Question

What assumptions were made in the derivation of the design equation for: (a) the batch reactor? (b) the CSTR? (c) the plug-flow reactor (PFR)? (d) the packed-bed reactor (PBR)? (e) State in words the meanings of \(-r_{A} .-r_{A}^{\prime} .\) and \(r_{A}^{\prime} .\) Is the reaction rate \(-r_{A}\) an extensive quantity? Explain.

Short answer

In summary, the design equations and assumptions for different reactors are: (a) Batch reactor: \(V = C_{A0}\int_{0}^{t}dt/(-r_{A})\), assuming perfect mixing and no flow. (b) CSTR: \(V = F_{A0}X/(-r_{A})\), assuming perfect mixing, steady state and no radial gradients. (c) PFR: \(V = \int_{0}^{X}F_{A0}dX/(-r_{A})\), assuming plug flow and no radial gradients. (d) PBR: \(W = \int_{0}^{X}F_{A0}dX/(-r_{A}^{'})\), assuming plug flow and constant pressure drop. \( -r_{A}\) is the reaction rate per unit volume, \( -r_{A}^{\prime}\) is the reaction rate per unit weight of the catalyst, and \(r_{A}^{\prime}\) is the production rate per unit weight of the catalyst. The reaction rate \( -r_{A}\) is an intensive quantity as it does not depend on the amount of material in the system.

Step-by-step solution

Derivation of Batch Reactor Design Equation and Assumptions

The basic design equation for a batch reactor is: \( V = C_{A0}\int_{0}^{t}dt/(-r_{A}) \). The assumptions made while deriving this equation include: 1. The system is perfectly mixed: This ensures a uniform concentration of reactants throughout the reactor. 2. There is no flow of material in or out of the system during the reaction: This is because the batch reactor is shut for a certain amount of time for the reaction to take place.

Derivation of CSTR Design Equation and Assumptions

The design equation for a CSTR is: \( V = F_{A0}X/(-r_{A}) \). The assumptions for this are: 1. Perfect mixing: Like the batch reactor, the CSTR is also considered perfectly mixed. 2. Steady state operation: The reaction rate and reactant concentrations are constant over time. 3. No radial gradients: This assumes that there are no variations in concentration or temperature across different radial locations in the reactor.

Derivation of PFR Design Equation and Assumptions

The design equation for a PFR is: \( V = \int_{0}^{X}F_{A0}dX/(-r_{A}) \). Assumptions include: 1. Plug Flow: There is no mixing in the direction of flow, i.e., the flow is 'plugged'. 2. No radial gradients: There is uniform concentration and temperature across any cross-section of the reactor.

Derivation of PBR Design Equation and Assumptions

The design equation for a PBR is similar to the PFR, but incorporates the catalyst weight: \( W = \int_{0}^{X}F_{A0}dX/(-r_{A}^{'}) \). Assumptions include: 1. Plug Flow: Like the PFR, there is no mixing in the direction of flow. 2. Constant pressure drop along reactor length: It assumes that the pressure gradient caused by packing material is relatively constant.

Meanings of \( -r_{A} \), \( -r_{A}^{\prime} \) and \( r_{A}^{\prime} \)

\( -r_{A} \) refers to the molar rate of consumption of species A per unit volume (e.g. mol/(L·s)). \( -r_{A}^{\prime} \) refers to the molar rate of consumption of species A per unit weight of catalyst (e.g. mol/(kgcat·s)). \( r_{A}^{\prime} \) refers to the rate of production of species A per unit weight of the catalyst. It's basically \( -r_{A}^{\prime} \) for a product species (since the production rate is positive).

Is Reaction Rate Extensive?

Reaction rate \( -r_{A} \) is considered an intensive quantity. An intensive property doesn't depend on the amount of material in the system. Thus, the reaction rate, which is a measure of how quickly a reaction proceeds, does not directly depend on the total volume or mass of the reactants, but rather their concentration or partial pressures. This is why reaction rates are often given in terms of amount of substance per unit volume (or mass) per unit time.

Key concepts

Batch Reactor Design

The batch reactor is a fundamental piece of equipment in chemical reaction engineering, utilized for carrying out a variety of chemical reactions. It is characterized as a closed system where reactants are mixed together and allowed to react over a period of time without any influx or efflux to the system. Understanding the design and operation of batch reactors is paramount for chemical engineering students and professionals alike.

Let's delve into the core of batch reactor design. The primary equation governing the design is given by: \( V = C_{A0}\int_{0}^{t}dt/(-r_{A}) \). This equation is a reflection of two fundamental assumptions: the reactor ensures a perfectly uniform concentration of reactants due to thorough mixing and operates in a sealed condition, disallowing material flow during the reaction phase. The design's basis revolves around calculating the volume \(V\) for a given initial concentration \(C_{A0}\) and the reaction's progress over time \(t\), considering the rate of consumption of reactant A, \( -r_{A} \).

Continuous Stirred-Tank Reactor (CSTR)

A key concept in chemical engineering is the continuous stirred-tank reactor, or CSTR, which provides a steady-state environment for chemical reactions to occur. Unlike the batch reactor, a CSTR operates with continuous input and output flows, maintaining constant reactant concentrations and reaction rates over time. The cornerstone of CSTR design rests on the equation: \( V = F_{A0}X/(-r_{A}) \).

The assumptions behind this equation include perfect mixing, leading to no gradient of reactant concentrations across the reactor, and steady-state operation where the reaction conditions remain unchanged. Additionally, it presumes there's no concentration or temperature variation radially. This design is central for implementing reactions that need to be scaled up from lab to industrial scales, ensuring constant product quality and process efficiency.

Plug-Flow Reactor (PFR)

The plug-flow reactor design is a staple in chemical processes, known for its simplicity and effectiveness. Embodied by the equation \( V = \int_{0}^{X}F_{A0}dX/(-r_{A}) \), the PFR assumes that the reactant mixture flows through the reactor as distinct plugs, with each plug having a uniform concentration and temperature.

This lack of axial mixing allows for straightforward scaling of reaction time with reactor length and ensures that reactions exhibiting a high degree of conversion can be efficiently completed. However, this design typically necessitates additional considerations such as precise flow control and heat management. The unique characteristic of PFRs is their ability to provide large conversion levels in reactions that require longer residence times, due to their differential processing of reactant plugs.

Packed-Bed Reactor (PBR)

In industries dealing with catalyst-based reactions, a packed-bed reactor (PBR) is often the reactor of choice. The PBR's design integrates both chemical reaction and catalyst properties, which is represented by the equation: \( W = \int_{0}^{X}F_{A0}dX/(-r_{A}^{\prime}) \). Here, the flow of reactants is similar to that in a PFR, but the reaction occurs over a solid catalyst.

Primary assumptions hold that the flow is 'plugged' and free of mixing longitudinally, and that the pressure drop remains consistent along the reactor's length. This platform offers optimized contact between reactants and catalysts, enhancing reaction rates and making PBRs suitable for reactions that are limited by the catalyst's surface area. For students looking to apply their knowledge practically, understanding the dynamics of PBR is crucial, as it merges the principles of reaction kinetics with those of catalyst effectiveness.