Ideal Batch Reactors

A batch reactor is initially filled with all reagents. Diuing the reaction no substances are introduced into or removed from the reactor. Additionally, it is normally assumed that the volume remains constant during reaction. 

6) nj denotes the number of moles of substance i and, Vjj the stoichiometric number of component i in reaction j (positive for created substances and negative for removed ones), V the volume of the reacting mixture in m. Introducing the conversion X 

8) is integrated it enables one to calculate the time until a certain degree of conversion, X, is reached 

Equation (3.10) holds for an adiabatic system, which does not exchange heat with its surroundings. Multiplying Eq. (3.10) with the reactor volmne V and assuming for simplicity s sake that the material coefficients 

The maximum final temperature is reached if all of the reactant is consumed, i.e. n(t) = 0. The result is the adiabatic temperature rise AT  

A batch reactor is defined as a reactor in which there is no flow of mass across the system boundaries, once the reactants have been charged. The reaction is assumed to begin at some precise point in time, usually takoi as r = 0. This time may correspond, fw exarrqile, to when a catalyst or initiator is added to the batch, or to when the last reactant is added. 

Hgure 3 lb The top of die reactor in Hgute 3-la. The view port in the left front of the picture pennits the contents of the reactor to be observed, and can be opened to pennit solids to be charged to the reactor. Amotor that drives an agitator is located in the top crater of the picture, andacharging line and valve actuator coonected to a valve are on the 1 (Photo used with permissioo of Syngrata Crop Protection, Inc.) 

Batch reactors are used extensively diroughout the chemical and phaimaceutical industries to manufacture products on a relatively small scale. Properly equipped, these reactors are very flexible. A single reactor may be used to produce many different products. 

Batch reactors usually are mechanically agitated to ensure that the contents are well mixed. Agitation also increases the heat-transfer coefficient between the reactor contents and any heat-transfer surface in the reactor. In multiphase reactors, agitation may also keep a solid catalyst suspended, or may create surface area between two liquid phases or between a gas phase and a liquid phase. 

Vtty few reactions are thermally neutral (AHr = 0), so it frequently is necessary to either supply heat or remove heat as the reaction proceeds. Ihe most common means to transfer heat is to circulate a hot or cold fluid, either through a coil that is immersed in the reactor, or through a jacket that is attached to the wall of the reactor, or both. 

In Section 4.2, we derive the design equations for the first three ideal reactor models. Other reactor configurations are discussed in Chapter 9. 

2 SPECIES-BASED DESIGN EQUATIONS OF IDEAL REACTORS 

If the reactor is well mixed, the same conditions (concentration and temperature) exist everywhere hence, (r/) is the same throughout the reactor. Thus, Eq. 4.1.3 becomes 

Equation 4.2.1 is the species-based design equation of an ideal batch reactor, written for species j. To obtain the operating time, we separate the variables and integrate Eq. 4.2.1  

Equation 4.2.2 is the integral form of the species-based design equation for an ideal batch reactor, written for species j. It provides a relation between the operating time, t, the amount of the species in the reactor, Nj(t) and Nj(0), the species formation rate, (rj), and the reactor volume, V. Note that when the reaetor volume does not change during the operation, Eq. 4.2.2 reduces to 

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Heat and mass transfer limitations are rarely important in the laboratory but may emerge upon scaleup. Batch reactors with internal variations in temperature or composition are difficult to analyze and remain a challenge to the chemical reaction engineer. Tests for such problems are considered in Section 1.5. For now, assume an ideal batch reactor with the following characteristics ... 

The most important characteristic of an ideal batch reactor is that the contents are perfectly mixed. Corresponding to this assumption, the component balances are ordinary differential equations. The reactor operates at constant mass between filling and discharge steps that are assumed to be fast compared with reaction half-lives and the batch reaction times. Chapter 1 made the further assumption of constant mass density, so that the working volume of the reactor was constant, but Chapter 2 relaxes this assumption. 

The feed is charged all at once to a batch reactor, and the products are removed together, with the mass in the system being held constant during the reaction step. Such reactors usually operate at nearly constant volume. The reason for this is that most batch reactors are liquid-phase reactors, and liquid densities tend to be insensitive to composition. The ideal batch reactor considered so far is perfectly mixed, isothermal, and operates at constant density. We now relax the assumption of constant density but retain the other simplifying assumptions of perfect mixing and isothermal operation. 

The component balance for a variable-volume but otherwise ideal batch reactor can be written using moles rather than concentrations ... 

Chapter 2 treated multiple and complex reactions in an ideal batch reactor. The reactor was ideal in the sense that mixing was assumed to be instantaneous and complete throughout the vessel. Real batch reactors will approximate ideal behavior when the characteristic time for mixing is short compared with the reaction half-life. Industrial batch reactors have inlet and outlet ports and an agitation system. The same hardware can be converted to continuous operation. To do this, just feed and discharge continuously. If the reactor is well mixed in the batch mode, it is likely to remain so in the continuous mode, as least for the same reaction. The assumption of instantaneous and perfect mixing remains a reasonable approximation, but the batch reactor has become a continuous-flow stirred tank. 

The design equations for a chemical reactor contain several parameters that are functions of temperature. Equation (7.17) applies to a nonisothermal batch reactor and is exemplary of the physical property variations that can be important even for ideal reactors. Note that the word ideal has three uses in this chapter. In connection with reactors, ideal refers to the quality of mixing in the vessel. Ideal batch reactors and CSTRs have perfect internal mixing. Ideal PFRs are perfectly mixed in the radial direction and have no mixing in the axial direction. These ideal reactors may be nonisothermal and may have physical properties that vary with temperature, pressure, and composition. 

Figure 2.4. Schematic drawings of a cylindrical flow reactor and a batch reactor. In the ideal case the flow reactor operates as a plug-flow reactor in which the gas moves as a piston down through the tube, whereas the ideal batch reactor is a well-mixed Tank Reactor...

Ideal-batch reactor. Consider a batch reactor in which the feed is charged at the beginning of the batch and no product is withdrawn until the batch is complete. Given that ... 

It should be noted that the analysis for an ideal-batch reactor is the same as that for a plug-flow reactor (compare Equations 5.43 and 5.61). All fluid elements have the same residence time in both cases. Thus... 

Figure 5.4a compares the profiles for a mixed-flow and plug-flow reactor between the same inlet and outlet concentrations, from which it can be concluded that the mixed-flow reactor requires a larger volume. The rate of reaction in a mixed-flow reactor is uniformly low as the reactant is instantly diluted by the product that has already been formed. In a plug-flow or ideal-batch reactor,... 

High reaction rate in Equation 5.71 is favored by a high concentration of enzymes (CE ) and high concentration of feed (CA). This means that a plug-flow or ideal-batch reactor is favored if both the feed material and enzymes are to be fed to the reactor. 

As with continuous processes, the heart of a batch chemical process is its reactor. Idealized reactor models were considered in Chapter 5. In an ideal-batch reactor, all fluid elements have the same residence time. There is thus an analogy between ideal-batch reactors and plug-flow reactors. There are four major factors that effect batch reactor performance ... 

A simulation model needs to be developed for each reactor compartment within each time interval. An ideal-batch reactor has neither inflow nor outflow of reactants or products while the reaction is carried out. Assuming the reaction mixture is perfectly mixed within each reactor compartment, there is no variation in the rate of reaction throughout the reactor volume. The design equation for a batch reactor in differential form is from Chapter 5 ... 

The temperature and composition of the contents of an ideal batch reactor are uniform at any instant, but the concentration changes with time. Since the composition is uniform, the mass balance may be performed over the whole reactor. 

The simplest reactor configuration for any enzyme reaction is the batch mode. A batch enzyme reactor is normally equipped with an agitator to mix the reactant, and the pH of the reactant is maintained by employing either a buffer solution or a pH controller. An ideal batch reactor is assumed to be well mixed so that the contents are uniform in composition at all times. 

In chemical reaction engineering, an ideal batch reactor is defined as a closed reactor, meaning there is no addition and no removal of any components during the reaction time. The prototype of this reactor is the autoclave, where all reactants are charged into the reactor at the beginning of the operation (Figure 6.3). The reactor is then closed and heated to reaction temperature, the temperature at which the reaction is allowed to complete or at which a catalyst is added. After the reaction is completed, the reactor is cooled and discharged. It is now ready for a new cycle. 

A more quantitative analysis of the batch reactor is obtained by means of mathematical modeling. The mathematical model of the ideal batch reactor consists of mass and energy balances, which provide a set of ordinary differential equations that, in most cases, have to be solved numerically. Analytical integration is, however, still possible in isothermal systems and with reference to simple reaction schemes and rate expressions, so that some general assessments of the reactor behavior can be formulated when basic kinetic schemes are considered. This is the case of the discussion in the coming Sect. 2.3.1, whereas nonisothermal operations and energy balances are addressed in Sect. 2.3.2. 

In Chaps. 5 and 6 model-based control and early diagnosis of faults for ideal batch reactors have been considered. A detailed kinetic network and a correspondingly complex rate of heat production have been included in the mathematical model, in order to simulate a realistic application however, the reactor was described by simple ideal mathematical models, as developed in Chap. 2. In fact, real chemical reactors differ from ideal ones because of two main causes of nonideal behavior, namely the nonideal mixing of the reactor contents and the presence of multiphase systems. 

The principles and methods of scale-up can be applied to chemical reactors. In the absence of significant thermal effects, i.e., when the ratio <2r/ Vr may be considered negligible, ideal batch reactors do not show any problem of scale-up, because the volume Vr does not appear in the mathematical model (2.17), so that their performance is only determined by chemical kinetics (see Sect. 2.3). On the contrary, a very complex behavior is expected for real reactors in fact, this behavior cannot be analyzed in terms of mathematical models, and the design procedures must be largely based on semi-empirical rules of scale-up. 

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