# How can I determine density of a gas only given temperature?

I have a homework problem where exhaust is traveling through an exhaust system (assumed to be air for simplicity) from and engine and then released into the atmosphere. The exhaust is at a temperature of 250F. I need to find the pressure at the beginning of the exhaust system.

I've been able to set this up and created an equation to solve for the pressure. I assumed that the pressure of the atmosphere where it it exits is 0 atm. But the problem is that the pressure depends on the density of the exhaust (air). I was only given temperature of the exhaust in the problem and I need a second property to determine density. Is there another way I can determine density? I am also given the volumetric flow rate and the area of the piping system

• "I assumed that the pressure of the atmosphere where it it exits is 0 atm" You know, the unit "atmosphere" mean the normal atmosphere pressure. So the exit pressure is 1 atm since the gas is "released into the atmosphere". – Yrogirg Sep 9 '12 at 15:58
• oh, it turns out you won't need the pressure at the exhaust. – Yrogirg Sep 9 '12 at 16:10
• I assumed that the pressure of the atmosphere where it exits is 0 gauge pressure. I am not using absolute pressure. And yes I do need the pressure at the beginning of the exhaust system.... that't what I am solving for – Greg Harrington Sep 9 '12 at 21:58
• I've edited the answer. – Yrogirg Sep 10 '12 at 4:22

The key's is the Bernoulli's equation for the compressible flow:

$$\frac{v^2}{2} + \frac{p}{\rho} + u = \text{const}$$

$u$ is internal energy per unit mass, or using enthalpy $h$ per unit mass:

$$\frac{v^2}{2} + h = \text{const}$$

The other equation to find $v$ you'll get from the definition of the volumetric flow.

You have two equations to solve the system. I let it to you to find the $h$ for the ideal gas, you'll find that it depends only on the temperature.

So you'll find the temperature in the engine. To find the pressure in the engine you'll need also the density. Here mass conservation comes in hand:

$$\rho v S = \rho Q = const$$

where $Q = vS$ is the volumetric flow rate. All you need is the density at the exhaust which you find form the ideal gas law, knowing the exhaust temperature and pressure (1 atm or 0 atm gauge).

Maybe I am missing something, but it looks like you can do the following: 1) determine the gas velocity based on volumetric flow and pipe size; 2) write down the Bernoulli equation; 3) write down the equation of state for the (ideal) gas. As a result, you'll have two equations for pressure and density.

• akhmeteli: Could you please tell me if "T" is in Celsius? – user43300 Mar 26 '14 at 17:51
• @veronica: For example, in the ideal gas law $PV=nRT$, $T$ is in Kelvin. – akhmeteli Mar 26 '14 at 23:35

I am a exhaust system studier!

Exhaust gas density could be approximated by $r = 352.5/(T+273)$, in $Kg/m^3$. Where $T$ is the exhaust gas temperature after the turbo in °C.