D. P. Mishra
2.3 THERMODYNAMIC LAWS
The various components of rocket engines can be analyzed using the laws of thermodynamics. The first, second, and third laws of thermo- dynamics are briefly discussed in the following. For the details of these laws and derivations, one can refer to some standard thermodynamic books [1–3].
2.3.1 First Law of Thermodynamics
The first law of thermodynamics states that energy can be neither created nor destroyed but only transforms from one form to another. In other words, whenever energy transfer takes place between a system and its sur- rounding, there will be a change in the system. To illustrate this, let us con- sider gas confined in a piston cylinder as a closed mass system, as shown in Figure 2.1. Let δQ be the heat added to the system and δW be the work done by the system on the surrounding by the displacement of boundary.
The heat added to the system and work done by the system cause a change in energy in the system. According to the first law of thermodynamics, this is given by
dEt =δQ W dE dU dKE dPE−δ ; t = + + (2.1) where
dEt is the total energy content of the system
Et is the internal energy (U) + kinetic energy (KE) + potential energy (PE) Note that dEt is the exact differential that exists for state function and its value depends on the initial and final states of the system. In contrast, δQ and δW depend on the path followed by the process. Keep in mind that the first law of thermodynamics is an empirical relation between heat, work, and internal energy, first put forward by Joule in 1851 through conduct- ing a series of experiments. Of course, it has been verified thoroughly by subsequent experimental results. To date, nobody has managed to disprove this law.
Piston
System
CV
i e
(a) Energy (b)
FIGURE 2.1 Schematic of thermodynamic systems: (a) piston cylinder arrange- ment and (b) rocket engine.
2.3.2 First Law for Control Volume
In order to analyze various components of rocket engine, we need to con- sider a control volume (CV), as shown in Figure 2.1b. Hence the control volume approach rather than control mass approach is used for analyzing the various components of rocket engine, namely, combustion chamber and nozzle in the present book. Before applying the first law of thermody- namics, let us consider the assumptions being made for deriving this law:
• The control volume is fixed with respect to the coordinate system.
• No need to consider the work interactions associated with the moving body.
• Uniform flow conditions over the inlet and the outlet flow areas.
By invoking these assumptions, the first law of thermodynamics for the CV is given by
dE
dt =m h Vi
(
i+ i2/2+gZi)
−m h Vo(
o+ o2/2+gZo)
+ −Q W sh (2.2) wherem is the mass flow rate h is the enthalpy V is the velocity Z is the height E is the total energy
Q is the energy transfer rate as heat Wsh is the shaft power
This form of the first law of thermodynamics for CV is very useful for the analysis of components in the rocket engine. The reader can refer to certain standard thermodynamics books [3–5] to get acquainted with this impor- tant thermodynamic law.
2.3.3 Second Law of Thermodynamics
We know that the first law of thermodynamics does not say anything about the feasibility of direction in which the process may proceed. However, the second law of thermodynamics stipulates the direction of the process.
According to the famous scientist Kelvin Plank, the second law of ther- modynamics states that it is impossible to construct a cyclically operat- ing device such that it produces no other effects than absorbing energy as
heat from a single thermal reservoir and performs an equivalent amount of work. In other words, it is impossible to have a heat engine with thermal efficiency of 100%. Further, the second law of thermodynamics defines an important property of a system, known as entropy, which is expressed as
dS Q
=δT
(
reversible process)
(2.3) where dS is the change of entropy of the system during an incremental reversible heat exchange δQ, when the system is at temperature T. The entropy being a state variable can be used either for reversible or irrevers- ible process. An alternative, more general, relation for the entropy isdS Q T dSirrev
=δ +
(
irreversible process)
(2.4) whereδQ is the actual amount of heat added to the system during which entropy change
dSirrev is generated due to the irreversible, dissipative phenomena
Note that irreversible processes are caused due to friction, heat transfer with finite temperature gradient, and mass transfer with finite concentra- tion gradient. These irreversible processes due to their dissipative nature always result in increase of the entropy as given in the following:
dSirrev≥0 (2.5)
Combining Equations 2.4 and 2.5, we have dS Q
≥ δT (2.6)
For adiabatic process (δQ = 0), Equation 2.6 becomes
dS≥0 (2.7)
Note that this expression derived from the second law of thermodynamics (Equation 2.7) indicates the direction in which the process can proceed. A process can proceed either in the direction of increasing entropy or constant entropy of the system and its surrounding. The process cannot proceed in
the direction of decreasing entropy. Engineers attempt to reduce the irre- versibility to a large extent to enhance the performance of any practical device. The change in specific entropy of a system, ds, can be determined by assuming the heat interaction to be taking place in reversible manner by using the following relation:
δq Tds= (2.8) where
q is the heat per unit mass s is the entropy per unit mass
Substituting Equation 2.8 in the energy equation (Equation 2.1), we get
Tds du Pdv= + (2.9)
From the definition of enthalpy, we have
dh du Pdv vdP= + + (2.10) For thermally perfect gas, we can assume dh = Cp dT. Substituting this rela- tion in Equation 2.10 and using Equation 2.9, we get
ds C dT T R dP
p P
= − (2.11)
For an isentropic (reversible and adiabatic) process, ds = 0, we can inte- grate Equation 2.11 between state (1) and (2):
P P
T T
C Rp 2
1 2 1
=
/
(2.12) However, for a calorically perfect gas, specific heat can be expressed in terms of specific heat ratio γ, as follows:
C R
p =
− γ
γ 1 (2.13)
Using Equation 2.13, we may express Equation 2.12 in the form P
P T T
2 1
2 1
= 1
−
γ
γ (2.14)
By employing the perfect gas law, we can write the isentropic relation as P
P
T
2 T
1 2 1
2 1
= 1
=
− ρ
ρ
γ γ
γ (2.15)
We will use this equation very often while dealing with rocket engine. This expression for isentropic process can be easily applicable for the flow out- side the boundary layer. As in the case of nozzle flow, since the thickness of boundary layer is very thin in comparison to entire flow domain, one can easily analyze it by assuming the flow to be isentropic except in its bound- ary layer. Thus, this isentropic relation (Equation 2.15) can be used for the analysis of a wide range of practical problems in rocket engine.