Jet Propulsion/Fluid Mechanics

Adiabatic Processes
An adiabatic process is a thermodynamic process in which no heat is transferred to or from the working fluid. In an ideal gas turbine (Brayton cycle) the compression and expansion processes are adiabatic. We define &theta; as the pressure ratio in a process relating to the ambient conditions:


 * $$ \theta_2 = \left(\frac{P_2}{P_0}\right)$$

Then for adiabatic compression the temperature ratio &tau; is:


 * $$ \tau_2 = \frac{T_2}{T_0} = \theta_2^\frac{\gamma-1}{\gamma}$$

For air &gamma; = 1.4  so (&gamma;-1)/ &gamma;= 0.286 

A log-log plot simplifies the analysis for quick engineering calculations.


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Kinetic energy to fluid temperature

Changing the velocity of the fluid by pressure changes simultaneously changes the temperature. Compression work raises the apparent temperature of the fluid. We can relate Mach number to the internal energy of the fluid:

Aerodynamic Analysis
Aerodynamic analysis attempts to progressively analyze the flows in the aerodynamic stages. Practical design includes substantial theoretical, computational and experimental analysis.

The stagnation or total temperatures and pressures are needed to measure the energy additions in high speed gas flows that occur in gas turbines. Using the Mach number allows us to factor in the compressibility of the gas.

Compressible fluid flow equations
See [NACA 1135: Equations, Tables, and Charts for Compressible Flow].

Continuity
In a steady flow, for any two sections of the flow on a stream tube


 * $$ \rho_1 u_1 A_1 = \rho_2 u_2 A_2 $$

or in differential form


 * $$ \frac{d}{dx} (\rho u A) = 0 $$

or


 * $$ \frac{d\rho}{\rho} + \frac{du}{u} + \frac{dA}{A} = 0 $$

Where $$\rho$$ is the density of the fluid

$$u$$ is the internal energy per unit mass of the fluid

and

$$A$$ is the cross-sectional Area for the tube or channel

Momentum
The net force on the control volume matches the momentum change in the fluid


 * dp=-&rho; u du

Energy
The change in enthalpy is balanced by the change in kinetic energy


 * dh + u du=0

Where h is enthalpy per unit mass, u + pv, pv being the product of pressure and volume

Enthalpy
The enthalpy of a gas h at temperature T is


 * $$h = c_p T$$

where $$c_p$$ is the constant pressure specific heat of the gas. For air $$c_p$$ is about 1.005 kJ/kg K.

Entropy
Entropy change in a process can be expressed as


 * $$ s_2-s_1 = c_v \ln \left(\frac{T_2} {T_1}\right) - R \ln \left(\frac{\rho_2} {\rho_1}\right) $$

or in more conveniently terms of pressures


 * $$ s_2-s_1 = c_p \ln \left(\frac{T_2} {T_1}\right) - R \ln \left(\frac{P_2} {P_1}\right) $$

Stagnation temperature
Stagnation temperature is the temperature of the gas if it is brought to rest adiabatically. Adding the kinetic energy to the internal energy of the gas we get the relation


 * $$h_t = c_p T_t = c_p T + \frac{u^2}{2}$$


 * $$T_t = T + \frac{u^2}{2 c_p} $$

where Tt is the total(stagnation temperature of the flow.

The total enthalpy relation encapsulates the energy changes in isentropic compressors and turbines. To add energy to the flow the gas is put through a relative deceleration process against the compressor and diffuser surfaces and gains energy. To extract energy the gas is accelerated against nozzles and turbine buckets.

The Mach number of the flow is


 * $$ M = \frac{u}{a}=\frac {u}{\sqrt{\gamma R T}}$$

Substituting


 * $$T_t = T + \frac {\gamma R T M^2 }{ 2 c_p} $$

since  R= cp - cv  and  &gamma; = cp / cv 


 * $$T_t = T\left(1 + \frac {\gamma M^2 }{2} (c_p-c_v)/c_p\right) $$


 * $$\frac{T_t}{T}=\left(1 + \frac {\gamma-1}{2}M^2\right) $$

Where $$\gamma$$ (greek letter gamma) is the adiabatic expansion coefficent between pressure and volume $$p_1V_1^\gamma = p_2V_2^\gamma$$

This is the temperature if the gas is brought to rest adiabatically.

Isentropic stagnation pressure

 * $$\frac{p_t}{p}=\left(\frac{T_t}{T}\right)^{\gamma/(\gamma-1)} $$


 * $$\frac{p_t}{p}=\left(1 + \frac {\gamma-1}{2}M^2\right)^{\gamma/(\gamma-1)} $$

Isentropic stagnation density

 * $$\frac{\rho_t}{\rho}=\left(\frac{T_t}{T}\right)^{1/(\gamma-1)} $$


 * $$\frac{\rho_t}{\rho}=\left(1 + \frac {\gamma-1}{2}M^2\right)^{1/(\gamma-1)} $$

Duct flow
A steady inviscid adiabatic quasi-one dimensional flow obeys the following equations:

Differential continuity equation


 * d (&rho; u A) =0 

Differential momentum equation


 * dp=-&rho; u du

Differential energy equation


 * dh + u du=0

Rearranging continuity


 * $$\frac{d\rho}{\rho}+\frac{d u}{u}+\frac{d A}{A}=0$$

Rewrite momentum equation


 * $$\frac{d p}{\rho}=\frac{dp}{ d \rho }\frac{d \rho}{\rho} = -u du$$


 * $$\frac{dp}{ d \rho } \equiv \frac{\partial p}{\partial \rho}$$

The velocity of sound is:


 * a =(dp / d&rho;)1/2

Rearranging and substituting:


 * a2=(dp / d&rho;)


 * a2 d&rho; / &rho; = -u du


 * $$\frac{d \rho}{\rho}= -\frac{u du}{ a^2 }= -\frac{u^2}{ a^2 } \frac{du}{u} = -M^2 \frac{du}{u}$$

Substituting into continuity equation


 * $$M^2 \frac{du}{u}-\frac{d u}{u}-\frac{d A}{A}=0$$

We get the area velocity equation:


 * $$\frac{d A}{A} =(M^2-1) \frac{du}{u}$$

Thus for acceleration (positive du/u) the area must decrease for Mach numbers below 1 and increase for Mach numbers above 1.

The relationship between Mach number and duct area related to the throat area A* is:


 * $$\frac{A}{A^*}= \frac {1}{M} \left[ \frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M^2\right) \right]^{\frac{\gamma+1}{2(\gamma-1)}}$$

The temperature relation is


 * $$\frac{T}{T_t}= \left[ \frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M^2\right) \right]^{-1}$$

the pressure relation


 * $$\frac{p}{p_t}= \left[ \frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M^2\right) \right]^{-\frac{\gamma}{2(\gamma-1)}}$$

and the density relation


 * $$\frac{\rho}{\rho_0}= \left[ \frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M^2\right) \right]^{-\frac{1}{2(\gamma-1)}}$$

The figure below shows these relationships for air with &gamma; of 1.4.



A fully expanded gas would approach a Mach number of infinity as it's temperature drops to absolute zero.

Mass flow
The figure above shows this exchange for a fluid with &gamma;=1.4 undergoing an adiabatic expansion. Sonic velocity (Mach 1) is achieved when the pressure drops to 0.528 and the area for a particular mass flow is minimum at this Mach number. The flow at this condition is said to be choked and any further reductions in duct area will not produce acceleration of the stream. The mass flow per unit area is


 * $$ \dot{m} = A \rho v $$


 * $$ \dot{m} = A \rho_0 \left[ \frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M^2\right) \right]^{-\frac{1}{\gamma-1}} V $$

Nozzles
A nozzle converts internal energy of the gas into directed kinetic energy by expanding along a pressure gradient.

As the gas expands initially the volume increment is smaller than the velocity increment and the stream tube converges. A M=1 the effects balance and for M>1 the differential volume increase is greater than the velocity increase and a divergent stream is needed. The narrowest section of the nozzle is called the "throat".

Decreasing the pressure at the exit of a nozzle of fixed geometry increases the exit velocity until the velocity in the smallest section of the nozzle becomes sonic. The nozzle is then said to be "choked" and further reduction of the exit pressure has no effect on the flow upstream of the throat.

The maximum exit velocity depends on the energy content of the source gas.

Choked flow
Choked flow is the maximum flow that can pass through a passage for a given initial total conditions. Boundary layer effects further limit the flow through real nozzles.

Diffusers
A diffuser converts relative kinetic energy into pressure.

An ideal diffuser would recover the stagnation pressure, but practical diffusers cannot bring the fluid velocity to zero and have losses. The pressure recovered by such a diffuser is:


 * $$ \pi_d = p_{t2} / p_{t0} $$

A subsonic diffuser is a divergent passage. Diffusers operate in an adverse pressure gradient regime and the boundary layer development must be carefully managed to avoid flow separation. Boundary layers can be energized by extraction or aspiration but this has energy and complexity costs.

Achieving stable supersonic diffusion without shockwaves is almost impossible, since instabilities become rapidly magnified as the flow can rapidly snap to become subsonic via a normal shockwave and accelerate in the convergent passage. Usually multiple inclined shockwaves are employed to minimize entropy rise.

Shocks
The shock is a thin boundary across which heat transfer and viscous heating make the flow subsonic. The isentropic relations above are not applicable across a shock wave. The total temperature across a shock (normal to the shock surface) remains constant but the total pressure is lost. The loss depends on the incident Mach number.

Normal shock
The Mach number M2 after the shock is:


 * $$M_2^2=\frac{1+[(\gamma-1)/2]M_1^2}{\gamma M_1^2-(\gamma-1)/2}$$

A higher incident Mach number will transition to a smaller downstream subsonic Mach number.

The density &amp; velocity relation


 * $$\frac{\rho_2}{\rho_1}=\frac{u_1}{u_2}=\frac{(\gamma+1)M_1^2}{2+(\gamma-1)M_1^2}$$

the pressure relation


 * $$\frac{p_2}{p_1}=1+\frac{2 \gamma }{\gamma+1}(M_1^2-1)$$

and the temperature relation


 * $$\frac{T_2}{T_1}=\frac{h_2}{h_1}=\left[1+\frac{2 \gamma }{\gamma+1}(M_1^2-1)\right] \frac{2+(\gamma-1)M_1^2}{(\gamma+1)M_1^2}$$