If η_gs is gross stage efficiency; ηb is blade efficiency and η_n is nozzle efficiency of a steam turbine, then which of the following relations is correct?

Explanation:

Gross or Stage Efficiency:

It is the ratio of work done on the blades per kg of steam to the total energy supplied or heat drop per stage per kg of steam. It is also defined as the product of Blade efficiency and Nozzle efficiency.

i.e., ηStage = ηBlade ×  ηNozzle ...(i)

Now, Blade or diagram efficiency (ηBlade):

\({η _{Blade}} = \frac{{Work\;done\;on\;the\;blades}}{{Energy\;supplied\;on\;the\;blades}} = \frac{{mu{V_w}}}{{\frac{{mV_1^2}}{2}}}\)  

Where Vw = Whirl velocity, u = Linear Velocity of moving blade (m/s), m = Mass of steam/sec, V1 = Absolute velocity of steam entering moving blade (m\s)

Nozzle efficiency (ηNozzle): 

\({\eta _{Nozzle}} = \frac{{Increase\;in\;K.E\;in\;nozzle}}{{Isentropic\;heat\;drop}} = \frac{{V_1^2}}{{2\left( {{h_1} - {h_2}} \right)}}\)

Using Equation (i) we have

\({\eta _{Stage}} = \frac{{u{V_w}}}{{{h_1} - {h_2}}}\)

Where h1 = Total heat before expansion through a nozzle, h2 = Total heat after expansion through a nozzle

so, h1 - h2 = Heat drop through a stage of fixed blades ring and moving blade rings = Total energy supplied

∴ \({\eta _{Blade}} = \frac{{Work\;done\;on\;blades}}{{Total\;energy\;supplied}}\)