Question 8.10: Dry air is travelling in an insulated constant area duct of ......

Engineering Fluid Mechanics [1114568]

Dry air is travelling in an insulated constant area duct of 60 mm diameter. The flow Mach number at entry to the duct is 2.5. What is the maximum possible length of the pipe to achieve smooth flow without discontinuity if the average friction factor for the flow in the duct is 0.003.

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Given: $M=2.5 ; D=0.06 m ; f=0.003$

Refer Figure 8.21. Since the duct is of constant area and is insulated, the flow can be modelled as Fanno flow. For Fanno flow, the maximum possible length for the duct corresponds to the length of the pipe required to choke the flow (M = 1 at exit). In Fanno tables (Appendix A2), the maximum resistance parameter to reach sonic condition at exit ( $\left(4 f L_{\max } / D\right)$ ) is tabulated as a function of γ and M.

From Fanno flow tables (γ = 1.4), corresponding to M = 2.5

$4 f \frac{L_{\max }}{D}=0.43197$

Hence, the length of the pipe required to achieve sonic condition at exit (maximum possible length)

$\begin{aligned}L_{\max}&=\frac{0.43197 \times D}{4 \times f}=\frac{0.43197 \times 0.06}{4 \times 0.003}\\&=2.16 m\end{aligned}$

Fanno Flow Tables for Perfect Gas (y = 1.4)
M	F/F*	T/T*	p₀/p₀*	p/p*	4f L_max/D	M
0	Indef	1.200000	Indef	Indef	Indef	0
0.02	22.83364	1.199904	28.94213	54.77006	1778.450	0.02
0.04	11.43462	1.199616	14.48149	27.38175	440.3522	0.04
0.06	7.642847	1.199137	9.665910	18.25085	193.0311	0.06
0.08	5.752883	1.198466	7.261610	13.68431	106.7182	0.03
0.10	4.623634	1.197605	5.821829	10.94351	66.92156	0.10
0.12	3.874734	1.196554	4.864318	9.115592	45.40796	0.12
0.14	3.343168	1.195314	4.182400	7.809317	32.51131	0.14
0.16	2.947427	1.193887	3.672739	6.829072	24.19783	0.16
0.18	2.642227	1.192274	3.277926	6.066183	18.54265	0.18
0.20	2.400397	1.190476	2.963520	5.455447	14.53327	0.20
0.22	2.204644	1.188495	2.707602	4.955370	11.59605	0.22
0.24	2.043440	1.186333	2.495562	4.538289	9.386481	0.24
0.26	1.908803	1.183992	2.317287	4.185054	7.687567	0.26
0.28	1.795031	1.181474	2.165554	3.881988	6.357214	0.28
0.30	1.697941	1.178782	2.035065	3.619057	5.299253	0.30
0.32	1.614396	1.175917	1.921851	3.388741	4.446743	0.32
0.34	1.541997	1.172883	1.822876	3.185286	3.751953	0.34
0.36	1.478876	1.169682	1.735778	3.004218	3.180118	0.36
0.38	1.423560	1.166317	1.658696	2.842004	2.705448	0.38
0.40	1.374868	1.162791	1.590140	2.695819	2.308493	0.40
0.42	1.331845	1.159107	1.528905	2.563377	1.974366	0.42
0.44	1.293709	1.155268	1.474005	2.442804	1.691525	0.44
0.46	1.259814	1.151278	1.424629	2.332557	1.450911	0.46
0.48	1.229620	1.147140	1.380097	2.231346	1.245341	0.48
0.50	1.202676	1.142857	1.339844	2.138090	1.069060	0.50
0.52	1.178596	1.138434	1.303388	2.051873	0.917418	0.52
0.54	1.157054	1.133873	1.270321	1.971916	0.786625	0.54
0.56	1.137771	1.129178	1.240294	1.897550	0.673571	0.56
0.58	1.120503	1.124353	1.213007	1.828199	0.575683	0.58
0.60	1.105041	1.119403	1.188200	1.763364	0.490822	0.60
0.62	1.091203	1.114330	1.165646	1.702610	0.417197	0.62
0.64	1.078828	1.109139	1.145148	1.645558	0.353299	0.64
0.66	1.067774	1.103834	1.126535	1.591871	0.297853	0.66
0.68	1.057919	1.098418	1.109655	1.541257	0.249775	0.68
0.70	1.049150	1.092896	1.094373	1.493452	0.208139	0.70
0.72	1.041372	1.087272	1.080571	1.448227	0.172149	0.72
0.74	1.034494	1.081549	1.068144	1.405372	0.141122	0.74
0.76	1.028441	1.075731	1.056999	1.364704	0.114468	0.76
0.78	1.023140	1.069824	1.047053	1.326055	0.091672	0.78
0.80	1.018528	1.063830	1.038230	1.289277	0.072290	0.80
0.82	1.014549	1.057753	1.030464	1.254233	0.055932	0.82
0.84	1.011151	1.051598	1.023696	1.220803	0.042256	0.84
0.86	1.008285	1.045369	1.017871	1.188875	0.030965	0.86
0.88	1.005910	1.039069	1.012941	1.158349	0.021795	0.88
0.90	1.003987	1.032702	1.008863	1.129133	0.014512	0.90
0.92	1.002480	1.026273	1.005597	1.101143	0.008913	0.92
0.94	1.001356	1.019784	1.003108	1.074302	0.004815	0.94
0.96	1.000586	1.013240	1.001365	1.048540	0.002057	0.96
0.98	1.000143	1.006644	1.000337	1.023792	0.000495	0.98
1.00	1.000000	1.000000	1.000000	1.000000	0.000000	1.00
1.02	1.000135	0.993312	1.000330	0.977108	0.000459	1.02
1.04	1.000527	0.986582	1.001305	0.955066	0.001769	1.04
1.06	1.001156	0.979816	1.002907	0.933827	0.003838	1.06
1.08	1.002003	0.973015	1.005119	0.913347	0.006585	1.08
1.10	1.003052	0.966184	1.007925	0.893588	0.009935	1.10
1.12	1.004287	0.959325	1.011312	0.874510	0.013823	1.12
1.14	1.005694	0.952441	1.015267	0.856080	0.018188	1.14
1.16	1.007259	0.945537	1.019780	0.838265	0.022977	1.16
1.18	1.008970	0.938615	1.024840	0.821035	0.028142	1.18
1.20	1.010815	0.931677	1.030440	0.804362	0.033638	1.20
1.22	1.012783	0.924727	1.036572	0.788219	0.039426	1.22
1.24	1.014864	0.917768	1.043229	0.772582	0.045471	1.24
1.26	1.017049	0.910802	1.050406	0.757428	0.051739	1.26
1.28	1.019330	0.903832	1.058100	0.742735	0.058201	1.28
1.30	1.021697	0.896861	1.066305	0.728483	0.064832	1.30
1.32	1.024144	0.889891	1.075018	0.714652	0.071607	1.32
1.34	1.026662	0.882924	1.084239	0.701224	0.078504	1.34
1.36	1.029247	0.875964	1.093964	0.688183	0.085503	1.36
1.38	1.031891	0.869011	1.104193	0.675513	0.092586	1.38
1.40	1.034588	0.862069	1.114926	0.663198	0.099738	1.40
1.42	1.037334	0.855139	1.126162	0.651224	0.106943	1.42
1.44	1.040123	0.848224	1.137903	0.639577	0.114189	1.44
1.46	1.042950	0.841326	1.150150	0.628245	0.121462	1.46
1.48	1.045811	0.834446	1.162904	0.617216	0.128753	1.48
1.50	1.048702	0.827586	1.176167	0.606478	0.136050	1.50
1.52	1.051619	0.820749	1.189943	0.596021	0.143346	1.52
1.54	1.054558	0.813935	1.204234	0.585833	0.150631	1.54
1.56	1.057517	0.807146	1.219044	0.575906	0.157899	1.56
1.58	1.060490	0.800384	1.234376	0.566229	0.165143	1.58
1.60	1.063477	0.793651	1.250235	0.556794	0.172357	1.60
1.62	1.066474	0.786947	1.266626	0.547593	0.179535	1.62
1.64	1.069478	0.780275	1.283553	0.538617	0.186673	1.64
1.66	1.072486	0.773635	1.301021	0.529858	0.193766	1.66
1.68	1.075498	0.767028	1.319037	0.521310	0.200811	1.68
1.70	1.078510	0.760456	1.337606	0.512966	0.207803	1.70
1.72	1.081521	0.753920	1.356735	0.504817	0.214740	1.72
1.74	1.084528	0.747421	1.376430	0.496859	0.221620	1.74
1.76	1.087531	0.740960	1.396698	0.489086	0.228438	1.76
1.78	1.090526	0.734538	1.417546	0.481490	0.235195	1.78
1.80	1.093514	0.728155	1.438982	0.474067	0.241886	1.80
1.82	1.096492	0.721813	1.461013	0.466811	0.248512	1.82
1.84	1.099460	0.715512	1.483648	0.459717	0.255070	1.84
1.86	1.102415	0.709253	1.506894	0.452781	0.261560	1.86
1.88	1.105357	0.703037	1.530761	0.445996	0.267979	1.88
1.90	1.108285	0.696864	1.555257	0.439360	0.274328	1.90
1.92	1.111198	0.690735	1.580391	0.432867	0.280606	1.92
1.94	1.114096	0.684650	1.606172	0.426513	0.286812	1.94
1.96	1.116976	0.678610	1.632611	0.420295	0.292946	1.96
1.98	1.119838	0.672616	1.659717	0.414208	0.299008	1.98
2.00	1.122683	0.666667	1.687500	0.408248	0.304997	2.00
2.02	1.125508	0.660764	1.715971	0.402413	0.310913	2.02
2.04	1.128314	0.654907	1.745139	0.396698	0.316756	2.04
2.06	1.131100	0.649098	1.775017	0.391100	0.322526	2.06
2.08	1.133866	0.643335	1.805614	0.385616	0.328225	2.08
2.10	1.136610	0.637620	1.836944	0.380243	0.333851	2.10
2.12	1.139334	0.631951	1.869016	0.374978	0.339405	2.12
2.14	1.142035	0.626331	1.901843	0.369818	0.344887	2.14
2.16	1.144715	0.620758	1.935437	0.364760	0.350299	2.16
2.18	1.147372	0.615233	1.969810	0.359802	0.355639	2.18
2.20	1.150007	0.609756	2.004975	0.354940	0.360910	2.20
2.22	1.152619	0.604327	2.040944	0.350173	0.366111	2.22
2.24	1.155208	0.598946	2.077731	0.345498	0.371243	2.24
2.26	1.157774	0.593613	2.115349	0.340913	0.376306	2.26
2.28	1.160316	0.588328	2.153811	0.336415	0.381302	2.28
2.30	1.162835	0.583090	2.193131	0.332002	0.386230	2.30
2.32	1.165331	0.577901	2.233323	0.327672	0.391092	2.32
2.34	1.167803	0.572760	2.274401	0.323423	0.395888	2.34
2.36	1.170252	0.567666	2.316381	0.319253	0.400619	2.36
2.38	1.172677	0.562620	2.359275	0.315160	0.405286	2.38
2.40	1.175079	0.557621	2.403100	0.311142	0.409889	2.40
2.42	1.177456	0.552669	2.447870	0.307197	0.414429	2.42
2.44	1.179811	0.547765	2.493602	0.303324	0.418907	2.44
2.46	1.182141	0.542908	2.540309	0.299521	0.423324	2.46
2.48	1.184448	0.538097	2.588010	0.295787	0.427680	2.48
2.50	1.186732	0.533333	2.636719	0.292119	0.431977	2.50
2.52	1.188993	0.528616	2.686453	0.288516	0.436214	2.52
2.54	1.191230	0.523944	2.737228	0.284976	0.440393	2.54
2.56	1.193444	0.519319	2.789063	0.281499	0.444514	2.56
2.58	1.195634	0.514739	2.841972	0.278083	0.448579	2.58
2.60	1.197802	0.510204	2.895975	0.274725	0.452588	2.60
2.62	1.199947	0.505715	2.951089	0.271426	0.456541	2.62
2.64	1.202069	0.501270	3.007331	0.268183	0.460441	2.64
2.66	1.204169	0.496870	3.064720	0.264996	0.464286	2.66
2.68	1.206246	0.492514	3.123274	0.261863	0.468079	2.68
2.70	1.208301	0.488202	3.183011	0.258783	0.471819	2.70
2.72	1.210334	0.483933	3.243951	0.255755	0.475508	2.72
2.74	1.212345	0.479708	3.306113	0.252777	0.479146	2.74
2.76	1.214334	0.475526	3.369515	0.249849	0.482735	2.76
2.78	1.216302	0.471387	3.434179	0.246970	0.486274	2.78
2.80	1.218248	0.467290	3.500123	0.244138	0.489765	2.80
2.82	1.220173	0.463235	3.567368	0.241352	0.493208	2.82
2.84	1.222076	0.459221	3.635934	0.238612	0.496604	2.84
2.86	1.223959	0.455249	3.705842	0.235917	0.499953	2.86
2.88	1.225821	0.451318	3.777113	0.233265	0.503257	2.88
2.90	1.227662	0.447427	3.849768	0.230655	0.506516	2.90
2.92	1.229483	0.443577	3.923829	0.228088	0.509731	2.92
2.94	1.231284	0.439767	3.999317	0.225561	0.512902	2.94
2.96	1.233065	0.435996	4.076255	0.223074	0.516030	2.96
2.98	1.234826	0.432264	4.154664	0.220627	0.519115	2.98
3.00	1.236568	0.428571	4.234568	0.218218	0.522159	3.00
3.10	1.244989	0.410678	4.657311	0.206723	0.536777	3.10
3.20	1.252951	0.393701	5.120957	0.196080	0.550444	3.20
3.30	1.260477	0.377596	5.628647	0.186209	0.563232	3.30
3.40	1.267592	0.362319	6.183699	0.177038	0.575207	3.40
3.50	1.274320	0.347826	6.789621	0.168505	0.586429	3.50
3.60	1.280682	0.334076	7.450111	0.160554	0.596955	3.60
3.70	1.286701	0.321027	8.169066	0.153133	0.606836	3.70
3.80	1.292398	0.308642	8.950585	0.146199	0.616119	3.80
3.90	1.297791	0.296883	9.798973	0.139710	0.624849	3.90
4.00	1.302899	0.285714	10.71875	0.133631	0.633065	4.00