Question

Saturated water vapor is condensing on a \(0.5 \mathrm{~m}^{2}\) vertical flat plate in a continuous film with an average heat transfer coefficient of \(7 \mathrm{~kW} / \mathrm{m}^{2} \cdot \mathrm{K}\). The temperature of the water is \(80^{\circ} \mathrm{C}\left(h_{f g}=2309 \mathrm{~kJ} / \mathrm{kg}\right)\) and the temperature of the plate is \(60^{\circ} \mathrm{C}\). The rate at which condensate is being formed is (a) \(0.0303 \mathrm{~kg} / \mathrm{s}\) (b) \(0.07 \mathrm{~kg} / \mathrm{s}\) (c) \(0.15 \mathrm{~kg} / \mathrm{s}\) (d) \(0.24 \mathrm{~kg} / \mathrm{s}\) (e) \(0.28 \mathrm{~kg} / \mathrm{s}\)

Short answer

Answer: (a) 0.0303 kg/s

Step-by-step solution

Determine the temperature difference between water and plate

Given, the temperature of the water \(T_w = 80^{\circ} \mathrm{C}\) and the temperature of the plate \(T_p = 60^{\circ} \mathrm{C}\). First, we will find the temperature difference between the water vapor and the plate: $$\Delta T = T_w - T_p = 80 - 60 = 20 \mathrm{~K}$$

Calculate the heat transfer rate

We will now find the heat transfer rate using the formula: $$Q = hA\Delta T$$ Where \(Q\) is the heat transfer rate, \(h\) is the heat transfer coefficient, \(A\) is the area, and \(\Delta T\) is the temperature difference. Given, \(h=7\ \mathrm{kW}/{\mathrm{m}^2\cdot\mathrm{K}}\), \(A=0.5\ \mathrm{m}^2\), and \(\Delta T = 20\ \mathrm{K}\): $$Q = 7 \times 0.5 \times 20 = 70\ \mathrm{kW}$$

Convert heat transfer rate to Watts

Now, we need to convert the heat transfer rate from kilowatts to watts: $$Q = 70\ \mathrm{kW} \times 1000\ \frac{\mathrm{W}}{\mathrm{kW}} = 70000\ \mathrm{W}$$

Calculate the rate of condensate formation

Finally, we will find the rate of condensate formation using the formula: $$\dot{m} = \frac{Q}{h_{fg}}$$ Where \(\dot{m}\) is the rate of condensate formation and \(h_{fg}\) is the latent heat of vaporization. Given, \(h_{fg} = 2309\ \mathrm{kJ/kg} = 2309000\ \frac{\mathrm{J}}{\mathrm{kg}}\): $$\dot{m} = \frac{70000}{2309000} = 0.0303\ \frac{\mathrm{kg}}{\mathrm{s}}$$ The rate at which condensate is being formed is (a) \(0.0303\ \mathrm{kg/s}\).

Key concepts

Heat Transfer Coefficient

Understanding the heat transfer coefficient is crucial when studying how heat moves from one medium to another. It represents the amount of heat that flows per unit area, per unit time, for a temperature difference of one degree Celsius or Kelvin. In the condensation example, a high heat transfer coefficient means heat transfers efficiently from the water vapor to the cold plate, leading to rapid condensation.The coefficient depends on various factors, including the nature of the fluid, the flow regime (laminar or turbulent), and the properties of the surface where the heat exchange takes place. The heat transfer coefficient is measured in units of power per area per temperature difference, usually expressed as watts per square meter per Kelvin For students looking to deepen their understanding, remembering that this coefficient serves as a proportional constant in the basic heat transfer equation, can be beneficial. If two materials have the same temperature difference and area for heat transfer, but one has a higher heat transfer coefficient, more heat will flow through that material in a given time.

Temperature Difference

The concept of temperature difference lies at the heart of thermodynamics and heat transfer. It's the driving force behind the flow of heat energy; heat naturally flows from a hotter substance to a cooler one until thermal equilibrium is reached. The greater the temperature difference, the larger the potential for heat transfer.In the context of condensation on a plate, the temperature difference between the saturated water vapor and the cooler plate surface directly influences the rate of energy transfer. A larger difference results in a faster heat transfer rate, which in turn increases the condensate formation rate because more vapor can be cooled and condensed per unit time. Highlighting the real-world implications of temperature difference, such as in heating systems or refrigeration processes, where manipulating this difference can improve efficiency or cooling rates, can be quite enlightening for students. This practical application demonstrates the importance of understanding temperature difference in engineering and environmental control systems.

Latent Heat of Vaporization

Latent heat of vaporization is the amount of thermal energy required to change a substance from liquid to vapor phase without a temperature change. This is a form of hidden energy, as it doesn't raise the temperature, but is consumed to break the intermolecular bonds. It's crucial in phase change processes like boiling, condensation, and evaporation.In the case of the condensation on the plate, knowing the latent heat of vaporization is essential to calculate the mass of water that condenses per second. It takes a lot of energy to vaporize water; hence, a lot of energy is released when vapor condenses back into liquid. The latent heat value is usually quite substantial, indicating why cooling systems and power plants must manage phase changes very carefully.To put this in perspective, while resurfacing certain points may be repetitive, it is important for learners to recognize that the high value of water's latent heat of vaporization means that even small amounts of condensation release significant amounts of heat. Comprehending this concept is pivotal for various applications, such as designing HVAC systems, understanding weather patterns, or calculating energy requirements in phase-change materials.