FLOW ANALYSIS USING CONTROL VOLUMES

FM 4.1 A 1.5 m³ rigid tank contains air whose density is 1.18kg m/ ³. A high-pressure air is allowed to enter the tank until the density in the tank rises to 7.20kg m/ ³. The mass of air that entered the tank is

(A) 10.8 kg (B) 9.03 kg

FM 4.2 The wind blows through a 2.2m#3m garage door with a speed of 1.5m s as / shown in figure. What will be the average speed V of the air through the two 0.91m#1.22m windows ?

(A) 1.125m s (B) / 2.23m s/

FM 4.3 Water is being pumped into a bathtub whose cross-section 3m#4m. The bathtub has a 5 cm diameter orifice and water is discharged through orifice at a constant velocity of 5m s. If the water level in the tub rises at a rate of 2/ cm/min , the rate at which water supplied to the pool in m s³/ , is

(A) 0.0139 (B) 0.0269

FM 4.4 Three pipes steadily delivers water at 20 Cc to a large exit pipe as shown in figure below. The velocity V = m s and the exit flow rate vo = . m s. If increase in vo by 20% would increase vo by 10%, the velocities V V1, and V are

(A) V =12m s,V = . m s,V = . m s

(B) V = V = V =

(D) V = V = V =

FM 4.5 Water at 20 Cc flows through the piping junction, enters at section 1 with flow rate of 1.26#10⁻³m s³/ as shown in figure below. A portion of the flow is diverted through the shower-head, which contains 50 holes of 1 mm diameter at section 3.

The average velocity at section 2 is 2.5 m/s. If flow through the shower is uniform , the exit velocity from the shower head jet is

(A) 121m s (B) / 605m s/

FM 4.6 An oil having a specific gravity of 0.85 is pumped with a water jet pump as shown in figure. The water and oil mixture has an average specific gravity of 0 90 and . water flow rate is 0.5m s³/ . What will be the flow rate in m s³/ at which the pump moves oil ?

(A) 1.5 (B) 0.5

FM 4.7 Water flowing through an 8 cm diameter pipe enters a porous section as shown in figure below, which allows a uniform radial velocity Vw through the wall surfaces for a distance of 1.2 m. If the entrance average velocity is 12 m/s and the exit velocity is 9m s, what will be the / Vw in cm/s ?

(A) 0.5 (B) 5

FM 4.8 Air at the steady rate of 25m³/min is drawn into a compressor at standard atmospheric conditions. The compressor pressure ratio, pexit pinlet is 10 to 1 and through the compressor p ρⁿ remains constant with n =1. . If the average velocity in the compressor discharge pipe is not to exceed 25m s, what will be / the minimum discharge pipe diameter ?

(A) 6.4 m (B) 0.0064 m

FM 4.9 Consider the river flowing towards a sea at mean velocity of 3m s with a rate / of 250m s³/ at a location 90 m above the sea surface. The power generation potential of the entire river at that location is

(A) 230 MW (B) 225 MW

Common Data For Linked Answer Q. 10 and 11

A pump motor unit is used to supply water to a storage tank from a lake at a elevation of 20 m. The pump can supply water at a rate of 0.070m s³/ with a expanse of 20.4 kW electric power. Any frictional losses in pipes and any changes in kinetic energy is neglected.

FM 4.10 The overall efficiency of the pump-motor unit is

(A) 67% (B) 50%

FM 4.11 The pressure difference between the inlet and the exit of the pump is

(A) 201 kPa (B) 192 kPa

Common Data For Linked Answer Q. 12 and 13.

Air at standard atmospheric condition enters the compressor at a rate of 0.28m s³/ . It comes out to the tank through a 3 cm diameter pipe with a density of 1.8kg m/ ³ and a uniform speed of 214m s. (Take / ρ =air 1. kg m )

FM 4.12 What is the mass flow rate in kg s at which the mass of air in the tank is / increasing ?

(A) 0.65 (B) 0.073

FM 4.13 What is the average time rate of change of air density in kg m s/ ³ within the tank

(A) 1.56 (B) 1.30

FM 4.14 The pipe flow in shown figure below, fills a cylindrical tank. At time t=0, the water depth in the tank is 65 cm. What will be the time required to fill the remainder of the tank ?

(A) 41.5 s (B) 23 s

FM 4.15 Water is being pumped from a lake to a reservoir of 15 m height by a 7 hp (shaft power) pump. If the mechanical efficiency of the pump is 82%, what will be the maximum volume flow rate of water ?

(A) 35.5 /L s (B) 29 /L s

FM 4.16 Water enters the bottom of the cone as shown in figure at a uniformly increasing average velocity V=at. If d is very small and h=0 at t=0, the expression for the water surface rise is

(A) h t( )= : at d cot qD (B) h t( )= : at d cot qD (C) h t( )= : at d cot qD (D) h t( )= : at d cot qD

FM 4.17 The velocity distribution is uniform at the entrance of a 0.9 m wide channel with a velocity V as shown in figure. Further at the downstream the velocity profile is given by u= y− y , where u is in m s and y/ is in meter. What will be the value of V in m s ?/

(A) 0.70 (B) 0.35

FM 4.18 A fluid flowing past incompressibly over a flat plate as shown in figure below, with a uniform inlet profile u=U_o and a exit profile u U 3 2

o 3

, < η η- F, where _{η δ}⁼^Y. What will be the expression for volume flow rate vo across the top surface of the control volume ?

(A) vo=U b_o d (B) v 5U b

o d

= o (C) v 3U b

o d

o (D) v 3U b

o d

= o

FM 4.19 A syringe plunger is moved forward at the steady rate of 10mm s and the vaccine / leaks pass the plunger at 0 1 of the volume flow rate, out the needle opening. . The inside diameters of the syringe and the needle are 15 mm and 0.525 mm, respectively. What will be the average velocity of the needle exit flow ?

(A) 5.56m s (B) / 7.42m s/

FM 4.20 Water flows through a horizontal pipe at a rate of 35 /L s. The pipe diameter is reduced from 15 cm to 8 cm by a reducer. The pressure difference at the centre line, between the two sections of pipe is measured to be 30kPa. If the kinetic energy correction factors to be 1.05, the irreversible head loss in the reducer is

(A) 5.44 m (B) 6.75 m

FM 4.21 Consider an incompressible steady flow between two parallel plates as shown in

figure below. The uniform upstream velocity is, u=Uo= m s, while downstream velocity profile is u=az zo−z , where a is a constant. If zo= cm and the fluid is gasoline at 20 Cc , what is the value of uma ?

(A) 3cm s (B) / 12cm s/

FM 4.22 A fire hose nozzle is to deliver water that will rise 40 m vertically. What is the stagnation pressure required at the nozzle inlet if (a) no loss is there (b) a loss of 30N m kg⁻ / is there, respectively ?

(A) 392 kPa, 422 kPa (B) 316 kPa, 294 kPa (C) 294 kPa, 316 kPa (D) 422 kPa, 392 kPa

FM 4.23 The water level in a tank is 16 m above the ground. A hose is connected to the bottom of the tank and the Nozzle at the end of the hose is pointed straight up.

The tank is at sea level and water surface is open to the atmosphere. There is a pump in the line leading from the tank to the Nozzle, which increases the pressure of water. If the water jet rises to a height of 30 m from the ground, the minimum pressure rise supplied by the pump to the water line is

(A) 139.4 kPa (B) 135 kPa

FM 4.24 The test section wall in figure shown below contains 12064 holes of 5 mm diameter each. The suction velocity through each hole is Vr =4 0m min and the entrance velocity V0=215m min. For incompressible steady flow of air at 20 Cc , what will be the final velocity Vf ?

(A) 4.63m s (B) / 4.13m s/

FM 4.25 The velocity distribution in an open channel flow is characterise by the relation m s

U y h

V= i

where U = free-surface velocity, y= perpendicular distance from the channel bottom in meter and h= depth of the channel in meter.

What is the average velocity of the channel stream as a function of U ?

(A) 0.833 (B) 0.0625

FM 4.26 Consider a water jet that is deflected by a stationary cone such as shown in figure below. If the jet velocity and diameter are 30m s/ and 5 cm, respectively and the jet is deflected by 45c, what amount of force is required to hold the cone against the water stream ?

(A) FRx=−5 8 ,FRy =0 (B) FRx=5 8 ,FRy =0 (C) FRx=0,FRy=−5 8 (D) FRx=−5 8 ,FRy =5 8

FM 4.27 The vane turns water jet completely around as shown in figure below. If the water has pressure p and temperature TcC, the maximum jet velocity is

(A) V

D F 2 ²

= ; rp E (B) V

D F 2 ²

= ; rp E (C) V

D F 2

= ;rp E (D) V

D 2F

2 2

= ;rp E

FM 4.28 A jet of water with velocity V is directed in the positive x direction and it is deflected by a flat plate. The plate is moving towards the on coming water jet with velocity 0.5 V. If the jet cross-sectional area is A and a force F is required to maintain the plate stationary then the magnitude of force required to move the plate towards the jet, as shown in figure below, is

(A) ρAV (B) 0 25. ρAV²

FM 4.29 Air enters in a jet engine at 20 Cc and 1 atm, where A1= m and V1= m s and leaves at 1 atm, where A = m and V =1 m s as shown in figure below. If the air-fuel ratio is 30 1, the test stand support reaction R: x will be

(A) 281 kN (B) 356 kN

Common Data For Q. 30 and 31

The water flows steadily from a tank mounted on a cart as shown in figure.

After the water jet leaves the nozzle of the tank, it falls and strikes a vane attached to another cart. Consider the cart’s wheels are frictionless and the fluid is inviscid.

FM 4.30 What is the tension in rope A ?

(A) 592 N (B) 320 N

FM 4.31 What is the tension in rope B ?

(A) 490 N (B) 636N

FM 4.32 Water at 20 Cc flows steadily through a reducing pipe bend as shown in figure below. Known conditions are p1= k a gage, A1= 1 cm , p = k a gage,

A = and mo = g s. Neglecting bend and water weight, the total force which must be resisted by the flange bolts will be

(A) 9.8 kN (B) 14.5 kN

FM 4.33 A 300 mm diameter circular plate is held perpendicular to an axisymmetric horizontal jet of air having a velocity of 30m s and a diameter of 85 mm as / shown in figure. A hole at the centre of the plate is provided which results in a discharge jet of air having a velocity of 40m s and a diameter of 35 mm. What / will be the horizontal component of force required to hold the plate stationary ?

(A) 9.90 N (B) 8.53 N

FM 4.34 Water flows through a reducing section of pipe as shown in figure below. All fluids are at 20 Cc . If D1=8 cm, D2=5 cm, p2=1 atm, V2=1 m s, and the manometer reading is h=58 cm. What will be the horizontal force resisted by each bolt when number of bolts is 4 ?(γ =Hg 1 28 N m )

(A) 167 N (B) 200 N

FM 4.35 A water tank is drained through a hole of area Ao with fluid velocity V = gh as shown in figure below, where h is the depth of water above the hole and the cylindrical tank have bottom area Ab. Expression for the time to drain the tank from an initial depth of ho is

(A) t A A

g h

o# o

= (B) t

A A

g h

b# o

= (C) t

A gh

b # o

= (D) t

A gh

o# o

FM 4.36 Water is pumped from a reservoir as shown in figure. The head loss is known to be 1 2. V²/2g, where V is the average velocity in the pipe. The relationship between the pump head and the flow rate is hp= − vo, where hp is in the meters and vo is in m s³/ . What will be the flow rate vo in m s³/ ?

(A) 0.052 (B) 0.52

FM 4.37 Water exists to the standard sea-level atmosphere through the split nozzle as shown in figure below. The weight flow rate at section 2 and 3 is equals to 748 N/s. If D = cm, D2=D3= 0 cm and p = 35 k a (absolute), the force on the flange bolts at section 1 is

(A) 802 N (B) 1768 N

FM 4.38 Water enters a pump impeller radially and leaves the impeller with a tangential component of absolute velocity of 10m s. The impeller exit diameter is 60 mm / and the impeller speed is 1800 rpm. If the stagnation pressure rise across the impeller is 45 kPa, the loss of available energy across the impeller and the hydraulic efficiency of the pump respectively, are

(A) 8.7N m kg⁻ / , 0 597 (B) . 11.6N m kg⁻ / , 0 796. (C) 14.5N m kg⁻ / , 0 995 (D) . 5.8N m kg⁻ / , 0 398.

FM 4.39 The water enters a horizontal, circular cross-sectional, sudden contraction nozzle at section (1) with a uniformly distributed velocity of 7.5m s and a pressure of / 517 kPa as shown in figure. The water exits from the nozzle into the atmosphere at section (2) with the velocity of 30.5m s. What will be the axial component of / the anchoring force required to hold the contraction in place ?

(A) 1521N (B) 759 N

FM 4.40 Water is flowing through a U-section pipe as shown in figure below. At flange (1), flange (2) and Location (3) the pressures are 100 kPa, 50 kPa and 100 kPa, respectively. If the momentum flux correction factor to be 1.03, the total x and z forces at the two flanges connecting the pipe are

(A) FRx=− 3.11 ,FRz =112 (B) FRx= 3.11 ,FRz =−112 (C) FRx=112 ,FRz =− 3.11 (D) FRx=−112 ,FRz = 3.11

FM 4.41 A nozzle is connected to a vertical pipe and discharges water into the atmosphere at a rate of 0.01m s³/ as shown in figure. The gage pressure at the flange is 40 kPaand the nozzle has a weight of 200 N. If the volume of water in the nozzle is 0.012 m³, what will be the vertical component of the anchoring force required to hold the nozzle in place ?

(A) 1140 N (B) 1072 N

FM 4.42 Water at 20 Cc flows through a 5 cm diameter pipe as shown in figure, which turns the water flow direction completely around. The pressure at flange 1 is

1 5 a

p1= (abs), at flange 2 is p2=1 a (abs) and mass flow rate is 23.45 kg/s. What will be the total force which flanges must withstand ?

(A) Fx = 53 (B) Fx=115

Common Data For Linked Answer Q. 43 and 44.

The water flows through a horizontal bend and discharges into the atmosphere as shown in figure. When the pressure gage reads 69 kPa, the resultant x

-direction anchoring force, FAx, in the horizontal plane required to hold the bend in place is shown in figure and the flow is not frictionless.

FM 4.43 The flow rate through the bend is

(A) 0.20m s³/ (B) 0.02m s³/

FM 4.44 What is the anchoring force FAy in y-direction, required to hold the bend in place ?

(A) 3251 N (B) 3041 N

FM 4.45 A liquid jet of velocity Vj and area Aj strikes a single 180c bucket on a turbine wheel rotating at angular velocity ω as shown in figure below. What will be the expression for maximum power in terms of ρ, Aj and Vj ?

(A) Pmax= rAjVj (B) Pmax= rA Vj j

FM 4.46 Consider a free jet of fluid which strikes a wedge as shown in figure. A portion of the total flow is deflected by 30c and the remainder is not deflected. The horizontal and vertical components of force needed to hold the wedge stationary are FH and FV, respectively. If the effect of gravity is neglecting and the fluid speed remains constant, the force ratio FH FV is

(A) 2 7. (B) −2 7.

FM 4.47 A liquid jet of velocity V_j and diameter Dj strikes a fixed cone of θ =60c and deflects back as a conical sheet at the same velocity. What will be the restraining force F ?

(A) F= rA V (B) F= . rA V

FM 4.48 A vertical circular cross section jet of air strikes a conical deflector as shown in figure. A vertical anchoring force of 0.1 N is required to hold the deflector in the place. If the magnitude of velocity of the air remains constant and ρ =air . kg m , what will be the mass of the deflector ?

(A) 0.108 kg (B) 1.08 kg

FM 4.49 The box in figure shown below has three 1.27 cm holes on the right side. The volume flows of 20 Cc water from top and bottom hole is votop=vobottom=2 2 cm s and from middle is vomiddle= cm s. What will be the force, which this water flow causes on the box ?

(A) 316 N (B) 126 N

FM 4.50 Water flows through the horizontal tee connection as shown in figure. The flow of water is considered frictionless, incompressible and one-dimensional. If each pipe has an inside diameter of 1 m, the x and y components of the force exerted by the tee on the water respectively, are

(A) −185 kN, −45.8 kN (B) −185 kN, 45.8 kN (C) 185 kN, −45.8 kN (D) 185 kN, 45.8 kN

FM 4.51 The tank in figure given below has a mass of 51 kg when it is empty and contains 600 L of water at 20 Cc . Pipes 1 and 2 have area A =2. #10 m⁻ ² and

0.0 m s

vo= . What should be the scale reading W ?

(A) 6374 N (B) 8800 N

FM 4.52 Water is flowing in a 0.1 m diameter uniform pipe as shown in figure. The pipe puts the force on the fluid in the 6 m section. What will be the axial and normal components of the force, respectively ?

(A) 19 N, 327N (B) 27 N, 218N

FM 4.53 Water enters in a elbow at diameter D1=10 cm, p1=233 kPa (gage) and exits to atmosphere at D2=3 cm as shown in figure below. At a weight flow rate of 150 N/s the force on the flange bolts at section 1 is (Neglect the weight of water and elbow)

(A) 2094 N (B) 264 N

Common Data For Linked Answer Q. 54 and 55.

Two water jets collide and form one homogeneous jet as shown in figure below.

Neglect the effect of gravity.

FM 4.54 The speed V at the exit and direction θ of the combined jet respectively, are (A) 5.36m s, / 21 5c (B) . 2.15m s, / 8 6c.

FM 4.55 What is the head loss for a fluid particle flowing from (1) to (3) and from (2) to (3) ?

(A) 558N m s⁻ / (B) 419N m s⁻ /

Common Data For Q. 56 and 57.

Water enters the rotor at a rate of 0.005m s³/ along the axis of rotation as shown in figure. The cross section area of each of the three nozzle exits normal to the relative velocity is 18 mm² and θ =30c.

FM 4.56 What will be the resisting torque required to hold the rotor stationary ?

(A) 225 N m⁻ (B) 150 N m⁻

FM 4.57 How fast will be the rotor spin steadily if the resisting torque is reduced to zero ?

(A) 200rad s (B) / 120rad s/

FM 4.58 In a pipe flow of water, the distribution of axial direction velocity u is linear from zero at the wall to maximum of uc at the centerline. What will be the average velocity u and the kinetic energy coefficient α, respectively ?

(A) uc, 5 4 (B) . uc, 1.1

c, 2.7 (D) 3uc, 8.1

FM 4.59 Consider inward flow radial turbine which involves a nozzle angle α =1 60c and an inlet rotor tip speed U1= m s. The absolute velocity leaving the rotor at section (2) is radial with a magnitude of 12m s and the ratio of rotor inlet to / outlet diameters is 1 8. If the fluid is water, what will be the energy transfer per . unit mass of fluid flowing through this turbine ?

(A) 77.7N m kg⁻ / (B) 58.3N m kg⁻ / (C) 38.8N m kg⁻ / (D) 97.1N m kg⁻ /

FM 4.60 The velocity profile in a turbulent pipe flow may be approximated with the expression

u u

R 1 r

1 n

= −a k

where u= local velocity in the axial direction, uc= centerline velocity in the axial direction, R= pipe inner radius from pipe axis, r= local radius from pipe axis and n= constant.

What will be the kinetic energy coefficient α for n= ?

(A) 111 1 (B) . 11 1.

FM 4.61 Water flows vertically upward in a circular cross-sectional pipe as shown in figure.

The velocity profile over the cross-sectional area at section (1) is uniform and at section (2) the velocity profile is characterise by the relation:

V w

Rr k

c 1

= a − k

where V= local velocity vector, wc= centerline velocity in the axial direction, R= pipe radius and r= radius from pipe axis.

What will be the expression for the fluid pressure drop between sections (1) and (2) ?

(A) p p

R^z gh w

p r r

− = − − (B) p p

R^z gh w

p r r

− = + +

R^z gh w

p r r

− = + + (D) p p

R^z gh w

p r r

− = − +

Common Data For Q. 62 and 63.

A hydraulic jump forms near the downstream end of a river spillway as shown in figure. The velocity of the channel flow is reduced abruptly across the jump.

Consider the conservation of mass and linear momentum principles and the width of flow is unity.

FM 4.62 What will be the expression for h , shown in figure above ?

(A) h h h

g V h

= − + b l + (B) h h h

V g

= − + b l + h (C) h h

h V h

= − c m + (D) h h

h V h

= − c m −

FM 4.63 If energy conservation is considered, the expression for the loss of available energy across the jump will be

(A) h hg (h h )

4 2 1 1− 2 3 (B) ( )

h h

g h −h

h h h −h (D) ( )

h hg h h 4 2 1 2− 1 3

***********

SOLUTIONS

FM 4.1 Option (B) is correct.

From mass balance

Mass of air entered =Final mass oftank−Initial mass of tank ma =mf −mi

v v

f i

r r

= − =(rf−ri) v v=constant fortan

. . .

7 30 1 18 # 1 5

=] − g ] g =9.03 kg

FM 4.2 Option (B) is correct.

Consider steady incompressible flow of wind, then vogarage door =vowindow+vowindow

Agarage doorVgarage door =AwindowV+AwindowV= AwindowV where Vgarage door =Normal velocity to the garage door

. sin 1 5 30c

Thus 2 2. #3#1 5. sin30c =2#0 91. #1 22. #V

V .

. 2.2 m s/ 2 22

4 95 3

= =

FM 4.3 Option (A) is correct.

From mass conservation to the bathtub.

The rate of increase in the amount of water

= The difference between water supply rate and water discharge rate

mopool =mosupply−modischarge

vopool =vosupply−vodischarge mo =rgvo

vosupply =vopool+vodischarge ...(i)

Then vodischarge =Vdischarge#Aorifice

( . )

0.00982m s/ 4

5#p# 0 05 ² ₃

= =

vo_pool =V_{level rise}#A_pool = . # # 0.24m³/min 0.004m s³/

= =

Substituting in equation (i), we get

vosupply =0 004. +0 00982.

0.01382,0.0139m s³/

FM 4.4 Option (A) is correct.

For steady flow, we have

vo + +vo vo = ov

or V A +V A +V A =V A ...(i)

since vo =V A

or 0 05 . =V #p #

A control volume around sections (1, 2, 3) yields

vo1 = +vo2 vo3 ...(i)

And with V2=2. m s, flow rate

vo₂ A V2 2 2. ( . )2 2

#p #

= = =0.000785m s³/

Thus from equation (i), vo3 = −vo1 vo2

. .

1 26#10 ³ 0 785#10 ³

= ⁻ − ⁻ =0.475#10⁻³m s³/ Each hole carries v

50³

For steady flow process

mo_inlet = om_outlet

On combining equation (i) and (ii), we have

v v

v S Go −S G =v S Go −

For a suction velocity of Vw, the cylindrical suction surface area Aw =2pR#L

2#p#0.04#1.2 0.3016 m²

= =

Since for steady flow

vo =vow+vo It is steady flow process, so mass flow remains constant.

mo = om

Here p & p denotes the inlet an exit pressures respectively.

Now from equation (i) and (ii), we get

d pp

The total mechanical energy of the river water per unit mass becomes emech =P E. .+K E. . The power generation potential of the river water is

Pma = omemech

The mass flow rate of water

mo=rvo =1000#0.07=70kg s/ Then the Energy output of the pump

Eout

Δ o =mo#g#h= # . # =13.7 kW The overall efficiency of pump-motor unit

pump motor

The change in pressure at inlet to exit must be equal to the useful mechanical

In document ME Vol 2 FM (Page 128-176)