The terms in Eq. 1.47 are as follows:
Pressure p is in N/m2 or Pa.
Volume V is in m3.
Mass m is in kg.
Temperature T is in K.
Specific volume v is in m3/kg.
Molar amount of gas n is in kmole (1 kmole of any gaseous substance occupies a volume of 22.41 m3 at the standard temperature 0°C and pressure 101.325 Pa).
Gas constant in J/kg. K.
Molecular mass of any gas μ is in kg/kmole.
Universal gas constant
1.3.3.1 Adiabatic Processes
An adiabatic process is one where the energy of the system changes only by means of work transfer, and there is no heat crossing the boundary. The relationship between the state properties can be written as
(1.48)
The volume V can be replaced by the specific volume v = V/m, which yields the additional equation p1/p2 = (v2/v1)γ. The exponent γ is the ratio of specific heat capacities. The specific heat capacity (the word capacity will be dropped in future references) is defined as the amount of heat energy required to raise the temperature of a unit quantity of matter by one degree Celsius (on a mass basis c in J/kg. K and on a mole basis C in J/kmole. K). The specific heat at constant pressure is written as cp in J/kg. K or Cp in J/kmole. K and at constant volume as cv in J/kg. K or Cv in J/kmole. K. Both cp and cv increase with temperature. Table A.1 in Appendix A shows the molar specific heats at constant pressure of some gases as a function of temperature. The specific heat at constant volume can be determined from the following equations, assuming the gases behave as ideal gases:
1.3.3.2 Heat‐Only Process
A hot object tends to cool to the temperature of colder surroundings, and a cold object is warmed to the temperature of hotter surroundings. The phenomenon is caused by the heat‐transfer process, in which the energy of a system changes while no work is done on or by the system (no work exchange with the surroundings). This process can occur with the volume remaining constant, and the change of energy in the system for an ideal gas is then
(1.49)
ΔU is the internal energy of the system.
If the pressure remains constant during the process, the change of energy in the system is
(1.50)
The source of the heat in both these cases could be external or internal. Examples of processes with internal heat sources are the spark ignition engine (constant‐volume combustion of the fuel) and the gas turbine combustor (constant‐pressure combustion of the fuel).
1.3.3.3 Isothermal Process
An isothermal process takes place at constant temperature, and the equation of state can be written as
(1.51)
1.3.3.4 Isochoric Process
An isochoric process is a constant‐volume process for which the equation of state is reduced to
(1.52)
1.3.3.5 Polytropic Process
A process is referred to as polytropic when it deviates from the adiabatic as a result of heat crossing the boundary in addition to work. The relationship is similar to the adiabatic with the adiabatic exponent γ replaced by a polytropic exponent n (n < γ):
(1.53)
The volume V can also be written in terms of specific volumes v = V/m, which yields the additional equation p1/p2 = (v2/v1)n.
1.3.4 Cycles
When a fluid undergoes a series of processes and then returns to its initial state, the fluid executes a thermodynamic cycle. A cycle that consists only of reversible processes is a reversible cycle.
To illustrate the application of some of the processes discussed earlier when calculating cycles, let us consider the theoretical thermodynamic cycle shown in Figure 1.8, which is known as the Diesel cycle in honour of the German inventor Rudolf Diesel.
Figure 1.8 Application of process equations in theoretical cycles: (a) Diesel cycle; (b) calculation scheme for compression and expansion work.
The cycle has two heat‐only processes and two polytropic processes:
Process 2 − 3: Constant‐pressure heat addition Qin = mcp(T3 − T2).
Process 4 − 1: Constant‐volume heat rejection Qout = mcv(T4 − T1).
Process 1 − 2: Adiabatic compression PVγ = C (C is the constant in Eq. 1.48).
The magnitude of the compression work done on the gas can be determined as shown in Figure 1.8b:
