Loss coefficients

A loss can be interpreted as any mechanism that leads to entropy generation and reduces/increases the power output/consumption of the turbomahcinery, such as viscous friction in boundary layers, wake mixing, or shock waves ([Denton, 1993]). In meanline analysis, the losses are characterized in terms of a loss coefficient. This parameter has different possible definitions, which are provided here for reference.

The stagnation pressure loss coefficient is defined as the reduction in stagnation pressure from inlet to outlet relative to a reference dynamic pressure. For compressor blades, the inlet conditions are used for reference, while the exit conditions are used for turbine blades:

\[\begin{split}\mathrm{Y}= \begin{cases} \frac{p_{0, \mathrm{in}}-p_{0, \mathrm{out}}}{p_{0, \mathrm{in}} - p_\mathrm{in}}, & \text{for compressor blades} \\ \frac{p_{0, \mathrm{in}}-p_{0, \mathrm{out}}}{p_{0, \mathrm{out}} - p_\mathrm{out}}, & \text{for turbine blades}. \end{cases}\end{split}\]

Subscript 0 refer to the stagnation property in the relative frame of reference.

The kinetic energy loss coefficient is the ratio of the enthalpy increase due to irreversibility to the isentropic total-to-static enthalpy change. Also for this loss coefficient, the inlet conditions ar used for reference for compressore blades, and exit conditions for turbine blades:

\[\begin{split}\Delta \phi^2 = \begin{cases} \frac{h_{\mathrm{out}}-h_{\mathrm{out},s}}{h_{0,\mathrm{in}}-h_{\mathrm{in},s}} = \frac{h_{\mathrm{out}}-h_{\mathrm{out},s}}{\frac{1}{2}v_{\mathrm{in},s}^2} =1 - \left(\frac{v_{\mathrm{out}}}{v_{\mathrm{in},s}}\right)^2 = 1- \phi^2, & \text{for compressor blades} \\ \frac{h_{\mathrm{out}}-h_{\mathrm{out},s}}{h_{0,\mathrm{out}}-h_{\mathrm{out},s}} = \frac{h_{\mathrm{out}}-h_{\mathrm{out},s}}{\frac{1}{2}v_{\mathrm{out},s}^2} =1 - \left(\frac{v_{\mathrm{out}}}{v_{\mathrm{out},s}}\right)^2 = 1- \phi^2, & \text{for turbine blades}. \end{cases}\end{split}\]

Here, \(\phi^2\) is the ratio of actual to ideal kinetic energy at the cascade’s exit, commonly interpreted as the efficiency of the cascade.

The enthalpy loss coefficient is analogously defined, but it utilizes the actual total-to-static enthalpy change in the denominator:

\[\begin{split}\zeta= \begin{cases} \frac{h_{\mathrm{out}}-h_{\mathrm{out},s}}{h_{0,\mathrm{in}}-h_{\mathrm{in}}} = \frac{h_{\mathrm{out}}-h_{\mathrm{out},s}}{\frac{1}{2}v_{\mathrm{in}}^2} = \left(\frac{v_{\mathrm{out},s}}{v_{\mathrm{in}} }\right)^2 - 1 = \frac{1}{\phi^2}-1, & \text{for compressor blades} \\ \frac{h_{\mathrm{out}}-h_{\mathrm{out},s}}{h_{0,\mathrm{out}}-h_{\mathrm{out}}} = \frac{h_{\mathrm{out}}-h_{\mathrm{out},s}}{\frac{1}{2}v_{\mathrm{out}}^2} = \left(\frac{v_{\mathrm{out},s}}{v_{\mathrm{out}} }\right)^2 - 1 = \frac{1}{\phi^2}-1, & \text{for turbine blades}. \end{cases}\end{split}\]

The entropy loss coefficient is the product of exit temperature and the entropy increase across the cascade, divided by the kinetic energy at the cascade’s exit:

\[\begin{split}\varsigma = \begin{cases} \frac{T_\mathrm{out}(s_{\mathrm{out}}-s_{\mathrm{in}})}{\frac{1}{2}v_{\mathrm{in}}^2}, & \text{for compressor blades} \\ \frac{T_\mathrm{out}(s_{\mathrm{out}}-s_{\mathrm{in}})}{\frac{1}{2}v_{\mathrm{out}}^2}, & \text{for turbine blades}. \end{cases}\end{split}\]

When the proposed model is evaluated, the loss coefficient computed from its definition, as given above, and the loss coefficient computed using the loss model, as described in Loss models, may not have the same value. Therefore, the loss coefficient error is given by

\[Y_{\mathrm{error}} = Y_{\mathrm{definition}} - Y_{\mathrm{loss\,model}}\]

The evaluation of the model involves measures to ensure that the loss coefficient error is zero. For performance analysis, this is ensured by thr root-solver algorithm, while for design optimization this is implemented as an equality constraint for the same purpose.