A metal boiler base has a thermal conductivity of 200\text{ W}\cdot\text{m}^{-1}\cdot\text… — JAMB UTME

A metal boiler base has a thermal conductivity of $200\text{ W}\cdot\text{m}^{-1}\cdot\text{K}^{-1}$ , a thickness of $6.0\text{ mm}$ , and an effective surface area of $0.15\text{ m}^2$ . If heat is conducted through the base at a rate of $500\text{ kW}$ , what is the temperature difference across the two faces of the boiler base?

A
$10\text{ }^\circ\text{C}$
$100\text{ }^\circ\text{C}$ Answer
C
$373\text{ }^\circ\text{C}$
D
$2.5\text{ }^\circ\text{C}$

Answer

The temperature difference across the two faces of the boiler base is

100\text{ }^\circ\text{C}

Using Fourier's law of thermal conduction,

P = \frac{k A \Delta T}{d}

. Substituting

P = 500,000\text{ W}

d = 0.006\text{ m}

k = 200\text{ W}\cdot\text{m}^{-1}\cdot\text{K}^{-1}

, and

A = 0.15\text{ m}^2

yields

\Delta T = \frac{500,000 \times 0.006}{200 \times 0.15} = 100\text{ }^\circ\text{C}

Step-by-Step Solution

Convert given quantities into standard SI units

Heat transfer rate

P = 500\text{ kW} = 500,000\text{ W}

, thickness

d = 6.0\text{ mm} = 0.006\text{ m}

, area

A = 0.15\text{ m}^2

, thermal conductivity

k = 200\text{ W}\cdot\text{m}^{-1}\cdot\text{K}^{-1}

All parameters must be in SI base units before substitution into the thermal conduction equation.

State the equation for rate of heat conduction and rearrange for temperature difference (

\Delta T

)

P = \frac{k A \Delta T}{d} \implies \Delta T = \frac{P \cdot d}{k \cdot A}

To isolate the unknown temperature gradient component.

Substitute the values and compute the result

\Delta T = \frac{500,000 \times 0.006}{200 \times 0.15} = \frac{3000}{30} = 100\text{ }^\circ\text{C}.

Direct algebraic simplification yields the temperature difference.

Key Concept

Thermal Conduction Rate Formula

Estimated Time:1m 30s

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