A simplified simulation of gas turbine engine operation John Olsen & John Page School of Mechanical and Manufacturing Engineering, The University of New South Wales, Sydney, N.S.W., 2052. [email protected] & [email protected] Abstract. This work outlines the analysis used in simulating the operation of a gas turbine engine. Its starting point is the analysis of Cumpsty (2006). Cumpsty’s work suggests that it is possible to simplify the behavior of gas turbine engines enormously by taking into account aspects that are common to all turbine maps, i.e., the collapse of curves representing differing constant non-dimensional spool rotational speed. As a result, the relationship between the pressure ratio and the non-dimensional mass flow rate as well as the pressure ratio and the isentropic efficiency are essentially independent of the rotational speed of the spool. Following Cumpsty, we develop equations to enable us to calculate both pressure ratios and normalised mass flow rates through compressors so that we can plot the behavior of these engines on compressor maps. numbering regime adopted in this work, i.e., station 2 at 1. INTRODUCTION the inlet to the compressor, station 3 at the inlet to the combustion chamber, station 4 at the inlet to the turbine For straight and level flight, aircraft require a constant and station 5 at the inlet to the propelling nozzle. throughput of energy to maintain altitude and speed. In many aircraft applications, gas turbine engines are utilised to provide forward thrust. It can be shown that the net thrust Tn is: n & ( j −= VVmT ) (1) where m& is the mass flow rate of air through the engine (we ignore the mass flow rate of fuel through the engine), Vj is the magnitude of the velocity of the jet issuing from the propelling nozzle with respect to the aircraft and V is the magnitude of the velocity of the incoming airflow with respect to the aircraft. To maximise propulsive efficiency, Vj must be kept low meaning that the mass flow rate must be kept high. For an aircraft powered by gas turbine engines, the rate at which the flow issuing from the jet nozzles can transfer energy to the aircraft is equal to the product of the thrust and velocity of the incoming airflow with respect to the aircraft. If we increase rate at which the jet transfers energy to the aircraft so that it exceeds the energy required for straight and level unaccelerated flight, the aircraft will climb. Similarly, if we decrease the rate at which the jet transfers energy to the aircraft to a level below that required for straight and level unaccelerated flight, the aircraft will descend. Changing the pressure ratio across the engines propelling nozzles controls the rate at which the jet transfers energy to the aircraft. This is the job of the gas turbine engine. Figure 1. Basic engine configurations a) single-spool turbojet, b) two-spool turbospool and c) two-spool 1.1. Basic operation of gas turbines turbofan, given with the numbering regime that is used in this work. The numbering regime is the same as that To understand the simulation, we need to offer a brief used by Cumpsty (2006). explanation of how gas turbine engines work. For further information we refer to reader to Gunston In a single spool engine, all compressor and turbine (2006) and Hünecke (2003). Figure 1 shows diagrams stages connect to the same shaft which is commonly of the basic engine configurations for the single-spool called the spool. We distinguish between engine turbojet, the two-spool turbojet and the more common compressor operation at design and off design two-spool turbofan engine. The figure also shows the conditions. The design condition for a commercial airliner typically corresponds to engine operation when k the aircraft cruises at a pressure height of around ⎡ ⎤ ()−1k ⎛ − 1k ⎞ 2 (3) 35,000ft and an airspeed specified as a Mach 0 ⎢1PP += ⎜ ⎟ M ⎥ . number ⎣ ⎝ 2 ⎠ ⎦ of 0.85. At this design condition, the rotational velocity of the spool and the velocity of the incoming For most gas turbine engine operating conditions, the air to the engine, are such that the inlet air impinges on flow from the propelling nozzles is choked. This is the compressor blades at zero degrees of incidence. At emphasised by Cumpsty (2006) and supported by El- this condition, the isentropic efficiency of the Sayed (2008) and Crane (2005). This means that the compressor is maximised. This ideal flow arrangement velocity at the nozzle throat has reached the local speed is not met at other engine operating conditions such as of sound. Changing the stagnation pressure and take off and landing, where there is a significant temperature through it can only alter the flow from a reduction in compressor efficiency. The practical need fixed convergent nozzle. Therefore the purpose of the to operate engines at off design conditions limits the gas generator (i.e. the compressor, combustor and the pressure ratio that a multiple stage compressor can turbine) is to provide the propelling nozzle with a flow attain if connected to a single-spool. of gas sufficient to choke the propelling nozzles. If we control the stagnation pressure and/or temperature of As the thermodynamic efficiency of an engine is a the flow through the propelling nozzles, we can vary function of pressure ratio, a second and sometimes third the energy transfer to the aircraft. spool is required to drive compressors. This minimises the pressure that the compressor fitted to a single spool It is normal practice to convert turbine and compressor needs to achieve. Modern multi-spool compressors can isentropic efficiencies into polytropic efficiencies for achieve pressure ratios of around 45:1. reasons that need not concern us here (see for example, Cumpsty (2006), Cumpsty (2004) or Dixon (2008)). In this work, we consider two-spool rather three-spool This polytropic efficiency will also remain constant engines. Turbofan engines require a minimum of two- throughout the turbine or compressor. Unlike the spools. Typically, the low-pressure spool drives the fan. isentropic efficiency, it is included in the relationship The role of the fan is two fold. Firstly, it acts as a between the temperature ratio and the pressure ratio compressor pressurising air prior to entering the engine across turbine and compressors. core where a multiple stage compressor driven by the high-pressure spool further compresses the air. 2. ANALYSIS USED IN THE PRESENT Secondly, the fan accelerates a much larger mass flow SIMULATION BASED ON CUMPSTY”S of air, which improves the propulsive efficiency of the REASONING engine including reducing energy lost do to noise. The intention of our present work is to simulate the 1.2. A note on some of the terminology used in this behaviour of different configurations of gas turbine work engines as an aid to teaching aerospace propulsion to It will be noted that in this paper that we specify undergraduate students at our university. Our approach temperatures and pressures as stagnation quantities T differs from the industry codes in that it takes advantage 0 of the simplifications suggested in the analysis of gas and P0 rather than as the more common static quantities T and P. For example, we specify the stagnation turbines by Cumpsty (2006). temperature of air at the inlet to the compressor as T02. The stagnation temperature is the temperature at a gas 2.1. The simplifications can achieve if bought to rest without loss of energy These simplifications arise from aspects common to all through transfers of either heat or work in either a turbine maps. Turbine maps are basically graphs which reversible or irreversible manner. Atkins (2007) depict the relationship between the pressure ratio across provides a good introduction to the concepts of heat, a turbine with either the non-dimensional mass flow work and reversibilility. The airflow through a gas rate through the turbine or the isentropic efficiency of turbine engine is always compressible and so we the turbine. specify its velocity as a Mach number M. The Mach number is a dimensionless number, which expresses the The first simplification arises from the turbine maps ratio of the flow velocity to the local speed of sound. which show the relationship between pressure ratio and Stagnation temperature is therefore expressed as: non-dimensional mass flow rate per unit area. Cumpsty argues that curves plotted at differing constant non- ⎡ ⎛ − 1k ⎞ 2 ⎤ dimensional spool rotational speed, practically collapse 0 ⎢1TT += ⎜ ⎟ M ⎥ (2) ⎣ ⎝ 2 ⎠ ⎦ on top of one another. This means that the relationship between the pressure ratio and the non-dimensional where k is the ratio of specific heat capacities. The mass flow rate is essentially independent of the stagnation pressure relies on the assumption that the rotational speed of the spool. Although properly flow is bought to rest in a reversible manner and is designed turbines and nozzle guide vanes do not expressed as: operate in the choked condition, we simplify the analysis greatly if we accept that they behave as though and they are effectively choked. Therefore the non- 2 η p ()k− 1 dimensional mass flow rate per unit area m is taken to T ⎛ A ⎞ 2k −η p ()k − 1 05 = ⎜ 4 ⎟ . (8) be approximately constant, since for choked flow, the ⎜ ⎟ T04 ⎝ A9 ⎠ normalised mass flow rate per unit area is dependent only on the isentropic index of the gas k = ( cp / cv ) As a result, we can express both the stagnation pressure where c and c are the specific heat capacities of the p v ratio (P04 /P05) and the stagnation temperature ratio gas at constant pressure and volume (see Çengel & (T04 /T05) across the turbines as functions of the ratio of Boles, 2006 for a review of these thermodynamic the turbine inlet area to the nozzle exit area A4 /A9 .