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Isentropic Duct Flows
In this section we examine the behavior of isentropic flows, continuing the development of the relations in section (Bob). First it is important to identify the set of basic relations that will be used for isentropic, reversible flow and those that are relevant to non-isentropic, irreversible flows that will be the subject of the next section (Bof). In the latter case, one-dimensional flows are governed by equations for
- Continuity
- Momentum
- Energy
- State
and these are to be solved for the four unknowns, u, p, ρ and T with specific versions of the equations described in section (Bof). However, with steady, reversible, isentropic flows, the momentum and energy equations have already been used in section (Bob) to derive the isentropic relations and hence the simpler set of four equations that will be further utilized in this section to document the steady, isentropic flow of velocity, u, pressure, p, temperature, T , and density, ρ, in a duct of cross-sectional area, A(s), are
- Continuity: ρuA = constant
- Energy: total enthalpy, h∗^ = constant which becomes cpT + u^2 /2 = constant
- State: p = ρRT
- Isentropic Relation: pρ−γ^ = constant
Reworked using the definition of the speed of sound and the Mach number, M = u/c, this set of equations can be written as Continuity: ρuA = constant (Boe1)
Energy: T
(γ − 1) 2
M^2
= constant = T 0 (Boe2)
State and Isentropic Relations:
p
(γ − 1) 2
M^2
}γ/(γ−1) = constant = p 0 (Boe3)
ρ
(γ − 1) 2
M^2
} 1 /(γ−1) = constant = ρ 0 (Boe4)
where T 0 , p 0 and ρ 0 are called stagnation or reservoir reference quantities since they pertain to conditions where the velocity is zero.
To further develop these relations we write the continuity equation (Boe1) as
ρuA = ρMcA = ρM(γRT )
1 (^2) A = constant (Boe5)
and therefore, using the energy equation (Boe2) to substitute for T ,
1 AM
[
(γ − 1) 2
M^2
](γ+1)/2(γ−1) = constant (Boe6)
Thus we can compare two locations along the duct denoted by the subscripts 1 and 2 to write
A 1 A 2
M 2
M 1
[
1 + (γ − 1)M 12 / 2 1 + (γ − 1)M 22 / 2
](γ+1)/2(γ−1) (Boe7)
which directly relates the areas and Mach numbers of the flows at those two locations.
We have already introduced one reference state, namely the stagnation or reservoir state and, in assessing or calculating isentropic duct flows, it is often convenient to evaluate that reference state as a part of the calculation whether or not that particular state actually occurs in the flow. In other words we evaluate T 0 , p 0 and ρ 0 whether or not there is a stagnation point or reservoir in the flow. Furthermore, there is a second reference state which is useful to establish namely the state at the point where the Mach number is unity and the flow is sonic. We shall later discover the physical relevance of this possible location or reference point. For the present, we only need establish that if there is a location within the duct where M = 1 then the pressure, temperature, density and area at that location are denoted by p∗, T ∗, ρ∗^ and A∗^ respectively. For reasons that will emerge later this state is called the throat reference state and the conditions at that location are termed the throat pressure, temperature and density. By setting M = 1 in equations (Boe2) to (Boe4) the throat conditions can be established as
p∗ p 0
(γ + 1)
}γ/(γ−1) ;
T ∗
T 0
(γ + 1)
ρ∗ ρ 0
(γ + 1)
} 1 /(γ−1) (Boe8)
and for air with γ = 1.4 these yield p∗/p 0 = 0.528, T ∗/T 0 = 0.634 and ρ∗/ρ 0 = 0.833. Furthermore, it is often useful to relate the conditions at some arbitrary location given by a Mach number, M, to the throat conditions and the relations for this purpose also follow from equations (Boe2) to (Boe4) and (Boe7) as
T T ∗^
[
(γ + 1) 2
] [
(γ − 1) 2
M^2
]− 1
(Boe9)
p p∗^
[
(γ + 1) 2
]γ/(γ−1) [ 1 +
(γ − 1) 2
M^2
]−γ/(γ−1) (Boe10)
ρ ρ∗^
[
(γ + 1) 2
] 1 /(γ−1) [ 1 +
(γ − 1) 2
M^2
]− 1 /(γ−1) (Boe11)
A
A∗^
= M−^1
[
(γ − 1) 2
M^2
](γ+1)/2(γ−1) [ γ + 1 2
]−(γ+1)/2(γ−1) (Boe12)
u u∗^
ρ∗ ρ
A∗
A
(Boe13)
where the last result follows from the continuity equations and equations (Boe11) and (Boe12).
In Figures 1 and 2 the quantities A/A∗, T /T 0 , p/p 0 , ρ/ρ 0 , T /T ∗, p/p∗, ρ/ρ∗, and u/u∗^ are tabulated against M for air (γ = 1.4, R = 280 m^2 /s^2 Ko). The values of T /T 0 , p/p 0 , and ρ/ρ 0 are also plotted against M in Figure 3 and the value of A/A∗^ is plotted in Figure 4.
Figure 2: Table of ratios for isentropic duct flows.
Figure 3: Graphs of T /T 0 , p/p 0 , and ρ/ρ 0 against Mach Number, M.
Figure 4: Graph of A/A∗^ against Mach Number, M.
