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Chapter 10: Problem 13

Water is to be boiled at atmospheric pressure in a mechanically polished steel pan placed on top of a heating unit. The inner surface of the bottom of the pan is maintained at \(110^{\circ} \mathrm{C}\). If the diameter of the bottom of the pan is \(30 \mathrm{~cm}\), determine \((a)\) the rate of heat transfer to the water and \((b)\) the rate of evaporation.

Short Answer

Expert verified
Question: Given that the temperature of the bottom surface of a steel pan is 110°C and its diameter is 30 cm, determine the rate of heat transfer to the water in the pan and the rate of evaporation of the water. Answer: The rate of heat transfer to the water is approximately 3,535.5 W, and the rate of evaporation is approximately 8.663 × 10⁻⁶ kg/s.

Step by step solution

01

List the given information

The given information in the problem is as follows: - The temperature of the inner surface of the bottom of the pan, \(T_s = 110^{\circ} \mathrm{C}\) - The diameter of the bottom of the pan, \(D = 30 \mathrm{~cm}\)
02

Calculate the heat transfer area

In order to determine the rate of heat transfer, we first need to calculate the area of the bottom of the pan, through which heat transfer occurs from the heating source to the water. The area, \(A\), can be calculated using the formula for the area of a circle: \(A = \pi (\frac{D}{2})^2\) Here, \(D = 30 \mathrm{~cm} = 0.3 \mathrm{~m}\). Plugging in the values, we have: \(A = \pi (\frac{0.3}{2})^2 = 0.0707 \mathrm{~m}^2\)
03

Calculate the rate of heat transfer

The formula for the rate of heat transfer, \(Q\), is as follows: \(Q = h A (T_s - T_w)\) Here, \(h\) is the heat transfer coefficient, \(A\) is the area calculated in step 2, \(T_s\) is the temperature of the pan's bottom surface, and \(T_w\) is the temperature of water, which is the boiling point of water (\(100^{\circ} \mathrm{C}\)) at atmospheric pressure. Assuming a heat transfer coefficient for convection boiling of water, \(h = 5000 \mathrm{~W/(m^2K)}\). Plugging in the values, we get: \(Q = 5000 \cdot 0.0707 \cdot (110 - 100) = 3,535.5 \mathrm{~W}\) The rate of heat transfer to the water is approximately \(3,535.5 \mathrm{~W}\).
04

Calculate the rate of evaporation

The rate of evaporation, \(m_{evap}\) can be found using the following formula: \(m_{evap} = \frac{Q}{L_v}\) Here, \(L_v \approx 40.79 \times 10^5 \mathrm{~J/kg}\) is the approximate heat of vaporization of water at the atmospheric pressure, and \(Q\) is the rate of heat transfer that we calculated in step 3. Plugging in the values, we get: \(m_{evap} = \frac{3,535.5}{40.79 \times 10^5} = 8.663 \times 10^{-6} \mathrm{~kg/s}\) The rate of evaporation is approximately \(8.663 \times 10^{-6} \mathrm{~kg/s}\). In summary, the rate of heat transfer to the water is \(3,535.5 \mathrm{~W}\), and the rate of evaporation is \(8.663 \times 10^{-6} \mathrm{~kg/s}\).

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Most popular questions from this chapter

Saturated steam at \(100^{\circ} \mathrm{C}\) condenses on a 2-m \(\times 2-\mathrm{m}\) plate that is tilted \(40^{\circ}\) from the vertical. The plate is maintained at \(80^{\circ} \mathrm{C}\) by cooling it from the other side. Determine (a) the average heat transfer coefficient over the entire plate and \((b)\) the rate at which the condensate drips off the plate at the bottom. Assume wavy-laminar flow. Is this a good assumption?

Name the different boiling regimes in the order they occur in a vertical tube during flow boiling.

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