Static flow and Its Application Text book, Chapter 2

Fluids: Statics vs Dynamics

Fluid Statics Basic Principles: Fluid is at rest : no shear forces Pressure is the only force acting What are the forces acting on the block? Air pressure on the surface - neglect Weight of the water above the block Pressure only a function of depth

Section 1: Pressure

Example 2.4 Consider a tank containing mercury, water, benzene, and air as shown. Find the pressure (gauge) over the benzene. If an opening is made in the top of tank. Given SG of mercury, benzene and water are 13.55, 0.879 and 1.00 respectively. Use the density of water equal to 999 Kg/m3

Solution

First what is the direction of the fluid? Example 2.5 First what is the direction of the fluid?

Solution

Example: Find PA- PB

What is h?

A B

x

Hydrostatic Equation : Compressible Fluids By the Ideal gas law: Hydrostatic Equation : Compressible Fluids R is the Gas Constant T is the temperature ρ is the density Then, Note: γ = ρg and not a constant, then Using special R Gases such as air, oxygen and nitrogen are thought of as compressible, so we must consider the variation of density in the hydrostatic equation: 24

Case of uniform temperature: T=To Text Book Example 2.2 At an altitude z, of 11,000 m, the atmospheric temperature T is -56.6 0C and The pressure p is 22.4 KN/m2 .Assuming that the temperature remains the Same at higher altitudes, calculate the density of the air at an altitude of z2 of 15,000 m. Assume R = 287 J/Kg.K. 25

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Case of Non-uniform temperature Now, for the Troposphere, Temperature is not constant: Substitute for temperature (T) and Integrate: β is known as the lapse rate, 0.00650 K/m (z=0~11 km), and Ta is the temperature at sea level, 288.15 K. pa is the pressure at sea level, 101.33 kPa, R is the gas constant, 286.9 J/kg.K Starting from, 27

Pressure Distribution in the Atmosphere 28

For Adiabatic atmosphere let = = 29

Also from Thermodynamic (for adiabatic) 30 30

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For Isentropic Process An isentropic flow is a flow that is both adiabatic and reversible We can extended from the adiabatic equation 32 32

Text Book Example 2.3 Calculate the Pressure, Temperature and density of atmosphere at an altitude z, of 1200 m, if zero altitude temperature is 15 0C and Pressure 101 KN/m2 . Assume that conditions are adiabatic (k = 1.4) and R = 287 J/Kg.K. 33

P1 = 87.32 kN/m-2 T1 = 3.3 oC 34