Question

A feedwater heater operates at steady state with liquid water entering at inlet 1 at 7 bar, \(42^{\circ} \mathrm{C}\), and a mass flow rate of \(70 \mathrm{~kg} / \mathrm{s}\). A separate stream of water enters at inlet 2 as a two-phase liquid-vapor mixture at 7 bar with a quality of \(98 \%\). Saturated liquid at 7 bar exits the feedwater heater at 3 . Ignoring heat transfer with the surroundings and neglecting kinetic and potential energy effects, determine the mass flow rate, in \(\mathrm{kg} / \mathrm{s}\), at inlet 2 .

Short answer

The mass flow rate at inlet 2 is 13.84 kg/s.

Step-by-step solution

Determine enthalpy of water at inlet 1

Liquid water enters at inlet 1 at 7 bar and 42°C. Using steam tables, find the enthalpy, \(h_1\), of water at these conditions. From the steam tables, the enthalpy of sub-cooled liquid water at 7 bar and 42°C is \(h_1 = 176.8 \, \mathrm{kJ/kg}\).

Determine enthalpy of the two-phase mixture at inlet 2

The second inlet has a two-phase liquid-vapor mixture at 7 bar with a quality of 98%. Using the quality and steam tables, find the enthalpy, \(h_2\), of this mixture. The enthalpy can be calculated using the equation: \[h_2 = h_f + x \, (h_{fg})\] where \(h_f\) is the enthalpy of saturated liquid and \(h_{fg}\) is the enthalpy of vaporization. From the steam tables at 7 bar, \(h_f = 721.2 \, \mathrm{kJ/kg}\), \(h_{fg} = 2087.1 \, \mathrm{kJ/kg}\). Thus, \[h_2 = 721.2 + 0.98 \, (2087.1) = 2766.96 \, \mathrm{kJ/kg}\].

Determine enthalpy at outlet 3

Saturated liquid exits at outlet 3 at 7 bar. Using steam tables, find the enthalpy, \(h_3\), at these conditions. From the steam tables, the enthalpy of saturated liquid (which corresponds to \(h_f\) at 7 bar) is \(h_3 = 721.2 \, \mathrm{kJ/kg}\).

Apply energy balance to the feedwater heater

Since the system operates at steady state and ignoring heat transfer with the surroundings, apply the steady-state energy balance: \[ m_1 h_1 + m_2 h_2 = m_3 h_3 \] Where \( m_3 \) is the total mass flow rate out of the heater (sum of \( m_1 \) and \( m_2 \)). This yields: \[ m_1 h_1 + m_2 h_2 = (m_1 + m_2) h_3 \]

Solve for \(m_2\)

Rearrange the energy balance equation and solve for \(m_2\): \[ m_2 (h_2 - h_3) = m_1 (h_3 - h_1) \] \[ m_2 = \frac{m_1 (h_3 - h_1)}{h_2 - h_3} \] Substitute the values: \[ m_2 = \frac{70 \, \mathrm{kg/s} \, (721.2 - 176.8)}{2766.96 - 721.2} = 13.84 \, \mathrm{kg/s} \]

Key concepts

steady state

In the context of thermodynamics, a system is said to be in a steady state if its properties (such as pressure, temperature, and mass flow rate) do not change over time.
This means that the amount of energy entering the system is equal to the amount of energy leaving the system.

For example, in a feedwater heater operating at steady state, the mass and energy entering through the inlets must balance the mass and energy exiting.
Hence, the mass flow rate and enthalpy (energy per unit mass) at the inlet and outlet points remain constant over time.

• Mass Flow Balance: The mass entering the system must equal the mass leaving the system.
• Energy Flow Balance: The energy entering through the water streams must match the energy exiting the system.

enthalpy

Enthalpy represents the heat content of a system per unit mass and is denoted as 'h'.
It encompasses both the internal energy of the system and the work needed to make space for it by displacing its surroundings.
This makes it especially useful in energy balance calculations.

For instance, in a feedwater heater problem, we deal with three sets of enthalpy values:

• Inlet 1: Liquid water at 7 bar and 42°C has an enthalpy determined using steam tables.
• Inlet 2: A two-phase mixture at 7 bar with a 98% quality, calculated based on the formula for mixture enthalpies.
• Outlet: Saturated liquid at 7 bar, with its enthalpy directly obtained from steam tables.

Using these enthalpy values, we can then set up our energy balance to find the unknown quantities, like the mass flow rate at inlet 2 in this example.

energy balance

The energy balance equation is foundational for analyzing thermodynamic systems.
It ensures that all the energy entering and leaving the system is accounted for, adhering to the principle of the conservation of energy.

For a feedwater heater operating at steady state, the energy balance can be expressed as:
m_1 h_1 + m_2 h_2 = m_3 h_3
This signifies that the energy in the system's inlets (flow rate times enthalpy) equals the energy at the outlet.

• Where
• m_1 is the mass flow rate at inlet 1,
• h_1 is the enthalpy at inlet 1,
• m_2 is the mass flow rate at inlet 2,
• h_2 is the enthalpy at inlet 2, and
• m_3 is the total mass flow rate at the outlet, which is the sum of m_1 and m_2.

In the given exercise, rearranging this equation allows us to solve for the unknown mass flow rate at inlet 2.

two-phase mixture

A two-phase mixture contains both liquid and vapor phases coexisting at equilibrium.
The 'quality' or 'x' of such a mixture denotes the fraction of the mass that is in the vapor phase.

In thermodynamic problems, understanding and calculating the properties of two-phase mixtures is crucial.
For the mixture at inlet 2 in our feedwater heater problem, the quality is given as 98%, or 0.98.

We use this quality in calculating the enthalpy of the mixture as follows:
h_2 = h_f + x (h_{fg})
Where:

• h_f is the enthalpy of saturated liquid.
• h_fg is the enthalpy of vaporization (the energy required to convert saturated liquid into dry vapor).
• x is the quality of the mixture.

By inputting the correct values from the steam tables, we find the specific enthalpy needed for the energy balance equation.