-Ac= cross sectional area available for flow
Figure 4.2 Plug-flow reactor (PFR).
dt Ax where is the cross-sectional area of the reactor, x the distance from the reactor entrance, Ax the length of the infinitesimal volume, and F CA| and F-C J A the mass flow rates of A evaluated at the distances x and x -I- Ax from the reactor entrance. In the limit as Ax » 0, the first term on the right side in Eq. 4.8 becomes the partial derivative of FCA with respect to distance and Eq. 4.8 reduces to aC\ ¿(F'C,)
At constant flow rate, the reactor will achieve steady-state, at which there is no change with respect to time in the concentration at any point within the reactor, reducing the equation to dCv
Hence, it is theoretically possible to calculate the rates by determining the concentration gradient through the reactor. In practice, however, this is difficult to do so plug-flow reactors are seldom used to generate rate data in this fashion.
The main purpose for which the mass balance equation is used is to determine the size of reactor required to achieve a desired conversion from a feed stream of given composition and How rate. If the rate expression is known, Eq. 4.11 can be rearranged and integrated over the length of the reactor, L, to give f
Hence, the required ratio of reactor volume to feed flow rate, V/F, can be obtained.
Butch Reactor. The CSTR and the PFR are both continuous flow reactors. As stated previously, however, environmental engineers sometimes use batch reactors which do not receive flow throughout the entire operational cycle. For the simplest cycle, they are rapidly charged with feed, allowed to react, and the treated effluent removed. Under such conditions a batch reactor has no in or out terms in the mass balance equation during the reaction period. As in a CSTR, a batch reactor is assumed to be perfectly mixed so that the entire reactor content is homogeneous at any given time. Hence, the rate is independent of position in the reactor and the concentration varies only with time. Therefore, it is appropriate to take the whole reactor volume as the control volume. The mass balance on reactant A around a batch reactor is d(CvV) = 0 -0 4- r,-V (4.13)
For constant volume, this simplifies to which states that the rate is equal to the time rate of change of the reactant concentration.
Equation 4.14 suggests that by running a batch reactor experiment, measuring the concentrations of appropriate reactants over time, and taking the time derivatives of the concentration history, one can obtain numerical values of the rate, rA, associated with various concentrations of reactants. Having obtained the data of rate versus reactant concentration, it is then possible to deduce the reaction rate expression, which can be used in the mass balance equations for any other batch or continuous flow reactor to determine the holding time or size required to achieve a given degree of reaction.
An analogy between a batch and a plug-flow reactor can be seen by comparing Eq. 4.14 to Eq. 4.11. Noting that the term F/A, in Eq. 4.11 is the axial velocity through the PFR and that distance divided by velocity is time, it can be seen that Eqs. 4.14 and 4.11 are the same when distance x is translated into the time required to reach it. Thus, it can be seen that each element of fluid passing through a PFR may be thought of as an infinitesimal batch reactor.
Ideal continuous flow reactors are useful for experimental purposes and for understanding how factors like the flow rate and reactor volume influence performance. In fact, a significant portion of this book will be devoted to modeling the performance of ideal reactors as a way of gaining understanding about biochemical operations. It is important to recognize, however, that few wastewater treatment reactors are ideal. There are several reasons for this. One is size. The larger reactors are, the harder it is to achieve ideal mixing conditions. Another is the effect of other requirements. For example, the need for oxygen transfer to aerobic systems imparts considerable axial mixing in reactors that have been designed to approximate plug flow. Yet another is practicality. Site requirements and other such factors may prevent the economical construction of a reactor with a configuration that even approximates ideality.
Residence Time Distribution. Since most full-scale reactors in biochemical operations are nonideal, how might their mixing characteristics be identified and represented? One method is through measurement of the residence time distribution (RTD).
Equations 4.4 and 4.12 both contain the term V/F. This term represents the average length of time that an element of fluid (and therefore a dissolved constituent) stays in a reactor of constant volume receiving a constant flow rate of fluid with constant density. Hence, it has been given the name mean hydraulic residence time, or simply hydraulic residence time. It will be given the symbol t for use in equations and the acronym HRT for use in the text. It represents the time required to process one reactor volume of feed:
Although the HRT is the mean residence time of fluid elements in a reactor, it is not the actual residence time of all elements. Rather, different elements of fluid reside in a reactor for different lengths of time, depending on the routes they follow, and the distribution of those times depends on the reactor's mixing characteristics/ i: Let the function, £(t), be the fraction of the elements in the effluent stream having residence times less than t. With this definition, it is apparent that F(0) = 0 and F(^c) = 1. In other words, none of the fluid can pass through the reactor in zero time and all must come out eventually. This function, which is shown in Figure 4.3, is known as the cumulative distribution function, or F curve. Another function is the point distribution function, £(t), which is related to the cumulative distribution function by:
Thus, it follows that £(t)dt is the fraction of the effluent that has a residence time between t and t + dt, and thus is the RTD function of the fluid in the reactor. The area under the RTD curve (also called the E curve) between the limits of 0 and is unity since the entire fraction of fluid must have residence times between 0 and sc. A typical RTD (£) curve is shown in Figure 4.4.
The mixing characteristics of the two ideal continuous flow reactors represent the extremes between which all others lie. In a perfect PFR, all fluid elements stay in the reactor for exactly the mean residence time. Thus, there is no distribution of residence times; they are all the same. In contrast, in a CSTR all fluid elements have equal probability of leaving the reactor at any moment, regardless of how long they have been in it. This means that the residence time of a fluid element in a CSTR is not fixed but is subject to statistical fluctuations." In particular, the RTD is a negative exponential:
Integrating Eq. 4.17 from t = 0 —» ^ confirms that the area under the £(t) curve for a CSTR is indeed unity. Furthermore, Eq. 4.17 shows that the most probable residence time for fluid elements in a CSTR is zero. The mean residence time, t, is also the standard deviation of the distribution of residence times.
Experimental Determination of Residence Time Distribution. The RTD for a given reactor and flow rate may be determined experimentally by introducing an inert tracer into the reactor input and observing the time response of the tracer concentration in the reactor effluent. The two most convenient types of tracer inputs are step and impulse signals."
Imagine a reactor receiving influent at constant flow rate, F. At time zero a continuous flow of a soluble tracer is added to that stream, instantly changing the tracer concentration from zero to ST<) and maintaining it at that concentration thereafter, i.e., a step change in concentration. We immediately start measuring the output tracer concentration, ST, and plot ST/Sm versus time. How does the resulting curve relate to the RTD? To answer that, imagine that we could divide the effluent stream into two fractions, one that has spent less time than t in the reactor, £(t), and one that has spent more, 1 — £(t), where t is measured from the time that the tracer was introduced. Any flow that has been in the reactor less than t will have the tracer in
Figure 4.3 Cumulative residence time distribution function, F(t).
it, and thus, the fraction .F(t) will have a tracer concentration S, = ST(). Conversely, any fluid that has been in the reactor more than t was present before the tracer was introduced and thus has no tracer in it. Consequently, the fraction 1 — F(t) has ST = 0. Since mass flow rate is the product of liquid flow rate and concentration, the total mass flow rate of tracer in the effluent at time t, F-S,(t), will be:
Equation 4.20 states that the cumulative distribution function, F(t), is identical to the normalized tracer concentration response due to the step input. Thus, imposition of a step input of tracer to a reactor is a convenient way of experimentally determining the cumulative distribution function. According to Eq. 4.16, it may be differentiated to obtain the RTD curve.
Differentiation of experimental data is risky, and thus it would be better to have a direct way of determining the RTD function. A similar analysis can be used to
establish that the point distribution function, £(t), is identical to the normalized effluent concentration curve resulting from an impulse input into the feed.1' An impulse input is one in which a slug of tracer is added instantaneously to the reactor feed. Consequently, in this case, the measured output concentration is normalized by dividing each measured concentration, ST, by the total area under the curve of S, versus time. The area can be obtained by Simpson's rule or by graphical integration. Furthermore, the area is equivalent to the total mass of tracer added divided by the flow rate to the reactor, if both of those quantities are known. Equation 4.16 also suggests that the F(t) function can be obtained from the impulse input response by integration. Hence, either a step input or an impulse input may be used to obtain both £(t) and F(t). However, the more commonly used technique is to obtain the £(t) function from an impulse input, and then obtain the F(t) function by integration if it is needed.
Use of a tracer test on a reactor will allow determination of its mixing characteristics. If the RTD function resulting from the test conforms to either of the ideal reactor types, the performance of the reactor can be predicted or simulated by application of mass balance equations for all relevant constituents with appropriate reaction rate expressions. However, if the RTD function deviates from the ideal, more involved techniques must be used.
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