Work and flow coefficient

If we consider the form of the ψ-Φ relationship for a single stage turbine with symmetrical velocity triangles in simple terms, we can conclude that the work coefficient ψ is a linear function of the flow coefficient Φ. This follows from the fact that the flow leaves a blade or vane row in the direction given by the trailing edge geometry (see Figure 1).

Figure 1.

Turbine work and flow coefficient.

The work output H is a straight line when plotted as Ψ over Φ, whereas work input H_is – calculated from pressure ratio - is not (see Figure 2).

Figure 2.

Peak efficiency point of the ψ-Φ correlation.

The losses in the expansion process, described by efficiency η = H/H_is, are smallest at the peak efficiency point.

The only prerequisite for the validity of Equation 1 is that the (relative) flow direction downstream of the blades and vanes is enforced by the geometry of the blades and vanes. Furthermore, in an incompressible fluid (constant density), there is only one curve for ψ_is = f(Φ) and this is valid for any speed.

Figure 3 shows data published by Broichhausen (1994) with lines for Ψ plotted against flow coefficient Φ. The lower the speed, the more the Ψ-Φ correlation is a straight line.

Figure 3.

Ψ-Φ correlations from a turbine.

Efficiency

Figure 4 shows the efficiency values corresponding to Figure 3. For the relative speed values 0.4, 0.6 and 0.8 the efficiency lines collapse in the region where flow coefficients are less than Φ ≈ 1.2, where the peak efficiency values occur.

Figure 4.

Efficiency corresponding to Figure 3.

Another efficiency correlation originates from theoretical considerations about impulse turbines. In such a turbine the inlet guide vane works like a nozzle which produces the jet velocity V₁. There is no velocity change in the rotor of an impulse turbine (W₂ = W₁) as can be seen in Figure 5. Maximum efficiency is achieved when the direction of the absolute velocity V₂ at the turbine exit is axial. The ratio of U/V₁ for optimum efficiency is approximately 0.47.

Figure 5.

Impulse turbine.

Experience from working with the maps of many single- and multi-stage turbines has shown that efficiency generally correlates well with the speed/jet-velocity ratio. Figure 6 demonstrates this for the same turbine as used for Figure 3.

Figure 6.

Efficiency = f(N/V_jet).

Note that Figure 6 does not employ the absolute circumferential speed U – it uses relative speed N/N_ref. The x-axis numbers in Figure 6 therefore differ in magnitude from the value shown in Figure 5.

The correlations of efficiency with flow coefficient and speed ratio are useful only in the region with positive effective work H_eff. Efficiency is zero if effective work is zero and drops to minus infinity when pressure ratio is 1, see Figure 7.

Figure 7.

Efficiency around pressure ratio 1.

Torque

Turbine power can be expressed as the product of flow and specific work as well as the product of angular speed and torque:

Rearrangement and insertion of Equation 1 yields

This equation is valid where flow velocity V_ax is proportional to mass flow m, in the incompressible flow region. Under this condition, Trq/m² is a linear function of 1/Φ and Trq/m is – for a given circumferential speed U – a linear function of m:

Figure 8 shows that Equation 4 is in line with measured data in the low mass flow parts of the speed lines. The dashed lines are exactly parallel, the horizontal distance between the lines is independent of speed. This is remarkable because the data are from a three-stage turbine while Equation 4 has been derived for a single stage machine.

Figure 8.

Torque/Flow = f(Flow).

In addition to the scale for flow, Figure 8 shows a scale for Mach number. The torque/flow lines bend upwards at Mach numbers higher than 0.4… 0.45, which is due to compressibility effects.

The zero-speed line

For a locked rotor, the turbine may be considered as a pipe with restrictions. Total temperature is constant because no work is transferred. Total pressure decreases from the inlet to the exit of the turbine. For incompressible flow, the zero-speed line – i.e. the pressure ratio = f(flow) correlation - is a parabola.

The locked rotor changes the direction of the fluid and downstream of the rotor this is determined by the rotor blade exit geometry. Torque is proportional to the force in the circumferential direction which is exerted by the fluid on the locked rotor. Equation 4 is also valid for the locked rotor.

The zero-flow line

Reverse flow never happens in turbines during starting and windmilling simulations. The zero-flow line is a lower pressure ratio limit for the turbine map extension discussed. The velocity triangles yield interesting insights about operation at zero flow. Figure 9 shows how the design point velocity triangles and the enthalpy-entropy diagram look typically.

Figure 9.

Velocity triangles and enthalpy-entropy diagram, turbine design point.

If the mass flow is reduced at the same speed to very low values and finally to zero, then the incidence to the rotor becomes highly negative, see Figure 10.

Figure 10.

Velocity triangles for low mass flow and mass flow <ε.

The enthalpy-entropy diagrams for the two low mass flow velocity triangles are shown in Figure 11. From the diagram on the right one can conclude that for the zero-flow case, the effective specific work H_eff is twice as big as the isentropic specific work H_is since both W₁ and V₂ are equal to circumferential speed U.

Figure 11.

Enthalpy-entropy diagrams corresponding to Figure 10.

Corrected isentropic work H_is/T₁ is proportional to corrected speed squared:

H_is/T₁ relates to pressure ratio:

Thus, pressure ratio relates to corrected speed:

Relating the true corrected speed to a reference value makes it easy to determine the constant in this equation because then the speed term equals unity. The difference between specific enthalpy for the temperatures T_2,is and T₁ for zero flow is W₁²/2 as the right part of Figure 11 shows.

Since W₁ equals U (Figure 10) it holds that

The circumferential Mach number M_U,ref at the map reference point needs to be estimated. The constant in Equation 7 follows from Equation 9:

Pressure ratio for any other corrected speed follows from

Flow at pressure ratio 1

Figure 12 shows the velocity triangles and the enthalpy – entropy diagram for pressure ratio 1. The rotor entry triangle is symmetrical which makes the total temperature in the relative system T_1,rel equal to the inlet temperature T₁. The total pressure losses of the inlet guide vane P₀ - P_1,rel are compensated by the rotor, which works as a compressor.

Figure 12.

Pressure ratio 1.

The shape of the velocity triangles remains the same when spool speed changes, the triangles remain similar. All enthalpy differences change in proportion to circumferential speed squared and mass flow changes in proportion to circumferential speed. Therefore, the pressure ratio 1 line is linear in a plot of flow vs. speed.

Baseline map

The turbine map shown previously in Figures 3, 4, 6 and 8 serves to illustrate the map extension method. This map is from a three-stage low pressure turbine designed for a business jet engine.

The map as shown in Figure 14 has been created using 30 ß-lines with the standard procedure described in the Smooth T manual. The ß-line grid encloses all the given data points. The ß-numbers (between 0 and 1) have no physical meaning. The map reference point is properly placed at the peak efficiency location of the speed line 1.

Figure 14.

Baseline map.

Extension to zero flow

A new ß-line grid is required for extending the map to zero flow. The upper ß-line (ß = 1) remains unchanged, the lower ß-line, however, will now get a physical meaning: it will represent zero-flow. This line passes through the origin (relative speed = 0, pressure ratio = 1) and bends downwards when speed increases. The shape of the zero-flow line is given by Equation 10 and is defined implicitly by the circumferential Mach number M_U for relative speed 1.

Note that the number of ß-lines needs to be increased because the pressure ratio range is bigger. 50 ß-lines have been used in Figure 15.

Figure 15.

Revised ß-line grid.

Start the map extension with a guess for the ß = 0 line by choosing M_U = 0.5, for example. All the extended flow lines begin with mass flow zero on the ß = 0 line and approach the measured data points smoothly.

Modify the assumption for M_U if the lines flow = f(pressure ratio) do not merge smoothly with the part of the line which is defined by measured data. Special attention should be paid to comparing the shape of the line for the lowest speed with that for zero speed.

The discussion of Figure 12 lead to the conclusion that in a plot flow = f(speed) with contour lines for pressure ratio, the line 1.0 must be a straight line. As Figure 16 shows, this is true.

Figure 16.

Constant pressure ratio lines in the plot of flow = f(speed).

Also check the correlation between isentropic work coefficient and flow coefficient (Figure 17). All speed lines must collapse in the low flow coefficient region.

Figure 17.

Ψ_is = f(Φ).

Next check if the torque/flow = f(flow) lines are straight in the region of low flow. That corresponds to low Mach number and is where incompressible flow theory is applicable. Figure 18 shows how the extended map looks now.

Figure 18.

Map extension to low flow.

Equation 1 indicates that in incompressible flow the correlation between work and flow coefficients is linear and all lines collapse. Figure 19 shows that this is truly the case in our map example.

Figure 19.

Ψ = f(Φ) with added speed lines.

Extension to low speed

Extending the map to low speed uses the same ß-line grid as before. Add one speed line after the other and adapt the mass flow line first in such a way that it matches Figure 20 reasonably well.

Figure 20.

Torque/Flow = f(Flow).

Next check the torque/flow = f(flow) plot – in the beginning it will probably not look good. Correct the torque/flow lines in such a way that efficiency lines up with the other data in the plot efficiency = f(speed/jet velocity), as in Figure 6.

If it is not feasible to get both efficiency and torque/flow lines right, then the form of the function flow = f(pressure ratio) needs to be modified. Go back and modify the flow lines in such a way that the torque/flow lines have the desired shape.

To get all the correlations right requires only a few iterations. The final result of the map extension process is shown in Figures 20, 21 and 22.

Figure 21.

Flow and specific work.

Figure 22.

Map with extrapolated and interpolated speed lines.