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1 INTRODUCTION TO MASS TRANSFER
(Incropera Ch 14; Cengel Ch 14; Incropera Ch 24)
1.1 LEARNING OBJECTIVES
Short Term: After completing this course, you should be able to Apply the fundamentals of heat and mass transfer to

• solve problems;
• quantify transfers of heat and mass à CHE2162;
• design simple equipment à CHE3165;
• communicate results effectively to other engineers (state
assumptions, significant figures, units);
• define the assumptions (and hence limitations) on which
your calculations depend; and
• comment on the validity of your calculations.
Long-term: In 10 years time you should still be able to:
• identify the mode/s of heat and mass transfers;
• assess how these might be affected by environmental
conditions (temperature, fluid properties, velocity, pressure
• perform basic calculations to quantify heat and mass
transfer rates and/or parameters;
• set out/set up problems so they can be solved (draw
diagram, state assumptions, and work out how to proceed);
• interpret your answer: level of confidence? under/over-
predict? implications for operation? validity of assumptions?

CHE2163: HEAT AND MASS TRANSFER INTRODUCTION TO MASS TRANSFER WEEK 1
Note: due to deficiencies in textbooks – readings are from a combination of Incropera, Cengel, and WWWR (see Moodle for PDF copies of relevant material from Cengel and WWWR)
Learning Objectives, Tasks, Readings, and Learn ChemE Videos
1. Explain the key differences between heat and mass transfer a. Cengel p 796-799
b. Unit conversions and the need for equilibrium equations c. LearnChemE – “Deriving Molar Transfer Flux Equations” d. Start filling in the Table starting on p2 of the study guide.
2. Convert fluently between common units of concentration (mass basis, mole basis, concentration, fractions, ratios, partial pressures)
a. Incropera p785-786; Cengel p 799-801; WWWR p 400-403 b. Complete unit conversion worksheet
3. Calculate a diffusive flux using Fick’s Law, understanding the assumptions and limitations associated with its use
a. Incropera p786-790; Cengel p 801-802; WWWR p 399-406 b. Fill in the blanks – Section 1.5
c. LearnChemE – “Hydrogen Diffusion Through Palladium”
By the end of this week, you should be able to:
• Use Fick’s law to solve simple diffusion problems, where concentration data is provided in a range of different units

MASS TRANSFER SUMMARY
MASS TRANSFER SYMBOLS AND UNITS
Actions for students:
• complete the units column in the table of symbols, and indicate which variables are vectors; and
• update table of symbols as course progresses.
• Please note every text is different – we will endeavor to follow a standard set of nomenclature as much as possible, but every textbook is slightly different; if you encounter more than one nomenclature for a term, why not define and reference them separately! (e.g. N’s and n’s and j’s and J’s are different everywhere!)
TABLE OF MASS TRANSFER SYMBOLS AND UNITS Definition
Mass density of species A
Total mass density of the mixture
Total moles per unit volume for mixture Mass fraction of species i in mixture
Molar concentration of species A in mixture
kmol A m-3
C=Σ Ci mA= ρA/ ρ

xA= molesi/total moles (liquid)
Molar fraction of species A in liquid mixture
yA= molesA/total moles (gas)
Molar fraction of species A in gaseous mixture
XA= CA/CB (liquid)
Molar ratio of species A to species B in a
kmol A/kmol B
YA= CA/CB (gas)
Molar ratio of species A to species B in a
Partial pressure of species A in mixture Total pressure of mixture
Molecular diffusivity of A in B

1.2 EXERCISE ON COMPARISONS BETWEEN HEAT AND MASS TRANSFER (CENGEL P796-799):

1.3 WHAT IS MASS TRANSFER? We have already learnt that:
• for heat transfer, we now consider the temperature within a phase B, or crossing an interface between phases B and C (phases can be solid, liquid, or vapour), which could involve diffusive, convective, or radiative modes
• for heat transfer within a single phase, local temperature differences drive energy (heat) transfer down the temperature gradient. We can often assume fluids are well mixed in the bulk, and transfer happens between bulk and interface temperatures (convection). For solids, we often assume a linear temperature gradient for conductive heat transfer.
• for heat transfer across a phase boundary, temperatures on either side of the interface are equal, because thermal equilibrium implies a continuous temperature distribution.

EXAMPLE: Comparison of temperature and oxygen profiles in a lake and the air above a lake.
EXAMPLE: Dissolution of a drug from a tablet into liquid.

Using similar logic, but with crucial differences:
for mass transfer, we now consider the concentration of a solute, A, within a phase B, or crossing an interface between phases B and C (phases can be solid, liquid, or vapour) which could involve diffusive and/or convective modes only.
for mass transfer within a single phase (B or C), local concentration differences drive mass transfer down the concentration gradient. We can often assume fluids are well mixed in the bulk, and transfer happens between bulk and interface concentrations (convection). For solids or stagnant fluids, we might assume a linear concentration gradient for (diffusive) mass transfer. (note – within a single phase, concentration difference is equivalent to chemical potential (fugacity) difference; but these are not equal between 2 phases). But there is a problem here; if we try to formulate an equation for a mass transfer rate for solute A between 2 phases:
for mass transfer, we use additional steps in our calculations which involves (a) unit conversions to work with common concentration units, and (b) equilibrium expressions to determine solute A concentration on one side of the interface, given a known solute concentration on the other side (next lecture!)

TYPES OF MASS TRANSFER
Mass transfer may occur in the absence of fluid motion via molecular diffusion, analogous to the heat transfer mode ___________________, described by ________________
Mass transfer may be enhanced by fluid motion, analogous to the heat transfer mode ___________________, directly related to _______________.
Fluid motion may be external, i.e. forced convection, or generated by a concentration gradient, i.e. _________________.
If fluid is stagnant, we can be faced with situations where molecular diffusion and fluid motion can occur at the same time (e.g. balloon filled with solution containing food dye….. in a river),
EXAMPLES (also see Figures 14-1 through to 14-4 on Cengel p796- 798):

How do we choose concentration units?? o _________________________
o _________________________ o _________________________
• Does this mean we have __ different equations for Fick’s Law? NO – there is one equation, yet it can be expressed different based on the units of the terms involved
• Therefore in mass transfer, we often need to include an extra step in our calculations in comparison to heat transfer in order to account for the:
o ______________________________

1.4 SUMMARY AND REVISION OF COMPLICATIONS FOR MASS TRANSFER – STUDENTS TO COMPLETE:
COMPLICATON 1: UNITS AND UNIT CONVERSIONS
• In heat transfer, the units of Q, k, h, A, T, etc, remain ________ regardless of the nature of the medium through which heat is transferred
• Given that mass transfer must always occur in mixtures, the units of concentration change based on:
o the phase (e.g. _______, ________, or ________) of the medium through which mass is being transferred
o the units of the data provided for both the solute and the medium, e.g. mass data could be provided on a _______ or _______ basis
• As the units of concentration change, then there are often multiple forms of the one equation, to account for __________________
• If problems involve mass transfer in the gas phase, concentration is often provided in units of ___________
• _____________________________ states that the sum of component pressures in a system equals the total pressure of that system
• If we can assume that the ideal gas equation applies, then we can convert units of __________ to units of __________ using the ideal gas equation:

COMPLICATON 2: DISCONTINUITIES AT PHASE BOUNDARIES
• In heat transfer, temperature distributions are considered to be _______________ across phase boundaries, e.g. ______________________
• In fact, to reach this conclusion, we have applied a ___________________, which is required to solve the general heat equation (or the equation left over after we have simplified it)
• In mass transfer, we also require boundary conditions, but we cannot use the same ones as in heat transfer.
• At phase boundaries, we need to know the concentrations of the species of interest (A) on both sides, which is governed by _____________________, such as
o __________ for dilute gas/liquid systems
o __________ for non-dilute gas/liquid systems o __________ for systems involving solids

1.5 EQUILIBRIUM EQUATIONS REQUIRED TO DETERMINE CONCENTRATIONS (INCROPERA P795-798; CENGEL P807- 810; WWWR P438-439)
Concentration gradients can be determined using equilibrium relationships: (these will form the boundary conditions of many problems) For example:
1. Concentration at gas-liquid interface for a dilute solution:
Henry’s law (low solubility)
xA (interface) =pA (interface) /H
H= Henry’s Law constant
2. Partial pressure at gas-liquid boundary for an ideal liquid mixture (could even be pure liquid)
Raoult’s law
pA(interface) =xA(interface) PA,sat
PA,sat = saturation vapour pressure of A 3. Concentrations at gas- solid boundary:
solubility relationship
CA (interface) = S pA (interface)
S = solubility ( kmol/m3kPa) pA = partial pressure (Pa)

i. Determine the mole fraction of air dissolved in water at the surface of a lake whose temperature is 17C and atmospheric pressure is 92 kPa.

ii. A nickel plate is in contact with hydrogen gas. Determine the molar and mass density of hydrogen in nickel at the interface where hydrogen is at 358K, 300 kPa.

iii. Consider a glass of water in a room at 15C and 97 kPa. If the relative humidity is 100% determine
1. Mole fraction of water vapour in air 2. Mole fraction of air dissolved in water

In each case – what is the driving force for mass transfer? Can you calculate it? What would happen if you tried to do this WITHOUT using equilibrium equations?
Pressurised hydrogen gas is stored at 400K in a 4.8m outer diameter spherical container made of Nickel. The shell of the container is 6cm thick. The hydrogen is stored at 10 bar pressure. Outside the container, the concentration of hydrogen is negligible. Determine the mass flow rate of hydrogen from the container.

1.6 SO IN ORDER TO SOLVE A MASS TRANSFER PROBLEM
• We follow the procedure that you have been using in Heat Transfer (definition, diagram…..)
• When it comes to analysis, we often need to add two more steps:
1) Basedontheproblemdataandequilibriumdata,choosea set of units for the concentrations involved and work out the correct version of the Fick’s Law equation
2) Useanequilibriumequation(Henry/Raoult/Solubility/X)to determine concentrations either side of the interface, so that driving force terms are in the same units.
• If you understand how to use unit conversions and equilibrium data in order to calculate mass transfer rates, then you can apply this knowledge to the other problems that we will encounter in this course (“stagnant film diffusion” and “equimolar counterdiffusion” (Cengel p 825-830)), convection, 2-film theory, etc)

Therefore, when approaching mass transfer problems, we can use the same procedure as for heat transfer (i.e. draw diagram, problem definition, assumptions, etc), BUT, there are additional steps required
• UNIT CONVERSIONS and choice of the correct form of the relevant rate equation: you must be fluent in unit conversions to learn mass transfer!!!! (complete Section1.5 and Tutorial 1).
• EQUILIBRIUM EQUATIONS: you must understand diffusion and boundary conditions based on equilibrium relationships (Section 2) in order to determine the concentration driving force across a phase boundary

1.7 UNITS CONVERSION WORKSHEET – STUDENTS TO COMPLETE
You need to learn the following conversions and become fluent in them. Practise, practise, practise!
GAS UNIT CONVERSIONS
(refer to Table 24.1 in WWWR et al. for further information)
Data for the concentration of component A in gas mixtures can be given in many forms, including:
pA: partial pressure of A (kPa, psi, atm etc)
CA: molar concentration of A (k mols m-3 or gmol L-1) ρA: mass density of A (kg m-3, μg L-1 etc)
yA: molar fraction of A (kmol A/total kmol)
YA: molar ratio of A (kmol A/kmol mixture excluding A).
You need to be able to convert fluently between these units. For example, given the molar fraction of gas A, yA, in a gaseous mixture and the molar concentration of the gaseous mixture, C, we can calculate the molar concentration of gas A, CA.
C émolesAù = _______étotalmolesù ́_______é molesA ù A êë m 3 úû êë m 3 úû êë t o t a l m o l e s úû
To find the molar concentration in terms of pressures we first need to take the Ideal Gas Law.
The Ideal Gas Law is

Which is, by definition, the overall molar concentration, C, that is
C = étotalmolesù = _______ Vêëm3 úû
This can then be substituted into equation (1) to give the molar concentration of gas A in terms of total pressure, PT and molar fraction of A, yA:
(Hint: Substitute equation (3) into equation (1)):
C émolesAù = ________étotalmolesù ́________é molesA ù A êë m 3 úû êë m 3 úû êë t o t a l m o l e s úû
For Ideal gases, the partial pressure pA is related to the total pressure:
Gas Unit Conversion Summary:
LIQUID UNIT CONVERSIONS
Similarly, data for the concentration of component A in gas mixtures can be given in many forms, including:
P =P ́________ AT
Convert partial pressure to concentration (substitute equation (5) into equation
Cé ù=yé ù ́Cé ù=Pé ù ́yé ù=P ́p=p AêúAêúêúTêúAêúT A A
RT RT P RT ëûëûëûëûëû T

CA: molar concentration of A (k mols m-3 or gmol L-1) ρA: mass density of A (kg m-3, μg L-1 etc)
mA (or ωA) : mass fraction of A (kg A/total kg)
xA: molar fraction of A (kmol A/total kmol)
XA: molar ratio of A (kmol A/kmol mixture excluding A).
Given the molar fraction of liquid A, xA, in a mixture and the molar density of
the mixture, C, we can calculate the molar density of liquid A, CA: C émolesAù = _______é ù ́Cé ù (7)
Aêë m3 úû êë úû êë úû By definition
rékgù=Cétotalmolesù ́mw é kg ù êëm3 úû êë m3 úû T êëtotalmolesúû
Convert molar fraction to molar density (substitute for C in equation 7) (9)
Molar fraction can be related to mass fraction using molecular mass:
x é molesA ù=m é kgA ù ́mw é totalkg ù (10)
Convert molecular mass to molar density (substitute equation (10) into equation (9)):
C émolesAù = _________ Aêëm3 úû
A êëtotalmoles úû A êëtotalkg úû mwA ê kgA
ë molesA û
T êtotalmolesú
C émolesAù = _______ Aêëm3 úû

Liquid Unit Conversion Summary:
(Include units)
CA=xAC= r xA=rmA
C émolesAù = ______________________________ Aêëm3 úû

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