Calculating Outlet Temperatures of Water and Air in a Heat Exchanger

Question:

How can we determine the outlet temperatures of the water and air in a double-pipe counter flow heat exchanger?

Answer:

To determine the outlet temperatures of the water and air in the double-pipe counter flow heat exchanger, we can use the principle of energy conservation. First, let's calculate the heat transferred from the hot air to the water. The heat transferred can be found using the equation:

Q = m_air * C_air * (T_out_air - T_in_air)

where: Q is the heat transferred, m_air is the mass flow rate of the air, C_air is the specific heat capacity of the air, and T_out_air and T_in_air are the outlet and inlet temperatures of the air, respectively. Given data:

- Mass flow rate of the air (m_air) = 0.3 kg/s - Specific heat capacity of the air (C_air) = 1010 J/kg.K - Inlet temperature of the air (T_in_air) = 90°C

Now, we need to determine the outlet temperature of the air, T_out_air. However, we don't have enough information to directly calculate it. We will need to use the overall heat transfer coefficient, length, and diameter of the tube to find the heat transfer rate, which can be used to find the outlet temperature.

The heat transfer rate can be calculated using the equation:

Q = U * A * ΔT_lm

Given data: - Overall heat transfer coefficient (U) = 80 W/m².K - Internal diameter of the tube (D) = 0.012 m - Length of the tube (L) = 12 m

To calculate ΔT_lm, we need the outlet and inlet temperatures of both the air and water. However, we only have the inlet temperature of the water. Without the outlet temperature of the water, we cannot determine the log mean temperature difference and, therefore, cannot calculate the heat transfer rate and outlet temperature of the air. Thus, based on the given information, we cannot determine the outlet temperatures of the water and the air.

Detail Explanation:

In a double-pipe counter flow heat exchanger, the objective is to transfer heat from a hot fluid (in this case, air) to a colder fluid (water). The specific heat capacities of both fluids and their flow rates play a crucial role in determining the outlet temperatures. The overall heat transfer coefficient represents the efficiency of heat transfer between the fluids.

To calculate the outlet temperatures, we initially look at the heat transferred from the hot air to the water based on their respective mass flow rates, specific heat capacities, and temperature differences. However, the outlet temperature of the air depends on the heat transfer rate, which is determined by the overall heat transfer coefficient, tube dimensions, and temperature differences.

The log mean temperature difference is essential for accurately calculating the heat transfer rate. Yet, without knowing the outlet temperature of the water, we cannot determine this crucial parameter. As a result, the outlet temperatures of both the water and air remain undetermined with the provided information.