Chapter Seven: Entropy

WHAT IS ENTROPY Entropy is a measure of molecular disorder, or molecular randomness. As a system becomes more disordered, the positions of the molecules become less predictable and the entropy increases. The entropy of a substance is lowest in the solid phase and highest in the gas phase. Unlike energy, entropy is a nonconserved property, and there is no such thing as a conservation of entropy principle.

Entropy Change of a System Note that entropy is a property, and the value of a property does not change unless the state of the system changes. Therefore, the entropy change of a system is zero if the state of the system does not change during the process.

Third Law of Thermodynamics Entropy is the measure of molecular disorder or randomness. As a system becomes more disordered, the position of the molecules becomes less predictable and the entropy increases. Entropy is the lowest in solids because molecules are held in place and simply vibrate and highest in gases where the molecules are free to move in any direction. Third Law of Thermodynamics States that: “Entropy of a pure crystalline substance at absolute zero temperature (zero Kelvin) is zero since the state of each molecule is known”.

Isentropic Process A process that is both adiabatic and reversible is referred to as isentropic, and for a closed ystem:

Assume that the system is at steady state ( )

ENTROPY BALANCE The second law of thermodynamics states that entropy can be created but it cannot be destroyed. The entropy change of a system during a process is greater than the net entropy transfer by an amount equal to the entropy generated during the process within the system, and the increase of entropy principle for any system is expressed as:

Mechanisms of Entropy Transfer 1. Heat Transfer 2. Mass Flow Note that there is no entropy change due to work

Entropy Generation Irreversibilities such as friction, mixing, chemical reactions, heat transfer through a finite temperature difference, unrestrained expansion or compression always cause the entropy of a system to increase, and entropy generation is a measure of the entropy created by such effects during a process. For irreversible process: For a reversible process, the entropy generation is zero and thus the entropy change of a system is equal to the entropy transfer. For reversible process:

Change in Entropy and Its Rate

Open System (Control Volumes) A control volume involves mass flow across its boundaries, and its entropy change is a combination of entropy transfer accompanying heat and mass transfer and the entropy generation within the system boundaries.

Special Cases 1. Steady state process 2. Steady state process, Single-stream 3. Steady state process, Single-stream, Adiabatic

Closed Systems A closed system involves no mass flow across its boundaries, and its entropy change is simply the difference between the initial and final entropies of the system. The entropy change of a closed system is due to the entropy transfer accompanying heat transfer and the entropy generation within the system boundaries. 19

Steps for Solving Entropy Problems Sketch the process List data you may need to solve the problem and highlight what is known and what is unknown. Perform Mass Balance. Perform Energy Balance. Perform Entropy Balance. 20

EXAMPLE: Entropy Generation in a Mixing Chamber Water at 20 psia and 50°F enters a mixing chamber at a rate of 300 lbm/min where it is mixed steadily with steam entering at 20 psia and 240°F. The mixture leaves the chamber at 20 psia and 130°F, and heat is lost to the surrounding air at 70°F at a rate of 180 Btu/min. Neglecting the changes in kinetic and potential energies, determine the rate of entropy generation during this process.

We take the mixing chamber as the system We take the mixing chamber as the system. This is a control volume since mass crosses the system boundary during the process. We note that there are two inlets and one exit. Under the stated assumptions and observations, the mass and energy balances for this steady-flow system can be expressed in the rate form as follows:

Data =

Entropy Balance: The rate of entropy generation during this process can be determined by applying the rate form of the entropy balance on an extended system that includes the mixing chamber and its immediate surroundings so that the boundary temperature of the extended system is 70°F = 530 R:

For ideal gas:

End of Chapter Seven

Tutorial on Chapter Seven (Entropy)

Problem 1 Water at 200 kPa and 20°C is heated in a chamber by mixing it with steam at 200 kPa and 300°C. Water enters the mixing chamber at a rate of 2.5 kg/s, and the chamber is estimated to lose heat to the surrounding air at 25°C at a rate of 600 kJ/min. If the mixture leaves the mixing chamber at 200 kPa and 60°C, determine: the mass flow rate of the steam, and the rate of entropy generation during this mixing process. Assume steady state operation. T1= T3= T2=

Solution 1 Assumptions: 1. This is a steady-flow process since there is no change with time. 2. Kinetic and potential energy changes are negligible. 3. There are no work interactions. Properties: Noting that T < Tsat @ 200 kPa = 120.23C, the cold water and the exit mixture streams exist as a compressed liquid, which can be approximated as a saturated liquid at the given temperature. From tables: 31

Solution 1 Analysis: (a) We take the mixing chamber as the system, which is a control volume. The mass and energy balances for this steady-flow system can be expressed in the rate form as: Mass balance: Energy balance: Combining the two relations gives and substituting, the mass flow rate of the superheated steam is determined to be 32

Solution 1 Also, (b) The rate of total entropy generation during this process is determined by applying the entropy balance on an extended system that includes the mixing chamber and its immediate surroundings so that the boundary temperature of the extended system is 25C at all times. It gives

Solution 1 Substituting, the rate of entropy generation during this process is determined to be

Problem 2 Steam is to be condensed in the condenser of a steam power plant at a temperature of 50°C with cooling water from a nearby lake, which enters the tubes of the condenser at 18°C at a rate of 101 kg/s and leaves at 27°C. Assuming the condenser to be perfectly insulated and the operation is steady state, determine (a) the rate of condensation of the steam and (b) the rate of entropy generation in the condenser. Steam (Sat. vapor) T3=50C T1=18C Water T4=50C (Sat. liquid) T2=27C

Solution 2

Solution 2

Solution 2 Then the heat transfer rate to the cooling water in the condenser becomes The rate of condensation of steam is determined to be

Solution 2

Solution 2

Problem 3 A well-insulated heat exchanger is to heat water (Cp= 4.18 kJ/kg · °C) from 25°C to 60°C at a rate of 0.50 kg/s. The heating is to be accomplished by geothermal (Brine) water (Cp= 4.31 kJ/kg · °C) available at 140°C at a mass flow rate of 0.75 kg/s. Assuming steady state operation, determine: (a) the rate of heat transfer from hot to cold water, and (b) the rate of entropy generation in the heat exchanger. Water T1=25C Brine T3=140C T2=60C T4=?

Solution 3

Solution 3

Solution 3

Solution 3

Solution 3 Noting that both fresh and geothermal water are incompressible substances, the rate of entropy generation is determined to be

End of Tutorial