A hot dog can be considered to be a cylinder 5 in long and...

A hot dog can be considered to be a cylinder 5 in long and 0.8 in in diameter whose properties are $\rho=61.2 \mathrm{lbm} / \mathrm{ft}^3, c_p=0.93 \mathrm{Btu} / \mathrm{lbm} \cdot{ }^{\circ} \mathrm{F}, k=0.44 \mathrm{Btu} /$ $\mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}$, and $\alpha=0.0077 \mathrm{ft}^2 / \mathrm{h}$. A hot dog initially at $40^{\circ} \mathrm{F}$ is dropped into boiling water at $212^{\circ} \mathrm{F}$. If the heat transfer coefficient at the surface of the hot dog is estimated to be 120 $\mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^2$ - ${ }^{\circ} \mathrm{F}$, determine the center temperature of the hot dog after 5,10 , and 15 min by treating the hot $\operatorname{dog}$ as (a) a finite cylinder and (b) an infinitely long cylinder.

Key Concepts

Transient Heat Conduction

This concept refers to the time-dependent conduction of heat within a solid object. In problems where the temperature of an object changes with time due to differences between its initial temperature and the imposed boundary temperatures, transient conduction analysis is used. It involves solving the heat equation with appropriate initial and boundary conditions to predict the temperature distribution within the object at any given time.

Convective Boundary Conditions

In many heat transfer problems, including this one, the object is exposed to a fluid at a different temperature, and heat is transferred at the surface by convection. Newton's law of cooling is applied at the surface, linking the heat transfer coefficient with the temperature difference between the object's surface and the fluid. This boundary condition is crucial in accurately modeling the heat exchange between the solid and its surroundings.

Biot Number

The Biot number (Bi) is a dimensionless parameter that compares the internal thermal resistance of the object to the external convective heat transfer resistance. It is defined as Bi = hL_c/k, where h is the convective heat transfer coefficient, L_c is the characteristic length, and k is the thermal conductivity. The magnitude of the Biot number indicates whether the lumped capacitance method (for Bi << 1) is appropriate or if spatial temperature variations must be considered.

Finite vs Infinite Cylinder Approximation

This concept relates to the treatment of the geometric boundaries of cylindrical objects. In a finite cylinder, all dimensions are considered and the effects at the ends of the cylinder can influence the temperature distribution. In contrast, an infinitely long cylinder assumes that the axial effects are negligible due to the cylinder extending indefinitely in one direction, simplifying the analysis to radial conduction only. The choice between these models affects the complexity and accuracy of the temperature predictions.

Dimensionless Analysis (Fourier Number)

The Fourier number (Fo) is a dimensionless time parameter defined as Fo = ?t/L^2, where ? is the thermal diffusivity, t is the time, and L is a characteristic length. It is used to characterize the rate of transient heat conduction and to compare the progression of thermal response across different geometries and materials. This parameter is essential for nondimensionalizing the heat conduction equations and for the use of analytical solutions or correlations in transient conduction problems.

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