where ρ is the density and υ is the specific volume (that is, the inverse of the density).
For example, the density of air is ρ ¼ 1.29kgm
À3 at a pressure of one atmosphere
(equal to 1.013 Â 10
5 Nm
À2 ) and at a temperature of 273 K according to Eq. (1.3).
1.3.2 The First Law of Thermodynamics
The first law of thermodynamics states that the energy of an isolated system is
conserved; however, the energy may be transformed from one form to another but
the energy cannot be created nor destroyed. The mathematical form of the first law of
thermodynamics for an infinitesimal change is given by the equation,
dQ ¼ dE þ dW,
ð1:4Þ
where dQ is the quantity of heat exchanged between the system and its surroundings,
dE
2 is the change in the internal energy of the system and dW is the work done. For a
system of constant mass m that exerts a uniform pressure p on its surroundings, the
first law can be written as
dQ ¼ mde þ pdV,
where de is the change in the specific internal energy, that is, the change in the
internal energy per unit mass and dV is the change in volume. Dividing across by m,
gives,
dq ¼ de þ pdυ,
ð1:5Þ
where dq is the quantity of heat transferred per unit mass and dυ is the change in the
specific volume.
1.3.3 Heat Capacity
If a system undergoes a temperature increase of dT following a transfer of heat dq,
the heat capacity of the system is defined as
2 The symbol E is used to represent the internal energy of a system rather than the more commonly
used symbol U, as this symbol is used in this text to represent the velocity of air motion and the
velocity of shock waves.
4
1 Brief Outline of the Equations of Fluid Flow
For example, the density of air is ρ ¼ 1.29kgm
À3 at a pressure of one atmosphere
(equal to 1.013 Â 10
5 Nm
À2 ) and at a temperature of 273 K according to Eq. (1.3).
1.3.2 The First Law of Thermodynamics
The first law of thermodynamics states that the energy of an isolated system is
conserved; however, the energy may be transformed from one form to another but
the energy cannot be created nor destroyed. The mathematical form of the first law of
thermodynamics for an infinitesimal change is given by the equation,
dQ ¼ dE þ dW,
ð1:4Þ
where dQ is the quantity of heat exchanged between the system and its surroundings,
dE
2 is the change in the internal energy of the system and dW is the work done. For a
system of constant mass m that exerts a uniform pressure p on its surroundings, the
first law can be written as
dQ ¼ mde þ pdV,
where de is the change in the specific internal energy, that is, the change in the
internal energy per unit mass and dV is the change in volume. Dividing across by m,
gives,
dq ¼ de þ pdυ,
ð1:5Þ
where dq is the quantity of heat transferred per unit mass and dυ is the change in the
specific volume.
1.3.3 Heat Capacity
If a system undergoes a temperature increase of dT following a transfer of heat dq,
the heat capacity of the system is defined as
2 The symbol E is used to represent the internal energy of a system rather than the more commonly
used symbol U, as this symbol is used in this text to represent the velocity of air motion and the
velocity of shock waves.
4
1 Brief Outline of the Equations of Fluid Flow