Simulation of Diffusion and Convection in Human Capillaries

Submitted by:

Group T4

Amit Mehta

Ofer Sagiv

Kathryn Guterl

Dominic Mangiardi

Abstract

In this experiment, the phenomena of diffusion and convection were examined. Two major trends were seen from the results: it was found that average diffusion flux decreases as the viscosity of the medium increases, and that the average flux also decreases as time passes, due to decreasing concentration gradient. For water as the medium within which dye (1 g/L) was diffusing, the flux was measured at 1.30 x 10-6 kg/m2s, + 2.15 %. For the highest viscosity solution used (25% sucrose), the dye was found to diffuse at a flux of 2.89 x 10-7 + 7.23% - 7.15% kg/m2s. These numbers were within the same order of magnitude that mass transfer theory predicts. When convection and diffusion were run simultaneously, Peclet numbers ranging from 351 down to 14 were simulated. The total flux at which the Peclet number equals 1, which is the case in the capillary, was determined to be 6.305 x 10 –7 kg/m2s for water in the tubing used.

Background

In Vitro Relevance

In the human body, it is necessary at times to transport substances against the direction of blood flow. The main mechanism by which the body can accomplish this is diffusion. For instance, oxygen, hormones, nutrients, and many other substances travel throughout the body in the blood. The elements that flow through the arteries, to the arterioles, and then down to the capillaries, are absorbed into the body as needed. Sometimes, however, a situation develops where a certain part of the body is in need of an element that is in greater concentration downstream in the body. We are able to supply that part of the body with what it needs through the ability to transport substances against the direction of flow. The main mechanism by which this is accomplished is diffusion, or the motion of particles from an area of higher concentration to an area of lower concentration. Additionally, based on the fact that the velocity of fluid is slowest in the capillaries and the diameter is the least (compared to other blood vessels), the capillaries would be expected to be the site of maximum diffusion.

Diffusion

The mass flux of a substance is the amount of substance that passes per given increment of time through a unit area normal to the direction of mass flux. Fick’s First Law of Diffusion is used to describe isothermal, isobaric diffusion through a stationary liquid. Fick’s First Law is the following:

JA,z = - DAB * (dCA/dz)(Eq. 1)

where JA,z is the flux, DAB is the diffusion coefficient, and (dCA/dz) is the concentration gradient in the z direction is used for binary diffusion, or a dilute mixture containing a large excess of all components, or a mixture where all diffusion coefficients are considered equal. Calculation of the flux, JA,z , which describes the amount of substance which passes through unit surface area in unit time will be the primary focus of this experiment. In addition to the concentration gradient, other physical conditions such as temperature differences, pressure differences, and differences in external field forces such as gravity, electric, and magnetic fields affect the mass flux of a substance.

The second law of thermodynamics, which can be used to derive Fick’s First Law, describes how systems not in equilibrium will move toward equilibrium with time. The driving force of substance’s movement is the change in chemical potential with distance, (-duA/dz), where uA is the chemical potential of substance A. The flux term can be related to the chemical potential as follows:

JA,z = CA (vA – Vz)=- CA*( DAB /R*T)* (duA/dz)(Eq. 2)

where CA is the concentration of substance A, (vA – Vz) is the velocity of species A relative to the molar average velocity, R is the gas constant, and T is the temperature. The chemical potential of component A can be described as:

uA = uo + R*T*ln(CA)(Eq. 3)

where uo is a constant. If one takes the derivative of Equation 3 and uses this result in Equation 2, Fick’s Law (Equation 1) is obtained.

Diffusion Coefficient

The Stokes-Einstein equation can describe liquid diffusion coefficients of nonelectrolytes in low concentration solutions:

DAB = (k*T)/(6**r*uB)(Eq. 4)

where k is the Boltzman constant, T is the temperature, r is the solute particle radius, and uB is the solvent viscosity. These liquid diffusion coefficients depend on the concentration due to changes in the viscosity with concentration and changes in the ideality of the solution.

Governing Equations of Diffusion and Convection

The mass flux has both a concentration gradient and a bulk motion contribution when describing a system with fluid flow. The equation of continuity for component A is:

•nA + dp/dt - rA = 0(Eq. 5)

where •nA is the dot product of the gradient and the mass flux vector, dp/dt is the change in density with time, and rA is the rate at which component A is produced. The mass flux relative to fixed spatial coordinates can be further described as:

nA= -p*DAB wA + wA*( nA + nB)(Eq. 6)

where wA is the gradient of the mass fraction of component A. In the situation of no fluid motion, no production of component A, and no variation in the diffusivity or density, the above equations combine and reduce to:

dcA/dt= DAB 2 *cA(Eq. 7)

where dcA/dt is the change in concentration of component A with time and 2 *cA is the gradient of the concentration of substance A squared.

When both convection and diffusion are present, a dimensionless relationship known as the Peclet number is used to determine whether convection or diffusion is dominating in a system. Mathematically, this relation involves the mean velocity of fluid flow,V, the length parameter of the tube, L, and diffusivity of fluid, . (see Equation 8)

Pe = V*L/(Eq. 8)

For a tube, if the length is much greater than the diameter, the length parameter simplifies to just the diameter of the tube. If the Peclet number is less than one, diffusion will dominate over convection in the system.

Procedure

First, four solutions of dye were made using the Fisher Bio-Tech R-250 dye powder at concentrations of 1.0 gram per liter in pure water, 5%, 15%, and 25% sucrose solutions. For each of these mixtures, which were used as diffusion mediums, a calibration curve relating the concentration of dye in solution and the absorbance was constructed using the Spectrophotometer. This was performed by measuring the absorbance of a range of known concentration of dye solutions.

Next, the apparatus used to collect the diffusion only data was constructed, and is shown below in Figure 1:

A horizontal tube of radius 1/8” was set up with a total of four valves: two inner valves two inches apart, and two outer valves 3 inches from the inner valves. Initially, the solution used in the trial (either the water or one of the sugar solutions) was run through the tube until all the air bubbles were flushed out. Then, the bottom clamp was closed, followed by the upper clamp. Then, the two inner clamps were closed, isolating the inner section of the tube. Next, two 0.5 cc needle syringes were used to puncture the tube at either end of the middle section. One syringe contained the dye solution (made with the same solution that the tube already contained), while the other syringe was empty. As the dye was injected into the tube section using the syringe, the other syringe filled, thus keeping the amount of solution in the tube and, therefore, the pressure in the tube, constant. Once all the dye was injected, the solution in the other syringe was pumped back into the section as the other syringe filled. This was repeated several times to ensure that the dye solution in the tube was thoroughly mixed. Then, the two syringes were taken out and immediately Vaseline was used to plug the holes in the tube. This was done to ensure the pressure in the isolated section of the tube matched the pressure throughout the other sections. The two inner valves were slowly and carefully opened simultaneously so as to avoid forming a pressure gradient between the sections of the tube. The tube was then left sitting to allow the dye to diffuse away from the middle section of tube. During data collection, the middle clamps were again closed, isolating the dye that had diffused into the outer regions of the tube. The dye in this outer region was collected by inserting a syringe filled with the solution in the tube into the tube at a point directly after the clamp. The solution in the syringe was injected into the tube, flushing the dye into a pre-massed beaker. This was repeated until all the dye had been collected. Then, this collected solution was diluted with 2 ml of the solution that was used in the tube to ensure that enough solution was available to fill the spectrophotometer couvette. The mass of the beaker and solution was measured, as well as the absorbance of the solution. With this information the exact amount of dye that had diffused was calculated. For each medium, six trials were performed.

The second aim in the experiment was to measure diffusion in the case of forced convection. To accomplish this, the original diffusion setup was expanded, as can be seen in Figure 2.

The tube was filled with solution as before, but this time another tube was connected to the original tube to form a closed loop. This was done to ensure the pressure throughout the tube was equal. Both tubes were completely filled with solution, and the ends were connected immersed in solution so as to prevent any air bubbles from disturbing the mass transport. The closed loop also passed through a flow pump, which was used to create various flow rates (from 0.42 microliters/min to 105 microliters/min ). Additionally, the loop was kept level to minimize any error associated with potential differences. The valves were opened in the same manner as described earlier, and the forced convection caused the dye to both flow and diffuse downstream. After the desired time interval, the valves were closed and the amount of upstream diffusion was calculated in the same manner as for the diffusion trials.

Results

Calibration Curves

Since the spectrophotometer was used to calculate the concentration of dye diffused, calibration curves had to be constructed. This was done for the four different mediums used in the trials. The spectrophotometer absorbance reading was zeroed using distilled water for each curve. Consequently, a different equation was used to calculate the concentration for each medium (see Figure 3).

Results, Diffusion

Table 1 below is a summary of the results from the diffusion trials for all mediums (6 trials were conducted per medium). For each set of trials for a given medium, the original concentration of dye injected was held constant, but the time of each trial did vary. It was observed that as the viscosity of the medium increases, the flux seemed to generally decrease (there seems to be an inconsistency between 5% and 15%; this will be addressed later). As can be seen in the table, the average flux decreases by an order of magnitude from water, whose viscosity is 1, to 25% sucrose solution, whose viscosity is 1.86 cps. Note the relatively large standard deviations for the fluxes (See error analysis for calculation details).

Time Dependence of Flux

The relatively high standard deviations in the average fluxes for each solution (ranging from 27 to 95.4 % of the average flux) led to the closer examination of the only non-constant variable among the trials (for a given viscosity), being time. As can be seen in Figure 4 below for the water trials, for a given viscosity, a clear time dependence of flux was observed. As the run time of each trial was increased, the average flux calculated was decreased. The same (linear) dependence of flux on time was seen for the three sucrose solutions.

Flux Dependence on Viscosity

The general trend relating viscosity and the flux observed in the experiment was further explored. As the flux was found to also be dependent upon time, the comparison between flux and viscosity was only made among trials run for the same amount of time. Figure 5 below shows a plot of flux versus viscosity for only the trials that were run for 60 minutes.

Since the range of times was not constant for the trials of each viscosity, the average flux’s standard deviations seen in Table 1 is not necessarily indicative of the accuracy of these results. Therefore as a result of these findings, a summary of results, separated by time of trial as well as viscosity, was constructed. As would be expected from the trends mentioned, the mass flux values in Table 2 below should decrease with increased viscosity and time (see Table 2).

Statistical basis for Results

In order to see if the differences between flux results among trials of different viscosities were statistically significant, T-tests were conducted. As can be seen from Table 3 below, where the mean fluxes are compared between the 25% sucrose solution trials and the water trials, the t–statistical value was much greater than the t-critical. Since the initial hypothesis was set at 0, or no difference, the t-test indicates that the fluxes are significantly different and therefore verify seen in the results. As can be seen from Table 4, where all other mean fluxes are compared, only the comparison between the 15% sucrose and the water, in addition to the 25% sucrose and water comparison, was statistically different. The reason for this might be because of the small sample size. If time permitted to take more data, the statistical significance could have be seen with all trials. Note that as the difference in viscosities increases, the t-stat increases and approaches t-critical. This might indicate that the mean fluxes would be significantly different if the number of trials was increased.

Diffusion with Forced Convection

The results for the diffusion trials with forced convection can be seen in Table 5 below. The overall (experimentally measured) flux is assumed to be due to a combination of diffusion and forced convection. Based on the setup, there should be no contribution from a pressure gradient or height difference. The flux due to convection was calculated based on the flow rate set on the flow pump, which was found to be accurate to approximately 4.7%. Therefore total flux minus the flux due to convection is the flux due to diffusion. Water was used as the medium in all trials, and the time of trial was maintained at 60 minutes. The Peclet numbers were calculated by using the mean velocity of flow as the flow rate multiplied by the cross-sectional area.

Based on the findings from the trials with only diffusion, it was seen that for a given medium, the same concentration of dye, and the same tubing (which would influence cross-sectional area, and consequently flux), the flux due to diffusion is constant. This phenomenon is reflected in the percent contribution of diffusion to flow, which increases as the flow rate (i.e., forced convection) decreases (see Table 5).

Estimation of Flux at Peclet Number of 1

Based on the definition of Peclet number, this number will equal 1 at the point where diffusion and convection are both equally contributing to the overall flux. The actual value at which this equality occurs can be quantified by extrapolating to the 50% flux point. If this is done with the data presented in Table 5, the flux due to diffusion is 3.153 x 10 –7 kg/m2 s, which is within the same order of magnitude seen in the diffusion results. However, a more accurate extrapolation would require more trials over a wider range of flow rates. (see Figure 6)

Discussion & Conclusions

The initial assumption in this experiment which enabled one to calculate average flux (Table 1) was that the flux was relatively constant. In order for this to be true, the concentration gradient created by the bulk injection of the dye was assumed to remain constant for the time trials were run. Since trials were not run for very long times and since the diffusion of the dye did not advance more than an inch to an inch and a half, this assumption could be made to examine trends in the experiment. (Note this assumption had to be made since the quantity that was measured, bulk concentration in the upstream and downstream chambers, reflects the amount of mass of dye diffused after a certain amount of time but does not provide any information regarding instantaneous flux fluctuation with time nor instantaneous concentration gradient (dc/dx).)

It is now known, after tabulation of the results that the flux does vary with time because of a change in the effective concentration gradient that drives diffusion. Fick’s equation indicates that as the concentration gradient increases, with constant diffusivity, the flux should increase. As can be seen from Table 2, the flux does vary with time but the range of fluxes measured do not overlap over the different times (the exception was the 15 % and 5% sucrose medium trials). This can be best seen between the water and 25% sucrose trials, as the lowest flux for water (8.20 x 10-7 kg/m2s) is higher than the highest flux for 25% sucrose (2.04 x 10-7 kg/m2s). This might render the initial assumption valid for the purpose and method used in this experiment, as the effect of time was not dramatic enough to prevent one from seeing the general trend in viscosity.