1. The heat equation describes the flow of thermal energy (heat) in materials. The temperature V obeys: әу Ət k Cpp 2 where t is time, k is the thermal conductivity, c, is the specific heat capacity, and p is the mass density. (a) Assuming the problem is spherically symmetric, write the heat equation in spherical coordinates. (b) Apply the technique of separation of variables, assuming V = u(r) f(t) T = Find the the resulting differential equations for u and f, and provide their general solution. Note: (1) By factoring rout of u as we have, the spherically-symmetric Laplacian is simplified and the solution to the differential equation for u should be easy to recognize. (2) Here I would you to choose the sign of the separation constant so that the solutions for f(t) are exponentials that decay with increasing time. like (c) What is the form of the solution when the separation constant is zero? (d) Let's apply this equation to the cooking of a turkey,- maybe this will come in handy next month. Let's assume that our "turkey" is a sphere of radius R = 0.2 m and starts out thawed and at room temperature (To = 25° C) 180° C. In order to be quantitative, and that the oven is set at T₁ assume that turkey has the same thermal properties as water: thermal conductivity: k = 0.90 W/(m-K) Cpp = 4.18 x 106 J/m³. heat capacity: The recipe described above implies an initial condition of V(r, 0) = To and a spatial boundary condition of V(R, t) = T₁. Clearly, the solution for V must be regular at r = 0. Physical considerations also rule out choices of the separation constant that give rise to increasing exponentials. What is the most general form of the solution subject to these restrictions? (e) In order to more easily impose the boundary conditions, is useful to con- sider - T₁ so that (R, t) = 0. Note that (r, t) also satisfies the heat equation. What are the restrictions now on the radial solution? (f) By imposing the initial conditions, determine the complete solution (r, t). (g) Considering only the first non-vanishing time-dependent term, how long does it take to cook this turkey? Assume the turkey is "done" when the core temperature reaches 70° C.
1. The heat equation describes the flow of thermal energy (heat) in materials. The temperature V obeys: әу Ət k Cpp 2 where t is time, k is the thermal conductivity, c, is the specific heat capacity, and p is the mass density. (a) Assuming the problem is spherically symmetric, write the heat equation in spherical coordinates. (b) Apply the technique of separation of variables, assuming V = u(r) f(t) T = Find the the resulting differential equations for u and f, and provide their general solution. Note: (1) By factoring rout of u as we have, the spherically-symmetric Laplacian is simplified and the solution to the differential equation for u should be easy to recognize. (2) Here I would you to choose the sign of the separation constant so that the solutions for f(t) are exponentials that decay with increasing time. like (c) What is the form of the solution when the separation constant is zero? (d) Let's apply this equation to the cooking of a turkey,- maybe this will come in handy next month. Let's assume that our "turkey" is a sphere of radius R = 0.2 m and starts out thawed and at room temperature (To = 25° C) 180° C. In order to be quantitative, and that the oven is set at T₁ assume that turkey has the same thermal properties as water: thermal conductivity: k = 0.90 W/(m-K) Cpp = 4.18 x 106 J/m³. heat capacity: The recipe described above implies an initial condition of V(r, 0) = To and a spatial boundary condition of V(R, t) = T₁. Clearly, the solution for V must be regular at r = 0. Physical considerations also rule out choices of the separation constant that give rise to increasing exponentials. What is the most general form of the solution subject to these restrictions? (e) In order to more easily impose the boundary conditions, is useful to con- sider - T₁ so that (R, t) = 0. Note that (r, t) also satisfies the heat equation. What are the restrictions now on the radial solution? (f) By imposing the initial conditions, determine the complete solution (r, t). (g) Considering only the first non-vanishing time-dependent term, how long does it take to cook this turkey? Assume the turkey is "done" when the core temperature reaches 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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Part d-f
![1. The heat equation describes the flow of thermal energy (heat) in materials. The temperature Ψ obeys:
\[
\frac{\partial \Psi}{\partial t} = \frac{k}{c_p \rho} \nabla^2 \Psi
\]
where \( t \) is time, \( k \) is the thermal conductivity, \( c_p \) is the specific heat capacity, and \( \rho \) is the mass density.
(a) Assuming the problem is spherically symmetric, write the heat equation in spherical coordinates.
(b) Apply the technique of separation of variables, assuming \( \Psi = u(r)f(t) \).
Find the resulting differential equations for \( u \) and \( f \), and provide their general solution. Note: (1) By factoring \( r^{-1} \) out of \( u \) as we have, the spherically-symmetric Laplacian is simplified and the solution to the differential equation for \( u \) should be easy to recognize. (2) Here I would like you to choose the sign of the separation constant so that the solutions for \( f(t) \) are exponentials that decay with increasing time.
(c) What is the form of the solution when the separation constant is zero?
(d) Let’s apply this equation to the cooking of a turkey—maybe this will come in handy next month. Let’s assume that our “turkey” is a sphere of radius \( R = 0.2 \, \text{m} \) and starts out thawed and at room temperature (\( T_0 = 25^\circ \text{C} \)) and that the oven is set at \( T_1 = 180^\circ \text{C} \). In order to be quantitative, assume that turkey has the same thermal properties as water:
- thermal conductivity: \( k = 0.90 \, \text{W/(mK)} \)
- heat capacity: \( c_p\rho = 4.18 \times 10^6 \, \text{J/m}^3 \).
The recipe described above implies an initial condition of \( \Psi(r, 0) = T_0 \) and a spatial boundary condition of \( \Psi(R, t) = T_1 \). Clearly, the solution for \( \Psi \) must be regular at \( r = 0 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9b65ef36-cf51-4f81-80a6-74e205c9e9b1%2F829f0d2c-bbaf-41c5-891e-c17cd2f03386%2Fmza1cxf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. The heat equation describes the flow of thermal energy (heat) in materials. The temperature Ψ obeys:
\[
\frac{\partial \Psi}{\partial t} = \frac{k}{c_p \rho} \nabla^2 \Psi
\]
where \( t \) is time, \( k \) is the thermal conductivity, \( c_p \) is the specific heat capacity, and \( \rho \) is the mass density.
(a) Assuming the problem is spherically symmetric, write the heat equation in spherical coordinates.
(b) Apply the technique of separation of variables, assuming \( \Psi = u(r)f(t) \).
Find the resulting differential equations for \( u \) and \( f \), and provide their general solution. Note: (1) By factoring \( r^{-1} \) out of \( u \) as we have, the spherically-symmetric Laplacian is simplified and the solution to the differential equation for \( u \) should be easy to recognize. (2) Here I would like you to choose the sign of the separation constant so that the solutions for \( f(t) \) are exponentials that decay with increasing time.
(c) What is the form of the solution when the separation constant is zero?
(d) Let’s apply this equation to the cooking of a turkey—maybe this will come in handy next month. Let’s assume that our “turkey” is a sphere of radius \( R = 0.2 \, \text{m} \) and starts out thawed and at room temperature (\( T_0 = 25^\circ \text{C} \)) and that the oven is set at \( T_1 = 180^\circ \text{C} \). In order to be quantitative, assume that turkey has the same thermal properties as water:
- thermal conductivity: \( k = 0.90 \, \text{W/(mK)} \)
- heat capacity: \( c_p\rho = 4.18 \times 10^6 \, \text{J/m}^3 \).
The recipe described above implies an initial condition of \( \Psi(r, 0) = T_0 \) and a spatial boundary condition of \( \Psi(R, t) = T_1 \). Clearly, the solution for \( \Psi \) must be regular at \( r = 0 \
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