Uploaded by Issah Ismaila

# 4

```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 (  &middot;  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         



∴    

      

  
```