General Expression for Thrust

MAE 4261: AIR-BREATHING PROPULSION

TURBOFAN ANALYSIS

Overview

This document utilizes the same developmental procedure and key results of the turbojet engine thermodynamic cycle analysis model. The purpose of a turbofan engine is to improve on the rather poor propulsive efficiency of the turbojet engine at low flight Mach numbers. In essence there is too much kinetic energy in the exhaust jet (per unit mass). This is the motivating reason for the introduction of the turbofan engine. Some schematics, along with station numbering convention are shown below:

Figure 1: Cross-Section of PW4000 High Bypass Ratio Turbofan Engine (Ref: http://www.aircraftenginedesign.com/cepics1.html). The PW4000 112-inch fan engine is the second derivative model in the PW4000 engine family. The PW4084, certified at 86,760 pounds thrust, was the launch engine for the Boeing 777. For more information on this class of engines, consult:

http://www.pratt-whitney.com/prod_comm_pw4000_112.asp.

Figure 2: Engine Designed for Subsonic Cruise, Cross-Sectional Representation with Station Numbers

Figure 3: Engine Designed for Subsonic Cruise and for Supersonic Flight with Afterburning, Cross-Sectional Representation with Station Numbers

Our goal is to model these engines thermodynamically, in the same manner that we treated the turbojet engines. In this engine, we have two jets. We can write the total mass flow as follows:

In this expression, is the total mass flow entering the engine, is the mass flow passing through the core (compressor, burner and turbine), and is the mass flow by-passing the engine. To simplify, we may write Equation 1 as simply:

In this expression, b is the bypass ratio, or the ratio of the mass flow that passes through the fan relative to the core (remainder of the engine) mass flow. In the same way, the total thrust of this engine is the thrust generated from change in momentum flux through the core, Tc, and the thrust generated by change in momentum flux through the bypass duct, Tb.

Equation 8, developed in the turbojet analysis applies to the core jet:

Next we proceed exactly as we did in the turbojet analysis (see Equation 4). For the fan stream, there is no turbine, so we may write:

A similar expression (see Equation 5) for the total pressure ratio is:

For an ideally expanded exhaust nozzle, p8=p0, and we may write:

Solving for the ratio T8/T0, we have:

This equation says that there is no change in static temperature across the fan, which is the same result that you will find in the design of wind tunnels. Substituting this into our general result for thrust (from conservation of mass and momentum, see Equation 3):

We add this to the thrust of the core jet, to arrive at an expression for the total thrust of a turbofan engine:

Now, we again proceed as we did with the turbojet analysis in performing a power (or work) balance between the compressor, fan and turbine (examine Equation 9 for turbojet development). Recall that now the turbine (located within the core) must power the compressor as well as the fan.

Substituting this result into Equation 12 gives the result for the thrust of the turbofan engine. To do so at this point will result in a large algebraic equation that is completely accurate, but rather difficult to interpret. Instead, it makes sense to think about how to simplify this expression to make it more understandable. For this engine we have more parameters than in the turbojet. Namely, we now have b and tf, which are used to characterize the fan flow. We can relate some of these parameters to others by noting that the highest propulsive efficiency is realized when U8=U6 since this maximizes the ratio of momentum-to-energy in the jets. For this situation, the two terms under the square root of Equation 12 are equivalent, which requires that:

We can now solve for tf as a function of b using Equation 13.

When this equation is satisfied, the thrust equation becomes simply:

Now, as for the turbojet there is still the choice of compression ratio. Generally, we want to choose it for maximum power, which in this case for given turbine inlet temperature, qt, and bypass ratio, b, means maximum thrust. To maximize the thrust, T, we choose tc to maximize tf, since T increases monotonically with tf. From the expression for tf (Equation 15):

Notice that this is precisely the same result as for the turbojet. Substituting in the expression for tf:

Now we have three parameters:

Table 1: Summary of Governing Parameters for Turbofan Engines

We can again write an expression for the propulsive efficiency of the turbofan engine as:

1