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Mass Transfer – Chemical reactions 10-1
- Definitions: Reaction rate, order reaction & equilibrium constant
- Heterogeneous & Homogeneous reactions
- Diffusion-Controlled Reaction
- Diffusion and 1st Order Heterogeneous Reactions
- Mechanism of Irreversible Heterogeneous Reactions
- Heterogeneous Reactions of Unusual Stoichiometries
Chemical Reactions & Mass Transfer
A chemical reaction is a conversion of one set of chemical substances into another: AB + CD → AD + BC
Mass Transfer – Chemical reactions 10-2
The reaction rate is proportional to the concentration of reactants:(2)
With being the rate constant.
Many chemical reactions are reversible. Forward and reverse reactions finally lead to an equilibrium:
We can define (1) and (2) also for the reverse reaction:
The reaction rate is the decrease in reactant concentration or the increase in product concentration with time:
(1)
Mass Transfer – Chemical reactions 10-3
Now, the rate of the total reaction is the difference in the rates of the forward and reverse reactions:
the reaction has reached chemical equilibrium and the ratio of rate constants is the equilibrium constant of the chemical reaction:
If and
1
AD BC
AB CD
c cK
c c
Mass Transfer – Chemical reactions 10-4
Reaction orderApart from temperature and pressure, the reaction rate can depend on the concentration of the reactants.
A reaction rate depending on the concentration of one reactant is a first order reaction, e.g.:
one depending on two concentrations is of second order:
For an irreversible second order reaction
Mass Transfer – Chemical reactions 10-5
The forward reaction
is third order, since:
A reaction rate not depending on any concentration is called a zero order reaction.
Note: Reactions typically occur in a series of multiple reactions. One step is the rate determining one. Therefore, the overall reaction would be of second or first order. (depending also on concentration)
Mass Transfer – Chemical reactions 10-6
Reactions and mass transferChemical reactions are always coupled with mass transfer since the reactants have to travel to the location where the conversion takes place (“sink” for the reactants) while the products have to travel away (“source” for the products).
Comparing the speed of mass transfer with that of reaction allows to determine the rate-limiting step and if rxns have to be accounted for:
reaction rate >> mass transfer → neglect reactionsmass transfer >> reaction rate → include reactions
When reaction rate and mass transfer are comparable then we need to incorporate reaction into the models.
iiii rccD
tc
02
(see: Generalized Mass Balances)
Mass Transfer – Chemical reactions 10-7
Activation energyActivation energy is required for the reaction to start and to carry on. The activation energy is typically supplied as heat (higher temperature) but sometimes it can be other electromagnetic radiation, such as UV light.
A catalyst lowers the activation energy and increases the speed of the reaction at a certain temperature. The catalyst (typically a solid, but could also be a liquid) is not consumed after the reaction.
Examples of chemical reactions• Combustion (gaseous, liquid or solid fuels)• Production of chemicals (catalysts)• Fuel cells, batteries (catalysts)• Gas cleaning (e.g. removal of SO2 by CaO)• (Nano-) particle synthesis•…
Mass Transfer – Chemical reactions 10-8
Heterogeneous & Homogeneous reactions
Heterogeneous reactions take place AT an interface (e.g. solid catalyst surface, coal combustion) and diffusion and reaction occur by steps IN SERIES.
Homogeneous reactions take place THROUGHOUT the volume in the same phase (fuel combustion in engine). Diffusion and reaction occur by steps partially in PARALLEL.
For simplification of mass transfer problems some heterogeneous reactions can be treated as homogeneous ones, depending on the definition of the observation space.
Mass Transfer – Chemical reactions 10-9
For example, in combustion of coal particles in fluidized beds initially reaction takes place AT the coal surface – so heterogeneous reaction.
CO2
O2
O2CO2
O2
O2
O2
Coal
O2O2
CO2
O2
CO2
O2
Later on, however, a highly porous inorganic structure (e.g. fly ash) develops and combustion takes place THROUGHOUT the particle.
Mass Transfer – Chemical reactions 10-10
Thus, this case can be described as a homogeneous reaction and the “homogeneous reaction model” can be applied:
So the assignment of the reaction type can be subjective!
Combustion is an irreversible reaction, so
Even though microscopically the reaction still is a surface reaction, macroscopic observation of the entire particle gives the picture of reactions occurring homogeneously throughout the particle.
Mass Transfer – Chemical reactions 10-11
In summary, coal particle combustion can be described as:
a) Heterogeneous reaction and transport:
011
21
ccDt
c
The reaction is introduced at the boundary conditions!
b) Homogeneous (homogeneous-like) reaction model and transport:
10
1121 rccD
t
c
Mass Transfer – Chemical reactions 10-12
Examples of reaction assignment
Oxidation of lead sulfide (semiconductor) particles:
PbS particles → porous PbOheat
Removal of ammonia from off-gas:
NH3 / air + H2O → NH4OHscrubbing
Adsorption of sulfur dioxide (from combustion off-gas):
SO2 + Ca(OH)2/H2O CaSO4scrubbing
using Ca(OH)2 slurry
Mass Transfer – Chemical reactions 10-13
10.1 Diffusion-Controlled Reaction
Every diffusion-controlled process involves multiple steps. E.g. in dehydrogenation of C2H6 on Pt crystals
(1) C2H6 diffuses to Pt surface(2) C2H6 reacts on the surface(3) Product diffuses away from the
surface
(a)
When step 1 takes much longer than step 2 then we have a diffusion-controlled reaction.
Mass Transfer – Chemical reactions 10-14
In b) we have a porous particle and the reactant diffuses to catalytically active sites where it reacts and the product diffuses away while reaction continues. Very important in catalysis, scrubbing and extraction (homogeneous reaction model).
In a) we have a diffusion-controlled heterogeneous reaction where the reactant diffuses to a (e.g. hot or catalytic) surface, reacts and the product diffuses away.
(a)
(b)
Mass Transfer – Chemical reactions 10-15
In c) a slow-moving molecules react with more mobile ones facilitating the transport of the former across a membrane (example: oxygen uptake by hemoglobin in the lungs).
In case d), molecules are well mixed and react upon contact. The reaction rate depends on the Brownian motion of the molecules (acid/base reactions, CH4combustion etc.)
(d)
(c)
Mass Transfer – Chemical reactions 10-16
Goal: To find the overall rate of the process
(a)
(b)
Mass Transfer – Chemical reactions 10-17
10.2 Diffusion and 1st Order Heterogeneous Reactions (dilute solutions)
A first-order reaction with respect to a reactant is one where the reaction rate is proportional to the reactant concentration:
i ir = c
At the steady state the overall reaction rate/area equals the diffusion fluxes
2
2232
1111
r
)(
)(
cckn
cckn
i
i
Mass Transfer – Chemical reactions 10-18
The surface reaction is
Species 1 ↔ Species 2κ2
κ-2
And the reaction rate is given by
i22i122 ccr
Note that the equilibrium constant is
2
22K
To find the overall reaction we must eliminate the surface (or interfacial) concentrations that are hard to measure.
Mass Transfer – Chemical reactions 10-19
Following the analysis of the mass transfer across interfaces (“overall MTC”) we obtain:
2
2112 K
ccKnr
Where the overall MTC or “resistance” is:
2321 Kk11k11
K
Mass Transfer – Chemical reactions 10-20
Similarities/differences with interfacial mass transfer:
1. The resistances 1 / k1, 1 / κ2 and 1 / (k3K2) add to the total resistance 1/K
2 species of ionconcentrat bulk
existing the withmequilibriu in
1 species of ionconcentrat
/ 22 Kc
2. Henry’s law constant (partition coefficient) characterizes destribution between phases: K2 ↔ HK2 and H vary over a wide range. Here, similarly to H, the K2
determines the relative impact of k1 and k3.
Mass Transfer – Chemical reactions 10-21
Example 10.2.1: Limiting cases of 1st order reactions
(a) Fast stirringk1 and k3 are large, so K → κ2. Then,
2
2122 K
ccr
(b) High temperatureUsually at high T the reactions are fast so 1/κ2 is very small and can be neglected
2
21
2312 K
cc
Kk1k11
r
(c) Irreversible reaction κ-2 = 0 so K2 → 121
2 c1k1
1r
Note that ONLY in this case the resistances are simply additive.
Determine overall rate of the process when:
Mass Transfer – Chemical reactions 10-22
Example 10.2.2: Cholesterol solubilization in bile
Bile* is the body’s "detergent" for fat digestion through cholesterol excretion. Failure of bile leads to cholesterol gallstones. Typically, gallstones are removed by surgery. However, lab data show that gallstones can be dissolved by administering certain components of the bile.
*Bile = Galle (in German)
Goal: Find dissolution (“reaction”) rate of gallstoneAs a cholesterol gallstone is a solid, mass transfer/chemical reaction at a solid/fluid interface needs to be understood: rotating (spinning) disk apparatus for cholesterol dissolution rates.
Mass Transfer – Chemical reactions 10-23
cholesterol dissolution rate = 5.39 ꞏ 10-9 g/(cm2 s) = r2solubility = 1.48 ꞏ 10-3 g/cm3 = c1D = 2ꞏ10-6 cm2/sdisk diameter, d = 1.59 cmRe = 11200bile kinematic viscosity, = 0.036 cm2/scholesterol solution, density over the bile = 1ꞏ10-5 g/cm3
If the reaction is irreversible find:a) surface reaction rate constantb) dissolution rate of a 1 cm radius cholesterol gallstone
Mass Transfer – Chemical reactions 10-24
(a) For this case21 /1k1
1K
9 2
22 1 3 3
1
5.39 10 g cm srr K c K
c 1.48 10 g cm
Find k1 from the appropriate mass transfer coefficient using the correlation for the spinning disk:
s/cm1016.2D
ddD
62.0k 331212
1
So κ2 can be obtained from k1 and K as κ2 = 3.6ꞏ10-6 cm/s
Mass Transfer – Chemical reactions 10-25
(b) dissolution rate of a 1 cm radius cholesterol gallstone
The 1 cm cholesterol gallstone is assumed to stand still, but the density gradient between cholesterol and bile results in flow by free convection so the mass transfer coefficient is:
1/4 1/331
2
k d d g2 0.6
D D
3 5 3 21 31
6 2 3 2 2
k 1cm (1cm) 1 10 g / cm 980cm / s2 0.6 (18000)
2 10 cm / s 1g / cm (0.036cm / s)
51k 5.6 10 cm / s
Mass Transfer – Chemical reactions 10-26
As we are still dealing with irreversible reactions:
s/cm104.3
s/cm106.3/1s/cm106.5/1
1
/1k11
K
6
65
21
In a unstirred bile, the rate is also controlled by the reaction.
Mass Transfer – Chemical reactions 10-27
10.3 Finding the Mechanism of Irreversible Heterogeneous Reactions
Typically, the reaction mechanism is not known so it has to be inferred.
The surface reaction takes place as
products2species
solid
1species
gaseous
Mass Transfer – Chemical reactions 10-28
Physical situation Rate-controlling step
Size R = ƒ(time, reagent)
Size = ƒ(temperature) Size = ƒ(flow) Remarks
A Shrinking Reaction R c1t strong temperature Independant of flow Other reactionparticle variation stoichomerties can
be found easily
B Shrinking External diffusion R2 (c1t) small particles Weak temperature Independent for small The exact variation
particle R3/2 (c1t)larger particles variation particles only with flow dependson the mass transfercoefficient
C Shrinking Reaction R c1t strong temperature Independent of flow This is the same ascorea case A, except for
ash formation
D Shrinking External diffusion R c1t Weak Usually about square This case iscorea root of flow uncommon
E Shrinking Ash diffusion R (c1t)1/2 Weak Independant of flow This case is common,corea an interesting
contrast with theprevious one
Notes: aThis is often called the topochemical model. The size R refers to the cone
Mass Transfer – Chemical reactions 10-29
Cases A and C: surface reaction controls
2122 ccr
As the solid concentration (c2) is constant the reaction is 1st
order with respect to c1.
Mass balance per particle:
12
22122
2
222
3
cdtdr
r4ccdtdr
cr4
r4rcr34
dtd
Be careful, distinguish between radius r and reaction rate r2!
If the gas phase concentration c1 is constant and the particle has an initial radius R0
tcRr 120
Mass Transfer – Chemical reactions 10-30
Case B: Diffusion outside of a shrinking particle controls
Identify correlation (forced convection around solid sphere)
1/2 1/3
1k d dv=2+0.6
D D
B.1) For very small particles Re 0, sorD
kDdk
2
Mass balance:
tcDc
Rr
ccD
dtdr
r
crD
rdtdr
cr
ccrkcrdtd
i
2
120
2
12
12
22
112
23
2
44
)(434
Mass Transfer – Chemical reactions 10-31
B.2) For large particles
2/16/1
3/22/1
1
3/12/11
rDv
42.0k
Dvd
6.0D
dk
Similarly,
1/ 2 2 / 31/ 2
2 11/ 6
1/ 2 2 / 33 / 2 3 / 2 1
0 1/ 62
dr v Dc r -0.42 c
dt
c0.64 v Dr R - t
c
Mass Transfer – Chemical reactions 10-32
Cases C, D, and E: shrinking core model:Two diffusional resistances in series
Bulk fluid
Particle surface
Unreacted core
R0
Physical situation Rate-controlling step
A Shrinking Reactionparticle
B Shrinking External diffusionparticle
C Shrinking Reactioncorea
D Shrinking External diffusioncorea
E Shrinking Ash diffusioncorea
Mass Transfer – Chemical reactions 10-33
Case D: Diffusion in the surrounding bulk fluid controls
Mass balance:
tc
ckRr
kcr4dt
drcr4
kcr4cr3
4
dt
d
2
10
12
22
12
23
Similar dependence of r with t as with cases A and C. Dependence on flow (through k) but weak dependence on temperature.
Mass Transfer – Chemical reactions 10-34
Case E: Diffusion in the ash (or shell) controls
Here the thickness of the shell is important Film model:
rRD
k0
where D is the effective diffusivity through the shell
Mass balance:
2/1
2
10
0
122
3
tc
cD2Rr
rRcD
r4cr34
dtd
Mass Transfer – Chemical reactions 10-35
10.4 Heterogeneous Reactions of Unusual Stoichiometries
10.4.1 Irreversible second-order heterogeneous reaction (additivity of resistances)
Mass Transfer – Chemical reactions 10-36
Steady-state mass transfer to the surface:
)cc(knr i11111
Surface reaction:2i121 cr
Find the c1i by equating the above
1k
c41
2k
c
0ckckc
1
12
2
1i1
11i112i12
So the overall reaction rate is:
1 2 11 1 1
1 2 1
k 4 cr k c 1 1 1
2c k
Mass Transfer – Chemical reactions 10-37
Compare this with the corresponding first order reaction:
1 2 11 1 1
1 2 1
k 4 cr k c 1 1 1
2c k 121
1 /1/11
ck
r
first order reactionsecond order reaction
Mass Transfer – Chemical reactions 10-38
10.4.2 Heterogeneous Reactions in Concentrated Solutions
Until now we assumed that k1 ≠ f(r2). This is good for dilute solutions.
Cracking of hydrocarbons:
1 mol of species 1 n moles of species 2, n > 1
Example:
Reforming
1 mol of species 1 n moles of species 2, n < 1
κ2 κ2
In both cases we must account for CONVECTION in the calculation of mass transfer.
Mass Transfer – Chemical reactions 10-39
In cracking, convection is AWAY from the surface: The reacting species must diffuse AGAINST the current of product species.In reforming, convection is TOWARDS the surface as reaction "pulls" reactants. The product must diffuse away.
ν > 1 crackingν < 1 reformingν = 1 treat as before
Goal: To calculate the overall rate!
Mass Transfer – Chemical reactions 10-40
Overall rate of reaction across this film is:
2
1 1 2 1i
nr n c
We must calculate the flux n1 for concentrated solutions:
01
11 vc
dzdc
Dn
Now the flux from convection is
1210 n)1(nncv
So the equation for flux n1 can be written as: 0Dn
cnDc
1dzdc 1
111
Mass Transfer – Chemical reactions 10-41
B.C.’s: z = 0: c1 = c10z = l: c1 = c1i
Solving for n1 gives:
1 1 2
1 110
k c 1 ( 1)n /( c)r n ln
( 1) 1 ( 1)c / c
wherelD
k1
For diffusion control (κ2ꞏc10/n1 is large) the following diagram shows the rate relative to that of dilute solutions:
Mass Transfer – Chemical reactions 10-42
Consider studying cracking of oil by flowing it over a hot plate.
If the molecular weight of the product is only 25% of that of the oil, by how much the convection introduced mass transfer changes the cracking reaction rate?
1 molecule oil = ν x 0.25 molecules of the product. So ν ≈ 4
As only oil is flowing (c10/c) = 1. So 1 + (ν - 1) c10/c ≈ 4 From the above figure:
%40ratereaction1:1
rateactual
The convection REDUCES the reaction rate when the effect of reaction on the mass transfer coefficient is neglected.
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