3.2 Pressure Variation with Elevation l Basic Differential Equation
* Considering the pressure and gravitational force acting on the element in direction.
∑
sin
sin
Taking limit and considering sin
lim
→
∴
→
(Pressure varies only with the elevation within the fluid)
l Pressure Variation for a Uniform-Density Fluid
* With constant density and thus the specific weight of fluid, taking integration for eq. (1)
→
(incompressible static flow)
→ piezometric head, (incompressible static flow)
Thus, at two points in fluid with different pressure and elevation,
→
or
l Pressure Variation for Compressible Fluids
For compressible fluid ( or varies significantly), ideal gas, using the equation of state.
multiplying with g,
∴
Where R = gas constant ( · for dry air) T = absolute temperature
P = absolute pressure(Pa) According to
* US standard atmosphere :
* at sea level, the standard atmospheric pressure :101.3KPa at sea level, the standard atmospheric temperature : 288K
* atmosphere - troposphere : from sea level to 13.7km → temperature is (대류권) decreased linearly with elevation at a lapse late of 5.87K/km
- stratosphere : from troposphere to 16.8 km → ℃ then (성층권) temperature is increased to ℃ at 30.5km
l Pressure Variation in the Troposphere Let
where = temperature at a reference level
= lapse rate (고도가 높아짐에 따른 외기의 감률) Using eq.(1) and (4)
∴
substituting eq,(5) into eq.(6)
Separating the variable and taking integration,
′
→ ′ ∴
′
→ ′
→ ′
′
′
→ ln
ln ′
ln
ln
ln
∴
→
l Pressure Variation in the Stratosphere
* Temp in stratosphere = constant (Assumed as constant) Thus, Taking integration for eq.(6),
ln
ln
∴
