Contents. 1 Introduction 4. 2 Methods Results and Discussion 15

Transcription

1 Contents 1 Introduction 4 2 Methods 11 3 Results and Discussion 15 4 Appendices Variable Definitions Sample Calculations Data

2 List of Figures 1 Falling Liquid Level in Tray Three Drying Regimes Falling Drying Rate Regimes Marble tray within oven Pyronmeter within oven Dried Marble Drying experimental curve for trial Drying experimental curve for trial

3 List of Tables 1 Overall Weight Changes Through Drying Average Theoretical Heat and Mass Transfer Coefficients Average Experimental Heat and Mass Transfer Coefficients Drying Experiment Summary Data Collected from First Trial (10 on Both Heat and Wind Settings) Data Collected from Second Trial (10 Heat Setting 7 Wind Setting) First Trial Theoretical Parameters Second Trial Theoretical Parameters First Trial Experimental Parameters Second Trial Experimental Parameters

4 1 Introduction Drying is a separation process by phase creation and addition, with use of both an energy and mass separating agent [1]. The need for drying in many chemical operations is directly related to the common use of water within these operations; whether as an universal solvent, convenient coolant, or a reactant. For this reason, water is generally present in products at the end of numerous chemical processes and in many industries this water must be removed to form a solid product. Drying is frequently employed in the food industry, as the complete removal of moisture upon packaging prevents spoilage [2]. Even though drying occurs spontaneously under normal conditions, the high industrial throughput of material requires a faster separation process and therefore drying equipment is necessary. While the exact specifications of equipment may vary, the common goal is to increase the rate of mass transfer of a liquid species in a solid-liquid mixture to the surrounding vapor. In order for this mass transfer to occur, the vapor pressure of the liquid must be greater than its partial pressure in the vapor [1]. If water is being removed from a mixture, the surrounding air should therefore possess as low humidity as possible. To ensure this water concentration gradient between the mixture and air, heat is added, which simultaneously lowers the humidity of the surrounding gas through temperature-dependent saturation limits and increases the vapor pressure of the liquid in mixture through heat transfer. However, if the warm air is stagnant above the mixture, then the gas would become saturated and decrease the rate of drying. Therefore, a fan is often used to provide a constant supply of fresh air and effectively increase the rate of mass transfer through convection. While these conditions (high temperature, low humidity, and high air flow rate) are perhaps the most important factors in increasing mass transfer rate, additional conditions such as internal diffusion, capillary flow, equilibrium moisture content, and heat sensitivity must also be considered [1]. Optimizing all of these conditions often cannot be completed attempted through simulation alone; it requires many empirical tests on pilot-scale plants. 4

5 Figure 1: These three panels shows how the effect of liquid diffusion through the solid can greatly effect the rate of drying. This diffusion is difficult to model, and is often not explicitly included [1]. Because simulating all the facets of drying exactly is difficult, a simplistic model of drying which is still useful can be used. This model breaks down the entire operation into three regimes. 1. The mixture warming to the ambient temperature, a film of water remains at the mixture-air interface. 2. The mixture reaches a steady-state temperature, motivating water to the surface of the mixture. Vaporization is the limiting step in the separation process. 3. Nearly all of the starting water in the mixture has evaporated. The process becomes 5

6 dependent on the rate of diffusion of the water through the almost dry solid, causing the overall rate of drying to decrease. Figure 2: The three regimes in terms of the moisture content versus time. The first regime occurs from point A to B, the second regime from B to D, and the third regime from D to infinity [1]. The second regime has a constant time versus moisture content slope, and is most easily modeled mathematically. Two equations have been developed, one in terms of mass transfer (equation 1) and the other heat transfer (equation 2). R c = h (T g T w ) H vap (1) R c = M w k (H s H a ) (2) The variables for these equations are defined in the appendix. Measurements taken during the experimental process provide values for both temperatures and humidities, and the enthalpy of vaporization may be calculated through a DIPPR relationship [3]. However, this still leaves three unknowns: flux, mass transfer coefficient, and heat transfer 6

7 coefficient. The latter value may be estimated through an empirical relationship for a flat plate and parallel flow. h = G 0.8 (3) Empirical relations also exist for the estimation of the mass transfer coefficient. Based on a Reynolds number calculation, laminar flow was assumed to exist within the drier. This determination and the flat plate geometry of the mixture led to equation 4, below Re 1/2 Sc 1/3 = Sh = kl D (4) All of the calculations discussed so far have been mostly theoretical in nature because they use an empirically-derived relationships that do not take into account the actual rate of drying in this experiment. To remove these theoretical aspects, the actual rate of drying can be calculated, as shown in equation 5 below. R c = mol t A (5) This experimental rate of drying may then be substituted into equations 2 and 1. The experimental heat and mass transfer coefficients may then be determined through algebra. This evaluation is important because it allows for a comparison between the theoretical and actual results. If there is a large difference between these results, then secondary effects, such as capillary action, are likely influencing the rate of drying. While it is not possible to provide a good comparison of theoretical and empirical in the other two drying regimes, it is possible to incorporate this data into a total expected time 7

8 to dry calculation. This calculation begins by analyzing the differential equation which most generally models the drying process, and thereby governs the shape of figure 6. This equation is shown below in equation 6. R = m s A dx dt (6) To transform this rate based equation into a value of time, integration must take place, as shown in equation 7 below. dt = m s A dx R (7) To continue, an assumption is made that the first regime is very small (less than interval of measurement, or 4 minutes) and therefore does not need to be considered. To determine the time for the constant rate period, or the second regime, the rate term may be pulled out of the integral leading to equation below. t second regime = m s (X o X c ) AR c (8) To determine the time of the third regime, an assumption that the rate fell linear to the moisture content was made, as shown in equation 9. The substitution of the linear equation into equation 7 then leads to a possible integration and the final result as shown in equation 10, below. R = R cx Xc (9) 8

9 t third regime = m sx c AR c ln ( Xc X f ) (10) The motivation for this assumption was that it was simple and widely used. An illustration of the linear form is shown in figure 6, along with the two other common assumptions, parabolic and complex. Figure 3: The three graphs show the most commonly assumed drying rates for the third regime of a falling drying rate. From top to bottom are linear, parabolic, and complex drying assumptions. For this analysis the linear rate was chosen as it best modeled the data [1]. 9

10 By adding the assumed zero time of the first regime to the calculated times of the second and third regimes a total drying time is achieved, as shown below in equation 11. t T = m ( s (X o X c ) + X c ln AR c ( Xc X f )) (11) This equation does not provide a strict comparison between theory and empirical measurements, but it still provides a method to see if the first assumption of linear drying versus moisture is true. Furthermore, this analysis combined with the previous calculation of transfer coefficients enables a robust comparison between the ideal conditions and controlling factors in theory compared to the actual empirical results. 10

11 2 Methods The drying experiment involved preparing a mixture of water and dry marble, and observing the properties of this mixture as it sits in a hot, convective environment. Before the drying commences, the rectangular tray (10.75in. x 7.25in.) which holds the marble, was weighed. This initial measurement allows for the marble to be directly weighed in the tray. Approximately 300 grams of marble is used in each trial. The tray was subsequently removed from the balance, and approximately 100 grams of water is mixed until a homogeneous mixture was formed. This final mixture may then be weighed again so that the exact weight of the water added can be calculated. The specific measurements for the empty tray, marble, and water weights may be found in table 1 in the results section. Figure 4: Tray containing the marble sitting within the oven. The marble mixture is then placed on a hanging rack within the Armfield Tray Oven, model UOP8. This oven consists of a metal duct with a heater and fan at one end that supplies warm air to any object placed in the duct. Both the heater and fan are controlled by simple knobs that range from the lowest output of 1, to the highest of 10. To insure that drying finished within the lab period, the heater is always set to 10, while the fan is set to 10 in the first trial and 7 in the second trial. As the marble sat in the drier, 11

12 measurements of the total tray and mixture weight, and the mixture temperature are recorded every three to four minutes. The weight was measured by a scale sitting on top of the oven that held the hanging oven rack, and the temperature of the cake was measured by a Fisher Scientific Traceable infrared thermometer. At eight minute intervals, the dry and wet bulb temperatures may be recorded from a pyronmeter that is placed in a slot of the duct, either directly upwind or downwind of the marble tray. After the temperature measurement, the pyronmeter is moved to the alternative position, and eight minutes again pass to ensure the wet bulb temperature has reached steady-state. The actual values of the humidity are later calculated from the online app provided by Vaisala [5]. The wind speed was also measured every eight minutes, with the Airflow LCA 6000 anemometer, held within an inch of the air duct s end to insure consist measurements. Finally, the temperature of the air in the drier was measured by a thermometer taped to the oven rack. Since this temperature did not fluctuate at any time, the measurement was only taken at the beginning of the drying process. 12

13 Figure 5: Pyronmeter placed within the oven. Both the wet and dry bulb thermometers are held within the metal case. After the marble was able to maintain its weight for 12 minutes (and therefore likely did not contain any water) the tray was removed from the oven and placed on the original balance. If the weight of the marble was within 1 gram of the marble before any water was added to it then the drying process was considered complete. If the weight was greater than 1 gram, then the tray was placed back in the oven. At the conclusion of the experiment, the oven was turned off, and the marble was disposed of in a bucket. This procedure was originally set forth in the Cooper Union Lab Manual [?]. 13

14 Figure 6: Tray and parts of the marble cake after drying has completed. 14

15 3 Results and Discussion To investigate the heat and mass transfer characteristics of the drying experiment, the theoretical transfer coefficients were compared to empirical values calculated from the collected data, a summary of which can be found in table 1. The conditions for each trial were kept relatively similar except for the air speed; the air speed of trial 1 is greater than that of trial 2. Table 1: Overall Weight Changes Through Drying Trial 1 Trial 2 empty tray (g) marble (g) water (g) marble after drying (g) atmosphere pressure (mm Hg) ambient temperature ( F) The theoretical mass transfer coefficient was first determined using the relationships illustrated in equation 1 and equation 2. The parameters required to solve for the coefficient, namely the absolute and saturation humidities, the enthalpy of vaporization, and the heat transfer coefficient, were calculated for each measurement using their dependence on the temperature of the solid-liquid mixture as well as the dry and wet bulb temperatures. To calculate the heat transfer coefficients and the mass transfer coefficients, equation 3 and equation 4, respectively, were employed; these values are displayed in the appendix. Table 2 displays the average empirical heat and mass transfer coefficients for drying regimes two and three for both trials. Table 2: Average Theoretical Heat and Mass Transfer Coefficients Trial Drying Regime h (W/m 2 /K) k (m/s) 1 2 II III II III

16 As seen from their values, the transfer coefficients are greater for regime two, as expected, since the drying rate is quicker for the second drying regime. Because trial 1 has a greater air speed, the drying rate of trial 1, as well as its transfer coefficients, should be greater than those of trial 2 which the data supports for regime two. Regime three for both trials should be similar as they only depend on the rate of diffusion of water through the marble mixture. Subsequently, another method was utilized to determine the mass and heat transfer coefficients using experimental data. The change in moisture content of the solid-liquid mixture per unit time calculated for each measurement was used to determine the drying rate and empirical transfer coefficients by applying equation 5, the results of which are shown in the appendix; the average transfer coefficients for each drying regime and for both trials are shown in table 3. Table 3: Average Experimental Heat and Mass Transfer Coefficients Trial Drying Regime h (W/m 2 /K) k (m/s) 1 2 II III II III With both a theoretical value and an experimental one for the mass and heat transfer coefficients, a percent error can be calculated for both coefficients for each trial. A theoretical total time for drying can be calculated using equation 11 for each measurement and averaged to estimate an actual time for drying. Table 4 displays a summary of the values determined by the calculations. Table 4: Drying Experiment Summary theoretical empirical Abs. % Diff. trial h (W/m 2 /K) k (m/s) t T (min) h (W/m 2 /K) k (m/s) t T (min) h k

17 As seen from the table, the transfer coefficients calculated from the collected data are greater than those generated from equations. This is probably due to the fact that the water loss determined from the humidities of the inlet and outlet doesn t properly account for all the water lost from the mixture. Water vapor may escape from the various holes such as those used to set the psychrometer in the drying unit, and therefore utilizing a mass balance can draw an incomplete picture of the situation. Measuring the mass of the mixture itself to generate moisture content at each interval should be a far more accurate representation of the water lost of the mixture. With a greater amount of water loss accounted for, the calculated drying rate and transfer coefficients should be greater in the empirical scenario. Another important distinction to note is the similar and different coefficients for each trial. The heat transfer coefficients for both trials are relatively similar as expected since the heater setting was set to the maximum of 10 for both trials. The mass transfer coefficients for the first trial noticeably exceeds those of the second trial, again as expected; the fan setting and therefore air speed are all greater in the first trial than the second one. Faster air speeds means an increased rate of convective mass transfer and mass transfer coefficient. This would also effect the total time of drying for both theoretical and actual values; the second trial has greater drying times as expected. While the percent errors between theoretical and empirical values are relatively large, they are still within the margin of error to be good predictors for future experiments. The calculated mass and heat transfer coefficients may provide good estimates for rates of drying processes. The estimated drying times, however, are nearly equal and should be very valuable information for similar drying processes. Figure 7 and figure 8 show the plots of the time drying versus the moisture content of the mixture for both trials. While it is expected to observe three distinct regimes of drying on the plots, only two are seen for both figures. When the mixture first begins the drying process, the drying rate should steadily increase as the liquid reaches a constant temperature inside the oven, regime two should occur afterwards when the temperature of the mixture 17

18 and also the drying rate are approximately constant and should result in a linear section of the drying curve. For the plots generated from experimental data, the curves seem to trend linearly from the beginning; the first regime of drying likely happens too quickly for the plots to display. As seen from the raw data in the appendix, the temperature of the mixture reaches a constant value after only one or two measurements. Figure 7: Drying experimental curve for trial 1 18

19 Figure 8: Drying experimental curve for trial 2 When all the water at the surface of the mixture exposed to the air evaporates, the third drying regime starts. The rate of drying in this regime is no longer dependent on the rate of vaporization of the water; the diffusion of water through the marble is the rate limiting step, which severely decreases the rate of drying. This can be seen in both figure 7 and figure 8 as the drying curves approach zero moisture content asymptotically. 19

20 References [1] Seader, J. D.; Henley, E. J.; Roper, D. K. Seader. Separation process principles. Hoboken, NJ: Wiley, [2] Boyer, R.; Huff, K. Using Dehydration to Preserve Fruits, Vegetables, and Meats. Virginia Cooperative Extension, [3] Green, Don W., and Robert H. Perry. Perry s Chemical Engineering Handbook. 8th ed. New York, NY: McGraw Hill, [4] Drying. ChE 372 Senior Chemical Engineering Laboratory. The Cooper Union; New York, NY. [5] "Vaisala Humidity Calculator 5.0." Vaisala Humidity Calculator 5.0. Accessed February 22,

21 4 Appendices 4.1 Variable Definitions Symbol Units Meaning R c mol s 1 m 2 Drying rate, constant R mol s 1 m 2 Drying rate, general h W m 2 K 1 heat transfer coefficient k m s 1 mass transfer coefficient T g K temperature, ambient gas T w K temperature, wet bulb M w mol g 1 molar mass of water H s g m 3 humidity, saturated H a g m 3 humidity, absolute G g s 1 m 2 flux of air though drier Re - Reynolds Number Sc - Schmidt Number Sh - Sherwood Number L m length of tray D m 2 s 1 Mass diffusivity t s time A m 2 area of tray mol mol moles of mixture in tray ms g mass of marble in tray X - moisture content, general X c - moisture content, end second regime X o - moisture content, start second regime X f - moisture content, end third regime H V ap J mol 1 enthalpy of water vaporization 21

22 4.2 Sample Calculations To reach values for the transfer coefficients and time to completion, calculations were completed that roughly model the equations set out in Separation Process Principles [1]. To begin these calculations absolute humidity, relative humidity and vapor pressures were calculated from the gathered wet and dry bulb temperatures, and the online tool provided by Viasala [5]. From this point on, equations gathered from Perry s Chemical Engineering Handbook [3] and Separation Process Principles [1] were used to complete the analysis. A description of these calculations are shown below. Saturation Humidity Saturation Humidity = H a relative humidity times0.01 (12) 47.7 g m 3 = (13) Enthalpy of Vaporization Enthalpy V aporization = ( ) (1 T ) ( T T crit T crit T T crit (14) 2 ) 43, 783( J mol ) = ( ) ( )( ) (15) 647 Theoretical Heat Transfer Coefficient ( P h = T ) (16) 22

23 20.17 W ( ) , 592 m 2 K = (17) Theoretical Mass Transfer Coefficient k = v air L 1/2 viscosity viscosity D 1/3 D L (18) k m s /2 1/ = (19) Rate of Drying R c = mol t A (20) mol sm 2 = (21) Moisture Content X = m wet m dry m dry (22) = (23) Time of Second Regime t second regime = mol s (X o X c ) AR c (24) 23

24 7, 385s = ( ) (25) Time of Third Regime t third regime = mol sx c ln AR c ( Xc X f ) (26) 13, 375s = ( ) ln 0.03 (27) Total Time to Dry t T = (t second regime + t third regime ) 1 60 (28) 346min = (t second regime + t third regime ) 1 60 (29) 24

25 4.3 Data Table 5: Data Collected from First Trial (10 on Both Heat and Wind Settings) t (min) mass total (g) T ( F) air speed (m/s) T d ( F) T w ( F) psychrometer location outlet inlet outlet inlet outlet inlet outlet inlet outlet inlet outlet inlet outlet inlet outlet inlet outlet inlet outlet 25

26 t (min) mass total (g) T ( F) air speed (m/s) T d ( F) T w ( F) psychrometer location inlet outlet inlet outlet inlet outlet inlet outlet inlet outlet Table 6: Data Collected from Second Trial (10 Heat Setting 7 Wind Setting) t (min) mass total (g) T ( F) air speed (m/s) T d ( F) T w ( F) psychrometer location outlet inlet outlet inlet outlet inlet outlet 26

27 t (min) mass total (g) T ( F) air speed (m/s) T d ( F) T w ( F) psychrometer location inlet outlet inlet outlet inlet outlet inlet outlet inlet outlet inlet outlet inlet outlet inlet outlet inlet 27

28 Table 7: First Trial Theoretical Parameters t (min) T (K) H a (g/m 3 ) H s (g/m 3 ) H vap (J/mol) h (W/m 2 /K) k (m/s) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

29 t (min) T (K) H a (g/m 3 ) H s (g/m 3 ) H vap (J/mol) h (W/m 2 /K) k (m/s) , , , , , , , , , , , , , , , , Table 8: Second Trial Theoretical Parameters t (min) T (K) H a (g/m 3 ) H s (g/m 3 ) H vap (J/mol) h (W/m 2 /K) k (m/s) , , , , , , , , , , , , , , , , , , , , , , ,

30 t (min) T (K) H a (g/m 3 ) H s (g/m 3 ) H vap (J/mol) h (W/m 2 /K) k (m/s) , , , , , , , , , , , , , , , , , , , , , , , , ,

31 Table 9: First Trial Experimental Parameters t (min) X h (W/m 2 /K) k (m/s) t T (min)

32 t (min) X h (W/m 2 /K) k (m/s) t T (min) Table 10: Second Trial Experimental Parameters t (min) X h (W/m 2 /K) k (m/s) t T (min)

33 t (min) X h (W/m 2 /K) k (m/s) t T (min)