Isothermal and Non-Isothermal Fluids

Isothermal Fluid (Constant Temperature Solution)

Recap equation (1.7)

The equation of state for a compressible system generally relates its density to its pressure and temperature. If the fluid is a perfect gas at rest at constant temperature, it can be written from eq (1.7) that

Since ,                                                                  

= constant
	(3.17)

Where p_o and r_o are the pressure and density at some reference horizontal plane. With the help of Eq. (3.17), Eq. (3.14) becomes,

	(3.18)
	(3.19)

where z and z₀ are the vertical coordinates of the plane concerned for pressure p and the reference plane respectively from any fixed datum.

Non-isothermal Fluid

The temperature of the atmosphere up to a certain altitude is frequently assumed to decrease linearly with the altitude z as given by

(3.20)

where T₀ is the absolute temperature at sea level and the constant a is known as lapse rate.

For the standard atmosphere, a = 6.5 K/km and T₀= 298 K.

With the help of p =rRT and (3.20), the Eq. (3.14) can be written as,

(3.21)

Integration of Eq. (3.21) yields

Hence,

(3.22)

The altitude z in Eq. (3.22) is measured from the sea level where the pressure is p_o.