Question

By writing an energy balance on the heat exchanger of a binary vapor power cycle, obtain a relation for the ratio of mass flow rates of two fluids in terms of their enthalpies.

Short answer

Answer: The relationship for the ratio of mass flow rates of two fluids in a binary vapor power cycle heat exchanger using an energy balance is given by: m_1/m_2 = (h_2_out - h_2_in)/(h_1_in - h_1_out) where m_1 and m_2 are the mass flow rates of fluid 1 and fluid 2, respectively, and h_1_in, h_1_out, h_2_in, and h_2_out are the enthalpies of fluid 1 and fluid 2 at the heat exchanger's inlet and outlet, respectively.

Step-by-step solution

Write the energy balance equation for the heat exchanger

The energy balance equation states that the energy entering the system equals the energy leaving the system. For the heat exchanger in a binary vapor power cycle, we consider two fluids: the working fluid (fluid 1) and the secondary fluid (fluid 2). The exchange of heat between the fluids in the heat exchanger is the only form of energy transfer in this system. Therefore, we can write the energy balance equation as: Q_in = Q_out Where Q_in is the heat transferred from fluid 1 to fluid 2, and Q_out is the heat absorbed by fluid 2.

Express Q_in and Q_out in terms of enthalpy and mass flow rate

As we know, heat transfer is related to enthalpy change and mass flow rate. So we can rewrite the energy balance equation as: m_1 * (h_1_in - h_1_out) = m_2 * (h_2_out - h_2_in) Where m_1 and m_2 are the mass flow rates of fluid 1 and fluid 2, respectively. Similarly, h_1_in, h_1_out, h_2_in, and h_2_out are the enthalpies of fluid 1 and fluid 2 at the heat exchanger's inlet and outlet, respectively.

Find the ratio of mass flow rates

We are now ready to find the required ratio of mass flow rates of two fluids in terms of their enthalpies. Divide both sides of the equation by the product of mass flow rates: (m_1 * (h_1_in - h_1_out))/(m_1 * m_2) = (m_2 * (h_2_out - h_2_in))/(m_1 * m_2) Cancel out m_1 * m_2 from both sides: (h_1_in - h_1_out) / m_2 = (h_2_out - h_2_in) / m_1 Finally, set the left-hand side equal to the inverse of the right-hand side value to obtain the ratio of mass flow rates: m_1 / m_2 = (h_2_out - h_2_in) / (h_1_in - h_1_out) This equation represents the required relationship of the mass flow rates in terms of enthalpies for the heat exchanger in the binary vapor power cycle.

Key concepts

Enthalpy

Enthalpy is a crucial concept in thermodynamics, particularly in the realm of heat exchanges and energy transfer systems. It represents the total heat content of a system and is a measure of the energy that is contained within a substance, not just in the form of sensible heat, but also including the energy required for phase changes, such as melting or evaporation.

In the context of a heat exchanger, enthalpy changes become important because they account for the heat absorbed or released when a fluid enters or leaves the system. To make sense of this in practice, consider a steam generator: steam's enthalpy will change significantly as it condenses water inside a heat exchanger. This is because the enthalpy of vaporization (the energy needed to turn water into steam) is released during the condensation process.

Understanding enthalpy allows us to calculate how much energy is transferred between fluids, which is essential for designing efficient heat exchangers and producing accurate energy balance equations. In turn, this understanding feeds into calculating the efficiency and output of systems such as power plants, refrigerators, and air conditioning units.

Mass Flow Rate

The mass flow rate is a fundamental parameter in the analysis of fluid systems, which refers to the amount of mass passing through a given surface per unit of time. It is commonly expressed in units such as kilograms per second (kg/s) and is denoted as the symbol 'm' with subscripts used to differentiate between different fluids or points in the system.

In the exercise concerning a heat exchanger, the mass flow rate is closely tied to the concept of enthalpy. The relationship between these two parameters is pivotal in the analysis of energy transfers involving fluids. This relationship showcases how the quantity of heat transfer in a system is not merely a function of temperature difference, but is also dependent on the mass flow rate of the fluids exchanging heat.

Furthermore, in practical applications, the mass flow rate is paramount for determining the size and rating of components such as pumps, pipes, and valves. It also influences the design decisions of process engineers when it comes to ensuring adequate flow and preventing issues like erosion or insufficient cooling/heating in systems.

Heat Exchanger

A heat exchanger is a device designed to efficiently transfer heat from one medium to another, often with the intention of either heating or cooling a fluid. They are essential components in various industrial processes and are found in everyday equipment such as boilers, refrigerators, and air conditioners.

The main role of a heat exchanger is to facilitate the exchange of thermal energy between two or more fluids without mixing them together. It can operate through different mechanisms like conduction, where heat is transferred through a solid barrier, or convection, where it's carried away by a moving fluid. In our textbook problem, the heat exchanger uses the heat from a binary vapor cycle and transfers it efficiently, which is vital for the system's overall energy balance.

When evaluating a heat exchanger's performance, factors like the temperatures and mass flow rates of entering and exiting fluids have to be considered to ensure that energy conservation principles are adhered to. These dynamics, governed by the energy balance and the principle of enthalpy, dictate the size and type of heat exchanger required for a given application, ensuring that the energy transfer is optimized for efficiency and cost-effectiveness.