Question

A 5 kW water heater uses a copper wire resistance that is 50 cm long and has a radius of 0.15 cm. To heat the water, the resistance is completely submerged. Determine the convection heat transfer coefficient when heating pure water to its boiling point using a resistance whose surface temperature is 150°C.

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Answer to a math question A 5 kW water heater uses a copper wire resistance that is 50 cm long and has a radius of 0.15 cm. To heat the water, the resistance is completely submerged. Determine the convection heat transfer coefficient when heating pure water to its boiling point using a resistance whose surface temperature is 150°C.

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Hester
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116 Answers
1. Identificar as variáveis relevantes para o problema:

- Potência elétrica: P = 5 \, \text{kW} = 5000 \, \text{W}

- Comprimento do fio: L = 0.5 \, \text{m}

- Raio do fio: r = 0.15 \, \text{cm} = 0.0015 \, \text{m}

- Temperatura da superfície: T_s = 150 \, \text{°C}

- Temperatura da água em ebulição: T_\infty = 100 \, \text{°C}

2. Determinar a área de superfície do fio de cobre:

A_s = 2 \pi r L = 2 \pi (0.0015) (0.5) \approx 0.00471 \, \text{m}^2

3. A utilização da fórmula da transferência de calor por convecção:

A relação básica para transferência de calor por convecção é dada por:

Q = h A_s (T_s - T_\infty)

4. Determinar a taxa de transferência de calor, que é também a potência fornecida:

Q = P = 5000 \, \text{W}

5. Substituir na equação de convecção para calcular o coeficiente de convecção:

5000 = h \times 0.00471 \times (150 - 100)

6. Resolver para h :

h=\frac{5000}{0.00471 \times50}\approx21222\,\text{W/m}^2\text{K}

Portanto, o coeficiente de transferência de calor por convecção é aproximadamente 21222\,\text{W/m}^2\text{K} .

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