Thermodynamics Statements , Definitions and Formulae

A thermodynamic system is a collection of matter that is enclosed by a closed boundary(system boundary).

A closed system is one in which matter does not cross the system boundary.

An open system is one in which matter crosses the system boundary.

Systems can have two types of boundary interactions. These are activities that allow work or heat to cross the system boundary.

A heat interaction happens due to a temperature difference across the boundary.

A work interaction  can be due to the action of a force or a torque , addition of electrical charge , magnetization etc. In theory , any kind of work interaction could be fully converted into another kind of work interaction by using a suitable device such as a motor (electrical work to rotational mechanical work) , a pulley attached to a rotating shaft (rotational mechanical work to work by linear displacement of a force) or any such arrangement.

The state of a system refers to the overall condition of a system. If the state of a system is known , all the physical properties of the system are known.

For a pure substance , it is enough to know two suitable properties to determine its thermodynamic state. This rule is known as the two-property rule.

In thermodynamics , all pure chemical substances are pure substances. Uniform mixtures of fluids , liquid solutions and alloys can also be treated as pure substances.

Intensive and extensive properties of a system :

Any property of a system that can be defined at a point within the system is an intensive property.          A property that can be defined only for a whole system is an extensive property.

Temperature(T) , pressure(p) and specific volume(v) , viscocity(µ) , are examples of intensive properties.

Mass(m,M) , and volume of a system are extensive properties. 

A simple thermodynamic system(simple system) is one in which only the energy stored in the form of internal energy (heat energy or sometimes heat energy + chemical energy) is important.

This is because the change in energy contained in other forms like kinetic , potential , electrical and elastic energy is negligible.

A process is an activity that causes the state of a system to change. 

Process usually involve work and/or heat interactions and sometimes no interaction.

To be able to determine theoretically the heat and work interaction during a process , the state of a system should be known during the whole process. This is made possible in some thermodynamic processes by assuming the process to be quasi-static(almost static and in equilibrium for all practical purposes). A quasi-static expansion (or compression of a fluid) is also a fully resisted process.  

A cyclic process( cycle , thermodynamic cycle) is one in which a system is returned to its initial state at the end of the process. A cyclic process may consist of a sequence of processes.

Two systems are in thermal equilibrium if no heat will flow between them when they are brought into contact. The zeroth law of Thermodynamics states that if each of two systems can be separately at thermal equilibrium with a third , then they can be at equilibrium with each other.

From this law we learn that systems in equilibrium have a property in common and we identify this property as temperature.

Heat interaction across a system boundary take place only when there is a temperature difference across the system boundary.

The First law of Thermodynamics can be stated in several ways. The following applies to a closed system :

If a system is taken through a cyclic process the algebraic sum of the heat received by the system and the work done on the system is zero.

So , is not possible to have a device that will continue to produce work without receiving any form of energy.(Such a device is called a perpetual motion machine of first kind or PMM1)   

The First law leads to another important result that the internal energy , U of a system is a property of the system and therefore its value depends only on the state of the system.
For a closed system ;

ΔU = Q +W

Where ΔU is the change in internal energy of the system and Q and W are , respectively , the heat supplied to it and the work done on it.

For a process that takes the system from state 1 to state 2 the equation becomes ;

U₂ – U₁ = Q_1-2 + W_1-2 

In a cyclic process , the state at the end of the process is the same as that at the start. thus we have ;

φ dQ + φ dW = 0  

Or simply ;                                                      Q + W = 0

where Q and W are the heat and work going into the system during the cycle.

In an open system , equations are written for the conservation of matter and conservation of energy( A modified version of the first law equation for closed system).

M_f –M_i = ∑ m_j         ( equation of continuity)

Where M is mass inside the system. subscripts i and f refer to the start and end of the period of flow considered. subscript j refers to the stream through which flow occurs. 

E_f - E_i = Q + W + ∑m_j (h_j - c_j²/2 + gz_j)

(unsteady flow energy equation , UFEE) When the energy (E) inside the system is only internal energy(U) , the equation is written as , 

 U_f - U_i = Q + W + ∑m_j (h_j - c_j²/2 + gz_j)

Where m , h , c , g and z refer to the mass crossing the system boundary at a given location , the specific enthalpy , velocity , acceleration due to gravity and elevation above a reference level.


Enthalpy (H = U + pV) is the sum of internal energy and 'flow work' (pV) , the work to transfer the fluid across the system boundary. specific enthalpy (h =pv) is enthalpy per unit mass.


When a flow is steady ,  U_f  =  U_i  and the UFEE changes to ;
Q + W + ∑m_j (h_j + c_j²/2 + gz_j) = 0    (steady flow energy equation , SFEE)
The continuity equation , the UFEE and SFEE becomes fewer. For a single stream through which mass m enters an open system ;

M_f - M_i = m

U_f - U_i = Q + W + m (h + c²/2 + gz) 

The equations are used in problems of charging and discharging of pressure vessels.

For steady flow arrangement with two streams (one inlet and one outlet) , if matter entering the leaving is m , the SFEE is ;

m { (h₂ – h₁) + (c₂² – c₁²) / 2 + g (z₂ – z₁) } + Q + W = 0

Work

In mechanics , work is defined in mechanics as the scalar product of force and displacement of its point of application.

Thermodynamics recognizes other forms of work too. Work is defined as any boundary interaction whose entire effect could be converted fully into that of a force undergoing displacement. 

Boundary expansion work is important in thermodynamics : If the volume V of closed system S at pressure p increases by dV , the boundary expansion work done on the system is ;

dW = -p dV

For a process taking a system from state 1 to state 2 ;

W_1-2 = - ₁∫² p dV

For a quasi-static process taking a system from state 1 to state 2 , and following the law                 pVⁿ = constant ;

W_1-2 = (p₂V₂ – p₁V₁) / (n-1)

(take care about the case of n=1).

If the relationship between p and V is in the form of a table or graph a suitable numerical or graphical method may be used.

For a cyclic process ;

 φ dQ + φ dW = 0

Working fluids:

For a perfect gas , the equation of state is ;

 pV = mRT

For a quasi-static process following the law ;

pVⁿ = constant

the work done during process 1-2 is ;

W_1-2 = mR(T₂ – T₁) / (n-1)

And ;

W_1-2 = mRT ln(V₂/V₁)

For an isothermal process.

For real gases and vapors , the equation of state may be written in a suitable mathematical form , or a table of properties or a property chart may be used.

Joule’s law states that the internal energy of a perfect gas is a function of its temperature only. In a simple system comprising a gas , the perfect gas assumption is used and , taking specific heat at the constant volume , C_v to be constant ,

U₂ – U₁ = m(u₂ – u₁) = mC_v(T₂ – T₁)

(for a gases with variable specific heat , a formula for C_v or a table of properties will be used.)

The specific heat at constant pressure , C_p is given by ;

C_p = C_v + R

The ratio of specific heats ,

  γ = C_p / C_v

(γ is taken as constant in many gas process.)

Solids and liquids are much less compressible than gases and their expansion due to heat is also smaller. Often , we neglect their change in volume and their internal energy is considered to be a function of temperature only.

Vapours are gaseous substances that are easily liquefied by compression. In engineering , we deal with vapour containing droplets of liquid. Here , the liquid and vapor parts are in equilibrium. For a pure chemical substance , there is a unique relationship between the pressure and temperature under such conditions. So , we need a property besides temperature or pressure to know the state. No such property is easy to measure in practical situations. A special property called dryness fraction (symbol x) is used for a pure substance that is part liquid and part vapour (wet vapor).

Steam with water droplets is wet steam. Steam at saturation temperature without water is dry saturated steam or saturated steam. Water at saturation temperature is saturated water.

Properties of wet steam like volume (V , v) , internal energy (u , U) , enthalpy (H , h) and entropy (S , s)can be given in the form ;

ɸ = (1-x) ɸ_f  +  x ɸ_g

Where ɸ is the property of the wet steam and subscripts f and g refer to the saturated liquid and saturated vapor.

Vapor at temperature higher than saturation temperature is super-heated vapor. Other properties are given in the tables as functions of temperature and pressure.

Liquid at temperature less than saturation temperature is sub-cooled liquid. Properties of liquid can be taken to be approximately those corresponding to saturated liquid at the temperature of the liquid.