A francis turbine has the following specifications.

Inlet radius 300 mm, outer radius 150 mm, flow rate $0.05 {m^{3}}/{s}$ , inlet guide blade $30^{\circ}$ , angle the absolute velocity at exit makes an angle of $80^{\circ}$ with the axis inlet absolute velocity $6 {m}/{s}$ , exit absolute velocity $3 {m}/{s}$ ; overall efficiency $80\%$ , hydraulic efficiency $90\%$ . Determine head and power output if angular velocity is $25 {rad}/{s}$ .

Question

A francis turbine has the following specifications.

Inlet radius 300 mm, outer radius 150 mm, flow rate [latex]0.05 {m^{3}}/{s}[/latex], inlet guide blade [latex]30^{\circ}[/latex], angle the absolute velocity at exit makes an angle of [latex]80^{\circ}[/latex] with the axis inlet absolute velocity [latex]6 {m}/{s}[/latex], exit absolute velocity [latex]3 {m}/{s}[/latex]; overall efficiency [latex]80\%[/latex], hydraulic efficiency [latex]90\%[/latex]. Determine head and power output if angular velocity is [latex] 25 {rad}/{s}[/latex].

Accepted Answer

Refering to velocity diagrams shown in Fig. 6.15, we have

[latex] R_{1} = 300 mm, R_{2} = 150 mm, Q = 0.05 {m^{3}}/{s}[/latex]

[latex] \alpha_{1} = 30^{\circ}, \alpha_{2} = 80^{\circ}, V_{1} = 6 {m}/{s}, V_{2} = 3 {m}/{s}[/latex]

[latex] \eta_{0} = 80\%, \eta_{h} = 90\%, w = 25 {rad}/{s}[/latex]

[latex] u_{1} = wR_{1} = \frac{300}{1000} imes 25 = 7.5 {m}/{s}[/latex]

[latex] u_{2} = wR_{2} = \frac{150}{1000} imes 25 = 3.75 {m}/{s}[/latex]

[latex] V_{1w} = V_{1} \cos \alpha_{1} = 6 imes \cos 30 = 6 imes 0.866 = 5.2 {m}/{s}[/latex]

[latex] V_{2w} = V_{2} \cos \alpha_{2} = 3 imes \cos 80 = 3 imes 0.172 = 0.52 {m}/{s}[/latex]

Euler's head [latex] E = \frac{u_{1} V_{1w} - u_{2} V_{2w}}{g}[/latex]

[latex] E = \frac{7.5 imes 5.2 - 3.75 imes 0.52}{9.8} = 3.7[/latex]

[latex] E = 3.7 m[/latex]

[latex] \eta_{h} = \frac{E}{H}[/latex]

[latex] H = \frac{3.7}{0.9} = 4.1 m[/latex]

Power input [latex] = \gamma QH = \frac{9800 imes 0.05 imes 4.1}{1000} = 2 kw[/latex]

Power output [latex] = \eta_{0} imes power input = 2 imes 0.8 = 1.6 kw[/latex]

[latex] P = 1.6 kw[/latex]

Question 6.2: A francis turbine has the following specifications. Inlet ra...

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