Mechanical Engineering Principles

262 MECHANICAL ENGINEERING PRINCIPLES

Problem 10. Some air at a temperature of

40 °C and pressure 4 bar occupies a volume

of 0.05 m^3. Determine the mass of the air

assuming the characteristic gas constant for

airtobe287J/(kgK).

From above,pV = mRT, wherep = 4bar=
4 × 105 Pa (since 1 bar= 105 Pa — see Chapter 21),
V = 0 .05 m^3 ,T =( 40 + 273 )K=313 K, and
R=287 J/(kg K).

Hence ( 4 × 105 )( 0. 05 )=m( 287 )( 313 )

from which,mass of air,

m=

( 4 × 105 )( 0. 05 )

( 287 )( 313 )

= 0 .223 kgor223 g

Problem 11. A cylinder of helium has a

volume of 600 cm^3. The cylinder contains

200 g of helium at a temperature of 25°C.

Determine the pressure of the helium if the

characteristic gas constant for helium is

2080 J/(kg K).

From the characteristic gas equation,pV=mRT,
whereV=600 cm^3 = 600 × 10 −^6 m^3 ,
m=200 g= 0 .2 kg,T=( 25 + 273 )K=298 K
andR=2080 J/(kg K).

Hence (p)( 600 × 10 −^6 )=( 0. 2 )( 2080 )( 298 )

from which,pressure,p =

( 0. 2 )( 2080 )( 298 )

( 600 × 10 −^6 )

=206 613 333 Pa

= 206 .6MPa

Problem 12. A spherical vessel has a

diameter of 1.2 m and contains oxygen at a

pressure of 2 bar and a temperature of

− 20 °C. Determine the mass of oxygen in the

vessel. Take the characteristic gas constant

for oxygen to be 0.260 kJ/(kg K).

From the characteristic gas equation,pV=mRT

where V=volume of spherical vessel

=

4

3

πr^3 =

4

3

π

(

1. 2

2

) 3

= 0 .905 m^3 ,

p=2bar= 2 × 105 Pa,

T=(− 20 + 273 )K=253 K

and R= 0 .260 kJ/(kg K)=260 J/(kg K).

Hence ( 2 × 105 )( 0. 905 )=m( 260 )( 253 )

from which,mass of oxygen,

m=

( 2 × 105 )( 0. 905 )

( 260 )( 253 )

= 2 .75 kg

Problem 13. Determine the characteristic

gas constant of a gas which has a specific

volume of 0.5 m^3 /kg at a temperature of

20 °C and pressure 150 kPa.

From the characteristic gas equation,pV=mRT

from which, R=

pV

mT

where p= 150 × 103 Pa,

T=( 20 + 273 )K

=293 K

and specific volume, V/m= 0 .5m^3 /kg.

Hence thecharacteristic gas constant,

R=

(p

T

)(V

m

)

=

(

150 × 103

293

)

( 0. 5 )

=256 J/(kgK)

Now try the following exercise

Exercise 123 Further problems on the

characteristic gas equation

1. A gas occupies a volume of 1.20 m^3
   when at a pressure of 120 kPa and
   a temperature of 90°C. Determine the
   volume of the gas at 20°C if the pressure
   is increased to 320 kPa. [0.363 m^3 ]


2. A given mass of air occupies a volume
   of 0.5 m^3 at a pressure of 500 kPa and
   a temperature of 20°C. Find the volume
   of the air at STP. [2.30 m^3 ]