Mass Transfer Principles for Vapor-Liquid Unit Operations (1 of 3)

www.ChemicalEngineeringGuy.com
1. Introduction
2. Equilibrium Fundamentals
3. Molecular Diffusion
4. Convective Mass Transfer & MT Coefficients
5. Interphase Mass Transfer
6. Conclusion

1. Introduction
2. Equilibrium Fundamentals
1. Basics
2. Vapor-Liquid Equilibrium
3. Equilibrium Diagrams
4. VLE Cases
5. Deviations of VLE
6. Bonus – Getting VLE Data
7. Gas solubility in Liquids
3. Molecular Diffusion
1. Introduction to Mass Transfer
2. Fick’s Law & Cases
4. Convective Mass Transfer & MT
Coefficients
1. Convective MT
2. MT Coefficients
3. Correlations
5. Interphase Mass Transfer
1. Introduction to MT in Interphases
2. Theories for Diffusion between Phases
3. Two Film Theory Applied to Industrial
Processes
6. Conclusion

 Understand the principles behind Unit Operations involving
Vapor-Liquid Phases.
 Know which Industrial Applications are based in this concept:
 distillation, gas absorption, stripping, scrubbing, etc…
 Model Molecular Diffusion via Fick’s Law and its common cases
(UMD & EMD)
 Know the relevance of Equilibrium & Mass Transfer

 Understand Correlations for Mass Transfer Cases and why they are
used in Engineering Applications
 Understand the importance of films & interphases
 Learn about the different Theories behind Interphase Mass
Transfer
 Solve common engineering problems found in Gas absorption and
Distillation

 It is extensively used in the industry.
 You will most likely model/design/operate/control/simulate a Unit
Operation or Process involving this topic
 Many students fail to understand the concepts behind:
 Absorption
 Distillation
 Humidification & Drying
 Even though they might understand, modeling and further
analysis is weak due to lack of theoretical fundaments
 These units are the core fundamentals of Separation Processes,
understand these and it will be easier to understand other
processes

 The size of a distillation or absorption column depends on two things
 The quantity of material to be processed
 The rate at which material can move from one phase to another.
 The First  Mass/Energy Balance Related
 The Second One  Mass Transfer Phenomena!

 Common Examples we will analyze:
 (1) diffusion in a quiescent medium
 Concentration Gradient from Point A to B
 (2) mass transfer in laminar flow
 Flow in pipes and Concentration Distribution
 (3) mass transfer in the turbulent flow
 Mixing in an Agitation Vessel
 (4) mass exchange between phases
 Gas-Liquid Absorption
 Vapor-Liquid Distillation

 Vapor is below its critical point
 Typically, will condense
 Examples
 Water Vapor
 Benzene Vapor
 Diesel/Gasoline Vapors
 Gas is above its critical point
 Typically, will NOT condense
 Examples:
 Nitrogen
 CO2
 Helium

 Check out my old video on Gas vs. Vapor
 https://www.youtube.com/watch?v=fqXXe9wnVFQ
 Like it 

 Vapor-Liquid
 Distillation
 Batch
 Flash
 Fractional
 Reactive
 Vacuum
 Steam
 Humidification & Drying
 Gas-Liquid
 Absorption
 Gas Absorption
 Gas Desorption
 Stripping
 Scrubbing

 The process by which one or more solutes are
absorbed by another solvent, typically in a gas-
liquid interaction in which the material goes form
gas to liquid
 If you want to get technical:
 What is Gas Absorption? (Lec041)
 https://www.youtube.com/watch?v=luRgTdLnxgg

 The process by which one or more solutes are absorbed by another solvent, typically
in a gas-liquid interaction in which the material goes form liquid to gas

 It is a physical separation process where one or more
components are removed from a liquid stream by a
vapor stream.
 In industrial applications the liquid and vapor streams
can have co-current or countercurrent flows.
 Stripping is usually carried out in either a packed or
trayed column

 A scrubber is a waste gas treatment installation in
which a gas stream is brought into intensive contact
with a liquid
 The goal is to allow certain gaseous components to
pass/dissolve from the gas to the liquid.
 The Gas Scrubber removes traces of liquid droplets
from gas streams to protect downstream equipment
from damage and failure.

 It is typically used upstream of gas treating
equipment that contains dry desiccants or
mechanical equipment such as compressors.
 Examples:
 Amine gas treating
 Carbon dioxide scrubbing

 Check out this video:
 https://www.youtube.com/watch?v=LaQ26JEFuec

 It is the process of separating the components
or substances from a liquid mixture by using
selective boiling and condensation.
 There are many types:
 Batch
 Flash
 Simple
 Fractional
 Reactive
 Vacuum
 Steam

 Batch  Discontinuous, Discrete Separation
 Flash  Instantaneous Vap-Liq Separation
 Simple  Single-Stage Separation
 Fractional  Multi-Stage Separation
 Reactive  Separation with Reaction
 Vacuum  Pressure Decrease
 Steam  Steam Addition for sensitive Separations

 Check this video out:
 https://www.youtube.com/watch?v=BaBMXgVBQKk|

Separation
Process
Engineering
2nd Edition
by Phillip Wankat
Unit Operations of Chemical Engineering
7th edition
by W. McCabe (Author), J. Smith (Author),
Harriott Emeritus (Author)
Principles and Modern Applications
of Mass Transfer Operations
2nd Edition
by Jaime Benitez

Mass-Transfer
Operations
3rd Edition
by Robert E.
Treybal
Transport Processes and
Separation Process Principles
5th Edition
By Geankoplis, Hersel ,Lepek
Separation Process Principles
2nd Edition
by J. D. Seader, E. J. Henley,
D. K. Roper (Author)

 Teachable Notes
 Is my lecture slow? Do you want to get shorter lectures?
  Use Faster Playback (1.25x or so)
 Scrolling to fast? Moving the pointer very quickly?
  Use Slower Playback (0.5x or 0.25x)
 Do you need extra video resolution?
 Ensure you select high qualities…
 Typically, the website will adjust to your best needs according to your Internet Service Provider
 Ensure to select:
 Common Quality  720p
 Best quality  1080p
 Extra Information below video
 Always ensure to scroll down of the lecture to verify more content:
 Diagrams, Screenshots, Articles, links, etc…

 UDEMY Notes
 Is my lecture slow? Do you want to get shorter lectures?
  Use Faster Playback (1.25x or so)
 Scrolling to fast? Moving the pointer very quickly?
  Use Slower Playback (0.5x or 0.25x)
 Do you need extra video resolution?
 Ensure you select high qualities…
 Typically, the website will adjust to your best needs according to your Internet Service Provider
 Ensure to select:
 Common Quality  720p
 Best quality  1080p

 For all my students, I created this Course:
 ChemEngGuy Student Group
 If you are in a Process Simulation Course:
 Aspen Plus & HYSYS Forum (unofficial)
 Linkedin Q&A Support - Aspen Plus & HYSYS (unofficial)

Contact me if needed!
Please contact me if required (doubts, questions, comments, suggestions)
Contact@ChemicalEngineeringGuy.com
You can also check out more content here:
My Youtube Channel 
My Fan Page 
The LinkedIn
My website:

1. Basic Topics
 Ideal Gas and Solution, Real Gas/Solution, Phase Equilibrium, Solubility
2. Vapor-Liquid Equilibrium
 VLE Pure, VLE Binary, Volatility
3. Equilibrium Diagrams (Txy, Pxy, XY)
4. Vapor-Liquid Equilibrium Thermodynamics
 Raoult’s Law for IG-IS, Models for Real Cases
5. Deviations of VLE: Azeotropes
6. BONUS - Getting VLE Data on Aspen Plus ®
7. Gas Solubility in Liquids
 Overview, Equilibrium Diagrams, Analysis and Henry’s Law

 Ideal Solution & Gas, Equilibrium
 Concept of Ideality vs. Reality
 Non-ideal solution
 Vapor & Partial Pressures
 Phases & Equilibrium between Phases
 Solubility

 The probability distribution of particles by velocity or energy
is given by the Maxwell speed distribution.
 The ideal gas model depends on the following assumptions:
 Molecules are indistinguishable, small, hard spheres
 Collisions are elastic
 All motion is frictionless
 no energy loss in motion or collision
 Newton's laws apply
 The average distance between molecules is much larger than the
size of the molecules
Do you need the Full Version?
https://courses.chemicalengineeringguy.com/courses
My Fan Page 
The LinkedIn
My website:

 Continuation…
 The molecules are constantly moving in random directions with
a distribution of speeds
 No attractive or repulsive forces between the molecules apart
from those that determine their point-like collisions
 The only forces between the gas molecules and the surroundings
are those that determine the point-like collisions of the
molecules with the walls (the so called PRESSURE)
 In the simplest case, there are no long-range forces between
the molecules of the gas and the surroundings
 Such as gravitational, magnetic, electric, etc.. forces
My Fan Page 
The LinkedIn
My website:

 Best definition:
 They follow:
 PV = nRT

 The average intermolecular forces of attraction and repulsion in the solution are
unchanged on mixing the pure liquids
 The volume of the solution varies linearly with composition
 Ex. D1 = 1000 kg/m3; D2 = 800 kg/m3
 What is the density of a 50-50% mix?
 D3 = (1000+800)/2 = 900 kg/m3  only true for ideal solutions
 Same for Tb, Pb, viscosities, etc…
 There is neither absorption nor evolution of heat in mixing the liquids
 Convenient for Energy balances

 Ideality requires that molecules be:
 similar in size
 Structure
 chemical nature
 nearest approach is the optical isomers.
 In practice, for engineering purposes, many solutions or organic compounds in a
homologous series considered approximately ideal.
 Thermodynamically speaking:
 An ideal solution is a solution in which the enthalpy of solution ( ΔHsolution=0 ) is zero
 with the closer to zero the enthalpy of solution  the more "ideal" it becomes

 Typically, will fit our math models
 Will not deviate strongly
 Is based on “simple”
 Assumptions are generally ok, but specific case might not fit this
My Fan Page 
The LinkedIn
My website:

 In real life, we will see that “ideal” cases are not always common
 Its your task as an engineer to understand when we can apply and when we can’t
apply ideal cases.
 When an ideal case is not valid, we tend to state “real” case

 Does not fit properly the Ideal Gas Law Model

 A Tank containing Oxygen Gas Marks 500g of content.
 Verify if this is true with ideal gas law
 Apply Ideal Gas Law
Note 
This is the actual
content!
15
10
0.082
130
atm L
K mol
PV nRT
P atm
V L
R
T K







My Fan Page 
The LinkedIn
My website:

Note 
This is the actual
content!
2
2
(15 )(10 )
(0.082 )(130 )
14.07
(14.07 )(32g/ mol)
450
atmL
Kmol
PV nRT
PV atm L
n
RT K
n mol
mass O mol MW mol
mass O g

 

 


https://demonstrations.wolfram.com/CompressibilityFactorCharts/

 AKA: Real Solution
 Solutions of liquids which do not obey Raoult's Law are called non-ideal solutions.
 Most solutions show, at least to some extent, non-ideal behaviour.
 In other words, these solutions show deviation from ideal behaviour.
 EXAMPLE:
 Ethanol and water, form a non-ideal solution.
 When 50 ml of ethanol is mixed with 50 ml of water, the total solution volume is 96.4 ml.
 Typical Models for “Real solution – Ideal Gas” diluted case  Henry’s Law

 Diagrams for non-ideal solutions
My Fan Page 
The LinkedIn
My website:

 Molecules that are dissimilar enough from each
other will exert repulsive forces.
 For example, polar water molecules are strongly
repulsed by organic hydrocarbon molecules.
 The repulsive forces result in activity
coefficients greater than unity, since the
molecules tend to leave the liquid phase.

 A greater partial pressure will be exerted, leading to a positive
deviation from ideality.
 On the other hand, if the molecules of the
components attract each other strongly, the activity
coefficients will be less than unity (but still greater than zero),
and less molecules will leave the liquid phase.
 The mixture will exert lower partial pressure, and producing
a negative deviation from ideality.
My Fan Page 
The LinkedIn
My website:

 Vapor pressure or equilibrium vapor pressure is defined as:
 the pressure exerted by a vapor in thermodynamic equilibrium with its
condensed phases (solid or liquid) at a given temperature in a closed system.
 The equilibrium vapor pressure is an indication of a liquid's evaporation
rate
 Low vapor pressure  low rate, high Tvap
 It relates to the tendency of particles to escape from the liquid (or a
solid).
 A substance with a high vapor pressure at normal temperatures is often
referred to as volatile.

 The pressure exhibited by vapor present above a liquid surface
is known as vapor pressure.
 As the temperature of a liquid increases, the kinetic energy of
its molecules also increases.
 As the kinetic energy of the molecules increases, the number
of molecules transitioning into a vapor also increases, thereby
increasing the vapor pressure.
 Typical Models:
 Antoine Equation

 In a mixture of gases, each gas has a partial pressure which is the notional pressure
of that gas if it alone occupied the entire volume of the original mixture at the same
temperature.
 The total pressure of an ideal gas mixture is the sum of the partial pressures of the
gases in the mixture.
 Best Modelled by Dalton’s Law
My Fan Page 
The LinkedIn
My website:

 Dalton's law expresses the fact that:
 the total pressure of a mixture of gases is equal to the sum of the partial pressures of the
individual gases in the mixture.
 This equality arises from the fact that in an ideal gas the molecules are so far apart
that they do not interact with each other.

 For example, given an ideal gas mixture of nitrogen (N2), hydrogen (H2) and
ammonia (NH3):
 where:
 P = total pressure of the gas mixture
 PN2= partial pressure of nitrogen (N2)
 PH2= partial pressure of hydrogen (H2)
 PNH3= partial pressure of ammonia (NH3)

 Are Partial Pressure and Vapor Pressure the same concept?
 Can they be equal, numerically speaking?
 How to differ them?
My Fan Page 
The LinkedIn
My website:

 Dalton's Law of Partial Pressures
https://demonstrations.wolfram.com/GasAbsorptionWithChemicalReactions/

 Ammonium nitrite, , decompose on heating to form N2 gas:
 When a sample of this species is decomposed in the equipment shown 
 is collected over water at
 A) Find the total grams of decomposed
4 2NH NO
     4 2 2 22NH NO s N g H O l
2511V mL N 26 , 745T C P torr  
4 2NH NO

 Note that the total GAS “measured” includes:
 Vapor pressure of water
 Nitrogen gas
 Using ideal gas law
total nitrogen vaporP P P 
@ 26 = 25.21vapor
T CP torr 
– 745 25.21 719.79nitrogen total vaporP P P torrrr   
  
  
2
2
719.79 0.511
=
62.36 26 273
0.019726
0.019726 28
0.5523
torr L
atm K
g
mol
PV nRT
torr LPV
N
RT
N mol of N
Mass mol x MW x
M g N
K
ass of






 

My Fan Page 
The LinkedIn
My website:

 Check out this dynamic Lecture in your free time! You will enjoy!
 Ensure to verify differences in Partial Pressure and Vapor Pressure!

 A phase of matter is characterized by having
relatively uniform chemical and physical properties.
 Phases are different from states of matter.
 The states of matter (e.g., liquid, solid, gas) are
phases, but matter can exist in different phases yet
remain in the same state of matter.
 For example:
 liquid mixtures can exist in multiple phases
 such as an oil phase and an aqueous phase.

 Solubility is the property of a solid, liquid or gaseous chemical
substance called solute to dissolve in a solid, liquid or gaseous
solvent.
 The solubility of a substance fundamentally depends on:
 the physical and chemical properties of the solute and solvent
 Temperature
 Pressure
 Presence of other chemicals (i.e. pH)
My Fan Page 
The LinkedIn
My website:

 The extent of the solubility of a substance in a
specific solvent is measured as the saturation
concentration
 where adding more solute does not increase the
concentration of the solution and begins to precipitate
the excess amount of solute.

 The most familiar example of solubility is:
 A solid acting as a solute dissolves in a liquid acting as a solvent
 In this specific course, we will assume solubility as:
 A gas acting as a solute dissolves in a liquid acting as a solvent

https://demonstrations.wolfram.com/DissolvingASolute/
My Fan Page 
The LinkedIn
My website:

 Broad Definition: a state in which influences are balanced.
 There are many types of Equilibriums:
 Force Eq.
 Dynamic Eq.
 Static Eq.
 Thermal Eq.
 Chemical Eq.
 Phase Eq.
 Solubility Eq.

 Phase Equilibrium
 Vap-Liq
 Liq1-Liq2
 Sol-Liq
 Sol-Sol
 Vap-Sol
 Vap-Liq1-Liq2
Typical Pure Component Phase Diagram
Focus: S-L-G-V-SCF
From now on, we will assume
“Phase Equilibrium” as “Vapor-
Liquid Equilibrium”

 Vap-Liq
 Liq1-Liq2
 Sol-Liq
 Sol-Sol
 Vap-Sol
 Vap-Liq1-Liq2
Typical Binary Component Phase Diagram
Focus: Liquid - Liquid
My Fan Page 
The LinkedIn
My website:

 Vap-Liq
 Liq1-Liq2
 Sol-Liq
 Sol-Sol
 Vap-Sol
 Vap-Liq1-Liq2
Typical Binary Component Phase Diagram
Focus: Solid-Solid

 Solubility Equilibrium
 Solid (ST) – Solid (SV)
 Solid (ST) – Liquid (SV)
 Solid (ST) – Gas (SV)
 Liquid (ST) – Solid (SV)
 Liquid (ST) – Liquid (SV)
 Liquid (ST) – Gas (SV)
 Gas (ST) – Solid (SV)
 Gas (ST) – Liquid (SV)
 Gas (ST) – Gas (SV)
ST: Solute
SV: Solvent Our main focus in Absorption/Stripping/Scrubbing are gases

1. VLE Pure Phase Equilibrium
2. VLE Binary Mixture Equilibrium
3. Volatility & Relative Volatility
4. Introduction to the K-Value

 Vapor–liquid equilibrium (VLE) describes the distribution of a chemical species
between the vapor phase and a liquid phase.
 The concentration of a vapor in contact with its liquid is often expressed in terms of
vapor pressure
 which will be a partial pressure (a part of the total gas pressure) if any other gas(es) are
present with the vapor.
 The equilibrium vapor pressure of a liquid is (in general):
 strongly dependent on temperature.
Recall: we will assume
“Phase Equilibrium” as
“Vapor-Liquid Equilibrium”
My Fan Page 
The LinkedIn
My website:

 If a vapor with components at certain concentrations or partial
pressures is in vapor–liquid equilibrium with its liquid:
 The component concentrations in the liquid will be determined
dependent on:
 the vapor concentrations
 on the temperature.
 The equilibrium concentration of each component in the liquid
phase:
 Is different from its concentration (or vapor pressure) in the vapor
phase
 But there is a relationship***

 This relationship can be observed/predicted/calculated
 The VLE concentration data can be determined:
 Experimental Data
 Calculated with Raoult's law, Dalton's law, and Henry's law.

 Technically, Vapor-Liquid Phase Equilibrium… Form now on just VLE
 Binary  Two species, this is a Mixture, typically in a homogeneous phase
 Examples:
 Acetone + Water
 Ethanol + Water
 Sulfuric Acid at 98% (Water+ H2SO4)
 Hexane + Octane
 Compositions vary (0-100% and 100-0%)
 Each mix has its unique Vap. Pressure, Tboil
My Fan Page 
The LinkedIn
My website:

 Typical Data collected
 xi  Composition of “i” in liquid composition
 yi  Composition of “i” in liquid composition
 Pi  Vapor Pressure of “i”
 Ti  Temperature of “i”
 PT  Total Pressure of the system
 TT  Total Temperature of the system
Tfixed  Pvaries
Pfixed  Tvaries
Water – Ethanol
100°C, 78°C

 A typical equilibrium curve for a binary mixture on x-y
plot is shown here
 The 45° line is commonly added for reference
 This VLE plot expresses the bubble-point and the dew-
point of a binary mixture at constant pressure.
My Fan Page 
The LinkedIn
My website:

 NOTE:
 This particular VLE plot shows a binary ideal mixture that
has a uniform vapor-liquid equilibrium that is relatively
easy to separate.
 On the other hand, the VLE for non-ideal systems are
more difficult separation.
 NOTE: We will see more Diagrams! See Section 2.3

https://demonstrations.wolfram.com/PXYAndTXYDiagramsForVaporLiquidEquilibriumVLE/

 Volatility is:
 the tendency of a substance to vaporize.
 It is related to a substance's vapor pressure.
 At a given temperature:
 a substance with higher vapor pressure vaporizes more readily than a
substance with a lower vapor pressure
 In order to separate a binary mixture using distillation process,
there must be a difference in volatilities of the components.
 The greater the difference, the easier it is to do so.

 Which is easier to “Separate” at 1 atm
 Methyl Chloride – Fluorobenzene
 Butane Neo-Penante
 Which one liquifies 1st at 25°C
 Butane
 Methyl Acetate
My Fan Page 
The LinkedIn
My website:

 Relative volatility is a measure of the differences in volatility between two
components, and hence their boiling points.
 A measure for this is termed the relative volatility.
 We define volatility of component “A” as
 Partial pressure of component-A divide by mole fraction component-A in liquid

 For a binary mixture of A and B, therefore:
 Volatility of A = pA / xA
 Volatility of B = pB / xB
 Where,
 p is the partial pressure of the component
 x is the liquid mole fraction.
 Relative volatility is the ratio of volatility of A (MVC) over volatility of B (LVC):
𝛼 𝐴𝐵 =
ൗ
𝑝 𝐴
𝑥 𝐴
ൗ
𝑝 𝐵
𝑥 𝐵
=
𝑝 𝐴 𝑥 𝐵
𝑝 𝐵 𝑥 𝐴

 Given that
 XB= (1-XA)
 PB = yBPT
 Then, 𝛼 𝐴𝐵 =
ൗ
𝑝 𝐴
𝑥 𝐴
ൗ
𝑝 𝐵
𝑥 𝐵
=
𝑝 𝐴 𝑥 𝐵
𝑝 𝐵 𝑥 𝐴
𝛼 𝐴𝐵 =
ሻ𝑝 𝐴(1 − 𝑥 𝐴
൫𝑦 𝐵 𝑝 𝑇ሻ𝑥 𝐴
𝛼 𝐴𝐵 =
ሻ𝑝 𝐴(1 − 𝑥 𝐴
൫1 − 𝑦 𝐴ሻ𝑝 𝑇 𝑥 𝐴
=
ሻ𝑝 𝑇 𝑦 𝐴(1 − 𝑥 𝐴
൫1 − 𝑦 𝐴ሻ𝑝 𝑇 𝑥 𝐴
𝛼 𝐴𝐵 =
ሻ𝑦 𝐴(1 − 𝑥 𝐴
ሻ𝑥 𝐴(1 − 𝑦 𝐴
𝛼 𝐴𝐵 =
ሻ𝑦(1 − 𝑥
ሻ𝑥(1 − 𝑦
My Fan Page 
The LinkedIn
My website:

 When a = 1.0, no separation is possible
 WHY?
 both component-A and component-B are equally volatile.
 They will vapourise together when heated.
 Solving the above equation for a = 1.0, we obtain: y = x.
 The larger the value of a above 1.0, the greater the
degree of separability, i.e. the easier the separation.
 Recall that when a system has reached equilibrium, no
further separation can take place - the net transfer rate
from vapour to liquid is exactly balanced by the transfer
rate from liquid to vapour.

 Solving the equation for a = 1.0, we obtain: y = x.
 NOTES:
 The larger the value of a above 1.0  the greater the
degree of separability
 i.e. the easier the separation.

 Recall that when a system has reached equilibrium, no
further separation can take place
 The net transfer rates are ZERO:
 Rate from vapour to liquid = Rate from liquid to vapour.
 IMPORTANT:
 Separation by distillation is only feasible within the region
bounded by:
 the equilibrium curve
 AND the 45o diagonal line.

 From the equilibrium curve:
 the greater the distance between the equilibrium curve
and the diagonal line (where y = x):
 the greater the difference in liquid and vapour
compositions and therefore the easier the
separation by distillation.
 Just keep in mind:
 in general  relative volatility of a mixture changes
with the mixture composition.
 i.e. it is not a fixed value!
My Fan Page 
The LinkedIn
My website:

 If the value of a is known and is constant, we can use it to obtain the equilibrium
curve.
 This can be done by rearranging the equation for relative volatility, to obtain the
function y = f(x) or x = f(y).
 The above are non-linear relationships between x and y.
 These equations can be used to determine the equilibrium relationship ( y vs. x )
provided a (or more usually, the average relative volatility, aave) is known.

 Here are some selected examples of binary mixtures whose VLE can be accurately
represented with a constant relative volatility values.


 Mathematical solutions if available for calculating
the relative volatility of a binary mixture if it is
constant.
 This shows several equilibrium curves with different
values of the relative volatility.
 Note that the greater the value of a, the greater is
the separation area.
http://www.separationprocesses.com/Distillation/DT_Chp01i.htm

https://demonstrations.wolfram.com/FlashDistillationOfAConstantRelativeVolatilityMixture/
My Fan Page 
The LinkedIn
My website:

 When talking about volatility, it is common to use K-Values
 A K-value is nothing more than the ratio between y-x compositions in equilibrium.
 The interesting part on K-Value is that they may be obtained experimentally
 The main goal of thermodynamic calculations, is to obtain a function of yi with
respect to xi…
 This way, only “xi” is required as an input for Ki
 Ki helps in equilibrium calculations, specially for distillation

 For the given graph… Calculate Ki for the given point.
1. x = 0.00
2. x = 0.10
3. x = 0.30
4. x = 0.55
5. x = 0.70
6. x = 0.95
7. x = 0.99

 For the given graph… Calculate Ki for the given point.
1. x = 0.00, y = 0.00; Ki = yi/xi = N/A
2. x = 0.10, y = 0.41; Ki = yi/xi = 4.10
3. x = 0.30, y = 0.55; Ki = yi/xi = 1.83
4. x = 0.55, y = 0.61; Ki = yi/xi = 1.11
5. x = 0.70, y = 0.64; Ki = yi/xi = 0.94
6. x = 0.95, y = 0.98; Ki = yi/xi = 1.03
7. x = 0.99, y = 0.99; Ki = yi/xi = 1.00
My Fan Page 
The LinkedIn
My website:

https://demonstrations.wolfram.com/KValueOfSeveralHydrocarbonsVersusTemperatureAndPressure/

1. Phase Diagrams:
 Phase Rules
 Binary Diagrams
 Txy, Pxy, XY

 We have been showing Diagrams involving relationship between Equilibrium
My Fan Page 
The LinkedIn
My website:

 It is important to know how to read them
 NEVER assume their units/dimensions
 Example:
 Mol per Liter (solute/solution, aka Molarity)
 Mol per Mol (solute/solution, aka Mol Fraction)
 Mol per Mol (solute/solvent, aka Molality)
 mg per mL (solute/solution, aka solubility)
 The diagrams are unique, meaning that they describe a
relationship in equilibrium at given conditions of solute,
solvent, pressure, temperature, etc…

 Technically speaking, Gibbs Phase Rule:
 Is a general principle governing systems in thermodynamic equilibrium.
 If given:
 F as the number of degrees of freedom
 C as the number of components
 P as the number of phases
 Then this is true:
 F = C-P+2
 For a Binary system, C = 2 (2 components, A and B)
 F = C-P+2= 2-P+2 = 4-P
 For a Vapor-Liquid Equilibrium, P = (2 phases, vapor and liquid)
 F = C-P+2= 2-2+2 = 4-2=2
 F = 2
Only Applies for
Phase Equilibrium!

 Recall that Gibb’s Phase Rule  F = C-P+2
 What is the meaning of F?
 The total amount of parameters (variables) that you must set up in order to get a UNIQUE
value
 For the Binary System, C = 2; and Vapor-Liquid System, P = 2, we need 2 variables:
 Typically:
 Temperature
 Pressure
 Composition of Species “i” in Liquid (xi)
 Composition of Species “j” in Liquid (xj)
 Composition of Species “i” in Vapor (yi)
 Composition of Species “j” in Vapor (yj)

 WHY?
 We now that 1 = xi+xj and 1 = yi+yj
 We also know that for a given P-T condition, we will get a single value if given
 Example (Valid only for Binary VLE)
 Given Pressure and xi  Only T is required to get yi *** Most common CASE!
 Given xi and yi  T and P can be obtained!
 Given Temperature and Pressure  Only xi is required to get yi (not so common)
 Given Pressure and yi  Only yi is required to calculate T (not so common)
 The idea:
 Once we complete the degrees of freedom, we can “FIX” the system
My Fan Page 
The LinkedIn
My website:

 This can be drawn in a Diagram…
 Given:
 P is fixed to 1 atm…
 We propose xi from 0.0 to 1.0
 We then need 2 variables (2 DOF)
 T to calculate yi
 yi to calculate T
 If we know the x-y & T at P… we can
draw this:

https://demonstrations.wolfram.com/GibbsPhaseRuleForOneAndTwoComponentSystems/

 Case 1: Case 2: Case 3:
 Prove that you do not need more information to draw a diagram
T & Pi, Py given T, xi, yi given
T, xi, yi given
My Fan Page 
The LinkedIn
My website:

 We will consider only 2-component mixture
 A (more volatile)
 B (less volatile).
 Recall that VLE Data has plenty of value
(compositions, pressures, temperatures, etc.)
 There are 3 types of phase diagram
 X vs. Y
 T vs. x-y
 P vs. x-y

 It shows the typical equilibrium curve for a binary
mixture on x-y plot.
 It contains less information than the phase diagram (i.e.
temperature is not included), but it is most commonly
used.
 It is useful for graphical design in determining the
number of theoretical stages required for a distillation
column.

 These occur when there is a deviation from ideal solution behavior.

 Equilibrium curve for any binary mixture can
be obtained from the corresponding diagram
 In the above analysis, the pressure is assumed
constant.
 When the pressure is changed, the entire VLE
is changed as well.
My Fan Page 
The LinkedIn
My website:

 Find:
 X = ? , Y = 0.50
 X = 0.75, Y = ?

 The Figure 5.1 shows a constant pressure phase diagram for an
ideal solution (one that obeys Raoult's Law).
 At constant pressure, depending on relative concentrations of each
component in the liquid, many boiling point temperatures are
possible for mixture of liquids (solutions) as shown in phase disgram
diagram (Figure 5.1).
 For mixture, the temperature is called bubble point temperature
when the liquid starts to boil and dew point when the vapor starts
to condense.
Constant Pressure Phase Diagram
My Fan Page 
The LinkedIn
My website:

 Boiling of a liquid mixture takes place over a range of boiling
points.
 Likewise, condensation of a vapor mixture takes place over a range
of condensation points.
 The upper curve in the boiling point diagram is called the dew-
point curve (DPC) while the lower one is called the bubble-point
curve (BPC).
 At each temperature, the vapor and the liquid are in equilibrium.
 The constant pressure phase diagram is more commonly used in the
analysis of vapor-liquid equilibrium.
Constant Pressure Phase Diagram

My Fan Page 
The LinkedIn
My website:

 Consider a container whereby a fixed amount of
liquid:
 benzene-toluene mixture
 Its gonna be gradually heated.
 Describe Steps:
 A – Liquids both, below bubble point
 B – The liquid gets saturated, Bubble point
 C – Partial evaporation
 D – The vapor gets saturated, Dew Point
 E – Vapors both, above their dew point

 We will study the changes in:
 composition (mole fraction benzene)
 Temperature
 P = constant
 From point (a) to point (e)
 The Initial Concentration:
 Benzene = 0.40 mole fraction
 Toluene = 0.60 mole fraction

 Point (a)
 Liquid-phase
 no vapour
 T 86°C
 Concentration of benzene Liquid:
 x= 0.40; Vapour: y = 0.00
My Fan Page 
The LinkedIn
My website:

 Point (b)
 Liquid-phase
 first bubble of vapour produced
 Temperature 95.2 oC
 (Bubble Point - solution about to boil)
 x = 0.40; Vapour: y = 0.61

 Point (c):
 Vapour-Liquid Mixture;
 continued vaporization of liquid as heat is added
 Temperature 98.0 oC
 x = 0.31; Vapour: y = 0.52

 Point (d):
 Vapour-phase;
 last droplet of liquid remains
 Temperature 101.6 oC (dew point)
 Concentration of benzene
 Liquid: x = 0.21; Vapour: y = 0.40 Point
My Fan Page 
The LinkedIn
My website:

 (e): Vapour-phase only,
 no liquid
 Temperature 108 oC

 Identify Composition of Points A, B, C
 Find dew point Pressure at T = 60°C

https://demonstrations.wolfram.com/VaporLiquidLiquidEquilibriumVLLE/

https://demonstrations.wolfram.com/SolidSolidLiquidEquilibrium/
My Fan Page 
The LinkedIn
My website:

 The constant temperature phase diagram is useful in
the analysis of solution behaviour.
 The more volatile liquid will have a higher vapor
pressure (i.e. pA at xA = 1.0)
Constant Temperature (Isothermal) Phase Diagram

 Although most distillations are carried out
at atmospheric or near atmospheric pressure, it is
not uncommon to distill at other pressures.
 High pressure distillation (typically 3 - 20 atm)
usually occurs in thermally integrated processes
 They also occur when the normal boiling point of
the vapour product is lower than the temperature
of the cooling water required to condense it.
My Fan Page 
The LinkedIn
My website:

 In such cases, it is normally cheaper to pressurize
the column
 This raises the boiling point of the vapour product
 Cheaper than to install a refrigeration system to
condense it.
 As shown here:
 the phase diagram becomes narrower at higher
pressures; and the corresponding temperatures also
becomes higher.
 WHY?

 Separability becomes less at higher
pressures (height decreases)
 At elevated pressures, the vapour phase
deviates from ideal gas behavior
 Note that this might require modifications
to the VLE data is required

 At pressure P3, the critical pressure of
the more volatile component is exceeded
 There is no longer a distinction between
vapour and liquid.
 See  P4, P5
 Distillation is no longer possible beyond
this point.
 Therefore:
 The vast majority of distillations are
carried out at pressures below 70% of the
critical pressure.
My Fan Page 
The LinkedIn
My website:

 For the given system:
 What is the given Temperature
 Pmix at x = 0.3 toluene
 If x = 0.5, P = 40mmHG  phase?

1. Fundamentals
 Equilibrium Models (EOS & Activity Models)
2. Cases
 Case 1: IS – IG
 Case 2: IS - RG
 Case 3: RS - IG
 Case 4: RS - RG
My Fan Page 
The LinkedIn
My website:

 Recall from Thermodynamics II, in order to have “phase” equilibrium, we require
the following to be true:
The concept of Fugacity
is not within the Scope
of the Course!

 Analysis of the equation 
 Logically:
 Left side is vapor related properties
 Right side is liquid related properties
 Mathematically
 We can manipulate variables for our convenience
 Multiply/Divide
 Ignore or Neglect
 Note that this must be done based on real life

 See CASE 1: Ideal Gas with Ideal Solution
 All idealities are valid
 The model simplifies a lot!
 At low P, fugacity coefficients are near 1.
 If no rea solution, then activity must be near 1.
 Then, we get our “ideal” model… We will see
this on Case 1.
My Fan Page 
The LinkedIn
My website:

 Sometimes, the activity coefficient is the MOST relevant concept to solve
 These are called Activity Models. We need to calculate activity coefficient
 Typically in polar-polar mixtures
 Models:
 Margules
 Van Laar
 Wilson
 NRTL
 UNIFAC
 UNIQUAC

 Check out how we solve the following models:
 Margules  https://www.youtube.com/watch?v=moEGVoZp9zg
 Van Laar  https://www.youtube.com/watch?v=lE7Wh6XPv0o
 NRTL  https://www.youtube.com/watch?v=FmWN-suDZuc
 UNIFAC  https://www.youtube.com/watch?v=zoSblGYqoaM

 Sometime, modeling the GAS/VAPOR phase is the most important aspect of the
model, typically at high pressures.
 Activity can be neglected
 The common EOS:
 Peng Robinson (PR)
 Soave Redlich Kwong (SRW)
My Fan Page 
The LinkedIn
My website:

 Fitting Experimental Data to Peng Robinson 
https://www.youtube.com/watch?v=nbyTspnN2bU
My Fan Page 
The LinkedIn
My website:

 Ideal Solution = IS, Ideal Gas = IG
 Real Solution = RS, Real Gas = RG
 Case 1: IS – IG *most commonly analysed
 Case 2: IS - RG
 Case 3: RS - IG
 Case 4: RS - RG

 As stated, the easiest model is the Ideal Solution with Ideal Gas Case.
 Ideal Solution:
 No interaction between A+B
 Ideal Gas
 No interaction between A+B
 The formal definition of equilibrium, for any case will always be:
 Where,
 fi
V = fugacity of species “i” in the vapor phase
 fi
L = fugacity of species “i” in the liquid phase
The concept of Fugacity
is not within the Scope
of the Course!
My Fan Page 
The LinkedIn
My website:

 The Ideal Case:

 Raoult got to this observation experimentally.
 He worked:
 with nonpolar-nonpolar substances (ideal solutions)
 Low Pressure (ideal gases)
 Raoult’s Observation
 The Total Pressure of the system was related to the partial pressure
of A and partial pressure of B
 It was a proportion related to the molar fraction of each
 The vapor pressure of the mixture was never:
 Lower than the lowest vapor pressure component (B)
 Higher than the highest vapor pressure component (A)
My Fan Page 
The LinkedIn
My website:

 Recall that in equilibrium:
 Raoult’s Law States:
 That the partial vapor pressure of each component of an ideal mixture of
liquids is equal to the vapour pressure of the pure component multiplied
by its mole fraction in the mixture.
 Essentially:
This will apply only for:
Ideal solution & Ideal gas
Sat
i i iy P x P

 Mathematically, for a binary system:
 And given that
 Then:
 We get:
This will apply only for:
Ideal solution & Ideal gas
Sat
A A A
Sat
B B B
y P x P
y P x P


1
1
A B
A B
x x
y y
 
 
(1 ) (1 )Sat Sat
B B B A A By P x P y P x P    
(1 ) (1 )
Sat
A A A
Sat
A A B
y P x P
y P x P

   Partial Pressure  T
Variables  xA, yA, T, P
My Fan Page 
The LinkedIn
My website:

 Further Manipulation:
 Replacing for the partial pressures:
T A BP P P 
sat
T B
A sat sat
A B
P P
x
P P




 From
 We have the equation for determining y after x is known:
 By repeating the calculations for several vapour pressures
 We can obtain a series of x and y values from which the equilibrium curve can be plotted.
sat
A
A A
T
P
y x
P
 
  
 
Sat
A A Ay P x P
y mx b
y mx
 

My Fan Page 
The LinkedIn
My website:

sat
A
A A
T
P
y x
P
 
  
 

 Calculate the vapor-liquid compositions (P = 1atm) in equilibrium at 95°C for Benzene-Toluene.
 P°benzene 95°C= 155.7 kPa
 P°toluene 95°C = 63.3 kPa
My Fan Page 
The LinkedIn
My website:

 If in equilibrium  Raoult’s Law is valid, then:
(1 )
( )
101.32 63.
0.411
3
155.7 63.3
4
A A B B T
A A A B T
A A B A B T
A A B T B
T B
A
A B
A
A
x P x P P
x P x P P
x P P x P P
x P P P P
P P
x
P P
kPa kPa
x
kPa kPa
x
   
    
     
     
 

  





 Get all compositions in equilibrium:
 For vapor compositions, get partial pressures
0.41141
5
1
0.588
B A
B
x x
x
   

(0.4114)(155.7 ) (101.3 )
(0.4114)(155.7 )
101.3
0.6323
1 1 0.6323
0.3676
A A T A
A
A
A
B A
B
x P P y
kPa kPa y
kPa
y
kPa
y
y y
y
 



   

My Fan Page 
The LinkedIn
My website:

 Make a conclusion…
0.4114
0.5885
0.6323
0.3676
A
B
A
B
y
x
y
x





 Recall that we defined K-Value as:
 Our task will be to define the ratio with mathematical models… not just by
experimental data!
 We will do this for:
 Raoult’s Model
 Henry’s Law
 More rigorous Models
 Equation of State
 Activity Models
My Fan Page 
The LinkedIn
My website:

 For Raoult’s Law:
 Applying Raoult’s Law, then substituting for Ki:
 We can get Pi-sat using Antoine Equations  this requires Temperature!

 For Henry’s Law
 Given that
 According to Henry’s Law:
 Then 
 Solving for Ki
 In this specific case, K-value is fixed for supercritical components:
( )( )A A totalp y P
AHA Ap x
A( )( ) HA total Ay P x
A A
A total
A
A
A total
A
total
y H
x P
y H
K
x P
H
K
P

 


 Case 1: IS – IG *most commonly analyzed
 Case 2: RS - IG
 Case 3: IS - RG
 Case 4: RS - RG
My Fan Page 
The LinkedIn
My website:

 See Case 4… it is commonly the one used as hardcore calculations are already needed
 Typically:
 Ideal Gas  Easy to calculate
 Real Solution  hard to calculate
 Recall that in equilibrium 
 Fugacity of Liquid & Vapor:
 Equation:
 Forcing Ki   
V
i i i
L sat
i i i i
f P y P
f x P 
 

sat
i i i iy P x P 
sat
i i i
i
y P
x P



 Case 1: IS – IG *most commonly analysed
 Case 2: RS - IG
 Case 3: IS - RG
 Case 4: RS - RG

 Using an Equation of State (EoS) Approach (focus on gas)
 The fugacity of each component is determined by an EoS.
 In other words, both phases are described by only one EoS.
 It is a powerful tool and relatively accurate if used appropriately.
 This approach is widely used in industry for light hydrocarbon and non-polar systems.
 Under these conditions the fugacities are expressed by
My Fan Page 
The LinkedIn
My website:

 Using an EoS-Activity Coefficient Approach
 The approach is based on an EoS which describes the vapor phase non-ideality through the
fugacity coefficient
 This will also consider an activity coefficient model which accounts for the non-ideality of the
liquid phase.
 This approach is widely used in industry for polar systems exhibiting highly non-ideal behavior.
 Under these conditions the fugacities are expressed by

 Note that at low pressures… we can further assume 
 Note that this will be the Case 2 
sat V
i i 
My Fan Page 
The LinkedIn
My website:

 Make no worries! This is typically calculated by a computer & Process Simulator!
 Its important to understand how it is calculated
 What are the effects and changes
 Why are Deviations caused!

1. Deviations
2. Azeotropes
 Positive Boiling Point Azeotrope
 Negative Boiling Point Azeotrope
My Fan Page 
The LinkedIn
My website:

 If no ideal solution/gas, the model deviates:
 Positive Deviations
 Negative Deviations
 (Stronger) Azeotropic formation

 Deviation from ideal solution behaviour can be
classified as:
 Positive
 negative
 This depends on whether the total pressure exerted by
the solution is:
 greater or less than, that predicted by Raoult's Law.
My Fan Page 
The LinkedIn
My website:

 In ideal solution:
 the total pressure varies linearly with liquid mole fraction
 i.e. a straight-line relationship.
 The greater the departure from a straight line
 the greater is the deviation from ideal behaviour,
 The Non-Ideal behaviour is best demonstrated
using constant temperature phase diagrams.

 When the total pressure of a system at equilibrium is less
than the ideal value:
 the system is said to deviate negatively from Raoult's Law.
 the activity coefficients for both A and B are less than 1.0
 The P vs. x lines are located below the straight lines
provided by Raoult's Law.
 The partial pressures of each component are large than
ideal
 The total pressure curve PT vs. x is located below the
straight line for ideal solution.

 A mixture whose total pressure is greater than that
computed for ideality is said to show positive
deviations from Raoult's Law.
 Most mixtures fall into this category.
 The activity coefficients are greater than 1.0
 The P vs. x lines are located above the straight lines
provided by Raoult's Law.
 The partial pressures of each component are larger
than ideal
 The total pressure curve PT vs. x is located above the
straight line for ideal solution.
My Fan Page 
The LinkedIn
My website:

https://demonstrations.wolfram.com/VaporLiquidEquilibriumDiagramForNonIdealMixture/
My Fan Page 
The LinkedIn
My website:

 Very large deviations from ideality lead to a special class of
mixtures known as:
 Azeotropes
 Azeotropic mixtures
 Constant-boiling mixtures.
 It is a special class of liquid mixture that boils at a constant
temperature at a certain composition.
 It behaves as if it was one component with one constant boiling
point.
 A boiling liquid mixture produces a vapour of exactly the same
composition
 The liquid does not change its composition as it evaporates.

 Two types of azeotropes are known
 minimum-boiling
 maximum-boiling (less common).
 One of the best known minimum-boiling azeotrope is
the ethanol-water system:
 at 1 atm occurs at 89.4 mole percent ethanol and 78.2 oC.

https://demonstrations.wolfram.com/AzeotropesOfBinaryMixturesContainingEthanol/

 When the positive deviations from ideality are sufficiently
large
 the mixture is said to form a minimum-boiling azeotrope.
My Fan Page 
The LinkedIn
My website:

 The Figures below show the constant
temperature phase diagram
My Fan Page 
The LinkedIn
My website:

 Next, constant pressure phase diagram plus
equilibrium curve for a minimum-boiling
azeotropic

 Ethanol-water system
 (at 1 atm, 89.4 mole %, 78.2 oC)
 Carbon-disulfide – acetone
 (61.0 mole% CS2, 39.25 oC, 1 atm)
 Benzene – water
 (29.6 mole% H2O, 69.25 oC, 1 atm) are minimum-
boiling azeotropes.

 The characteristic of such mixture is that the total
pressure goes through a maximum (constant
temperature phase diagram)
 Therefore the temperature goes through
a minimum (constant pressure phase diagram), shown
as point L.
My Fan Page 
The LinkedIn
My website:

 At point L
 the concentration in the vapour phase is the same as the
concentration in the liquid phase ( y = x ), and a = 1.0.
 This concentration is known as the azeotropic
composition (0.61 mole fraction CS2).
 At this point, the mixture boils at a constant
temperature (39.25 oC under 1 atm) and without change
in composition.
 On the equilibrium diagram, it can be seen that at this
point, the equilibrium curve crossed the 45o diagonal.
My Fan Page 
The LinkedIn
My website:

 It occurs when the negative deviations are very large, and
the total pressure curve in this case passes through
a minimum
 It gives rise to a maximum in the temperature
 i.e. boiling point
 Shown next, the constant temperature phase diagram and
constant pressure phase diagram plus equilibrium curve for a
maximum-boiling azeotrope mixture of acetone and
chloroform.

 The azeotropic composition is 0.345 mole fraction acetone.
 Point L
 It is now a minimum on the constant temperature phase diagram
 A maximum (64.5 oC, under 1 atm) on the constant pressure phase
diagram.
 These azeotropes are less common than the minimum type.

 For the following systems (1-6)
 Identify whether or not they are azeotropes
 If they are, select if it is a MAX. or MIN Point Azeotrope
 Identify the Composition, Temperature and Pressure for the given azeotrope
 Example:
 (1) benzene - toluene at 334.15 K and 101 kPa;
 This is IDEAL, no azeotrope
 No Max/Min Point
 (x,y) and T,P not required
My Fan Page 
The LinkedIn
My website:

 (2) methanol - water at 298.15 K and 101 kPa;
My Fan Page 
The LinkedIn
My website:

 (3) 1-propanol–water at 323.07 K and 101 kPa;

 (4) 1-butanol (1)–water (2) at 298.15 K and 101 kPa;
My Fan Page 
The LinkedIn
My website:

 (5) ethyl acetate – dichloromethane at 348.15 K and 101 kPa;

 (6) acetone –chloroform at 308.32 K and 101 kPa.
My Fan Page 
The LinkedIn
My website:

 Azeotropic mixtures cannot be easily
separated by ordinary distillation methods.
 For example, in the case of acetic acid - water
 Azeotropic distillation must be used.
 Often the equipment and set-up used is unique
for each mixture.
 See setup 

 Getting Data From Aspen Plus – Existing Components
 Getting Data from NIIST Database
My Fan Page 
The LinkedIn
My website:

 Aspen Plus (AP for short) is the leading Chemical Process
Simulator in the market (or at least in the Chemical
Engineering World).
 It will allow the user to build a process model and then
simulate it using complex calculations:
 models, equations, math calculations, regressions, etc…
 According to AP® website:
 (it will) Maximize profits using a plant-wide simulation solution
that combines unparalleled accuracy and engineering
collaboration with time-saving workflows.(1)

 Read more here:
 https://www.chemicalengineeringguy.com/the-blog/process-simulation/what-is-aspen-plus/

 Aspen Plus has pre-loaded databanks which can be used directly
 Steps:
1. Open Aspen Plus, let it load and open a NEW simulation
2. In the physical environment  add the two components (water thanol)
3. Select an appropriate Property Method
4. Go to Physical Analysis
5. Verify Vapor-Liquid of Binary Mixture
My Fan Page 
The LinkedIn
My website:

 Get a T-xy Diagram given:
 VLE (We use VLL for example)
 P = 1 atm
 Systems:
 Ethane-Pentane
 Water – Phenol
 Methods:
 Ideal
 UNIFAC

 Aspen Plus may not have our required data pre-loaded. Sometime, we need to
retrieve them from experimental data or from NIST Database
 Steps:
1. Open Aspen Plus, let it load and open a NEW simulation
2. In the physical environment  add the two components (water ehantol)
3. Select an appropriate Property Method
4. Go to NIST Databanks
5. Download Vapor-Liquid of Binary Mixture
6. Verify the required data
7. Select it and Plot it
My Fan Page 
The LinkedIn
My website:

 Retrieve data for:
 Binary System
 Water-Ethanol
 Save data
My Fan Page 
The LinkedIn
My website:

Mass Transfer Principles for Vapor-Liquid Unit Operations (1 of 3)

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to Mass Transfer Principles for Vapor-Liquid Unit Operations (1 of 3)

Similar to Mass Transfer Principles for Vapor-Liquid Unit Operations (1 of 3) (20)

More from Chemical Engineering Guy

More from Chemical Engineering Guy (20)

Recently uploaded

Recently uploaded (20)

Mass Transfer Principles for Vapor-Liquid Unit Operations (1 of 3)