Second Law
The second law of thermodynamics states that the total entropy of a system can not decrease. This statement applies to systems which are independent from other systems. There are two important historical statements which helped to establish the second law of thermodynamics.
Kelvin-Planck Statement
It is impossible to create a cycle that solely transfers heat into work. In other words, it is impossible to have a heat engine with a thermal efficiency of 100%.
Clausius Statement
It is impossible to create a cycle that solely transfers heat from a lower temperature thermal reservoir to a higher temperature thermal reservoir. In other words, it is impossible to create a heat pump/refrigeration cycle without any input work/energy.
Second Law for Cycles
The ideas of reversibility, efficiency, and thermal reservoirs are most important when discussing the second law for cycles. Entropy is a property more clearly defined when applying the second law to processes. Combining the second law for cycles and processes allows for entropy analysis of systems.
Thermal Efficiency and Coefficient of Performance
Efficiency is a measure of how much useful energy can be extracted from a system for a given input energy. Efficiency is commonly expressed as a percentage. We must keep in mind what effect you desire out of a system. For instance, the useful energy out of a heat engine is work, while the input energy is heat. Conversely, the useful energy for a refrigeration cycle or heat pump is to move heat, while the input is work. When using heat to create work we measure efficiency as thermal efficiency, \eta. When refrigerating we use a coefficient of performance, \beta. Likewise, when heating a space with a heat pump we use a similar coefficient of performance, \beta^\prime.
\eta=\dfrac{\text{Benefit}}{\text{Cost}}=\dfrac{\dot{W}_{net}}{\dot{Q}_{in}}
\beta=\dfrac{\text{Benefit}}{\text{Cost}}=\dfrac{\dot{Q}_{cooling}}{\dot{W}_{compressor}}
\beta^\prime=\dfrac{\text{Benefit}}{\text{Cost}}=\dfrac{\dot{Q}_{heating}}{\dot{W}_{compressor}}
Reversibility
A process is reversible when it can be performed in reverse, leaving no trace of the original process and returning the system to its original state. Reversible processes are nearly impossible in real life due to factors such as friction, mixing, and heat transfer. The reversible process is considered the theoretical ideal process, acting as the limiting case for the efficiency of a cycle. Any cycle must have an efficiency less than or equal to the efficiency of the reversible version of that cycle.
Carnot Cycle
The Carnot cycle is a theoretical heat engine or heat pump which is constructed solely of reversible processes. As such the Carnot cycle is the cycle with the highest possible efficiency. The Carnot cycle operates between two thermal reservoirs, a high temperature reservoir of temperature T_H, and a low temperature thermal reservoir of temperature T_L. Application of the first law on a Carnot cycle shows that work is the difference between heat flow in and heat flow out of the cycle.
W=Q_H-Q_L
The efficiency of a Carnot cycle is solely dependent on the temperature of the relevant thermal reservoirs, T_L and T_H. For any real cycle the thermal efficiency or COP must be less than the Carnot efficiency/COP.
\eta<\eta_{rev}
\beta<\beta_{rev}
\beta^\prime<\beta^\prime_{rev}
In a Carnot cycle the heat transferred to/from the thermal reservoirs is directly related to the temperature of the reservoirs, in Kelvin.
\left(\dfrac{Q_H}{Q_L}\right)_{rev}=\dfrac{T_H}{T_L}
Therefore the Carnot cycle efficiency/COP can be calculated using the temperatures of the thermal reservoirs.
\eta=\dfrac{W}{Q_H}=\dfrac{T_H-T_L}{T_H}
\beta=\dfrac{Q_L}{W}=\dfrac{T_L}{T_H-T_L}
\beta^\prime=\dfrac{Q_H}{W}=\dfrac{T_H}{T_H-T_L}
Second Law for Processes
Entropy describes the tendency for systems to become disordered over time. The amount of thermal energy that can be converted into work will decrease over time, entropy is a measure of how much thermal energy is unavailable. Entropy is measured in \frac{J}{K}, and is a property of substances that can be found in tables.
Inequality of Clausius
If we consider a Carnot heat engine we find that the following relation is true:
\displaystyle\oint\delta Q=Q_H-Q_L>0
Because of the relation between Q_H, Q_L, T_H, and T_L for reversible cycles, it is clear that:
\displaystyle\oint\dfrac{\delta Q}{T}=\dfrac{Q_H}{T_H}-\dfrac{Q_L}{T_L}=0
For an irreversible heat engine:
Q_{L, irr}>Q_{L, rev}
As such,
\dfrac{Q_{H}}{T_H}<\dfrac{Q_{L, irr}}{T_L}
Therefore:
\boxed{\displaystyle\oint\dfrac{\delta Q}{T}=\dfrac{Q_H}{T_H}-\dfrac{Q_{L, irr}}{T_L}<0}
A similar proof exists for refrigeration cycles, therefore leading to the Inequality of Clausius:
\displaystyle\oint\dfrac{\delta Q}{T}=0\to\text{reversible}
\displaystyle\oint\dfrac{\delta Q}{T}<0\to\text{irreversible}
\displaystyle\oint\dfrac{\delta Q}{T}>0\to\text{impossible}
Entropy
It can be shown that the integral \oint\frac{\delta Q}{T} for a reversible system is independent of path, proving that it is a property based on the state of a substance. In fact, this property is called entropy, S, and has the following relation:
\displaystyle S_2-S_1=\int_1^2 \left(\dfrac{\delta Q}{T}\right)_{rev}
Similar to specific volume and specific work we can define specific entropy as:
s=\dfrac{S}{m}
It is also evident that heat transfer during a process can be defined by entropy.
\displaystyle Q_{1\to 2}=\int_1^2 TdS
Therefore, just as specific work can be thought of as the area under a P–\nu curve, heat can be thought of as the area under a T–S curve.
Finally, using the first law we can relate all substance properties to one another with whats called the Gibbs equation:
TdS=dU+PdV
TdS=dH-VdP
Entropy for Pure Substances
Solids and Incompressible Liquids:
d\nu = 0 and du=CdT
Tds=CdT
s_2-s_1=C\ln\left(\dfrac{T_2}{T_1}\right)
Compressible Liquids:
s\approx s_f
Saturated Mixtures:
s=s_f+xs_{fg}
Superheated Vapours:
Look up in relevant substance tables.
Ideal Gases:
du=C_{v0}dT, dh=C_{p0}dT, and P\nu=RT
Tds=C_{v0}dT+Pd\nu
s_2-s_1=C_{v0}\ln\left(\dfrac{T_2}{T_1}\right)+R\ln\left(\dfrac{\nu_2}{\nu_1}\right)
Tds=C_{p0}dT-\nu dP
s_2-s_1=C_{p0}\ln\left(\dfrac{T_2}{T_1}\right)-R\ln\left(\dfrac{P_2}{P_1}\right)
Ideal gas processes
| Polytropic Exponent | Condition | Result |
|---|---|---|
| n=0 | Isobaric | P=\text{constant} |
| n=1 | Isothermal | T=\text{constant} |
| n=k=\dfrac{C_{p0}}{C_{v0}} | Isentropic | s=\text{constant} |
| n=\infty | Isochoric | \nu=\text{constant} |
Overall Entropy Change
The entropy change for an irreversible process must be larger than that for a reversible process. Therefore, it is clearly true that the following holds for an irreversible process:
\displaystyle S_2-S_1\ge\int_1^2\dfrac{\delta Q}{T}
As such, we will define a term called entropy generation, S_{gen}. Entropy generation represents the entropy created by the factors leading to irreversibility, such as friction and mixing.
\displaystyle S_2-S_1=\int_1^2\dfrac{\delta Q}{T}+S_{gen}
For a control mass, denoted with subscript c.m., the entropy generation can be defined as:
S_{gen}=m_{c.m.}(s_2-s_1)_{c.m.}-\dfrac{Q_{1\to 2}}{T_{surr}}