THRUST EQUATION

THRUST EQUATION

Momentum Thrust: If the condition (area A, pressure P and velocity V) at the engine intake and exhaust are designated with the subscripts 'a' and 'j' respectively, then a mass of 'air (m) flowing per unit time through the engine will experience an:

 Increase in velocity   = (V_j - V_a).

The momentum gain = m (V_j - V_a), where m is the mass flow rate of air through the engine under steady condition.              

= rate of change of momentum

= Applied force to the air mass flow as per Newton’s 2^nd Law of motion.

According to Newton's Third Law, for every action, there is an equal and opposite reaction. Therefore as the air mass is accelerated through the engine, there will be an equal and opposite reaction (thrust) acting on the engine in the forward direction. Since the force is obtained due to a change in momentum of the air, this is called the Momentum Thrust of the engine.

Momentum Thrust         = m (V_j - V_a)

=  m V_j - m V_a

Consideration may be given to the fuel mass flow rate (m_f) that is mixing with air at combustion chamber with initial zero velocity relative to the engine, the thrust equation may be modified as follows:

Momentum Thrust         =   (m + m_f )V_j –  m V_a

= m (V_j- V_a ) + m_f V_j

Pressure Thrust: Considering the engine as a physical body in the air, it will be subjected to pressures acting at the intake (P_a) and the exhaust (P_j). The pressures will produce a pressure force of (P_j - P_a)A_j acting on the engine in the forward direction. This force is the result of an unbalanced pressure and is called the Pressure Thrust. Hence,

Pressure Thrust   = (P_j - P_a)A_j

In most practical cases, pressure thrust exists because all of the pressure of the engine cannot be converted into velocity at the exhaust (i.e. gas does not fully expanded to atmospheric pressure). It becomes more pronounced and significant as the speed of the aircraft becomes supersonic and the exhaust nozzle becomes choked. At choked nozzle condition, velocity of exhaust gas cannot exceed M =1, unless it is a C-D duct and invariably there remains significant amount of unconverted pressure.

Total Thrust: The Total Thrust on a jet engine will be the sum of the momentum thrust and the pressure thrust.

Total Thrust         = Momentum Thrust + Pressure Thrust

T_t                = m (V_j- V_a ) + m_f V_j + (P_j – P_a) A_j

In actual practice, fuel flow is usually neglected when net thrust is computed, because the weight of the air that leaks from various section of the engine is assumed to the approximately equivalent to the weight of the fuel consumed. Therefore, the final equation for computing the thrust by a turbo-jet engine becomes:

T_t                = m (V_j- V_a ) +  (P_j – P_a) A_j

This is a general thrust equation and is applicable for all kinds of jet propulsion.

1.5.4 Gross Thrust, Momentum-Drag and Net Thrust: An analysis of the total thrust of a jet engine will show that it can be grouped into two parts.

T_t                = [m V_j + (P_j - P_a)A_j] – [mV_a]

The forward part composed of the exhaust jet momentum [mV_j] and the pressure thrust (P_j-P_a)A_j and is called the Gross Thrust of the engine, i.e. thrust developed by the engine. The rear part is the momentum force of the incoming air impinging on the engine intake and is called the Momentum Drag. Hence the total thrust is the difference of the gross thrust and the momentum drag and it is also called the Net Thrust (actual thrust) of the engine. Hence,

T_gross            = m V_j + (P_j - P_a)A_j

D_momentum         =  mV_a

Net Thrust = Gross Thrust - Momentum Drag

Gross thrust is actually the thrust at the static aircraft, with aircraft speed zero.

Power of aircraft gas turbine engines:  Turbojet engines are rated on the basis of takeoff thrust generated at standard atmospheric conditions. This is conventional, because output of turbojet engines for the aircraft is THRUST (propulsive force).

Gas turbine engines for turboprop are the torque turbine engines and the output of the engine is in the form of TORQUE on the shaft. Hence, the rating of the engine is the Shaft Horse Power expressed in BHP.

For comparison purpose, thrust of the turbojets may be converted into horse power, called Thrust Horse Power (THP).

THP =  

For turboprop aircraft, total power is the summation of BHP at the engine output shaft (input to the propeller) and the THP from the exhaust thrust. The summation of these two is termed as ESHP (equivalent shaft horsepower).