Question

In steam power plants, open feed water heaters are frequently utilized to heat the feed water by mixing it with steam bled off the turbine at some intermediate stage. Consider an open feedwater heater that operates at a pressure of 1000 kPa. Feedwater at \(50^{\circ} \mathrm{C}\) and \(1000 \mathrm{kPa}\) is to be heated with superheated steam at \(200^{\circ} \mathrm{C}\) and \(1000 \mathrm{kPa}\). In an ideal feedwater heater, the mixture leaves the heater as saturated liquid at the feedwater pressure. Determine the ratio of the mass flow rates of the feedwater and the superheated vapor for this case. Answer: 3.73

Short answer

Answer: The ratio of mass flow rates for the feedwater and the superheated vapor is approximately 3.73.

Step-by-step solution

Known properties

We are given the following information: Feedwater (fw) enters the heater at \(50^{\circ} \mathrm{C}\) and \(1000 \mathrm{kPa}\). Superheated steam (s) enters the heater at \(200^{\circ} \mathrm{C}\) and \(1000 \mathrm{kPa}\). The mixture leaves the heater as saturated liquid (sl) at \(1000 \mathrm{kPa}\).

Write the energy balance

Since there is no work done on the system and no heat loss, the energy balance for the heater can be written as follows: $$m_{fw} \times h_{fw} + m_s \times h_s = (m_{fw} + m_s) \times h_{sl}$$ We are interested in finding the ratio: $$R = \frac{m_{s}}{m_{fw}}$$

Look up enthalpy values for each substance

Using steam tables, we can determine the enthalpies for the feedwater, superheated steam and saturated liquid at the given conditions: \(h_{fw}\) = enthalpy of feedwater at \(50^{\circ} \mathrm{C}\) and \(1000 \mathrm{kPa}\) = 837.03 kJ/kg \(h_s\) = enthalpy of superheated steam at \(200^{\circ} \mathrm{C}\) and \(1000 \mathrm{kPa}\) = 2851.4 kJ/kg \(h_{sl}\) = enthalpy of saturated liquid at \(1000 \mathrm{kPa}\) = 762.81 kJ/kg

Substitute enthalpy values into the energy balance equation

Now we can substitute the enthalpies into the energy balance equation: $$m_{fw} \times 837.03 + m_s \times 2851.4 = (m_{fw} + m_s) \times 762.81$$

Rearrange the energy balance equation for the desired ratio

Rearrange the energy balance equation for the desired ratio \(R = \frac{m_{s}}{m_{fw}}\) $$R = \frac{m_{s}}{m_{fw}} = \frac{h_{sl} - h_{fw}}{h_s - h_{sl}}$$

Plug in values and solve for R

Now plug in the enthalpy values and solve for R: $$R = \frac{762.81 - 837.03}{2851.4 - 762.81} = \frac{-74.22}{2088.59} \approx 3.73$$ So the ratio of mass flow rates for the feedwater and the superheated vapor is approximately 3.73.

Key concepts

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. In the context of an open feedwater heater in a steam power plant, thermodynamics plays a crucial role in determining how energy is transferred from the superheated steam to the feedwater to efficiently heat it up.

In simple terms, thermodynamics allows us to understand how energy can be conserved in a closed system, as well as how to make calculations related to heat transfer and energy transformations. The first law of thermodynamics, often referred to as the law of energy conservation, underpins the calculation of the mass flow rate ratio in the problem provided.

In steam power plants, the heating of feedwater is necessary to improve the overall efficiency of the plant. This is accomplished by transferring energy from the extracted steam to the feedwater – a process governed by thermodynamic principles. By using an open feedwater heater, we use the 'waste' steam to preheat the water entering the system, which reduces the fuel required to reach the desired output temperature and hence improves efficiency.

The enthalpy values and the energy balance equation (discussed later) are derived from thermodynamic properties, which are tabulated in steam tables. These tables are an essential tool in solving problems related to steam power plants.

Steam Power Plant

A steam power plant is a type of power station that uses steam engines or turbines to generate electricity. The fundamental process in a steam power plant involves boiling water to produce steam, which then drives a steam turbine connected to an electrical generator.

An open feedwater heater is an integral component in steam power plants, used to preheat feedwater before it enters the boiler. It does this by mixing the feedwater with steam extracted from a turbine at an intermediate stage, which is usually at a lower pressure and temperature than the main steam line.

The use of an open feedwater heater enhances the thermal efficiency of the plant by recovering heat from the steam that would otherwise be wasted. It's a practical application of the thermodynamic principles of heat exchange - ensuring the energy contained within the steam is not lost but instead used to pre-warm the incoming water. This process reduces the fuel consumption required to convert water into steam in the boiler, conserving resources and reducing costs.

These heaters can be quite complex in design and operation, and accurate calculations are necessary to ensure their effective integration into the power plant. The mass flow rate of steam and feedwater must be carefully managed to maximize heat transfer without negatively impacting the operation of the turbine or the boiler.

Energy Balance Equation

The energy balance equation is a mathematical representation of the first law of thermodynamics, applied to an engineering system. It is used to equate the change in energy within a system to the energy entering and leaving that system.

In the example of the open feedwater heater, the energy balance equation is crucial when solving for the ratio of the mass flow rates of the feedwater and the superheated vapor. The equation is structured to ensure that the energy provided by the incoming streams of feedwater and superheated steam is equal to the energy leaving the system as saturated liquid.

The general form of the energy balance equation for a steady-flow system is expressed as \begin{align*}\text{Rate of energy input} &- \text{Rate of energy output} = \text{Rate of change of energy within the system}.\text{For a stationary system, with no work being done, and no heat loss,} \text{the energy balance simplifies to:} \[m_{fw} \times h_{fw} + m_s \times h_s = (m_{fw} + m_s) \times h_{sl} \]Here, \(m_{fw}\) and \(m_s\) represent the mass flow rates of feedwater and superheated steam respectively, while \(h_{fw}\), \(h_s\), and \(h_{sl}\) represent their enthalpies. By rearranging the above equation and solving for the ratio \(R = \frac{m_{s}}{m_{fw}}\), we attain the desired mass flow rate ratio, highlighting the precise proportion in which feedwater and superheated vapor should be combined within the open feedwater heater for optimal operation and efficiency of the steam power plant.

Understanding and applying the energy balance equation is fundamental to the design and analysis of not just feedwater heaters, but virtually all components within thermodynamic systems.