# Isentropic Flow Calculator

Created by Rahul Dhari
Reviewed by Steven Wooding
Last updated: Feb 23, 2022

Our entropic flow calculator returns fluid properties for an isentropic flow process. This isentropic flow process forms the basis of compressible flow and gas dynamics that contribute to the development of several cutting-edge technologies like jet engines, isentropic flow through nozzles in a rocket, and turbines. These machines and flow conditions are critical to achieving sufficient thrust for human flight, in the atmosphere, or escape velocity for space travel.

This calculator for compressible flow covers the condition (pressure, density, and temperature) of gas at different stages, such as static pressure, stagnation pressure, and critical flow properties. Read on to understand what is static pressure and how to calculate isentropic flow properties.

## What are isentropic flow relations?

Before getting into isentropic flow relations, let's look at the broader picture, i.e., the field of gas dynamics. It is the study of the dynamic conditions of high-velocity gas flow. Say, for instance, in a jet engine, the gas or air is pushed through air intakes and into the engine nozzle at high speed. These speeds are often faster than sound, i.e, supersonic. The air begins moving at slow speeds, i.e., subsonic, and gradually breaks the sound barrier to achieve supersonic speeds.

Additionally, this change in velocity leads to changes in flow conditions such as pressure, density, and temperatures, and in some cases, even normal or oblique shock waves.

Thermodynamically, an isentropic process is one in which no heat is added or removed from the system, i.e, the entropy change is zero. Also, Reynold's number for such flows is very high, in compliance with the kinetic theory of gases, considering mean free path and mean velocity.

The air is forced to flow through a nozzle or diffuser to achieve supersonic conditions. The most convenient way to cause this change in flow conditions is to change the cross-sectional area. Different nozzle and diffuser designs can be made with variable cross-sectional areas to extract different flow velocities and conditions. To this end, isentropic flow relations play a critical role in relating the air's conditions at different stages. The stages are — stagnation, static, dynamic, and critical flow.

## Mach number, static pressure, and stagnation temperature

The Mach number connects area, pressure ($p$), temperature ($T$), and density ($ρ$) across different stages of isentropic flow. Mach number, $M$ is the ratio of flow velocity $c$ and the speed of sound $a$ at temperature, $T$. The speed of sound, $a$ is the function of specific heat ratio, $\gamma$, specific gas constant, $R$, and temperature, $T$.

$\quad a = \sqrt {\gamma R T}$

The Mach number equation is:

$\quad M = \frac{c}{a} = \frac{c}{ \sqrt {\gamma R T} }$

As the airflow turns supersonic and hypersonic, the disturbances in the air move downstream. These disturbances are sound waves. At speeds faster than sound ($M>1$), the flow strikes as a cone, with flow velocity moving faster than sound and sound waves striking at the edge of the cone (see figure). The angle formed by this cone is known as the Mach angle. The Mach angle, denoted by the symbol $\mu$, is:

$\quad \mu = \sin^{-1}(\frac{1}{M})$

Stagnation condition: The properties of air at zero local fluid velocity. The pressure and temperature at this point are called stagnation pressure and temperature, respectively. The parameters at this condition are denoted by the subscript $0$.

Static condition: The properties of air at any point in the flow are defined by static parameters such as static pressure and temperature. The parameters at this condition are denoted by the subscript $s$.

The pressure at the two conditions above is related using the stagnation pressure formula or also static pressure formula:

$\quad \frac{p_0}{p_s} = \bigg(1 + \frac{\gamma - 1}{2} M^2\bigg)^{\frac{\gamma}{\gamma-1}}$

Similarly, the definition of stagnation temperature and static temperature relation is also a function of Mach number and specific heat ratio. Mathematically:

$\quad \frac{T_0}{T_s} = 1 + \frac{\gamma - 1}{2} M^2$

Also, the density ratio is:

$\quad \frac{\rho_0}{\rho_s} = \bigg(1 + \frac{\gamma - 1}{2} M^2\bigg)^{\frac{1}{\gamma-1}}$

The three equations mentioned above are denoted as the isentropic flow equations or compressible flow equations and are used to estimate the static conditions from stagnation conditions and vice versa.

Pressure and temperature ratio
You can find the value of the ratio of the static conditions and stagnation conditions in the advanced mode of this compressible flow calculator.

What is dynamic pressure? is the fluid's kinetic energy. In other words, the pressure due to the fluid's velocity is called dynamic pressure. The units of dynamic pressure are force per area or $\text{N/m}^2$ (bar). The formula for dynamic pressure, $q$, relates the density of the fluid and flow velocity:

$\quad q = \frac{1}{2} \rho v^2$

## Isentropic flow relations — critical flow and stagnation pressure equation

The nozzle or diffuser's cross-sectional area reduces up to a point and then increases again. This narrowest section is known as the throat. The parameters at this state are critical parameters as the flow reaches the critical flow velocity. The Mach number at the throat is 1. The parameters at this point are denoted by a superscript $^*$. The conditions at different states are shown in the figure below:

Consider the above isentropic relations, with Mach number M = 1. The isentropic flow equations become:

\quad \begin{align*} \frac{p^*}{p_0} &= \bigg(\frac{2}{\gamma + 1}\bigg)^{\frac{\gamma}{\gamma-1}}\\ \\ \frac{T^*}{T_0} &= \bigg(\frac{2}{\gamma + 1}\bigg)\\ \\ \frac{\rho^*}{\rho_0} &= \bigg(\frac{2}{\gamma + 1}\bigg)^{\frac{1}{\gamma-1}} \end{align*}

The first equation is critical to stagnation pressure ratio, and considering the specific heat ratio of 1.4, the value of this ratio becomes 0.528. Similarly, values of 0.833 and 0.634 are obtained for temperature and density ratios. The cross-sectional area at the throat is also a function of Mach number, such that:

$\!\footnotesize\frac{A}{A^*} = \frac{1}{M} \bigg(\frac{2}{\gamma+1} + \frac{\gamma - 1}{\gamma+1}M^2\bigg)^{\frac{\gamma+1}{2(\gamma-1)}}$

Critical flow velocity ($c^*$) and mass flow rate: Considering the Mach number as 1, you can calculate the flow velocity and mass flow rate $\dot{m}$ of the fluid using the throat area, critical density and critical flow velocity as:

\quad \begin{align*} c^* &= a^* = \sqrt{\gamma R T^*}\\ \dot{m} &= \rho^* A^* c^* \end{align*}

Next time you see a rocket or sitting in a passenger flight, notice the exhaust. The air coming through it is undergoing the process you learned here. These flow equations are the lynchpin of most gas dynamics and jet propulsion-based studies employed to design and study these complex systems. Now let's take a look at how you can use this isentropic flow calculator.

## How to calculate isentropic flow properties

In most cases of isentropic flow, designers first calculate the Mach number and then refer to the isentropic flow table with the appropriate specific heat ratio to estimate the pressure, temperature, and density ratios. Here, this calculator is a substitute for the isentropic flow tables. To calculate the isentropic flow properties:

1. Enter the Mach number or flow velocity for the flow to obtain the Mach angle (for Mach > 1).
2. Fill in the stagnation pressure.
3. Insert the stagnation temperature.
4. The calculator will determine the ratios for pressure, temperature, and density for static and critical conditions and return the flow condition at these points.
5. Enter the cross-sectional area of the throat.
6. The calculator will return the mass flow rate and the cross-sectional area of the nozzle/diffuser.

## Example: Using the Isentropic flow calculator

Perform the calculation for static pressure, temperature, and density for a flow having Mach number 2 and the following stagnation conditions:

• Stagnation pressure, $p_0$ = 0.69 bar;
• Stagnation temperature, $T_0$ = 310 K; and
• Throat area, $A^*$ = 0.1 m2.

To estimate conditions for isentropic flow through nozzle:

1. Enter the Mach number, $M$ = 2.
2. Fill in the stagnation pressure, $p_0$ = 0.69 bar.
3. Insert the stagnation temperature, $T_0$ = 310 K.
4. Using the static pressure formula:
\quad \begin{align*} \frac{p_0}{p_s} &= \bigg(1 + \frac{1.4 - 1}{2} 2^2\bigg)^{\frac{1.4}{1.4-1}}\\ p_s &= 1.278p_0 = 0.08819 \text{ bar} \end{align*}

Similarly, the following values of temperature, and density were obtained for static and critical state:

• Static temperature, $T_0$ = 172.22 K;
• Critical pressure, $p^*$ = 0.365 bar;
• Critical temperature, $T^*$ = 258.3 K; and
• Critical velocity, $c^*$ = 322.2 m/s;

## FAQ

### What is stagnation pressure?

The pressure at the point at which the local fluid velocity is zero is called stagnation pressure. The critical to stagnation pressure ratio (p*/p0) is used to estimate the stagnation pressure. For a give specific heat ratio of 1.4, p0 = p* / 0.528.

### How do I calculate stagnation temperature?

To calculate stagnation temperature:

1. Subtract the 1 from the specific heat ratio, γ.
2. Divide the resultant by 2.
3. Multiply the number by the square of Mach number, M.
4. Add 1 to the product to obtain the temperature ratio.
5. Multiply the temperature ratio by the static temperature to obtain stagnation temperature.

### How do I calculate critical flow velocity?

To calculate the critical flow velocity:

1. Multiply the ratio of specific heats, γ by the specific gas constant, R.
2. Multiply the product by the critical temperature, T*.
3. Find the square root of the product to obtain the critical flow velocity.

### How do I calculate dynamic pressure?

To calculate the dynamic pressure:

1. Find the density of the fluid, ρ.
2. Find the fluid velocity, v.
3. Multiply the square of fluid velocity by the density of fluid.
4. Divide the produce by 2 to obtain the dynamic pressure. The dynamic pressure equation is:
q = 0.5 × ρ × v²
Rahul Dhari
Mach number
Flow velocity (c)
ft/s
Speed of sound (a)
ft/s
Mach number (M)
Stagnation
Pressure (P₀)
psi
Temperature (T₀)
°F
Density (ρ₀)
lb/cu ft
Static
Pressure (Ps)
psi
Density (ρs)
lb/cu ft
Temperature (Ts)
°F
Area (A)
in²
Throat
Pressure (P*)
psi
Temperature (T*)
°F
Density (ρ*)
lb/cu ft
Area (A*)
in²
Flow velocity (c*)
ft/s
Flow rate
Mass flow rate
kg/s
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