A centrifugal impeller of 40-cm diameter is used to pump hydrogen at 15 C and 1-atm pressure. Estimate the maximum allowable impeller rotational speed to avoid compressibility effects at the blade tips.

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Accepted Answer

• Assumptions: The maximum fluid velocity is approximately equal to the impeller tip speed:

[latex]V_{max} \approx \Omega r_{max} where r_{max} = D/2 = 0.20 m[/latex]

• Approach: Find the speed of sound of hydrogen and make sure that [latex]V_{max}[/latex] is much less.

• Property values: From Table A.4 for hydrogen, R = 4124 [latex]m^2/(s^2 - K)[/latex] and k = 1.41. From Eq. (1.39) at 15°C = 288K, compute the speed of sound:

[latex]\frac{dx}{u} = \frac{dy}{v} = \frac{dz}{w} = \frac{dr}{V}[/latex] (1.39)

[latex]a_{H_2} = \sqrt{kRT} = \sqrt{1.41[4124 m^2/(s^2 - K)](288 K)} \approx 1294 m/s[/latex]

Table A.4 Properties of Common Gases at 1 atm and 20°C (68°F)
Gas	Molecular weight	[latex]R, m^2/(s^2 \cdot K)[/latex]	[latex] ho g, N/m^3[/latex]	[latex]\mu, N \cdot s/m^2[/latex]	Specific-heat ratio	Power-law exponent [latex]n^*[/latex]
[latex]H_2[/latex]	2.016	4124	0.822	9.05 E-6	1.41	0.68
He	4.003	2077	1.63	1.97 E-5	1.66	0.67
[latex]H_2O[/latex]	18.02	461	7.35	1.02 E-5	1.33	1.15
Ar	39.944	208	16.3	2.24 E-5	1.67	0.72
Dry air	28.96	287	11.8	1.8 E-5	1.40	0.67
[latex]CO_2[/latex]	44.01	189	17.9	1.48 E-5	1.30	0.79
CO	28.01	297	11.4	1.82 E-5	1.40	0.71
[latex]N_2[/latex]	28.02	297	11.4	1.76 E-5	1.40	0.67
[latex]O_2[/latex]	32.00	260	13.1	2.00 E-5	1.40	0.69
NO	30.01	277	12.1	1.90 E-5	1.40	0.78
[latex]N_2O[/latex]	44.02	189	17.9	1.45 E-5	1.31	0.89
[latex]Cl_2[/latex]	70.91	117	28.9	1.03 E-5	1.34	1.00
[latex]CH_4[/latex]	16.04	518	6.54	1.34 E-5	1.32	0.87

[latex]^*[/latex]The power-law curve fit, Eq. (1.27), [latex]\mu / \mu_{293K} \approx (T/293)^n[/latex], fits these gases to within ±4 percent in the range 250 ≤ T ≤ 1000 K. The temperature must be in kelvins.

[latex]\frac{\mu}{\mu_0} \approx \begin{cases} \left(\frac{T}{T_0} ight)^n & power law \ \frac{(T/T_0)^{3/2} (T_0 + S)}{T+S} & Sutherland law \end{cases}[/latex] (1.27)

• Final solution step: Use our rule of thumb, Eq. (4.18), to estimate the maximum impeller speed:

Ma ≤ 0.3 (4.18)

[latex]V = \Omega r_{max} \leq 0.3a or \Omega(0.2 m) \leq 0.3(1294 m/s)[/latex]

Solve for [latex]\Omega \leq 1940\frac{rad}{s} \approx 18,500\frac{rev}{min}[/latex] Ans.

• Comments: This is a high rate because the speed of sound of hydrogen, a light gas, is nearly four times greater than that of air. An impeller moving at this speed in air would create tip shock waves.

Question 4.4: A centrifugal impeller of 40-cm diameter is used to pump hyd...

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