.. _loss_model_ko1982: Kacker-Okapuu (1982) ======================================= This section describes the loss model proposed by :cite:`kacker_mean_1982` to compute aerodynamic losses in axial turbines. Overview of the method ---------------------- The :cite:`kacker_mean_1982` loss system is a refinement of the correlations proposed by :cite:`ainley_method_1951`, :cite:`ainley_examination_1951`, and by :cite:`dunham_improvements_1970`. The general form of the loss model is given by: .. math:: Y = f_{\mathrm{Re}} \, f_{\mathrm{Ma}} \, Y_{\mathrm{p}} + Y_{\mathrm{s}} + Y_{\mathrm{cl}} + Y_{\mathrm{te}} The expressions used to compute each term as a function of the cascade geometry and flow variables are presented in the next sections. Some of the signs from the original correlations were modified to comply with the angle conventions used in this work. These modifications are explicitly mentioned in the text. Reynolds number correction factor --------------------------------- The term :math:`f_{\mathrm{Re}}` accounts for the effects of the Reynolds number and it is computed according to the following equation: .. math:: f_{\mathrm{Re}} = \begin{cases} (\frac{\mathrm{Re}}{2 \cdot 10^5} )^{-0.40} & {\text{for }} \mathrm{Re} < 2 \cdot 10^5\\ 1 & {\text{for }} 2 \cdot 10^5 < \mathrm{Re} < 1 \cdot 10^6\\ ( \frac{\mathrm{Re}}{1 \cdot 10^6})^{-0.20} & {\text{for }} \mathrm{Re} > 1 \cdot 10^6 \end{cases} where the Reynolds number is given in terms of the chord length and the density, viscosity, and relative velocity at the outlet of the cascade: .. math:: \mathrm{Re} = \frac{\rho_{\mathrm{out}} \, w_{\mathrm{out}} \, c}{\mu_{\mathrm{out}}} Mach number correction factor ----------------------------- The term :math:`f_{\mathrm{Ma}}` accounts for losses associated with supersonic flows at the trailing edge of the blades and it is computed by: .. math:: f_{\mathrm{Ma}} = \begin{cases} 1 & {\text{for }} \mathrm{Ma_{\,out}^{\, rel}} \leq 1 \\ 1+60\cdot(\mathrm{Ma_{\,out}^{\, rel}}-1)^2 & {\text{for }} \mathrm{Ma_{\,out}^{\, rel}} > 1 \end{cases} The Mach number is defined by the relative velocity and the speed of sound at the outlet of the cascade: .. math:: \mathrm{Ma_{\,out}^{\, rel}} = w_{\mathrm{out}}/a_{\mathrm{out}} .. _profile_loss_KO: Profile loss coefficient ------------------------- The profile loss coefficient :math:`Y_{\mathrm{p}}` is computed according to: .. math:: Y_{\mathrm{p}} = 0.914 \cdot \left( \frac{2}{3} \cdot Y_{\mathrm{p}}' \cdot K_{\mathrm{p}} + Y_{\mathrm{shock}} \right). The term :math:`Y_{\mathrm{p}}'` is given by: .. math:: Y_{\mathrm{p}}' = \left[ Y_{\mathrm{p, \, reaction}} - \left( \frac{\theta_{\mathrm{in}}}{\beta_{\mathrm{out}}} \right) \left| \frac{\theta_{\mathrm{in}}}{\beta_{\mathrm{out}}} \right| \cdot (Y_{\mathrm{p, \, impulse}}- Y_{\mathrm{p, \, reaction}}) \right] \cdot \left(\frac{t_{\mathrm{max}}/c}{0.20}\right)^{-\frac{\theta_{\mathrm{in}}}{\beta_{\mathrm{out}}}} where the terms, :math:`Y_{\mathrm{p, \, reaction}}` and :math:`Y_{\mathrm{p, \, impulse}}` are be obtained from graphical data by :cite:`aungier_turbine_2006`. The subscript *reaction* refers to blades with zero inlet metal angle (i.e., axial entry) and the subscript *impulse* refers to blades that have an inlet metal angle with the same magnitude but opposite sign as the exit relative flow angle. The second term of the right-hand side is a correction factor that accounts for the effects of the maximum blade thickness. In this equation, the sign of :math:`\beta_{\mathrm{out}}` was changed with respect to the original work of Kacker--Okappu to comply with the angle convention used in this work. The factor :math:`K_{p}` accounts for compressible flow effects when the Mach number within the cascade is subsonic and approaches unity. These effects tend to accelerate the flow, make the boundary layers thinner, and decrease the profile losses. :math:`K_{p}` is a function on the inlet and outlet relative Mach numbers and it is computed from the following set of equations: .. math:: K_{\mathrm{p}} = 1-K_{2} \cdot \left(1-K_{1} \right) .. math:: K_{1} = \begin{cases} 1 & \text{for } \mathrm{Ma_{\,out}^{\, rel}} < 0.20 \\ 1-1.25 \cdot(\mathrm{Ma_{\,out}^{\, rel}}-0.20) & \text{for } 0.20 <\mathrm{Ma_{\,out}^{\, rel}} < 1.00 \\ 0 & \text{for } \mathrm{Ma_{\,out}^{\, rel}} > 1.00 \\ \end{cases} .. math:: K_{2} = \left( \frac{\mathrm{Ma_{\,in}^{\, rel}}}{\mathrm{Ma_{\,out}^{\, rel}}} \right)^2 The term :math:`Y_{\mathrm{shock}}` accounts for the relatively weak shock waves that may occur at the leading edge of the cascade due to the acceleration of the flow. After some algebra, the equations proposed in the Kacker--Okapuu method can be condensed to: .. math:: Y_{\mathrm{shock}} = 0.75 \cdot \left(f_{\mathrm{hub}} \cdot\mathrm{Ma_{\,in}^{\, rel}} -0.40 \right)^{1.75} \cdot \left( \frac{r_{\mathrm{hub}}}{r_{\mathrm{tip}}} \right)_{\mathrm{in}} \cdot \left( \frac{p_{\mathrm{0rel,in}}-p_{\mathrm{in}}}{p_{\mathrm{0rel,out}}-p_{\mathrm{out}}} \right) where :math:`f_{\mathrm{hub}}` is given graphically in :cite:`kacker_mean_1982` and it is a function of the hub-to-tip ratio only. .. _secondary_loss_KO: Secondary loss coefficient -------------------------- The secondary loss coefficient :math:`Y_{\mathrm{s}}` is computed according to: .. math:: Y_{\mathrm{s}} = 1.2 \cdot K_{\mathrm{s}} \cdot \left[0.0334 \cdot f_{\mathrm{AR}} \cdot Z \cdot \left( \frac{\cos(\beta_{\mathrm{out}})}{\cos(\theta_{\mathrm{in}})} \right) \right] The factor 1.2 is included to correct the secondary loss for blades with zero trailing edge thickness. Trailing edge losses are accounted independently. The factor :math:`K_{\mathrm{s}}` accounts for compressible flow effects present when the Mach number approaches unity. These effects tend to accelerate the flow, make the end wall boundary layers thinner, and decrease the secondary losses. :math:`K_{\mathrm{s}}` is computed from: .. math:: K_{\mathrm{s}} = 1-K_{3} \cdot \left(1-K_{\mathrm{p}} \right) where :math:`K_{\mathrm{p}}` is given similarly as described in :ref:`profile_loss_KO` and :math:`K_{3}` is given as a function of the axial aspect ratio :math:`H/b` only: .. math:: K_{3} = \left(\frac{1}{H/b}\right)^2 :math:`f_{\mathrm{AR}}` accounts for the blade aspect ratio :math:`H/c` and it is given by: .. math:: f_{\mathrm{AR}} = \begin{cases} \frac{1-0.25\cdot \sqrt{2-H/c}}{H/c} & \text{for } H/c < 2\\ \frac{1}{H/c} & \text{for } H/c > 2 \end{cases} The Ainley-Mathieson loading parameter :math:`Z` is given by the following set of equations: .. math:: Z = \left(\frac{C_{\mathrm{L}}}{s/c}\right)^2 \, \frac{\cos(\beta_{\mathrm{out}})^2}{\cos(\beta_{\mathrm{m}})^3} .. math:: \left(\frac{C_{\mathrm{L}}}{s/c}\right) = 2 \cos(\beta_{\mathrm{m}}) \, \left[\tan(\beta_{\mathrm{in}}) - \tan(\beta_{\mathrm{out}})\right] .. math:: \tan(\beta_{\mathrm{m}}) = \frac{1}{2}\left[\tan(\beta_{\mathrm{in}}) + \tan(\beta_{\mathrm{out}})\right] where the sign of :math:`\beta_{\mathrm{out}}` was changed with respect to the original work to comply with the angle convention used in this work. .. _tip_clearance_KO: Tip clearance loss coefficient ------------------------------ The tip clearance loss coefficient :math:`Y_{\mathrm{cl}}` is computed according to: .. math:: Y_{\mathrm{cl}} = B \cdot Z \cdot \left(\frac{c}{H}\right) \cdot \left( \frac{t_{\mathrm{cl}}}{H}\right)^{0.78} In this equation, :math:`Z` is given similarly as decribed in :ref:`secondary_loss_KO`. The Kacker-Okapuu loss system proposes :math:`B=0.37` for rotor blades with shrouded tips, and :math:`B=0.00` for stator blades. In addition, Kacker and Okapuu warn that using :math:`B=0.47`, as suggested by :cite:`dunham_improvements_1970`, over-predicts the loss for rotor blades with plain tips. .. _trailing_edge_KO: Trailing edge loss coefficient ------------------------------ The trailing edge loss coefficient :math:`Y_{\mathrm{te}}` is computed according to: .. math:: Y_{\mathrm{te}} \approx \zeta = \frac{1}{\phi^2}-1 = \frac{1}{1-\Delta \phi^2}-1 Where the pressure loss coefficient :math:`Y` was approximated by the enthalpy loss coefficient :math:`\zeta` and then related to the kinetic energy loss coefficients :math:`\phi^2` and :math:`\Delta \phi^2`. See the work by :cite:`dahlquist_investigation_2008` for details about the definitions of the different loss coefficients and the relations among them. The parameter :math:`\Delta \phi^2` is computed by interpolation of impulse and reaction blades according to: .. math:: \Delta \phi^2 = \Delta \phi_{\mathrm{reaction}}^2 - \left( \frac{\theta_{\mathrm{in}}}{\beta_{\mathrm{out}}} \right) \left| \frac{\theta_{\mathrm{in}}}{\beta_{\mathrm{out}}} \right| \cdot ( \Delta \phi_{\mathrm{impulse}}^2 -\Delta \phi_{\mathrm{reaction}}^2) The sign of :math:`\beta_{\mathrm{out}}` was changed with respect to the original work of Kacker--Okappu to comply with the angle convention used in this work. :math:`\Delta \phi_{\mathrm{reaction}}^2` and :math:`\Delta \phi_{\mathrm{impulse}}^2` are the kinetic energy loss coefficients of reaction and impulse blades and they are a function of the trailing edge thickness to opening ratio :math:`t_{\mathrm{te}}/o` only. The functional relation was published in graphical form. Final remarks ------------- The Kacker--Okapuu loss model was developed to estimate the performance of competent turbine designs and its predictions will not be accurate if the input parameters are outside the range of the experimental data used to develop the correlations. This situation is often encountered before the optimization algorithm converges since, in general, it is not possible to satisfy constraints for each iterate of a nonlinear programming problem. For this reason, some of the variables used within the Kacker--Okapuu loss model were bounded to avoid numerical problems that might prevent the convergence to a feasible solution. For instance, some variables were forced to be non-negative because the correlations were not developed to cover such cases. These modifications do not affect the final results of the optimization and they are not reported in this paper although they are documented in detail within the code.