Question

A composite wall consists of two plates A and B placed in series normal to the flow of heat. The thermal conductivities are \( k_A \) and \( k_B \) and the specific heat capacities are \( C_{pA} \) and \( C_{pB} \), for plates A and B respectively. Plate B has twice the thickness of plate A. At steady state, the temperature difference across plate A is greater than that across plate B when:

• A. \( C_{pA} > C_{pB} \)
• B. \( C_{pA} < C_{pB} \)
• C. \( k_A < 0.5 \, k_B \)
• D. \( k_A > 2 \, k_B \)

Hint:

For steady-state series heat flow, use thermal resistance analogy: - Heat flux is constant: \( q = \frac{\Delta T}{R_{th}} \). - Higher temperature drop occurs across the plate containing the higher thermal resistance. - \( R_{th} = \frac{L}{k} \). For \( \Delta T_A > \Delta T_B \), we need \( R_{th,A} > R_{th,B} \Rightarrow \frac{L_A}{k_A} > \frac{L_B}{k_B} \). Substituting \( L_B = 2L_A \) immediately yields \( k_A < 0.5 k_B \).

Answer 1

The Correct Option is C

Concept: Under steady-state heat conduction through a composite wall arrangement where layers or plates are connected in a series orientation, the rate of heat transfer through each individual layer must remain completely identical. This is governed by Fourier's Law of Heat Conduction, which states: \[ Q = -k \cdot A \cdot \frac{\Delta T}{\Delta x} \] Where:
• \( Q \) is the total steady-state rate of heat flow across the cross-sectional area.
• \( k \) represents the unique physical parameter known as the thermal conductivity of the material.
• \( A \) represents the surface area perpendicular to the direction of heat flow.
• \( \Delta T \) represents the temperature drop across the corresponding thickness.
• \( \Delta x \) or \( L \) represents the spatial thickness of the physical plate.

Step 1: Setting up variables and identifying constraints from the problem description.
Let us establish and explicitly define the parameter metrics for both Plate A and Plate B:
• Thickness of Plate A: \( L_A \)
• Thickness of Plate B: \( L_B \)
• Thermal conductivity of Plate A: \( k_A \)
• Thermal conductivity of Plate B: \( k_B \)
• Temperature difference across Plate A: \( \Delta T_A \)
• Temperature difference across Plate B: \( \Delta T_B \) The text directly provides a structural constraint regarding the thickness relation: "Plate B has twice the thickness of plate A." Mathematically, this gives: \[ L_B = 2 \cdot L_A \]

Step 2: Equating the steady-state heat flow rates for a series network.
Since the configuration is in series and normal to the flow of heat, the heat flux (\( q = Q/A \)) through Plate A matches the heat flux through Plate B perfectly: \[ q_A = q_B \] Applying Fourier's equation to each individual layer independently, we can write: \[ \frac{k_A \cdot \Delta T_A}{L_A} = \frac{k_B \cdot \Delta T_B}{L_B} \] Now, let us substitute the expression for thickness, \( L_B = 2 \cdot L_A \), into the right-hand side of our steady-state balance equation: \[ \frac{k_A \cdot \Delta T_A}{L_A} = \frac{k_B \cdot \Delta T_B}{2 \cdot L_A} \] We can eliminate the common denominator term \( L_A \) from both sides by multiplying the entire mathematical equation by \( L_A \): \[ k_A \cdot \Delta T_A = \frac{k_B \cdot \Delta T_B}{2} \] Rearranging this algebraic formulation to express the ratio of the temperature differences across both individual components gives: \[ \frac{\Delta T_A}{\Delta T_B} = \frac{k_B}{2 \cdot k_A} \]

Step 3: Evaluating the conditional limit for temperature variations.
The core objective is to determine the specific criteria required so that "the temperature difference across plate A is greater than that across plate B." Mathematically, this condition is written as: \[ \Delta T_A > \Delta T_B \] Dividing both sides by the positive quantity \( \Delta T_B \), this statement transforms into: \[ \frac{\Delta T_A}{\Delta T_B} > 1 \] Now, substitute our previously derived ratio involving thermal conductivities into this inequality: \[ \frac{k_B}{2 \cdot k_A} > 1 \] Since the thermal conductivity coefficients are strictly positive engineering constants, we can perform standard algebraic cross-multiplication without flipping the direction of our inequality sign: \[ k_B > 2 \cdot k_A \] To isolate the parameter \( k_A \), divide both sides of this final inequality constraint by the integer constant 2: \[ \frac{k_B}{2} > k_A \quad \Rightarrow \quad k_A < \frac{1}{2} \, k_B \] Rewriting this fraction in standard decimal notation yields: \[ k_A < 0.5 \, k_B \] Note that under steady-state operation, the specific heat capacity parameters (\( C_{pA} \) and \( C_{pB} \)) play absolutely no operational role because temperature distribution profiles across the system are time-invariant (\( \partial T / \partial t = 0 \)). Hence, variations in heat capacity parameters do not impact steady temperature differentials.