 Ch 6. Entropy th Multimedia Engineering Thermodynamics Entropy Tds Relations EntropyChange IsentropicProcess IsentropicEfficiency EntropyBalance (1) EntropyBalance (2) ReversibleWork
 Chapter 1. Basics 2. Pure Substances 3. First Law 4. Energy Analysis 5. Second Law 6. Entropy 7. Exergy Analysis 8. Gas Power Cyc 9. Brayton Cycle 10. Rankine Cycle Appendix Basic Math Units Thermo Tables Search eBooks Dynamics Statics Mechanics Fluids Thermodynamics Math Author(s): Meirong Huang Kurt Gramoll ©Kurt Gramoll THERMODYNAMICS - THEORY

In the second law analysis, it is useful to plot the process on diagrams for which has one coordinate is entropy. The two diagrams commonly used in second law analysis are temperature-entropy (T-s) and enthalpy-entropy (h-s) diagrams. For some pure substance, like water, the entropy is tabulated with other properties.

The T-s Diagram The Total Heat Transfer Equals the Total Area under the Process Curve on the T-s Diagram
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On a P-v diagram, the area under the process curve is equal, in magnitude, to the work done during a quasi-equilibrium expansion or compression process of a closed system. On a T-s diagram, the area under an internally reversible process curve is equal, in magnitude, to the heat transferred between the system and its surroundings. That is, Note the area has no meaning for irreversible processes. T-s Diagram
of a Carnot Cycle

The T-s diagram of a Carnot cycle is shown on the left. The area under process curve 1-2 (area 1-2-B-A-1) equals the heat input from a source (QH). The area under process curve 3-4 (area 4-3-B-A-4) equals the heat rejected to a sink (QL). The area enclosed by the 4 processes (area 1-2-3-4-1) equals the net heat gained during the cycle, which is also the net work output. How to Use the Mollier Diagram
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 Mollier Diagram for Water: SI Units Mollier Diagram for Water: US Units
The h-s Diagram

The enthalpy-entropy (h-s) diagram is valuable in the analysis of steady-flow devices such as nozzle, turbine, and compressor.

The h-s diagram is also called Mollier diagram after the German scientist R. Mollier. The Mollier diagram of water is shown on the left. The Mollier diagram for water contains constant-quality lines, constant-pressure lines, and constant-temperature lines. The temperature lines in the mixture region are straight.

The Tds Relations The Tds Relations for Closed System

In the previous section, the definition of entropy is given by Rearranging the above equation gives (1)

The entropy change during an internally reversible process (1-2) is Only when the relation between δQ and T is known, the entropy change can be determined. The relations between δQ and T can be found by considering the energy balance of a closed system.

The differential form of the energy balance for a closed system, which contains a simple substance and undergoes an internally reversible process, is given by

dU = δQrev - δWrev             (2)

The boundary work of a closed system is

δWrev = PdV                     (3)

Substituting equations (1) and (3) into equation (2) gives

dU = TdS- PdV
TdS = dU + PdV

or

Tds = du +Pdv                   (4)

where
s = entropy per unit mass

Equation (4) is known as the first relation of Tds, or Gibbs equation. The Tds Relations for Open System

The definition of enthalpy gives

h = u + Pv

differential the above equation yields

dh = du +Pdv + vdP

Replacing du + Pdv with Tds yields

dh = Tds + vdP
Tds = dh -vdP                   (5)

Equation (5) is known as the second relation of Tds.

Although the Tds equations are obtained through an internally reversible process, the results can be used for both reversible or irreversible processes since entropy is a property.

Rewriting equations (4) and (5) in the following form

ds = du/T + Pdv/T
ds = dh/T + vdP/T

The entropy change during a process can be determined by integrating the above equations between the initial and the final states.