Question

An air conditioner condenser in an automobile consists of \(2 \mathrm{~m}^{2}\) of tubular heat exchange area whose surface temperature is \(30^{\circ} \mathrm{C}\). Saturated refrigerant-134a vapor at \(50^{\circ} \mathrm{C}\) \(\left(h_{f g}=152 \mathrm{~kJ} / \mathrm{kg}\right)\) condenses on these tubes. What heat transfer coefficent must exist between the tube surface and condensing vapor to produce \(1.5 \mathrm{~kg} / \mathrm{min}\) of condensate? (a) \(95 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) (b) \(640 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) (c) \(727 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) (d) \(799 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) (e) \(960 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\)

Short answer

Answer: The required heat transfer coefficient is 95 W/m²·K.

Step-by-step solution

Calculate Heat transfer rate (Q_dot)

As we are given the mass of condensate produced per minute, we need to calculate the heat transfer rate (Q_dot) from the given mass and specific enthalpy (h_fg). Formula to calculate heat transfer rate (Q_dot) is: Q_dot = (mass_flow_rate) * (h_fg) Given values are: mass_flow_rate = 1.5 kg/min h_fg = 152 kJ/kg mass_flow_rate needs to be converted to kg/s: mass_flow_rate = 1.5 kg/min * (1 min / 60 s) = 0.025 kg/s Now, substitute the values in the formula: Q_dot = (0.025 kg/s) * (152000 J/kg) = 3800 J/s or 3800 W

Calculate Heat transfer coefficient (h)

Now, we will use the formula to find the heat transfer coefficient (h): h = Q_dot / (A * ∆T) Given values are: Q_dot = 3800 W A = 2 m² ∆T = (50°C - 30°C) = 20°C Now, substitute the values in the formula: h = 3800 W / (2 m² * 20°C) = 95 W/m²·K Therefore, the heat transfer coefficient must exist between the tube surface and condensing vapor to produce 1.5 kg/min of condensate is: (a) 95 W/m²·K

Key concepts

Condensate Production

When dealing with heat exchangers, particularly in air conditioning systems, condensate production is a critical concept. This is the process where a gas or vapor changes its state to liquid due to cooling. In the case of air conditioner condensers like the one in our original exercise, condensate is produced when saturated refrigerant-134a vapor cools down and changes into liquid form on the surface of the tubular heat exchanger.

The rate at which condensation occurs is vital for the efficiency of the system. More specifically, the exercise asked us to calculate the heat transfer coefficient necessary to achieve a specific condensate production rate, which in our scenario is 1.5 kg/min. This measure is directly related to the cooling capacity of the air conditioner.

To make this concept clearer, envision water vapor forming on the outside of a cold beverage glass. That's a similar principle, where a decrease in temperature causes the vapor in the air to lose energy and become liquid, which is essentially the production of condensate.

Refrigerant-134a Properties

Understanding the properties of the refrigerant-134a is fundamental when calculating heat transfer in air conditioning systems. Refrigerant-134a, also known as tetrafluoroethane (C2H2F4), is commonly used in the automotive and refrigeration industries. Its properties, such as boiling point, specific heat, and enthalpy, are crucial to designing and evaluating refrigeration cycles and the associated equipment.

In our case, the saturation temperature and specific enthalpy (\(h_{fg}\) = 152 kJ/kg) determine how much heat is removed during the condensation process. Recall that enthalpy is the total heat content of a system, and the vapor's specific enthalpy of phase change (\(h_{fg}\)) is important as it tells us how much heat needs to be removed to convert the vapor into a liquid at a constant temperature.

This property of refrigerant-134a facilitates the calculation of the heat transfer rate, leading us to determine the amount of energy involved in producing the condensate from the refrigerant vapor. Comprehending these properties of refrigerant-134a in the exercise allows students to understand how the refrigerant's characteristics play a pivotal role in the heat transfer process within the system.

Tubular Heat Exchanger

Tubular heat exchangers are essential components that facilitate the heat transfer process. They are widely used in various applications, including HVAC systems, power plants, and chemical processing. In a tubular heat exchanger, fluids at different temperatures flow through tubes and exchange heat without mixing with each other.

In our exercise, the surface area of these tubes (2 m²) and the temperature difference between the tube surface and the refrigerant (\( \triangle T \) = 20°C) are given. The efficiency of the heat exchanger can be described by the heat transfer coefficient. This coefficient is a measure of the exchanger's ability to transfer heat from one fluid to another.

The higher the heat transfer coefficient, the more efficient the heat exchanger is at transferring heat. In simpler terms, think of the coefficient as a score that tells you how good the tube material and design are at passing along the baton of heat from the refrigerant vapor to the surrounding environment.

Understanding how tubular heat exchangers work, and the role they play in systems like air conditioners, helps students grasp why the heat transfer coefficient is crucial for specifying and sizing equipment in thermodynamic applications.