An ordinary egg can be approximated as a 5.5-cm-diameter sphere with a thermal conductivity of \( k = 0.6 \, \frac{W}{m \cdot K} \), an overall density of \( \rho = 1000 \, \frac{kg}{m^3} \), and a heat capacity of \( C_p = 3000 \, \frac{J}{kg \cdot K} \).The egg starts at an initial uniform temperature \( T_i = 10^\circ C \) and is placed into boiling water at \( T_\infty = 90^\circ C \). With a convective heat transfer coefficient \( h = 10 \, \frac{W}{m^2 \cdot K} \), determine how long it will take for the egg to reach \( T = 70^\circ C \).This problem uses the One-term approximation model to compute the time needed for the center and the surface of the egg to reach 70°C.### Steps for Solving the Problem:#### For the One-term approximation:1. **Compute the Biot number**: Use the egg radius (not diameter) as the characteristic length.2. **Compute the non-dimensional excess temperature** \( \theta = \frac{T - T_{\infty}}{T_i - T_{\infty}} \).3. **Lookup Values**: Use the table to find \( A_1 \) and \( \lambda_1 \) for the closest Biot number related to a sphere.4. **Center Temperature Calculation**: Calculate \( \tau \) using \( \theta = A_1 e^{-\lambda_1^2 \tau} \), then find the time using \( \tau = \frac{at}{r_0^2} \).5. **Surface Temperature Calculation**: First calculate \( \tau \) using \( \theta = A_1 e^{-\lambda_1^2 \tau} \sin\left(\frac{\lambda_1 r}{r_0}\right) \) with \( r = r_0 \), then use the definition of \( Fo = \tau \).### Table 1 Explanation:This table provides coefficients for the one-term approximate solution of transient one-dimensional heat conduction in various shapes:- **Plane Wall**, **Cylinder**, and **Sphere** configurations have different \( A_1

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An ordinary egg can be approximated as a 5.5-cm-diameter sphere whose thermal conductivity of roughly W = : 0.6 overall density of p = 1000 k and heat capacity of Cp = 3000; mk m² kgk k I 90°C The egg is initially at a uniform temperature of T; = 10°C and is dropped into boiling water at T Taking the convective heat transfer coefficient to be h = 10. determine how long it will take for the egg to W m²K reach T = 70°C. = In solving this problem, please use the One-term approximation model to compute the time needed by the center of the egg and the surface of the egg to reach the desired temperature of 70 C

An ordinary egg can be approximated as a 5.5-cm-diameter sphere whose thermal conductivity of roughly W = : 0.6 overall density of p = 1000 k and heat capacity of Cp = 3000; mk m² kgk k I 90°C The egg is initially at a uniform temperature of T; = 10°C and is dropped into boiling water at T Taking the convective heat transfer coefficient to be h = 10. determine how long it will take for the egg to W m²K reach T = 70°C. = In solving this problem, please use the One-term approximation model to compute the time needed by the center of the egg and the surface of the egg to reach the desired temperature of 70 C

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Question

An ordinary egg can be approximated as a 5.5-cm-diameter sphere with a thermal conductivity of \( k = 0.6 \, \frac{W}{m \cdot K} \), an overall density of \( \rho = 1000 \, \frac{kg}{m^3} \), and a heat capacity of \( C_p = 3000 \, \frac{J}{kg \cdot K} \).

The egg starts at an initial uniform temperature \( T_i = 10^\circ C \) and is placed into boiling water at \( T_\infty = 90^\circ C \). With a convective heat transfer coefficient \( h = 10 \, \frac{W}{m^2 \cdot K} \), determine how long it will take for the egg to reach \( T = 70^\circ C \).

This problem uses the One-term approximation model to compute the time needed for the center and the surface of the egg to reach 70°C.

### Steps for Solving the Problem:

#### For the One-term approximation:

1. **Compute the Biot number**: Use the egg radius (not diameter) as the characteristic length.
2. **Compute the non-dimensional excess temperature** \( \theta = \frac{T - T_{\infty}}{T_i - T_{\infty}} \).
3. **Lookup Values**: Use the table to find \( A_1 \) and \( \lambda_1 \) for the closest Biot number related to a sphere.
4. **Center Temperature Calculation**: Calculate \( \tau \) using \( \theta = A_1 e^{-\lambda_1^2 \tau} \), then find the time using \( \tau = \frac{at}{r_0^2} \).
5. **Surface Temperature Calculation**: First calculate \( \tau \) using \( \theta = A_1 e^{-\lambda_1^2 \tau} \sin\left(\frac{\lambda_1 r}{r_0}\right) \) with \( r = r_0 \), then use the definition of \( Fo = \tau \).

### Table 1 Explanation:

This table provides coefficients for the one-term approximate solution of transient one-dimensional heat conduction in various shapes:

- **Plane Wall**, **Cylinder**, and **Sphere** configurations have different \( A_1

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