Fraction unreacted

It is convenient to define the fraction of reacted functional groups in a reaction mixture by a parameter p, called the extent of reaction. Thus p is the fraction of A groups which have reacted at any stage of the process, and 1 - p is the fraction unreacted ... 

Equations 8-148 and 8-149 give the fraction unreacted C /C o for a first order reaction in a closed axial dispersion system. The solution contains the two dimensionless parameters, Np and kf. The Peclet number controls the level of mixing in the system. If Np —> 0 (either small u or large [), diffusion becomes so important that the system acts as a perfect mixer. Therefore,... 

Equation (1.24) is arguably the most important result in chemical reaction engineering. It shows that the concentration of a reactant being consumed by a first-order batch reaction decreases exponentially. Dividing through by Oq gives the fraction unreacted,... 

Reactor Performance Measures. There are four common measures of reactor performance fraction unreacted, conversion, yield, and selectivity. The fraction unreacted is the simplest and is usually found directly when solving the component balance equations. It is a t)/oo for a batch reaction and aout/ciin for a flow reactor. The conversion is just 1 minus the fraction unreacted. The terms conversion and fraction unreacted refer to a specific reactant. It is usually the stoichiometrically limiting reactant. See Equation (1.26) for the first-order case. 

Batch reactors give the lowest possible fraction unreacted and the highest possible conversion for most reactions. Batch reactors also give the best yields and selectivities. These terms refer to the desired product. The molar yield is the number of moles of a specified product that are made per mole... 

This gives = 0.8047 h. The molecular weight of the monomer, Ma, is not actually used in the calculation. Extrapolation of the first-order kinetics to a 4-h batch predicts that there will be 900exp(-3.22) = 36kg or 4% by weight of monomer left unreacted. Note that the fraction unreacted, Ya, must be defined as a ratio of moles rather than concentrations because the density varies during the reaction. 

This messy result apparently requires knowledge of five parameters k, (A )o> Poo, and po- However, conversion to dimensionless variables usually reduces the number of parameters. In this case, set Y = Na/(Na)o (the fraction unreacted) and r = t/thatch (fractional batch time). Then algebra gives... 

Example 3.1 Find the fraction unreacted for a first-order reaction in a variable density, variable-cross-section PER. 

Solution It is easy to begin the solution. In piston flow, molecules that enter together leave together and have the same residence time in the reactor, t. When the kinetics are first order, the probabiUty that a molecule reacts depends only on its residence time. The probability that a particular molecule will leave the system without reacting is exp(— F). For the entire collection of molecules, the probability converts into a deterministic fraction. The fraction unreacted for a variable density flow system is... 

Example 3.2 Consider the reaction 2A B. Derive an analytical expression for the fraction unreacted in a gas-phase, isothermal, piston flow reactor of length L. The pressure drop in the reactor is negligible. 

The fraction unreacted is /< > . Set z = L to obtain it at the reactor outlet. Suppose = 1 and that kai /Ui = 1 in some system of units. Then the variable-density case gives z = 0.3608 at = 0.5. The velocity at this point is 0.75m . The constant density case gives z = 0.5 at a = 0-5 and the velocity at the outlet is unchanged from The constant-density case fails to account for the reduction in u as the reaction proceeds and thus underestimates the residence time. 

Thus, we can make a reasonably accurate initial guess for Qp. This guess is used to calculate the conversion in a tubular reactor of the given dimensions. When the right guess is made, the mean residence time will be 2h and the fraction unreacted will be 20%. The following code follows the general procedure for liquid-phase PFRs. The fraction unreacted is calculated as the ratio of which is denoted as Phi/Philn in... 

Suppose an inert material is transpired into a tubular reactor in an attempt to achieve isothermal operation. Suppose the transpiration rate q is independent of and that qL = Qtrms- Assume all fluid densities to be constant and equal. Find the fraction unreacted for a first-order reaction. Express your final answer as a function of the two dimensionless parameters, QtranslQin and kVIQm where k is the rate constant and... 

Equation (4.10) uses the general definition of fraction unreacted in a flow system. It is moles out divided by moles in. The corresponding, general definition of conversion is... 

For the numerical example, QoutlQin = 0.949 and the fraction unreacted is 0.487 compared with 0.5 if there were no change. Thus, the density change causes a modest increase in conversion. 

Solution In a real problem, the individual values for k, V, and Qi would be known. Their values are combined into the dimensionless group, kVIQi . This group determines the performance of a constant-density reactor and is one of the two parameters needed for the variable-density case. The other parameter is the density ratio, r = Pmommer/Ppoiymer- Setting kVIQtn = 1 gives T = 0.5 as the fraction unreacted for the constant-density case. The individual values for k, V, Qi , Pmommer, and Ppoiymer can be assigned as convenient, provided the composite values are retained. The following... 

Thus, the combination reactor gives intermediate performance. The fraction unreacted from the composite reactor will be lower than that from a single CSTR with F = q + F2 but higher than that from a single PFR with F = Fi + F2. 

Determine the fraction unreacted for a second-order reac-composite reactor consisting of two equal-volume... 

Example 5.5 Ingredients are quickly charged to a jacketed batch reactor at an initial temperature of 25°C. The jacket temperature is 80°C. A pseudo-first-order reaction occurs. Determine the reaction temperature and the fraction unreacted as a function of time. The following data are available ... 

FIGURE 5.2 (a) Temperature and (b) fraction unreacted in a nonisothennal batch reactor with... 

FIGURE 8.1 Fraction unreacted versus dimensionless rate constant for a first-order reaction in various isothermal reactors. The case illustrated with diffusion is for = 0.1. 

Solution For a first-order reaction, we can arbitrarily set = 1 so that the results can be interpreted as the fraction unreacted. The choices for 7 and J ... 

Example 8.6 Generalize Example 8.5 to determine the fraction unreacted for a first-order reaction in a laminar flow reactor as a function of the dimensionless groups and kt. Treat the case of a parabolic velocity profile. 

Y=aoi,tlain = 0.02. There is a safety concern about the premixing step. One proposal is to feed A and B separately. Component A would be fed into the base of the bed using a central tube with diameter 0.212m and component B would be fed to the annulus between the central tube and the reactor wall. The two streams would mix and react only after they had entered the bed. The concentrations of the entering components would be increased by a factor of 2, but the bed-average concentrations and Us would be unchanged. Determine the fraction unreacted that would result from the proposed modification. 

Expressing this result in terms of the fraction unreacted gives a simpler form ... 

Example 15.9 Use residence time theory to predict the fraction unreacted for an isothermal, homogeneous, first-order reaction occurring in a CSTR and aPFR. 

Thus, the fraction unreacted is the Laplace transform with respect to the transform parameter k of the differential distribution function. 

Equation (15.44) shows that Uoutik) has already been normalized by and is thus equal to the fraction unreacted, Y. ... 

Given k fit) for nny reactor, you automatically have an expression for the fraction unreacted for a first-order reaction with rate constant k. Alternatively, given ttoutik), you also know the Laplace transform of the differential distribution of residence time (e.g., k[f(t)] = exp(—t/t) for a PER). This fact resolves what was long a mystery in chemical engineering science. What is f i) for an open system governed by the axial dispersion model Chapter 9 shows that the conversion in an open system is identical to that of a closed system. Thus, the residence time distributions must be the same. It cannot be directly measured in an open system because time spent outside the system boundaries does not count as residence but does affect the tracer measurements. 

Set ttin = 1 SO that a ut is the fraction unreacted. Then Zwietering s differential equation becomes... 

Thus, from part (a) we know that the fraction unreacted lies somewhere between 0.167 and 0.358. 

Thus, knowledge of the residence time distribution has narrowed the possible range on the fraction unreacted. It is now known to be between 0.233 and 0.287. 

Part (c) considers the mixing extremes possible with the physical arrangement of two tanks in series. The two reactors could be completely segregated so one limit remains 0.233 as calculated in part (b). The other limit corresponds to two CSTRs in series. The first reactor has half the total volume so that Uinkii = 2.5. Its output is 0.463. The second reactor has (ai )2ki2 = 1.16, and its output is 0.275. This is a tighter bound than calculated in part (b). The fraction unreacted must lie between 0.233 and 0.275. 

Thus Marvel and Levesque found that from 79 to 85 percent of the oxygen was removed in this process, to be compared with the theoretically calculated figure of 81.6 percent (fraction unreacted equal to 1/c) for intramolecular reaction of this type in a head-to-tail polymer. A head-to-head, tail-to-tail arrangement, consisting of 1,4-diketone structures, should be expected to yield furan rings... 

The reactor effluent is distilled and unreacted EB is recycled. The EB hydroperoxide is then reacted with propylene at 250°F and pressure in the range of 250-700 psi in the presence of a metal catalyst to produce propylene oxide and methylbenzyl alcohol B in Figure 8-7). The reactor mixture is separated by multiple fractionators. Unreacted propylene and EB are recycled. PO is recovered overhead. The methyl benzyl alcohol is easily dehydrated in the vapor stage at 450—500° F and 500 psi pressure over a titanium dioxide or silica gel catalyst to form styrene. Acephenone is one of the by-products. 

Recall that for first-order kinetics a plot of In (fraction unreacted) versus time has a slope — k. Also note that the reaction reaches an equilibrium characterized by an absorbance 0.115 the data must be corrected for this. For both the anionic and cationic micelles, qualitatively sketch, emphasizing the charge state, the micelle, the solubilized substrate, and the approaching OH" reactant. Indicate how these pictures are consistent with the experimental rate constants. 

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