Thermodynamics: An Engineering Approach
Thermodynamics: An Engineering Approach
8th Edition
ISBN: 9780073398174
Author: Yunus A. Cengel Dr., Michael A. Boles
Publisher: McGraw-Hill Education
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Chapter 17.7, Problem 128RP
To determine

To derive an expression for the speed of sound based on van der Walls’s equation of state P=RTvbav2 and using this relation determine the speed of sound in carbon dioxide at 80°C and 320kPa, and compare the result to that obtained by assuming ideal-gas behavior.

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Answer to Problem 128RP

The speed of sound in carbon dioxide by using the relation obtained from the van der Walls equation is 289.43m/s.

The speed of sound in carbon dioxide by assuming as ideal gas is 292.04m/s.

Explanation of Solution

Write the given equation of state.

P=RTvbav2 (I)

Here, the pressure is P, the gas constant is R, the temperature is T, the specific volume is v, and the van der Walls constants are a,b.

Partially differentiate the Equation (I) with respect to specific volume v by the keeping the temperature T as constant.

(Pv)T=v(RTvbav2)T=v(RTvb)Tv(av2)T=RTv(1vb)av(1v2)=RT[1(vb)2]a(2v3)

=RT(vb)2+2av3 (II)

Write the relation of speed of the sound.

c2=k(Pρ)T (III)

Here, the specific heat ratio is k, the pressure is P, the density is ρ, and the symbol indicates the partial derivative of variables.

Write the relation between density and specific volume.

ρ=1v

Partially differentiate the ρ with respect to specific volume v.

ρv=v(1v)ρv=1v2ρ=v(1v2)ρ=vv2

Substitute vv2 for ρ in Equation (III).

c2=k(Pvv2)T=kv2(Pv)T (IV)

Substitute RT(vb)2+2av3 for (Pv)T in Equation (IV).

c2=kv2[RT(vb)2+2av3]=(kv2)RT(vb)2+(kv2)2av3=v2kRT(vb)22akv (V)

Write the formula for velocity of sound at the given conditions of CO2 by assuming it as ideal gas.

c=kRT (VI)

Refer Table A-1, “Molar mass, gas constant, and critical-point properties”.

The molar mass (M) of carbon dioxide is 44.01kg/kmol44kg/kmol.

The gas constant (R) of carbon dioxide is 0.1889kJ/kgK or 0.1889kPam3/kgK.

Refer Table A-2, “Ideal-gas specific heats of various common gases”.

The specific heat ratio (k) of carbon dioxide is 1.289.

Conclusion:

Express the van der Walls constant a and b per unit mass as follows.

a=364.3kPam6/kmol2×1(44kg/kmol)2=364.3kPam6/kmol21936kg2/kmol2=0.1882kPam6/kg2

b=0.0427m3/kmol×144kg/kmol=9.70×104m3/kg

Substitute 320kPa for P, 0.1889kPam3/kgK for R, 80°C for T, 9.70×104m3/kg for b, and 0.1882kPam6/kg2 for a in Equation (I).

320kPa=(0.1889kPam3/kgK)(80°C)v9.70×104m3/kg0.1882kPam6/kg2v2320kPa=(0.1889kPam3/kgK)(80+273)Kv0.00097m3/kg0.1882kPam6/kg2v2320kPa=66.6817kPam3/kgv0.00097m3/kg0.1882kPam6/kg2v2 (VII)

By using Equation solver or online calculator solve the Equation (VII) and the value of v is obtained as follows.

v=0.2065m3/kg

Substitute 0.2065m3/kg for v, 1.279 for k, 0.1889kPam3/kgK for R, 80°C for T, 9.70×104m3/kg for b, and 0.1882kPam6/kg2 for a in Equation (V).

c2=[(0.2065m3/kg)2(1.279)(0.1889kPam3/kgK)(80°C)(0.2065m3/kg9.70×104m3/kg)22(0.1882kPam6/kg2)(1.279)0.2065m3/kg]c2=[(0.05454m6/kg2)(0.1889kPam3/kgK)(80+273)K0.04224m6/kg22.3313kPam3/kg]c2=[(0.05454m6/kg2)(66.6817kPam3/kg)0.04224m6/kg22.3313kPam3/kg]c2=86.0989kPam3/kg2.3313kPam3/kg

c2=83.7676kPam3/kg×1000m2/s21kPam3/kgc=83767.6m2/s2c=289.4263m/s289.43m/s

Thus, the speed of sound in carbon dioxide by using the relation obtained from the van der Walls equation is 289.43m/s.

When the carbon dioxide is assumed as ideal gas, the velocity of sound is determined as follows.

Substitute 0.1889kJ/kgK for R, 1.279 for k, and 80°C for T in Equation (VI).

c=1.279×0.1889kJ/kgK×80°C=1.279×0.1889kJ/kgK×(80+273)K=85.2859kJ/kg×1000m2/s21kJ/kg=292.0375m/s

292.04m/s

Thus, the speed of sound in carbon dioxide by assuming as ideal gas is 292.04m/s.

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Chapter 17 Solutions

Thermodynamics: An Engineering Approach

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