A brine solution of salt flows at a constant rate of 6 L/min into a large tank that initially held 100 L of pure water. The solution inside the tank is kept well stirred and flows out of the tank at a rate of 5 L/min. If the concentration of salt in the brine entering the tank is 0.5 kg/L, determine the mass of salt in the tank after t min. When will the concentration of salt in the tank reach 0.4 kg/L? Determine the mass of salt in the tank after t min. mass=kg
A brine solution of salt flows at a constant rate of 6 L/min into a large tank that initially held 100 L of pure water. The solution inside the tank is kept well stirred and flows out of the tank at a rate of 5 L/min. If the concentration of salt in the brine entering the tank is 0.5 kg/L, determine the mass of salt in the tank after t min. When will the concentration of salt in the tank reach 0.4 kg/L? Determine the mass of salt in the tank after t min. mass=kg
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Problem Statement:
A brine solution of salt flows at a constant rate of 6 L/min into a large tank that initially held 100 L of pure water. The solution inside the tank is kept well stirred and flows out of the tank at a rate of 5 L/min. If the concentration of salt in the brine entering the tank is 0.5 kg/L, determine the mass of salt in the tank after \( t \) minutes. When will the concentration of salt in the tank reach 0.4 kg/L?
---
### Calculation Task:
Determine the mass of salt in the tank after \( t \) minutes.
\[ \text{mass} = \boxed{ } \, \text{kg} \]
---
### Explanation:
This problem involves a dynamic system where a brine solution with a known concentration of salt is flowing into a tank while the mixed solution flows out at a different rate. Understanding this system requires basic knowledge of differential equations and principles of mass balance.
1. **Input/Output Calculation**:
- **Input Salt Concentration**: 0.5 kg/L
- **Inflow Rate**: 6 L/min
- **Outflow Rate**: 5 L/min
2. **Initial Conditions**:
- **Initial Volume of Water**: 100 L
- **Initial Mass of Salt**: 0 kg
3. **Differential Equation Formulation**:
- The volume of the liquid in the tank changes over time due to the difference between the inflow and outflow rates.
- The concentration changes as salt is added and removed.
4. **Solution Steps**:
- Define the volume of the solution in the tank as a function of time.
- Set up and solve the differential equation incorporating the rates of inflow and outflow and the concentration of the entering solution.
- Calculate the mass of salt in the tank at any time \( t \).
By proceeding through these steps, one can derive a function for the concentration of salt over time and identify when it reaches 0.4 kg/L. This type of problem is typical in engineering, particularly in chemical and environmental engineering disciplines, where process dynamics are analyzed.
### Graphs and Diagrams:
The prompt does not include any specific graphs or diagrams. However, a detailed explanation using a graph might include:
- A graph showing the volume of liquid in the tank over time.
- A graph](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe57d9078-3a26-4ab9-a5c6-f67f173c178d%2F93afc3b9-0bf0-4b43-8c34-672ab888777b%2Fyywxke_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
A brine solution of salt flows at a constant rate of 6 L/min into a large tank that initially held 100 L of pure water. The solution inside the tank is kept well stirred and flows out of the tank at a rate of 5 L/min. If the concentration of salt in the brine entering the tank is 0.5 kg/L, determine the mass of salt in the tank after \( t \) minutes. When will the concentration of salt in the tank reach 0.4 kg/L?
---
### Calculation Task:
Determine the mass of salt in the tank after \( t \) minutes.
\[ \text{mass} = \boxed{ } \, \text{kg} \]
---
### Explanation:
This problem involves a dynamic system where a brine solution with a known concentration of salt is flowing into a tank while the mixed solution flows out at a different rate. Understanding this system requires basic knowledge of differential equations and principles of mass balance.
1. **Input/Output Calculation**:
- **Input Salt Concentration**: 0.5 kg/L
- **Inflow Rate**: 6 L/min
- **Outflow Rate**: 5 L/min
2. **Initial Conditions**:
- **Initial Volume of Water**: 100 L
- **Initial Mass of Salt**: 0 kg
3. **Differential Equation Formulation**:
- The volume of the liquid in the tank changes over time due to the difference between the inflow and outflow rates.
- The concentration changes as salt is added and removed.
4. **Solution Steps**:
- Define the volume of the solution in the tank as a function of time.
- Set up and solve the differential equation incorporating the rates of inflow and outflow and the concentration of the entering solution.
- Calculate the mass of salt in the tank at any time \( t \).
By proceeding through these steps, one can derive a function for the concentration of salt over time and identify when it reaches 0.4 kg/L. This type of problem is typical in engineering, particularly in chemical and environmental engineering disciplines, where process dynamics are analyzed.
### Graphs and Diagrams:
The prompt does not include any specific graphs or diagrams. However, a detailed explanation using a graph might include:
- A graph showing the volume of liquid in the tank over time.
- A graph
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