The image presents a derivation of the common expressions for radiative heat transfer rates between two surfaces, based on their geometrical configurations. Below is a transcription of the content for an educational context:---**Radiative Heat Transfer Rate Derivations****a) Infinite Parallel Plates:**- *Diagram Description:* Two parallel plates, identified as \( A_1, T_1, \varepsilon_1 \) and \( A_2, T_2, \varepsilon_2 \).- *Equation:* \[ q_{12} = \frac{A\sigma(T_1^4 - T_2^4)}{\frac{1}{\varepsilon_1} + \frac{1}{\varepsilon_2} - 1} \]**b) Infinitely Long Concentric Cylinders:**- *Diagram Description:* Two concentric cylinders with radii \( r_1 \) and \( r_2 \).- *Equation:* \[ q_{12} = \frac{A_1\sigma(T_1^4 - T_2^4)}{\frac{1}{\varepsilon_1} + \frac{1 - \varepsilon_2}{\varepsilon_2} \left(\frac{r_1}{r_2}\right)} \]**c) Concentric Spheres:**- *Diagram Description:* Two concentric spheres with radii \( r_1 \) and \( r_2 \).- *Equation:* \[ q_{12} = \frac{A_1\sigma(T_1^4 - T_2^4)}{\frac{1}{\varepsilon_1} + \frac{1 - \varepsilon_2}{\varepsilon_2} \left(\frac{r_1}{r_2}\right)^2} \]**Explanation:**- \( A \) or \( A_1 \) are the areas of the surfaces.- \( T_1 \) and \( T_2 \) represent the temperatures of the first and second surfaces, respectively.- \( \varepsilon_1 \) and \( \varepsilon_2 \) are the emissivities of the surfaces.- \( \sigma \) is the Stefan-Boltzmann constant.- The expressions calculate the rate \( q_{12

Consider heat transfer by radiation takes place between two opaque and non black bodies (say body-1 and body-2).Draw thermal network circuit for the above arrangement.Write expression for radiation heat transfer from body-1 to body-2.Here is Stefan-Bolzmann constant, are surfaces areas of bodies exchanging heat, are emmisivities of bodies 1 and 2, and is shape factor of with respect to .For infinite parallel plates, surface areas are same. Thus, .Since plates are parallel, shape factor of body with respect to itself is zero. That is, and From principle of conservation, Consider equation (1).Substitute all known values.Equation (1a) represents the expression for radiation heat transfer between two parallel infinite plates.For infinite concentric cylinders, surface areas are and . Here are radii of inner and outer cylindrical surfaces and L is length of each cylinder.Body-1 (inner cylinder) is a convex surface. For convex surface, shape factor of the surface with respect to itself is zero. That is, .From principle of conservation, Consider equation (1).Substitute all known values.Equation (1b) represents the expression for radiation heat transfer between two concentric cylidners.For concentric spheres, surface areas are and . Here are radii of inner and outer spherical surfaces.Body-1 (inner sphere) is a convex surface. For convex surface, shape factor of the surface with respect to itself is zero. That is, .From principle of conservation, Consider equation (1).Substitute all known values.Equation (1c) represents the expression for radiation heat transfer between two concentric spheres.Hence verified with the given values.

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Verify (i.e., derive) the common expressions for the radiative heat transfer rate between two surfaces below. a) Infinite parallel plates: A₁, T₁, E1 c) Concentric spheres: A2, T2, E2 b) Infinitely long concentric cylinders: 12 ۲۱ 12 912 912 912 Ar(T4 – Tâ) 1 1 + E1 E2 A₁ (T₁-T₂) 1-82 (G) + E1 E2 A₁ (T₁ – T₂) +1=52 (12) ² 1 E1

Verify (i.e., derive) the common expressions for the radiative heat transfer rate between two surfaces below. a) Infinite parallel plates: A₁, T₁, E1 c) Concentric spheres: A2, T2, E2 b) Infinitely long concentric cylinders: 12 ۲۱ 12 912 912 912 Ar(T4 – Tâ) 1 1 + E1 E2 A₁ (T₁-T₂) 1-82 (G) + E1 E2 A₁ (T₁ – T₂) +1=52 (12) ² 1 E1

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

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Question

The image presents a derivation of the common expressions for radiative heat transfer rates between two surfaces, based on their geometrical configurations. Below is a transcription of the content for an educational context:

---

**Radiative Heat Transfer Rate Derivations**

**a) Infinite Parallel Plates:**

- *Diagram Description:* Two parallel plates, identified as \( A_1, T_1, \varepsilon_1 \) and \( A_2, T_2, \varepsilon_2 \).

- *Equation:*

\[
q_{12} = \frac{A\sigma(T_1^4 - T_2^4)}{\frac{1}{\varepsilon_1} + \frac{1}{\varepsilon_2} - 1}
\]

**b) Infinitely Long Concentric Cylinders:**

- *Diagram Description:* Two concentric cylinders with radii \( r_1 \) and \( r_2 \).

- *Equation:*

\[
q_{12} = \frac{A_1\sigma(T_1^4 - T_2^4)}{\frac{1}{\varepsilon_1} + \frac{1 - \varepsilon_2}{\varepsilon_2} \left(\frac{r_1}{r_2}\right)}
\]

**c) Concentric Spheres:**

- *Diagram Description:* Two concentric spheres with radii \( r_1 \) and \( r_2 \).

- *Equation:*

\[
q_{12} = \frac{A_1\sigma(T_1^4 - T_2^4)}{\frac{1}{\varepsilon_1} + \frac{1 - \varepsilon_2}{\varepsilon_2} \left(\frac{r_1}{r_2}\right)^2}
\]

**Explanation:**

- \( A \) or \( A_1 \) are the areas of the surfaces.
- \( T_1 \) and \( T_2 \) represent the temperatures of the first and second surfaces, respectively.
- \( \varepsilon_1 \) and \( \varepsilon_2 \) are the emissivities of the surfaces.
- \( \sigma \) is the Stefan-Boltzmann constant.
- The expressions calculate the rate \( q_{12

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Step 1: Develop the expression for radiation heat transfer between two opaque and non black bodies.

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Step 2: a) Infinite parallel plates

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Step 3: b) Infinite concentric cylinders.

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