Find the maximum permissible head under which a Francis turbine [latex]\left(N_{s}=70\right)[/latex] can operate if it is set [latex]10 \mathrm{ft}[/latex] above tailwater at an elevation of [latex]5000 \mathrm{ft}[/latex] with water temperature at [latex]60^{\circ} \mathrm{F}[/latex].

Table A.3 at [latex]5000 \mathrm{ft}[/latex] elevation: [latex]\frac{p_{\text {atm }}}{\gamma}=\frac{12.23(144)}{62.4}=28.22 \mathrm{ft}[/latex] Table A. 1 for water at [latex]60^{\circ} \mathrm{F}: \quad \frac{p_{v}}{\gamma}=0.59 \mathrm{ft}[/latex] Fig. [latex]16.17[/latex] for Francis turbine with [latex]N_{s}=70: \quad \sigma_{c}=0.32[/latex] Eq. (16.23): [latex]\quad h_{\max }=\frac{28.22-0.59-10}{0.32}=55.1 \mathrm{ft} \quad[/latex] This is the maximum permissible head to assure against cavitation. [latex] h_{\max }=\frac{p_{\mathrm{atm}} / \gamma-p_v / \gamma-z_B}{\sigma_c} [/latex] (162.3)

Question 16.SP.12: Find the maximum permissible head under which a Francis turb......

Fluid Mechanics With Engineering Applications [917590]

Find the maximum permissible head under which a Francis turbine $\left(N_{s}=70\right)$ can operate if it is set $10 \mathrm{ft}$ above tailwater at an elevation of $5000 \mathrm{ft}$ with water temperature at $60^{\circ} \mathrm{F}$ .

Step-by-Step

The 'Blue Check Mark' means that this solution was answered by an expert.
Learn more on how do we answer questions.

Table A.3 at $5000 \mathrm{ft}$ elevation: $\frac{p_{\text {atm }}}{\gamma}=\frac{12.23(144)}{62.4}=28.22 \mathrm{ft}$

Table A. 1 for water at $60^{\circ} \mathrm{F}: \quad \frac{p_{v}}{\gamma}=0.59 \mathrm{ft}$

Fig. $16.17$ for Francis turbine with $N_{s}=70: \quad \sigma_{c}=0.32$

Eq. (16.23): $\quad h_{\max }=\frac{28.22-0.59-10}{0.32}=55.1 \mathrm{ft} \quad$

This is the maximum permissible head to assure against cavitation.

$h_{\max }=\frac{p_{\mathrm{atm}} / \gamma-p_v / \gamma-z_B}{\sigma_c}$ (162.3)

TABLE A.1 Physical properties of water at standard sea-level atmospheric pressure ${ }^a$
Temperature,	Specific weight,	Density,	Absolute viscosity ${}^b$	Kinematic viscosity, ${}^b$	Surface tension,	Saturation vapor pressure,	Satur’n vapor pressure head,	Bulk modulus of elasticity,
$\boldsymbol{T}$	$\boldsymbol{\gamma}$	$\boldsymbol{\rho}$	$\boldsymbol{ \mu}$	$\boldsymbol{\nu}$	$\boldsymbol{\sigma}$	$\boldsymbol{ p_v}$	$\boldsymbol{ p_v}/ \boldsymbol{\gamma}$	$\boldsymbol{E_v}$
${ }^{\circ} \mathbf{F}$	$\mathbf{l b} / \mathbf{f t}^3$	$\boldsymbol{ slugs/ft { }^3}$	$10^{-6} \mathbf{lb} \cdot \mathrm{sec} / \mathbf{ft}^2$	$10^{-6} \mathbf{ft}^2 / \mathbf{sec}$	$\mathbf{lb} / \mathbf{ft}$	psia	ft abs	psi
32 ${ }^{\circ} \mathrm{F}$	62.42	1.940	37.46	19.31	0.00518	0.0885	0.204	293,000
40 ${ }^{\circ} \mathrm{F}$	62.43	1.940	32.29	16.64	0.00514	0.122	0.281	294,000
50 ${ }^{\circ} \mathrm{F}$	62.41	1.940	27.35	14.10	0.00509	0.178	0.411	305,000
60 ${ }^{\circ} \mathrm{F}$	62.37	1.938	23.59	12.17	0.00504	0.256	0.592	311,000
70 ${ }^{\circ} \mathrm{F}$	62.30	1.936	20.50	10.59	0.00498	0.363	0.839	320,000
80 ${ }^{\circ} \mathrm{F}$	62.22	1.934	17.99	9.30	0.00492	0.507	1.173	322,000
90 ${ }^{\circ} \mathrm{F}$	62.11	1.931	15.95	8.26	0.00486	0.698	1.618	323,000
100 ${ }^{\circ} \mathrm{F}$	62.00	1.927	14.24	7.39	0.00480	0.949	2.20	327,000
110 ${ }^{\circ} \mathrm{F}$	61.86	1.923	12.84	6.67	0.00473	1.275	2.97	331,000
120 ${ }^{\circ} \mathrm{F}$	61.71	1.918	11.68	6.09	0.00467	1.692	3.95	333,000
130 ${ }^{\circ} \mathrm{F}$	61.55	1.913	10.69	5.58	0.00460	2.22	5.19	334,000
140 ${ }^{\circ} \mathrm{F}$	61.38	1.908	9.81	5.14	0.00454	2.89	6.78	330,000
150 ${ }^{\circ} \mathrm{F}$	61.20	1.902	9.05	4.76	0.00447	3.72	8.75	328,000
160 ${ }^{\circ} \mathrm{F}$	61.00	1.896	8.38	4.42	0.00441	4.74	11.18	326,000
170 ${ }^{\circ} \mathrm{F}$	60.80	1.890	7.80	4.13	0.00434	5.99	14.19	322,000
180 ${ }^{\circ} \mathrm{F}$	60.58	1.883	7.26	3.85	0.00427	7.51	17.84	318,000
190 ${ }^{\circ} \mathrm{F}$	60.36	1.876	6.78	3.62	0.00420	9.34	22.28	313,000
200 ${ }^{\circ} \mathrm{F}$	60.12	1.868	6.37	3.41	0.00413	11.52	27.59	308,000
212 ${ }^{\circ} \mathrm{F}$	59.83	1.860	5.93	3.19	0.00404	14.69	35.36	300,000
${ }^{\circ} \mathbf{C}$	$\mathbf{kN} / \mathbf{m}^3$	$\mathbf{~kg} / \mathbf{m}^3$	$\mathbf{~N} \cdot \mathbf{s} / \mathbf{m}^2$	$10^{-6} \mathbf{~m}^2 / \mathbf{s}$	$\mathbf{N} / \mathbf{m}$	$\mathbf{kN} / \mathbf{m}^2 \text { abs }$	$\mathbf{m} \text { abs }$	$10^6 \mathbf{kN} / \mathbf{m}^2$
0 ${ }^{\circ} \mathrm{C}$	9.805	999.8	0.001781	1.785	0.0756	0.611	0.0623	2.02
5 ${ }^{\circ} \mathrm{C}$	9.807	1000.0	0.001518	1.519	0.0749	0.872	0.0889	2.06
10 ${ }^{\circ} \mathrm{C}$	9.804	999.7	0.001307	1.306	0.0742	1.230	0.1255	2.1
15 ${ }^{\circ} \mathrm{C}$	9.798	999.1	0.001139	1.139	0.0735	1.710	0.1745	2.14
20 ${ }^{\circ} \mathrm{C}$	9.789	998.2	0.001002	1.003	0.0728	2.34	0.239	2.18
25 ${ }^{\circ} \mathrm{C}$	9.777	997.0	0.000890	0.893	0.072	3.17	0.324	2.22
30 ${ }^{\circ} \mathrm{C}$	9.765	995.7	0.000798	0.800	0.0712	4.24	0.434	2.25
40 ${ }^{\circ} \mathrm{C}$	9.731	992.2	0.000653	0.658	0.0696	7.38	0.758	2.28
50 ${ }^{\circ} \mathrm{C}$	9.690	988.0	0.000547	0.553	0.0679	12.33	1.272	2.29
60 ${ }^{\circ} \mathrm{C}$	9.642	983.2	0.000466	0.474	0.0662	19.92	2.07	2.28
70 ${ }^{\circ} \mathrm{C}$	9.589	977.8	0.000404	0.413	0.0644	31.16	3.25	2.25
80 ${ }^{\circ} \mathrm{C}$	9.530	971.8	0.000354	0.364	0.0626	47.34	4.97	2.2
90 ${ }^{\circ} \mathrm{C}$	9.467	965.3	0.000315	0.326	0.0608	70.10	7.40	2.14
100 ${ }^{\circ} \mathrm{C}$	9.399	958.4	0.000282	0.294	0.0589	101.33	10.78	2.07
${ }^a$ In these tables, if (for example, at $32^{\circ} \mathrm{F}$ ) $\mu$ is given as $37.46$ and the units are $10^{-6} \mathrm{lb} \cdot \mathrm{sec} / \mathrm{ft}^2$ then $\mu=37.46 \times 10^{-6} \mathrm{lb} \cdot \mathrm{sec} / \mathrm{ft}^2$ . ${ }^b {\text {For viscosity, see also Figs. A.1 and A.2. }}$ .

TABLEA.3 The ICAO ${}^a$ standard atmosphere ${}^b$
$\textbf { Elevation above sea level }$	$\textbf { Temperature } \\ T$	$\textbf { Absolute pressure } \\ p$	$\textbf { Specific weight } \\ \gamma$	$\textbf { Density, } \\ \rho$	$\textbf { Absolute viscosity } \\ \mu$	$\textbf { Kinematic viscosity } \\\nu$	$\textbf { Speed of sound } \\ c$	$\textbf { Gravitational acceleration } \\ g$
$\mathbf{ft}$	${ }^{\circ} \mathbf{F}$	$\mathbf{psia}$	$\mathbf{lb} / \mathbf{ft}^3$	$\mathbf{sl} / \mathbf{ug} / \mathbf{ft}^3$	$10^{-6} \mathbf{lb} \cdot \mathbf{sec} / \mathbf{ft}^2$	$10^{-3} \mathbf{ft}^2 / \mathbf{sec}$	$\mathbf{ft} / \mathbf{sec}$	$\mathbf{ft} / \mathbf{sec}^2$
0	59.000	14.6959	0.076472	0.0023768	0.37372	0.15724	1116.45	32.1740
5,000	41.173	12.2283	0.065864	0.0020481	0.36366	0.17756	1097.08	32.158
10,000	23.355	10.1083	0.056424	0.0017555	0.35343	0.20133	1077.4	32.142
15,000	5.545	8.2970	0.048068	0.0014961	0.34302	0.22928	1057.35	32.129
20,000	-12.255	6.7588	0.040694	0.0012672	0.33244	0.26234	1036.94	32.113
25,000	-30.048	5.4607	0.034224	0.0010663	0.32166	0.30167	1016.11	32.097
30,000	-47.832	4.3726	0.028573	0.00089065	0.31069	0.34884	994.85	32.081
35,000	-65.607	3.4676	0.023672	0.00073819	0.29952	0.40575	973.13	32.068
40,000	-69.700	2.7300	0.018823	0.00058726	0.29691	0.50559	968.08	32.052
45,000	-69.700	2.1489	0.014809	0.00046227	0.29691	0.6423	968.08	32.036
50,000	-69.700	1.6917	0.011652	0.00036391	0.29691	0.81589	968.08	32.020
60,000	-69.700	1.0488	0.007218	0.00022561	0.29691	1.3160	968.08	31.991
70,000	-67.425	0.6509	0.004449	0.0001392	0.29836	2.1434	970.9	31.958
80,000	-61.976	0.4063	0.002737	0.000085707	0.30182	3.5215	977.62	31.930
90,000	-56.535	0.2554	0.001695	0.000053145	0.30525	5.7436	984.28	31.897
100,000	-51.099	0.1616	0.001058	0.000033182	0.30865	9.3018	990.91	31.868
$\mathbf{km}$	${ }^{\circ} \mathbf{C}$	$\mathbf{kPa} a b s$	$\mathbf{~N} / \mathbf{m}^3$	$\mathbf{~kg} / \mathbf{m}^3$	$10^{-6} \mathbf{~N} \cdot \mathbf{s} / \mathbf{m}^2$	$10^{-6} \mathbf{~m}^2 / \mathbf{s}$	$\mathbf{m} / \mathbf{s}$	$\mathbf{m} / \mathbf{s}^2$
0	15.000	101.325	12.0131	1.2250	17.894	14.607	340.294	9.80665
1	8.501	89.876	10.8987	1.1117	17.579	15.813	336.43	9.8036
2	2.004	79.501	9.8652	1.0066	17.260	17.147	332.53	9.8005
3	-4.500	70.121	8.9083	0.90925	16.938	18.628	328.58	9.7974
4	-10.984	61.66	8.0250	0.81935	16.612	20.275	324.59	9.7943
5	-17.474	54.048	7.2105	0.73643	16.282	22.110	320.55	9.7912
6	-23.963	47.217	6.4613	0.66011	15.949	24.161	316.45	9.7882
8	-36.935	35.651	5.1433	0.52579	15.271	29.044	308.11	9.7820
10	-49.898	26.499	4.0424	0.41351	14.577	35.251	299.53	9.7759
12	-56.500	19.399	3.0476	0.31194	14.216	45.574	295.07	9.7697
14	-56.500	14.170	2.2247	0.22786	14.216	62.391	295.07	9.7636
16	-56.500	10.352	1.6243	0.16647	14.216	85.397	295.07	9.7575
18	-56.500	7.565	1.1862	0.12165	14.216	116.86	295.07	9.7513
20	-56.500	5.529	0.8664	0.08891	14.216	159.89	295.07	9.7452
25	-51.598	2.549	0.3900	0.04008	14.484	361.35	298.39	9.7300
30	-46.641	1.197	0.1788	0.01841	14.753	801.34	301.71	9.7147
$a$ International Civil Aviation Organization; see Sec. 2.9. $b$ In these tables, if (for example, at $0 \mathrm{ft}$ ) $\mu$ is given as $0.37372$ and the units are $10^{-6} \mathrm{lb} \cdot \mathrm{sec} / \mathrm{ft}^2$ then $\mu=0.37372 \times 10^{-6} \mathrm{lb} \cdot \mathrm{sec} / \mathrm{ft}^2$