Tutorial 2 Fluid pressure

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Tutorial 2
Fluid pressure
1. A cylinder contains a fluid at a gauge pressure of 360 KN/m2. Express this pressure in terms of a head
of (a) water, and (b) mercury of sp gr = 13.6
What would be the absolute pressure in the cylinder if atmospheric pressure is 760mm Hg.
Solution:
Pressure (P) = 360 KN/m2 = 360x103 N/m2
Head (h) = ?
where ρ = Density of fluid
a) Head in terms of water (ρ = 1000 kg/m3)
= 36.7m
b) Head in terms of mercury
ρ = sp gr x density of water = 13.6x1000 = 13600 kg/m3
= 2.7m
Atmospheric pressure (h) = 760mmhg = 0.76m hg
)
Atmospheric pressure (
Absolute pressure (Pabs) = ?
Pabs = Pgauge + Patm = 360+101.3 = 461.3KN/m2
= 101396N/m2 = 101.3KN/m2
2. What would the pressure in kN/m2 be if the equivalent head is measured as 400mm of (a) mercury (sp
gr 13.6) (b) water (c) oil specific weight 7.9 kN/m3 (d) a liquid of density 520 kg/m3?
Solution:
Head (h) = 400mm = 0.4m
Pressure (P) =?
where ρ = Density of fluid
a) In terms of mercury, ρ = sp gr x density of water = 13.6x1000 = 13600 kg/m3
= 13600x9.81x0.4 = 53366 N/m2 = 53.366 KN/m2
b) In terms of water, ρ = 1000 kg/m3
= 1000x9.81x0.4 = 3924 N/m2 = 3.924 KN/m2
c) In terms of oil of sp. wt. ( ) = 7.9 kN/m3
= 7.9x.4 = 3.16 KN/m2
d) In terms of liquid with ρ = 520 kg/m3
= 520x9.81x0.4 = 2040 N/m2 = 2.04 KN/m2
3. A manometer connected to a pipe indicates a negative gauge pressure of 50mm of mercury. What is
the absolute pressure in the pipe in N/m2 if the atmospheric pressure is 1 bar?
Solution:
Atmospheric pressure (Patm) = 1 bar = 1x105 N/m2
Head (h) = -50mmhg = -0.05m hg
Absolute pressure (pabs) = ?
ρ of mercury = sp gr x density of water = 13.6x1000 = 13600 kg/m3
Gauge pressure (Pgauge)
= -13600x9.81x0.05 = -6671 N/m2
Pabs = Pgauge + Patm = -6671+1x105 = 93329 N/m2 = 93.3KN/m2
4. An open tank contains 5.7m of water covered with 2.6m of kerosene (sp wt = 8KN/m 3). Find the
pressure at the interface and at the bottom of the tank.
Solution:
Height of kerosene (h) = 2.6m
Height of water (h1) = 5.7m
Sp wt of kerosene ( ) = 8KN/m3
Pressure at interface (Pint) = ?
Pressure at bottom (Pbottom) = ?
= 8x2.6 = 20.8KN/m2
= 20.8+9.81x5.7 = 76.7 KN/m2
5. The closed tank in the fig. is at 200 C. If the pressure at point A is 96 Kpa absolute, what is the absolute
pressure at point B? What percent error results from neglecting the specific weight of air? (Take sp wt of
air = 0.0118 KN/m3)
A
5m
B
Air
Air
3m
D
C
5m
3m
Solution:
Water
Sp wt of air (
) = 0.0118 KN/m3
Sp wt of water (
) = 9.81 KN/m3
Starting from A,
PA + PAC -PCD-PDB = PB
PB = 96 + 0.0118x5 – 9.81x2 – 0.0118x3 = 76.404 Kpa
Neglecting air,
PB = 96 – 9.81x2 = 76.38 Kpa
Error = (76.404-76.38)/76.404 = 0.00031 = 0.031%
6. In the fig., the pressure at point A is 2900 N/m2. Determine the pressures at points B, C and D. (Take
density of air = 1.2 kg/m3)
Air
Air
1m
C
0.8m
B
A
Air
1.5m
2m
X
Water
0.7m
D
Solution:
Density of water (ρ) = 1000kg/m3
PA = 2900 N/m2
Density of air (
) = 1.2 kg/m3
PB = ?, PC =?, PD = ?
Starting from A,
= 2900 – 1000x9.81x0.2 = 938 N/m2
= 2900 + 1000x9.81x1.3 -1.2x9.81x2.3 = 15626 N/m2
= 2900 + 1000x9.81x2 = 22520 N/m2
7. In the fig., the absolute pressure at the bottom of the tank is 233.5 Kpa. Compute the sp gr of olive oil.
Take atmospheric pressure = 101.3 Kpa.
Oil (sp gr = 0.9)
1.5m
Water
2.5m
Olive oil
2.9m
Mercury
0.4m
Solution:
Absolute pressure at bottom (Pabs) =233.5 Kpa
Atmospheric pressure (Patm) = 101.3 Kpa
Sp wt of water ( ) = 9.81 KN/m3
Sp wt of oil ( ) = 0.9x9.81 KN/m3 = 8.829 KN/m3
Sp wt of mercury ( ) = 13.6x9.81 KN/m3 = 133.416 KN/m3
Sp gr of olive oil (S) = ?
Pabs = Patm + Pgauge
Pabs = Patm + Poil + Pwater + Polive oil + Pmercury
KN/m3
= 1.44
8. The tube shown in the fig. is filled with oil of sp gr 0.82. Determine the pressure heads at A and B in
meters of water.
oil
A
2.1m
Air
B
0.5m
X
oil
oil
Solution:
sp gr of oil = 0.82
Sp wt of oil ( ) = 0.82x9810 = 8044.2 N/m3
Head in terms of water at a and B (hA and hB) = ?
Take atmospheric pressure to be 0 for gauge pressure.
= - 8044.2x2.6 = -20914.9 Pa
= -2.132 m
= -20914.9 + 8044.2x2.1= -4022.1 Pa
= -0.41 m
9. Calculate the pressures at A, B, C and D in the fig.
Air
Air
A
C
0.4m
X
0.4m
Oil
Sp gr 0.9
B
0.5m
1m
Water
Solution:
Sp wt of water ( ) = 9810 N/m3
sp gr of oil = 0.9
Sp wt of oil ( ) = 0.9x9810 = 8829 N/m3
Take atmospheric pressure to be 0 for gauge pressure.
= - 9810x0.8 = 7848 Pa
= 9810x0.5 = 4905 Pa
Neglecting air, PC =PB = 4905 Pa
= 4905 + 8829x1.9 = 21680 Pa
D
10. The tank in the fig. contains oil of sp gr 0.75. Determine the reading of gauge A in N/m2.
Air
Oil
0.2m
X
3m
Hg
A
Solution:
sp gr of oil = 0.75
Sp wt of oil ( ) = 0.75x9810 = 7357.5 N/m3
Sp wt of mercury ( ) = 13600x9.81 N/m3 = 133416 N/m3
Take atmospheric pressure to be 0 for gauge pressure.
Starting from X and neglecting air,
PA = -133416x0.2+7357.5x3 = -4610.7 N/m2
11. In the left hand of the fig., the air pressure is -225mm of Hg. Determine the elevation of the gauge
liquid in the right hand column at A.
20.4KN/m
2
Air
110.2m
Air
Oil
107.8m
Sp gr = 0.8
Water
106m
h
A’
A
Liquid (sp gr. = 1.6)
Solution:
Air pressure at the left hand tank = -225mm Hg = -0.225m Hg
Sp wt of water ( ) = 9810 N/m3
Sp wt of oil ( ) = 0.8x9810 = 7848 N/m3
Sp wt of mercury ( ) = 13600x9.81 KN/m3 = 133416 KN/m3
Sp wt of liquid (
) = 1.6x9810 = 15696 N/m3
= -0.225x133416 = -30018.6 N/m2
PA’= PA
-30018.6 + 7848x(110.2-106) + 15696 xh = 20400 + 9810 x(107.8-106+h)
h = 5.96m
Elevation at A = 106-5.96 = 100.04m
12. Compartments B and C in the fig. are closed and filled with air. The barometer reads 99.98 Kpa.
When gages A and D read as indicated, what should be the value of x for gage E? (Hg in each tube)
206.8Kpa
A
C
254mm
B
x
D
E
Solution:
PA = 206.8 KPa = 206800 Pa
Sp wt of mercury ( ) = 13.6x9810 N/m3 = 133416 N/m3
Starting from A and neglecting air,
206800 – 133416 X + 133416x0.254 = 0
X = 1.8m
13. In the fig., the areas of the plunger A and cylinder B are 38.7 cm2 and 387 cm2, respectively, and the
weight of B is 4500 N. The vessel and the connecting passages are filled with oil of specific gravity 0.75.
What force F is required for equilibrium, neglecting the weight of A?
F
A
4.8m
XL
Oil (sp gr = 0.75)
XR
B
Solution:
Area of A (AA) =38.7 cm2, Area of B (AB) =387 cm2
Weight of B (WB) = 4500N
Sp wt of water ( ) = 9810 N/m3
Sp wt of oil ( ) = 0.75x9810 N/m3 = 7357.5 N/m3
Pressure at XL = Pressure at XR
PA + 7357.5x4.8 = 4500/387x10-4
PA = 80963 N/m2
F = PA AA = 80963 x 38.7x10-4 = 313N
14. For the open tank, with piezometers attached on the side, containing two different immiscible
liquids as shown in the fig., find (a) the elevation of liquid surface in piezometer A, (b) the elevation of
liquid surface in piezometer B, and (c) the total pressure at the bottom of the tank.
A
Elv 2.1m
B
Liquid A
(sp gr = 0.72)
Elv 0.3m
Liquid B
(sp gr = 2.36)
Elv 0m
Solution:
Sp wt of liquid A ( ) = 0.72x9810 N/m3 = 7063.2 N/m3
Sp wt of liquid B ( ) = 2.36x9810 N/m3 = 23151.6 N/m3
a) Elevation of liquid surface in piezometer A = elevation of liquid A in the tank = 2.1 m
b) Pressure due to A at the interface (PA) =
= 7063.2 x (2.1-0.3) = 12713.8 N/m2
Equivalent head for liquid B due to PA is
= 12713.8/23151.6 = 0.55m
Elevation of liquid surface in piezometer B = Elv of liquid at B + hA = 0.3 + 0.55 = 0.85m
c) Total pressure at the bottom =
= 7063.2 x (2.1-0.3) + 23151.6x0.3 = 19659 N/m2
15. In the fig., gage A reads 290Kpa abs. What is the height of water h? What does gage B read?
Air : 175Kpa abs
Water
Hg
h
B
A
Solution:
Sp wt of water ( ) = 9.81 KN/m3
Sp wt of mercury ( ) = 13.6x9.81 = 133.416 KN/m3
PA =290Kpa abs
h=?
PB =?
290 = 175 + 9.81h+133.416x0.7
h = 2.2m
(
)
PB = 175 + 9.81 (2.2+0.7) = 203.4 KN/m2
70cm
16. A manometer is attached to a tank containing three different fluids as shown in fig. What will be the
difference in elevation of the mercury column in the manometer (i.e. y)?
Elv 6m
Air pressure = 35Kpa
Elv 5m
Oil (sp gr = 0.82)
Elv 2m
Water
Elv 0m
1m
A
B
Hg
Solution:
Sp wt of water ( ) = 9.81 KN/m3
Sp wt of mercury ( ) = 13.6x9.81 = 133.416 KN/m3
Sp wt of oil ( ) = 0.82x9.81 = 8.0442 KN/m3
PA = PB
35+ 8.0442x3+9.81x3= 133.416y
y = 0.66m
y
17. Determine the pressure difference between two points A and B in the fig.
Kerosene
Sp. gr. = 0.82
Air
3
Sp. wt. = 11.77 N/m
Benzene
Sp. gr. = 0.88
B
10cm
A
41cm
21cm
15cm
9cm
Water
Mercury
Sp. gr. = 13.6
Solution:
Sp wt of water ( ) = 9.81 KN/m3
Sp wt of mercury ( ) = 13.6x9.81 = 133.416 KN/m3
Sp wt of benzene ( ) = 0.88x9.81 = 8.6328 KN/m3
Sp wt of kerosene ( ) = 0.82x9.81 = 8.0442 KN/m3
Sp wt of air (
) = 0.01177 KN/m3
Starting from A,
(
PA – PB = 10.22 Kpa
)
(
)
18. Fig. below shows a conical vessel having its outlet at A to which a U-tube manometer is connected.
The reading of the manometer given in the fig. shows when the vessel is empty. Find the reading of the
manometer when the vessel is completely filled with water.
2m
3m
A
Initial
h
y
Water
Final
20 cm
h
Hg
Solution:
Sp wt of water ( ) = 9.81 KN/m3
Sp wt of mercury ( ) = 13.6x9.81 = 133.416 KN/m3
When the vessel is empty, equating pressure at initial level
9.81 y = 133.416x0.2
y = 2.72m
When the vessel is filled with water, let us say the mercury moves by height h
Equating pressure at final level
(
)
(
)
(
)
(
)
h = 0.1145m
Deflection of mercury = 2h+0.2 =2x0.1145+0.2 = 0.429m
19. Compute the absolute pressure at point A in the fig.
Water
Oil
Sp. gr. = 0.88
15cm
A
23cm
17.5cm
Water
Mercury
Sp. gr. = 13.6
Solution:
Sp wt of water ( ) = 9.81 KN/m3
Sp wt of mercury ( ) = 13.6x9.81 = 133.416 KN/m3
Sp wt of oil ( ) = 0.88x9.81 = 8.6328 KN/m3
Starting from A,
PA = 27.7 Kpa
Absolute pressure at A = Patm +27.7 = 101.3+27.7 = 129Kpa
20. The fig. shows a 1cm diameter U-tube containing mercury. If now 20cc of water is poured into the
right leg, find the levels of the free liquid surfaces in the two tubes.
Water
h
y
y
E
18cm
Mercury
F
Solution:
With the addition of 20ml of water,
Drop in mercury level in right leg = y
Rise in mercury level in left leg = y
Depth of water column = h
(
h= Vol of water added/Cross sectional area =
) = 0.254m
PE = PF
13.6x9810x2y = 9810x0.254
y = 0.0093m = 0.93cm
Height of free mercury level in the left leg = 18+0.93 = 18.93cm
Height of free water level in the right leg = 18-0.93+25.4 = 42.47cm
21. The diameters of the limbs A and B of a U-tube shown in fig. are 5mm and 20mm respectively. The
limb A contains a liquid of sp. gr. 0.9 while the limb B contains a liquid of sp.gr. 1.3. The fig. shows the
position of the liquids in the two limbs. Find what pressure should be applied on the surface of the
heavier liquid in limb B so that the rise in level in the limb A is 10mm.
A
B
10mm
P
liquid of
sp.gr. = 0.9
ha
h
hb
E1
E
F
liquid of
sp.gr. = 1.3
No external pressure
Solution:
Sp gr of liquid in A = 0.9
Sp. gr. of liquid in B = 1.3
Diameter of limb A = 5mm
Diameter of limb B = 20mm
10mm
External pressure P
F1
a. When no external pressure is applied
PE = PF
(a)
b. When external force p is applied on the surface of liquid B
Rise in limb A = 10mm
Fall in limb B = h
Level EF shifts to new position is E1F1.
Volume of liquid transferred to left limb = Volume of liquid fell in right limb
c/s area of left limbx10mm = c/s of right limbxh
(
(
) )
(
h = 0.000625m
PE1 = PF1
(
(
) )
)
(
)
(
(
)
)
(b)
From a and b
(
P = 135.5 N/m
)
2
22. Find the pressure difference between containers A and B shown in the figure.
A
20cm
Air
0
45
Water
B
50cm
30cm
E
Mercury
F
15cm
Solution:
Pressure at E = pressure at F
= -82633 N/m2
23. An 8cm diameter piston compresses manometer oil into an inclined 7mm diameter tube, as shown
in figure below. When a weight W is added to the top of the piston, the oil rises an additional distance of
10cm up the tube. How large is the weight, in N?
W
10cm
D = 8cm
h
Piston
L1
A
Initial level
X
X
dh
Y
Y
Final level
Oil, sp gr
= 0.827
D = 7mm
0
15
Solution:
Diameter of piston (D) = 8cm = 0.08m
Diameter of tube (d) = 7mm = 0.007m
h = 0.10Sin15 = 0.0258m
When the manometer is not connected to the container, the mercury in the reservoir is at original level
and at level A in the tube.
Equating pressure at XX
(a)
Due to compression, the fluid in the container moves down by dh and the fluid in the tube moves up by
10cm.
Volume of fluid fallen = Volume of fluid risen
dh = 0.000766m
Equating the pressure at new level (YY)
(
From a and b
)
(b)
(
)
Neglecting dh
W = 1.05 N
24. Figure below shows a pipe containing a liquid of sp gr 0.8 connected to a single column
micromanometer. The area of reservoir is 60 times that of the tube. The manometer liquid is mercury.
Find the pressure in the pipe.
A
Liquid of
sp gr 0.8
0.25m
y
A
h1
X
dy
Z
X
Z
Hg
Solution:
When the manometer is not connected to the container, the mercury in the reservoir is at original level
and at level A in the tube.
Equating pressure at XX
(a)
Due to pressure, manometric liquid in the reservoir drops by dy and it will travel a distance of 0.25m in
the tube.
Volume of fluid fallen = Volume of fluid risen
Ady=ax0.25
60ady = ax0.25
dy = 0.0041m
Equating pressure at new level (ZZ)
(
)
(
)
(b)
From a and b
(
)
(
)
= 33868 N/m2
25. Find the gauge readings at A and B if the atmospheric pressure is 755mmHg.
Closed
A
Air
Oil, sp.
gr. = 0.8
Hg
Vapor
1.5m
Water
0.6m
C
2m
B
D
0.2m
Hg
Solution:
Atm. pr. =
= 100729 N/m2
Neglect pressure due to mercury vapor and air (small)
Writing pressure equation for gauge A,
= 13.6x9810x0.6 = 80049.6 N/m2 (abs. pr.)
Gauge pressure at A = Abs. pr. - Atm. pr. = 80049.6-100729 = -20679.5 N/m2
Writing pressure equation for gauge B starting from gauge A
= 8750.5 N/m2
Gauge pressure at B = 8750.5 N/m2

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