Fundamental Equations

Gas Law

The relation between pressure, temperature, density and molecular weight is expressed as follows:

Eq. 2.1.1

Here the variables are:

r is the gas density (kg/m3 in the program).

P is the pressure in absolute units (kPa in the program).

T is the absolute temperature (K in the program).

R is the universal gas constant (R = 8.314 kJ/kmole K).

Z is a compressibility factor to correct for deviation from the ideal gas law.

Speed of Sound in a Gas

Sound is by definition an isentropic process. It occurs too fast for allowing heat transfer between the gas molecules (hence it is adiabatic), and it is also nearly reversible otherwise sound would not propagate very far.

It is an elastic property relating pressure and density at constant entropy and by definition it is:

Eq. 2.2.1

If we assume that then the formula becomes:

Eq. 2.2.2

This is the formula which is often quoted but one needs to remember that it is only true if the gas follows the law. Also the isentropic exponent is often called the ratio of specific heats because for most gases at low pressures . This law is not always true when the pressures get higher, and simply using the ratio of specific heat and applying it as the isentropic exponent can lead to errors. The k which is required is that which expresses the gas isentropic compression best.

Bernoulli's Equation for Gases

The equation simply states that between two sections 1 and 2, for steady flow (i.e. the flow rate remains constant at every point). The change in energy per unit of mass of fluid depends only on the amount of work, W, and heat, Q, that has entered as the fluid goes from 1 to 2.

Eq. 2.3.1

g is the gravitational acceleration (9.81 m/s2).

Z is the elevation at sections 1 and 2 respectively.

v is the speed at sections 1 and 2 respectively.

U is the gas internal energy at sections 1 and 2 respectively.

P is the pressure at sections 1 and 2 respectively.

r is the density at sections 1 and 2 respectively.

W is the work entering between sections 1 and 2.

Q is the heat entering between sections 1 and 2.

Special Case when Heat and Work are not Involved

In this instance, W and Q are both zero.

In addition if the flow is horizontal, or the change of elevation is small enough to be insignificant, then = 0, so equation 2.3.1 becomes

Eq. 2.4.1

The internal energy for an ideal gas is

And the thermodynamic definition of Enthalpy is

So equation 2.4.1 can also be written in term of Enthalpy:

Eq. 2.4.2

This equation is currently not used much for sizing relief valves but the improvement arising from the various thermodynamic models show that in the future this equation will be able to be used directly for solving relief valve sizing problems. This is especially true when the problem is far away from conventional problems, for instance for compressible fluids close to critical conditions or when two phase flow is involved, where current international standards fall far short of offering sensible solutions.