1. Chapter One: Mole Balances
1.1 Rate of Reaction -rA
Heterogeneous reactions involve more than one phase.
In heterogeneous reaction systems, reaction rate expressed in measures other than volume, such as reaction surface area or catalyst weight.

2. For a gas-solid catalytic reaction, gas molecules must interact with solid catalyst surface for reaction to take place.
Dimensions of heterogeneous reaction rate (mol A/s. g catalyst).

3. Rate equation is solely a function of
properties of reacting materials
& reaction conditions (e.g., species concentration, temperature, pressure, or type of catalyst, if any) at a point in system.

4. Rate equation is independent of type of reactor in which reaction is carried out.

5. For example, algebraic form of rate law for -rA for reaction
Chemical reaction rate law is an algebraic equation involving concentration, (not a differential equation).
For example, algebraic form of rate law for -rA for reaction
A → Products
may be a linear function of concentration,

6. maybe some other algebraic function of concentration. such asOr
For a given reaction, particular concentration dependence that rate law follows, must be determined from experimental observation.

7. 1.2 General Mole Balance Equation
To perform a mole balance on any system, system boundaries must first be specified.
Volume enclosed by these boundaries is referred to as “system volume”.

8. Perform a mole balance on species j in a system volume:
A mole balance on species j at any instant in time, t, yields following equation:

9. where Nj represents number of moles of species j in system at time t.

10. If all system variables (e. g
If all system variables (e.g., temperature, catalytic activity, concentration of chemical species) are spatially uniform throughout system volume,

11. Total rate of generation within system volume is sum of all rates of generation in each of subvolumes.
If total system volume is divided into M subvolumes, total rate of generation is

12. Taking appropriate limits (i. e
Taking appropriate limits (i.e., let M → ∞ & V → 0) & making use of definition of an integral, rewrite foregoing equation in form
replace Gj in Eq.(1-3)

13. From this general mole balance equation, we can develop design equations for various types of industrial reactors: batch, semibatch, & continuous-flow.

14. Upon evaluation of these equations we can determine time (batch) or
reactor volume (continuous-flow) necessary to convert a specified amount of reactants into products.

15. 1.3 Batch Reactors A batch reactor is used for small-scale operation,
testing new processes that have not been fully developed,
manufacture of expensive products, &
processes that are difficult to convert to continuous operations.

16. Reactor can be charged (i. e
Reactor can be charged (i.e., filled) through holes at top (Figure 1-5[a]).

17. but it also has disadvantages of high labor costs per batch,
Batch reactor has advantage of high conversions by leaving reactant in reactor for long periods of time,
but it also has disadvantages of high labor costs per batch,
variability of products from batch to batch,
&difficulty of large-scale production.

18. With batch reactor Fj0=Fj =0 while reaction is being carried out.
Resulting general mole balance on species j is:

19. If reaction mixture is perfectly mixed (Figure 1-5 [b]) then, there is no variation in rate of reaction throughout reactor volume, take rj out of integral, integrate, & write mole balance in form

20. Let's consider the isomerization of species A in a batch reactor
As reaction proceeds, NA decreases & NB increases, as shown in Figure 1-6.

21. We might ask what time, t1, is necessary to reduce initial number of moles from NA0 to a final desired number NA1. Applying Eq.(1-5) to isomerization
rearranging,

22. This equation is integral form of mole balance on a batch reactor.
Integrating at t =0, NA =NA0 , & at t =t1, NA = NA1,
This equation is integral form of mole balance on a batch reactor.
t1, necessary to reduce number of moles from NA0 to NA1 & also to form NB1 moles of B.

23. 1.4 Continuous-Flow Reactors
Continuous flow reactors are almost always operated at steady state.
3 types:
Continuous Stirred Tank Reactor (CSTR),
Plug Flow Reactor (PFR), &
Packed Bed Reactor (PBR).

24. 1.4.1 Continuous-Stirred Tank Reactor
Used commonly in industrial processing with operated continuously (Figure 1-7).
Referred to as continuous-stirred tank reactor (CSTR) or backmix reactor, & used primarily for liquid phase reactions.

26. Normally operated at steady state & assumed to be perfectly mixed; consequently, there is no time dependence or position dependence of temperature, concentration, or reaction rate inside CSTR.
That is, every variable is same at every point inside reactor.

27. Because T & C are identical everywhere within reaction vessel, they are same at exit point as they are elsewhere in tank.
Thus T & C in exit stream are modeled as being same as those inside reactor.

28. No spatial variations in rate of reaction (i.e., perfect mixing),
Applying general mole balance equation to a CSTR operated at steady state:
No spatial variations in rate of reaction (i.e., perfect mixing),
Design equation for a CSTR:

29. CSTR design equation gives reactor V necessary to reduce from Fj0 to exit flow rate Fj , when species j is disappearing at a rate of –rj.
Fj is just product of concentration & volumetric flow rate :

30. Consequently, combine Eqs
Consequently, combine Eqs. (1-7) & (1-8) to write a balance on species A as:

31. 1.4.2 Tubular Reactor
In addition to CSTR & batch reactors, another type of reactor commonly used in industry is tubular reactor.
cylindrical pipe & normally operated at steady state, as CSTR.
Tubular reactors are used most often for gas-phase reactions.

32. A schematic & a photograph of industrial tubular reactors are shown in Figure 1-8.

33. In tubular reactor, reactants are continually consumed as they flow down length of reactor.
In modeling tubular reactor, we assume that concentration varies continuously in axial direction only (not radial) through reactor.

34. Consequently, reaction rate, which is a function of concentration for all but zero-order reactions, will also vary axially.

35. Consider systems in which flow field may be modeled by that of a plug flow profile (e.g., uniform velocity as in turbulent flow), as shown in Figure 1-9.

36. Equation, to design PFRs at steady state can be developed in two ways:
directly from Equation (1-4) by differentiating with respect to volume V or
from a mole balance on species j in a differential segment of reactor volume ∆V.

37. Let choose second way to arrive at different form of PFR mole balance.
Differential volume, ∆V, shown in Fig. 1-10, chosen sufficiently small such that there are no spatial variations in reaction rate within this volume.
Thus generation term, ∆G , is

38. Dividing by ∆V & rearranging
term in brackets resembles definition of derivative

39. Again consider isomerization A → B, in a PFR
Again consider isomerization A → B, in a PFR. As reactants proceed down reactor, A is consumed by chemical reaction & B is produced. Consequently, FA decreases & that of B increases, as shown in Figure 1-12.

40. Rearranging Eq. (1-12) in form
Ask what is reactor volume V1 necessary to reduce entering molar flow rate of A from FA0 to FA1.
Rearranging Eq. (1-12) in form
& integrating with limits at V =0, FA=FA0 & at V =V1, FA =FA1.
V1 is volume necessary to reduce FA0 to some specified value FA1 & also volume necessary to produce a molar flow rate of B of FB

41. 1.4.3 Packed-Bed Reactor
Principal difference between reactor design calculations involving homogeneous reactions & those involving fluid-solid heterogeneous reactions is that for latter, reaction takes place on surface of catalyst.

42. Consequently, reaction rate is based on mass of solid catalyst, W, rather than on reactor volume, V.
For a fluid-solid heterogeneous system, rate of reaction of a substance A is defined as

43. Mass of solid catalyst is used because amount of catalyst is important to rate of product formation. Reactor volume that contains catalyst is of secondary significance.

44. Fig shows a schematic of an industrial catalytic reactor with vertical tubes packed with catalyst.

45. Derivation of design equation for a packed-bed catalytic reactor (PBR) will be carried out in a manner analogous to development of tubular design equation.

46. To accomplish this derivation, we simply replace volume coordinate in Eq. (1-10) with catalyst weight coordinate W (Figure 1-14).

47. As with PFR, PBR is assumed to have no radial gradients in concentration, temperature, or reaction rate. Generalized mole balance on species A over catalyst weight ∆W results in equation:

48. Dimensions of generation term in Equation (1-14) are
After dividing by ∆W & taking limit as ∆W→0, we arrive at differential form of mole balance for a packed-bed reactor:
Integral form of packed-catalyst-bed design equation can be used to calculate catalyst weight.

49. Example 1-1 How Large is it?
Consider Liquid phase cis -trans isomerzation of 2-butene
which we will write symbolically as A→B
1st order (-rA= kCA) reaction carried out in a tubular reactor in which volumetric flow rate is constant, i.e., v=vo.

50. 1. Sketch concentration profile.
2. Derive an equation relating reactor volume to entering & exiting concentrations of A, rate constant k, & volumetric flow rate ν.
3. Determine reactor volume necessary to reduce exiting concentration to 10% of entering concentration when volumetric flow rate is 10 dm3/min (i.e.. liters/min) & specific reaction rate, k, is 0.23 min-1.

51. Solution
1. Species A is consumed as we move down reactor, & as a result, both molar FA & concentration CA will decrease as we move.
Because volumetric flow rate is constant, ν=ν0, one can use Eq. (1-8) to obtain concentration of A. CA=FA/ν0, & then by comparison with Figure 1-12 plot concentration of A as a function of reactor volume as shown in Figure E1-1.l.

52. For a tubular reactor, mole balance on species A given by Eq. (1-11).
For a first-order reaction, rate law is:
Because volumetric flow rate,ν, is constant (ν=ν0), as it is for most liquid phase reactions,

53. Multiplying both sides of Eq. (E1-1. 2) by minus one & substituting Eq
Multiplying both sides of Eq. (E1-1.2) by minus one & substituting Eq. (E 1-1.1) yields:

54. Using conditions at entrance of reactor that when V= 0, then CA = CA0,
Rearranging gives
Using conditions at entrance of reactor that when V= 0, then CA = CA0,
Carrying out integration of Eq. (El-l.4) gives

55. Substituting CA0, CA, ν0 & k in Equation (El-1.5), we have
3. To find volume, V1 at which CA = (1/10) CA0 for k = 0.23 min-1 & ν0= 10 dm3/min.
Substituting CA0, CA, ν0 & k in Equation (El-1.5), we have

56. We see that a reactor volume of 0
We see that a reactor volume of 0.1 m3 is necessary to convert 90% of species A entering into product B for parameters given.