Question

A composite slab of unit area is made up of three material layers having thickness (x) and thermal conductivities (k) as (x1, k1), (x2, k2), and (x3, k3) respectively. Its thermal resistance (Rt) would be,

• A. \(R_t = [(x_1/k_1) + (x_2/k_2) + (x_3/k_3)]\)
• B. \(R_t = [(k_1/x_1) + (k_2/x_2) + (k_3/x_3)]\)
• C. \(R_t = [(k_1 x_1) + (k_2 x_2) + (k_3 x_3)]\)
• D. \(R_t = [(k_1 x_1) - (k_2 x_2) - (k_3 x_3)]\)

Hint:

Composite Wall Resistance Formula: For unit area in series: $\mathbf{R_t = \frac{x_1}{k_1} + \frac{x_2}{k_2} + \frac{x_3}{k_3}}$ ($R = \frac{\text{thickness}}{\text{conductivity}}$).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
One-dimensional steady-state heat conduction in series composite walls: according to Fourier's law and electrical analogy, the total conductive thermal resistance of multilayer walls in series per unit area is the arithmetic sum of individual layer resistances ($R_t = \sum rac{x_i}{k_i}$).
Key Formula or Approach:
\[ R_{\text{cond}} = \frac{x}{k \cdot A} \quad \xrightarrow{A = 1\text{ m}^2} \quad \mathbf{R_t} = R_1 + R_2 + R_3 = \mathbf{\left[ \left(\frac{x_1}{k_1}\right) + \left(\frac{x_2}{k_2}\right) + \left(\frac{x_3}{k_3}\right) \right]} \]

Step 2: Detailed Explanation:
In thermal engineering and heat transfer calculations for dairy cold storage walls:
- By Fourier's law of 1D steady-state heat conduction, heat flow through a layer of thickness $x$, thermal conductivity $k$, and cross-sectional area $A$ is:
\[ q = \frac{k \cdot A \cdot \Delta T}{x} = \frac{\Delta T}{R_{\text{cond}}} \quad \text{where } R_{\text{cond}} = \frac{x}{k \cdot A} \]
- For a 3-layer composite wall in series with unit area ($A = 1\text{ m}^2$):
1. Thermal resistance of layer 1: $R_1 = \frac{x_1}{k_1}$.
2. Thermal resistance of layer 2: $R_2 = \frac{x_2}{k_2}$.
3. Thermal resistance of layer 3: $R_3 = \frac{x_3}{k_3}$.
- The total equivalent thermal resistance ($R_t$) is the direct sum of series resistances:
\[ \mathbf{R_t = \left[ \left(\frac{x_1}{k_1}\right) + \left(\frac{x_2}{k_2}\right) + \left(\frac{x_3}{k_3}\right) \right]} \]

Step 3: Final Answer:
Therefore, total thermal resistance is \(R_t = [(x_1/k_1) + (x_2/k_2) + (x_3/k_3)]\), corresponding to option (A).