Properties of Foamed Asphalt for Warm Mix Asphalt Applications (2015)

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83 A P P E N D I X A A validated physical model for binder foaming can be an extremely useful tool to understand and explain the impact of various factors on binder foaming. For example, a physical model can be used to qualitatively explain the effects of temper- ature, binder type, water content, additive, and foaming noz- zle (through water droplet size during mixing of binder with water) on the foaming characteristics. Eventually, this knowl- edge can be used to (1) identify the factors that have the most significant impact on foaming characteristics and (2) optimize the water content for effective foaming of different binders. To this end, a physical model that explains expansion and foaming in binders is presented in the following section. Analytical Background Foamed binder is produced through the injection of small droplets of cold water into hot binder. When a droplet of water comes in contact with hot binder, it turns into steam and expands to form a bubble. Binder that forms the skin of the bubble holds the pressurized steam within it by balancing the difference between the internal and atmospheric pressure with its surface tension. This process occurs for each droplet, resulting in a foamed binder. The relationship between external atmospheric pressure, internal pressure due to steam, and sur- face tension is given by the Laplace equation (Equation A-1). The internal pressure due to steam can also be calculated using the universal gas law (Equation A-2). The two equations (Laplace and universal gas law) can be combined to obtain a relationship between the bubble diameter, droplet size, tem- perature, and surface tension of the binder, as shown in Equa- tion A-3. The main assumptions in these two equations are that the gas (steam) in each of the bubbles is ideal, the bubbles are spherical, and every droplet of water converts into steam and is effective in forming a bubble. The only unknown param- eters to employ in these equations are the surface tension of the binder at the foaming temperature and the size distribution of the water droplets. The size distribution of water droplets dif- fusing within the binder is in turn dictated by factors such as the water content and design of the foaming nozzle. P P D 4 (A-1)bubble atm− = γ Where: Pbubble = pressure inside the bubble (Pa). Patm = atmospheric pressure (Typical value: 101325 Pa). g = surface tension of the binder (N/m). D = bubble diameter (m). P V nRT (A-2)bubble = Where: Pbubble = pressure inside the bubble (Pa). V = volume of the bubble (m3). n = number of moles (mass/atomic mass of compound). R = the universal gas constant (8.314 J/mole. Kelvin). T = temperature (Kelvin). P D D nRT 6 2 3 0 (A-3) atm 3 2pi + γpi − = Where: Patm = the atmospheric pressure (Pa). g = the surface tension of the binder (N/m). D = the diameter of the bubble (m). n = number of moles (mass/atomic mass of compound). R = the universal gas constant (8.314 J/mole. Kelvin). T = foaming temperature (Kelvin). The aforementioned physical relationships were used to (1) develop a physical model to theoretically determine ERmax using water content, initial bubble size distribution, and sur- face tension of base binders as inputs, and (2) employ the theoretical analysis in conjunction with experimental data to explore the impact of water content, initial bubble size distri- bution, and surface tension of binders on ERmax. Influence of Binder Properties on Binder Foam Expansion

84 Materials and Test Method To accomplish the aforementioned objectives, a testing pro- gram was designed to measure ERmax, bubble size distribution on the surface of the foam, and surface tension of base binders. Three binders were used for this study: N6, N7, and O7. Each binder was foamed in the laboratory using the Accufoamer foaming unit at three water contents that varied from 1% to 3% by weight of the binder. All foaming was carried out at 160°C. Test procedures included the use of the following: a laser sensor to measure ER as a function of time, a digital camera to measure bubble size distribution on the surface of the foam, and a maximum bubble pressure device (manufactured by SensaDyne) to measure surface tension of the binders. Results of the Parametric Analysis Influence of Surface Tension on Bubble Size Distribution The surface tensions of the three binders were measured using the differential maximum bubble pressure method. This method measures the pressure difference between two capil- laries of different radii as bubbles are produced by injecting a gas through these capillaries when immersed in the binder. In this study, argon was injected through the binder via two ori- fices of different diameters. The pressure differential between the bubbles from the two orifices was measured using a dif- ferential pressure transducer. Water at different temperatures was used to calibrate the transducer. The maximum differ- ential bubble pressure is directly proportional to the surface tension of the binder. The surface tension of the binder was continually measured and recorded over a range of tempera- tures. A bubble flow rate of 1 bubble/s was used for these tests. Figure A-1 illustrates the surface tension of the three binders as a function of temperature. The test results demonstrate that the surface tension of binders is linearly related to tempera- ture; this is true for most liquids. The surface tension of the binders was used with the Laplace equation to determine the internal pressure within the foam bubbles. The most important feature of the Laplace equation is that the pressure required to maintain the bubble is inversely proportional to its diameter. This means that smaller bubbles have greater internal steam pressures. The expected inter- nal pressure within the bubbles was calculated for different bubble diameters assuming the binder surface tension to be 45 mN/m. Figure A-2 illustrates the results from this analysis. These results show that bubble size does not significantly affect the internal bubble pressure for binder bubbles that are greater than 100 micrometers in diameter. In other words, the contribution of surface tension to internal steam pressure is negligible (for bubbles greater than 100 microm- eters). A corollary to this is that the bubble size distribu- tion of the foamed binder is not significantly affected by the surface tension of the base binder, and internal steam pressure is approximately equal to atmospheric pressure. Figure A-1. Surface tension of binders as a function of temperature. Figure A-2. Influence of surface tension on binder foam internal pressure. Contribution of surface tension to bubble pressure Total bubble pressure Ratio = Patm Pbubble

85 Image analysis of the binder foam surface at different times showed that bubbles have sizes that are much higher than 100 micrometers. Influence of Bubble Size Distribution on ERmax An analysis was conducted to determine whether the ini- tial distribution of water droplets in the mixing unit of the foaming device and the subsequent initial distribution of the bubble diameters had an impact on ERmax. ERmax was calcu- lated for several initial water droplet/bubble size distributions for a given water content, and surface tension of the binder (g = 45 mN/m) was kept as a constant. The methodology used for this analysis was as follows. The mass of the binder (mbinder) used for this analysis was 200 g, which is also the mass used in the experimental mea- surements. The atmospheric pressure (Patm) was taken as 101,325 N/m2, and it was assumed that mwater grams of water were added by the foaming unit by dispensing and mixing it with the binder in the form of N number of water drop- lets. Consequently, N numbers of binder foam bubbles are created. Using the Laplace equation, the following can be obtained for the internal pressure of any bubble j: P P D j j 4 (A-4)bubble atm= + γ The following equation can be derived by solving Equa- tion A-3 for nj: n D RT D RT j j f j f 101325 6 2 3 (A-5) 3 2 = pi + γpi The number of water droplets, N, can be obtained as: 18 (A-6) water avg N m n = Substituting Equation A-4 into the ideal gas law equation, individual bubble volume (Vj) can be calculated as follows: V n RT P P D D P D j j f j j j j 6 2 3 4 (A-7) bubble atm 3 2 atm = = pi + γpi + γ The cumulative volume of foam, Vt, for N number of water droplets is the sum of the individual bubble volume and the volume of the binder: (A-8)binder 1 V V Vt j j N ∑= + = ER (A-9)max binder binder1 binder V V V V V t jj N∑ = = + = Volume of the binder (without foam), Vbinder, can be deter- mined using mass (mbinder) and density of the unfoamed binder, taken as 1.034 g/cm3: 1.034 0.967 (A-10)binder binder binderV m m= = Where: Tf = the temperature of the foam (not the same as the foaming temperature of the binder). When the binder at 160°C mixes with water at 25°C (room temperature), the foam will have a temperature lower than the temperature of the binder. The temperature of the foam can be determined using thermal equilibrium, assuming no heat is lost during the foaming process. If Tf is the final temperature of the foam, the quantity of excess heat, Q, available from the binder to vaporize water can be determined as: 160 (A-11)binder binderQ m C Tf( )= − Where: Cbinder = the specific heat of binder (Cbinder = 2.093 J/g/°C). The excess heat, Q, from the binder is equivalent to the amount of heat needed to vaporize mwater at 25°C and raise its temperature to Tf, which can be determined as: 25 (A-12)water water waterQ m C T m Lf v( )= − + Where: Cwater = the specific heat of water (Cwater = 4.185 J/g/°C). Lv = latent heat of vaporization (Lv = 2256 J/g). Combining Equations A-11 and A-12 and solving for Tf: 160 25 (A-13) binder binder water water water binder binder water water T m C m C m L m C m C f v  = + − + Substituting these values for the 200 g binder foamed with 1% (2 g) water content and solving Equation A-13, Tf = 146.8°C. A typical average bubble size of 4.2-mm diameter was assumed for the analysis. The number of moles, n, for the average bubble size was back-calculated using Equation A-5. N was then determined using Equation A-6. That is, water is dispensed in the form of 101,756 droplets with each weigh- ing 19.65 micrograms to mix with the binder and produce foam. The volume of the bubbles for each of the droplets and then the total volume of the foam were computed. For a mean of 19.65 micrograms, the total volume of the foam was

86 calculated as 4,147 cm3. This translates into an ERmax of 20.7 for the 200 g of binder used in these calculations. Similarly, ERmax was computed for several different bubble diameters and numbers of bubbles. Figure A-3 presents a sum- mary of results from this analysis. Note that for a given water content, only one of the two parameters, initial average bubble diameter or number of bubbles, needs to be assumed, while the other can be calculated. Results of the analysis demonstrate that ERmax remains the same irrespective of the bubble size and num- ber of bubbles as long as the total water content turning into steam and forming the bubbles was constant. In other words, for a given volume of binder and water content, bubble size distribution does not significantly affect the maximum expansion ratio of the foam for bubbles that are more than 100 micrometers in diameter. The theoretical maximum expansion ratio was also computed by varying the surface tension of the binder between 40 and 60 mN/m for a fixed water content and an assumed initial bubble/water droplet size distri- bution. The results from this analysis show that the surface ten- sion of the binders (in the range specified previously) does not significantly affect the maximum expansion ratio. This is con- sistent with the analysis shown in Figure A-2. The bubble size distribution affects rate of foam decay because of Stokes’ law. Hence, for large bubbles (bubbles that are more than 100 micrometers in diameter) where the surface tension of the binder does not affect the bubble volume significantly, the equation for ERmax can be simplified as follows. From Equation A-9, ERmax is given by: ER (A-14)max binder bubble binder binder V V V V V t = = + Total volume of the bubble, Vbubble, can be determined using the ideal gas law equation: 18 18 (A-15)bubble atm water atm water atm V nRT P m RT P m RT P f f f = = = Hence, ERmax can be simplified to: V V V m RT P m m m RT m P m P f f ER 18 0.967 0.967 ER 17.41 17.41 (A-16) max bubble binder binder water atm binder binder max water binder atm binder atm = + = + = + The excess heat available from the binder can also be used to determine the theoretical maximum amount of water that can be added to the binder for foaming. If water exceeding this threshold is added, a portion of the foaming water will not be converted to steam due to lack of heat to achieve this. Combining Equations A-11 and A-12, and solving for mwater: 160 25 (A-17)water binder binder water m m C T C T L f f v ( ) ( )= − − + If the minimum foam temperature, Tf, is set to a typical WMA mixing temperature, for example, to 135°C, the maxi- mum limit will be 3.85 g. This value translates to 1.93% water content. Also, theoretically a binder at a temperature of 160°C has enough heat to convert a maximum value of 4.89% water by weight, resulting in a foam at a temperature of 100°C (boil- ing point of water). In other words, if one were to mix more than approximately 5% of water at room temperature with the binder at 160°C, the temperature of the resulting foam would fall to 100°C, preventing conversion of excess water to steam. Effect of Foaming Water Content on Foaming Efficiency By comparing the theoretical ERmax to the measured ERmax, it is possible to determine the percentage of water that is effective in foaming the binder. This comparison for the three binders at 1%, 2%, and 3% water content reveals that the measured expansion ratio is consistently much lower than the theoretical maximum expansion ratio. The ratio between the measured and theoretical expansion ratios decreases as the water content increases (Figure A-4). Despite the ideal conditions assumed for the theoretical ERmax, the results clearly indicate that not all water added to the binder is effec- tive in foaming the binder. It is also clear that as the water content is increased, the percentage of water that is effec- tive in foaming decreases. The reduced foaming may be due to incomplete mixing of the water droplets with the binder during the production of the foam. Other factors, such as the dis- pensing mechanism of the foaming unit, can also affect the percentage of water that is effective in foaming. Although results from Figure A-2 illustrate that the surface tension of the binder does not influence the size of large foamed bubbles, the surface tension of the binder may still affect the Figure A-3. Influence of bubble size on maximum expansion ratio.

87 mixing characteristics of water droplets with the binder, con- sequently affecting the percentage of water that is effective in foaming. For example, Berthier (2008) showed that on a microscopic scale, surface tension and capillary forces domi- nate the fluid mechanics of micro-droplets. Surface tension and capillary forces dominate gravity or inertia forces, which are dominant on a macroscopic scale. The surface tension and capillary forces play a key role in determining the behavior of micro-droplets on different substrates and geometry in micro- systems (Berthier 2008). Uhlig (1937) also investigated the relationship between surface tension and solubility of gases and developed a theory to describe the solubility of gases based on energy change in transferring a solute molecule of radius r to a solvent of surface tension s as follows: 4 (A-18) 2 ln r E kT γ = − pi σ + Where: E = the interaction energy of solute and solvent. g = solubility (ratio of concentration of solute in the sol- vent to that in the gas). K = the Boltzmann’s constant. T = the absolute temperature. Solubility is defined as the ratio of concentration of sol- ute in the solvent to that in the gas. The equation derived by Uhlig (1937) demonstrates that solubility decreases as the surface tension of the solvent increases. Therefore, the sur- face tension of binders at the foaming temperature may affect the efficiency with which the binder and water mix with each other to produce foam. To investigate this, the surface ten- sions of three binders (N6, N7, and O7) at the foaming tem- peratures were compared to solubility (ratio of water effective in foaming to that of water wasted). The surface tensions of the binders determined using the differential maximum bub- ble pressure method are presented in Table A-1. The percent of water effective in foaming was back-calculated using the lab-measured maximum expansion ratio, ERmax. The percent effective water content and solubility values for the three binders at three water contents are presented in Table A-2. Figure A-5 compares the surface tensions of the binders to solubility and shows a strong correlation between solubility and surface tension of binders. Figure A-6 shows a similar trend between percent of water effective in foaming to sur- face tension of binders. These empirical results indicate that the surface tension of the binder dictates the efficiency with which water mixes with the binder to produce foam. Once this relationship is established, the maximum expansion ratio can be theoretically estimated using the equations developed in the previous section. However, more data need to be collected for a thorough theoretical investigation of this phenomenon. The following conclusions were drawn on the basis of the results from the theoretical analysis: 1. The surface tension of the binder affects ERmax significantly, not by influencing the internal pressure of bubbles but by affecting mixing characteristics of binders with water. 2. The initial bubble size (or water droplet size) distribu- tion does not affect the maximum expansion ratio of the Figure A-4. Percent of water effective in foaming as a function of water content. Water Content (%) Foam Temperature (Tf), °C Surface Tension, mN/m N6 N7 O7 1 146.8 45.8 48.4 55.9 2 134.1 50.9 52.2 62.3 3 121.9 55.8 55.8 68.5 Table A-1. Surface tension of base binders and final temperature of binder foams at various water contents. Table A-2. Effective water content and solubility of binder foams. Water Content (%) Effective Water Content (%) SolubilityN6 N7 O7 N6 N7 O7 1 0.49 0.36 0.26 0.979 0.559 0.355 2 0.32 0.22 0.17 0.474 0.283 0.208 3 0.26 0.17 0.14 0.359 0.210 0.164

88 foam as long as initial bubble diameter is more than about 100 micrometers. This corresponds to an initial water droplet size of 0.016 mm or more. Based on experimental results, it is clear that significant expansion is achieved due to bubbles that exceed this diameter. 3. A comparison of the theoretical to the measured maxi- mum expansion ratio reveals that only a small fraction of water added to the binder is effective in foaming. This fraction of water decreases as the water content increases. The effective water content is an important consideration because it may be possible to optimize the water content used to produce foamed binder by using less water and thus reducing the risk of a higher humidity environment in the drum mix plant. 4. The physical model presented can assess the effect of water content and binder type on foam quality. Figure A-5. Relationship between solubility and surface tension of the binder. Figure A-6. Relationship between effective water content and surface tension of the binder.