Benner-Sjolander-Moustapha (2006)

This section describes the loss model proposed by [Benner et al., 1997, Benner et al., 2006, Benner et al., 2006] to compute aerodynamic losses in axial turbines.

Overview of the method

The loss system follows from a series of publications [Benner et al., 1997, Benner et al., 2006, Benner et al., 2006] and builds on the previous loss models proposed by [Ainley and Mathieson, 1951, Kacker and Okapuu, 1982, Tremblay et al., 1990].

The general form of the loss model is given by:

\[\mathrm{Y_{tot}} = (\mathrm{Y_{p}} + \mathrm{Y_{te}} + \mathrm{Y_{inc}})(1-Z/H) + \mathrm{Y_{s}} + \mathrm{Y_{cl}}\]

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

The correlation for the secondary loss coefficient (\(\mathrm{Y_{s}}\)) takes two different forms depending on the aspect ratio:

If aspect ratio \(\leq 2.0\)

\[\mathrm{Y_{s}} = \frac{0.038 + 0.41 \tanh(1.20\delta^*/H)}{\sqrt{\cos(\xi)}CR(H/c)^{0.55}\frac{\cos(\beta_\mathrm{out})}{\cos(\xi)}^{0.55}}\]

wheras if aspect ratio \(>2.0\)

\[\mathrm{Y_{s}} = \frac{0.052 + 0.56 \tanh(1.20\delta^*/H)}{\sqrt{\cos(\xi)}CR(H/c)\frac{\cos(\beta_\mathrm{out})}{\cos(\xi)}^{0.55}}\]

where:

• \(CR = \frac{\cos(\beta_\mathrm{in})}{\cos(\beta_\mathrm{out})}\) is the convergence ratio.

• \(H\) is the mean height.

• \(c\) is the chord.

• \(\delta^*\) is the inlet endwall boundary layer displacement thickness.

• \(\xi\) is the stagger angle.

• \(\beta_\mathrm{out}\) is the exit relative flow angle.

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:

\[\chi = \frac{\mathrm{d_{le}}}{s}^{-0.05}\mathrm{We_{le}}^{-0.2}\frac{\cos\theta_\mathrm{in}}{\cos\theta_\mathrm{out}}^{-1.4}(\beta_\mathrm{in} - \beta_\mathrm{des})\]

where:

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

• \(s\) is the pitch.

• \(\mathrm{We_{le}}\) is the leading edge wedge angle.

• \(\theta_\mathrm{in}\) 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:

1. For \(\chi \geq 0\):

\[\Delta \phi_{p}^2 = \sum_{i=1}^{8} a_i \, \chi^i\]

with coefficients:

\[\begin{split}a_1 = -6.149 \times 10^{-5} \\ a_2 = +1.327 \times 10^{-3} \\ a_3 = -2.506 \times 10^{-4} \\ a_4 = -1.542 \times 10^{-4} \\ a_5 = +9.017 \times 10^{-5} \\ a_6 = +1.106 \times 10^{-5} \\ a_7 = -5.318 \times 10^{-6} \\ a_8 = +3.711 \times 10^{-7}\end{split}\]

2. For \(\chi < 0\):

\[\Delta \phi_{p}^2 = \sum_{i=1}^{2} b_i \, \chi^i\]

with coefficients:

\[\begin{split}b_1 = -8.720e-4 \times 10^{-4} \\ b_2 = +1.358e-4 \times 10^{-4}\end{split}\]

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.

Penetration depth

The penetration depth (\(Z\)) refer to the spanwize penetration of the vortices originating from the cascade inlet endwall boundary layer. The penetration depth relative to hte mean blade height is calcluated by:

\[\frac{\mathrm{Z_{te}}}{H} = \frac{0.10F_t^{0.79}}{\sqrt{CR}\left(\frac{H}{c}\right)^{0.55}} + 32.70\frac{\delta^*}{H}^2\]

where:

• \(CR = \frac{\cos(\beta_\mathrm{in})}{\cos(\beta_\mathrm{out})}\) is the convergence ratio.

• \(H\) is the mean height.

• \(c\) is the chord.

• \(\delta^*\) is the inlet endwall boundary layer displacement thickness.

• \(\beta_\mathrm{out}\) is the exit relative flow angle.

• \(F_t\) is the tangential loading coefficient.

The tangiential loading coefficient is calculated by:

\[F_t = 2\frac{s}{c_\mathrm{ax}}\cos^2(\beta_m)(\tan(\beta_\mathrm{in} - \beta_\mathrm{out}))\]

where:

• \(s\) is the pitch.

• \(c_\mathrm{ax}\) is the axial chord.

• \(\beta_m = \tan^{-1}(0.5(\tan(\beta_\mathrm{in}) + \tan(\beta_\mathrm{out})))\) is the mean vector angle.

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