermic polymerization reaction takes place between polyol and isocyanate thereby increasing the temperature of the reaction mixture:

d(mCO2,g T ) dt dXOH dX W + ( −ΔHW )C W0 dt dt

(1)

Reaction Kinetics. Polymerization reaction ⎡ E ⎤ dXOH = A OHexp⎢ − OH ⎥COH0(1 − XOH) ⎣ RT ⎦ dt (rNCO − 2rWXW − XOH)

(2)

and blowing reaction ⎡ E ⎤ dX W = AW exp⎢ − W ⎥(1 − XW ) ⎣ RT ⎦ dt

(3)

Concentration of CO2 in the Liquid Phase. By mass balance, the increase in the concentration of CO2 in the liquid phase is given by the difference between the amount generated by reaction and that diffused into the gas phase: mCO2,g CCO2 = XW (t )rWCOH0 − MCO2VL

∼OH + ∼NCO → ∼NHCOO∼ + ΔHOH

Mass Balance for CO2 in the Gas Phase. If C*CO2 is the equilibrium concentration of CO2 at the gas−liquid interface,

Water, used as the chemical blowing agent, reacts exothermically with isocyanate to produce carbon dioxide: 10521

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Bubble Size Distribution. The population balance equation for bubbles having size between l and (l + dl), in the absence of breakage and coalescence, is given by10

the rate of mass transfer of CO2 into a bubble of diameter l is given by * ) = k mπl 2ΔCCO NCO2(l) = k mπl 2(CCO2 − CCO 2 2

(4)

∂ ∂ (fVL) + (fGVL) = 0 ∂t ∂l

If f(l) dl is the number of bubbles per unit volume of liquid phase, in size range between l and l + dl, the rate of increase of mass of CO2 in the gas phase can be expressed as dmCO2,g dt

=

∫l

The method for solving the entire set of equations of the mathematical model is exactly the same as that of Niyogi et al.8 except that the calculation of bubble size distribution has been simplified. The method for solving eq 15 needed to obtain bubble size distribution is very similar to that of Kumar and Ramkrishna,11 and hence only a few details are given here. A numerical approach is followed to determine the discrete bubble size distribution where bubbles are grouped into separate sets depending upon their time of birth. The present model assumes a certain concentration (treated as a model parameter) of seed bubbles (nb,0) to be present initially. Therefore, at time t = 0, the moments of the distribution can be calculated as described below.

∞

MCO2[NCO2(l)f (l)VL] dl

N

(5)

Nucleation Rate. The expression for nucleation rate (J), according to the classical nucleation theory, has been used here. ⎡ −ΔF * ⎤ J = MBexp⎢ ⎣ nkT ⎦⎥

(6)

where ΔF * =

16πσ 3 3(PV,CO2 − PL)2

(7)

Here M denotes the total number of molecules of CO2 in the liquid phase. The frequency factor B in eq 6, is treated as an adjustable parameter due to the lack of experimental data. The vapor pressure exerted by the liquid phase, PV,CO2, is calculated using Henry’s law: PV,CO2 = HCCO2

(8)

(9)

Growth Rate. The growth rate G(l) of a bubble of size l is obtained from the mass balance equation, assuming the vapor phase to be ideal: d ⎡π 3 P ⎤ 2 * )] ⎢ l ⎥ = NCO2(l) = k mπl [(CCO2 − CCO 2 dt ⎣ 6 RT ⎦

2k mRT ΔCCO2 dl l dT =G= + dt P 3T dt

(10)

2D l

(11)

dt

= 2πDΔCCO2MCO2

f1 VL|t = 0 = nb,0lN

(17)

f2 VL|t = 0 = nb,0lN 2

(18)

f3 VL|t = 0 = nb,0lN 3

(19)

t ≥ tc + K Δt

(12)

∫l

lmax

∑ nb,i i=0

(21)

K

∑ nb,ili i=0

lfVL dl

N

(22)

K

f2 VL =

(13)

∑ nb,ili 2 i=0

The growth rate is obtained from eq 11 by substituting km: 4RTDΔCCO2 dl l dT G= = + dt Pl 3T dt

(20)

K

f0 VL =

f1 VL =

= 2πDΔCCO2MCO2f1 VL

i = 1, ... K

where tc is the cream time. K increases by unity at each time step, as long as nucleation occurs, and remains constant otherwise. When the number and size for each set of bubbles are known, the moments of the distribution can be calculated at any instant of time as follows:

The diffusion coefficient D is assumed to be constant. The rate of increase of mass of CO2 in the gas phase is obtained from eq 5 using eq 12: dmCO2,g

(16)

nb, i(t ) = nb, i(tc + iΔt ) = J(tc + iΔt )VLΔt

Mass Transfer Coefficient. The liquid to gas mass transfer coefficient km is considered to be purely diffusional and controlled by liquid-side resistance. Under these conditions, the Sherwood number is 2, and km becomes

km =

f0 VL|t = 0 = nb,0

Beginning with the first instance of nucleation, the bubbles born during each successive small period of time Δt are grouped into different sets indexed by the time interval in which they are born after the cream time. As the value of Δt is sufficiently small, it is assumed that within a single set, the bubbles will experience the same kind of external conditions throughout, and thus have the same size. During the time interval Δt, the number of nuclei formed at any given time t is given by J(t)VLΔt. As coalescence is assumed to be absent, if there are K such sets at any time t, the ith set contains nb,i number of bubbles given by the equation

The Henry’s constant H is evaluated by using data on the solubility of CO2 in the liquid phase at atmospheric pressure: * = 1 atm HCCO 2

(15)

(23)

K

f3 VL =

(14)

∑ nb,ili 3 i=0

10522

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Industrial & Engineering Chemistry Research where li is the size of the ith bubble set. Here, the 0th set (i = 0), corresponds to the set of the initially added seeds to the system. The mass balance for CO2 can now be written as dmCO2,g dt

K

=

∑ 2πDMCO nb, iliΔCCO 2

2

i=0

(25)

The above are discrete versions of eq 13. In order to use the above equation, it is necessary to calculate the size of the ith set of bubbles at any time. Equation 14, can be written for each set i as G=

4RTDΔCCO2 dli l dT = + i dt Pli 3T dt

(26)

Initial condition for this equation is given by t = ri ,

li = lN

(27)

where ri represents the time of birth of ith set of bubbles. Thus, the size of the ith set of bubbles at any instant of time is calculated using the above equation. The set of eqs 1−3 and 21−26 are solved simultaneously to obtain the bubble size distribution and other properties of foam. The overall foam density is calculated by using the following equation: mp + m W0 ρall = m RT mp m CO2, g + ρ + ρW PM

■

CO2

p

W

Figure 1. Variation of bulk temperature with time for different initial bubble counts.

(28)

PARAMETERS FOR COMPUTATION The data used for computations are same as given by Niyogi et al. (Tables 1−5 of ref 8). The value of B in eq 6 was adjusted such that the experimental temperature and density profiles of Baser and Khakhar7 are reproduced by the simulation results.8 The same fitted value of B = 10−10 s−1 has been adopted here. As mentioned earlier, the size of seed bubbles is assumed to be equal to that of critical nucleus (lN). Its value is8 0.4 × 10−6 m. Numerical Solution. The set of ordinary differential equations is solved simultaneously using an IMSL subroutine which implements Adams Moulton method for initial value problems. Simulation is carried out until the gel point is reached, i.e., XOHgel = 0.5.

Figure 2. Variation of partial pressure of CO2 with time for different initial bubble counts.

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in the liquid phase is determined by two opposing factors: the rate of generation of CO2 due to reaction and the rate of depletion of CO2 due to diffusion into both the seeded and nucleated bubbles. Consider at first the scenario when seed bubbles are absent. The reaction generates CO2, and this has the effect of increasing PV,CO2. Exothermic rise in temperature also tends to increase it. As more and more CO2 is generated, the reaction mixture becomes supersaturated and primary nucleation starts. PV,CO2 is now reduced by the increase in the number of bubbles due to continuing nucleation, and the mass transfer of CO2 into nucleated bubbles. Area increases due to growth, and consequent accelerated mass transfer enhances the reduction in PV,CO2. Though initially the production rate of CO2 is larger, a point is reached when its rate of depletion exceeds production rate, and PV,CO2 reaches a maximum and declines. As PV,CO2 continues to decrease, primary nucleation slows down to negligible rates after some time. Consequently, the depletion rate declines, and PV,CO2 attains a minimum. At this juncture, the rates of production and depletion of CO2 are of similar

RESULTS AND DISCUSSION In the present study, model simulations are carried out using different combinations of initial conditions for nb,0, T0, and WW,0 in order to find a regime where multimodal distribution of bubble size can be achieved. The mold pressure is taken to be atmospheric. As mentioned earlier, B was fitted to secure agreement with data of Baser and Khakhar7 for the case when seed bubbles are absent. A comparison of those predictions with observations was already reported8 and is not reproduced here. However, it may be mentioned that the agreement was found to be satisfactory. Effect of Initial Bubble Count (nb,0). Figure 1 shows the temperature profile with time, for various initial bubble counts. The temperature rises due to the exothermic polymerization and blowing reaction. Since the heat generation is entirely dominated by the reactions, the temperature history is found to be independent of nb,0. Evolution of partial pressure of CO2, (PV,CO2), with time is shown in Figure 2. The rise or fall of the partial pressure of CO2 10523

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Industrial & Engineering Chemistry Research order of magnitude, and a plateau is observed. Subsequently, while increasing area for mass transfer due to growth and exhaustion of reactants tends to decrease PV,CO2, the rise in temperature tends to increase it. It appears that the latter is a little more effective for some time, and PV,CO2 increases slowly. However, effect of exhaustion of reactants eventually dominates, and a second peak is observed. This general behavior is observed in Figure 2 not only for nb,0 = 0 but for all other values of nb,0 too. Noting that PV,CO2 > PL, it can be seen from Figure 2 that, but for a very short time in the beginning, the liquid remains supersaturated until the end. However, contribution from secondary nucleation is not observable for nb,0 = 0. This can also be seen from Figure 3. It is because the nucleation rate is a

level at which a plateau of PV,CO2 is observed as well as the subsequent peak reached are higher. This indicates the possibility of secondary nucleation. The trends described continue until nb,0 is increased up to 4 × 1012. When the value of nb,0 is increased beyond this level, a reverse trend appears. The first peak reached in the partial pressure continues to decrease, indicating a reduced extent of nucleation. But the peak becomes narrower suggesting that the reduced nucleation rate is being compensated by a higher value of nb,0. For the same reason, when nb,0 is increased beyond 4 × 1012, the minima observed, and the level of the plateau as well as the second peak reached are reduced. From Figure 2, it can be seen that the minimum is the highest when nb,0 = 4 × 1012. At the instant when the minimum has been reached, the number of seeds and nuclei generated are just sufficient to significantly slow down the first burst of nucleation, but a fairly high concentration of CO2 is maintained in the liquid phase. Subsequently, the rate of generation of CO2 becomes overwhelming, leading to the highest increase in PV,CO2. Among the set investigated, this value of nb,0 is then the optimum for creating secondary nucleation. However, the conditions that can create perceptible secondary nucleation can be ascertained only by examining the nucleation rate and the number density of bubbles, i.e., the zeroth moment of the bubble size distribution. Profiles of both primary and secondary nucleation rate are shown in Figure 3, and those of zeroth moment are shown in Figure 4. As discussed earlier, the first peak in PV,CO2 decreases

Figure 3. Variation of nucleation rate with time for different initial bubble counts.

very sensitive nonlinear function of supersaturation, and the second peak in PV,CO2 is not large enough to cause significant nucleation for nb,0 = 0. Hence, secondary nucleation will be substantial only if the second peak in PV,CO2 is enhanced, and this can be done by reducing the intensity of primary nucleation. Addition of seed bubbles achieves this purpose. The number density of bubbles created by primary nucleation in the absence of seed bubbles is of the order of 1013. If the seed density is of the same order of magnitude as this, secondary nucleation will be ineffective. Hence, the seed density has to be lower than this. When nb,0 is increased, the magnitude of the first peak of the partial pressure curve is seen in Figure 2 to decrease since CO2 diffuses into the already existing bubbles. Here the nucleation rate is suppressed and thus the depletion of CO2 is slower. Therefore, bubble nucleation continues, but at a reduced rate, until the total number and area of bubbles in the system is large enough to exhaust the supersaturation to a level to bring nucleation to negligible levels. With an increase in nb,0, the peak is broader due to sustenance of nucleation for a longer time. Once again, a minimum in PV,CO2 is reached for reasons enumerated above. But the minimum is higher indicating less consumption of CO2 since the total number of bubbles is less compared to the case when seed bubbles are absent. Thus, the

Figure 4. Variation of zeroth moment with time for different initial bubble counts.

as nb,0 is increased, and it translates into decreased rate of primary nucleation. It is also seen that secondary nucleation is not observable for nb,0 = 0. As nb,0 is increased, it is seen in Figure 3 that the rate of primary nucleation decreases, and this is in consonance with the observation that the first peak in PV,CO2 decreases. For seed bubble densities less than 4 × 1012, primary nucleation occurs at a reduced rate but for longer duration before it becomes negligible. For larger seed densities, the reduced rate of primary nucleation is compensated for by the extra area provided by the large number of seeds, and reduced nucleation occurring for a shorter period itself is enough to render primary nucleation insignificant. As already seen, the second peak in PV,CO2 increases until nb,0 increases up 10524

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Industrial & Engineering Chemistry Research to 4 × 1012 but decreases thereafter. In accordance with this, it can be observed from Figure 3 that secondary nucleation does occur for all values of nb,0 but its rate is maximum for nb,0 = 4 × 1012. It is also seen that the rates are very small compared to that of the primary nucleation. These conclusions are corroborated by profiles of zeroth moment (f 0VL), which represents the total bubble count, shown in Figure 4. It is seen that nucleation does not start until about a second for all values of nb,0. The bubble count is zero until that time when nb,0 = 0. The total number of bubbles is highest when bubbles are not present initially since nucleation rate for this condition is at least an order of magnitude greater than when seeds are present. As argued previously, the primary nucleation rate decreases with increased number of seed bubbles, and hence the bubbles added to the seeds by nucleation decreases. However, the zeroth moment is the sum of seeds and the nucleated bubbles. Hence, the zeroth moment decreases with increasing number of density of seeds until nb,0 = 4 × 1012 and increases thereafter since nb,0 is itself greater than the number of nucleated bubbles. In fact, for values of nb,0 greater than 4 × 1012, the number of additional bubbles created by nucleation is insignificant, and zeroth moment does not change significantly with time. As indicated by eq 13, the rate of mass transfer is directly proportional to the first moment, f1VL. The first moments were computed for all values of nb,0 and found to be always less when seeds are present than when they are absent. Hence, mass transfer rates are also lower when seeds are added. For nb,0 = 4 × 1012, the difference was found to be maximum, and thus the buildup of supersaturation was also maximum. Hence, the secondary nucleation rate can be expected to be maximum for nb,0 = 4 × 1012 as already seen in Figure 3. As the number of additional bubbles created by nucleation is small and also since their sizes are small compared to those created by primary nucleation, no significant change in the rate of increase of the first moment was observed after secondary nucleation occurs. All these expected trends were confirmed by results of computations, and hence a graph of the profiles of the first moment (f1VL) is not presented. The third moment, (f 3VL), represents the total gas volume in the foam. As CO2 generated by reaction is very large compared to the volume of seeded bubbles and almost all the gas generated diffuses into the bubbles, profiles of the third moment were found to be unaffected by the presence of initially added bubbles, and are not shown graphically. The variation of mean bubble diameter is shown in Figure 5. As the total gas volume is constant, the mean diameter is expected to be inversely proportional to the total number of bubbles. Hence, the mean bubble size is found to be maximum for nb,0 = 4 × 1012, as the count of bubbles is least in this case. It may be noted that computations were carried out at several values of nb,0, and a smooth curve has been drawn through all the points. Figure 6 shows the time at which various events occurred as nb,0 is varied. It is seen that the cream time (tc), rise time (tr), and gel time (tgel) are unaffected by nb,0. However, for reasons elaborated earlier, the time at which the primary nucleation almost stops passes through a maximum with an increase in seed bubble count. However, as the supersaturation left after the arrest of primary nucleation is maximum at nb,0 = 4 × 1012, the time at which the secondary nucleation is almost halted also goes through a maximum. Additionally, the gel time and time at which the secondary nucleation stops coincide for nb,0 = 4 × 1012. With the increase in nb,0, the starting time of significant

Figure 5. Variation of mean bubble size with initial bubble counts.

Figure 6. Variation of nucleation period for different initial bubble counts.

second nucleation decreases, passes through a minimum and then starts increasing. This is also along expected lines since the supersaturation left after the arrest of primary nucleation is maximum at nb,0 = 4 × 1012. The effect of nb,0 on the final bubble size distribution is shown in Figure 7. As expected, a unimodal distribution is obtained when bubbles are not present initially. For all other cases, either a bi- or trimodal distribution is found. A bimodal distribution suggests absence of significant secondary nucleation. The seeded bubbles grow into bubbles of uniform size, and they had the maximum time to grow. They constitute the population of biggest bubbles, and they are represented by vertical lines in Figure 7 in all the curves with a finite value of nb,0. The positioning of the size distribution along the size axis as nb,0 is varied is in accordance with the dependence of the mean size on nb,0. The decrease in the peak in the bubble size distribution corresponding to only the primary nucleation seen in Figure 7 is to be expected from the decrease in the rate of primary nucleation seen when seeds are added. In principle, curves for all nonzero values of nb,0 must show a trimodal distribution, one each for primary and secondary nucleation and one for the seeds. However, secondary nucleation is significant only for 4 × 1012 and 6 × 1012. Thus, a trimodal 10525

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the volume occupied by the bubbles created by secondary nucleation is only four percent of the total volume. The range of parameters investigated here does not create enough secondary nucleation to occupy volume comparable to that of added bubbles and or primary nucleation. Effect of Initial Weight Fraction of Water (WW,0). The influence of initial water concentration is studied at an initial temperature T0 of 315 K and for the initial bubble count of nb,0 = 4 × 1012. Figure 8 shows the effect on the temperature

Figure 7. Final bubble size distribution for different initial bubble counts.

distribution is found only for these two seed densities. The numbers contributed by secondary nucleation for the other two values of nb,0 are too small and do not show up in Figure 7. The secondary nucleation rate is maximum for nb,0 = 4 × 1012. Correspondingly, the numbers contributed by secondary nucleation are also maximum at 4 × 1012. Effect of Seed Bubble Size. Results of calculations performed by varying size of the seed bubbles are presented in Table 1. All results correspond to the gel point. The volume of gas occupied by seed bubbles and bubbles created by primary as well as secondary nucleation are also given in the table. Seed bubbles create secondary nucleation by reducing the intensity of primary nucleation by absorbing the CO2 generated. Their effectiveness therefore depends upon the rate of mass transfer of CO2 into them, and their capacity to absorb CO2. First, for seed size = lN, it can be seen that the primary nucleation is reduced by a factor of 3 when nb,0 = 2 × 1012. However, secondary nucleation is not significant. The volume occupied by seed bubbles is only half of that occupied by the bubbles created by primary nucleation. When nb,0 is raised to 4 × 1012, where the secondary nucleation is maximum, primary nucleation is reduced significantly. Now, the volume occupied by these bubbles is only 10% of that of seed bubbles. Thus, primary nucleation is suppressed to a great extent by the added bubbles and secondary nucleation is not still significant enough. When bubble size is increased, the two effects of addition of seeds oppose each other and the net effect is not significant until bubble size is increased to unreasonably large size. It can be seen that the secondary nucleation increases significantly when the seed size is 50 times that of lN. But even for this case,

Figure 8. Variation of bulk temperature with time for different initial water concentrations.

profile. Temperature rise is sharper and maximum temperature attained at gel point is higher for higher WW,0. Decreasing water content slows down the blowing reaction and hence the rate of rise of temperature. The slower rise in temperature in turn slows down the exothermic polymerization reaction as well. Higher temperature attained at higher water contents is a result of the greater extent to which the exothermic reaction with water proceeds. Figure 9 shows the variation of partial pressure of CO2 with time as initial water content is varied. The peak value decreases with decrease in WW,0, as less amount of CO2 is produced. Also, after the initial peak, the curve tends to flatten out more for lesser initial weight fraction of water because the smaller amount of CO2 produced is not enough for amplification of the supersaturation. The variation of nucleation rates with time is depicted in Figure 10. The nature of the curves directly follows from the trends of PV,CO2 discussed above. With less amount of blowing agent available, the second peak of nucleation rate diminishes and secondary nucleation in not visible for WW,0 = 0.1. The final

Table 1. Effect of Seed Size and Other Properties total number of bubbles nb,0

seed size lN lN lN 10lN 50lN

0 2 4 4 4

× × × ×

1012 1012 1012 1012

primary 1.34 × 1013 5.04 × 1012 3.29 × 1011 3.1 × 1011 1.19 × 1011

volume of secondary 0 1.63 6.43 6.82 1.57

× × × ×

10526

104 109 109 1010

seed

primary

secondary

0 50.6 157.4 157.9 164.8

169.8 119.1 12.13 11.66 4.67

0 ∼0 0.0243 0.0259 0.0638

DOI: 10.1021/acs.iecr.5b01198 Ind. Eng. Chem. Res. 2015, 54, 10520−10529

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Industrial & Engineering Chemistry Research

Figure 9. Variation of partial pressure of CO2 with time for different initial water concentrations.

Figure 11. Final bubble size distribution for different initial water concentrations.

was depicted in Figure 3 and therefore are not presented graphically. The final bubble size distribution is shown in Figure 12. It is observed that lowering the initial temperature results in smaller

Figure 10. Variation of nucleation rate with time for different initial water concentrations.

bubble size distribution, for various WW,0 is plotted in Figure 11. With decrease of WW,0, the peak values of bubble counts as well as the bubble sizes decrease, and the multimodality of the distribution also vanishes. Again, the primary reason lies with the decreased availability of blowing agent. Effect of Initial Temperature (T0). The effect of initial temperature is studied for three different values, but at nb,0 = 4 × 1012, and WW,0 = 0.30. Reaction kinetics of the exothermic reactions is faster at higher temperatures, and hence a steeper rate of increase of temperature is expected as the initial temperature is increased. It is also to be noted, that the foaming period is less for higher T0, because gelling condition is reached earlier. Results were found to be along the expected lines and are not shown graphically. Variation of nucleation rate with time, for the three different T0 were evaluated. Both the peaks for the primary and second nucleation were found to increase with increase in T0, as higher temperature leads to faster generation of CO2, and attainment of higher degree of supersaturation. The second peak was found to shift toward higher times for lower values of T0 due to slower reaction kinetics. The general shape of curves is similar to what

Figure 12. Final bubble size distribution for different initial temperatures.

number of bubbles with bigger sizes. This is to be expected since slower reaction kinetics gives longer period of time for bubble growth. Further, as lowering temperature suppresses the secondary nucleation, the multimodality of the distribution tends to disappear. Final Comment. The parametric investigation shows that secondary nucleation and hence trimodal bubble size distribution can be achieved by varying initial temperature and water content and by seeding the feed to a reaction injection mold with small bubbles when only a chemical blowing agent is used. It is observed that effect of secondary nucleation is maximum when initial water content is quite high. However, under the conditions investigated, the number of bubbles added by secondary nucleation is small compared to that created by primary nucleation. In the present investigation, only the number density of seed bubbles was varied while the size of the seeds was kept constant. Further, addition of a 10527

DOI: 10.1021/acs.iecr.5b01198 Ind. Eng. Chem. Res. 2015, 54, 10520−10529

Article

Industrial & Engineering Chemistry Research

lN = critical nucleus size, m m = mass of component indicated by subscript, kg M = molecular weight of component indicated by subscript, kg/kmol M = number of molecules of CO2/unit volume of polymer solution, number/m3 n = total number of molecules of CO2 in a critical nucleus nb,i = number of bubbles nucleated during the ith time interval, i = 0 indicates initial addition NCO2(l) = mass transfer rate of CO2 into bubble of size l, kmol/s P = pressure, N/m2 PL = pressure in the liquid phase, N/m2 PV,CO2 = partial pressure of CO2 in the liquid phase, N/m2 R = gas law constant, J/kmol K r = ratio of equivalents of component indicated by subscript to those of polyol ri = time of birth of ith set of bubbles t = time, s tc = cream time, s tgel = gel time, s tr = rise time, s T = temperature, K VL = volume of the liquid phase, m3 W = weight fraction of water in polyol X = fractional conversion of reactive species XOH,gel = chemical conversion of polyol to gelation

physical blowing agent with suitable boiling characteristics may result in improved control over the phenomena. The effect of these parameters has to be investigated to find optimal conditions to make secondary nucleation more effective.

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CONCLUSIONS Multimodal bubble size distribution in polyurethane foams is desired over unimodal as the former leads to superior properties of interest. Multiple nucleation events separated by long enough time intervals can produce multimodal bubble size distributions. It is hypothesized that initially added bubbles can create multiple nucleation events. An existing mathematical model for predicting bubble size distribution in water blown free rise polyurethane foams is extended to include initially added bubbles. The model affirms the hypothesis. It is found that there exists a window of number density of initially present bubbles to obtain nucleation for a second time after the first nucleation dies down. This gives rise to a bimodal bubble size distribution when the initially added bubbles are ignored. The effects of initial weight fraction of water, as well as that of initial temperature are also studied. It is observed that highest initial concentration of water and highest initial temperature give rise to the most favored conditions for the occurrence of the second nucleating event.

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AUTHOR INFORMATION

Corresponding Authors

Greek Letters

*E-mail: [email protected] (R.K.). *E-mail: [email protected] (K.S.G.).

ρall = overall foam density at gel point, kg/m3 ρL = density of the liquid phase, kg/m3 ρp = density of polymer and prepolymer, kg/m3 ρW = density of water, kg/m3 σ = surface tension, N/m

Notes

The authors declare no competing financial interest.

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NOTATION AOH = frequency factor for polymerization kinetics, m3/ kequiv s AW = frequency factor for isocyanate-water reaction kinetics, s−1 B = frequency factor of gas bubbles joining the nucleus, s−1 C = concentration of component indicated by subscript in bulk liquid, kmol/m3 CCO * 2 = equilibrium concentration of CO2 in liquid phase, kmol/m3 ΔCO2 = defined in eq 4 CpCO2,g = specific heat of CO2 in gas phase, J/kg K Cpp = specific heat of polymer and prepolymer, J/kg K D = diffusion coefficient, m2/s EOH = activation energy for polymerization kinetics, J/kmol EW = activation energy for isocyanate-water reaction kinetics, J/kmol f(l) = number density function of bubbles in the size range l to l + dl, 1/m4 f nVL = nth moment of bubble size distribution, mn ΔF* = minimum free energy change for the formation of critical nucleus (classical nucleation theory), J/K mol G = growth rate of bubbles, m/s H = constant in Henry’s law, N m/kmol −ΔHOH = heat of polymerization reaction, J/kequiv −ΔHW = heat of isocyanate−water reaction, J/kmol J = nucleation rate, number/m3 s k = Boltzmann constant, J/molecule K km = mass transfer coefficient, m/s l = bubble diameter, m

Subscripts

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o = initial condition CO2 = carbon dioxide g = gas L = liquid phase NCO = isocyanate OH = polyol p = polymer, prepolymer W = water

REFERENCES

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DOI: 10.1021/acs.iecr.5b01198 Ind. Eng. Chem. Res. 2015, 54, 10520−10529

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DOI: 10.1021/acs.iecr.5b01198 Ind. Eng. Chem. Res. 2015, 54, 10520−10529