chemical reactions & mass transfer - eth z...mass transfer –chemical reactions 10-2 the...

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


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)


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 cKc c


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


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)


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)


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•…


Heterogeneous & Homogeneous reactionsHeterogeneous 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.


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.


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.


In summary, coal particle combustion can be described as:

a) Heterogeneous reaction and transport:0

1121

ccD

tc

The reaction is introduced at the boundary conditions!

b) Homogeneous (homogeneous-like) reaction model and transport:

10

1121 rccD

tc


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

Heterogeneous

Homogeneous

Not sure


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.


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)


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)


Goal: To find the overall rate of the process

(a)

(b)


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 )(

)(

cckncckn

i

i


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.


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 Kk11k11K


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 bulkexisting the withmequilibriu in

1 species of ionconcentrat/ 22 Kc

2. Henry’s law constant (partition coefficient) characterizes distribution between phases: K2 ↔ HK2 and H vary over a wide range. Here, similarly to H, the K2determines the relative impact of k1 and k3.


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

ccKk1k1

1r

(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:


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.


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


(a) For this case21 /1k1

1K

9 22

2 1 3 31

5.39 10 g cm srr K c Kc 1.48 10 g cm

Find k1 from the appropriate mass transfer coefficient using the correlation for the spinning disk:

s/cm1016.2D

ddD62.0k 3

31212

1

So κ2 can be obtained from k1 and K as κ2 = 3.6ꞏ10-6 cm/s


(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

2k d d g2 0.6D D

3 5 3 21 31

6 2 3 2 2k 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


As we are still dealing with irreversible reactions:

s/cm104.3

s/cm106.3/1s/cm106.5/11

/1k11K

6

65

21

In a unstirred bile, the rate is also controlled by the reaction.


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

solid1species

gaseous


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 easilyB 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 formationD Shrinking External diffusion R c1t Weak Usually about square This case is

corea root of flow uncommonE 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


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

r4ccdtdrcr4

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


Case B: Diffusion outside of a shrinking particle controls

Identify correlation (forced convection around solid sphere)

1/2 1/31k d dv=2+0.6D D

B.1) For very small particles Re 0, sorDk

Ddk

2

Mass balance:

tcDcRr

ccD

dtdrr

crDr

dtdrcr

ccrkcrdtd

i

2

120

2

12

12

22

112

23

2

44

)(434


B.2) For large particles

2/16/1

3/22/1

1

3/12/11

rDv42.0k

Dvd6.0

Ddk

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 cdt

c0.64 v Dr R - tc


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


Case D: Diffusion in the surrounding bulk fluid controls

Mass balance:

tcckRr

kcr4dtdrcr4

kcr4cr34

dtd

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.


Case E: Diffusion in the ash (or shell) controls

Here the thickness of the shell is important Film model:

rRDk0

where D is the effective diffusivity through the shell

Mass balance:

2/1

2

10

0

122

3

tc

cD2Rr

rRcDr4cr

34

dtd


10.4 Heterogeneous Reactions of Unusual Stoichiometries

10.4.1 Irreversible second-order heterogeneous reaction (additivity of resistances)


Steady-state mass transfer to the surface:)cc(knr i11111

Surface reaction: 2i121 cr

Find the c1i by equating the above

1k

c412kc

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 12c k


Compare this with the corresponding first order reaction:

1 2 11 1 1

1 2 1

k 4 cr k c 1 1 12c k 1

211 /1/1

1 ck

r

first order reactionsecond order reaction


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.


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!


Overall rate of reaction across this film is:

2

1 1 2 1inr n c

We must calculate the flux n1 for concentrated solutions:

01

11 vc

dzdcDn

Now the flux from convection is

1210 n)1(nncv

So the equation for flux n1 can be written as: 0Dncn

Dc1

dzdc 1

111


B.C.’s: z = 0: c1 = c10z = l: c1 = c1i

Solving for n1 gives:

1 1 21 1

10

k c 1 ( 1)n /( c)r n ln( 1) 1 ( 1)c / c

wherelDk1

For diffusion control (κ2ꞏc10/n1 is large) the following diagram shows the rate relative to that of dilute solutions:


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.

chemical reactions & mass transfer - eth z...mass transfer –chemical reactions 10-2 the...

Documents