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Chapter 4: Problem 65

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

Expert verified
The mass flow rate at inlet 2 is approximately 18.32 kg/s.

Step by step solution

01

- Identify Known Variables

Identify the given values for the exercise: - Inlet 1: - Pressure, \(P_1 = 7\) bar - Temperature, \(T_1 = 42^{\bullet}C\) - Mass Flow Rate, \( \dot{m}_1 = 70 ~ kg/s \)- Inlet 2: - Pressure, \(P_2 = 7\) bar - Quality, \(x_2 = 0.98 \)- Outlet (State 3): - Pressure, \(P_3 = 7\) bar (Saturated Liquid)
02

- Determine Enthalpies

To find the enthalpy values, use steam tables.For Inlet 1: - \(h_1 = 179.9 ~ kJ/kg\) (for liquid water at 7 bar and 42掳C)For Inlet 2: - Find \(h_f\) and \(h_fg\) at 7 bar from the steam tables. - \(h_f = 697.2 ~ kJ/kg\) - \(h_fg = 2030.3 ~ kJ/kg\) - Calculate \( h_2 \) using the quality: \[ h_2 = h_f + x_2 h_fg = 697.2 + (0.98 \times 2030.3) = 2673.5~ kJ/kg \]For State 3 (Saturated Liquid at 7 bar): - \(h_3 = h_f = 697.2 ~ kJ/kg\)
03

- Apply Mass and Energy Balances

Since the system is in a steady state and there is no heat transfer with the surroundings, the mass and energy balances are as follows:Mass Balance: - \( \dot{m}_1 + \dot{m}_2 = \dot{m}_3 \)Energy Balance: - \( \dot{m}_1 h_1 + \dot{m}_2 h_2 = \dot{m}_3 h_3 \)
04

- Express Mass Balance in Terms of Mass Flow Rates

Since the exit stream is a combination of both inlets, the mass balance equation becomes:\[ \dot{m}_3 = \dot{m}_1 + \dot{m}_2 \]
05

- Solve for Mass Flow Rate at Inlet 2

First, find \( \dot{m}_3 \) using the energy balance equation:\[ \dot{m}_1 h_1 + \dot{m}_2 h_2 = (\dot{m}_1 + \dot{m}_2) h_3 \]Rearrange to solve for \( \dot{m}_2 \):\[ \dot{m}_2 (h_2 - h_3) = \dot{m}_1 (h_3 - h_1) \]\[ \dot{m}_2 = \frac{\dot{m}_1 (h_3 - h_1)}{h_2 - h_3} \]Substitute the values:\[ \dot{m}_2 = \frac{70 ~kg/s (697.2 - 179.9)~kJ/kg}{2673.5 - 697.2} ~kJ/kg \]\[ \dot{m}_2 = \frac{70 \times 517.3}{1976.3} \dot{m}_2 = 18.32 ~kg/s \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

steady state process
In thermodynamic systems, a steady state process means that the properties of the system do not change with time. This implies that the variables such as pressure, temperature, and mass flow rates are constant during the observation period. In the context of the feedwater heater problem, this means that the mass flow entering the heater equals the mass flow exiting it. Importantly, since the process is steady state, we assume no accumulation of mass or energy in the system. This helps simplify our mass and energy balance equations.
enthalpy calculation
Enthalpy is a measure of the total energy of a thermodynamic system, and it includes both internal energy and the energy required to make room for it by displacing its environment. To solve the feedwater heater problem, we need to calculate the enthalpy at different states using steam tables:

For inlet 1, with given pressure and temperature, the enthalpy (\(h_1\)) is directly looked up from the steam tables for water at 7 bar and 42掳C.
For inlet 2, a two-phase mixture at 7 bar with a quality of 98%, we use the formula: \[ h_2 = h_f + x_2 h_fg \] Here, \(h_f\) is the enthalpy of saturated liquid and \(h_fg\) is the enthalpy change from liquid to vapor at the given pressure.
For outlet 3, which is saturated liquid at 7 bar, the enthalpy (\(h_3\)) is simply the enthalpy of the saturated liquid at 7 bar, i.e., \(h_f\).
mass and energy balances
In the steady state operation of the feedwater heater, the principle of conservation of mass and energy applies.

**Mass Balance Equation:**
Since the system is said to have no mass accumulation, the mass entering the heater must equal the mass leaving:
\[\dot{m}_1 + \dot{m}_2 = \dot{m}_3\]
**Energy Balance Equation:**
Similarly, since there is no heat transfer with the surroundings and ignoring kinetic and potential energy effects, the energy entering must equal the energy leaving:
\[\dot{m}_1 h_1 + \dot{m}_2 h_2 = \dot{m}_3 h_3\]
With the mass flow rates and enthalpies defined, these equations can be used to determine unknown variables such as the mass flow rate at inlet 2.
steam tables
Steam tables are an essential tool in thermodynamics and are used to find the properties of water and steam at various temperatures and pressures. For the feedwater heater problem, steam tables are used to find:

- The enthalpy (\(h_1\:and\:h_3\)) of water and saturated liquid at 7 bar.
- The enthalpy (\(h_f\:and\:h_fg\)) values for calculating the enthalpy of the two-phase mixture at inlet 2.

Using steam tables simplifies the calculation process since the needed thermodynamic properties are pre-tabulated for quick reference based on pressure and temperature. This allows for an accurate determination of enthalpies, which are crucial for solving mass and energy balances accurately.

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Most popular questions from this chapter

Generating power by harnessing ocean tides and waves is being studied across the globe. Underwater turbines develop power from tidal currents. Wave-power devices develop power from the undulating motion of ocean waves. Although tides and waves long have been used to meet modest power generation needs, investigators today are aiming at large-scale power generation systems. Some see the oceans as providing a potentially unlimited renewable source of power. Critically evaluate the viability of tidaland wave power, considering both technical and economic issues. Write a report, including at least three references.

Steam at 160 bar, \(480^{\circ} \mathrm{C}\), enters a turbine operating at steady state with a volumetric flow rate of \(800 \mathrm{~m}^{3} / \mathrm{min} .\) Eighteen percent of the entering mass flow exits at 5 bar, \(240^{\circ} \mathrm{C}\), with a velocity of \(25 \mathrm{~m} / \mathrm{s}\). The rest exits at another location with a pressure of \(0.06\) bar, a quality of \(94 \%\), and a velocity of \(400 \mathrm{~m} / \mathrm{s}\). Determine the diameters of each exit duct, in \(\mathrm{m}\).

Carbon dioxide gas is heated as it flows steadily through a 2.5-cm-diameter pipe. At the inlet, the pressure is 2 bar, the temperature is \(300 \mathrm{~K}\), and the velocity is \(100 \mathrm{~m} / \mathrm{s}\). At the exit, the pressure and velocity are \(0.9413\) bar and \(400 \mathrm{~m} / \mathrm{s}\), respectively. The gas can be treated as an ideal gas with constant specific heat \(c_{p}=0.94 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\). Neglecting potential energy effects, determine the rate of heat transfer to the carbon dioxide, in \(\mathrm{kW}\).

Wind turbines and hydraulic turbines develop mechanical power from moving streams of air and water, respectively. In each case, what aspect of the stream is tapped for power?

A large pipe carries steam as a two-phase liquid-vapor mixture at \(1.0 \mathrm{MPa}\). A small quantity is withdrawn through a throttling calorimeter, where it undergoes a throttling process to an exit pressure of \(0.1 \mathrm{MPa}\). For what range of exit temperatures, in \({ }^{\circ} \mathrm{C}\), can the calorimeter be used to determine the quality of the steam in the pipe? What is the corresponding range of steam quality values?
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