Chapter One Mole Balances 1 1 Rate of

Chapter One: Mole Balances 1. 1 Rate of Reaction -r. A 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.

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).

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.

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

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

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

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”.

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:

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

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

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

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)

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

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

1. 3 Batch Reactors A batch reactor is used for Øsmall-scale operation, Øtesting new processes that have not been fully developed, Ømanufacture of expensive products, & Øprocesses that are difficult to convert to continuous operations.

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

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.

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

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

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

We might ask what time, t 1, is necessary to reduce initial number of moles from NA 0 to a final desired number NA 1. Applying Eq. (1 -5) to isomerization rearranging,

Integrating at t =0, NA =NA 0 , & at t =t 1, NA = NA 1, This equation is integral form of mole balance on a batch reactor. t 1, necessary to reduce number of moles from NA 0 to NA 1 & also to form NB 1 moles of B.

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).

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.

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.

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.

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: 30 -1 -2010

CSTR design equation gives reactor V necessary to reduce from Fj 0 to exit flow rate Fj , when species j is disappearing at a rate of –rj. Fj is just product of volumetric flow rate : concentration &

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

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 gasphase reactions.

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

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.

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

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.

Equation, to design PFRs at steady state can be developed in two ways: (1) directly from Equation (1 -4) by differentiating with respect to volume V or (2) from a mole balance on species j in a differential segment of reactor volume ∆V.

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

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

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.

Ask what is reactor volume V 1 necessary to reduce entering molar flow rate of A from FA 0 to FA 1. Rearranging Eq. (1 -12) in form & integrating with limits at V =0, FA=FA 0 & at V =V 1, FA =FA 1. V 1 is volume necessary to reduce FA 0 to some specified value FA 1 & also volume necessary to produce a molar flow rate of B of FB 1. 1 -2 -2010

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.

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

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.

Fig. 1 -13 shows a schematic of an industrial catalytic reactor with vertical tubes packed with catalyst.

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.

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

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:

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.

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

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 dm 3/min (i. e. . liters/min) & specific reaction rate, k, is 0. 23 min-1.

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=ν 0, one can use Eq. (1 -8) to obtain concentration of A. CA=FA/ν 0, & then by comparison with Figure 1 -12 plot concentration of A as a function of reactor volume as shown in Figure E 1 -1. l.

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=ν 0), as it is for most liquid phase reactions,

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

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

3. To find volume, V 1 at which CA = (1/10) CA 0 for k = 0. 23 min-1 & ν 0= 10 dm 3/min. Substituting CA 0, CA, ν 0 & k in Equation (El-1. 5), we have

We see that a reactor 3 volume of 0. 1 m is necessary to convert 90% of species A entering into product B for parameters given.