https://file.aiquickdraw.com/mathbot/file/1726570794650p3zigd22.jpg

To solve the problem step by step, we need to find the equivalent thermal resistance for the composite wall shown in the diagram. The wall consists of four sections, labeled A, B, C, and D, with different thermal conductivities (\(k_a, k_b, k_c, k_d\)) and thicknesses (\(L_a, L_b, L_c, L_d\)).

### Step 1: Analyze the configuration

The diagram shows that sections A and D are in series, while sections B and C are in parallel. The sections are bounded by temperatures \(T_1\) and \(T_2\).

- **Series arrangement (A and D):** Heat flows sequentially through A and D.
- **Parallel arrangement (B and C):** Heat splits between paths B and C.

### Step 2: Thermal resistance of each section

Thermal resistance (\(R\)) of a material is given by the formula:

\[
R = \frac{L}{kA}
\]

Where:
- \(L\) is the thickness of the material.
- \(k\) is the thermal conductivity.
- \(A\) is the cross-sectional area through which heat flows.

Now, let's write the thermal resistance for each section:

- For section **A**: \(R_A = \frac{L_a}{k_a A_A}\)
- For section **B**: \(R_B = \frac{L_b}{k_b A_B}\)
- For section **C**: \(R_C = \frac{L_c}{k_c A_C}\)
- For section **D**: \(R_D = \frac{L_d}{k_d A_D}\)

### Step 3: Combine resistances for parallel and series sections

- **For parallel resistances (B and C):** The total resistance for resistances in parallel is given by:

\[
\frac{1}{R_{BC}} = \frac{1}{R_B} + \frac{1}{R_C}
\]

This gives:

\[
R_{BC} = \left( \frac{1}{R_B} + \frac{1}{R_C} \right)^{-1}
\]

- **For series resistances (A, BC, and D):** The total resistance for resistances in series is simply the sum of the individual resistances:

\[
R_{\text{total}} = R_A + R_{BC} + R_D
\]

### Step 4: Substitute the expressions for each resistance

Substitute the expressions for \(R_A\), \(R_B\), \(R_C\), and \(R_D\) into the total resistance formula:

\[
R_{\text{total}} = \frac{L_a}{k_a A_A} + \left( \frac{1}{\frac{L_b}{k_b A_B} } + \frac{1}{\frac{L_c}{k_c A_C} } \right)^{-1} + \frac{L_d}{k_d A_D}
\]

This is the expression for the equivalent thermal resistance for the composite wall.

Would you like further details on this derivation or a simplified final expression?

---

### Relative questions:
1. How does thermal resistance change with material thickness?
2. What is the significance of the thermal conductivity \(k\) in thermal resistance?
3. How do parallel and series resistances affect the total thermal resistance?
4. How can we experimentally determine the thermal conductivities of the materials used?
5. What is the impact of varying the cross-sectional areas on thermal resistance?

### Tip:
Remember that in thermal systems, resistances in series add directly, while resistances in parallel combine inversely.

Write down the expression for equivalent thermal resistance for the composite wall.

Learn how to calculate the equivalent thermal resistance for a composite wall with sections in series and parallel. This step-by-step solution involves calculating individual thermal resistances using the formula R = L / kA, then combining them based on series and parallel rules. Suitable for undergraduate engineering students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation