METHOD OF DESIGN OF A TURBINE The present invention relates to a method of design of a turbine of an engine system with an exhaust gas circulation (EGR) system. The method relates particularly to a dual entry turbine used with exhaust gas recirculation. The invention further relates to an engine system incorporating a turbocharger having a turbine. Turbochargers are well-known devices for supplying air to the intake of an internal combustion engine at pressures above atmospheric pressure (boost pressures). A conventional turbocharger essentially comprises an exhaust gas driven turbine wheel mounted on a rotatable shaft within a turbine housing. Rotation of the turbine wheel rotates a compressor wheel mounted on the other end of the shaft within a compressor housing. The compressor wheel delivers compressed air to the inlet manifold of the engine, thereby increasing engine power. The turbocharger shaft is conventionally supported by journal and thrust bearings, including appropriate lubricating systems, located within a central bearing housing connected between the turbine and compressor wheel housing.

In known turbochargers, the turbine stage comprises a turbine chamber within which the turbine wheel is mounted; an annular inlet passageway defined between facing radial walls arranged around the turbine chamber; a volute (a spiral flow passage) having an inlet for receiving exhaust gas from the engine exhaust manifold, and communicating with the inlet passageway around the radially outer portion of the inlet passageway; and an outlet passageway extending from the turbine chamber. The passageway, volute and chamber communicate such that pressurised exhaust gas admitted to the inlet flows through the inlet passageway to the outlet passageway via the turbine and rotates the turbine wheel. The intersection between the volute and the inlet passageway is known as the "critical area". It is known to improve turbine performance by providing vanes, referred to as nozzle vanes, in the inlet passageway so as to deflect gas flowing through the inlet passageway towards the direction of rotation of the turbine wheel. Turbines may be of a fixed or variable geometry type. Variable geometry turbines differ from fixed geometry turbines in that the size of the inlet passageway can be varied to optimise gas flow velocities over a range of mass flow rates so that the power output of the turbine can be varied to suite varying engine demands. For instance, when the volume of exhaust gas being delivered to the turbine is relatively low, the velocity of the gas reaching the turbine wheel is maintained at a level which ensures efficient turbine operation by reducing the size of the annular inlet passageway.

Oxides of nitrogen (NO _x ), which are recognised to be harmful to the environment, are produced during the combustion process in an engine. In order to meet legislation intended to limit emissions exhaust gas recirculation (EGR) systems are used, in which a portion of the engine exhaust gas is recirculated through the combustion chambers. This is typically achieved by directing an amount of the exhaust gas from the exhaust manifold to the inlet manifold of the engine. The recirculated exhaust gas partially quenches the combustion process of the engine and hence lowers the peak temperature produced during combustion. Because NO _x production increases with increased peak temperature, recirculation of exhaust gas reduces the amount of undesirable NO _x formed. A turbocharger may form part of an EGR system. In some known internal combustion engines a variable geometry turbine (which forms part of a turbocharger) is used to increase the pressure (also known as back pressure) of the exhaust gas. This creates a pressure differential between the exhaust gas and the engine intake such that the exhaust gas will flow via an exhaust gas recirculation channel to the engine intake. However, the creation of back pressure by the variable geometry turbine can impair the operating performance of the internal combustion engine.

Known types of turbine include double flow turbine and twin flow turbine. Double flow turbines and twin flow turbines have an inlet which includes two separate flow passages separated by a dividing wall. The two separate flow passages which define at least part of the volute meet at the generally annular inlet passageway. In the case of a twin flow turbine, the two separate flow passages meet at the generally annular inlet passageway such that each flow passage supplies a respective portion of the inlet passageway, the two respective portions being axially spaced from one another. In the case of a double flow turbine, the two separate flow passages meet at the generally annular inlet passageway such that each flow passage supplies a respective portion of the inlet passageway, the two respective portions being substantially in the same plane perpendicular to the axis, but being circumferentially separate (which may also be referred to as circumferentially segmented).

When EGR is used to control criteria pollutants, an engine system with a divided exhaust manifold and a twin entry turbocharger can improve the fuel efficiency by reducing the pumping work needed to drive the EGR. In this configuration, the EGR is drawn from one manifold, relieving the need to maintain the exhaust manifold pressure above the inlet manifold pressure in the second manifold. This second manifold is referred to as the Lambda manifold. Respective volutes (the "EGR volute" and the "Lambda volute") are provided in the turbocharger to receive the exhaust gas from the the EGR and Lambda manifolds, and the recirculation line of the exhaust-gas recirculation system branches off the flowpath from the EGR manifold into the EGR volute. By drawing the exhaust flow from the EGR manifold, the flows from the EGR and Lambda manifolds into the respective turbocharger volutes will be different. Because of the need to maintain a higher exhaust manifold pressure (EMP) and lower flow rate into the turbine, the critical area of the EGR volute should, in general, be smaller than that of the Lambda volute. That is, the turbine housing should be asymmetric. The ratio of the EGR:Lambda volute critical flow areas has a strong impact on the ability to achieved the desired EGR flows, air to fuel ratios and brake fuel efficiency.

Commonly, a valve assembly may be provided to regulate a pressure of exhaust within EGR flow. The valve assembly may include a balance valve and/or a wastegate valve, The balance valve may be configured to selectively allow exhaust from EGR volute to pass to the Lambda volute. The wastegate valve may be configured to selectively allow exhaust from one of the volutes (most commonly, the Lambda volute but optionally the EGR volute) to bypass the turbine wheel, and pass straight to the outet of the turbine. One or more actuators are provided to control the balance valve and the wastegate valve.

Current methods of designing the asymmetric housing to suit specific application parameters are time consuming and labour intensive. The present invention seeks to address this problem. The present invention also provides an alternative method of design of a turbine.

Summary of the Invention

The present inventors have developed a system for selection of parameters of the turbine. Surprisingly, it was discovered that a desirable choice for an asymmetry factor F, which measures the ratio of the mass flow parameters in the two flow paths of a turbocharger with EGR, can be obtained as a function of the pressure drop across the EGR path, and a pressure ratio which is a ratio of the ambient pressure and the pressure at the turbine outlet.

In general terms, the invention proposes a method of designing a turbine of a turbocharger of an engine system, in which at at least one engine speed, a pressure drop along an EGR path is determined, and then the turbine design is selected such that the asymmetry factor is according to the function of the pressure drop. The engine speed may be chosen to be peak torque. The invention further proposes engines in which, at least one engine speed, the asymmetry factor has a relationship to the pressure drop along the EGR path according to the function.

In one possibility the asymmetry factor may be selected by choosing a property of the geometry of the turbine housing. For example, the asymmetry factor may be produced by a suitable choice of the critical areas for the EGR volute and the Lambda volute. Alternatively or additionally, the turbocharger design may incorporate at least one balance valve and/or at least one wastegate valve (preferably the wastegate valve(s) comprise at least one wastegate value located so as to draw exhaust gas from the Lambda volute, although optionally the wastegate valve(s) may comprise at least one wastegate valve arranged to draw exhaust gas from the EGR volute), and a control system for the balance valve and/or wastegate valve(s). The control system may be chosen to control the valve(s) such that the asymmetry factor F is according to the function. By adopting a suitable control of the balance valve and/or the wastegate valve, the asymmetry factor may be made to obey the function for a plurality of engine speeds. According to a first aspect of the invention there is provided a method of designing a turbine of a turbocharger of an engine system, the engine system comprising:

an internal combustion engine comprising at least one cylinder defining a respective bore, within which a piston is arranged to reciprocate, the cylinder having a gas inlet, a gas outlet and a fuel inlet;

the internal combustion engine having an inlet and an outlet including an EGR manifold and a Lambda manifold;

an exhaust gas recirculation system comprising an exhaust gas recirculation path arranged to pass at least a portion of the gas exhaust from the engine outlet back to the engine inlet;

a turbocharger comprising a compressor and a turbine;

the turbine comprising a housing, the housing defining a turbine outlet, a turbine chamber between the at least one turbine inlet and the turbine outlet, an annular inlet passageway arranged around the turbine chamber, an EGR volute having a first turbine inlet for receiving exhaust gas from the engine exhaust manifold, and a Lambda volute having a second turbine inlet for receiving exhaust gas from the Lambda manifold,

the EGR volute and Lambda volute communicating with the inlet passageway around the radially outer portion of the inlet passageway and defining respective flowpaths from the respective turbine inlets to the inlet passageway, and

the turbine further comprising a turbine wheel rotatably mounted within the turbine chamber for rotation about an axis such that it is rotated by gas exhaust from the engine outlet passing from the turbine inlets to the turbine outlet;

the compressor comprising a housing, the housing defining a compressor inlet, in gas communication with an air source, a compressor outlet in gas communication with the engine inlet, a chamber between the compressor inlet and the compressor outlet and an impeller wheel rotatably mounted within the chamber for rotation about an axis such that rotation of the impeller wheel compresses air from the compressor inlet and passes the compressed air to the compressor outlet;

the turbine wheel being coupled to the impeller wheel such that the rotation of the turbine wheel drivably rotates the impeller wheel;

wherein the method comprises the steps of:

(i) for at least one engine speed determining an EGR loop pressure drop (x) along the recirculation path; (ii) determining an asymmetry ratio representing the ratio of mass flow parameters along the flowpaths, from an expression substantially of the form: y=a In (x) + b (A) where the parameter y is the asymmetry ratio multiplied by a pressure ratio Z of the ambient pressure and the turbine outlet pressure, and a and b are real valued constants;

(iii) selecting a turbine design with a ratio of the mass flow parameters at the at least one engine speed according to the determined asymmetry ratio.

In a preferred form, the formula (A) is given by:

y = -0.1311n(x) + 0.8523. (B) The values of x and y may be such that expressions (A) and (B) are exactly observed, at least to within the typical accuracy of a numerical computation. Alternatively, it is to be understood that in certain embodiments of the first aspect of the invention, a value for the asymmetry ratio may be chosen such that, for each value of x, / differs from the exact value given by Equations (A) and (B) by a small respective tolerance. For example, for each of the at least one engine speeds, a value for y may be used which differs from by value given by Eqn. (A) or (B) by no more than a tolerance value δ. The tolerance value may have a magnitude less than 0.1 , less than 0.05, less than 0.04, less than 0.02, less than 0.01 , less than 0.005, less than 0.001 , less than 0.0005 or even less than 0.0001 .

In one option, the step of selecting the turbine design comprises selecting the areas of the respective "critical areas" at which the EGR volute and the Lambda volute join the inlet passageway. The method provides a relatively fast and simple method of sizing the two critical areas of a turbine, of an engine system which comprises exhaust gas recirculation, based on certain design parameters of the engine system.

Optionally, the method comprises the step of manufacturing a turbine having first and second inlets with the selected critical areas of the first and second turbine inlets. According to a second aspect of the invention there is provided an engine system manufactured according to the method of the first aspect of the invention.

According to a third aspect the invention provides an engine system produced according to the above principles.

Specifically, the third aspect of the invention provides an engine system comprising: an internal combustion engine comprising at least one cylinder defining a respective bore, within which a piston is arranged to reciprocate, the cylinder having a gas inlet, a gas outlet and a fuel inlet, the internal combustion engine having an inlet and an outlet including an EGR manifold and a Lambda manifold;

an exhaust gas recirculation system comprising an exhaust gas recirculation path arranged to pass at least a portion of the gas exhaust from the engine outlet back to the engine inlet;

a turbocharger comprising a compressor and a turbine;

the turbine comprising a housing, the housing defining at least one turbine inlet in gas communication with the engine outlet, a turbine outlet, a turbine chamber between the at least one turbine inlet and the turbine outlet, an annular inlet passageway arranged around the turbine chamber, an EGR volute having a first turbine inlet for receiving exhaust gas from the engine exhaust manifold, a Lambda volute having a second turbine inlet for receiving exhaust gas from the Lambda manifold,

the EGR volute and Lambda volute communicating with the inlet passageway around the radially outer portion of the inlet passageway and defining respective flowpaths from the respective turbine inlets to the inlet passageway, and

the turbine further comprising a turbine wheel rotatably mounted within the turbine chamber for rotation about an axis such that it is rotated by gas exhaust from the engine outlet passing from the at least one turbine inlets to the turbine outlet;

the compressor comprising a housing, the housing defining a compressor inlet, in gas communication with an air source, a compressor outlet in gas communication with the engine inlet, a chamber between the compressor inlet and the compressor outlet and an impeller wheel rotatably mounted within the chamber for rotation about an axis such that rotation of the impeller wheel compresses air from the compressor inlet and passes the compressed air to the compressor outlet; the turbine wheel being coupled to the impeller wheel such that the rotation of the turbine wheel drivably rotates the impeller wheel;

wherein, for at least one engine speed, the relationship between the EGR loop pressure drop (x) along the recirculation path, and the asymmetry ratio representing the ratio of mass flow parameters along the flowpaths, is according to the expression y = -0.131 ln(x) + 0.8523 + δ. (C) where the parameter y is the asymmetry ratio multiplied by a pressure ratio Z of the ambient pressure and the turbine outlet pressure, and δ is a tolerance parameter having a magnitude no greater than 0.1 .

More preferably, the value of δ for each value of x has a magnitude less than 0.05, less than 0.04, less than 0.02, less than 0.01 , less than 0.005, less than 0.001 , less than 0.0005, or even less than 0.0001. The at least one engine speed may include the peak torque engine speed for the engine system, and/or the rated power speed for the engine system, and/or one or more engine speeds which are a first engine speed n _to plus a respective proportion X of the difference between the first engine speed ni ₀ and a second engine speed n _hi , where is selected from the group comprising 15%, 25%, 50% and 75%, and n _to and n _hi are respectively the lowest engine speed for which a first predetermined power value is achievable by the engine system, and a highest engine speed for which a second predetermined power value is achievable by the engine system.

Preferably, the engine system further comprises a balance valve connecting the EGR volute and the Lambda volute and/or a wastegate gate for diverting exhaust gas from the Lambda volute to the turbine outlet. The engine system further preferably comprises a valve control system for controlling the balance valve and/or the wastegate valve. The control system is preferably arranged so that for a plurality of engine speeds, the asymmetry ratio is according to Eqn. (C). The range of engine speeds preferably includes the peak torque engine speed. It may further include the rated engine speed. It may include any one or more of an engine speed which is higher than the peak torque speed by 50% of the difference between the rated engine speed and peak torque speed, and an engine speed which is higher than the peak torque speed by 70% of the difference between the rated engine speed and the peak torque speed. It may further include any one or more of the engine speeds defined above using the values X, Π| ₀ and n ₁₀ .

Specific embodiments will now be described with reference to the accompanying drawings:

Figure 1 schematically depicts an axial cross-section through a variable geometry turbocharger; Figure 2 schematically depicts an engine system having a turbine that is the subject of a method of design described herein;

Figure 3 is a flowchart showing the steps of the method of design of a turbine described herein ;

Figures 4a to 4c are each a graph showing the values of pressure ratio and a mass flow parameter of the compressor of the engine system shown in Figure 2, that are calculated to provide a required mass flow rate of air through the engine of the engine system shown in Figure 2, plotted over a compressor map;

Figure 5 is a graph showing the variation of turbine expansion ratio (P ₃ /P ₄ ) with a turbine mass flow parameter for a number of different sizes of turbine;

Figure 6 schematically depicts a second engine system having a turbine that is the subject of a method of design described herein;

Figure 7 shows a flowchart showing the steps of the method of design of a turbine described herein; Figure 8 schematically depicts a third engine system having a turbine that is the subject of a method of design described herein;

Figure 9 shows a flowchart showing the steps of the method of design of a turbine described herein; Figure 10 is a graph showing the variation of turbine expansion ratio (P ₃ /P ₄ ), for first and second inlets of the turbine, with a turbine mass flow parameter;

Figure 1 1 is a graph illustrating results from carrying out methods according to Figures 7 and 9; and

Figure 12 is a flowchart showing the steps of a method of design of a turbine according to an embodiment of the present invention. Figure 1 illustrates a turbocharger 100 comprising a turbine 1 and a compressor 2 interconnected by a central bearing housing 3. A turbocharger shaft 4 extends from the turbine 1 to the compressor 2 through the bearing housing 3. The turbine 1 and compressor 2 each comprise a respective housing 101 , 102. A turbine wheel 5 is mounted on one end of the shaft 4 for rotation within the turbine housing 101 , and a compressor wheel 6 is mounted on the other end of the shaft 4 for rotation within the compressor housing 102. The shaft 4 rotates about turbocharger axis V-V on bearing assemblies located in the bearing housing 3. The turbine housing 101 defines an inlet 7 to which gas from an internal combustion engine (83- see below) is delivered. The exhaust gas flows from the inlet 7 to an outlet 81 via an axial outlet passageway, an annular inlet passageway 9 and turbine wheel 5.

The gas inlet 7 is made up of a first volute 9a, and a second volute 9b. These volutes each have respective critical areas 15a, 15b at their most radially-inward portion, where they direct the exhaust gas at the turbine wheel 5. In this example, the first and second volutes 9a, 9b are symmetric, but in other turbochargers, they are differently dimensioned from each other. Optionally, a wastegate (not shown) may be provided in the turbine housing 101 to divert a portion of the exhaust gas from the exhaust gas inlet 7 to the exhaust gas outlet 81 without passing the turbine wheel 5.

Gas flowing from the inlet 7 to the outlet 81 passes over the turbine wheel 5 and as a result torque is applied to the shaft 4 to drive the compressor wheel 6. Rotation of the compressor wheel 6 within the compressor housing 102 draws air through an inlet 22 of the compressor and delivers the pressurised air to an outlet 82 via an air outlet volute 23 from which it is fed to an internal combustion engine 83. We now describe three detailed possible realisations of the turbo-charger, shown in Figures 2, 6 and 8, and for each one explain a design methodology for the turbo- charger. Figure 2 shows a symmetric case with EGR but no cooler. Figure 3 shows a symmetric case with EGR and a cooler. Figure 8 shows an asymmetric case with EGR and a cooler. Results obtained from using this methodology in relation to Figure 8 are shown in Figure 1 1 , and this leads to a method of design which is an embodiment of the invention, and shown in Figure 12.

First example of a turbo-charger design method Referring to figure 2 there is shown a schematic view of a known engine system 70, incorporating the turbocharger 100 of Figure 1 . The engine system 70 comprises an internal combustion engine 83, and said turbocharger 100.

The internal combustion engine 83 comprises six cylinders 40. Each cylinder defines a bore, within which a piston (not shown) is arranged to reciprocate. Each cylinder has an inlet and an outlet. Each cylinder is substantially identical.

The internal combustion engine 83 further comprises an inlet manifold 41 which connects each inlet of the cylinders 40 to an entry port 42 of the inlet manifold 41.

The entry port 42 of the inlet manifold 41 is connected via path 43 to the outlet 82 of the compressor 2. The compressor wheel 6 is driven to rotate by the rotation of the turbine wheel 5, The compressor wheel 6 draws air in from an air source (not shown) at ambient pressure (P _amb ) and ambient temperature (T _amb ) via a path 91 to the compressor inlet 22. A filter 89 is provided in the path 91 , between the air source and the compressor inlet 22.

At the inlet 22 of the compressor 2 the air is at a pressure P- _\ . At the outlet 82 of the compressor 2 the air is at a pressure P ₂ . The ratio of the air pressure at the compressor outlet 82 (P ₂ ) to the air pressure at the compressor inlet (P^ is the compressor pressure ratio (PR).

The compressor 2 delivers compressed air from its outlet 82 via the path 43 to the inlet manifold 41 of the internal combustion engine 83 and thus to the inlets of the cylinders 40.

A cooler 84 (referred to as a charge air cooler) is provided in the path 43, between the compressor outlet 82 and the inlet manifold 41 . The cooler 84 cools the compressed air prior to the compressed air being delivered to the inlet manifold 41 . There is a pressure drop (AP _CAC ) across the cooler 84.

The air pressure at the engine inlet manifold 41 (the 'inlet manifold pressure') will be referred to as P| _M -

The internal combustion engine 83 further comprises an exhaust manifold assembly 85.

The exhaust manifold assembly 85 connects the outlets of the cylinders 40 to the turbine inlet 7. Exhaust from the cylinders 40 thus drives the turbine wheel 5 to rotate, which in turn rotates the compressor wheel 6 via the shaft 4. As mentioned above, the compressor 2 delivers compressed air to the inlet manifold 41 . On exiting the turbine 1 , the exhaust gas is released to the atmosphere from an outlet after travelling along an exhaust outlet path 86. Optionally, an after-treatment system (not shown) may be provided on the exhaust outlet path 86.

An optional wastegate valve 192 is provided which allows a proportion of the exhaust gas to be diverted away from the turbine wheel 5 to the exhaust outlet path 86. The pressure and temperature of the exhaust gas from the engine 83 at the turbine inlet 7 will be referred to as P ₃ and T ₃ respectively. The mass flow rate of the exhaust gas at the turbine inlet 7 will be referred to as rh _exh .

The pressure and temperature of the exhaust gas at the turbine outlet 81 will be referred to as P ₄ and T ₄ respectively. A method of calculating the required critical area of the turbine 5, for certain properties of the internal combustion engine 83 will now be described.

With reference to Figure 3, the method comprises the following steps:

(i) Calculate a required engine mass flow rate of air through the engine (rh _air ) for a plurality of speeds of the engine. These preferably include the peak torque engine speed and the rated (i.e. maximum power) engine speed;

(ii) Calculate the compressor pressure ratio (PR) required, to provide the

calculated engine mass flow rate of air (rh _air ), at each engine speed;

(iii) Calculate a compressor mass flow parameter, at each engine speed;

(iv) Select a compressor design that provides both the required compressor pressure ratio (PR) and compressor mass flow parameter at each engine speed;

(v) Determine the efficiency of the compressor;

(vi) Determine the power required by the compressor (P _∞ m _P ) at each engine speed;

(vii) Calculate the power needed to be generated by the turbine in order to

provide the required power to the compressor;

(viii) Calculate the expansion ratio (ER) of the turbine needed to provide the power required by the compressor;

(ix) Calculate a turbine mass flow parameter based on the flow rate through the turbine;

Select a turbine design with a critical inlet area that provides the required values of expansion ratio (ER) and turbine mass flow parameter and determining the critical inlet area.

Consider waste-gate requirements.

Manufacture a turbine having the critical inlet area, and any required wastegate.

The steps of this method will now be described in more detail. In relation to step (i), at each of a plurality of required engine speeds (N) the mass flow rate m _air (g/s) of air through the engine 83 is calculated from required values of engine power (P _En g). brake specific fuel consumption (BSFC) and the air to fuel ratio (AFR) of the engine 83, at that engine speed, from equation 1 . mai = P _E ng X BSFC X AFR

Equation 1

Where:

rh _air = mass flow rate of air through the internal combustion engine 83

P _En g = Power of the internal combustion engine (kW)

BSFC = Brake specific fuel consumption of the internal combustion engine (g/kW.hr)

AFR = air to fuel ratio of the internal combustion engine

The plurality of engine speeds (operating points) typically include at least the peak torque engine speed, the rated engine speed. They preferably include at least one low engine speed, at which it is challenging to get sufficient air flow. In relation to step (ii), the mass flow rate of air m _air through the engine 83 can be expressed as equation 2.

Equation 2 Where: m _air = mass flow rate of air through the engine 83

D = Engine swept volume (m ³ )

N = Engine speed (rev/sec)

NRPC = Number of Revs Per Cycle of the engine

η _νο ι = Engine Volumetric Efficiency

P _amb = Ambient pressure (Pa) APfiiter = Pressure change across the filter (Pa)

P ₂ = Pressure at compressor outlet (Pa)

Pi = Pressure at compressor inlet (Pa)

ΔΡΟΑΟ = Pressure change across the charge air cooler (Pa)

R = Gas constant for air (= 0.287 J/g.K)

IMT = Inlet manifold temperature (K)

The ratio of the pressure at the compressor outlet to the pressure at the compressor inlet (P ₂ /Pi) is known as the pressure ratio (PR) across the compressor 2.

Re-writing equation 2 in terms of the compressor pressure ratio (PR) gives: m _air X NRPC x R x IMT AP _CAC

PR = — 1 7Z 7 +

N X D X r\ _vol X [Pamb ~ ^Pfilter) [Pamb ~ ^P filter)

Equation 3

This equation provides a means of calculating the compressor pressure ratio (PR) required, to provide the required engine mass flow rate of air (m _air ), at each engine speed (the engine mass flow rate (rh _air ) at each engine speed, is calculated from equation 1 ).

The values of the number of revs per cycle of the engine (NRPC), the gas constant for air (R), the inlet manifold temperature (IMT), the engine swept volume (D), the engine volumetric efficiency (η _ν0 ι), the pressure change across the charge air cooler (AP _CAC ), the ambient pressure (P _am b) and the pressure change across the filter (AP _fi |, _er ), at each engine speed, are pre-selected values that are required by the design.

In relation to step (iii), a compressor mass flow parameter is calculated as:

m _nir ^T _x

l M ^{vl 1} F ^r Pcomp = —— p

1

Equation 4

Where:

MFP _comp = Compressor mass flow parameter

m _air = mass flow rate of air through the engine Pi = Pressure at compressor inlet (Pa)

J-i = Temperature (absolute) at compressor inlet (K)

As with equation 3, the engine mass flow rate (rh _air ) at each engine speed is calculated from equation 1

In relation to step (iv), a design of compressor is selected, from a range of known designs, that provides both the required compressor pressure ratio (PR) and compressor mass flow parameter at each engine speed.

The compressor design is selected by matching known compressor maps to the required compressor pressure ratios (PR) and compressor mass flow parameters at the engine speeds, as calculated above. In this regard, a compressor map is a plot of compressor pressure ratio (PR) (on the y- axis) against compressor mass flow parameter (on the x-axis) for a certain compressor design. Examples of compressor maps for three different compressors are shown in Figures 4a to 4c. The compressor maps may be obtained from empirical data, from measurements of the performance of known compressor designs, and/or from analytical methods that calculate the performance of a compressor based on its design. Such analytical methods are known in the art and so will not be described in any detail here. In this example, three different engine speeds (N) are used and, at each engine speed, the compressor pressure ratio (PR) required, to provide the calculated engine mass flow rate of air (rh _air ), is calculated from equation 3 the mass flow parameter is calculated from equation 4. These values of compressor pressure ratio (PR) and mass flow rate of air (m _air ) are plotted on the same axes used for a compressor map, as shown in Figures 4a to 4c. These values are known as 'running points' of the engine 83.

A compressor map is then chosen that the running points fit into. This is illustrated in Figures 4a to 4c. In Figure 4a the compressor map is too 'small' as the running points at higher values of mass flow rate parameter are not contained on, or within, the compressor map.

In Figure 4b the compressor map is too 'large' as the running points at lower values of mass flow rate parameter are not contained on, or within, the compressor map.

In Figure 4c the compressor map is a selected map that 'matches' the engine 83 since all of the running points are contained on, or within, the compressor map. This compressor map is then selected as a compressor map that is 'matched' to the engine 83. In relation to step (v), the compressor map corresponds to a particular compressor, with a compressor efficiency (r) _∞ mp)- In this regard, the compressor efficiency (n _∞ mp) can be ascertained from the contours of a compressor map (not shown). The compressor efficiency is the ratio of the isentropic change in enthalpy over the actual change in enthalpy. An efficiency of less than 100%, for a given realized increase in the pressure ratio, will result in a higher compressor outlet temperature than the ideal.

In relation to step (vi), from the compressor efficiency (r) _∞ mp), the compressor power (Powercomp) required for these running points is calculated according to the following equation:

Power _Comp

Equation 5

Where:

Power _comp = Compressor power (kW) rh _air = mass flow rate of air through the engine (kg/s)

Cp air = specific heat at constant pressure for air (~ 1005 KJ/kg.K)

L = compressor inlet temperature (K)

PR = compressor pressure ratio at the running point

ncomp = compressor efficiency (fraction, e.g. 0.7)

γ = isentropic expansion factor In relation to step (vii), the power of the turbine 1 required to drive the compressor 2, taking into account the bearing losses in the system, is calculated according to:

Power, comp

Power _turb =

bearing

Equation 6

Where:

Power, _urb = Turbine power (kW)

Power _∞mp = Compressor power (kW)

Hbearing = Bearing efficiency (fraction, e.g. 0.97) turbine power (Power, _urb ) can be expressed according to

Power _turb

Equation 7

Where:

turbine efficiency (fraction, e.g. 0.7)

m _exh = mass flow rate of exhaust from the engine (kg/s) (which equals the mass flow rate through the turbine)

T ₃ = Turbine inlet temperature (K)

P ₃ = Turbine inlet pressure (Pa)

P ₄ = Turbine outlet pressure (Pa)

In relation to step (viii), the ratio of turbine inlet pressure (P3) to turbine outlet pressure (P4) is known as the turbine expansion ratio (ER).

The mass flow rate (m _exh ) exhaust from the engine 83 (which equals the mass flow rate through the turbine 1 ) is the sum of the mass flow rate (m _air ) of air through the engine 83 and the mass flow rate of fuel (m _fuel ) (kg/s) through the engine 83. This is calculated according to:

Equation 8 Where:

m _exh = mass flow rate of exhaust from the engine (kg/s)

rh _air = mass flow rate of air through the engine (kg/s)

AFR = air to fuel ratio

Substituting equation 8 into equation 7, and rearranging for turbine expansion ratio (P3/P4) gives:

l-Y

Power _turb

ER =

Equation 9

The engine mass flow rate (m _air ) at each engine speed is calculated from equation 1 . The values of air to fuel ratio (AFR) and turbine efficiency (¾ _ur6 ) are pre-selected values that are required by the design. The value of specific heat at constant pressure for the exhaust gas (C _Pexh ) is known. Note that the exhaust gas comprises not only air but C0 ₂ , H ₂ 0, etc. The value of the turbine inlet temperature (T ₃ ) is estimated from the heat balance, since the fuel energy is partly converted to work, partly discharged to the coolant, and partly heats up the intake air, resulting in T ₃ . The turbine power required, to drive the compressor, at each engine speed, is calculated from the compressor power required to provide the required air mass flow rate (m _air ) at each engine speed, calculated from equation 5, in conjunction with equation 6. Equation 9 is then used to calculate the turbine expansion ratio required at each engine speed.

In relation to step (ix), a turbine mass flow parameter (MFP, _urb ) is calculated according to:

MFP _turb = ^ex _n ^h

Equation 10

Where:

MFP _turb = turbine mass flow parameter

m _exh = mass flow rate of exhaust from the engine (kg/s) T ₃ = Turbine inlet temperature (K)

P ₃ = Turbine inlet pressure (Pa)

As with equation 9, the value of the turbine inlet temperature (T ₃ ) is estimated from known values.

From a consideration of the conditions at the turbine outlet, the turbine exhaust pressure (P ₄ ) can be calculated according to:

^4 ^— ^ambient ^exh BP

Equation 11

Where:

P ₄ = Turbine exhaust pressure

Pam ient = Ambient pressure downstream of the turbine outlet (Pa) Pexh BP = Turbine exhaust back pressure (Pa)

The value of turbine exhaust back pressure (P _eX h BP) and the ambient pressure downstream of the turbine outlet (Pambiem) are estimated from known values. The exhaust backpressure depends on the operating conditions (e.g. it is higher for a higher exhaust flow rare). For example, it may vary as the square of the exhaust volumetric flow rate. Its magnitude also depends on what (if anything) is downstream of the turbine; for example, the backpressure will be raised if an after-treatment system is provided.

From the calculated values of expansion ratio (P ₃ /P ₄ ) in equation 9, along with the calculated values of turbine exhaust pressure (P ₄ ), the vales of the turbine inlet pressure (P3) can then be calculated from:

P ₃ = ER x P ₄

Equation 12 The turbine mass flow parameter (MFP, _urb ), at each engine speed, can then be calculated from equation 10.

In relation to step (x), a turbine flow map is a plot of the variation of turbine expansion ratio (ER) (y-axis) with the variation in turbine Mass Flow Parameter (MFP, _urb ), as shown in Figure 5.

Figure 5 shows a number of different turbine flow maps, that each correspond to a different size of turbine inlet 7 (each curve is labelled with the critical area of the turbine that corresponds to that curve). In this regard, each size of turbine, and therefore of turbine inlet 7, corresponds to a different curved line, with the curved lines moving to the right on the x-axis with increasing turbine inlet 7 size. In this regard, with increasing turbine inlet 7 size, the turbine Mass Flow Parameter (MFP, _urb ) increases for a certain turbine expansion ratio (P ₃ /P ₄ ).

The turbine flow maps may be obtained from empirical data, from measurements of the performance of known turbine designs, and/or from analytical methods that calculate the performance of a turbine based on its design. Such analytical methods are known in the art and so will not be described in any detail here.

The required turbine size is selected from the turbine flow map by selecting a curve, and therefore a turbine with a certain size of inlet 7, that provides the desired turbine expansion ratio (P ₃ /P ), for a certain turbine Mass Flow Parameter (MFP, _urb ). The desired combination of values of turbine expansion ratio (A) and turbine Mass Flow Parameter (B) is shown as X on Figure 5. In this example, the curve that provides the required combination of turbine expansion ratio (P ₃ /P ) and turbine Mass Flow Parameter (MFP, _ur ) corresponds to a turbine with a critical area of 15cm ² .

In step (x), consideration is given to whether a wastegate 92 should be included in the design, and if so how it is controlled. Up to this point, the peak torque is often the limiting factor for both the air flow and the EGR. In step (x), a check is performed of whether we can meet the requirements (both air fuel ratio (AFR) and EGR)) for other operating points, such as rated power. Mechanical limited are also considered (e.g. exhaust manifold pressure and/or turbo speed). If the requirements of all operating points cannot be satisfied with a single compressor and turbine, a waste-gate may needed to protect for altitude.

In step (x), an engine system is constructed using the critical areas obtained in step (x) and the wastegate considered in step (x).

The above method provides a relatively fast and simple method of sizing the critical area of a turbine based on certain design constraints of an engine, namely the engine power, speed, efficiency, brake specific fuel consumption, air to fuel ratio and inlet manifold temperature. The method also takes into account pressure drops across other components of the engine system, such as the across a charge air cooler and a filter. It will be appreciated that the method is applicable to where the engine system does not comprise a charge air cooler and/or filter, which the relevant equations modified to remove these terms as appropriate.

The above method is suitable for sizing a turbine without exhaust gas recirculation. It will now be described how this method can be modified in order to size a turbine where it is part of an engine system that uses exhaust gas recirculation. Modification of the method to include EGR

Referring to Figure 6 there is a shown an engine system according to a further example. The engine system shown in Figure 6 is identical to the engine system shown in Figure 2 except for the differences described below. Corresponding features are given corresponding reference numerals, but incremented by 100.

The engine system 201 shown in Figure 6 differs from the engine system shown in Figure 2 in that an exhaust gas recirculation (EGR) path 203 passes a portion (m _egr ) of the mass of gas exhaust (m _exh ) from the internal combustion engine 183 back to the engine inlet manifold 141 .

An EGR valve 204 is provided in the exhaust gas recirculation path 203, which can be selectively opened and closed, as well as throttled between the open and closed positions, so as to vary the amount of gas recirculated ( _egr ). An exhaust gas cooler 202 is provided in the EGR path 203, between the EGR valve 204 and the engine inlet manifold 141 . Note that in the configuration of Fig. 6, the EGR valve is shown on the "hot" (i.e. upstream) side of the cooler 202, but it may alternatively be provided on the "cold" side.

Accordingly, the mass flow rate of gas that passes to the engine inlet manifold 141 is the sum of the mass flow rate of gas (m _air ) from the compressor 106 and the mass flow rate of the recirculated gas (m _egr ) . The pressure of the recirculated gas entering the EGR valve 204 is P ₃ '. The pressure of the recirculated gas exiting the EGR valve 204 is P ₃ ".

The pressure of the recirculated gas entering the EGR cooler 202 is P ₃ '". The pressure of the recirculated gas exiting the EGR cooler 202 is P ₃ '".

The fraction (EGR _frac ) of the EGR mass flow rate (m _egr ) relative to the mass flow rate (jhexh ) of gas into the internal combustion engine 183 is calculated from:

E GRfrac

megr ^m air

Equation 13

A method of calculating the required critical area of the turbine 105, for certain properties of the internal combustion engine 183 will now be described.

With reference to Figure 7, the method comprises the following steps:

Calculate a required engine mass flow rate of air through the engine (rh _ai . for a plurality of speeds of the engine, typically including the peak torque and the rated power;

Calculate the compressor pressure ratio (PR) required, to provide the calculated engine mass flow rate of air(m _air ) and EGR fraction.

Calculate the compressor outlet pressure (P2) from the calculated compressor ratio; (iv) Calculate the turbine inlet pressure (P3) from the compressor outlet pressure (P2) and any pressure changes across the EGR system components;

(v) Calculate the turbine expansion ratio (ER) from the calculated turbine inlet pressure (P3) and the turbine outlet pressure (P4);

(vi) Calculate a turbine mass flow parameter from the calculated turbine inlet pressure (P3);

(vii) Select a turbine design with a critical inlet area that provides the required values of expansion ratio and turbine mass flow parameter and determining the critical inlet area; (viii) Perform a constraint analysis at the peak torque engine speed. Check also requirements at other operating points (e.g. rated power engine speed) and add wastegate if necessary.

(ix) Manufacture a turbine having the calculated critical area, and with any

required wastegate;

In relation to step (i), at each of a plurality of required engine speeds (N) (typically including at least the peak torque and rated power engine speeds). The mass flow rate of air (rh _air ) from the compressor (106) is calculated from required values of engine power (P _En g). brake specific fuel consumption (BSFC) and the air to fuel ratio (AFR) of the engine 183, at that engine speed, from equation 14 (which corresponds to equation 1 )- m _ai r+= P _E ng X BSFC X AFR

Where:

rh _air = mass flow rate of air passing to the engine inlet from the compressor (kg/s)

P _En g = Power of the internal combustion engine (kW)

BSFC = Brake specific fuel consumption of the internal combustion engine

(g/kW.hr)

AFR = air to fuel ratio of the internal combustion engine In relation to step (ii), the mass flow rate of air m _air through the engine 183 can be expressed as equation 15.

EGRfrac) NRPC

Equation 15

Where: m _air = mass flow rate of air passing to the engine inlet from the compressor

(kg/s)

EGRfrac = the fraction of the mass flow rate of gas passing to the engine inlet from the exhaust gas recirculation system relative to the total mass flow rate of gas into the internal combustion engine

D = Engine swept volume (m ³ )

N = Engine speed (rev/sec)

NRPC = Number of Revs Per Cycle of the engine

η _νο ι = Engine Volumetric Efficiency

P _amb = Ambient pressure (Pa)

APfiiter = Pressure change across the filter (Pa)

ΔΡΟΑΟ = Pressure change across the charge air cooler(Pa)

P ₂ = Pressure at compressor outlet (Pa)

P-t = Pressure at compressor inlet (Pa)

R = Gas constant for air and gas mixture (for air alone, this is 0.287 J/g.K) IMT = Inlet manifold temperature (K)

Equation 15 corresponds to equation 2 but modified to take account of the exhaust gas recirculation.

Re-writing equation 15 in terms of the compressor pressure ratio (PR) gives: (m _air ) X NRPC X R X IMT AP _CAC

N x D x _nvol X (1 - EGR _frac ) X P ₁ P ₁

Equation 16

This equation provides a means of calculating the compressor pressure ratio (PR) required, to provide the required engine mass flow rate of air (m _air ) at each engine speed (the engine mass flow rate (rh _air ) at each engine speed, is calculated from equation 14).

The values of the number of revs per cycle of the engine (NRPC), the gas constant (R) for air and EGR (this property is calculated based on the input air and EGR flow requirements), the inlet manifold temperature (IMT), the engine swept volume (D), the engine volumetric efficiency (η _ν0 ι), the pressure change across the charge air cooler 184 (AP _CAC ), the ambient pressure (P _amb ) and the pressure change across the filter (AP _f ii, _er ), at each engine speed, are pre-selected values that are required by the design.

The compressor inlet pressure (P^ is calculated from:

Pi ⁼ Pamb ^~ ^P filter

Equation 17

In relation to step (iii), the compressor outlet pressure (P ₂ ) is then calculated from the calculated compressor pressure ratio (PR) and compressor inlet pressure by:

P ₂ = PR x p ₁

Equation 18 In relation to step (iv), the turbine inlet pressure (P ₃ ) is then calculated according to:

^3 ⁼ Pi + &P _CAC + P _e g _{r coo} i _er + P _e g _{r va} i _ve

Equation 19

In relation to step (v), the turbine outlet pressure (P ₄ ) is calculated according to:

P — Pambient Pexh BP Equation 20

Where:

P ₄ = Turbine exhaust pressure

Partem = Ambient pressure downstream of the turbine outlet (Pa) Pexh BP = Turbine exhaust back pressure (Pa)

The value of turbine exhaust back pressure (P _eX h BP) and the ambient pressure downstream of the turbine outlet (Pambiem) are estimated from known values. The exhaust backpressure is estimated based on the operating conditions (e.g. it is higher for a higher exhaust flow rare). For example, it may vary as the square of the exhaust volumetric flow rate. Its magnitude also depends on what (if anything) is downstream of the turbine; for example, the backpressure will be raised if an after-treatment system is provided. Note that when comparing two operating points (peak torque and rated power engine speeds) there will tend to be significant differences in backpressure, so exhaust backpressure is an important consideration.

The turbine expansion ratio (ER) is then calculated from :

The net mass flow rate (m _exh ) of exhaust from the engine 183 into the turbine 107 is the sum of the mass flow rate (m _air ) of air from the compressor 106 and the mass flow rate of fuel (rh _fuel ) (kg/s) through the engine 183. This is calculated according to:

Equation 22

Where:

m _exh = mass flow rate of exhaust into the turbine (kg/s)

rh _air = mass flow rate of air through the engine (kg/s)

AFR = air to fuel ratio In relation to step (vi), a turbine mass flow parameter (MFP, _urb ) is calculated according to:

MFP _turb = ^exh

3

Equation 23

Where:

MFP _turb = turbine mass flow parameter

m _exh = mass flow rate of exhaust from the engine into the turbine (kg/s) T ₃ = Turbine inlet temperature (K)

P ₃ = Turbine inlet pressure (Pa)

The value of mass flow rate of exhaust from the engine (m _exh ) into the turbine used in equation 23 is calculated from equation 22. The value of the turbine inlet pressure (P ₃ ) used is that calculated from equation 19. T ₃ is estimated using the heat balance, since the fuel energy is partly converted to work, partly discharged to the coolant, and partly heats up the intake air, resulting in T ₃ .

In relation to step (vii), as with the previous example (see Figure 5), the required turbine size is selected from the turbine flow map by selecting a curve, and therefore a turbine with a certain size of inlet, that provides the desired turbine expansion ratio (P3/P4), for a certain turbine Mass Flow Parameter (MFP, _urb ).

In step (ix) a constraint analysis is performed. The preference would be to select the critical area to meet air and EGR flow requirements with the EGR valve wide open based on peak torque requirements, but we need to check the compressor balance and the turbine power. If these cannot be made to balance, then there are three options:

a) Change the air flow and/or EGR flow requirements, or

b) If the turbine power is too low, partially close the EGR valve, or

c) If turbine power too high, add a waste gate

Before deciding which option is best, step (ix) includes considering other key operating points such as rated power, since closing the EGR valve or adding a wastegate at peak torque, will likely require this to be done even more at rated power, and this is inefficient.

Step (ix) usually ends up trading off acceptable parameters at the other key operating points, for lower EGR at peak torque. In other words, the trade-offs force a sub-optimal configuration.

In relation to step (ix), turbine having the calculated critical area and any required wastegate is then manufactured.

The above method provides a relatively fast and simple method of sizing the critical area of a turbine, that is part of an engine system that uses exhaust gas recirculation, based on certain design constraints of an engine, namely the engine power, speed, efficiency, brake specific fuel consumption, air to fuel ratio and inlet manifold temperature. The method also takes into account pressure drops across other components of the engine system. It will be appreciated that the method is applicable to where the engine system does not comprise these components, the relevant equations may be modified to remove these terms as appropriate. The above method is suitable for sizing a turbine with a single entry turbine inlet, and may also be used for a dual entry symmetrical turbine inlet in which the EGR is drawn equally from both manifolds. It will now be described how this method can be modified in order to size a turbine with a dual entry asymmetrical turbine inlet. Modification of the method to include EGR and an asymmetric turbine

Referring to Figure 8 there is a shown an engine system according to a further example. The engine system shown in Figure 8 is identical to the engine system shown in Figure 6 except for the differences described below. Corresponding features are given corresponding reference numerals, but incremented by 100.

The engine system 301 shown in Figure 8 differs from the engine system shown in Figure 6 in that the turbine 205 comprises an asymmetric twin entry inlet, with a 'large' turbine inlet and a 'small' turbine inlet. This is modelled by modelling the turbine 205 as two turbines in the form of a small turbine 201 ' and a large turbine 201 ". The pressure at the small turbine inlet 207' is P _{3 sma} ii. The pressure at the large turbine inlet 207" is P _{3 l} arge- A balancing valve 285 (or "balance valve") is provided between the turbine inlets. Note that the balancing valve 285 may be internal to the turbine housing.

The wastegate 292 in this case is shown on the large turbine 201 ", diverting exhaust gas away from the large turbine inlet 207". In other words, this represents a wastegate positioned for diverting gas from the Lambda volute. However, in a variation the wastegate 292 (or an additional wastegate) could be arranged to divert gas from the EGR volute, i.e. in effect to avoid the small turbine 201 '. A method of calculating the required critical area of each of the large 207" and small 207' inlets of the turbine 205, for certain properties of the internal combustion engine 283 will now be described.

With reference to Figure 9, the method comprises the following steps:

(i) Calculate a required engine mass flow rate of air through the engine (rh _air ) for a plurality of speeds of the engine (e.g. rated and peak torque engine speeds);

Calculate the compressor pressure ratio (PR) required, to provide calculated engine mass flow rate of air (m _air ) and EGR fraction;

(iii) Calculate the compressor outlet pressure (P2) from the calculated

compressor ratio;

(iv) Calculate the small turbine inlet pressure (P3small) from the compressor outlet pressure (P2) and any pressure changes across components in the engine system; (v) Calculate the small turbine expansion ratio (ER) from the calculated small turbine inlet pressure (P3small ) and the small turbine outlet pressure (P4);

(vi) Calculate the power delivered from the turbine due to the flow in the small turbine inlet;

(vii) Calculate the power delivered from the turbine due to the flow in the large turbine inlet;

(viii) Calculate the expansion ratio of the large turbine;

(ix) Calculate a mass flow parameter for the large and small turbine inlets; For each of the large and small turbine inlets, calculate the critical area of the turbine necessary to provide the require expansion ratio and mass flow parameter;

Consider the need for waste-gate and balance valves;

Manufacture an asymmetric turbine having large and small inlets with the calculated critical areas, and any wastegate and balance valve. In relation to step (i), at each of a plurality of required engine speeds (N) (e.g. rated power and peak torque), the mass flow rate of air (m _air ) from the compressor (106) is calculated from required values of engine power (P _Eng ), brake specific fuel consumption (BSFC) and the air to fuel ratio (AFR) of the engine 183, at that engine speed, from equation 14 (which corresponds to equation 1 ).

(m _air ) = P _Eng X BSFC X AFR

Equation 24

Where:

rh _air = mass flow rate of air through the internal combustion engine 283 P _En g = Power of the internal combustion engine (kW)

BSFC = Brake specific fuel consumption of the internal combustion engine (g/kW.hr)

AFR = air to fuel ratio of the internal combustion engine

In relation to step (ii), the mass flow rate of air m _air can be expressed as equation 25.

P ₇ + ΔΡ, CAC

D x N {Pamb ^Pfilter) ^{x 2}

^X ΆνοΙ ^x

(1 - EGR _frac NRPC ^wul R X IMT

Equation 25

Where: m _air = mass flow rate of air through the engine 283 EGRfrac = the fraction of the mass flow rate of gas passing to the engine inlet from the exhaust gas recirculation system relative to the total mass flow rate of gas into the internal combustion engine D Engine swept volume (m ³ )

N = Engine speed (rev/sec)

NRPC = Number of Revs Per Cycle of the engine

η _νο ι = Engine Volumetric Efficiency

P _amb = Ambient pressure (Pa)

APfiiter = Pressure change across the filter (Pa)

P ₂ = Pressure at compressor outlet (Pa)

Pi = Pressure at compressor inlet (Pa)

ΔΡΟΑΟ = Pressure change across the charge air cooler (Pa)

R = Gas constant for air and EGR

IMT = Inlet manifold temperature (K)

Equation 25 corresponds to equation 15 but modified to take account of the

asymmetric turbine inlet.

Re-writing equation 25 in terms of the compressor pressure ratio (PR) gives: p _{R =} (rh _air X NRPC X R X IMT AP _CAC

N x D x _nvol X (1 - EGR _frac ) X P P

Equation 26

This equation provides a means of calculating the compressor pressure ratio (PR) required, to provide the required engine mass flow rate of air at each engine speed (rated power and peak torque) (the engine mass flow rate at each engine speed, is calculated from equation 24).

The values of the number of revs per cycle of the engine (NRPC), the gas constant for air (R), the inlet manifold temperature (IMT), the engine swept volume (D), the engine volumetric efficiency (η _ν0 ι), the ambient pressure (P _amb ) and the pressure change across the filter (AP _fi | _ter ) and pressure change across the charge air cooler (AP _cac ), at each engine speed, are pre-selected values that are required by the design. The compressor inlet pressure (P^ is calculated from: i — Pamb ΔΡ ^r f, ilter

Equation 27

In relation to step (iii), the compressor outlet pressure (P ₂ ) is then calculated from the calculated compressor pressure ratio (PR) and compressor inlet pressure by:

P ₂ = PR x p ₁

Equation 28

In relation to step (iv), the pressure (P _{3 sma} n) at the small turbine inlet 207' is then calculated using a modified version of equation 19, namely:

^3 small ~ Pi + ^PcAC + ^Pegr cooler + ^Pegr valve

Equation 29

In relation to step (v), the small turbine outlet pressure (P ₄ smaii) is calculated according to:

P4 small ~ Pambient Pexh

Equation 30

P ₄ smaii = Small turbine exhaust pressure

Pam ient = Ambient pressure downstream of the turbine outlet (Pa)

Pexh BP = Turbine exhaust back pressure (Pa)

Note that P ₄ large is the same as P _{4 S} maii. The backpressure is calculated using the total exhaust flow through the exhaust pipes and the after-treatment system (not just the flow through the small turbine). The value of turbine exhaust back pressure (P _eX h BP) and the ambient pressure downstream of the turbine outlet (P _a mbient) are estimated, as in the first two examples explained above.

The small turbine expansion ratio (ER) is then calculated from:

„„ _ P3 small

small ~ ,

^R 4 small

Equation 31

In relation to step (vi), the calculated small turbine expansion ratio (ER) is then used in equation 32 to calculate the required power of the small turbine:

Power _{turb sma} ii

f+m _nir + r _f , _lM - m

— turb small x ^Λ r I ^{mair + m} ^ ^{fuel ~ megr} \ ) χ Γ ^p P x ^Λ T ¹ ■ 3 ^' small

Equation 32 In relation to step (vii), taking into account that the large and small turbines 201 ', 201 " are on the same shaft, the power of the large turbine is calculated by:

Power c.omp

Power _turb i _arge — Power _turb small

bearing

Equation 33

In relation to step (viii), The calculated value of the power of the Iarge turbine is then used in equation 34, to calculate the expansion ratio of the Iarge turbine, according to:

large

Equation 34

The large turbine outlet pressure (P ₄ large) is calculated according to:

^4 large ~ ^ambient ^ex BP

Equation 35

Where:

P ₄ large = Large turbine exhaust pressure

Pambient = Ambient pressure downstream of the turbine outlet (Pa) Pexh BP = Turbine exhaust back pressure (Pa)

As noted above, P ₄ large is the same as P _{4 S} maii.

The value of turbine exhaust back pressure (P _eX h BP) and the ambient pressure downstream of the turbine outlet (Pambient) are estimated from known values, in the same way as before.

The large turbine expansion ratio (ER) is then calculated from :

_ P3 large

^large ~ p

4 large

Equation 36

In relation to step (ix), the value of the large turbine mass flow parameter is calculated from :

m _p eg _nr r + ~ m '"- _n a _i i _r r + ' r 'h"-fuel

X 3 Large

MFPiarge ~

3 large

Equation 37

The value of the small turbine mass flow parameter is then calculated from :

MFP _small =

3 small

Equation 38 In relation to step (x), as with the previous examples, for each of the large and small turbine inlets, the required critical area of the turbine inlet is selected from a turbine flow map by selecting a curve, and therefore a certain size of inlet, that provides the desired turbine expansion ratio (P ₃ /P ₄ ), for the required turbine Mass Flow Parameter (MFP, _urb ). This is illustrated in Figure 10.

A critical parameter here is the "asymmetry ratio", which will be defined as the flow of the mass flow parameters in the two inlets:

Equation 39

Note that an alternative definition could be based on the ratios of the critical areas of the two inlets, but this would be equivalent only when the balance valve and wastegate are closed.

In step (x), a constraint analysis would be carried out, similar to step (viii) of Fig. 7. The preference would be to select the critical area to meet air and EGR flow requirements with the EGR valve wide open based on peak torque requirements. However, at least one second operating point (e.g. rated power engine speed) would also be considered, and a decision made about whether to add a wastegate 292 for protection and/or a balance valve 285 to provide greater flexibility. Furthermore, a sizing range for the balance value might be generated.

In relation to step (xi), an asymmetric turbine having large and small inlets with the calculated critical areas, and any wastegate and/or balance value as selected in step (x), is then manufactured.

The above method provides a relatively fast and simple method of sizing the critical areas of the inlets of an asymmetric turbine, that is part of an engine system that uses exhaust gas recirculation, based on certain design constraints of an engine, namely the engine power, speed, efficiency, brake specific fuel consumption, air to fuel ratio and inlet manifold temperature. The method also takes into account pressure drops across other components of the engine system. It will be appreciated that the method is applicable to where the engine system does not comprise these components, the relevant equations may be modified to remove these terms as appropriate.

The present inventors carried out many instances of the analysis of Fig. 9, and discovered that when the results were plotted as shown in Fig. 1 1 , a good fit could be found between the EGR loop pressure drop, which is the value plotted on the x-axis of Fig. 1 1 , and the value plotted on the y-axis which is the asymmetry ratio of Equation 39, multiplied by the ratio Z of the ambient pressure and the pressure at the turbine outlet. Note that the EGR loop pressure drop is defined as the pressure difference along the path 303, i.e. between the points 310 and 320 on Fig. 8. The EGR pressure loss drop includes the pressure loss due to the pipes in the path 303, the EGR valve 304, the EGR cooler 302, and the pressure loss across the port at the point 320 at which output of the EGR path 303 connects to the air intake manifold 242.

The line indicated on Fig. 1 1 corresponds to the equation: y = -0.1311n(x) + 0.8523

Equation 40 with an R ² value of 0.9241 .

The EGR loop pressure drop may thus, surprisingly, be regarded as a critical parameter in selecting the asymmetry ratio. This suggests a far simpler form of the method of Fig. 9, as shown in Fig. 12. This method is an embodiment of the invention.

In a first step (i) of the method, the EGR loop pressure drop is calculated for a plurality of engine speeds. The engine speeds may be chosen to include the peak torque speed and/or the rated power speed. Additionally or alternatively, the engine speeds may further include any one or more of the A, B and C speeds of the ESC regulatory cycle, and/or the lowest speed of the NTE zone. Conventionally, these four speeds are defined as follows

Engine speed in rpm = n _t0 + X(n _hi — n _to )

Equation 41 where the engine speeds A, B and C are respectively the value of the right of Equation 41 when X=0.25, X=0.50 and X=0.75, and the lowest speed of the NTE zone is the value of the right of Equation 41 when X=0.15. The high speed n _hi is determined by calculating 70% of the declared maximum net power. Specifically, n _hi is defined as the highest engine speed (i.e. above the rated power speed) for which 70% of the maximum engine power is still available (that is, the highest engine speed for which this power value occurs on the power curve). The low speed n _l0 is determined by calculating 50% of the declared maximum net power. Specifically, n _l0 is defined as the lowest engine speed (i.e. below the rated power speed) for which 50% of the maximum engine power is still available (that is, the lowest engine speed for which this power value occurs on the power curve). The NTE approach establishes a control area ("the NTE zone") which represents engine speeds and load expected to be encountered in normal vehicle operation and use by diesel heavy-duty engines. In step (ii), the turbine outlet pressure P _top and the ambient pressure P _amb are used to calculate -^ ²² -, accounting for any downstream restrictions (e.g. pipes and

aftertreatment) for each of the engine speeds.

In step (iii), the asymmetry ratio is calculated for each of the engine speeds using Equation 40.

In step (iv), critical areas for the small and large turbine inlets are calculated to achieve the calculated asymmetry ratio. Typically, a balance valve 285 and/or a wastegate valve 292 will be required to produce the calculated asymmetry ratio for all engine speeds, and the required parameters of the valves at each of the engine speeds are derived in step (iv).

In step (v), an engine system is manufactured incorporating an asymmetric turbine having large and small inlets with the calculated critical areas, and any wastegate and balance valve. If a wastegate and/or balance value is present, a valve control system is provided to control the wastegate and balance value such that at each of the engine speeds the asymmetry ratio is as calculated in step (iv). The turbine is incorporated into the engine system, resulting in a system similar to one designed according to the method of Figure 9. In either case, the resultant engine system will obey Equation 40 to within a certain tolerance (6) , at least for the engine speeds used in the method. Preferably, the asymmetry ratio of the engine system in operation is according to Equation 40 to within a certain tolerance for all the engine speeds used in the method, and indeed for all engine speeds within a range which includes all the engine speeds used in the method.