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Having sized the rotor inlet and outlet, the next design phase is to design the three- dimensional rotor passage. This rotor passage must be designed to minimise losses, and achieve a favourable blade loading distribution. This important aspect is often overlooked during preliminary design, and is often the result of more computationally expensive design methods such as CFD. The three-dimensional construction of the rotor consists of defining the meridional profiles, the camberline and the blade thickness distributions. These three curve sets, when superimposed onto each other, form the full rotor geometry. By con- trolling how these curves are defined, the shape and area distribution of the rotor passage can be controlled. Combining this geometrical construction with a method to predict the

resulting velocity distribution within the rotor allows the designer to quickly iterate and arrive at a suitable preliminary rotor passage design.

4.3.1 Rotor three-dimensional construction

Although the three-dimensional construction of ORC turbines has not been considered in detail in the literature, the three-dimensional construction of ideal gas turbines has been discussed. The definition of the relevant curve sets, in addition to the procedure to construct the rotor geometry, has been well discussed by both Atkinson (1998) and Aungier (2006), and the approach adopted here combines aspects from both methodologies. Subsequently, the key equations are described in Appendix B, whilst readers should refer to these references for more information.

4.3.2 Prediction of meanline parameters

After constructing the full rotor passage, the passage area distribution through the rotor can be determined, and this can be used to provide a preliminary estimate of the velo- city distribution within the rotor. The first process is to construct a number of meanline quasi-normals on the meridional profile. To construct these, the co-ordinates of the hub and shroud profile curves are determined at 5% intervals of the total meridional path length, and these are connected by straight lines. The mid point of each line is determ- ined and these locations correspond to the points at which the velocity distributions will be established. At each quasi-normal the meridional flow area, with and without blade blockage, can then be determined. The process of accounting for blade blockage is re- latively involved, but has been well documented by Atkinson (1998). After calculating the passage area at each quasi-normal Aq, the complete rotor passage area distribution

is then obtained. An example of the resulting meanline quasi-normals and passage area distribution is shown later on for the developed ORC turbine (Figure 4.9).

The approach adopted to predict the velocity distribution within the rotor is based on an approached suggested by Watson and Janota (1982). For this rotating machine the rothalpy I must remain constant, and this parameter is known from the rotor one- dimensional design. Therefore, the total relative enthalpy at each quasi-normal, h0′, can

be determined from the known rothalpy value, rotor rotational speed ω and the quasi- normal mid point radius rq (Equation 4.50). The total relative enthalpy is related to

the static enthalpy h and the relative velocity w. For this analysis the rotor isentropic static-to-static efficiency ηR is also required and is defined by Equation 4.51. This can be

determined from the rotor one-dimensional design.

h0′ = I + 1 2(rqω) 2_{= h +} 1 2w 2 _(4.50) ηR = h4− h5 h4− h5s (4.51)

To determine the relative velocity at each quasi-normal an iterative proceeded is re- quired since neither h or w is known. As an initial guess, the total relative density, ρ0′, is

calculated based on h0′, and assuming an isentropic expansion from the rotor inlet.

ρ0′ = EoS(h₀′, s₄, fluid) (4.52)

An initial guess for the relative velocity is then obtained by applying mass continuity (Equation 4.53). For this analysis it is assumed that the relative flow within the rotor remains aligned with the blade such that the relative flow angle β is equal to the quasi- normal blade angle βq. The applicability of this assumption will be analysed and confirmed

later.

w = m˙

ρ0′A_qcos β_q

(4.53)

The static enthalpy h follows from Equation 4.50, whilst the enthalpy associated with an isentropic expansion hs can be obtained by assuming that ηR remains constant as

the flow expands through the rotor (Equation 4.54). This in turn determines the static pressure P , then supplying a new estimate for the density ρ. This process is repeated until convergence, and then repeated for each quasi-normal.

hs= h4−h4− h

ηR

(4.54)

P = EoS(hs, s4, fluid) (4.55)

ρ = EoS(P, h, fluid) (4.56)

It should be clarified that the method outlined above cannot be relied upon to provide an accurate representation of the flow field within the rotor. This is because this meth-

odology applies one-dimensional flow theories to simulate a highly three-dimensional flow field. Furthermore, assuming that the flow remains aligned with the blade does not ac- count for the flow entering the rotor with a certain amount of incidence, or for the flow deviation that is known to occur at the rotor outlet (Moustapha et al., 2003). However, the intention behind implementing this procedure is to aid in preliminary design. By calculating the rotor passage area distribution, and then estimating the velocity distribu- tion, this procedure can be used to assess and then modify the passage geometry to avoid abrupt changes in the flow area, or unintentional areas of diffusion.