Читать книгу Introduction to Flight Testing - James W. Gregory - Страница 31

The development of the standard atmosphere directly results from the hydrostatic equation, which is derived here based on a control volume analysis. Figure 2.2 illustrates an arbitrary control volume, measuring dx × dy × dh_G, and the forces acting upon it (here, h_G is the geometric altitude, or height above mean sea level (MSL)). The forces due to pressure acting on all of the side walls balance one another out in this static equilibrium condition, and we will consider only the forces acting in the vertical direction. The force acting upward on the bottom surface of the control volume is the pressure, p, times the cross‐sectional area dx dy. Similarly, on the top surface, we have a force of (p + dp)dx dy acting downward. (Here, the differential pressure dp accounts for pressure changes in the vertical direction.) Finally, we have the weight of the air inside the control volume acting downward, W = mg, where g is the local gravitational acceleration and the mass of the air inside the control volume can be found from the product of density and the volume,

(2.1)

Figure 2.2 Forces acting on a hydrostatic control volume.

Summing all the forces in the vertical direction and setting equal to zero (from Newton's second law applied to a stationary control volume), we obtain

(2.2)

Canceling terms leads to

(2.3)

which is the hydrostatic equation as a function of geometric altitude. This expression mathematically expresses the physical explanation that we presented earlier for the variation of pressure with altitude. As altitude increases (positive dh_G), the minus sign indicates that the pressure decreases (negative dp). The ρg term is an expression of the weight of the air inside the control volume, which is the reason for the pressure difference.