Institute of etroleum Engineering Department of Reservoir Engineering ractical training of reservoir engineering Submitted by : Supervised from : Ahoua David Christian Hussain Yaled Dr.-Ing. Viktor Reitenbach Edited by : Ahoua David Christian Hussain Yaled Adress : Osteröder Straße 36 Study program : Semester : Energy and raw materials : etroleum engineering 5 Semester Matriculation number : Ahoua David Christian : 435 668 Submission date : 03.04.16 Hussain Yaled : 434 997
Content Introduction... 1 1 Vapor liquid equilibrium... 1 1.1 Definition... 1 1.2 Applications... 2 2 Raoult s law... 2 2.1 Derivation (Mathematical Expression)... 3 2.2 Constraints... 4 3 Bubble and Dew oint Calculations in Matlab... 4 3.1 Bubble oint Calculation... 6 3.2 Dew oint Calculations... 10 4 Conclusion (Discussion of the Results)... 14 Bibliography / References... V Annex... VI I
Introduction The main goal of the following report is to present the theory of vapor-liquidequilibrium (VLE) and apply this theory in a practical manner by calculating the bubble point and dew point pressures at given temperatures.for this purpose we will use Matlab as the calculating platform, and also in order to prevent complicated equations only consider ideal vapor and liquid phase behaviors by using the Raoult s law. 1 Vapor liquid equilibrium 1.1 Definition The vapor-liquid-equilibrium is a state at which the gas and the liquid phase are in balance with each other, which indicates that the rate of vaporization ( liquid to vapor ) and condenion ( vapor to liquid ) are equal, that means the amount of substance which condene is the same amount of substance which vaporize. In addition the chemical potential is in both phase the same. For a clearly description of the vapor liquid equilibrium of pure substances, only the temperature and pressure is needed. For gas mixtures also, the composition of the vapor and liquid phases must be known. The compositions are mostly in Mol fractions, x for the liquid phase and y for the vapor phase. The vapor liquid equilibrium exist in the area between the triple point and the critical point.the pressure at the vapor liquid equilibrium is referred to as the urated vapor pressure. Mixture vapor liquid equilibriums are characterized by the fact that the composition of the liquid phase differs from the composition of the vapor phase.this effect occurs through the different volatility and therefore through the different partial pressures of the involved substances and separation processes in particular the rectification to separate mixtures. ure substances equilibrium are mostly presented in T (pressure-temperature) diagram, however the logarithm of the pressure is often plotted against the reciprocal of the temperature, because this sort of presentation approximately results in a straight line. Mixture vapor liquid equilibrium are mostly experimentally determined at constant pressure or temperature. Therefore Vapor liquid temperature of mixture are commonly presented as pressure against composition at constant temperature and for the case of 1
constant pressure temperature against composition. In each case for the liquid and vapor phase. 1.2 Applications Vapor-liquid equilibrium (VLE) help securing the bulk of industrial separation processes, particularly in distillation processes. VLE informations are useful in designing columns for distillation, particularly fractional distillation, which is a particular specialty of chemical engineers. Distillation is a process that use the separation or partial separation of components in a mixture with the operation of (vaporization) followed by condenion. Distillation takes advantage of differences in concentrations of components in the liquid and vapor phases. 2 Raoult s law Raoult's law is a law of thermodynamics established by French chemist François-Marie Raoult in 1882. It states that the partial vapor pressure of each component of an ideal mixture of liquids is equal to the vapor pressure of the pure component multiplied by its mole fraction in the mixture. The mathematical form of the Raoult s law is given as: i = i*xi (ideal solution) or i = i Yi (ideal gas). Where i is the partial vapor pressure of the component i in the gaseous/solution mixture (above the solution), i* is the vapor pressure of the pure component i,i is the liquid pressure of the pure component i and Xi is the liquid mole fraction of the component i, Yi is the vapor mole fraction of the component i in the mixture (in the solution). Therefore, since the partial pressures must be equal at equilibrium, we have the Raoult's Law equation for each component: Raoult s law for the component i : i*xi = i Yi 2
2.1 Derivation (Mathematical Expression) Originally was the Raoult s law found out as an idealized experimental law. With the utilization of the Raoult s law as the definition of an ideal solution, it is possible to figure out that the chemical potential of each component of the liquid is given by : µi = µi* + RT lnxi where µi* is the chemical potential of the component i in the pure state. This equation for the chemical potential may then be used to derive other thermodynamic properties of an ideal solution. Moreover, supposing that the vapor mixture acts as an ideal gas, it is then possible to re-drive Raoult s law as follows. If the system is in balance (equilibrium), the chemical potential of the component i has to be the same in the liquid solution and the vapor above it, that means that µi,liq = µi,vap. Supposing the liquid is an ideal solution, and using the formula for the chemical potential of a gas, gives : µi,liq* + RT lnxi = µi,vap + RT ln ( fi p ), fi represents the fugacity of the vapor of i, indicates the reference state and * indicates the pure component. For a pure i in equilibrium with its (pure) vapor, the corresponding equation is : µ i,liq * = µi,vap + RT ln ( fi p ) By subtracting both equations, the following formula is obtained : RT lnxi = RT ln ( f i ), which re-arranges to fi = xifi*. f i The fugacities can be replaced by simple pressures if the vapor of the solution behaves ideally : pi = xipi* which is Raoult s law. To summarize, it is to notice that, for the application of Raoult s law, an ideal behavior of the solution must be assumed, this assumption comes 3
from the fact that intermolecular forces between different molecules are equal to those between similar molecules which is the condition for an ideal solution. 2.2 Constraints 1. In case of the solution of an electrolyte this law is not applicable. 2.Raoult s law is not applicable in case of volatile solvent. 3.Raoult s law is applicable only in case of an ideal solution 4- Intermolecular forces between the components in the solution should be similar to those intermolecular forces between individual molecules. If we have a solution of A and B forces between A-A = B-B = A-B 5-The gaseous phase are assumed to be act as an ideal gas where we can use ideal gas law. 3 Bubble and Dew oint Calculations in Matlab In this notes we will explain how to perform Bubble and Dew point calculations in Matlab. Vapor (urated) ressure First of all we need the Antoine coefficients of the components which we can get either from the task or from the internet. In the following we assume the units to be in mmhg for pressure, and degrees C for temperature. In Matlab, we must first declare the Antoine coefficients and the mole fractions of the components. 4
all clear x Hex =0.15 x cyhex =0.2 x Hep =0.25 x cyhep =0.1 x Oct =0.15 x Non =0.15 A Hex =6.87024 A cyhex =6.84130 A Hep =6.89385 A cyhep =6.85390 A Oct =6.90940 A Non =6.93440 B Hex =1168.7200 B cyhex =1201.53100 B Hep =1264.37000 B cyhep =1331.5700 B Oct =1349.8200 B Non =1429.4600 C Hex =224.2100 C cyhex =222.64700 C Hep =216.63600 C cyhep =216.3600 C Oct =209.38500 C Non =201.8200 To calculate the urated (vapor) pressures of the components we need to implement Antoine s equation. To return pressure in bar the units are converted by dividing the equation with 750. Hex = @(T) 10 (A Hex B Hex /(T+C Hex )))/750 cyhex = @(T) 10 (A cyhex B cyhex /(T+C cyhex )))/750 Hep = @(T) 10 (A Hep B Hep /(T+C Hep )))/750 cyhep = @(T) 10 (A cyhep B cyhep /(T+C cyhep )))/750 Oct = @(T) 10 (A Oct B Oct /(T+C Oct )))/750 Non = @(T) 10 (A Non B Non /(T+C Non )))/750 To plot the Saturated (vapor) pressures against the temperatures we need the following code. T = linspace(57,92,2) lot(t, arrayfun( Hex, T), T, arrayfun( cyhex, T), T, arrayfun( Hep, T), T, arrayfun( cyhep, T), T, arrayfun( Oct, T), T, arrayfun( Non, T)) xlabel( Temperature [deg C] ) 5
ylabel( ressure [bar] ) title( Saturation (Vapor) ressure ) legend( Hexane, Cyclohexane, Heptane, Cycloheptane, Octane, Nonane, Location grid 3.1 Bubble oint Calculation At given pressure: We begin the bubble point calculation by solving the following equation x i i (T) i 1 = 0 fbub = @(, T, x) (x Hex Hex(T) + x cyhep cyhep(t) + x cyhex cyhex(t) + x Oct Oct(T) + x Hep Hep(T) + x Non Non(T) ) 1 6
If we have given the pressure and x, the bubble point temperature is then the value of T at which fbub(, T, x) = 0 = 100 x Hex =0.15 x cyhex =0.2 x Hep =0.25 x cyhep =0.1 x Oct =0.15 x Non =0.15 Tbub = fzero(@(t)fbub(, T, x),360) After the bubble point temperature is calculated, we can determine the vapor composition by Raoult s law. y i = i (T bub ) x i y Hex = x Hex Hex (Tbub ) y cyhex = x cyhex cyhex y Hep = x Hep Hep (Tbub ) (T bub ) 7
y cyhep = x cyhep cyhep y Oct = x Oct Oct (Tbub ) y Non = x Non Non (Tbub ) (T bub ) To visualize the results, the following code must be typed in. disp([ ressure:,, bar ]) disp([ Bubble oint Temperature, Tbub, degree C ]) disp([ Liquid hase Composition: ]) disp(x) disp([ Vapor hase Composition: ]) disp(y) At given temperature: If we have given the temperature T and x, then the bubble point pressure is the value at which x i i (T) = i The goal here is to find a p that isfies this equation. After calculating the vapor (urated) pressure of every component, we can then determine the partial pressure of each component. The sum of these partial pressures is then the bubble point pressure of the mixture. 8
T = 57 x Hex =0.15 x cyhex =0.2 x Hep =0.25 x cyhep =0.1 x Oct =0.15 x Non =0.15 Hex = @(T) 10 (A Hex B Hex /(T+C Hex )))/750 cyhex = @(T) 10 (A cyhex B cyhex /(T+C cyhex )))/750 Hep = @(T) 10 (A Hep B Hep /(T+C Hep )))/750 cyhep = @(T) 10 (A cyhep B cyhep /(T+C cyhep )))/750 Oct = @(T) 10 (A Oct B Oct /(T+C Oct )))/750 Non = @(T) 10 (A Non B Non /(T+C Non )))/750 Hex = x Hex Hex cyhex = x cyhex cyhex Hep = x Hep Hep cyhep = x cyhep cyhep Oct = x Oct Oct Non = x Non Non 9
bub = Hex + cyhex + Hep + cyhep + Oct + Non y Hex = Hex bub y cyhex = cyhex bub y Hep = Hep bub y cyhep = cyhep bub y Oct = Oct bub y Non = Non bub disp([ < Temperature:, T, degree C ]) disp([ Bubble oint ressure, bub, bar ]) disp([ Liquid hase Composition: ]) disp(x) disp([ Vapor hase Composition: ]) disp(y) 3.2 Dew oint Calculations At given pressure: To calculate the dew point at given pressure, we must first solve the following equation y i i i (T) 1 = 0 10
f dew = @ (, T, x) ( y Hex Hex (T) + y cyhex cyhex (T) + y Hep Hep (T) + y cyhep cyhep (T) + y Oct Oct (T) + y Non Non (T) ) 1 At given ressure and vapor mole fractions y, the dew point temperature T dew is the value for which f dew (, T, y) = 0 = 100 y Hex = 0,15 y cyhex = 0,2 y Hep = 0,25 y cyhep = 0,1 y Oct = 0,15 y Non = 0,15 T dew = fzero(@(t) f dew (, T, y), 360) disp( Dew oint Temperature ) disp(tdew) After we calculating the dew point temperature, we can determine the liquid phase composition from Raoult s law as follows x i = y i i (T dew ) x Hex = y Hex x cyhex = y cyhex Hex (T dew ) cyhex (T dew ) 11
x Hep = y Hep x cyhep = y cyhep x Oct = y Oct x Non = y Non Hep (T dew ) cyhep (T dew ) Oct (T dew ) Non (T dew ) To show the results the following code must be typed in disp( ressure:,, bar ) disp ( Dew oint Temperature, T dew, degree K ) disp( Liquid hase Composition: ) disp(x) disp( Vapor hase Composition: ) disp(y) At given temperature: If we have given the temperature T and y, then the dew point pressure dew is the value at which y i i (T) i = 1 dew The goal here is to find a p that isfies these equation. With the given temperature we can therefore calculate the dew point pressure. After that the calculation of the liquid phase composition can be easily done as follows T = 57 y Hex = 0,15 y cyhex = 0,2 y Hep = 0,25 12
y cyhep = 0,1 y Oct = 0,15 y Non = 0,15 dew = y Hex + y cyhex Hex cyhex + y Hep Hep 1 + y cyhep cyhep + y Oct Oct + y Non Non x Hex = y Hex dew Hex x cyhex = y cyhex dew cyhex x Hep = y Hep dew Hep x cyhep = y cyhep dew cyhep x Oct = y Oct dew Oct x Non = y Non dew Non disp( Temperature:, T, degreec ) disp ( Dew oint ressure, dew, bar ) disp( Liquid hase Composition: ) disp(x) disp( Vapor hase Composition: ) disp(y) 13
4 Conclusion (Discussion of the Results) We have calculated the bubble point and dew point pressures of the mixture at the given temperature range 57-92 C as well as the compositions of the vapor and liquid phases at bubble point and dew point pressures.as you can see, the compositions of the liquid and vapor phases differentiate from each other as stated before in the theory. It is Obvious that at the same temperature, the dew point pressures are smaller than the bubble point pressures which indicates that to reach the dew point curve, it is necessary to increase the pressure, while on the contrary for the bubble point the pressure must be decreased.however these results are only valid for ideal mixtures where Raoult s law is applicable. But in reality most mixtures don t behave ideally and therefore we can conclude that Raoult s law cannot be often applied for VLE calculations without further modifications. 14
Bibliography / References http://www.nt.ntnu.no/users/skoge/septek/lectures/flash_from_skogestad_ book.pdf (01.04.2016) http://vlecalc.com/compound_properties.html?comp1=33&numofc=1&co mpnames=1 (31.03.2016) http://www.mycheme.com/calculating-bubble-dew-points-for-idealmixtures/ (30.03.2016) http://jckantor.github.io/cbe20255/pdf/bubble_and_dew_oint_calculatio ns.pdf (02.04.2016) http://matlab.cheme.cmu.edu/2011/09/15/calculating-a-bubble-pointpressure/#1 (31.03.2016) https://en.wikipedia.org/wiki/vapor liquid_equilibrium (01.04.2016) http://chemwiki.ucdavis.edu/core/hysical_chemistry/hysical_roperties _of_matter/solutions_and_mixtures/ideal_solutions/changes_in_vapor_ ressure%2c_raoult's_law (03.04.2016) https://en.wikibooks.org/wiki/introduction_to_chemical_engineering_roc esses/vapor-liquid_equilibrium (03.04.2016) Reservoir Engineering II Lectures by rof.l.ganzer Thermodynamics: An Engineering Approach" by Yunus A. Çengel, Michael A. Boles, p. 65 V
Annex E X L A N A T I O N We hereby assure that we have written this work independently and that no other than the stated sources and tools were used, and in addition that all references of work that were literally or in essence taken from other sources, are identified, and that the work in same or similar form has not yet been submitted under an examination of any authority. We confirm my agreement, with the review of this work in terms of this information through an information-technical verification test. Clausthal-Zellerfeld, 3. April 2016 Ahoua David, Hussain Yaled VI