fluid ans t_2

Tutorial (2) Dr.waleed, Mob. 0100 4444 149 Page (1)

Tutorial (2)

(1) A spherical balloon with a diameter of 6 m is filled with helium at 20 oC

and 200 𝑘𝑃𝑎. Determine the mole number and the mass of the helium in

the balloon

Solution

P Pressure Pa

V Volume m3

N Number of moles kmol

Ru Universal gas constant J/kmol.K

T Temperature K

Rg Gas constant J/kg.K

M Molar mass Kg/kmol

m Mass kg

(

)

For helium (M=4 kg/kmol)

m = 9.28 x 4 = 37.1 kg

Tutorial (2) Dr.waleed, Mob. 0100 4444 149 Page (2)

(2) The pressure in an automobile tire depends on the temperature of the air

in the tire. When the air temperature is 25 oC, the pressure gage reads 210

𝑘𝑃𝑎. If the volume of the tire is 0.025 𝑚3, Determine the pressure in the tire

when the air temperature in the tire rises to 50 oC. Also, determine the

amount of air that must be bled off to restore pressure to its original value

at this temperature. Assume the atmospheric pressure to be 100 𝑘𝑃𝑎.

Solution

The absolute pressure in the tire is

T1 = 25 +273 = 298 K

T2 = 50 +273 = 323 K

Assume the tire volume is constant (also the mass)

Tutorial (2) Dr.waleed, Mob. 0100 4444 149 Page (3)

(3) It is observed that the density of an ideal gas is increased by 10% when

compressed isothermally from 10 𝑎𝑡𝑚 to 11 𝑎𝑡𝑚. Determine the percent

increase in density of the gas if it is compressed isothermally from 100 𝑎𝑡𝑚

to 101 𝑎𝑡𝑚.

Solution

For ideal gas

(

) (

)

(

)

(

)

⁄

At P = 10 atm

At P = 100 atm

Tutorial (2) Dr.waleed, Mob. 0100 4444 149 Page (4)

(4) What pressure must be applied to water (𝐸𝑣 = 2.2 × 109 𝑝𝑎) to reduce its

volume by 1%?

Solution

⁄

⁄

⁄

Tutorial (2) Dr.waleed, Mob. 0100 4444 149 Page (5)

(5) Eight kilometers below the surface of the ocean, the pressure is 81.7𝑀𝑃𝑎.

Determine the specific weight of the sea water at this depth if the specific

weight at the surface is 10.06 𝐾𝑁/𝑚3 and the average bulk’s modulus of

elasticity is 2.34 𝐺𝑃𝑎. Assume that 𝑔 does not change significantly.

Solution

𝑘𝑁 𝑚 ⁄

Depth = 8 km

𝑃 𝑀𝑃𝑎

𝑣

𝑔

𝑣

𝑚 𝑘𝑔⁄

𝑣

𝑣

𝑃

𝐾

𝑣 𝑣 𝑣

𝑣 𝑣

𝑣

𝑃 𝑃

𝐾

𝑣 𝑣 𝑃 𝑃

𝐾

𝑣

𝑣 𝑣 𝑣

𝑣

𝑔

𝑣

𝑁 𝑚 ⁄

Tutorial (2) Dr.waleed, Mob. 0100 4444 149 Page (6)

(6) Calculate the velocity of sound in water at 20 oC ,(𝐾 = 2.2 10

9 𝑝𝑎).

Solution

√𝐾

√

𝑚 ⁄

(7) Calculate the velocity of sound in air at 0

oC, 𝑃abs= 101.3 𝑘𝑝𝑎.

Solution

√𝐾 √ 𝑚 ⁄

Tutorial (2) Dr.waleed, Mob. 0100 4444 149 Page (7)

(8) A thin plate moves between two parallel, horizontal flat surfaces at a

constant velocity of 5 𝑚/ . The two surfaces are spaced 4 𝑐𝑚 apart, the plate

is 1𝑐𝑚 apart from the top surface and the medium between them is filled

with oil whose viscosity is 0.9 𝑁. /𝑚2. the part of the plate immersed in oil

at any given time is 2 𝑚 long and 0.5 𝑚 wide. If the plate moves between the

surfaces. Determine the force required to maintain this motion if

(a) Both upper and lower surfaces are stationary.

(b) The upper surface moves at velocity = 2 𝑚/ in the same direction as Plat

where the lower surface remains stationary.

Solution

The magnitudes of shear forces acting on the upper and lower surfaces of the plate are

|

|

𝑁

𝑁

Tutorial (2) Dr.waleed, Mob. 0100 4444 149 Page (8)

𝑁

𝑁

Tutorial (2) Dr.waleed, Mob. 0100 4444 149 Page (9)

(9) Fluid flow through a circular pipe is one-dimensional, and the velocity

profile for Laminar flow is given by

, where is the

radius of the pipe, 𝑟 is the radial distance from the center of the pipe, and

max is the maximum flow velocity, which occurs at the center.

Given that = 0.08 𝑚, max= 3 𝑚/ and μ = 0.001𝑘𝑔/𝑚.

Determine:

(a) The shearing stress acting on the pipe wall.

(b) The shearing stress acting on a plane parallel to the pipe walls and passing

through the centerline (mid-plane).

(c) Given the pipe length= 30 𝑚, calculate the drag force applied by the fluid

on pipe.

Solution

𝑟 ( 𝑟

)

𝑟|

Tutorial (2) Dr.waleed, Mob. 0100 4444 149 Page (10)

𝑟(

𝑟

)

𝑟

𝑁 𝑚 ⁄

Along the mid-plane r = 0

so

(

)

𝑁

Tutorial (2) Dr.waleed, Mob. 0100 4444 149 Page (11)

(10) 10. A 25 mm-diameter shaft is pulled through a cylindrical bearing as

shown in the figure. The lubricant that fills the 0.3 𝑚𝑚 gap between the shaft

and bearing is an oil having a kinematic viscosity of 8.0 × 10−4

and a specific

gravity of 0.91. Determine the force 𝑃 required to pull the shaft at a velocity of

3 𝑚/ . Assume the velocity distribution in the gap is linear.

Solution

∑

𝑃

𝑣 𝑐 𝑡 𝑎 𝑡

𝑔𝑎 𝑡

𝑣

𝐺

𝑃

𝑣

𝑃

𝑃 𝑁

Tutorial (2) Dr.waleed, Mob. 0100 4444 149 Page (12)

(11) The viscosity of a fluid is to be measured by a viscometer constructed of

two 40 𝑐𝑚 long concentric cylinders as shown in the figure. The outer

diameter of the inner cylinder is 12 𝑐𝑚, and the gap between the two

cylinders is 10 𝑚𝑚. The inner cylinder is rotated at 300 𝑟𝑝𝑚, and the torque

is measured to be 1.8 N.m, Determine the viscosity of the fluid.

Solution

𝑣 𝑐 𝑡 𝑎 𝑡

𝑔𝑎 𝑡

𝑣

𝑣

(

)

(

)

(

)

(

⁄ ) 𝑁 𝑚 ⁄