Question

In the heat flow equation, \( Q = -kA \frac{\Delta T}{L} \), the expression for thermal resistance is given by (k - thermal conductivity, A - Area of cross section, L - thickness of the wall)

• A. \( \frac{L}{\Delta T} \)
• B. \( \frac{A}{kL} \)
• C. \( \frac{k}{LA} \)
• D. \( \frac{L}{kA} \)

Hint:

Remember the analogy: - Heat Flow \( Q \leftrightarrow \) Electrical Current \( I \) - Temperature Difference \( \Delta T \leftrightarrow \) Voltage Difference \( V \) - Thermal Resistance \( R_{th} \leftrightarrow \) Electrical Resistance \( R \) The formula for thermal resistance depends on the mode of heat transfer and the geometry of the system. For conduction through a plane wall, it is \( L/(kA) \).

Answer 1

The Correct Option is D

Step 1: Understand the analogy between heat flow and electrical current.
Heat transfer through conduction can be analogous to the flow of electrical current through a resistor. In electrical circuits, Ohm's law states \( V = IR \), which can be rearranged as \( I = \frac{V}{R} \), where \( V \) is the voltage difference, \( I \) is the current, and \( R \) is the electrical resistance. Step 2: Identify the analogous terms in heat transfer.
In the heat flow equation \( Q = -kA \frac{\Delta T}{L} \):
\( Q \) represents the rate of heat transfer (heat flow), which is analogous to electrical current \( I \).
\( \Delta T \) represents the temperature difference across the wall, which is analogous to voltage difference \( V \).
Step 3: Rearrange the heat flow equation to resemble Ohm's law.
We can rewrite the heat flow equation as:
$$Q = \frac{- \Delta T}{L/(kA)}$$Ignoring the negative sign (which indicates the direction of heat flow from higher to lower temperature), we can see the analogy more clearly:$$Q = \frac{\Delta T}{L/(kA)}$$ Step 4: Identify the term corresponding to thermal resistance.
Comparing this form with Ohm's law \( I = \frac{V}{R} \), we can see that the term in the denominator, \( \frac{L}{kA} \), plays the role of thermal resistance \( R_{th} \):
$$R_{th} = \frac{L}{kA}$$ where:
\( L \) is the thickness of the wall (length of the heat flow path)
\( k \) is the thermal conductivity of the material
\( A \) is the area of cross-section perpendicular to the heat flow
Therefore, the expression for thermal resistance in heat conduction through a plane wall is \( \frac{L}{kA} \).