### Liquid-Solid Solution Dynamics #### Problem 10.14 Ninety kilograms of sodium nitrate is dissolved in 110 kg of water. When the dissolution is complete (at time \( t = 0 \)), pure water is fed to the tank at a constant rate \( \dot{m} \) (kg/min), and the solution is withdrawn from the tank at the same rate. The tank may be considered perfectly mixed. --- **(a) Write a total mass balance on the tank and use it to prove that the total mass of liquid in the tank remains constant at its initial value.** --- **(b) Write a balance on sodium nitrate, letting \( x(t, \dot{m}) \) equal the mass fraction of \( \text{NaNO}_3 \) in the tank and outlet stream. Convert the balance into an equation for \( \frac{dx}{dt} \) and provide an initial condition.** --- **(c) On a single graph of \( x \) versus \( t \), sketch the shapes of the plots you would expect to obtain for \( \dot{m} = 50 \, \text{kg/min}, 100 \, \text{kg/min}, \) and \( 200 \, \text{kg/min} \). (Don't do any calculations.) Explain your reasoning, using the equation of Part b in your explanation.** --- **(d) Separate variables and integrate the balance to obtain an expression for \( x(t, \dot{m}) \). Check your solution. Then generate the plots of \( x \) versus \( t \) for \( \dot{m} = 50 \, \text{kg/min}, 100 \, \text{kg/min}, \) and \( 200 \, \text{kg/min} \) and show them on a single graph. (A spreadsheet is a convenient tool for carrying out this step.)** --- **(e) If \( \dot{m} = 100 \, \text{kg/min} \), how long will it take to flush out 90% of the sodium nitrate originally in the tank? How long to flush out 99%? 99.9%?** --- For the graphical representations: - **Graph Description for Part (c):** - The graph should depict three curves of \( x \) versus \( t \) corresponding to \( \dot
### Liquid-Solid Solution Dynamics #### Problem 10.14 Ninety kilograms of sodium nitrate is dissolved in 110 kg of water. When the dissolution is complete (at time \( t = 0 \)), pure water is fed to the tank at a constant rate \( \dot{m} \) (kg/min), and the solution is withdrawn from the tank at the same rate. The tank may be considered perfectly mixed. --- **(a) Write a total mass balance on the tank and use it to prove that the total mass of liquid in the tank remains constant at its initial value.** --- **(b) Write a balance on sodium nitrate, letting \( x(t, \dot{m}) \) equal the mass fraction of \( \text{NaNO}_3 \) in the tank and outlet stream. Convert the balance into an equation for \( \frac{dx}{dt} \) and provide an initial condition.** --- **(c) On a single graph of \( x \) versus \( t \), sketch the shapes of the plots you would expect to obtain for \( \dot{m} = 50 \, \text{kg/min}, 100 \, \text{kg/min}, \) and \( 200 \, \text{kg/min} \). (Don't do any calculations.) Explain your reasoning, using the equation of Part b in your explanation.** --- **(d) Separate variables and integrate the balance to obtain an expression for \( x(t, \dot{m}) \). Check your solution. Then generate the plots of \( x \) versus \( t \) for \( \dot{m} = 50 \, \text{kg/min}, 100 \, \text{kg/min}, \) and \( 200 \, \text{kg/min} \) and show them on a single graph. (A spreadsheet is a convenient tool for carrying out this step.)** --- **(e) If \( \dot{m} = 100 \, \text{kg/min} \), how long will it take to flush out 90% of the sodium nitrate originally in the tank? How long to flush out 99%? 99.9%?** --- For the graphical representations: - **Graph Description for Part (c):** - The graph should depict three curves of \( x \) versus \( t \) corresponding to \( \dot
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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Transcribed Image Text:### Liquid-Solid Solution Dynamics
#### Problem 10.14
Ninety kilograms of sodium nitrate is dissolved in 110 kg of water. When the dissolution is complete (at time \( t = 0 \)), pure water is fed to the tank at a constant rate \( \dot{m} \) (kg/min), and the solution is withdrawn from the tank at the same rate. The tank may be considered perfectly mixed.
---
**(a) Write a total mass balance on the tank and use it to prove that the total mass of liquid in the tank remains constant at its initial value.**
---
**(b) Write a balance on sodium nitrate, letting \( x(t, \dot{m}) \) equal the mass fraction of \( \text{NaNO}_3 \) in the tank and outlet stream. Convert the balance into an equation for \( \frac{dx}{dt} \) and provide an initial condition.**
---
**(c) On a single graph of \( x \) versus \( t \), sketch the shapes of the plots you would expect to obtain for \( \dot{m} = 50 \, \text{kg/min}, 100 \, \text{kg/min}, \) and \( 200 \, \text{kg/min} \). (Don't do any calculations.) Explain your reasoning, using the equation of Part b in your explanation.**
---
**(d) Separate variables and integrate the balance to obtain an expression for \( x(t, \dot{m}) \). Check your solution. Then generate the plots of \( x \) versus \( t \) for \( \dot{m} = 50 \, \text{kg/min}, 100 \, \text{kg/min}, \) and \( 200 \, \text{kg/min} \) and show them on a single graph. (A spreadsheet is a convenient tool for carrying out this step.)**
---
**(e) If \( \dot{m} = 100 \, \text{kg/min} \), how long will it take to flush out 90% of the sodium nitrate originally in the tank? How long to flush out 99%? 99.9%?**
---
For the graphical representations:
- **Graph Description for Part (c):**
- The graph should depict three curves of \( x \) versus \( t \) corresponding to \( \dot
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