Moustapha-Kacker-Tremblay (1990)

This section describes the loss model proposed by [Tremblay et al., 1990] to compute aerodynamic losses in axial turbines.

Overview of the method

The loss system by [Tremblay et al., 1990], builds on the previous loss models proposed by [Ainley and Mathieson, 1951, Kacker and Okapuu, 1982].

The general form of the loss model is given by:

\[\mathrm{Y_{tot}} = \mathrm{Y_{p}} + \mathrm{Y_{te}} + \mathrm{Y_{s}} + \mathrm{Y_{cl}} + \mathrm{Y_{inc}}\]

The expressions used to compute each term as a function of the cascade geometry and flow variables are presented in the next sections.

Profile loss coefficient

The profile loss coefficient (\(\mathrm{Y_{p}}\)) is calculated as described in Profile loss coefficient.

Secondary loss coefficient

[Tremblay et al., 1990] introduced a correction factor (\(\mathrm{Y_{corr}}\)) the secondary loss coefficient proposed by [Kacker and Okapuu, 1982], to correct for incidence. The corrrection factor depends on an incidence parameter calculated as:

\[\chi = \frac{\beta_\mathrm{in} - \theta_\mathrm{in}}{180 - (\theta_\mathrm{in} + \theta_\mathrm{out})}\left(\frac{\cos(\theta_\mathrm{in})}{\cos(\theta_\mathrm{out})}\right)^{-1.5}\left(\frac{\mathrm{d_{le}}}{c}\right)^{-0.3}\]

where:

  • \(\chi\) is the secondary flow incidence parameter.

  • \(c\) is the chord length.

  • \(\mathrm{d_{le}}\) is the leading edge diameter.

  • \(\theta_\mathrm{in}\) is the inlet metal angle.

  • \(\theta_\mathrm{out}\) is the exit metal angle.

  • \(\beta_\mathrm{in}\) is the inlet relative flow angle.

From this parameter, the correction factor coefficient can be determined:

\[\begin{split}Y_\mathrm{corr} = \begin{cases} e^{0.9\chi} + 13\chi^2 + 400\chi^4 & \text{if } \chi \geq 0 \\ e^{0.9\chi} & \text{if } \chi < 0 \end{cases}\end{split}\]

The total loss coefficient is then calculated as:

\[\mathrm{Y_{s}} = \mathrm{Y_{corr}}\mathrm{Y_{s, KO}}\]

where \(\mathrm{Y_{s, KO}}\) is the secondary loss coefficient calculated as described in Secondary loss coefficient.

Tip clearance loss coefficient

The tip clearance loss coefficient (\(\mathrm{Y_{cl}}\)) is calculated as described in Tip clearance loss coefficient.

Trailing edge loss coefficient

The trailing edge loss coefficient (\(\mathrm{Y_{te}}\)) is calculated as described in Trailing edge loss coefficient.

Incidence loss coefficient

The incidence loss coefficient (mathrm{Y_{inc}}) is based on the kinetic energy loss coefficient. The calculation involves to determine the increment in profile loss coefficient due to incidence. The increment in profile losses due to incidence is based on the incidence parameter \(\chi\). The formula used to compute \(\chi\) is given by:

\[\chi = \left(\frac{\mathrm{d_{le}}}{s}\right)^{-1.6}\left(\frac{\cos{\theta_\mathrm{in}}}{\cos{\theta_\mathrm{out}}}\right)^{-2}(\beta_\mathrm{in} - \beta_\mathrm{des})\]

where:

  • \(\mathrm{d_{le}}\) is the leading edge diameter.

  • \(s\) is the pitch.

  • \(\theta_\mathrm{in}\) and \(\theta_\mathrm{out}\) is the blade metal angle at the inlet and outlet respectively.

  • \(\beta_\mathrm{in}\) and \(\beta_\mathrm{des}\) is the inlet relative flow angle at given and design conditions respectively.

Depending on the value of \(\chi\), two equations are used for computing the increment in profile losses:

\[\begin{split}\Delta\phi^2_p = \begin{cases} -5.1734e^{-6}\chi + 7.6902e^{-9}\chi^2 & \text{if } -800 \leq \chi \leq 0 \\ 0.778e^{-5}\chi + 0.56e^{-7}\chi^2 + 0.4e^{-10}\chi^3 + 2.054e^{-19}\chi^6 & \text{if } 0 \leq \chi \leq 800 \end{cases}\end{split}\]

Subsequently, the increment in profile loss coefficient, in terms of kinetic energy loss, is converted to the total pressue loss coefficient definition:

\[\mathrm{Y} = \frac{\left(1-\frac{\gamma -1}{2}\mathrm{Ma_{out}}^2(\frac{1}{(1-\Delta\phi^2_p)}-1)\right)^\frac{-\gamma}{\gamma - 1}-1}{1-\left(1 + \frac{\gamma - 1}{2}\mathrm{Ma_{out}}^2\right)^\frac{-\gamma}{\gamma - 1}}\]

where:

  • \(\gamma\) is the specific heat ratio.

  • \(\mathrm{Ma_{out}}\) is the cascade exit relative mach number.

  • \(\Delta\phi^2_p\) is the kinetic energy loss coefficient increment due to incidence.