The nozzle of a rocket motor is designed to generate a thrust of 30,000 N when operating at an altitude of 20 km. The pressure and temperature inside the combustion chamber are 1000 kPa and 2500 K. The gas constant of the gas in the jet is R = 280 m²/(s² K), and γ = 1.4. Assuming that the flow in

Question

The nozzle of a rocket motor is designed to generate a thrust of 30,000 N when operating at an altitude of 20 km. The pressure and temperature inside the combustion chamber are 1000 kPa and 2500 K. The gas constant of the gas in the jet is R = 280 m²/(s² K), and γ = 1.4. Assuming that the flow in the nozzle is isentropic, calculate the throat and exit areas. Use the isentropic table (Table 15.1).

TABLE 15.1 Isentropic Flow of a Perfect Gas (γ = 1.4)
M	[latex]p/p_0[/latex]	[latex]\rho/\rho_0[/latex]	[latex]T/T_0[/latex]	A/A*	M	[latex]p/p_0[/latex]	[latex]\rho/\rho_0[/latex]	[latex]T/T_0[/latex]	A/A*
0	1	1	1	∞	0.62	0.7716	0.831	0.9286	1.1656
0.02	0.9997	0.9998	0.9999	28.9421	0.64	0.7591	0.8213	0.9243	1.1451
0.04	0.9989	0.9992	0.9997	14.4815	0.66	0.7465	0.8115	0.9199	1.1265
0.06	0.9975	0.9982	0.9993	9.6659	0.68	0.7338	0.8016	0.9153	1.1097
0.08	0.9955	0.9968	0.9987	7.2616	0.7	0.7209	0.7916	0.9107	1.0944
0.1	0.993	0.995	0.998	5.8218	0.72	0.708	0.7814	0.9061	1.0806
0.12	0.99	0.9928	0.9971	4.8643	0.74	0.6951	0.7712	0.9013	1.0681
0.14	0.9864	0.9903	0.9961	4.1824	0.76	0.6821	0.7609	0.8964	1.057
0.16	0.9823	0.9873	0.9949	3.6727	0.78	0.669	0.7505	0.8915	1.0471
0.18	0.9776	0.984	0.9936	3.2779	0.8	0.656	0.74	0.8865	1.0382
0.2	0.9725	0.9803	0.9921	2.9635	0.82	0.643	0.7295	0.8815	1.0305
0.22	0.9668	0.9762	0.9904	2.7076	0.84	0.63	0.7189	0.8763	1.0237
0.24	0.9607	0.9718	0.9886	2.4956	0.86	0.617	0.7083	0.8711	1.0179
0.26	0.9541	0.967	0.9867	2.3173	0.88	0.6041	0.6977	0.8659	1.0129
0.28	0.947	0.9619	0.9846	2.1656	0.9	0.5913	0.687	0.8606	1.0089
0.3	0.9395	0.9564	0.9823	2.0351	0.92	0.5785	0.6764	0.8552	1.0056
0.32	0.9315	0.9506	0.9799	1.9219	0.94	0.5658	0.6658	0.8498	1.0031
0.34	0.9231	0.9445	0.9774	1.8229	0.96	0.5532	0.6551	0.8444	1.0014
0.36	0.9143	0.938	0.9747	1.7358	0.98	0.5407	0.6445	0.8389	1.0003
0.38	0.9052	0.9313	0.9719	1.6587	1	0.5283	0.6339	0.8333	1
0.4	0.8956	0.9243	0.969	1.5901	1.02	0.516	0.6234	0.8278	1.0003
0.42	0.8857	0.917	0.9659	1.5289	1.04	0.5039	0.6129	0.8222	1.0013
0.44	0.8755	0.9094	0.9627	1.474	1.06	0.4919	0.6024	0.8165	1.0029
0.46	0.865	0.9016	0.9594	1.4246	1.08	0.48	0.592	0.8108	1.0051
0.48	0.8541	0.8935	0.9559	1.3801	1.1	0.4684	0.5817	0.8052	1.0079
0.5	0.843	0.8852	0.9524	1.3398	1.12	0.4568	0.5714	0.7994	1.0113
0.52	0.8317	0.8766	0.9487	1.3034	1.14	0.4455	0.5612	0.7937	1.0153
0.54	0.8201	0.8679	0.9449	1.2703	1.16	0.4343	0.5511	0.7879	1.0198
0.56	0.8082	0.8589	0.941	1.2403	1.18	0.4232	0.5411	0.7822	1.0248
0.58	0.7962	0.8498	0.937	1.213	1.2	0.4124	0.5311	0.7764	1.0304
0.6	0.784	0.8405	0.9328	1.1882	1.22	0.4017	0.5213	0.7706	1.0366
TABLE 15.1 Isentropic Flow of a Perfect Gas (γ = 1.4)-cont'd
M	[latex]p/p_0[/latex]	[latex]\rho/\rho_0[/latex]	[latex]T/T_0[/latex]	A/A*	M	[latex]p/p_0[/latex]	[latex]\rho/\rho_0[/latex]	[latex]T/T_0[/latex]	A/A*
1.24	0.3912	0.5115	0.7648	1.0432	1.88	0.1539	0.2627	0.5859	1.5308
1.26	0.3809	0.5019	0.759	1.0504	1.9	0.1492	0.257	0.5807	1.5553
1.28	0.3708	0.4923	0.7532	1.0581	1.92	0.1447	0.2514	0.5756	1.5804
1.3	0.3609	0.4829	0.7474	1.0663	1.94	0.1403	0.2459	0.5705	1.6062
1.32	0.3512	0.4736	0.7416	1.075	1.96	0.136	0.2405	0.5655	1.6326
1.34	0.3417	0.4644	0.7358	1.0842	1.98	0.1318	0.2352	0.5605	1.6597
1.36	0.3323	0.4553	0.73	1.094	2	0.1278	0.23	0.5556	1.6875
1.38	0.3232	0.4463	0.7242	1.1042	2.02	0.1239	0.225	0.5506	1.716
1.4	0.3142	0.4374	0.7184	1.1149	2.04	0.1201	0.22	0.5458	1.7451
1.42	0.3055	0.4287	0.7126	1.1262	2.06	0.1164	0.2152	0.5409	1.775
1.44	0.2969	0.4201	0.7069	1.1379	2.08	0.1128	0.2104	0.5361	1.8056
1.46	0.2886	0.4116	0.7011	1.1501	2.1	0.1094	0.2058	0.5313	1.8369
1.48	0.2804	0.4032	0.6954	1.1629	2.12	0.106	0.2013	0.5266	1.869
1.5	0.2724	0.395	0.6897	1.1762	2.14	0.1027	0.1968	0.5219	1.9018
1.52	0.2646	0.3869	0.684	1.1899	2.16	0.0996	0.1925	0.5173	1.9354
1.54	0.257	0.3789	0.6783	1.2042	2.18	0.0965	0.1882	0.5127	1.9698
1.56	0.2496	0.371	0.6726	1.219	2.2	0.0935	0.1841	0.5081	2.005
1.58	0.2423	0.3633	0.667	1.2344	2.22	0.0906	0.18	0.5036	2.0409
1.6	0.2353	0.3557	0.6614	1.2502	2.24	0.0878	0.176	0.4991	2.0777
1.62	0.2284	0.3483	0.6558	1.2666	2.26	0.0851	0.1721	0.4947	2.1153
1.64	0.2217	0.3409	0.6502	1.2836	2.28	0.0825	0.1683	0.4903	2.1538
1.66	0.2151	0.3337	0.6447	1.301	2.3	0.08	0.1646	0.4859	2.1931
1.68	0.2088	0.3266	0.6392	1.319	2.32	0.0775	1.1609	0.4816	2.2333
1.7	0.2026	0.3197	0.6337	1.3376	2.34	0.0751	0.1574	0.4773	2.2744
1.72	0.1966	0.3129	0.6283	1.3567	2.36	0.0728	0.1539	0.4731	2.3164
1.74	0.1907	0.3062	0.6229	1.3764	2.38	0.0706	0.1505	0.4688	2.3593
1.76	0.185	0.2996	0.6175	1.3967	2.4	0.0684	0.1472	0.4647	2.4031
1.78	0.1794	0.2931	0.6121	1.4175	2.42	0.0663	0.1439	0.4606	2.4479
1.8	0.174	0.2868	0.6068	1.439	2.44	0.0643	0.1408	0.4565	2.4936
1.82	0.1688	0.2806	0.6015	1.461	2.46	0.0623	0.1377	0.4524	2.5403
1.84	0.1637	0.2745	0.5963	1.4836	2.48	0.0604	0.1346	0.4484	2.588
1.86	0.1587	0.2686	0.591	1.5069	2.5	0.0585	0.1317	0.4444	2.6367
TABLE 15.1 Isentropic Flow of a Perfect Gas (γ = 1.4)-cont'd
M	[latex]p/p_0[/latex]	[latex]\rho/\rho_0[/latex]	[latex]T/T_0[/latex]	A/A*	M	[latex]p/p_0[/latex]	[latex]\rho/\rho_0[/latex]	[latex]T/T_0[/latex]	A/A*
2.52	0.0567	0.1288	0.4405	2.6865	3.16	0.0215	0.0643	0.3337	4.9304
2.54	0.055	0.126	0.4366	2.7372	3.18	0.0208	0.063	0.3309	5.0248
2.56	0.0533	0.1232	0.4328	2.7891	3.2	0.0202	0.0617	0.3281	5.121
2.58	0.0517	0.1205	0.4289	2.842	3.22	0.0196	0.0604	0.3253	5.2189
2.6	0.0501	0.1179	0.4252	2.896	3.24	0.0191	0.0591	0.3226	5.3186
2.62	0.0486	0.1153	0.4214	2.9511	3.26	0.0185	0.0579	0.3199	5.4201
2.64	0.0471	0.1128	0.4177	3.0073	3.28	0.018	0.0567	0.3173	5.5234
2.66	0.0457	0.1103	0.4141	3.0647	3.3	0.0175	0.0555	0.3147	5.6286
2.68	0.0443	0.1079	0.4104	3.1233	3.32	0.017	0.0544	0.3121	5.7358
2.7	0.043	0.1056	0.4068	3.183	3.34	0.0165	0.0533	0.3095	5.8448
2.72	0.0417	0.1033	0.4033	3.244	3.36	0.016	0.0522	0.3069	5.9558
2.74	0.0404	0.101	0.3998	3.3061	3.38	0.0156	0.0511	0.3044	6.0687
2.76	0.0392	0.0989	0.3963	3.3695	3.4	0.0151	0.0501	0.3019	6.1837
2.78	0.038	0.0967	0.3928	3.4342	3.42	0.0147	0.0491	0.2995	6.3007
2.8	0.0368	0.0946	0.3894	3.5001	3.44	0.0143	0.0481	0.297	6.4198
2.82	0.0357	0.0926	0.386	3.5674	3.46	0.0139	0.0471	0.2946	6.5409
2.84	0.0347	0.0906	0.3827	3.6359	3.48	0.0135	0.0462	0.2922	6.6642
2.86	0.0336	0.0886	0.3794	3.7058	3.5	0.0131	0.0452	0.2899	6.7896
2.88	0.0326	0.0867	0.3761	3.7771	3.52	0.0127	0.0443	0.2875	6.9172
2.9	0.0317	0.0849	0.3729	3.8498	3.54	0.0124	0.0434	0.2852	7.0471
2.92	0.0307	0.0831	0.3696	3.9238	3.56	0.012	0.0426	0.2829	7.1791
2.94	0.0298	0.0813	0.3665	3.9993	3.58	0.0117	0.0417	0.2806	7.3135
2.96	0.0289	0.0796	0.3633	4.0763	3.6	0.0114	0.0409	0.2784	7.4501
2.98	0.0281	0.0779	0.3602	4.1547	3.62	0.0111	0.0401	0.2762	7.5891
3	0.0272	0.0762	0.3571	4.2346	3.64	0.0108	0.0393	0.274	7.7305
3.02	0.0264	0.0746	0.3541	4.316	3.66	0.0105	0.0385	0.2718	7.8742
3.04	0.0256	0.073	0.3511	4.399	3.68	0.0102	0.0378	0.2697	8.0204
3.06	0.0249	0.0715	0.3481	4.4835	3.7	0.0099	0.037	0.2675	8.1691
3.08	0.0242	0.07	0.3452	4.5696	3.72	0.0096	0.0363	0.2654	8.3202
3.1	0.0234	0.0685	0.3422	4.6573	3.74	0.0094	0.0356	0.2633	8.4739
3.12	0.0228	0.0671	0.3393	4.7467	3.76	0.0091	0.0349	0.2613	8.6302
3.14	0.0221	0.0657	0.3365	4.8377	3.78	0.0089	0.0342	0.2592	8.7891
TABLE 15.1 Isentropic Flow of a Perfect Gas (γ = 1.4)-cont'd
M	[latex]p/p_0[/latex]	[latex]\rho/\rho_0[/latex]	[latex]T/T_0[/latex]	A/A*	M	[latex]p/p_0[/latex]	[latex]\rho/\rho_0[/latex]	[latex]T/T_0[/latex]	A/A*
3.8	0.0086	0.0335	0.2572	8.9506	4.42	0.0038	0.0187	0.2038	15.4724
3.82	0.0084	0.0329	0.2552	9.1148	4.44	0.0037	0.0184	0.2023	15.7388
3.84	0.0082	0.0323	0.2532	0.2817	4.46	0.0036	0.0181	0.2009	16.0092
3.86	0.008	0.0316	0.2513	9.4513	4.48	0.0035	0.0178	0.1994	16.2837
3.88	0.0077	0.031	0.2493	9.6237	4.5	0.0035	0.0174	0.198	16.5622
3.9	0.0075	0.0304	0.2474	9.799	4.52	0.0034	0.0171	0.1966	16.8449
3.92	0.0073	0.0299	0.2455	9.9771	4.54	0.0033	0.0168	0.1952	17.1317
3.94	0.0071	0.0293	0.2436	10.1581	4.56	0.0032	0.0165	0.1938	17.4228
3.96	0.0069	0.0287	0.2418	10.342	4.58	0.0031	0.0163	0.1925	17.7181
3.98	0.0068	0.0282	0.2399	10.5289	4.6	0.0031	0.016	0.1911	18.0178
4	0.0066	0.0277	0.2381	10.7188	4.62	0.003	0.0157	0.1898	18.3218
4.02	0.0064	0.0271	0.2363	10.9117	4.64	0.0029	0.0154	0.1885	18.6303
4.04	0.0062	0.0266	0.2345	11.1077	4.66	0.0028	0.0152	0.1872	18.9433
4.06	0.0061	0.0261	0.2327	11.3068	4.68	0.0028	0.0149	0.1859	19.2608
4.08	0.0059	0.0256	0.231	11.5091	4.7	0.0027	0.0146	0.1846	19.5828
4.1	0.0058	0.0252	0.2293	11.7147	4.72	0.0026	0.0144	0.1833	19.9095
4.12	0.0056	0.0247	0.2275	11.9234	4.74	0.0026	0.0141	0.182	20.2409
4.14	0.0055	0.0242	0.2258	12.1354	4.76	0.0025	0.0139	0.1808	20.577
4.16	0.0053	0.0238	0.2242	12.3508	4.78	0.0025	0.0137	0.1795	20.9179
4.18	0.0052	0.0234	0.2225	12.5695	4.8	0.0024	0.0134	0.1783	21.2637
4.2	0.0051	0.0229	0.2208	12.7916	4.82	0.0023	0.0132	0.1771	21.6144
4.22	0.0049	0.0225	0.2192	13.0172	4.84	0.0023	0.013	0.1759	21.97
4.24	0.0048	0.0221	0.2176	13.2463	4.86	0.0022	0.0128	0.1747	22.3306
4.26	0.0047	0.0217	0.216	13.4789	4.88	0.0022	0.0125	0.1735	22.6963
4.28	0.0046	0.0213	0.2144	13.7151	4.9	0.0021	0.0123	0.1724	23.0671
4.3	0.0044	0.0209	0.2129	13.9549	4.92	0.0021	0.0121	0.1712	23.4431
4.32	0.0043	0.0205	0.2113	14.1984	4.94	0.002	0.0119	0.17	23.8243
4.34	0.0042	0.0202	0.2098	14.4456	4.96	0.002	0.0117	0.1689	24.2109
4.36	0.0041	0.0198	0.2083	14.6965	4.98	0.0019	0.0115	0.1678	24.6027
4.38	0.004	0.0194	0.2067	14.9513	5	0.0019	0.0113	0.1667	25
4.4	0.0039	0.0191	0.2053	15.2099

Accepted Answer

At an altitude of 20 km, the pressure of the standard atmosphere (Section A.4 in Appendix A) is 5467 Pa. If subscripts 0 and e refer to the stagnation and exit conditions, then a summary of the information given is as follows:

[latex]p_e=5467 \mathrm{Pa},p_0=1000 \mathrm{kPa},T_0=2500 K,\quad ext{ and }\quad ext{Thrust}= ho_eu_e^2A_e=30 \mathrm{kN}[/latex].

Here, we have used the facts that the thrust equals mass flow rate times the exit velocity, and the pressure inside the combustion chamber is nearly equal to the stagnation pressure. The pressure ratio at the exit is:

[latex]\frac{p_e}{p_0}=\frac{5467 Pa}{10^6 Pa }=5.467 imes 10^{-3}[/latex]

For this ratio of [latex]p_e/p_0[/latex], the isentropic table (Table 15.1) gives:

[latex]M_e=4.15, A_e / A^*=12.2, \quad ext { and } \quad T_e / T_0=0.225[/latex].

The exit temperature and density are therefore:

[latex]T_e=0.225(2500 K)=562 K[/latex], and

[latex] ho_e=p_e/RT_e=(5467 \mathrm{Pa})/\left(280 \mathrm{m}^2s^{-2}K^{-1} ight)(562 K)=0.0347 \mathrm{kgm}^{-3}[/latex].

The exit velocity is:

[latex]u_e=M_e\left[\gamma R T_e ight]^{1 / 2}=4.15\left[1.4\left(280 \mathrm{m}^2 s ^{-2} K ^{-1} ight)(562 K ) ight]^{1 / 2}=1948 \mathrm{ms}^{-1}[/latex].

The exit area is found from the expression for thrust:

[latex]A_e=\frac{ ext { Thrust }}{ ho_e u_e^2}=\frac{30,000 N}{\left(0.0347 \mathrm{kgm}^{-3} ight)\left(1948 \mathrm{ms}^{-1} ight)^2}=0.0228 \mathrm{m}^2[/latex].

Because [latex]A_{e}/A^*=12.2[/latex], the throat area is:

[latex]A^*=\left(0.0228 \mathrm{m}^2 ight)/12.2=0.0187 \mathrm{m}^2[/latex].

Question 15.3: The nozzle of a rocket motor is designed to generate a thrus...

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