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A hot dog can be considered to be a 12 -cm-long cylinder whose diameter is 2 cm and whose properties are $\rho=980 \mathrm{~kg} / \mathrm{m}^3, c_p=3.9 \mathrm{~kJ} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}, k=0.76 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}$, and $\alpha=2 \times 10^{-7} \mathrm{~m}^2 / \mathrm{s}$. A hot dog initially at $5^{\circ} \mathrm{C}$ is dropped into boiling water at $100^{\circ} \mathrm{C}$. The heat transfer coefficient at the surface of the hot dog is estimated to be $600 \mathrm{~W} / \mathrm{m}^2$. ${ }^{\circ} \mathrm{C}$. If the hot dog is considered cooked when its center temperature reaches $80^{\circ} \mathrm{C}$, determine how long it will take to cook it in the boiling water.

   A hot dog can be considered to be a 12 -cm-long cylinder whose diameter is 2 cm and whose properties are $\rho=980 \mathrm{~kg} / \mathrm{m}^3, c_p=3.9 \mathrm{~kJ} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}, k=0.76 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}$, and $\alpha=2 \times 10^{-7} \mathrm{~m}^2 / \mathrm{s}$. A hot dog initially at $5^{\circ} \mathrm{C}$ is dropped into boiling water at $100^{\circ} \mathrm{C}$. The heat transfer coefficient at the surface of the hot dog is estimated to be $600 \mathrm{~W} / \mathrm{m}^2$. ${ }^{\circ} \mathrm{C}$. If the hot dog is considered cooked when its center temperature reaches $80^{\circ} \mathrm{C}$, determine how long it will take to cook it in the boiling water.
 
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Introduction To Thermodynamics and Heat Transfer
Introduction To Thermodynamics and Heat Transfer
Yunus A. Cengel 1st Edition
Chapter 11, Problem 94 ↓

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## Determining the cooking time for a hot dog in boiling water  Show more…

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A hot dog can be considered to be a 12 -cm-long cylinder whose diameter is 2 cm and whose properties are $\rho=980 \mathrm{~kg} / \mathrm{m}^3, c_p=3.9 \mathrm{~kJ} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}, k=0.76 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}$, and $\alpha=2 \times 10^{-7} \mathrm{~m}^2 / \mathrm{s}$. A hot dog initially at $5^{\circ} \mathrm{C}$ is dropped into boiling water at $100^{\circ} \mathrm{C}$. The heat transfer coefficient at the surface of the hot dog is estimated to be $600 \mathrm{~W} / \mathrm{m}^2$. ${ }^{\circ} \mathrm{C}$. If the hot dog is considered cooked when its center temperature reaches $80^{\circ} \mathrm{C}$, determine how long it will take to cook it in the boiling water.
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Key Concepts

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Convection Boundary Condition
This boundary condition models the heat transfer between the surface of a solid and the surrounding fluid using a convective heat transfer coefficient. It is crucial in transient conduction problems as it defines the heat exchange at the surface, driving the temperature change within the object.
Fourier Number
A dimensionless time parameter defined in terms of thermal diffusivity, time, and a characteristic length. It quantifies the ratio of heat conduction to energy storage and is used to characterize the transient thermal response of materials.
Transient Heat Conduction
This concept deals with how temperature in a material changes with time due to internal conduction. It is essential for analyzing the dynamic response of solid objects to thermal disturbances, where the temperature distribution evolves until reaching thermal equilibrium.
Biot Number
A dimensionless parameter that compares the internal resistance to heat conduction within a body with the external resistance to heat convection. It helps determine whether the lumped capacitance method can be applied, typically considered valid when the Biot number is less than 0.1.
Lumped Capacitance Method
An analytical approach in transient heat transfer problems which assumes the temperature within the object is uniform throughout due to negligible internal temperature gradients. This method simplifies calculations by reducing the spatially distributed problem to one ordinary differential equation in time.

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A hot dog can be considered to be a 12 cm long cylinder, whose diameter is 2 cm and whose properties are ρ = 980 kg/m3, cp = 3.9 kJ/kg K, k = 25 W/mK, and α = 2 * 10^-7 m2/s. Consider a hot dog initially at 5°C which is dropped into boiling water at 100°C. The heat transfer coefficient at the surface of the hot dog is estimated to be 500 W/m2 K, and the hot dog can be considered to be cooked when its center temperature reaches 90°C. (a) Determine if a lumped capacitance analysis is or is not appropriate. (b) Determine the cooking time using the lumped capacitance analysis. (c) Determine the cooking time accounting for spatial effects.

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