ALTITUDE AND DERIVATION OF THE HYDROSTATIC EQUATION

Atmosphere! study about basic of the atmosphere is important in aerodynamics as it will tell us about what is pressure, density, temperature, specific volume flow velocity and streamline with its most commonly used units in aerodynamics it will also brief us about which are the 6 different types of altitude, hydrostatic equation 

PRESSURE

 The pressure is a normal force per unit, area exerted on the surface due to the time rate of change of momentum of the gas molecules on that surface. It is usually defined at a point in the gas or a point on a surface and can vary from one to another.

Commonly used units of pressure are  

1. Newton per square meter (N/m²)
2. Dynes per square centimetre (Dyn/cm²)
3. Pounds per square foot (lb/ft²)
4.  Atmospheres (atm)

 DENSITY

  The density of a substance including gas is the mass of that substance per unit volume. Density will be designated by symbol r

Commonly used units of density are

1.  kilogram per cube meter (kg/m³) 
2.  slug per cube foot (slug/ft³)
3.  pound per cube foot (lb_m/ft³)     (1 slug=32.2Ib_m)

 TEMPERATURE

Temperature is the measure of the average kinetic energy of the particles in the gas.

Commonly used units of temperature are

1.  kelvin (K)
2.  degree Celsius (°C)
3.  degree Rankine (°R)
4.  degree Fahrenheit (°F)

SPECIFIC VOLUME

r is mass per unit volume. Inverse if r is called a specific volume i.e. volume per unit mass

Units

1.  m³/kg
2.  ft³/slug   

FLOW VELOCITY

  Velocity is the distance travelled by an object with respect to time, we all know what 40 m/s velocity of a vehicle means. When we talk about flow velocity it is more subtle as 40 m/s due to south in a horizontal plane. It is very important to designate both speed and direction of flow velocity. 

Each region of gas doesn't need to have the same velocity, it will vary point to point. Flow velocity varies, along with p (pressure), r(density), T (temperature) is a point property

   

The velocity at any fixed point in the flowing gas is the velocity of an infinitesimally a small fluid element as it sweeps through point.

STREAMLINES

 The path took by a moving fluid element is called as streamlines of the flow. There can be no flow across streamlines

DIFFERENT TYPES OF ALTITUDE

There are 6 types of altitude

1. Pressure altitude
2. Temperature altitude
3. Density altitude
4. Absolute altitude
5. Geometric altitude
6. Geopotential altitude

Absolute altitude (h_a): The distance measured from the centre of the earth

1

Geometric altitude: it is defined as the distance measured from the ground from the fig, we can say that

From fig 1 we can say that

....1

                

From newton’s law of gravitation (g varies inversely as the square of the distance from the centre of the earth)

                                              

  (From equation no 1)   

                                   


 Here

                     g= local acceleration of gravity

                     g₀= gravitational acceleration at sea level

                      r= radius of the earth

                      h_G=geometric altitude

                     h_a= absolute altitude

 We will take a model which will allow us to calculate the variation of r,p, T as the function of altitude      

Consider the small stationary fluid element in air as shown in fig 2.

2

We take rectangular faces where dh_G is the height of the side face of the rectangle which is infinitesimally small. 
We will take a model which will allow us to calculate the variation of r,p, T as the function of altitude      

Consider the small stationary fluid element in the air as shown in the fig.

We take rectangular faces where dh_G is the height of the side face of the rectangle which is infinitesimally small.

On the bottom face, the pressure p is felt which gives rise to upwards force of p×1×1 exerted on the fluid element & the top surface is slightly at higher altitude by the distance of dh_G,

 The pressure differs from the infinitesimal value of dp hence, the pressure felt by the top surface is p+dp. It gives rise to a downwards force of (p+dp)(1)(1)on the fluid element.

 The volume of the fluid element is (1)(1)dh_G = dh_G.

Mass of the fluid element is rdh_G. The weight of the fluid element is grdh_G.  

Pressure forces and weight must be balanced because the element is in the rest position

......2

The above equation is called as  HYDROSTATIC EQUATION

Equation number 2 is in a differential form we will integrate this equation to get the variation of pressure in terms of altitude assuming g is constant throughout the atmosphere which is equal to g₀

Altitude h in equation 3 must be different from h_G in equation 2 to compensate for the fact that g is slightly different from g₀. We have defined a new altitude h, which is geopotential altitude. In practical geopotential altitude is a "fictitious" altitude  

RELATION BETWEEN GEOPOTENTIAL AND GEOMETRIC ALTITUDES   

By convention, we will set both h and h_G equal to zero at sea level. Now, consider a given point in the atmosphere. This point is at a certain geometric altitude h_G , and associated with it is a certain value of h (different from h_G,) integrating equation 5 between the sea level and given point we get