Vodka Production Experiment

Vodka Distillation, Production, and Distribution

By:

Brett Koehn

Submitted March 17, 2015

Based on experimental work for Laboratory #6,

conducted between February 25, 2015 and March 13, 2015 with group members

Michael Kagan and James Makel

in Section 4 of ECH 145A

Table of Contents:

Abstract………………………………………………………………………………………………………………………………………………..2

Introduction……………………………………………………………………………………………………………………………………………3

Theory…………………………………………………………………………………………………………………………………………………5

                -Figure 1…………………………………………………………………………………………………………………………………...6

                -Figure 2. …………………………………………………………………………………………………………………………………7

Experimental Methods………………………………………………………………………………………………………………………………...9

                -6.1…………………………………………………………..……………………………………………………………………...........10

                -6.2………………………………………………………..………………………………………………………………………...........11

                -6.3……………………………………………………..….………………………………………………………………………..........13

Results…………………………………………………………………………………………………………………………………......................13

                -Figure 3……………………………………………..…………………………………………………………………………………..14

                -Figure 4……………………………………………..…………………………………………………………………………………..15

                -Figure 5……………………………………………..…………………………………………………………………………………..16

                -Figure 6……………………………………………..…………………………………………………………………………………..17

                -Figure 7……………………………………………..…………………………………………………………………………………..18

                -Figure 8……………………………………………..…………………………………………………………………………………..18

                -Figure 9…………………………………………..……………………………………………………………………………………..19

                -Figure 10………………………………………..………………………………………………………………………………………19

                -Figure 11………………………………………..………………………………………………………………………………………20

                -Figure 12………………………………………..………………………………………………………………………………………21

Discussion…………………………………………………………………………………………………………………………………................22

Feasibility Report………………………………………………………………………………………………………………………………….....25

                -Figure 13………..………………………………………………………………………………………………………………………26

                -Figure 14………..………………………………………………………………………………………………………………………27

Conclusion…………………………………………………………………………………………………………………………………...............28

Nomenclature Table………………………………………………………………………………………………………………………………….29

                -Table 1…………..……………………………………………………………………………………………………………………...29

References…………………………………………………………………………………………………………………………………................30

Appendix…………………………………………………………………………………………………………………………………..................31

                -Table 2………...………………………………………………………………………………………………………………………...31

                -Figure 15…………………………………………...……………………………………………………………………………………31

                -Figure 16-……………………………………………………………………………………………………………………………….32

                -Figure 17………………………………………………………………………………………………………………………………..33

                -Figure 18………………………………………………………………………………………………………………………………..34

Abstract:

            Vodka distillation, production, and distribution were all practiced during this lab. This lab involved creating binary mixtures of ethanol and water, and measuring the densities. The densities of these mixtures were then compared to the mole fraction of ethanol. The mole fraction of ethanol went down as the density of the binary mixture went down. These binary mixtures were also distilled by a batch distillation and column distillation. Binary distillation allowed us to examine the distillation of a binary mixture at a microscopic level. This is because the instantaneous and cumulative densities of the binary mixture were measured as the distillation proceeded. Column distillation allowed us to examine the distillation of a binary mixture a macroscopic level. This is because we examined the concentration and the rate of the production of the final binary mixture over a long a period of time. The concentration and rate of production allowed us to calculate theoretical costs and profits of making and selling vodka at the University of California, Davis. We found that distilling and selling vodka would be very profitable for University of California, Davis.

Introduction:

           The Vodka laboratory experiment is important for a few reasons. First off, this lab involved distillation. Distillation is a method that has been performed for over 1500 years. Berthelot says that the Greek Alchemist Zosimus began performing distillations in the 3^rd century.^[1] Distillation is also a very common practice in organic chemistry labs, and is performed by chemical engineers regularly. Furthermore, the distillation performed during this lab yielded vodka. The production and selling of vodka or any alcohol is a very common practice in the United States. Manufacturing alcohol is popular in the United States because it is very profitable. Over 17.3 billion dollars’worth of vodka was sold in the United States during the year of 2014.^[2] Furthermore, Boland and Brester discuss how profitable alcohol production is in “Vertical Integration in the Matling Barley Industry: A “Silver Bullet” for Coors?”. According to this work, the Miller Coors company brought in 3.4 billion dollars of revenue in 2010.^[3]  Therefore, understanding the process of distilling and creating alcohol is important because alcohol is a desirable product. Finally, this lab also required the application of alcohol gauging. The government requires vodka distributors by law to perform alcohol gauging before selling alcohol. Therefore this lab was important because it taught the proper way to prepare, distill, and gauge alcohol.

            There have been a few important published works completed in the past that are similar to this vodka lab. About 1300 years after Zosimus, Lord Rayleigh wrote “The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science”.^[4] Lord Rayleigh distilled water and ethanol, which are the two components of alcohol and vodka. Lord Rayleigh proved how effective distillation is because he was able to distill a mixture of 20% ethanol by volume to 89% ethanol by volume or a proof of 178. Proof is the volume percent of ethanol multiplied by 200. The most effective method of distilling vodka is column distillation. McCabe and Thiele discuss column distillation in “Graphical Design of Fractionating Columns”.^[5] This work details how to calculate operating lines from the composition of the lighter component, which was ethanol during the vodka lab, in the vapor and liquid phase. These operating lines are then plotted relative to mole composition of the lighter component in vapor and liquid phase. This plot is used to determine the theoretical composition of the product relative to the number of trays in the column.

Theory:

            Alcohol gauging is a method of correcting the apparent proof of alcohol. The apparent proof is an inaccurate measurement because it decreases with decreasing temperature. Alcohol gauging must be performed to change the proof to an adjusted proof at 60^oF. The government requires alcohol gauging to be performed on any alcohol that will be sold. Therefore, the government can raise alcohol taxes because of adjusted proof.

            This lab involved the use of several general equations. The definition of density is

where ρ is density, M is mass, and V is volume. The relationship between the temperatures of Celsius and Fahrenheit is

where T ^oF is the temperature in Fahrenheit, and T ^oC is the temperature in Celsius. The equation for the thermal expansion of glass is

where ρ(60) is the density after the effect of thermal expansion. The definition of specific gravity is

where SG is the specific gravity and .99904 is the density of water at 4 ^oF. The definition of apparent proof in a mixture of water and ethanol is

where Pf  is the apparent proof and Vet is the volume fraction of ethanol. The definition of a mole of a component is

where mol is the moles of a component and MW is the molecular weight of that component.

Figure 1: Schematic of a simple a distillation. The distilland is heated, condensed, and collected in a separate container as distillate. The separate container of the distillate can be switched in and out to perform a batch distillation.

            During a batch distillation, a batch of distillate is removed from the accumulation of distillate. Therefore, batch distillation provides data that shows how the distillation is proceeding. Batch distillation is used for distilling small amounts of mixtures. Batch Distillation produces very pure mixtures.

Figure 2: This column above used for continually distillation is shown above. The feed stream enters the boiler and then it is continually distilled through each tray. The mixture is then condensed and put into the distillate collector. The reflux drum then removes any of the mixture not collected and feeds mixture back through the column.

Column distillation is more effective than batch distillation, because column distillation continually distills a mixture multiple times without interrupting the distillate. Therefore, column distillation also creates a product that is richer in the lighter component of a mixture when compared to batch distillation. Column distillation can also be analyzed with a McCabe Thiele graph to calculate the theoretical number of plates needed to create a product of a certain molar concentration. However, a McCabe Thiele graph requires a couple of equations to create operating lines which are used to calculate the number of theoretical plates. The first operating line is the slope of the reflux ratio. The second or lower operating line is calculated when the following equation is true,

where Vi is the mole flux of vapor leaving plate i, per mole product, Li is the mole flux of liquid of liquid leaving plate i, per mole product, O is the reflux ratio, and F is the molar flux of feed per mole product. Once this equation is true, the lower operating line can be calculated by the equation.

where xi is the mole fraction of the lighter component in liquid and xw is the mole fraction of the lighter component in the waste.

Experimental Methods:

6.1 Pycnometry:

            This lab was begun by measuring the volume of the pycnometer. The pycnometer was cleaned with acetone and air-dried. Next the pycnometer was weighed while it was empty. The pycometer was filled with deionized water. The pycnometer was tapped to remove any bubbles, and then the pyconmeter was marked with a sharpie at the meniscus of the fluid. A thermometer was then placed inside the pycnometer. The pycnometer was then capped, wiped down with a Kimwipe, and the temperature of the pycnometer was recorded. The thermometer was removed from the pycnometer, and the pycnometer was weighed.

            We created 6 samples of ethanol and water to put in pycnometer. The samples were created in the fume hood using a pipet. The six samples contained the following volumes of water and ethanol in mL respectively: 7.2 and 4.8, 6.6 and 5.4, 5.4 and 6.6, 4.8 and 7.2, 3 and 9, 1.2 and 10.8. These samples were individually put into pycnometer, and the same procedure is followed that was used before when measuring the mass and temperature of the filled pycnometer. A densitometer was also used to measure the density at room temperature.  The data collected for this section of lab was 8 different masses of the pycnometer, 7 different temperatures inside the pycnometer, and 6 different densities of the water-ethanol samples.

            The 6 masses of the pycnometer samples were converted to 6 densities by use of equation 1. The densities and temperatures of the 6 samples were then converted to proof using the densitytoApparentProof.m function. This function adjusted the 6 densities to account for thermal expansion by using equation 3, converted density to specific gravity with equation 4, checked if the specific gravity was in the allowed range by referencing table 6, and then interpolated in table 6 to find the apparent proof at this specific gravity. Table 6 contained relationships between proof, volume fractions of ethanol and water, and specific gravities. The 6 measured temperatures were converted to Fahrenheit with equation 2. The apparent proof and temperature in Fahrenheit was then converted to adjusted proof using the secondinterpolation.m function. This function corrected the apparent proof with the 60^oF adjusted proof using table 6. Table 6 contained the relationship between proof at 60^oF and apparent proof. The function also had to double interpolate to find the adjusted proof. The volume fraction of ethanol was calculated with equation 5. The volume fraction was multiplied by the pycnometer volume and then put in equation 1 to give the mass of ethanol. Equation 6 was used to convert the mass of ethanol to moles of ethanol. The same process was followed to give the moles of water. The mole fraction of ethanol was then calculated by the dividing the moles of ethanol by the total moles. Theoretical values of density and proof were then extracted directly from table 6. Similar to how the adjusted values of proof were converted to mole fractions of ethanol, the theoretical values of proof were also changed to ethanol mole fractions. This theoretical data is known at TTB curve A plot was made of the mole fraction of ethanol versus the density for both experimental and theoretical values.  A second plot was made with the experimental and theoretical values of ethanol mole fractions and adjusted proof.

6.2 Batch Distillation

            A sample of 10% ethanol by volume was created. 25 mL of ethanol and 225 mL of water was prepared in a three-neck round bottom flask. This flask was clamped above a heater, and a thermometer was placed inside the flask. A simple condenser was then placed on the opposite side of the neck. The distillation was started. The sample was heated to 78^oC, and we began recording data. We recorded the time, the temperature above the round bottom flask, the instantaneous temperature, the cumulative temperature, the instantaneous density, cumulative density, and the volume of distillate. We recorded values for every 10 ml of distillate collected. The cumulative densities and temperatures were recorded by using the densitometer on the total distillate. The instantaneous densities and temperatures were recorded by collecting 3 mL of distillate separately from the cumulative distillate, and then measuring this 3 mL with the densitometer sample immediately. The 3 mL sample was then poured back into the cumulative distillate. The distillation stopped once 150 mL of distillate had been collected.

            The cumulative and instantaneous densities and temperatures were converted to adjusted proof by using the densitytoApparentProof.m and secondinterpolation.m. The adjusted proof was converted to volume fraction of ethanol by equation 5, the volume of ethanol was calculated by multiplying the volume fraction of ethanol by the total volume of distillate for the cumulative values and 3 mL for the instantaneous values, the volume fraction of ethanol was converted to grams of ethanol by equation 1, and grams of ethanol was converted to moles of ethanol by equation 6. The same process was repeated for water, and the mole fraction of ethanol was computed by dividing by the moles of ethanol by the total moles.

A mass balance was applied on the distilland. The initial grams of ethanol and water in the distilland was calculated by equation 1. The initial grams of ethanol was subtracted by the grams in the cumulative distillate to give the grams left in the distilland. The grams of ethanol and water in the distilland were then converted to moles with equation 6. The moles of ethanol were divided by the total moles to obtain the mole fraction of ethanol. The volume of distillate was plotted versus the mole fraction of ethanol in the cumulative distillate, the instantaneous distillate, and the distilland. A second graph was made that plotted the time of the distillation versus the adjusted proof of the cumulative distillate, the instantaneous distillate, and the distilland. Theoritcal curves were superimposed from Rayleigh data. The Rayleigh data was cacluated by estimating a theoritcal mole fraction of ethanol.

6.3 Column Distillation

            A feed of 10% ethanol by volume was feed into the column. For the first trial, the column was set to a feed rate of 1 cc/s, power output of .85 kW, and reflux ratio of 2. For the second trial the column was set to a feed rate of 1.2 cc/s, power output of .85 kW, and reflux ratio of 3. The column needed about 30 minutes to reach equilibrium for each trial. The temperatures of trays 1 to 8 were recorded, as well as the temperatures of the condenser and boilers which are trays 0 and 9 respectively.  Samples of vodka were also extracted from the column and cooled, and the densitometer was used to calculate the densities and cooled temperatures. The flow rate of cooling water was also measured.

            The densities were converted to adjusted proof by using the same two functions from labs 6.1 and 6.2. The mole fraction of ethanol was then calculated exactly as the mole fraction of ethanol had been calculated in labs 6.1 and 6.2. A graph was then created that plotted tray number versus mole fraction of ethanol was plotted for each trial. Another graph was created that plotted tray number versus temperature inside each tray. A McCabe Thiele graph was also created by plotting operating lines, a vapor liquid equilibrium curve, the molar fraction of the waste, the molar fraction of the feed, and molar fraction of the product. Horizontal lines were then drawn from molar fraction of product to waste to determine the number of theoretical plates.

Results:

            During lab 6.1 the density of the ethanol-water samples decreased at a parabolic rate as the mole fraction of ethanol increased. The density of the samples also decreased as proof increased. However, the density decreased at an inverted parabolic rate when compared to proof. The densities ranged from .9410 g/mL to .8210 g/mL. The mole fractions of ethanol ranged from .7563 to .1811. The apparent proof ranged from 83.4134 to 181.8799.

Figure 3: This figure was graphed with data from lab 6.1. This figures shows the density of vodka samples as a function of the mole fraction of ethanol that is in the vodka. The density of vodka of our red experimental data starts at an ethanol mole fraction of a .1811 with a corresponding density of .9410 g/mL. The red experimental data then linearly decreases to an ethanol mole fraction of .7563 and a density of .8210 g/mL. The blue theoretical data is the TTB curve and starts at ethanol mole fraction of 0 and a density of 1 g/mL. This data also decreases at a linear rate. This data eventually reaches an ethanol mole fraction of 1 a density of about .7950 g/mL.

Figure 4: This figure was plotted with data from lab 6.1. This figure shows the density of vodka samples as a function of the adjust proof. The red experimental data starts at a proof of 83.4134 with a density .9410 g/mL, decreases at an increasing rate, and ends at a proof 181.8799 with a density of .7563 g/mL. The blue theoretical data starts at 0 proof with a density of 1 g/mL, decreases at linear rate from 0 proof to 60 proof, decreases at parabolic rate from 60 proof to 200 proof, and ends at proof of 200 and a density of about .7950 g/mL.

            During lab 6.2, the mole fraction of ethanol in the cumulative distillate decreased as the volume of distillate collected increased for both trials. The mole fraction of ethanol in the instantaneous distillate decreased at a quicker rate when compared to the cumulative distillate for both trials. The ethanol mole fraction of the distilland initially decreased, but eventually increased for both trials. The experimental data did not compare well with the theoretical data for lab 6.2.

Figure 5: This figure was created with data from lab 6.2. This figure shows the volume of distillate collected versus the ethanol mole fraction in the cumulative distillate, the instantaneous distillate, and the distilland. The mole fraction of ethanol in the cumulative distillate decreases at a parabolic rate as the volume of distillate increases. The mole fraction of ethanol in the instantaneous distillate decreases at a steeper parabolic rate compared to the cumulative distillate. The mole fraction of ethanol in the distilland initially decrease, but eventually increases. None of the experimental data agrees with the theoretical data from the Rayleigh model calculation.

Figure 6: This figure was tabulated with data from lab 6.2. This lab shows the time versus the adjusted proof of the cumulative distillate, the instantaneous distillate, and the distilland. The cumulative distillate decreases about 70 proof within 2660 seconds. The instantaneous distillate decreases about 90 proof within 1172 seconds. The mole fraction of ethanol in the distilland initially decrease, but eventually increases. None of the experimental data agrees with the theoretical data from the Rayleigh model calculation.

During lab 6.3, the liquid mole fraction of ethanol decreased as the tray number increased for both trials. The temperature increased as the tray number decreased for both trials. The mole fraction of ethanol in the product stream calculated from trials 1 and 2 were .51 and .6 respectively. The theoretical number of plates calculated in trials 1 and 2 was 3 and 3.5 respectively. Therefore, the efficiency of the column for trials 1 and 2 was 33.33% and 38.88%.

Figure 7: This figure contains data that was collected during the first trial of lab 6.3. This figure plots the tray number versus the mole fraction of ethanol in each tray. The mole fraction of ethanol decreases by more than .3 from trays 2 to 4 and then decreases by less than .1 from trays 4 to 9.

Figure 8: This figure contains data that was collected during the first trial of lab 6.3. This figure plots the tray number versus the temperature of that tray. The temperature increases as the tray number increases, except from tray 0 to 1 and from tray 8 to 9. The temperature decreases in between these two sets of trays.

Figure 9: This figure contains data that was collected during the second trial of lab 6.3. This figure plots the tray number versus the mole fraction of ethanol in each tray. The mole fraction of ethanol decreases by .1 from trays 0 to 2, decreases by .3 from trays 2 to 7, and then decreases by .1 from trays 7 to 9.

Figure 10: This figure contains data that was collected during the second trial of lab 6.3. This figure plots the tray number versus the temperature of that tray. The temperature increases a little less than 20^oC from trays 1 to 8. The temperature increases less than 1^oC in between trays 0 to 1 and trays 8 to 9.

Figure 11: This McCabe Thiele table is from trial 1. The waste stream had ethanol mole fractions of .02 in the liquid phase and .015 in the vapor phase. The ethanol mole fractions in the feed stream were .04 in the liquid phase and .12 in the vapor phase. The ethanol mole fractions in the product stream were .51 in the liquid phases and .5 in the vapor phase.

Figure 12: This McCabe Thiele table is from trial 2. The waste stream had ethanol mole fractions of .02 in the liquid phase and .01 in the vapor phase. The ethanol mole fractions in the feed stream were .05 in the liquid phase and .13 in the vapor phase. The ethanol mole fractions in the product stream were .6 in the liquid phases and .6 in the vapor phase.

Discussion:

            In Figure 3, the density of vodka decreases as the ethanol mole fraction increases in both data sets because the density of ethanol at room temperature is .789 g/mL and the density of water at room temperature is .998 g/mL. Therefore, the overall density of vodka will decrease as more ethanol is added, because the density of ethanol is lower than the density of water. This agrees with the Wei and Rowley who performed the same experiment.^[6] Figure one shows a uniform trend between the theoretical data and the experimental data. However, the experimental data does not lie exactly on the TTB curve.

            In figure 4, density decreases as the adjusted proof increases. This is because the adjusted proof is representation of the volume percent of ethanol in vodka. Therefore, as the adjusted proof increases, the mole fraction of ethanol will also increase, and because ethanol has lower density than water, the overall density will also be lower. In figure 2, both data sets display the same trend, but the experimental data is not accurate compared to the TTB

The experimental data from lab 6.1 in both graphs is precise but inaccurate. This inaccuracy is probably from a systemic error during lab 6.1. For example, the volume of the pycnometer was assumed to be 10 mL; but if the pycnometer actually held a different volume, the experimental data would be systematically skewed if the pycnometer did not hold exactly 10 mL. Furthermore, the vodka samples were not immediately covered with Parafilm, so ethanol could have evaporated. There would be systematic error if each of the samples experienced evaporation of ethanol.

In figure 5 and 6, the cumulative ethanol mole fraction decreased as the distillation proceeded, because ethanol has a lower boiling point and density. Therefore, over time more water was distilled into the cumulative distillate and the mole fraction of ethanol decreased. The instantaneous ethanol mole fraction also decreased. However, only 8 points of the cumulative density are shown. This is because there was error and instantaneous densities recorded were too high. Therefore, the densities could not be converted to ethanol mole fraction because at these densities there was not an ethanol mole fraction. The densitometer did not record exact values because there was air in the densitometer. Furthermore, the mole fraction of ethanol in the distilland eventually increases. This error occurred because ethanol evaporated from the distillate. Therefore, when the mass balance was performed, the moles of ethanol increased in the distilland. In addition, these errors caused the experimental data from 6.2 to vary from the superimposed theoretical data. However, these errors are common as Langreth got a similar error from doing a batch distillation.^[7]

In Figures 7 and 9, molar concentration decreases as tray number increases, because as the mixture is being constantly distilled as it travels up the column. In Figures 8 and 10, temperature increases with tray number, because ethanol has a lower boiling point than water. Therefore, the sample becomes easier to boil as it travels up the column. Figures 11 and 12 respectively show that 3 and 3.5 theoretical plates are needed to reach a product stream with the desired molar concentration of ethanol. This yields an efficiency of 33.33% and 38.88%. The column could operate at a higher efficiency.

Feasibility Report:

            To maximize the efficiency of this column, the reflux ratio should be decreased to 2. The reflux ratio should be set to 12, because figure 11 shows that the more bottles of vodka are produced as the reflux ratio decreases. However, the reflux ratio was not tested at a lower rates, so a reflux ratio of 1 or lower could theoretically increase the efficiency even more. The power should be set to .85 kW and the flow rate should be set to 1 cubic centimeters per second. This is because these were the parameters using trial .If the maximizing parameters above are used, the yearly production of vodka bottles 25,000, the annual energy cost is $350, and the gross profit each year is $500,000, and the net profit was $229,950.

            I would recommend that the Chemical Engineering department of Davis pursue this method of creating of vodka. $229,950 is money that this department could use to construct another Chemical Engineering lab.

Figure 13: This pie chart shows the profit and costs of producing vodka annually. The net profit was $229,950, the additional staff salary was $40,000, the yearly energy cost was $350, the bottle cork, and label cost was $12,500, the state tax was $13,200, the federal tax was $54,000, and the raw material cost was $150,000.

Figure 14: The first reflux ratio plotted above was 3 and produced 18,000 bottles of vodka a year at an operating energy cost of $298 a year. The second reflux ratio plotted above was 2 and produced 22,000 bottles of vodka a year at an operating energy cost of $298 per year.

Conclusion:

            This vodka laboratory experiment detailed the different methods of examine and distilling binary mixtures. As a binary mixture is distilled, the product will become rich in the lighter component. Therefore, the initial sample distilled will be the richest in the lighter component. The fraction of the lighter component will then gradually decrease over time as the other mixture also begins to distill.  The methods of distillation must be followed very carefully to create a product of the correct concentration.

            Creating vodka from distillation is a very intricate process, weather the distillation type be batch or column. Column is distillation is much more effective than batch distillation. Producing vodka from column distillation would yield a theoretical yearly profit of $229,950. Therefore, chemical engineers have jobs in the world because of the applications of column distillation. For example, the absolute, most efficient values of reflux ratio, feed rate, and power output were not found, so a chemical engineer could further study and analyze column distillation to truly maximize the efficiency while creating vodka.

Nomenclature Table:

Table 1: This table contains all of the variables that were used for the calculation in this lab.

References:

1)      Bryan H. Bunch and Alexander Hellemans, “The History of Science and Technology,’ Houghton Mifflin Harcourt. 88, (2004).

2)      John Jacobs, “Where America’s Money Goes,” The History of Spending, 34, 432 (2015).

3)      Michael A. Boland and Gary W. Brester, “Vertical Integration in the Matling Barley Industry: A “Silver Bullet” for Coors?” Review of Agricultural Economics, 28, 272-282 (2011).

4)      Lord Rayleuigh, “The Distillation of Binary Mixtures,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 537 (1902)

5)      W.L. McCabe and E.W. Thiele, “Graphical Design of Fraction Columns,” Industrial and Engineering Chemistry, 17, 605-611 (1925).

6)      Chien I Wei and Richard L. Rowley, ‘Binary liquid mixture viscosities and densities’ Journal of Chemical and Engineering Data, 29, 312 (1984).

7)      David Langreth, “Structure of Binary Liquid Mixtures,’ Physical Review, 8, 456-495, (1967).