CHE2163: HEAT AND MASS TRANSFER
CONVECTIVE MASS TRANSFER WEEK 11
Learning Objectives, Tasks, Readings, and Learn ChemE Videos
1. Mass Transfer Coefficients (MTCs; eg Welty p558)
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a. Another way to think about mass transfer rate equations
b. Different concentration units – different MTCs
2. Mass Transfer Convection (solid/fluid; as per Heat Transfer!!)
a. Cengel p 831-840; 28; HT lecture notes
b. Use procedure from Week 5, convective heat transfer
c. Highlighting specific differences between convective heat
and mass transfer with respect to correlations
3. Mass Transfer Convection (fluid/fluid)
a. “2-film theory” describing convective mass transfer
between fluid phases
b. 29 only, note this content NOT covered in
Cengel or Incropera
By the end of this week, you should be able to:
Convert between common MTCs (for simplicity only)
Use Sherwood correlations to find MTC and calculate rates Apply 2-film theory to problems involving fluid/fluid contact
4.1 MASS TRANSFER COEFFICIENTS – ANOTHER VIEW
So far, we have looked at three mass transfer equations, for three different situations (Welty p558):
Case 1 2 3 Description Fick’s Law Stagnant Film EMCD
Assumptions
Equation (expressed as MTC*DF)
Each of these equations is simply the same rate equation with some additional terms that describe something about the geometry and assumptions of the system
The same could be said for all of the steady-state heat transfer problems as well, and can be applied to all modes of transfer (Note, the shape factors change with geometries (rectangular vs cylindrical vs spherical systems) and do not necessarily apply to transient systems in the same way.)
We could say that each of these equations contains a mass transfer coefficient just like we could say the heat transfer equations contain heat transfer coefficients
4.2 CONVECTION MASS TRANSFER – SOLIDS AND FLUIDS
For conduction/diffusion problems, we can usually express this transfer coefficient as a product of algebraic variables and constants (e.g. Heat – k, A, 1/x; Mass – DAB, A, 1/x)
For convection problems, we most often use empirical correlations to describe the transfer coefficient, because they are often functions of fluid mechanics (e.g. heat – h; mass – k)
GOOD NEWS: for convection mass transfer involving a fluid in contact with a solid surface, you can use the same empirical correlations as for heat transfer!!!
BAD NEWS: for connective mass transfer we will have a range of units for concentrations, so we also need to describe a range of mass transfer coefficients:
Go to the correlation tables in Heat Transfer lectures – simply replace Nu with Sh; Pr with Sc; (Note – there are numerous correlations in textbooks and literature beyond those which we will consider)
As for heat transfer, for NATURAL convection evaluate fluid properties at the FILM density; for FORCED convection evaluate fluid properties at the BULK density
Simplest form of convective mass transfer is from a solid to a fluid, it is the most common mode in chemical/environmental engineering.
Rate of mass transfer is analogous to Newton’s Law of Cooling:
𝑚̇ 𝑐𝑜𝑛𝑣 = h𝑚𝑎𝑠𝑠𝐴(𝜌𝐴,𝑠 − 𝜌𝐴,∞) (kg.s-1) 𝑚̇ = 𝜌h 𝐴(𝑤 − 𝑤 ) (kg.s-1)
𝑐𝑜𝑛𝑣 𝑚𝑎𝑠𝑠 𝐴,𝑠 𝐴,∞
Where hmass is the convective mass transfer coefficient with units of m.s-1, A,s is the density of solute A at the surface (kg.m-3) and A,∞ is the density of solute A in the bulk fluid. This is the Cengel notation, but please note that in most other texts, the convective mass transfer coefficient is labelled as k, as per Welty, Treybal ,etc:
4.3 DIMENSIONLESS NUMBERS AND CORRELATIONS
Film transfer coefficients are most reliably calculated using empirical expressions; you can even use the same correlations as in “Convection Correlation” in Heat Transfer.
Note that many other correlations exist (as per heat transfer) including Chilton-Coburn analogy, etc see p 836 Cengel.
SHERWOOD NUMBER, Sh 𝑆h = h𝑚𝑎𝑠𝑠𝐿𝑐
Where: hmass = convective mass transfer coefficient (m/s) Lc = characteristic length (m)
DAB = molecular diffusivity of A in B (m2/s)
Q. What is the physical meaning of the Sherwood number?
Q. Where do I find Sherwood number correlations?
Q. What are the differences between the Nusselt number and the Biot number?
REYNOLDS NUMBER, Re
Re du or Re du inertia. forces
viscous. forces
Where: d = characteristic length (m) u = average velocity (m/s)
μ, ρ – dynamic viscosity (Pa.s) & fluid density (kg/m3), respectively
Re is the property of fluid & flow conditions. It characterizes flow regimes. For flow in circular pipes, these transitions occur at:
– laminar flow – transition flow
– turbulent flow
Q What is the physical meaning of the Reynolds number?
SCHMIDT NUMBER, Sc
𝑆𝑐 = 𝜐 = 𝑀𝑜𝑚𝑒𝑛𝑡𝑢𝑚 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑣𝑖𝑡𝑦
𝐷𝐴𝐵 𝑀𝑎𝑠𝑠 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑣𝑖𝑡𝑦
ν = kinematic viscosity (m2/s), also equal to 𝜇/𝜌 DAB = molecular diffusivity of A in B (m2/s)
Sc represents the ratio of molecular momentum to mass diffusion in the boundary layers. It is a property of the fluid:
If Sc ≈ 1, momentum & diffusive mass transfer approx. equal, and velocity and concentration boundary layer thicknesses similar
Q. What is the significance of the Sc for mass transfer?
NUMBER, Gr
𝑔(𝜌∞ − 𝜌𝑠)𝐿𝑐3 𝑔(Δ𝜌/𝜌)𝐿𝑐3 𝐺𝑟= 𝜌𝜐2 = 𝜐2
Where: g – acceleration due to gravity (9.8 ms-2) s – surface concentration (A plus B)
∞ – bulk fluid concentration (A plus B)
bulk fluid density (“film” density)
Lc- characteristic length scale (depends on geometry) ν- kinematic viscosity (m2s-1)
Important notes for using the Gr number:
If fluid is homogeneous (i.e. fluids with no concentration gradients), then can be swapped with T as in heat transfer situation
If fluid is nonhomogeneous then you must use
Evaluate properties for Sherwood correlations at the film fluid
density for NATURAL convection, but at the bulk fluid density for FORCED convection
LEWIS NUMBER, Le
Le is the ratio of thermal diffusivity to mass diffusivity in the boundary layers. It is a property of the fluid:
Le ≈ 1, momentum & diffusive mass transfer approx. equal, and velocity and concentration boundary layer thicknesses similar
Q. What then are the meanings of Pr, Sc and Le and why could they be useful?
𝐿𝑒 = 𝑆𝑐 = 𝑇h𝑒𝑟𝑚𝑎𝑙 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑣𝑖𝑡𝑦 𝑃𝑟 𝑀𝑎𝑠𝑠 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑣𝑖𝑡𝑦
EXAMPLE: Raw rock salt is dissolved in water to refine the raw product into a pure salt for human consumption…
1) Assuming water temperature is constant at 24 oC, what is the maximum concentration (mol.L-1) of salt that could possibly be reached in the water?
2) What is the mass transfer coefficient for natural convection from the rock salt crystal (assuming the flow is over a sphere (salt crystal) of length diameter 0.2 cm)?
3) What is the mass transfer coefficient for forced convection from the rock salt crystal if the stirrer is set at 200 rpm in the beaker (assuming the crystal is a particle of diameter 0.2 cm and the beaker radius is 2 inches)?
4) What is the initial rate of dissolution of the rock salt in freshwater at 24 oC? How would you expect the dissolution rate to change:
. a) as a function of temperature?
. b) if the water contains salt initially?
. c) if the rock salt was crushed into smaller pieces?
5) A key assumption here is that the system is at steady state. Under what conditions might this be true and under what conditions might this be false?
4.4 CONVECTION MASS TRANSFER – TWO FLUIDS IN CONTACT
To date, we have considered systems in which mass transfer of solute A occurs from a phase boundary
o By diffusion through a solid material, without any bulk motion (Fick’s law)
o By diffusion through a fluid, where some bulk motion is possible (SFD, EMCD)
o By convection through a fluid (Sherwood correlations)
In each of these situations, we apply an individual, or “film”
mass transfer coefficient and associated equation to determine mass transfer within the key phase (e.g. carbon dioxide evaporation from coke in AIR).
In the above cases, solute A on the other side of the phase boundary is usually assumed to be homogenous in the other phase, thus no driving force for mass transfer, e.g.
o Evaporation of ethanol through a stagnant air film o Dissolution of salt into water
In chemical engineering, it is very common to encounter systems in which a gas and a liquid, or two immiscible liquids, are thoroughly mixed, with the intention of transferring a solute A from one phase to the other, requiring analysis of multiple mass transfer coefficients, or overall mass transfer coefficients.
EXAMPLE: In the early stages of petroleum refining, crude oil is mixed with water in order to transfer ammonia from the oil phase into the water phase (“de-salting”). This is a mass transfer operation involving convective mass transfer between two fluids in contact.
There are two fluids involved, separated by a phase boundary, which is different to the single-phase convection problems we encountered earlier.
In this problem, the rate of mass transfer FROM the oil phase is equal to the rate of mass transfer INTO the water phase
Driving forces can be identified on either side of the phase boundaries, between “bulk” and “interface” concentrations. How is this different to SFD/EMCD problems??
How could we measure the concentrations of ammonia in the bulk oil and bulk water phases?
How could we measure the concentrations of ammonia at the oil-side and the water-side of the interface?
Write down the equations to determine the mass transfer rates in the oil phase, and the water phase, respectively:
If the interface concentrations are unknown, write down new equations to determine the mass transfer rates in the oil phase, and the water phase, respectively:
Other 2-film diagrams, involving different phases and difference concentration units
b y o u r r
UNITS FOR DIFFERENT MASS TRANSFER COEFFICIENTS
𝐽𝐴 𝑘𝑚𝑜𝑙𝐴 𝑚3 𝑚 𝐾𝐿~𝑘𝐿=∆𝐶=𝑚2.𝑠 ×𝑘𝑚𝑜𝑙𝐴=𝑠
𝐾 ~𝑘 = 𝐽𝐴 = 𝑘𝑚𝑜𝑙𝐴 × 1 = 𝑘𝑚𝑜𝑙𝐴 (note pressure units could be in atm or kPa) 𝐺 𝐺 ∆𝑃 𝑚2.𝑠 𝑘𝑃𝑎 𝑚2.𝑠.𝑘𝑃𝑎
𝐾𝑥~𝑘𝑥 = 𝐽𝐴 =𝑘𝑚𝑜𝑙𝐴× 𝑘𝑚𝑜𝑙 = 𝑘𝑚𝑜𝑙
∆𝑥 𝑚2.𝑠 𝑘𝑚𝑜𝑙𝐴 𝑚2.𝑠 𝐾𝑦~𝑘𝑦 = 𝐽𝐴 =𝑘𝑚𝑜𝑙𝐴× 𝑘𝑚𝑜𝑙 = 𝑘𝑚𝑜𝑙
∆𝑦 𝑚2. 𝑠 𝑘𝑚𝑜𝑙𝐴 𝑚2. 𝑠
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