# Isentropic flow relations

### From CFD-Wiki

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+ | == Isentropic flow == | ||

+ | |||

Constant entropy flow is called Isentropic flow. | Constant entropy flow is called Isentropic flow. | ||

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''Below are the formulas... someone needs to add some explaining text here'' | ''Below are the formulas... someone needs to add some explaining text here'' | ||

- | <math>M | + | Incompressible flow does not apply if <math>M << 1</math> is violated. |

+ | Therin | ||

- | <math>a = \sqrt{\gamma R T}</math> | + | <math>M = \frac{u}{a}</math> |

+ | |||

+ | is called the '''Mach-number'''. It is the relation between the '''flow-velocity u''' (fluid or object) and the '''speed of sound a''' in the surrounding medium. | ||

+ | |||

+ | |||

+ | In a '''caloric ideal''' gas the speed of sound depends on the '''absolute temperature T''' (K), the '''adiabatic exponent <math>\gamma</math>''' and the '''specific gas constant R''' (J/(kg K)) | ||

+ | |||

+ | <math>a = \sqrt{\gamma R T}</math>. | ||

+ | |||

+ | In this case all thermodynamic properties (temperature, pressure, density and speed of sound) can be expressed with explicit formulas which are functions of the Mach-number! | ||

+ | |||

+ | |||

+ | In the following all thermodynamic properties are related to their '''properties ''at rest''''' (u=0). | ||

+ | From the energy equation for a frictionless flow | ||

+ | |||

+ | <math> \frac{u^2}{2}+h=h_t </math> | ||

+ | |||

+ | you will get that the enthalpy at rest h<sub>t</sub> is allways the same regardless of an isentropic or non-isentropic state change. | ||

+ | As a matter of fact - <math>h=c_p T</math> - the temperature is it as well. | ||

+ | In contrary the pressure depends on how the gas is brougt to rest. The pressure at rest is only obtained if the state change is isentropic. If the entropy changes the pressure at rest changes as well, e.g. when passing a shock. | ||

+ | |||

+ | == Pressure == | ||

<math>\frac{p_0}{p} = (\frac{T_0}{T})^\frac{\gamma}{\gamma - 1}</math> | <math>\frac{p_0}{p} = (\frac{T_0}{T})^\frac{\gamma}{\gamma - 1}</math> | ||

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<math>\frac{p_0}{p}=(1+\frac{\gamma-1}{2}M^2)^{\frac{\gamma}{\gamma-1}}</math> | <math>\frac{p_0}{p}=(1+\frac{\gamma-1}{2}M^2)^{\frac{\gamma}{\gamma-1}}</math> | ||

- | <math>\frac{T_0}{T}=1+\frac{\gamma-1}{2}M^2</math> | + | |

+ | == Temperature and speed of sound == | ||

+ | |||

+ | <math>\frac{T_0}{T}= \left( \frac{a_0}{a}\right)^2=1+\frac{\gamma-1}{2}M^2</math> | ||

+ | |||

+ | |||

+ | == Density == | ||

<math>\frac{\rho_0}{\rho}=(1+\frac{\gamma-1}{2}M^2)^{\frac{1}{\gamma-1}}</math> | <math>\frac{\rho_0}{\rho}=(1+\frac{\gamma-1}{2}M^2)^{\frac{1}{\gamma-1}}</math> | ||

- | <math>\frac{A}{A*}=\frac{1}{M}*(\frac{\gamma+1}{2})^{\frac{-\gamma+1}{2(\gamma-1)}}*(1+\frac{\gamma-1}{2}M^2)^{\frac{\gamma+1}{2(\gamma-1)}}</math> | + | == Critical values == |

+ | |||

+ | Values which can be found at M=1 (speed of sound) are sometimes called '''critical values'''. | ||

+ | They are marked with the superscript *. | ||

+ | |||

+ | They differ from the properties at rest only by a constant value and are therefore used as reference values as well. | ||

+ | |||

+ | In the special case for gases which contain two atoms (<math> \gamma=1.4 </math>) you get: | ||

+ | |||

+ | <math> \frac{a^*}{a_0} = 0.913 </math> | ||

+ | |||

+ | <math> \frac{p^*}{p_0} = 0.528 </math> | ||

+ | |||

+ | <math> \frac{\rho^*}{\rho_0} = 0.634 </math> | ||

+ | |||

+ | === Critical cross-section === | ||

+ | |||

+ | The cross-section in a nozzle where M=1 is often used as a reverence, even if M=1 is never reached. | ||

+ | |||

+ | <math>\frac{A}{A^*}=\frac{1}{M}*(\frac{\gamma+1}{2})^{\frac{-\gamma+1}{2(\gamma-1)}}*(1+\frac{\gamma-1}{2}M^2)^{\frac{\gamma+1}{2(\gamma-1)}}</math> | ||

[[category: Equations]] | [[category: Equations]] |

## Revision as of 11:12, 29 January 2008

## Contents |

## Isentropic flow

Constant entropy flow is called Isentropic flow.

From a consideration of the second law of thermodynamics, a reversible flow maintains a constant value of entropy

*Below are the formulas... someone needs to add some explaining text here*

Incompressible flow does not apply if is violated. Therin

is called the **Mach-number**. It is the relation between the **flow-velocity u** (fluid or object) and the **speed of sound a** in the surrounding medium.

In a **caloric ideal** gas the speed of sound depends on the **absolute temperature T** (K), the **adiabatic exponent ** and the **specific gas constant R** (J/(kg K))

.

In this case all thermodynamic properties (temperature, pressure, density and speed of sound) can be expressed with explicit formulas which are functions of the Mach-number!

In the following all thermodynamic properties are related to their **properties at rest** (u=0).
From the energy equation for a frictionless flow

you will get that the enthalpy at rest h_{t} is allways the same regardless of an isentropic or non-isentropic state change.
As a matter of fact - - the temperature is it as well.
In contrary the pressure depends on how the gas is brougt to rest. The pressure at rest is only obtained if the state change is isentropic. If the entropy changes the pressure at rest changes as well, e.g. when passing a shock.

## Pressure

## Temperature and speed of sound

## Density

## Critical values

Values which can be found at M=1 (speed of sound) are sometimes called **critical values**.
They are marked with the superscript *.

They differ from the properties at rest only by a constant value and are therefore used as reference values as well.

In the special case for gases which contain two atoms () you get:

### Critical cross-section

The cross-section in a nozzle where M=1 is often used as a reverence, even if M=1 is never reached.