### Thermodynamic

The Zeroth Law of Thermodynamics

This law states that if object A is in thermal equilibrium with object B, and object B is in thermal equilibrium with object C, then object C is also in thermal equilibrium with object A. This law allows us to build thermometers. For example the length of a mercury column (object B) may be used as a measure to compare the temperatures of the two other objects.

The First Law of Thermodynamics

Conservation of Energy

The principle of the conservation of energy states that energy can neither be created nor destroyed. If a system undergoes a process by heat and work transfer, then the net heat supplied, Q, plus the net work input, W, is equal to the change of intrinsic energy of the working fluid, i.e.

∆U = U2 - U1 = Q + W

where U1 and U2 are intrinsic energy of the system at initial and final states, respectively. The special case of the equation applied to a steady-flow system is known as steady-flow energy equation. Applying this general principle to a thermodynamic cycle, when the system undergoes a complete cycle, i.e. U1 = U2, results in:

∑Q + ∑W = 0

where:

∑Q= The algebraic sum of the heat supplied to (+) or rejected from (-) the system.

∑W= The algebraic sum of the work done by surroundings on the system (+) or by the system on surroundings (-).

Applying the rule to the power plant shown in figure below,

gives:

∑Q = Qin - Qout

∑W = Win - Wout

Qin + Win - Qout - Wout = 0

where,

Qin = Heat supplied to the system through boiler,

Win = Feed-pump work,

Qout = Heat rejected from the system by condenser,

Wout = Turbine work.

The Second Law of Thermodynamics

The second law of thermodynamics states that no heat engine can be more efficient than a reversible heat engine working between two fixed temperature limits (Carnot cycle) i.e. the maximum thermal efficiency is equal to the thermal efficiency of the Carnot cycle:

η < ηmax = ηc

or in other words If the heat input to a heat engine is Q, then the work output of the engine, W will be restricted to an upper limit Wmax i.e.

W < Wmax = Q ηc

It should be noted that real cycles are far less efficient than the Carnot cycle due to mechanical friction and other irreversibility. Related topic:

· Exergy

Exergy or Availability

Exergy of a system is defined as the theoretical maximum amount of work that can be obtained from the system at a prescribed state (P, T, h, s, u, v) when operating with a reservoir at the constant pressure and temperature P0 and T0. The specific exergy of a non-flow system is:

U + Po v – To s

and for a steady flow system:

h + C2/2 + z g – To S

where,

u= Specific internal energy,

h= Specific enthalpy,

v= Specific volume,

s= Specific entropy,

C= Velocity,

Z= Height of the system measured from a fixed datum,

g= Gravity constant.

Carnot Cycle

By using the second law of thermodynamics it is possible to show that no heat engine can be more efficient than a reversible heat engine working between two fixed temperature limits. This heat engine is known as Carnot cycle and consists of the following processes:

· 1 to 2: Isentropic expansion

· 2 to 3: Isothermal heat rejection

· 3 to 4: Isentropic compression

· 4 to 1: Isothermal heat supply

The supplied heat to the cycle per unit mass flow is:

Q1 = T1 ∆s

The rejected heat from the cycle per unit mass flow is:

Q2 = T2 ∆s

By applying the first law of thermodynamics to the cycle, we obtain:

Q1 - Q2 - W = 0

And the thermal efficiency of the cycle will be:

η= W/Q1 = 1 - T2/T1

Due to mechanical friction and other irreversiblities no cycle can achieve this efficiency. The gross work output of cycle, i.e. the work done by the system is:

Wg = W41 + W12

and work ratio is defined as the ratio of the net work, W, to the gross work output, Wg, i.e.

W / Wg

The Carnot cycle has a low work ratio. Although this cycle is the most efficient system for power generation theoretically, it can not be used in practice. There are several reasons such as low work ratio, economical aspects and practical difficulties.

Heat Engine

Heat engine is defined as a device that converts heat energy into mechanical energy or more exactly a system which operates continuously and only heat and work may pass across its boundaries.

The operation of a heat engine can best be represented by a thermodynamic cycle. Some examples are: Otto, Diesel, Brayton, Stirling and Rankine cycles.

Forward Heat Engine

LTER= Low Temperature Energy Reservoir

HTER= High Temperature Energy Reservoir

A forward heat engine has a positive work output such as Rankine or Brayton cycle. Applying the first law of thermodynamics to the cycle gives:

Q1 - Q2 - W = 0

The second law of thermodynamics states that the thermal efficiency of the cycle, , has an upper limit (the thermal efficiency of the Carnot cycle), i.e.

η < ηc < 1

It can be shown that:

Q1 > W

which means that it is impossible to convert the whole heat input to work and

Q2 > 0

which means that a minimum of heat supply to the cold reservoir is necessary.

Reverse Heat Engine

LTER= Low Temperature Energy Reservoir

HTER= High Temperature Energy Reservoir

A reverse heat engine has a positive work input such as heat pump and refrigerator. Applying the first law of thermodynamics to the cycle gives:

- Q1 + Q2 + W = 0

In case of a reverse heat engine the second law of thermodynamics is as follows: It is impossible to transfer heat from a cooler body to a hotter body without any work input i.e.

W > 0

Turbine

Turbines are devices that convert mechanical energy stored in a fluid into rotational mechanical energy. These machines are widely used for the generation of electricity. The most important types of turbines are: steam turbines, gas turbines, water turbines and wind turbines.

Steam Turbine

Steam turbines are devices which convert the energy stored in steam into rotational mechanical energy. These machines are widely used for the generation of electricity in a number of different cycles, such as:

· Rankine cycle

· Reheat cycle

· Regenerative cycle

· Combined cycle

The steam turbine may consists of several stages. Each stage can be described by analyzing the expansion of steam from a higher pressure to a lower pressure. The steam may be wet, dry saturated or superheated.

Consider the steam turbine shown in the cycle above. The output power of the turbine at steady flow condition is:

P = m (h1-h2)

where m is the mass flow of the steam through the turbine and h1 and h2 are specific enthalpy of the steam at inlet respective outlet of the turbine.

The efficiency of the steam turbines are often described by the isentropic efficiency for expansion process. The presence of water droplets in the steam will reduce the efficiency of the turbine and cause physical erosion of the blades. Therefore the dryness fraction of the steam at the outlet of the turbine should not be less than 0.9.

Thermodynamic Cycle

Thermodynamic cycle is defined as a process in which a working fluid undergoes a series of state changes and finally returns to its initial state. A cycle plotted on any diagram of properties forms a closed curve.

A reversible cycle consists only of reversible processes. The area enclosed by the curve plotted for a reversible cycle on a p-v diagram represents the net work of the cycle.

· The work is done on the system, if the state changes happen in an anticlockwise manner.

· The work is done by the system, if the state changes happen in a clockwise manner

State of Working Fluid

Working fluid is the matter contained within boundaries of a system. Matter can be in solid, liquid, vapor or gaseous phase. The working fluid in applied thermodynamic problems is either approximated by a perfect gas or a substance which exists as liquid and vapor. The state of the working fluid is defined by certain characteristics known as properties. Some of the properties which are important in thermodynamic problems are:

· Pressure(P)

· Temperature(T)

· Specific enthalpy(h)

· Specific entropy(s)

· Specific volume(v)

· Specific internal energy(u)

The thermodynamic properties for a pure substance can be related by the general relationship, f(P,v,T)=0, which represents a surface in the (P,v,T) space. The thermodynamic laws do not give any information about the nature of this relationship for the substances in the liquid and vapor phases. These properties may only be related by setting up measurements. The measured data can be described by equations obtained e.g. by curve fitting. In this case the equations should be thermodynamically consistent.

The state of any pure working fluid can be defined completely by just knowing two independent properties of the fluid. This makes it possible to plot state changes on 2D diagrams such as:

· pressure-volume (P-V) diagram,

· temperature-entropy (T-s) diagram,

· enthalpy-entropy (h-s) diagram.

Perfect Gas or Ideal Gas

Experimental information about gases at low pressures i.e. Charles's law, Boyle's law and Avogadro's principle may be combined to one equation:

P V=n R T

known as perfect gas equation. Where,

P= absolute pressure,

T= absolute temperature,

V= volume of the gas,

n= number of moles,

and R is a constant, known as gas constant.

R=8314.51 J/(kmol.K)

The surface of possible states, (P,V,T), of a fixed amount of a perfect gas is shown in figure.

Any gas that obeys the above mentioned equation under all conditions is called a perfect gas (or ideal gas). A real gas (or an actual gas), behaves like a perfect gas only at low pressures. Some properties of actual gases such as specific heat at constant pressure and specific enthalpy are dependent on temperature but the variation due to pressure is negligible. There are empirical relations that calculate gas properties. The following polynom is a good approximation for the specific enthalpy of gases:

h = R (a1 T + a2 T2/2 + a3 T3/3 + a4 T4/4 + a5 T5/5 + a6)

where a1 to a6 are constants depending only on the type of the gas. It should be noted that this formulation will agree with Joule's law and we obtain a set of thermodynamically consistent equations. The above equation can be used directly for calculation of specific heat capacity of the gas:

Cp = (∂h/∂t)p = R (a1 + a2 T + a3 T2 + a4 T3 + a5 T4)

By using the relationship:

∂s/∂T = Cp/T

The specific entropy of the gas, s, will be:

s = R (a1 ln(T) + a2 T + a3 T2/2 + a4 T3/3 + a5 T4/4 + a7 – ln(P/Po))

where a7 is a constant and P0 is a reference pressure. Related topics:

· Amagat's law

· Dalton's law

· Joule's law

· Gas turbine

· Compressor

Amagat's law of additive volumes

The volume of a gas mixture is equal to the sum of the volumes of all constituents at the same temperature and pressure as the mixture like this figure.

Dalton's law of additive pressures

The pressure of a gas mixture is equal to the sum of the partial pressure of the constituents. The partial pressure is that pressure which a constituent would exert if it existed alone at the mixture temperature and volume.

Joule's Law

Joule's law state that the internal energy of a perfect gas is a function of the temperature only, i.e.

u=f(t)

System

A system is a collection of matter within defined boundaries. The boundaries may be flexible. There are two types of system: closed systemand open system.

Closed System

In closed systems, nothing leaves the system boundaries. As an example, consider the fluid in the cylinder of a reciprocating engine during the expansion stroke. The system boundaries are the cylinder walls and the piston crown. Notice that the boundaries move as the piston moves.

Open System

In open systems there is a mass transfer across the system's boundaries; for instance the steam flow through a steam turbine at any instant may be defined as an open system with fixed boundaries.

Isobaric Process

An isobaric process is one during which the pressure of working fluid remains constant

Isothermal Process

An isothermal process is one during which the temperature of working fluid remains constant

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