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Chapter 28: Problem 11

A nonisothermal batch reactor can be described by the following equations: $$\begin{aligned} &\frac{d C}{d t}=-e^{(-10 /(T+273))} C\\\ &\frac{d T}{d t}=1000 e^{(-10 /(T+273))} C-10(T-20) \end{aligned}$$ where \(C\) is the concentration of the reactant and \(T\) is the temperature of the reactor. Initially the reactor is at \(15^{\circ} \mathrm{C}\) and has a concentration of reactant \(C\) of \(1.0 \mathrm{gmol} / \mathrm{L}\). Find the concentration and temperature of the reactor as a function of time.

Short Answer

Expert verified
In summary, to solve the given nonisothermal batch reactor problem, we should: 1. Identify the governing ODEs and initial conditions: \(\frac{d C}{d t}=-e^{(-10 /(T+273))} C\) and \(\frac{d T}{d t}=1000 e^{(-10 /(T+273))} C-10(T-20)\), with initial conditions: \(C(0) = 1\;\mathrm{mol/L}\) and \(T(0) = 288\;\mathrm{K}\). 2. Convert initial temperature to Kelvin: \(T_{(0)}^{K} = T_{(0)}^{\circ\mathrm{C}} + 273 = 15 + 273 = 288\;\mathrm{K}\). 3. Use a numerical method (e.g., Euler's method or 4th-order Runge-Kutta method) and a software tool (e.g., MATLAB, Python, or Wolfram Mathematica) to solve the coupled ODEs with given initial conditions. 4. Analyze and interpret the obtained concentration and temperature profiles over time to gain insights into the reactor's behavior and evaluate its efficiency or optimize its operation.

Step by step solution

01

Identify the governing ODEs and initial conditions

We are given the following ODEs that describe the system behavior: $$\begin{aligned} &\frac{d C}{d t}=-e^{(-10 /(T+273))} C,\;\;\text{for concentration}\\ &\frac{d T}{d t}=1000 e^{(-10 /(T+273))} C-10(T-20),\;\;\text{for temperature} \end{aligned}$$ And the initial conditions are: \(C(0) = 1\;\text{gmol/L}\) and \(T(0) = 15^{\circ}\mathrm{C}\).
02

Convert grams per liter to moles per liter and Celsius to Kelvin

To keep the units consistent, we should convert the initially given temperature from Celsius to Kelvin and convert the concentration from grams per liter to moles per liter. Since the temperature in the ODEs is given in Kelvin, we will convert the initial temperature from Celsius to Kelvin as follows: \(T_{(0)}^{K} = T_{(0)}^{\circ\mathrm{C}} + 273 = 15 + 273 = 288\;\mathrm{K}\) We are given concentration in gmol/L which is already in moles per liter, so we don't need to convert it. Thus, we have the initial conditions: \(C(0) = 1\;\mathrm{mol/L}\) and \(T(0) = 288\;\mathrm{K}\).
03

Solution of coupled ODEs

To find the concentration and temperature as functions of time, we need to solve the coupled ODEs numerically since they are nonlinear and cannot be solved analytically. The most common method to numerically solve this kind of problem is to use numerical integration methods such as Euler's method or Runge-Kutta methods (e.g., 4th-order Runge-Kutta method). There are also several software tools available for solving these equations such as MATLAB, Python (SciPy), or Wolfram Mathematica. Here, we will use a general approach and leave the implementation to the specific software package that the student is using. Using your preferred numerical method and software, set up the given coupled ODEs and the initial conditions, and solve for the concentration and temperature over time.
04

Interpret the solution

Once the numerical solution is obtained, the concentration and temperature profiles over time can be plotted. This will provide insights into the reactor's behavior and help to understand how the concentration and temperature change over time. Based on the solution, the efficiency of the reactor or any other operational aspect could be evaluated and optimized.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Nonisothermal Batch Reactor
A nonisothermal batch reactor is a type of chemical reactor where the temperature within the reactor changes during the reaction process. In such a reactor, both the concentration of reactants and the temperature are variables that evolve over time. Unlike isothermal reactors, which maintain a constant temperature, nonisothermal reactors experience heat that can be generated or absorbed by the chemical reactions. This heat change affects the reaction rate, leading to complex interactions between concentration and temperature, as seen in the given equations.
  • The concentration of a reactant is denoted as \(C\), representing how much reactant is present in the reactor.
  • The temperature \(T\) impacts the speed and direction of reactions.
  • Heat balance and reaction kinetics are critical aspects of nonisothermal reactor models.
Understanding the behavior of nonisothermal batch reactors is crucial in industries such as pharmaceuticals, where precise temperature control can affect product quality.
Numerical Solution
In solving complex equations that arise from nonisothermal batch reactors, analytical solutions may not always be possible due to the non-linearity of the equations. Instead, numerical solutions are often employed. Numerical solutions involve approximating the solution of equations using specific algorithms that compute values at discrete points.
  • Numerical methods provide approximate solutions when exact solutions can't be found analytically.
  • Software tools like MATLAB, Python's SciPy, or Wolfram Mathematica are commonly used for these calculations.
  • The accuracy of these methods depends on the step size and the specific numerical technique used.
The goal is to use these methods to calculate how the reactant concentration and temperature within the reactor change over time with a high degree of accuracy.
Runge-Kutta Methods
The Runge-Kutta methods are a family of iterative methods used to solve ordinary differential equations (ODEs), like those found in the nonisothermal batch reactor. Among these, the 4th-order Runge-Kutta method is particularly popular for its accuracy and simplicity.
  • Runge-Kutta methods are more accurate than simpler methods like Euler's method.
  • The 4th-order method provides a good balance between computational effort and accuracy.
  • These methods involve calculating intermediate slopes to estimate the future behavior of the system.
In practice, applying a Runge-Kutta method involves setting up the given ODEs, implementing the method in a software tool, and iterating through time steps to simulate the reactor's behavior.
Initial Conditions
Initial conditions are critical to solving differential equations because they provide the starting point for numerical simulations. For the nonisothermal batch reactor, initial conditions include the initial concentration of the reactant and the initial temperature of the reactor.
  • Initial conditions are given as \(C(0) = 1 \; \text{mol/L}\) and \(T(0) = 15^{\circ}\mathrm{C}\).
  • Temperature must be converted to Kelvin as the kinetic equations depend on this scale: \(T(0) = 15 + 273 = 288\;\mathrm{K}\).
  • These can significantly influence the path the system will take and the eventual outcomes of the simulation.
Precisely setting these conditions ensures that the numerical solution accurately reflects the physical system modeled by the ODEs.

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Most popular questions from this chapter

Just as Fourier's law and the heat balance can be employed to characterize temperature distribution, analogous relationships are available to model field problems in other areas of engineering. For example, electrical engineers use a similar approach when modeling electrostatic fields. Under a number of simplifying assumptions, an analog of Fourier's law can be represented in one- dimensional form as \\[ D=-\varepsilon \frac{d V}{d x} \\] where \(D\) is called the electric flux density vector, \(\varepsilon=\) permittivity of the material, and \(V=\) electrostatic potential. Similarly, a Poisson equation for electrostatic fields can be represented in one dimension as $$\frac{d^{2} V}{d x^{2}}=-\frac{\rho_{v}}{\varepsilon}$$ where \(\rho_{v}=\) charge density. Use the finite-difference technique with \(\Delta x=2\) to determine \(V\) for a wire where \(V(0)=1000, V(20)\) \(=0, \varepsilon=2, L=20,\) and \(\rho_{v}=30\).

If \(c_{\mathrm{in}}=c_{b}\left(1-e^{-0.12 t}\right),\) calculate the outflow concentration of a conservative substance (no reaction) for a single, completely mixed reactor as a function of time. Use Heun's method (without iteration) to perform the computation. Employ values of \(c_{b}=40 \mathrm{mg} / \mathrm{m}^{3}\) \(Q=6 \mathrm{m}^{3} / \mathrm{min}, V=100 \mathrm{m}^{3},\) and \(c_{0}=20 \mathrm{mg} / \mathrm{m}^{3} .\) Perform the computation from \(t=0\) to 100 min using \(h=2 .\) Plot your results along with the inflow concentration versus time.

The following equation can be used to model the deflection of a sailboat mast subject to a wind force: \\[ \frac{d^{2} y}{d z^{2}}=\frac{f}{2 E I}(L-z)^{2} \\] where \(f=\) wind force, \(E=\) modulus of elasticity, \(L=\) mast length and \(I=\) moment of inertia. Calculate the deflection if \(y=0\) and \(d y / d z=0\) at \(z=0 .\) Use parameter values of \(f=60, L=30, E=\) \(1.25 \times 10^{8},\) and \(I=0.05\) for your computation.

Seawater with a concentration of \(8000 \mathrm{g} / \mathrm{m}^{3}\) is pumped into a well-mixed tank at a rate of \(0.6 \mathrm{m}^{3} / \mathrm{hr}\). Because of faulty design work, water is evaporating from the tank at a rate of \(0.025 \mathrm{m}^{3} / \mathrm{hr}\). The salt solution leaves the tank at a rate of \(0.6 \mathrm{m}^{3} / \mathrm{hr}\) (a) If the tank originally contains \(1 \mathrm{m}^{3}\) of the inlet solution, how long after the outlet pump is turned on will the tank run dry? (b) Use numerical methods to determine the salt concentration in the tank as a function of time.

A biofilm with a thickness, \(L_{f}[\mathrm{cm}]\), grows on the surface of a solid (Fig. \(P 28.13\) ). After traversing a diffusion layer of thickness, \(L[\mathrm{cm}],\) a chemical compound, \(A,\) diffuses into the biofilm where it is subject to an irreversible first-order reaction that converts it to a product, \(B\) Steady-state mass balances can be used to derive the following ordinary differential equations for compound \(A\) : $$\begin{aligned} &D \frac{d^{2} c_{a}}{d x^{2}}=0 \quad 0 \leq x
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