Calculating Boiling Heat Transfer Coefficient

The problem asks for the boiling heat transfer coefficient ($h$) for a wire submerged in water under specific conditions.

Identify Given Parameters

• Fluid: Water
• Pressure: 5 bar
• Saturation Temperature ($T_{sat}$): $150^\circ C$
• Surface Temperature ($T_s$): $200^\circ C$
• Wire Length ($L$): 250 mm = 0.25 m
• Wire Diameter ($D$): 2 mm = 0.002 m
• Current ($I$): 100 A
• Voltage ($V$): 2.5 V

Determine Heat Transfer Rate

The heat generated by the electrical current in the wire is transferred to the water via boiling. The heat transfer rate ($Q$) is equal to the electrical power input.

Calculation:

$ Q = V \times I $

$ Q = 2.5\ V \times 100\ A = 250\ W $

Calculate Heat Transfer Surface Area

The heat transfer occurs over the cylindrical surface area of the wire.

Calculation:

$ A = \pi \times D \times L $

$ A = \pi \times (0.002\ m) \times (0.25\ m) = 0.0005\pi\ m^2 $

Calculate Heat Transfer Coefficient

The boiling heat transfer coefficient ($h$) is defined by Newton's law of cooling, adapted for boiling:

$ Q = h \times A \times (T_s - T_{sat}) $

Rearranging the formula to solve for $h$:

$ h = \frac{Q}{A \times (T_s - T_{sat})} $

Substitute the calculated values:

$ h = \frac{250\ W}{(0.0005\pi\ m^2) \times (200^\circ C - 150^\circ C)} $

$ h = \frac{250\ W}{0.0005\pi\ m^2 \times 50^\circ C} $

$ h = \frac{250\ W}{0.025\pi\ m^2\ ^\circ C} $

$ h = \frac{10000}{\pi}\ W/m^2\ ^\circ C $

Final Answer

The calculated boiling heat transfer coefficient is $\frac{10000}{\pi}\ W/m^2\ ^\circ C$. This matches Option C.