Chapter 11, Problem 11.85P

To determine

(a)

Determine the type of turbine in the prototype.

Answer to Problem 11.85P

The type of turbine in the prototype is propeller turbine.

Explanation of Solution

Given Information:

Head, $H_p=45ft$

Speed of turbine, $n_p=360r/\min$

Power, $P_p=4000bhp$

$=2200000ft.lbf/s$

Prototype diameter, $D_p=7ft$

Model diameter, $D_m=1ft$

Prototype

Speed of turbine, $n_p=360r/\min$

Power,

$P_p=4000bhp$

$=2200000ft.lbf/s$

Specific speed,

$N_{sp}=\frac{n_p\left(rpm\right)\sqrt{P_p\left(bhp\right)}}{{\left[H_p\left(ft\right)\right]}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}$

Here, $H_p=45ft$

$N_{sp}=\frac{360\sqrt{4000}}{{45}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}$

$N_{sp}=195$

Therefore, from specific speed range table, the type of turbine in the prototype is propeller turbine

Conclusion:

The type of turbine in the prototype is propeller turbine.

To determine

(b)

Determine the appropriate head for the model test.

Answer to Problem 11.85P

The appropriate head for the model test is $H_m=9.87ft$.

Explanation of Solution

Given Information:

Head, $H_p=45ft$

Speed of turbine, $n_p=360r/\min$

Power, $P_p=4000bhp$

$=2200000ft.lbf/s$

Prototype diameter, $D_p=7ft$

Model diameter, $D_m=1ft$

Model head is given by,

$\frac{g{H}_p}{n_p^2{D}_p^2}=\frac{g{H}_m}{n_m^2{D}_m^2}$

Here, model turbine speed, $n_m=1180r/\min$

$H_m=H_p\frac{n_m^2{D}_m^2}{n_p^2{D}_p^2}$

$=45\times \frac{{1180}^2\times 1^2}{{360}^2\times 7^2}$

$H_m=9.87ft$

Conclusion:

The appropriate head for the model test is $H_m=9.87ft$.

To determine

(c)

Determine the appropriate flow rate for the model test.

Answer to Problem 11.85P

The appropriate flow rate for the model test is $Q_m=8.5f{t}^3/s$.

Explanation of Solution

Given Information:

Head, $H_p=45ft$

Speed of turbine, $n_p=360r/\min$

Power, $P_p=4000bhp$

$=2200000ft.lbf/s$

Prototype diameter, $D_p=7ft$

Model diameter, $D_m=1ft$

Prototype flow rate,

$Q_p=\frac{P_p}{{\eta }_p\rho g{H}_p}$

Here, efficiency, $\eta =0.88$

$=\frac{2200000}{0.88\times 1.94\times 32.2\times 45}$

$Q_p=889.3f{t}^3/s$

Model flow rate is calculated using the relation below,

$\frac{Q_p}{n_p{D}_p^3}=\frac{Q_m}{n_m{D}_m^3}$

$Q_m=Q_p\frac{n_m{D}_m^3}{n_p{D}_p^3}$

$=889.3\times \frac{1180\times 1^3}{360\times 7^3}$

$Q_m=8.5f{t}^3/s$

Conclusion:

The appropriate flow rate for the model test is $Q_m=8.5f{t}^3/s$.

To determine

(d)

Estimate the power expected to be delivered by the turbine model.

Answer to Problem 11.85P

The turbine model delivers $P_m=8.38hp$ of power.

Explanation of Solution

Given Information:

Head, $H_p=45ft$

Speed of turbine, $n_p=360r/\min$

Power, $P_p=4000bhp$

$=2200000ft.lbf/s$

Prototype diameter, $D_p=7ft$

Model diameter, $D_m=1ft$

Power delivered by model is calculated by the below relation,

$P_m={\eta }_m{Q}_m\rho g{H}_m$

$=0.88\times 8.5\times 1.94\times 32.2\times 9.87$

$=4611.2ft.lbf/s$

$P_m=8.38hp$

Conclusion:

The turbine model delivers $P_m=8.38hp$ of power