MASS TRANSFER: A STUDY ON THE DISSOLUTION KINETICS OF

A POTASSIUM CHLORIDE PILL

GROUP M2:

MICHELLE KAMANDREW N. FRANKLIN

WILLIAM MARSHNOELIA A. PACHECO

SUHAIL TORGA

DECEMBER 16, 2004

HYPOTHESIS

The primary hypothesis is that potassium chloride pills dissolve according to the equation

.

Furthermore, we hypothesize that b and Kc are independent of pill mass.

ABSTRACT

This study analyzed the mass transport of solid potassium chloride through a series of beakers. The potassium chloride was initially introduced into the system as a pill. The beakers were connected via pumps and the dissolution of the pill was tracked via conductivity meters. From this experiment a mean value of 0.0060 +/- 0.0022sec-1

was derived for the degradation constant b along with a value of 0.0399 +/- 0.0050 for Kc. It was also determined that both constants were independent of the mass of the pill, supporting the hypothesis. Additionally the data supports that the above equation does describe the dissolution of the potassium chloride, which is justified by the relatively low percent error between the actual slope of the concentration vs. time graph and the slope calculated using the above equation for the same times. The average percent error between these two values was 48%.

BACKGROUND

Compartmental analysis is a powerful tool used to quantify the change in amount of a particular substance found in interconnected compartments that interact through the exchange of material. For example, the transport of drugs through the body can be modeled with a multi-compartmental system. In our experiment, the mass of KCl in two beakers was measured as a function of time. As the pill dissolved in a well-mixed solution in the first beaker, the solution was continually transferred into second beaker.

Our model of KCl assumes that the dissolution of the pill depends on the surface area of the pill, which decreases exponentially with time as the pill dissolves. The surface area of each pill is given by

A = Aoe-bt

where Ao is the initial surface are of the pill and b is a time constant. The dissolution of the pill also depends on the difference in concentration at the surface of the pill and the well-mixed liquid. Therefore, the rate of dissolution of the pill is given by

dn/dt = Aoe-btKc(Co-C1)

where Kc is a mass transfer coefficient, C0 is the concentration of KCl at the surface of the pill, and C1 the well-mixed concentration of KCl .

A differential mass balance performed on the first beaker gives:

Moles of KCl in beaker 1 = (Moles dissolved from pill) – (Moles leaving beaker 1)

Equation 1

Equation 2

Equation 3 will be used to find the concentration C1 of KCl in the first beaker as a function of time.MATERIALS AND METHODS

Experiment:This experiment involved the use of a pump to transfer solution from one

compartment to the next. A single peristaltic pump with multiple in and out flows was utilized. Equalizing the various flow rates of the pump proved to be a challenge during the experimental setup. Since we were most concerned with the volume in first compartment, we ascertained before each set of trials that the Q1 and Q2 were equal in this compartment in order to keep the volume constant (Refer to Figure 1 below.)

Two 500mL beakers were used with a magnetic stirrer in each. Compartment one (C1) and compartment two (C2) both had an inflow tube, outflow tube, and an electrode. A constant volume of 200mL was maintained in each compartment. Q1 flowed into C1 with distilled water. Q2 flowed out of C1 and into C2 with KCl solution. Q3 flowed out of C2 with waste. With the pump used all the flow rates were approximately equal and thus, Q1 = Q2 = Q3 = 32 mL/min. A small basket made from wire served to immerse the pill in C1 while keeping it safe from damage due to the stir bar.

Figure 1. Experimental set up1

A total of eight pills were used, with two pills of each mass: 0.5g, 1g, 1.5g, and 2g. The KCl salt was ground into a fine powder using a mortar and pestle to increase the cohesion of the pill. After grinding, the predetermined amount was weighed and formed into a pill. After obtaining the height of the pill using a caliper, it was placed in its basket and lowered into the solution.

1 Group R2 2004 Lab Report

Compartment 1 Compartment 2

Equation 3

Biopac and a conductivity meter were utilized to measure the change in conductivity of the solution over time. Before beginning the trials, the meter was calibrated using a one point calibration.

Analysis:The values for the constants Kc and b from equation 3 were determined using

slopes taken from the concentration vs. time graphs. This approach provided an analytical solution to the differential equation simplifying the mathematical computation. The equations used for this approach are provided below:

Kc = [(dc/dt)*V1]/[A0 * C0](assumes t and C1

= 0)

b = Ln[(AoKc(Co – C1) – Q2C1)/((dc/dt)V)]/t(taken for time = 100, uses Kc derived from above equation)

Figure 2. Illustrates the method behind the calculation of Kc and b. The formulas used were b= Ln[(Ao*Kc*(Co-C1)-Q2*C1)/((dC/dt)*V1)]/t, and Kc=V1(dCo/dt)/[Co*Ao]. All Concentrations (C) were from compartment 1, Q2 is the flow rate of KCL solution out of beaker 1, and Ao is the initial surface area of each pill.

Equation 3(rearranged)

RESULTS

Height (cm) Radius (cm) Mass (g)Volume

(cm3)Surface area

(cm2) Ratio (SA/V) Trial0.48 0.63 1.01 0.60 4.39 7.34 10.50 0.63 1.01 0.62 4.47 7.17 20.68 0.63 1.50 0.84 5.17 6.14 10.75 0.63 1.53 0.94 5.46 5.84 20.27 0.63 0.50 0.33 3.54 10.72 10.26 0.63 0.50 0.32 3.51 10.96 21.40 0.63 2.00 1.75 8.04 4.60 11.16 0.63 2.00 1.45 7.09 4.90 2

Table 1. Dimensions of each pill.

Pill (g) Trial Kc b (1/sec)0.5 1 0.0372 0.00600.5 2 0.0499 0.00881.0 1 0.0400 0.00271.0 2 0.0393 0.00641.5 1 0.0425 0.00911.5 2 0.0402 0.00362.0 1 0.0327 0.00582.0 2 0.0371 0.0061  Avg 0.0399 0.0060  St dev 0.0050 0.0022

Table 2. Comparison of experimentally derived Kc and b values for all pills.

Pill Time (sec)

predicted dc/dt

actual dc/dt (from slope of graph)

percent error

0.5 g 150 4.98E-05 9.00E-05 44.6670.5 g 350 -1.82E-05 -5.00E-05 63.6001.0 g 150 9.92E-06 2.00E-05 50.4001.0 g 350 -7.39E-05 -5.00E-05 47.8001.5 g 150 5.72E-05 0.0001 42.8001.5 g 350 -1.81E-05 -3.00E-05 39.6672.0 g 150 1.26E-04 0.0002 37.0002.0 g 350 -2.04E-06 -5.00E-06 59.200

      AVG 48.142      St Dev 9.282

Table 3. Comparison of the slope of the concentration vs. time graph, the slope derived from the actual graph is compared to the slope expected using Equation 3.

Channel 1 Calibration

y = 0.0098xR2 = 1

00.020.040.060.08

0.10.120.140.160.18

0 5 10 15 20

Conductivity (mV)

Con

cent

ratio

n (m

ol/L

)

Graph 1. One point calibration of conductivity meter.

b vs. Mass

y = -0.0005x + 0.0067R2 = 0.018

0.00000.00100.00200.00300.00400.00500.00600.00700.00800.00900.0100

0 0.5 1 1.5 2 2.5

Mass (g)

b (1

/sec

)

Graph 2. If b depended on pill mass, some kind of correlation would be seen. Although the b for each mass is not exactly the same, no trend exists among the different masses.

Kc vs. Mass

y = -0.0049x + 0.0459R2 = 0.3376

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.5 1 1.5 2 2.5

Mass (g)

Kc

Graph 3. If Kc depended on pill mass, this plot would show the correlation. A strong trend between the different masses is not observed, which supports the hypothesis that Kc

is not dependent on mass.

Concentration in C1 vs. Time0.5g Pill

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 500 1000 1500 2000 2500 3000 3500

Time (sec)

Con

cent

ratio

n (m

ol/L

)

Trial 1Trial 2

Graph 4. Both trials of the 0.5g pill are plotted. Data shows considerable overlap, as expected.

Concentration in C1 vs. Time1.0 g Pill

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0 500 1000 1500 2000 2500 3000

Time (sec)

Con

cent

ratio

n (m

ol/L

)

Trial 1

Trial 2

Graph 5. Both trials of the 1.0g pill are plotted. Data shows overlap, but there is an unexpected bump in trial 1.

Concentration in C1 vs. Time1.5 g Pill

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 500 1000 1500 2000 2500 3000

Time (sec)

Con

cent

ratio

n (m

ol/L

)

Trial 1

Trial 2

Graph 6. Both trials for the 1.5g pill are shown. There appears to be less overlap than there was for the smaller masses.

Concentration in C1 vs. Time2.0 g pill

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 500 1000 1500 2000 2500 3000 3500 4000

Time (sec)

Con

cent

ratio

n (m

ol/L

)

Trial 1

Trial 2

Graph 7. Both trials for the 2.0 pill are shown.

DISCUSSION

Most convective phenomena are thought to be related to heat transfer only. As such, convection involves the transfer of heat due to the random motion and mixing of macroscopic portions of a fluid. It is important to note though, that there are two types of convection phenomena: natural and forced. Natural convection occurs as a result of a concentration gradient across the fluid. In contrast, forced convection occurs when the fluid’s motion can be attributed to an external force such as a pump. 2 In either case, heat can be thought of as moving in “chunks.” If this process is compared to mass transfer, the two are analogous.

Convective mass transfer can be said to involve “the migration of matter from a surface into a moving fluid or stream of gas.”3 It involves the random movement of particles through the fluid as a result of density (or concentration) gradients, which can be termed as natural. The movement of particles can also be caused by the presence of a flow rate which prevents the system from coming to equilibrium. Unlike diffusion, which involves microscopic particles, convection involves macroscopic ones. Both cases are analogous to convective heat transfer, and for compartmental analysis, the convective mass transfer can be seen as a combination of these two phenomena: natural and forced convection. 2 Convection -Heat Transfer. Engineers Edge. http://www.engineersedge.com/heat_transfer/convection.htm3 Comaposada Beringues, Josep. Sorption Isotherms and WaterDiffusitivity in Muscles of Pork Ham at Different NaCl Contents. UPC. 1999. http://www.tdx.cesca.es/TESIS_UPC/AVAILABLE/TDX-0430101-081810/

Due to the set-up of the experiment, as the pill dissolved, the particles were able to nearly reach equilibrium within the solution. Simultaneously, this well-mixed solution was pumped through a tube to a second compartment, where the particles tried to reach equilibrium once again. As a result, the pill continued to dissolve in the medium until all of its mass had been exhausted. Some time after the complete dissolution of the pill, all of the KCl had been transferred to the second compartment and eventually to the waste container. This process of having KCl move from compartment one to compartment two to the waste container simulates processes that occur inside the body. In the body, various substances are transferred across systems (or compartments) until they are eventually discarded.

In order to understand how convection affected mass transfer, it became necessary to calculate the convective mass transfer coefficient from the surface of the pill. Since only the surface of the pill was in contact with the surrounding fluid, it is logical to assume that mass should not affect how the pill dissolved. Further more, surface area is a factor which the convective mass transfer coefficient depends on. As observed from Equation 3, if the term Q1C1 is omitted, Kc becomes a function of initial surface area Ao.

In this experiment, only one pill was studied at a time, and therefore, surface area was closely related to how much mass was present in the pill. As seen in Table 2, the data shows that mass does not appear to affect Kc since the calculated Kc ranged within a well defined region: 0.0399 + 0.0050. This is further supported by Graph 3, in which no obvious trend can be seen between mass and Kc. Thus, the data supports the hypothesis set forth at the beginning of the experiment: the convective mass transfer coefficient (Kc) is independent of pill mass. Additionally, that data shows no correlation between b and the mass of the pill, as can be seen in Graph 2.

Given the specified experimental set-up, this hypothesis implies that Kc is also independent of the pill’s surface area since surface area is related to mass, which disagrees with the statement made above that Kc does depends on Ao. This discrepancy can be explained due to the sizes of the pills. Regardless of mass, the Kc of each pill varied only slightly, thus it would be difficult to determine whether surface area played an important role in convective mass transfer. This issue could be addressed by altering the approach taken in the experimental setup. If one made three pills of 0.5g, their total mass would be 1.5g, but the surface area of these three smaller pills would be considerable larger than that of one large pill. The Kc of these multiple pills could then be compared to the Kc of one 1.5g pill. Using this method, one would be able to determine if Kc is affected by surface area.

Regarding the surface erosion or decay of the pill, it can be said that this follows the decay law as shown in Equation 1. This decay is independent of how much mass is originally present in the pill, similar to Kc. As with radioactive materials, where the decay constant remains unchanged regardless of the sample (as long as they are from the sample specimen), the decay constant for the surface area will remain unchanged throughout pills as long as their composition remains the same. The results of this experiment confirm such claim, where the decay constant b was estimated to be 0.0060 + 0.0022 1/s. Such a large standard deviation can be attributed to the fact that the number itself was very small (in the order of ten to the negative three), and that the sample size was relatively small. Nevertheless, no trends arise, and the decay constant can be confirmed to be independent of mass.

The data supports that equation three describes the dissolution of the potassium chloride. This is justified by the relatively low percent error between the actual slope of the concentration vs. time graph and the slope calculated using the above equation for the same times. The concentration at times 150 and 350 seconds were obtained from the graph. Substituting these values into equation 3, the corresponding slopes, dC1/dT, for the graphs at these times were calculated. These slopes were compared with the slopes measured from the actual graphs, and an average error of 48% was obtained. Considering the many possibilities of compounding error, such as an error in pill mass or pill height, this error is relatively small and thus it was conclude that equation 3 can be used to model the dissolution of a potassium chloride pill under the given experimental conditions.