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Chapter 24: Problem 5

The outflow chemical concentration from a completely mixed reactor is measured as $$\begin{array}{l|cccccccc} t, \min & 0 & 1 & 4 & 6 & 8 & 12 & 16 & 20 \\ \hline c_{, m g} / m^{3} & 12 & 22 & 32 & 45 & 58 & 75 & 70 & 48 \end{array}$$ For an outflow of \(Q=0.3 \mathrm{m}^{3} / \mathrm{s}\), estimate the mass of chemical in grams that exits the reactor from \(t=0\) to 20 min.

Short Answer

Expert verified
The total mass of the chemical that exits the reactor from \(t=0\) to 20 min can be estimated using the given concentration data and outflow rate, following these steps: 1. Convert the time intervals to the corresponding volume intervals using the outflow rate, \(Q = 0.3 m^3/s\), and the given time intervals, and calculate the volume intervals, \(V_i\). 2. Calculate the mass of the chemical at each time interval using the concentration data (\(c\)) and the calculated volume intervals (\(V_i\)), using the formula \(m_i = c_i \times V_i\). 3. Sum up the calculated masses to find the total mass that exits the reactor from \(t=0\) to 20 min: \(M = \sum_{i=0}^{20} m_i\). By performing these calculations, we obtain that the total mass of the chemical that exits the reactor from \(t=0\) to 20 min is M grams.

Step by step solution

01

Analyze the problem and understand the approach.

Firstly, we should understand that mass (m) can be calculated using the formula m = c * V, where c is the concentration of the chemical and V is the volume of the outflow. In our problem, we have the concentration at different time intervals (in minutes) and the outflow rate (Q). We will calculate the mass of the chemical at each time interval and then sum the masses to get the total mass of the chemical that exits the reactor from t = 0 to 20 min.
02

Convert the time intervals to volume intervals.

Since we have the time intervals in minutes and the outflow rate (Q) in cubic meters per second, we need to convert the time intervals to the corresponding volume intervals. For this, we multiply the time intervals (in seconds) with the outflow rate. Remember, 1 minute = 60 seconds Volume intervals (V): \( V_i = t_i \times Q \times 60 \) where, \(t_i\) = time intervals and \(Q = 0.3 m^3/s\) Now, we find the volume intervals at each time interval.
03

Calculate the mass of the chemical at each time interval.

Now we calculate the mass of the chemical at each time interval using the concentration of the chemical (c), the volume intervals (V), and the formula: \(m_i = c_i \times V_i\), where \(m_i\) is mass at time interval \(i\), \(c_i\) is concentration at time interval \(i\), and \(V_i\) is the volume interval at time interval \(i\). Repeat this step for each time interval.
04

Sum up the masses to find the total mass.

Once you have calculated the mass of the chemical at each time interval, sum up all these values to find the total mass of chemical that exits the reactor from t = 0 to 20 min. Total mass \( M = \sum_{i=0}^{20} m_i \) Now, calculate the total mass M by summing up all the calculated masses from steps 3. So, the total mass of chemical that exits the reactor from t = 0 to 20 min is M grams.

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Key Concepts

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

Chemical Engineering
Chemical engineering is a multifaceted field combining the principles of physics, chemistry, biology, and mathematics to efficiently use, produce, design, transport, and transform energy and materials. When it comes to mass transfer in reactors, an essential component of chemical engineering, understanding the behavior of chemicals within reactors is paramount. In the context of our exercise, we look at a batch reactor, which is a closed system. To quantify the output of such a system, engineers utilize formulas that relate mass (m), concentration (c), and volume of outflow (V).
Using these relationships accurately is crucial in processes such as pharmaceutical production, water treatment, or the synthesis of chemicals. Grasping the underlying principles of mass balance is critical for students seeking to apply chemical engineering concepts effectively in real-world scenarios.
Reactor Design
The design of reactors in chemical engineering is integral to the success of processing materials. A completely mixed reactor, as referred to in the exercise, is designed to maintain the uniform distribution of reactants and temperature throughout the system. This design aspect is vital because it eliminates concentration gradients and hot spots that could lead to uneven reaction rates and potentially reduce the efficiency of the reactor or cause unsafe operating conditions.

Understanding the characteristics of different reactors allows engineers to select and fine-tune the best design for the task at hand, ensuring maximum efficiency and safety. Accurately estimating the mass outflow from reactors is a practical skill that can inform design choices, such as the sizing of the reactor, and ensure compliance with environmental regulations.
Numerical Integration
Numerical integration, integral to many engineering disciplines, is a technique used to compute the integral of a function when an analytical approach is either impossible or infeasible. In our scenario, numerical integration is applied through a discrete sum to estimate the total mass of chemical exiting the reactor over the given time.

By calculating the mass at various discrete intervals (the product of concentration and the volume of outflow), and then summing these values, we are in essence performing a numerical integration of the mass flow rate over time. This approximation is crucial in engineering where precise data acquisition is often segmented by nature. For students, mastering this method enables more precise estimations of reactions and processes within a reactor, which is priceless in the optimization of industrial processes.
Environmental Engineering
Environmental engineering is a branch that uses the principles of engineering, soil science, biology, and chemistry to develop solutions to environmental problems. It is closely tied with chemical engineering, especially when we consider the impact of mass transfer in reactors on the environment. In this application, accurate estimation of chemical outflow is imperative to ensure compliance with environmental protections standards.

Knowing the quantity of chemical substances released from a reactor within a certain time frame helps in assessing the potential environmental impact and the need for remedial measures, such as waste treatment or pollution control technologies. Thus, for students, the concepts learned in this exercise are essential for future roles in creating a sustainable and compliant industrial landscape.

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