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Chapter 4: Problem 88

Steam with a quality of \(0.7\), pressure of \(1.5\) bar, and flow rate of \(10 \mathrm{~kg} / \mathrm{s}\) enters a steam separator operating at steady state. Saturated vapor at \(1.5\) bar exits the separator at state 2 at a rate of \(6.9 \mathrm{~kg} / \mathrm{s}\) while saturated liquid at \(1.5\) bar exits the separator at state 3 . Neglecting kinetic and potential energy effects, determine the rate of heat transfer, in \(\mathrm{kW}\), and its associated direction.

Short Answer

Expert verified
The rate of heat transfer is 4292.057 kW, and it is being removed from the system.

Step by step solution

01

- Understand the Given Information

The problem involves steam with a quality of 0.7, a pressure of 1.5 bar, and a flow rate of 10 kg/s entering a steam separator. The separator outputs saturated vapor at 1.5 bar with a flow rate of 6.9 kg/s and saturated liquid at 1.5 bar.
02

- Identify the States

State 1 (inlet): Steam with a quality of 0.7, State 2 (outlet vapor): Saturated vapor at 1.5 bar, State 3 (outlet liquid): Saturated liquid at 1.5 bar.
03

- Find Enthalpies from Steam Tables

Using steam tables, find the specific enthalpies: \[ h_f @ 1.5 \text{ bar} = 467.11 \text{ kJ/kg} \ h_g @ 1.5 \text{ bar} = 2163.7 \text{ kJ/kg} \ h_1 = h_f + x \times h_{fg} = 467.11 + 0.7 \times (2163.7 - 467.11) = 1611.037 \text{ kJ/kg} \ h_2 = h_g = 2163.7 \text{ kJ/kg} \ h_3 = h_f = 467.11 \text{ kJ/kg} \]
04

- Apply Mass Balance

Using mass balance: \[ 10 = 6.9 + m_3 \ m_3 = 3.1 \text{ kg/s} \]
05

- Apply Energy Balance

Using energy balance: \[ q_{in} - q_{out} = m_1 h_1 - m_2 h_2 - m_3 h_3 \ \Delta q = 10 \times 1611.037 - 6.9 \times 2163.7 - 3.1 \times 467.11 \ q_{net} = 4292.057 \text{ kW} \] Negative sign indicates heat is being removed from the system.

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Key Concepts

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

enthalpy calculation
In thermodynamics, enthalpy is a measure of the total heat content in a system. It's denoted by H and calculated as the sum of internal energy and the product of pressure and volume. In this exercise, the enthalpies for different states are identified using the steam tables.
For instance, for steam at a quality of 0.7 and a pressure of 1.5 bar, the specific enthalpy is found using: \(h_1 = h_f + x \times h_{fg} = 467.11 + 0.7 \times (2163.7 - 467.11) = 1611.037 \text{ kJ/kg}\) Here, \(h_f\) is the enthalpy of saturated liquid and \(h_g\) is the enthalpy of saturated vapor. Knowing these values allows us to proceed with energy calculations.
mass balance
Mass balance is a fundamental principle in engineering that ensures the mass entering a system equals the mass leaving it. It helps in tracking the mass flow rates. In the given problem, the mass balance equation is applied as follows:
The inlet mass flow rate \(m_1\) is 10 kg/s. The outlet mass flow rate of vapor \(m_2\) is 6.9 kg/s. Using these values, the mass flow rate of the liquid \(m_3\) can be determined:
\(10 = 6.9 + m_3\)
Solving for \(m_3\), we get:
\(m_3 = 3.1 \text{ kg/s}\) This ensures that all the mass is accounted for in the system.
energy balance
The energy balance principle ensures that the energy entering a system equals the energy leaving it plus any change in the energy stored in the system. In this problem, neglecting kinetic and potential energy, the energy balance is:
\(q_{in} - q_{out} = m_1 h_1 - m_2 h_2 - m_3 h_3\)
Plugging in the known values:
\(q_{in} - q_{out} = 10 \times 1611.037 - 6.9 \times 2163.7 - 3.1 \times 467.11\)
This simplifies to:
\(q_{net} = 4292.057 \text{ kW}\) The negative value implies that heat is being extracted from the system.
steam tables
Steam tables are essential tools in thermodynamics. They provide the thermodynamic properties of water and steam, such as pressure, temperature, enthalpy, entropy, and specific volume. In this exercise, we used steam tables to find:
  • The enthalpy of saturated liquid (\(h_f\)) at 1.5 bar as 467.11 kJ/kg.
  • The enthalpy of saturated vapor (\(h_g\)) at 1.5 bar as 2163.7 kJ/kg.
This data enables calculating the specific enthalpy for steam with different qualities and states.
thermodynamic cycles
Thermodynamic cycles describe processes where a working fluid undergoes a series of stages and returns to its initial state. Each cycle can involve multiple phases: evaporation, condensation, compression, and expansion. In the context of steam separators, these devices play a crucial role in separating the liquid and vapor phases, aiding in the analysis and optimization of cycles like the Rankine cycle. Understanding the behavior of steam and its properties at various stages ensures the efficiency and effectiveness of heat engines and other thermodynamic systems.

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Most popular questions from this chapter

During cardiac surgery, a heart-lung machine achieves extracorporeal circulation of the patient's blood using a pump operating at steady state. Blood enters the wellinsulated pump at a rate of 5 liters/min. The temperature change of the blood is negligible as it flows through the pump. The pump requires \(20 \mathrm{~W}\) of power input. Modeling the blood as an incompressible substance with negligible kinetic and potential energy effects, determine the pressure change, in \(\mathrm{kPa}\), of the blood as it flows through the pump.

Air enters a compressor operating at steady state with a pressure of \(14.7 \mathrm{lbf} /\) in. \({ }^{2}\), a temperature of \(80^{\circ} \mathrm{F}\), and a volumetric flow rate of \(18 \mathrm{ft}^{3} / \mathrm{s}\). The air exits the compressor at a pressure of \(90 \mathrm{lbf} / \mathrm{in}^{2}\) Heat transfer from the compressor to its surroundings occurs at a rate of \(9.7\) Btu per lb of air flowing. The compressor power input is \(90 \mathrm{hp}\). Neglecting kinetic and potential energy effects and modeling air as an ideal gas, determine the exit temperature, in \({ }^{\circ} \mathrm{F}\).

The procedure to inflate a hot-air balloon requires a fan to move an initial amount of air into the balloon envelope followed by heat transfer from a propane burner to complete the inflation process. After a fan operates for 10 minutes with negligible heat transfer with the surroundings, the air in an initially deflated balloon achieves a temperature of \(80^{\circ} \mathrm{F}\) and a volume of \(49,100 \mathrm{ft}^{3}\). Next the propane burner provides heat transfer as air continues to flow into the balloon without use of the fan until the air in the balloon reaches a volume of \(65,425 \mathrm{ft}^{3}\) and a temperature of \(210^{\circ} \mathrm{F}\). Air at \(77^{\circ} \mathrm{F}\) and \(14.7 \mathrm{lb} / 1 n^{2}\) surrounds the balloon. The net rate of heat transfer is \(7 \times 10^{6} \mathrm{Btu} / \mathrm{h}\). Ignoring effects due to kinetic and potential energy, modeling the air as an ideal gas, and assuming the pressure of the air inside the balloon remains the same as that of the surrounding air, determine (a) the power required by the fan, in hp. (b) the time required for full inflation of the balloon, in min.

Refrigerant \(134 \mathrm{a}\) at a flow rate of \(0.5 \mathrm{lb} / \mathrm{s}\) enters a heat exchanger in a refrigeration system operating at steady state as saturated liquid at \(0^{\circ} \mathrm{F}\) and exits at \(20^{\circ} \mathrm{F}\) at a pressure of \(20 \mathrm{lbf} / \mathrm{in}^{2}\) A separate air stream passes in counterflow to the Refrigerant \(134 \mathrm{a}\) stream, entering at \(120^{\circ} \mathrm{F}\) and exiting at \(77^{\circ} \mathrm{F}\). The outside of the heat exchanger is well insulated. Neglecting kinetic and potential energy effects and modeling the air as an ideal gas, determine the mass flow rate of air,

Oxygen gas enters a well-insulated diffuser at \(30 \mathrm{lbf} / \mathrm{in} .^{2}\), \(440^{\circ} \mathrm{R}\), with a velocity of \(950 \mathrm{ft} / \mathrm{s}\) through a flow area of \(2.0 \mathrm{in}^{2}\) At the exit, the flow area is 15 times the inlet area, and the velocity is \(25 \mathrm{ft} / \mathrm{s}\). The potential energy change from inlet to exit is negligible. Assuming ideal gas behavior for the oxygen and steady-state operation of the diffuser, determine the exit temperature, in \({ }^{\circ} \mathrm{R}\), the exit pressure, in \(\mathrm{lbf} / \mathrm{in}^{2}\), and the mass flow rate, in lb/s.
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