Book: Process Dynamics and Control, Third Edition or Fourth Edition, by D. E. Seborg, T. F. Edgar, D. A. Mellichamp, F. J. Doyle III, John Wiley & Sons, Inc, 2011 ### Heat Transfer and Freezing Dynamics in an Insulated Cylindrical Water TankIn this problem, we analyze the thermal processes in a vertical, cylindrical tank filled with water. The tank is initially at a steady-state temperature of 20°C. The top and bottom of the tank are insulated, while the side allows heat transfer. The key parameters are:- **Tank Geometry:** Diameter = 0.5 m, Height = 1.0 m- **Overall Heat Transfer Coefficient, \( U \):** 200 W/m²·K- **Density of Water, \( \rho \):** 1000 kg/m³- **Heat Capacity, \( C_p \):** 4180 J/kg·K- **Melting Point of Water:** 0°C- **Heat of Fusion, \( \lambda \):** 334 kJ/kg#### Task (a)**Objective:** Develop a dynamic model to calculate the temperature of the water in the tank as a function of the ambient air temperature around the tank.**Guidance:** Use the principles of heat transfer to develop this model. Consider the heat loss through the side of the tank as a function of the temperature difference between the water and the ambient air.#### Task (b)**Objective:** Calculate the time for the first crystal of ice to form in the tank, assuming it is placed in a large walk-in freezer with an air temperature of -25°C.**Guidance:** Determine the rate of heat loss and the time required for the water temperature to drop to 0°C.#### Task (c)**Objective:** Calculate the time required to freeze all the water in the tank, neglecting any volume expansion due to freezing and assuming uniform temperature in the tank.**Guidance:** This task will require calculating the time needed for the temperature to drop to the freezing point, followed by the time to freeze the water completely. Consider the latent heat required for the phase change from liquid to solid.### Diagrams and GraphsNote: There are no graphs or diagrams provided in the text. In practice, diagrams of the tank and temperature profiles over time could help visualize the processes.### References and Further Reading- **Heat Transfer Texts:** For detailed methods to solve heat transfer problems.- **Thermodynamics:** Key principles related to phase changes and energy balances.---This format provides clarity on the problem, methods, and objectives, suitable for educational purposes

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Answered: 1. A vertical, cylindrical tank is…

Introduction to Chemical Engineering Thermodynamics

8th Edition

ISBN:9781259696527

Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart

Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart

Chapter1: Introduction

Section: Chapter Questions

Problem 1.1P

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Book: Process Dynamics and Control, Third Edition or Fourth Edition, by D. E. Seborg, T. F. Edgar, D. A. Mellichamp, F. J. Doyle III, John Wiley & Sons, Inc, 2011

### Heat Transfer and Freezing Dynamics in an Insulated Cylindrical Water Tank

In this problem, we analyze the thermal processes in a vertical, cylindrical tank filled with water. The tank is initially at a steady-state temperature of 20°C. The top and bottom of the tank are insulated, while the side allows heat transfer. The key parameters are:

- **Tank Geometry:** Diameter = 0.5 m, Height = 1.0 m
- **Overall Heat Transfer Coefficient, \( U \):** 200 W/m²·K
- **Density of Water, \( \rho \):** 1000 kg/m³
- **Heat Capacity, \( C_p \):** 4180 J/kg·K
- **Melting Point of Water:** 0°C
- **Heat of Fusion, \( \lambda \):** 334 kJ/kg

#### Task (a)
**Objective:** Develop a dynamic model to calculate the temperature of the water in the tank as a function of the ambient air temperature around the tank.

**Guidance:** Use the principles of heat transfer to develop this model. Consider the heat loss through the side of the tank as a function of the temperature difference between the water and the ambient air.

#### Task (b)
**Objective:** Calculate the time for the first crystal of ice to form in the tank, assuming it is placed in a large walk-in freezer with an air temperature of -25°C.

**Guidance:** Determine the rate of heat loss and the time required for the water temperature to drop to 0°C.

#### Task (c)
**Objective:** Calculate the time required to freeze all the water in the tank, neglecting any volume expansion due to freezing and assuming uniform temperature in the tank.

**Guidance:** This task will require calculating the time needed for the temperature to drop to the freezing point, followed by the time to freeze the water completely. Consider the latent heat required for the phase change from liquid to solid.

### Diagrams and Graphs

Note: There are no graphs or diagrams provided in the text. In practice, diagrams of the tank and temperature profiles over time could help visualize the processes.

### References and Further Reading
- **Heat Transfer Texts:** For detailed methods to solve heat transfer problems.
- **Thermodynamics:** Key principles related to phase changes and energy balances.

---

This format provides clarity on the problem, methods, and objectives, suitable for educational purposes

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