It is of interest to compare the sonic velocity for air, carbon monoxide, and nitrogen at a specified temperature. (a) Determine the sonic velocity for air at 45^◦ C. (b) Determine the sonic velocity for helium at 45^◦ C. (c) Determine the sonic velocity for hydrogen at 45^◦ C.

Question

It is of interest to compare the sonic velocity for air, carbon monoxide, and nitrogen ata specified temperature. (a) Determine the sonic velocity for air at [latex]45^{\circ } C[/latex]. (b) Determine the sonic velocity for helium at [latex]45^{\circ } C[/latex]. (c) Determine the sonic velocity for hydrogen at [latex]45^{\circ } C[/latex].

Accepted Answer

The sonic velocities for the three common gases at [latex]45^{\circ } C[/latex] are computed by applying Equation 1.162 [latex]C=\sqrt{\frac{E_{\upsilon }}{ ho } } =\sqrt{\frac{k_{p}}{ ho } } =\sqrt{\frac{k_{p}RT }{ ho } } =\sqrt{kRT} [/latex] as follows:
[latex] C=\sqrt{kRT} [/latex]
where the values for k and R, which are each independent of both the pressure and the temperature of the gas, are read from Table A.5 in Appendix A.
(a) The sonic velocity for air at [latex]45^{\circ } C[/latex] is determined as follows:

[latex]N:=kg\frac{m}{sec^{2} } [/latex] [latex]T:=45^{\circ } C=318.15K[/latex]
[latex]k_{air}: =1.4[/latex] [latex]R_{air}: =287\frac{Nm}{kgK} [/latex] [latex]C_{air}: =\sqrt{k_{air}R_{air}T} =357.537\frac{m}{s} [/latex]

(b) The sonic velocity for helium at [latex]45^{\circ } C[/latex] is determined as follows:

[latex]k_{He} :=1.4[/latex] [latex]R_{He}:=2077\frac{Nm}{kgK} [/latex] [latex]C_{He}: =\sqrt{k_{He}R_{He}T} =1.047 imes 10^{3} \frac{m}{s} [/latex]
[latex]\frac{C_{He}}{C_{air}} =2.929[/latex]

Thus, the sonic velocity of helium is about three times the sonic velocity of air, both at a temperature of [latex]45^{\circ } C[/latex].
(c) The sonic velocity for hydrogen at [latex]45^{\circ } C[/latex]is determined as follows:

[latex]k_{H2} :=1.4[/latex] [latex]R_{H2}:=4120\frac{Nm}{kgK} [/latex] [latex]C_{H2}: =\sqrt{k_{H2}R_{H2}T} =1.355 imes 10^{3} \frac{m}{s} [/latex]
[latex]\frac{C_{H2}}{C_{air}} =3.789[/latex] [latex]\frac{C_{H2}}{C_{He}} =1.293[/latex]

Thus, the sonic velocity of hydrogen is almost four times the sonic velocity of air, and about 1.3 times the sonic velocity of helium, all at a temperature of [latex]45^{\circ } C[/latex].

Table A.5
Physical Properties for Some Common Gases at Standard Sea-Level Atmospheric Pressure at Room Temperature [latex](68^{\circ } or 20^{\circ }C )[/latex]
Gas at [latex]68^{\circ }[/latex]	Chemical Formula	Molar Mass (m) slug=slug- mol	Density (ρ) [latex]slug/ft^{3} [/latex]	Absolute (Dynamic) Viscosity (μ) [latex]10^{-6} Ib-sec/ft^{2} [/latex]	Gas Constant (R) [latex]ft-Ib/(slug-^{\circ }R )=ft^{2}/(sec^{2} -^{\circ }R )[/latex]	Specific Heat		Specific Heat Ratio, [latex]K=C_{ ho }/C_{\upsilon }[/latex]
						[latex]C_{ ho }[/latex]	[latex]C_{\upsilon } [/latex]
						[latex]ft-Ib/(slug-^{\circ }R )=ft^{2}/(sec^{2} -^{\circ }R )[/latex]
Air		28.960	0.002310	0.376	1715	6000	4285	1.40
Carbon dioxide	[latex]CO_{2} [/latex]	44.010	0.003540	0.310	1123	5132	4009	1.28
Carbon monoxide	CO	28.010	0.002260	0.380	1778	6218	4440	1.40
Helium	He	4.003	0.000323	0.411	12.420	31.230	31.230	1.66
Hydrogen	[latex]H_{2}[/latex]	2.016	0.000162	0.189	24.680	86.390	86.390	1.40
Methane	[latex]CH_{2}[/latex]	16.040	0.001290	0.280	3100	13.400	13.400	1.30
Nitrogen	[latex]N_{2}[/latex]	28.020	0.002260	0.368	1773	6210	4437	1.40
Oxygen	[latex]O_{2}[/latex]	32.000	0.002580	0.418	1554	5437	3883	1.40
Water vapor	[latex]H_{2}O[/latex]	18.020	0.001450	0.212	2760	11.110	8350	1.33
at [latex]20^{\circ } C[/latex]		kg/kg-mol	[latex]kg/m^{3} [/latex]	[latex]10^{-6} N-sec/m^{2} [/latex]	[latex]N-m/(kg-^{\circ}K )=m^{2} /(sec^{2}-^{\circ}K )[/latex]	[latex]N-m/(kg-^{\circ}K )=m^{2} /(sec^{2}-^{\circ}K )[/latex]
Air		28.960	1.2050	18.0	287	1003	716	1.40
Carbon dioxide	[latex]CO_{2} [/latex]	44.010	1.8400	14.8	188	858	670	1.28
Carbon monoxide	CO	28.010	1.1600	18.2	297	1040	743	1.40
Helium	He	4.003	0.1660	19.7	2077	5220	3143	1.66
Hydrogen	[latex]H_{2}[/latex]	2.016	0.0839	9.0	4120	14.450	10.330	1.40
Methane	[latex]CH_{2}[/latex]	16.040	0.6680	13.4	520	2250	1730	1.30
Nitrogen	[latex]N_{2}[/latex]	28.020	1.1600	17.6	297	1040	743	1.40
Oxygen	[latex]O_{2}[/latex]	32.000	1.3300	20.0	260	909	649	1.40
Water vapor	[latex]H_{2}O[/latex]	18.020	0.7470	10.1	462	1862	1400	1.33

Question 1.45: It is of interest to compare the sonic velocity for air, car...

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