TUTOR MARKED ASSIGNMENT - The People's University IIIrd year.pdf · 5 TUTOR MARKED ASSIGNMENT ET...

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TUTOR MARKED ASSIGNMENT

ET 102

MATHEMATICS-III

Note : All questions are compulsory and carry equal marks. This assignment is based on all Blocks of Mathematics-III.

Maximum Marks : 100 Weightage : 30%

Course Code : ET-102 Last Date of Submission : July 31, 2012

BTCM/BTWRE

Q.1 (a) Test the convergence or divergence of the series 2

1( 1)

nn n

∝

=

Σ + −

.

(b) For a function f (x) defined by ( ) ,f x x x= − π < < π , obtain a fourier series.

Q.2 (a) Prove that 2

2

21

cos4 ( 1) ,

3

n

n

n xx x

n

∞

=

π= + .

Hence, show that 2

2

1

6n

π .

(b) If 1

2 cos xx

θ = + and 1

2 cos yy

φ = + show that one of the values of 1m n

m nx y

x y+

is 2 cos ( )m nθ + φ .

Q.3 (a) If tan log ( )x iy a ib+ = + , where 2 2 1a b+ ≠ , show that

2 2

2 2

2tan log ( )

1

ax y

a b+ =

− −.

(b) Show that the polar form of Cauchy-Riemann equations are

1u v

r r

∂ ∂=

∂ ∂θ,

1v u

r r

∂ ∂−

∂ ∂θ

Deduce that 2 2

2 2 2

1 10

u u u

r rr r

∂ ∂ ∂+ + =

∂∂ ∂θ.

Q.4 (a) Find the bilinear transformation which maps the points z = 1, i, – 1 into the points w = 2, i, – 2 respectively.

(b) Expand 1

( )( 1) ( 2)

f zz z

=− −

in the region | z | > 2.

Q.5 (a) Determine the poles of the following functions and the residue at each pole : 2

2

1

2

z

z z

+

−.

(b) The number N of bacteria in a culture grew at a rate proportional to N. The value of N was initially 100 and increased to 332 in one hour. What would be the value of N after

11

2 hours?

4

Q.6 (a) Solve 2tan tan cos cosdy

y x y xdx

+ = .

(b) Solve ( ) ( )mz ny p nx lz q ly mx− + − = − .

Q.7 (a) Solve 2 2 2

2 23 2

z z zx y

x yx y

∂ ∂ ∂+ + = +

∂ ∂∂ ∂.

(b) Find the solution of 2

2

2

u uh

tx

∂ ∂=

∂∂ for which (0, ) ( , ) 0u t u l t= = , ( , 0) sin

xu x

l

π= by

method of variables separation.

Q.8 (a) Test the series 33( 1 )n n for its convergence or divergence.

(b) Find the Fourier series to represent 2( ) ,f x x x x= − − π < < π .

Hence, show that 2

2 2 2 2

1 1 1 1. . .

121 2 3 4

π− + − + = .

Q.9 (a) Find the bilinear transformation which maps the points z = 0, − 1, ∞ into the points

w = − 1, − 2 − i, i.

(b) Find the general solution of the partial differential equation

2 2 2( ) ( ) ( )x y z p y z x q z x y− + − = −

Q.10 (a) If 1

2 cos xx

θ = + and 1

2 cos yy

φ = + , show that one of the values of m n

n m

x y

y x+ is

2 cos ( )m nθ − φ .

(b) Test for convergence of the series 2 3 12 6 14 2 21 . . . . . .

5 9 17 2 1

nn

nx x x x −−

+ + + + + ++

,

(for x > 0).

5


ET 201 (Part A)

MECHANICS OF FLUIDS

Note : All questions are compulsory and carry equal marks. This assignment is based on all

Blocks of Mechanics of fluids.


Course Code : ET-201A Last Date of Submission : July 31, 2012

BTCM

Q.1 (a) If the equation of a velocity distribution over a plate is given by v = 2y – y2, in which v is

the velocity in m/s at a distance y, measured in meters above the plate, what is the velocity gradient at the boundary and at 7.5 cm and 15 cm from it? Also determine the stress at these points if absolute viscosity µ = 8.60 poise.

(b) An iceberg weighing 8976 N/m3 floats in the ocean with a volume of 600 m3 above the surface. Determine the total volume of the iceberg if specific weight of ocean water is 10 055 N/m3.

Q.2 (a) The Velocity components in the x- and y-directions are given as 3

22

3

xyu x y

= −

and

32 2

3

yxv xy

= −

. Indicate whether the given velocity distribution is a possible field of

flow or not.

(b) If the expression for the stream function is described by 3 23x xyψ = − , indicate whether

the flow is rotational or irrotational. If the flow is irrotational determine the value of the velocity potential.

Q.3 (a) If the volume of a liquid decreases by 0.2 per cent for an increase of pressure from 6.867 MN/m2 to 15.696 MN/m2, what is the value of the bulk modulus of the liquid?

(b) Oil of specific gravity 0.90 flows in a pipe 300 mm diameter at the rate of 120 litres per second and the pressure at a point A is 24.525 kPa (gage). If the point A is 5.2 m above the datum line, calculate the total energy at point A in terms of metres of oil.

Q.4 (a) A circular orifice of area 6.45 × 10– 4 m2 is provided in the vertical side of a large tank. The tank is suspended from a knife edge 1.53 m above the level of the orifice. When the head of water is 1.22 m, the discharge is 1161.5 N/min and a turning moment of 14.421 N-m has to be applied to the knife edges to keep the tank vertical. Determine Cv, Cd and Cc of the orifice.

(b) Water under a constant head of 3 m discharge through an external cylindrical mouthpiece 50 mm diameter, for which Cv = 0.82, determine (i) the discharge in cumec and (ii) absolute pressure at vena-contracta; and the maximum head for the mouthpiece to flow full.

Q.5 (a) Derive an expression for mean velocity for laminar flow (i) through a pipe and (ii) between parallel plates.

(b) What do you understand by hydro dynamically smooth and rough pipes.

6

Q.6 (a) The population of a city is 800 000 and it is to be supplied with water from a reservoir 6.4 km away. Water is to be supplied at the rate of 140 liters per head per day and half the supply is to be delivered in 8 hours. The full supply level of the reservoir is R. L. 180.00 and its lower water level is R. L. 105.00. The delivery end of the main at R. L. 22.50 and the head required there is 12 m. Find the diameter of the pipe. Take f = 0.04.

(b) A compound piping system consist of 1800 m of 0.50 m, 1200 m of 0.40 m and 600 m of 0.30 m new cast iron pipes connected in series. Convert the system to (i) an equivalent length of 0.40 m pipe and (ii) and equivalent size pipe 3600 m long.

Q.7 (a) The velocity distribution in the boundary layer is given as 23 1

2 2

v

V= η − η in which

y η = δ

compute *δ

δ and

θ δ

.

(b) Given that a laminar boundary layer at zero pressure gradient over a flat plate is

described by the velocity profile 0

sin2

y

V

ν π = δ

.

Determine the momentum correction coefficient (or factor) and the energy correction coefficient (or factor).

Also show that boundary layer thickness δ, wall shear stress τ0 and coefficient of drag CD

are given by 20

0

0.3284.795 1.312; ;

Re Re ReD

x x x

VxC

ρδ = τ = = , where symbols have their usual

meaning.

Q.8 (a) Find the ratio of skin friction drag on the front two-third and rear one-third of a flat plate kept in a uniform stream at zero incidence. Assume the boundary layer to be turbulent over the entire plate.

(b) For laminar flow of an oil having dynamic viscosity µ = 1.766 Pa.s in a 0.3 m diameter pipe, the velocity distribution is parabolic with a maximum point velocity of 3 m/s at the centre of the pipe. Calculate the shearing stresses at the pipe wall and within the fluid 50 mm from the pipe wall.

Q.9 (a) Show by method of dimensional analysis that the resistance R to the motion of a sphere

of diameter D moving with uniform velocity V through a fluid having density ρ and

viscosity µ may be expressed as :

2( )R D Vv D

µ= ρ φ ρ

Also show that the above expression reduces to R = k µ VD when the motion is through viscous fluid at low velocity, where k is a dimensionless constant.

(b) Find the viscosity in poise of a liquid through with a steel ball of diameter 1 mm falls, with a uniform velocity of 20 mm/s. The specific gravity of the liquid is 0.91 and that of the

steel is 7.8. Given that k = 3π.

(b) Assuming that rate of discharge Q of a centrifugal pump is dependent upon the mass

density ρ of fluid, pump speed N (rpm), the diameter of impeller D, the pressure p and

the viscosity of fluid µ, show using the Buckingham’s π-theorem that it can be represented by

3

2 2 2( ) ,

gHQ ND

N D ND

υ = φ

where H = head and ν = kinematic viscosity of the fluid.

7

Q.10 (a) An oil having viscosity of 1.43 poise and specific gravity 0.9 flows through a pipe 25 mm diameter and 300 m long at 1/10 of the critical velocity for which Reynolds number is 2500. Find (i) the velocity of flow through the pipe; (ii) the head in metres of oil across the pipe length required to maintain the flow and (iii) the power of the flow.

(b) Two parallel plates kept 75 mm apart have laminar flow of glycerine between them with a maximum velocity of 1 m/s. Calculate the discharge per metre width, the shear stress at the plates, the difference in pressure in pascals (or N/m2) between two points 25 m apart, the velocity gradients at the plates and velocity at 15 mm from the plate. Take viscosity of glycerine as 8.35 poise.

8

Q.1 (a) A fish freezing plant of 100 tonnes capacity is to be maintained at – 50oC when the outside atmosphere temperature is 40oC. The actual C.O.P. of the refrigeration system used is 1/5 of the theoretical Carnot refrigerator working between the same temperature limits. Calculate the power required to run the plant.

(b) A reversible engine is supplied with heat from two constant sources at 900 K and 600 K and rejects heat to a constant temperature sink at 300 K. If the engine executes of complete cycles while developing 100 kW, and rejecting 3600 kJ of heat per min. Determine the heat supplied by each source per minute and efficiency of the engine.

Q.2 (a) A heat pump working on the Carnot cycle takes in heat from a reservoir 5oC and delivers heat to a reservoir 60oC. The heat pump is driven by a reversible heat engine which takes heat from a reservoir at 840oC and rejects heat at 60oC. The reversible heat engine also drives a machine that absorbs 36 kW. If the heat pump extracts 17 kJ/sec from 5oC reservoir, determine,

(i) The rate of heat supply from 840oC source.

(ii) The rate of heat rejection to 60oC sink.

(b) Threee Carnot engines R1, R2, R3 operate in series between two heat reservoirs which are at temperatures of 1000 K and 300 K.

Calculate intermediate temperautres if amount of work produced by these engines is in the, peoportions of 5 : 4 : 3.

Q.3 (a) A reversible engine works between three thermal reservoirs A, B and C. The engine absorbs an equal amount of heat from the thermal reservoirs A and B kept at temperatures TA and TB respectively, and rejects heat to the thermal reservoir C kept at

temperature TC. The efficiency of the engine is α times the efficiency of the reversible engine, which works between the two reservoirs A and C. Prove that

( ) ( )2 1 2 1A A

B C

T T

T T= α − + − α

(b) A heat pump is used to maintain an auditorium hall at 24oC when the atmospheric temperature is 10oC. The heat lost from the hall is 1500 kJ/min. Calculate the power required to run the heat pump if its COP is 30 percent of Carnot machine, working between the same temperature limits.


ET 201 (Part B)

ENGINEERING THERMODYNAMICS


Blocks of Engineering Thermodynamics.


Course Code : ET 201 B Last Date of Submission : July 31, 2012

BTCM

9

Q.4 (a) A piston and cylinder machine contains 1 kg of air, initially v = 0.8 m3/kg and T = 290 K. The air is then compressed in a slow frictionless process to a specific volume 0.2 m3/kg and a temperature 580 K. The law for compression is PV1.5 = 0.75 with P in bar and v in m3/kg. Determine work done and heat transferred during the process. Assume for air Cp = 1.00 kJ/kg-K and Cv = 0.743 kJ/kg-K and R = 0.287 kJ/kg-K.

(b) 3 kg of air at a pressure of 150 kPa and temperature 360 K is compressed polytropically to 750 kPa according to law PV1.2 = constant. The air is then cooled to initial temperature at constant pressure. The air is then brought to state (1) by following PV = C. Draw the cycle on PV-diagram and determine net work done and heat.

Q.5 (a) 1 m3 of gas is filled in a closed tank. The initial condition of the gas is 3 bar and 50oC. The gas is heated until the pressure becomes 5 bar. Find the change in internal energy, work done, heat supplied, change in entropy.

Take R = 0.287 kJ/kg-K and Cv = 0.743 kJ/kg-K.

(b) Find the work done, in kilojoules by an ideal gas in going from state A to state C along the path shown in the PV diagram as shown in Figure 1.

Figure 1

Q.6 (a) A cylinder contains 0.12 m3 of air at 1 bar and 100oC. the air is compressed to 0.03 m3. The final pressure is 6 bar.

Determine :

(i) The value of index n

(ii) Mass of air in the cylinder

(iii) Increase in internal energy

Take γ = 1.4, R = 0.287 kJ/kg-K and Cv = 0.72 kJ/kg-K.

(b) Determine pressure of 1 kg of oxygen at 100oC if the specific column is 0.2. m3/kg using. (i) Van der Waals equation and (ii) Ideal gas laws.

Take for O2, a = 13.93 × 104 N.m4 (kg mole)2; b = 0.0314 m3/kg mole, and R = 8314 J/kg mole K.

Q.7 (a) Steam at 20 bar and 360oC expands in a steam turbine to 0.08 bar. It is then condensed in a condenser to saturated water. The pump feeds back the water to the boiler. Assume ideal Rankine cycle and determine.

(i) The net work done/kg of steam

(ii) Efficiency of the Rankine cycle

(b) Steam at 20 bar and 360oC is expanded in a steam turbine to 0.08 bar. It then enters a condenser where it is condensed to saturated liquid. It is then fed back to the boiler. Determine :

(i) Net work (shaft work) per kg of steam.

(ii) Dryness fraction of steam entering the condenser.

T = Constant

A

0 1 2 3

100

k Pa

P

V, m3

10

Q.8 (a) A single stage, single acting reciprocating air compressor has a bore of 200 mm and a stroke of 300 mm. It runs at a speed of 500 rpm. The clearance volume is 5% of swept volume and polytropic index is 1.3 throughout. Intake pressure and temperature are 97 kPa, 20oC and the compression pressure is 550 kPa. Determine :

(i) FAD in m3/min

(ii) Air deliver temperature

(iii) Cycle power

(iv) ηiso Neglecting clearance volume

(b) Calculate the volumetric efficiency of the compressor having a cylinder diameter 410 mm and stroke 610 mm. Compressor makes 420 rpm and delivers 30 kg/min of air at 1.01325 bar and 15oC.

Q.9 (a) In an air standard Diesel cycle, compression begins at 103 KPa and 300 K. After compression heat addition is of 545 kJ/kg of air, the peak pressure reached in the cycle is 4.7 MPa. Calculate :

(i) Fuel cut-off ratio

(ii) Compression ratio

(iii) Maximum temperature in the cycle

(iv) Air standard efficiency

Take γ = 1.4, and Cp = 1.004 kJ/kg-K.

(b) A vapour absorption cycle has generator temperature 120oC, evaporator temperature – 10oC and the ambient temperature 30oC. Estimate the maximum possible COP. The actual COP is 0.5 of the maximum COP. If the capacity of the plant is 100 TR, calculate the fuel consumption of the plant. The calorific value of the fuel is 40 MJ/kg.

Q.10 (a) A dense air machine operates on reverse Brayton cycle and is required for a capacity of 10 TR. The cooler pressure is 4.2 bar and refrigerator pressure is 1.4 bar. The air is cooled in the cooler at a temperature of 50oC and temperature of air at inlet to compressor is – 20oC. Determine for the ideal cycle :

(i) C.O.P.

(ii) Mass of air circulated per min.

(iii) Theoretical piston displacement of compressor and expander.

(iv) Net power per tonne of refrigeration.

(b) 28 tonnes of ice from and at 0oC is produced per day in an ammonia refrigeration plant. The temperature range in the compressor is from 25oC to – 15oC. The refrigerator is dry and saturated at the end of compression. If actual COP is 60% of the theoretical COP, calculate the power supplied or required to drive the compressor. Assume latent heat of ice = 335 kJ/kg. Use properties of refrigerant given below :

Temperature oC hf kJ/kg Hg kJ/kg sf kJ/kg-K Sg kJ/kg-K

25 100.04 1319.22 0.3473 4.4852

− 15 − 54.56 1304.99 − 2.1338 5.0585

11

Q.1 (a) Two forces of 10 kg and 15 kg act simultaneously at a point. Find the resultant force, if the angle between the two forces be (i) 30o, (ii) 45o, (iii) 60o, (iv) 90o, and (v) 180o. Draw the force diagram for each situation.

(b) Find the angle between two equal force P, when their resultant is equal to (i) P, (ii) P/2,

(iii) 2 P, (iv) 3 P, and (v) √2 P. Draw the force diagram for each case.

Q.2 (a) A particle is acted upon by three forces equal to 5 kg, 10 kg and 13 kg, along the sides of an equilateral triangle taken in order. Find graphically the magnitude and direction of the resultant force. Find the magnitude and direction of the resultant force if forces are acting along the sides of right angle triangle when one of the angle is 45o.

(b) Four forces of 2, 2.5, 1, and 3 kg are acting simultaneously along straight lines OA, OB,

OC and OD, such that ∠ OAB = 40o, ∠ BOC = 100o and ∠ COD = 125o. Find graphically the magnitude and direction of the resultant force.

Q.3 A body weighing 10 kg, is suspended by two strings AC and BC at the point C. The lengths of the strings AC and BC are 3 metres and 4 metres respectively and the horizontal distance BC is also 4 metres. Find the tensions in the strings AC and BC.

Q.4 (a) A fly wheel of weight 200 N and diameter 20 cm is made to rotate at 10 rotation per second. Determine the K.E. of the wheel. If the frictional couple at its bearing is 10 Nm, determine the number of revolution it will make before coming to rest. If the frictional couple at bearing is double, determine the numbers of revolutions it will make before coming to rest.

(b) A mass of 4 kg moving with a velocity of 10 m/sec along x direction follows another mass of 10 kg moving with 5 m/sec in the same direction. Determine the final velocities of the two masses (i) if e = 0.6, (ii) if the impact is fully plastic, determine also the loss in KE, and (iii) the impact is perfectly elastic, determine the final velocities of the two bodies.

Q.5 A metal rod A of 25 cm length expands by 0.05 cm when its temperature is raised from 0oC to 100oC. Another rod B of a different metal of length 40 cm expands by 0.04 cm for the same rise in temperature. A third rod C of 50 cm length, made up of pieces of rod A and B placed end to end, expands by 0.03 cm on heating from 0oC to 50oC. Find the lengths of each portion of the composite rod C.

Q.6 A shell of weight 7 kN is fired horizontally from a gun weighing 400 kN, with a velocity 450 m/sec. With what velocity will the gun recoil? What will be the average force of resistance to bring it to rest in a distance of 2 meters.


ET 202 (Part A)

ENGINEERING MECHANICS

Note : All questions are compulsory and carry equal marks. This assignment is based on all Blocks of Engineering Mechanics.



BTWRE

12

Q.7 (a) Consider a simply supported beam subjected to a uniformly distributed load on the left half of the span as shown in Figure 1.

Figure 1

Determine SF and BM diagrams for the beam and find the location of the magnitude of maximum BM.

(b) Find the forces in all the members of the truss shown in Figure 2.

Figure 2

Q.8 A boy throws a ball so that it may just clean a wall 3.6 m high. The boy is at a distance of 4.8 m from the wall. The ball was found to hit the ground at a distance of 3.6 m on the other side of the wall as shown in Figure 3. Find the least velocity with which the ball can be thrown.

Figure 3

Q.9 Locate the centroid for plane section shown in Figure 4 and determine the moment of inertia about AB.

Figure 4

4 m 4 m

A B

3 m

3 m

A

B

C

D

E

150 kN

6 m 6 m

A O X

B

α

V

4.8 m 3.6 m

3.6 m

C

A B

12 cm

4 cm

4 cm 4 cm

φ 2 cm

Hole

13

Q.10 Calculate the force required, P, to cause a block of weight W1 to slide under the another block

of weight W2. What will be the tension in the string AB. W1 = 2000 N, W2 = 1000 N, µ = 0.3.

Figure 5

W1

W2

P

A

30o

14


ET 202 (Part B)

PRINCIPLES OF ELECTRICAL SCIENCES

Note : All questions are compulsory and carry equal marks. This assignment is based on all Blocks of Principles of Electrical Sciences.


Course Code : ET-202B Last Date of Submission : July 31, 2012

BTWRE

Q.1 (a) Define the terms electrical energy and electrical power. Give their symbols and units for measurement.

(b) Explain the following terms :

(i) Linearity

(ii) Active circuit

(iii) Depended energy sources

(c) Find the resistance at 20oC of 5 km of copper wire of cross-sectional area 0.5 cm2, if the

specific resistance of copper at this temperature is 17.3 × 10– 9 Ω-m. What would be its

resistance at 40oC if α = 0.0043 peroC?

Q.2 (a) State and explain Thevenin’s Theorem. What is the Thevenin equivalent of an ideal DC voltage source?

(b) State and explain the following :

(i) Superposition theorems

(ii) Maximum power transfer theorems

(c) Explain the working of :

(i) PMMC instrument

(ii) Induction type energy meter.

Q.3 (a) Explain the phenomenon of resonance in series RLC circuit and derive the expression for resonant frequency.:

(b) Explain the terms ‘power factor’. What is the need for power factor correction?

(c) An inductive load draws 1000 W from a 200 V, 50 Hz single phase source. A capacitor

of 25.3 µF connected in parallel with the impedance raises the overall p. f. of the combination to unity. What is the p. f. of the inductive load?

Q.4 (a) Give advantages of negative feedback over positive feedback.

(b) Derive unit step response of a second order system and find its time constants.

(c) Why is the B-wave in a synchronous machine nearly sinusoidal? How is this achieved in a salient pole machine?

15

Q.5 (a) Explain the following with reference to 3-φ systems :

(i) Meaning of phase sequence, and

(ii) Function of ground wire in the supply system.

(b) Explain the following test in a transformer :

(i) Open circuit (O. C.), and

(ii) Short circuit (S. C.) tests.

(c) The following data were obtained on a 20 kVA, 50 Hz, 2000/200 V distribution transformer

Voltage Current Power

(V) (I) (P)

OC test with HV open circuited 200 4 120

SC test with LV short circuited 60 10 300

Draw the approximate equivalent circuits of the transformer referred to HV and LV sides, respectively.

Q.6 (a) Explain the construction and working of a CRO. What are its applications?

(b) How is power measured in a 3-phase circuit using 2-watt meter method?

(c) A balanced 3-phase capacitive load of power factor 0.9 draws 10 A from a 400V, 3-phase supply. Find the readings of the two watt meters.

Q.7 (a) What is the effect of reversing the polarity of supply voltage on the direction of rotation in the case of shunt, series and compound d. c. motors? Comment.

(b) Explain the torque-armature characteristics of :

(i) A d. c. series motor

(ii) A d. c. shunt motor

(c) A 250 V d. c. shunt motor has Rf = 150 Ω and Ra = 0.6 Ω. The motor operates on no-load with a full field flux at its base speed of 1000 rpm with Ia = 5 A. If the machine drives a load requiring a torque of 100 N-m, calculate armature current and speed of motor. If the motor is required to develop 10 kW at 1200 rpm, what is the required value of external series resistance in the field circuit? Neglect saturation and armature reaction

Q.8 (a) Explain construction and working principle of full wave rectifier.

(b) Give an account of numerous applications of semiconductor diodes.

(c) Explain the characteristic of ideal OP-AMP? Is the assumption of virtual ground valid in practical op-amps? Explain the concept of CMRR.

Q.9 (a) What are different kinds of gate and flip-flop? Explain their operation.

(b) What are registers, counters and memories?

(c) Explain the working of an ADC and a DAC.

Q.10 (a) What is bus and bus interface unit?

(b) Classify the various 8085 instructions microprocessor.

(c) How stack is used in 8085 microprocessor?

16


ET 302 (Part A)

COMPUTER PROGRAMMING AND NUMERICAL METHODS

Note : All questions are compulsory and carry equal marks. This assignment is based on all Blocks of Computer Programming and Numerical Methods.


Course Code : ET302A Last Date of Submission : July 31, 2012

BTCM/BTWRE

Q.1 (a) Explain the terms : Truncation and rounding errors, Flow chart, Counters and floating

point arithmetic principles.

(b) Two sides b and c and the included angle A in a triangle ABC are given. Prepare a flow chart and computer pgoramme to compute the third side, the sum of the three sides and the area of the triangle.

(3 + 5 = 8)

Q.2 (a) Prepare a flow chart and Fortran programme to compute and print out the value of cos x by use of series expansion, when x assumes the values from 20o to 80o in steps of 10o. Computational work is to be continued lift the values converge upto four places of decimal.

(b) Use logical IF-statement to write and print the values of f (x) given by :

≤≤+

<≤+=−

5.105.7forlog3cos

5.75.2for3)(

2

xxx

xxexf

x

The values of x varying from 3.2 to 10.2 in steps of 0.5.

(5 + 6 = 11)

Q.3 A rocket is functioned from the ground. Its acceleration u (t) measured every 5 seconds is tabulated below. The velocity and the position of the rocket at t = 40 seconds. Use trapezoidal as well as the Simpson’s rules. Compare also the answers.

t 0 3 10 15 20 25 30 35 40

U (t)

(cm/sec2)

40.0 45.25 48.50 51.25 54.35 59.48 61.50 64.30 68.70

(5)

Q.4 Prepare a program to solve the differential equation :

32.2)1(;sin22 =+= − yxyxedx

dy xy

The program should compute and print the values of y from x = 1.0 to x = 4.6 in steps of 0.2. Use Runge-Kutta method of order 4.

(6)

17

Q.5 Two one-dimensional arrays C and D have 25 elements each. Write a computer program to compute and print this quantities :

2)(;)(

25

1

25

1

∑ −=∑ +=== ii

iiiidCBdCA

∑ ×==

30

1iii

dcP

(5)

Q.6 The value of ψ is to be calculated form the formula :

)tan21(

)sin1(log3

2

xx

xe x

−+

+=ψ −

for x = 2.0 (0.1) 5 .0

Use Do loop, to compute and print ψ for each value of x in the given range.

(6)

Q.7 The resisting force on a body is given by :

)1(

)1()( 2

2 vf

vfvf

l

mvF

+

−+−=

where m = 20.45 and l = 0.83 and

>+−

≤++=

6.0,1

6.0,1)(

3

3

xxx

xxxxf

Using function sub-program for f (x), compute F for v = 0.1 (0.1) 1.0.

(7)

Q.8 (a) Use Euler’s-Maclaurin formula to evaluate the integral :

dxxsin2

0

∫

π

By taking 8

and2

ππ=h . Compare the approximate value with the exact value.

(5)

Q.9 Apply Newton-Raphson formula to prove that the recurrence formula for finding the pth root c is :

11

)1(−+

+−=

pn

pn

nxp

cxpx

Hence, find the value of 3 10 . Show also that Newton’s method has quadratic convergence.

(8)

Q.10 (a) Use Muller’s method to find the root of the equation cos x = ex which lies between 0 and 1.

(b) Using Horner’s method, find the largest real root of 0243 =+− xx correct to three

decimal places.

(5 + 5 = 10)

18

Q.11 A is a square matrix defined by

=

211

010

112

A . Find all its eigen value and the corresponding

eigen vectors. With the supposition that A satisfies its own, characteristic equation, reduce the

polynomial IAAAAAAAA +−+−+−+− 285375 345678 into a quadratic matrix

expression. Hence express the entire expression in terms of a single matrix.

(9)

Q.12 (a) Using Newton’s divided difference formula, evaluate f (x) for x = 8 and x = 15 given :

x 4 5 7 10 11 13

f (x) 48 100 294 900 1210 2028

(b) Employ Bessel’s formula to find the value of F at x = 1.95 given that

x 1.7 1.8 1.9 2.0 2.1 2.2 2.3

F 3.144 3.283 3.391 3.463 3.997 4.491 4.585

(5 + 5 = 10)

Q.13 Given that :

x 1.0 1.1 1.2 1.3 1.4 1.5 1.6

y 7.989 8.403 8.781 9.129 9.451 9.750 10.031

Find 2

2

anddx

yd

dx

dy at x = 1.1 and at x = 1.6.

(5)

Q.14 Given that xy

2xy

dx

dy

+

−=

2

2 and y (0) = 1, find y for x = 0.1 (0.1) 0.5. Use Runge-Kutta method.

(5)

19


ET 302 (Part B)

TECHNICAL WRITING

Note : All questions are compulsory and carry equal marks. This assignment is based on all Blocks of Technical Writing.


Course Code : ET302B Last Date of Submission : July 31, 2012

BTCM/BTWRE

Q.1 (a) Why audience analysis is necessary in technical writing? Discuss in detail the related

aspects?

(b) Describe the three stage process of technical writing.

Q.2 How the following considerations help in good technical writing :

(i) Choice of write words and phrases

(ii) Sentence structure and length

(iii) Use of headings and lists

(iv) Paragraph structure and length

(v) Maintaining coherence within and between paragraphs

Q.3 Discuss in detail various methods in researching.

Q.4 (a) Discuss the salient features of object description.

(b) Keeping in view the salient features of object description, write a description of concrete mixture.

Q.5 Describe different visual elements that may be used in technical writing.

Q.6 (a) Discuss the salient features of a manual.

(b) Prepare a manual for Vicat’s apparatus for testing of cement.

Q.7 (a) Describe formal elements of a report.

(b) A primary school building is under construction. Last night the newly constructed wall collapsed. You are given responsibility to visit the site and give a report. Write the incident report based on your observations.

Q.8 (a) Discuss the purpose and essential features of a progress report.

(b) Prepare a proposal for construction of a community centre in your locality.

Q.9 Prepare ten slides for oral presentation on the topic “low cost housing”. Use various graphic aids to make the presentation more effective.

Q.10 (a) Discuss various elements of an article.

(b) Write a technical article on “energy efficient building materials”.

20


ET 502 (Part A)

STRENGTH OF MATERIALS

Note : All questions are compulsory and carry equal marks. This assignment is based on all Blocks of Strength of Materials.



BTCM

Q.1 A steel bar is placed between two copper bars each having the same area and length as the steel bar at 15oC. At this stage they are rigidly connected together at both the ends. When the temperature is raised to 315oC, the length of bar increases by 1.50 mm. Determine the original

length and the final stresses in the bars. Take Es = 2.1 × 105 N/mm2; Ec = 1 × 105 N/mm2,

αs = 0.000012 peroC, αc = 0.0000175oC.

Q.2 A compressed tube is made by striking a thin steel tube on a thin brass tube. As and Ab are the sectional areas of steel and brass tubes, and Es and Eb are the corresponding values of Young’s Modulus. Show that for any tensile load the extension of the compound tube is equal to that of a single tube of the same length and total cross-sectional area, but having a Young’s modulus of (Es As + Eb Ab) / (As + Ab).

Q.3 A square prism of wood 50 mm × 50 mm in cross-section and 300 mm long is subjected to a tensile stress of 40 N/mm2 along its longitudinal axis and lateral compressive stresses of 20 N/mm2 on one set of lateral tensile stress of 10 N/mm2 on the other set of lateral forces. Find the changed dimensions of the prism. Take Poisson’s Ratio = 0.4 and Modulus of

elasticity of 1.5 × 104 N/mm.

Q.4 The intensity of loading on a simply supported beam of 5 meters span increases uniformly from 8 kN/m at the one end to 16 kN/m at the other end. Find the position and magnitude of the maximum bending moment. Also draw shear force and Bending Moment diagram.

Q.5 Two rectangular plates, one of steel and other of brass each 37.5 mm by 10 mm are placed together to form a beam 37.5 mm wide and 20 mm deep on two supports 750 mm apart. The brass component being on the top of the steel component. Determine the maximum central load if the plates are :

(i) separate and can bend independently, and

(ii) firmly secured throughout their length.

Permissible stresses for brass and steel are 70 N/mm2. Take Eb = 0.876 × 105 N/mm2 and

Es = 2.10 × 105 N/mm2.

Q.6 ABCDE is a continuous beam supported at B, C and D. The over hand AB and DE are each of length x. The span BC and CD are each of length l. The beam carried a uniformly distributed load of w per unit run over the whole length. Find the over hang length x for the condition, the reaction at the three supports are equal.

Q.7 A hollow steel shaft 240 mm external and 160 mm internal diameter is to be replaced by a solid alloy shaft. If the both shafts should have the same polar modulus, find the diameter of

the latter and the ratio of the torsional rigidities. Take C for steel = 2.4 × C for alloy. If alternatively, the two shafts should have the same torsional rigidity, find the ratio of their polar moduli.

21

Q.8 A composite shaft 7.50 m long consists of a steel shaft 250 mm in diameter surrounded by a closely fitting 30 mm thick bronze tube. If the shear in the steel shaft shall not exceed 15 N/mm2, find the maximum power transmitted by the shaft at 250 rpm.

Take Cs = 8.5 × 104 N/mm2 and Cb = 4.2 × 104 N/mm2.

Q.9 Two planes AB and BC, which are at right angles, carry shear stress of intensity 17.5 N/mm2 while these planes also carry tensile stress of 70 N/mm2 and compressive stress of 35 N/mm2 respectively. Determine the principal planes and principal stresses. Also determine the maximum shear stress and the planes on which it acts.

Q.10 If a circular shaft is subjected to an axial twisting moment T and a bending moment M, show that when M = 1.2 T, the ratio of maximum shearing stress to the greatest principal stress is approximately 0.566.

22


ET 502 (Part B)

STRUCTURAL ANALYSIS

Note : All questions are compulsory and carry equal marks. This assignment is based on all Blocks of Structural Analysis.



BTCM

Q.1 (a) Draw the influence line diagrams for the following components in the structure given in Figure 1(a).

Figure 1(a)

(i) Reaction at A (ii) Reaction at B

(iii) Reaction at D (iv) Bending Moment at P

(i) Bending Moment at B (vi) Shear at P

(b) Draw the influence line diagram for the forces in members CD, CJ, JK and JD of the truss shown in Figure 1(b). The load is moving on a floor system resting on the bottom cord of the truss.

Figure 1(b)

Find the maximum value of the axial force in the members due to the following moving load systems :

(i) A 2 m long moving uniform load of 12 kN/m.

A

B C D E F

G

H I J K L

6 @ 4 m

3 m

A B P D C

2 m 2 m 2 m 1 m

23

(ii) A long uniform moving load of 10 kN/m and a concentrated moving load of 80 kN.

(iii) A truck traveling from left to right, the wheel loads of which are shown in Figure 1(c).

Figure 1(c)

Q.2 (a) A three hinged unsymmetrical parabolic arch has a span of 90 m. The right hand springing B is 10 m above the left hand spring A. The crown C is at 50 m from A and 30 m above it. It carries two loads of 50 kN and 100 kN at 10 m and 20 m from left hand support A respectively and a uniformly distributed load of 3 kN/m over the right portion starting from crown C. Draw the bending moment diagram and calculate the bending moment, normal thrust and radial shear at a point 30 m from the right hand support.

(b) Find the forces in the members of the truss shown in Figure 2.

Figure 2

Q.3 (a) Analyse the continuous beam shown in Figure 3(a) using Three Moment Theorem and draw bending moment and shear force diagrams.

Figure 3(a)

A

B C D E

F

G H I J

5 @ 4 m

3 m

80 kN

120 kN

2 m 4 m 4 m 2 m

2 kN/m 20 kN 15 kN 50 kN

EI = Constant

80 kN 50 kN 120 kN 60 kN

2 m 2 m 2 m

24

(b) For a continuous beam shown in Figure 3(b), the support A settles by 5 mm and support B settles by 2 mm along with the given loading. Draw the bending moment diagram for the beam.

Figure 3(b)

Q.4 (a) Analyse the plane frame shown in Figure 4(a) using slope deflection method and draw bending moment diagram.

Figure 4(a)

(b) Analyse the plane frame shown in Figure 4(b) using moment distribution method.

Figure 4(b)

EI = Constant = 72 × 103 kNm

2

3 kN/m

80 kN 50 kN

2 m 2 m 2 m 1 m 1 m

A B

C

3 m

4 kN/m

100 kN

EI = Constant

A

B C

D

4 m

4 m

80 kN

120 kN

A

B C

D

1 m 1 m

2 m 3 m

4 m

25

Q.5 (a) Analyse the plane frame shown in Figure 5(a) using symmetry concept.

Figure 5(a)

(b) Analyse the plane frame shown in Figure 5(b).

Figure 5(b)

Q.6 (a) Find the deflection at B and slope at C in the beam shown in Figure 6(a) for the given loading.

Figure 6(a)

(b) Find the slope and deflection at point C and slope at Point B in the beam shown in Figure 6(b) for the given loading.

Figure 6(b)

4 kN/m

A

B C

D

5 m

2 m

2 m

80 kN

80 kN

A

B C

2 m 2 m

120 kN

3 m

3

6 m

I

2 I

2 m 2 m

4 kN/m

A

B

C

I 2 I

50

1 m 1 m 2 m

3 kN/m 30 kN

A

EI = Constant

B

C

Hing

26

A B

C

P

W

Q.7 (a) Analyse the portal frame shown in Figure 7(a) using unit load method.

Figure 7(a)

(b) A semi-circular arch of uniform flexural rigidity, having one end hinged and other end placed on roller subjected to a horizontal force P and a vertical downward load W at crown C in the middle. Find the horizontal displacement of the roller end.

Figure 7(b)

Q.8 Generate flexibility and stiffness matrix for the plane frame shown in Figure 8.

Figure 8

3 m

EI = Constant

A

B

C

D

4 m

5 m

70 kN

A

B C

D

2 m 3 m

4 m 5 m

2 I

2 I

I

27

Q.9 (a) A suspension bridge cable of span 100 m and central dip 10 m is suspended from the same level at two towers. The bridge cable is stiffened by a three hinged stiffening girder which carries two concentrated loads of 30 kN each at points 20 m from both the ends. Draw the Bending Moment and Shear Force diagrams for the girder and calculate the maximum tension in the cable.

(b) A hollow cylindrical cast iron column 175 mm external diameter and 25 mm thick is 7 m in length having both ends hinged. Find the load using Rankine’s formula. Compare this load with that given by Euler’s formula.

Take σc = 550 N/mm2 and a = 1/160.

Q.10 Find the collapse load for the frames and beam shown in Figure 9.

Figure 9(a)

Figure 9(b)

Figure 9(c)

2 m 2 m 1 m

2 MP MP

2 P P/2

A

B C

D

2 m 2 m

4 m

3 m

4 m

2 m

3 m

P 5 m

MP

2 M

MP

3 P

5 m

A

B

C

D

2 m

4 m

3 m

P

5 m

3 M

2 M

MP

4 P

28


ET 505

TRANSPORTATION AND TRAFFIC ENGINEERING


Blocks of Transportation and Traffic Engineering.


Course Code : ET 505 Last Date of Submission : July 31, 2012

BTWRE

Q.1 (a) Discuss the principal modes of transport and their shortcomings in India.

(b) Enumerate the various characteristics that determine the choice of a transport system.

Q.2 (a) Describe in brief the “System Approach to Transport Planning” with the help of a block diagram.

(b) There are two projects as detailed below :

Initial Cost of Construction

Annual Maintenance Cost

Project A Rs. 12.5 crores Rs. 10 lakhs

Project B Rs. 15 crores Rs. 3.0 lakhs

What is the more economical alternative over a period of 20 years, the interest rate being 10 percent?

Q.3 (a) How are roads classified in India today?

(b) According to 1991 census, the area of a State in India was 300000 km2. The number of towns with population above 5000 was 600. The total number of towns and villages was 36000. Determine the length of various road categories.

Q.4 (a) Determine the length of a summit curve at a Junction of two gradients of 2.5% and –1.5% for :

(i) Stopping sight distance of 120 m, and

(ii) Overtaking sight distance of 340 m.

(b) Calculate the safe overtaking distance for a two lane Major District Road in plain terrain. Assume :

(i) Design speed : 80 km/hr

(ii) Hesitation time : 3.5 seconds

(iii) Overtaking time : 10 seconds

(iv) Safety distance : 50 m

29

Q.5 (a) Explain in brief the consideration that govern the selection of the alignment of a railway line.

(b) Calculate the radius of the turnout curve for a turnout 1 in 8½. Calculate the value of lead. The heat divergene is 140 mm. The gauge is broad gauge. The front straight leg of the vee-crossing is 900 mm. the switch angle is 1o35’.

Q.6 (a) What is skidding in pavements? Describe the factors which are responsible for causing a skid prone road surface.

(b) Design a dowel bar system for the following conditions :

(i) Design wheel load : 50 kN

(ii) Design load transfer : 40%

(iii) Slab thickness : 20 cm

(iv) Joint width : 2 cm

(v) Permissible flexural stress in dowel bar : 140 MN/m2

(vi) Permissible shear stress in dowel bar : 100 MN/m2

(vii) Permissible bearing stress in dowel bar : 10M N/m2

(viii) K value of sub grade : 70M N/m3

(ix) E : 30000 MN/m2

(x) µ : 0.15

Q.7 (a) Discuss in brief the various factors responsible for causing road accidents.

(b) The speed and concentration of vehicles in a traffic stream are observed to be as under :

Speed (km/m) 35 45 55 65 75 85

Concentration (veh/km) 75 63 50 35 23 10

Fit a linear regression equation? Find out the r2, the coefficient of determination.

Q.8 (a) What is the purpose of providing traffic signals at road intersections?

(b) A fixed time 2-phase signal is to be provided at an intersection having four arms. The design hour traffic and saturation flow are as under :

North South East West

Design Hour Flow (q) 750 350 700 550

Saturation Flow (s) 2150 1750 2800 2750

The time lost per phase due to starting delays is 2 seconds. Calculate the optimum cycle time. Allocate the green times to the two phases.

Q.9 (a) Describe the various elements of the cost of an Airport project. Also explain the economic analysis of an Airport project.

(b) Determine the thickness of concrete overlay as per the US Corps of Engineers method over an existing concrete pavement 20 cm thick which is in a good condition. It has been found that a new slab of 25 cm is needed. Also calculate the overlay thickness if the slab is badly cracked.

Q.10 (a) Discuss the importance of ship dimensions in the design of port facilities.

(b) A berth, 300 m long, caters to a ship carrying 48000 tonnes of cargo. Design a transit shed to accommodate the cargo. The cargo has a weight of 1.5 tonne per m3.

30

Q.1 A masonry dam 8 m high, 1.5 m wide at the top and 5 m wide at the base, retains water to a depth of 7.5 m. The water face of the dam is vertical. Find the normal stress at the toe and heel of the dam. Assume unit weight of masonry as 2.2 t/m3 and water as 1 t/m3.

Q.2 (a) “The selection of suitable site plays a vital role in construction of any type of dam”. Discuss with suitable examples.

(b) How do waves affect the stability of dam? Draw pressure-wave diagram and show that total pressure Pw and the moment Mw is given by :

22w wP h= γ

23

4w wM h

= γ

where, hw = height of waves from trough to crest, and

γ = density of water.

Q.3 What do you mean by the arbitrary profile of a gravity dam? Considering the effect of hydrostatic pressure and uplift pressure, show that the base width (b) of the arbitrary profile of

the gravity dam for no tension to occur can be written as 1

hb

s c=

−.

Q.4 (a) What do you mean by reservoirs? Explain various types of reservoirs in brief

(b) Briefly discuss the empirical relations for estimating sedimentation rate of Indian rivers.

Q.5 (a) What do you mean by hydraulic jump? Explain its significance.

(b) Discuss in detail as to how you will test the stability of an earth dam constructed with

C – φ soils.

Q.6 (a) What are “earth dams” and under what circumstance are they preferred?

(b) What are the precautions that you would take while constructing an earth dam?

Q.7 (a) What are different types of weirs? Explain with neat sketches the circumstances under which each type is adopted.

(b) Briefly explain the salient features of Khosla’s theory and how it is used in the design of permeable foundations.


ET 536 (Part A)

HYDRAULIC STRUCTURE-I


Blocks of Hydraulic Structure-I.


Course Code : ET 536 A Last Date of Submission : July 31, 2012

BTWRE

31

Q.8 (a) What is meant by a “spillway” and what is its necessity in dam construction?

(b) A horizontal rectangular channel 1.6 m wide carries a discharge of 1.65 m3/sec. Determine whether a Jump may occur at an initial depth of 0.2 m or not. If a Jump occurs, determine the sequent depth of this initial depth. Also determine the energy loss in the Jump

Q.9 Differentiate between the following :

(i) Overflow dam and Non-over flow dam

(ii) Firm yield and Design yield

(iii) Diversion and Storage Headworks

(iv) Weir and Barrage

Q.10 Write short notes on the following :

(i) Uplift pressure

(ii) Multipurpose Reservoir

(iii) Fish Ladder

(iv) Exit Gradient

32


ET 536 (Part B)

HYDRAULIC STRICTURE-II

Note : All questions are compulsory and carry equal marks. This assignment is based on all Blocks of Hydraulic Structure-II.



BTWRE

Q.1 (a) What do you mean by Permanent Canals? Explain in brief the advantages and disadvantages of such canals.

(b) Describe various functions served by “Berms”.

Q.2 (a) Draw typical cross-sections of canal in :

(i) Cutting,

(ii) Filling, and

(iii) Partial cutting and filling.

(b) Discuss in brief the general principles of alignment of a canal.

Q.3 (a) Starting from basic equation, show that :

1.5

0.54980

fS

R=

(b) Design an irrigation channel to carry 40 cumecs discharge at a slope of 1 in 5000. Assume :

Manning’s N = 0.0225

Kennedy’s m = 0.9

Side slopes = 1/2 : 1.

Q.4 (a) How will you justify economically the necessity of lining an existing canal? What added benefits you will expect if the canal to be lined is new and yet to be constructed?

(b) Design a trapezoidal concrete lined channel to carry a discharge of 350 cumecs at a slope of 1 in 6400. The side slopes of the channel may be taken as 1½ : 1. The value of

Manning’s (N) for the lining material may be taken as 0.013. Assume the limiting D

B ratio

to be 5.

Q.5 (a) What is an outlet? Write down the requirements that an outlet should fulfill.

(b) What do you understand by flexibility of an outlet? Derive an expression for the same.

Q.6 (a) What is a cross regulator and what are its functions?

(b) What are the components of a silt ejector and where is it located? Explain the functions of a silt ejector.

33

Q.7 (a) What are the various types of Falls commonly adopted on canals? Discuss the suitability of each type.

(b) Design a Sarda type fall across a canal for the following data :

(i) Discharge = 15 cumecs

(ii) Drop = 1.0 m

(iii) Depth of flow = 1.8 m

(iv) Bed width = 10 m

Q.8 (a) Explain the various steps which are followed in the design of spurs relevant to a given situation.

(b) Describe the layout and cross-section of guide bunds.

Q.9 Write short notes on the following :

(i) Design parameters of cross drainage work

(ii) Level Crossing

(iii) Components of a diversion headworks

(iv) Stabilisation of a river channel

Q.10 Differentiate between the following :

(i) Lined and Unlined Canal

(ii) Aqueduct and Syphon Aqueduct

(i) Lacey’s and Kennedy’s theory of canal design

(ii) Flexibility and Sensitivity

34


ET 540 (Part B)

FLOW IN OPEN CHANNEL

Note : All questions are compulsory and carry equal marks. This assignment is based on all Blocks of Flow in Open Channel.



BTCM

Q.1 (a) How Open Channel flow is different from Pipe flow?

(b) What do you understand by piezometric head?

(c) Differentiate between the following :

(i) Steady Flow and Unsteady Flow

(ii) Uniform Flow and Varied Flow

Q.2 (a) Define the following :

(i) Prismatic Channel

(ii) Hydraulic Radius

(iii) Hydraulic Depth

(iv) Section Factor

(b) A trapezoidal channel is 4.0 m wide at bottom and side slope is 1.5 H : 1 V. If the depth of flow is 1.5 m then find the area of flow, wetted perimeter, hydraulic radius, hydraulic depth and section factor.

(c) A circular conduit (d0 = 2.5 m) has a depth of flow (i) 1.5 m, and (ii) 0.8 m. In each case find area of flow, wetted perimeter, hydraulic radius, hydraulic depth and section factor.

Q.3 (a) What is energy grade line? How is it different from hydraulic grade line?

(b) Define Reynold number and Froude number. Give the classification of flow in an open channel based on these numbers.

(c) A flow of 100 litres per second flows in rectangular flume of width 0.60 m and having adjustable bottom slope. If Chezy’s C is 55, determine the bottom slope necessary for uniform flow with a depth of flow 0.30 m. Also find the conveyance and the state of flow (sub-critical or supercritical).

Q.4 (a) In a rectangular channel 3.5 m wide laid at a slope of 0.0036, a uniform flow occurs at a depth of 2.0 m. Find how high can the hump be raised without causing afflux? If the upstream depth of flow is to be raised to 2.5 m, what should be the height of the hump? Take Manning’s n = 0.015.

(b) For the purpose of discharge measurement, the width of a rectangular channel is reduced gradually from 3 m to 2 m and the floor is raised by 0.30 m at a given section. When the approaching depth of flow is 2 m, what rate of flow will be indicated by a drop of 0.15 m in the water surface elevation at the contracted section?

35

Q.5 (a) What are alternate depths of flow? Show that relation between the alternate depths y1 and y2 in a rectangular channel can be expressed by :

2 2

3 1 2

1 2

2

2c

y yy

y y=

where yc is the critical depth of flow.

(b) A trapezoidal channel having bottom width 5m and side slope 1 : 1 carries a discharge of 12 m3/s. Compute the critical depth and the critical velocity. If Manning’s n = 0.02, determine the bottom slope required to maintain the critical depth.

Q.6 (a) A horizontal rectangular channel 4 m wide carries a discharge of 16 m3/s. Determine whether a jump may occur at an initial depth of 0.5 m or not. If a jump occurs, determine the sequent depth to this depth. Also determine the energy loss in the jump.

(b) Show that the head loss in a hydraulic jump formed in a rectangular channel may be expressed as :

3

1 2

1 2

( )

2 ( )

V VE

g V V

−∆ =

+

Q.7 (a) State and discuss the assumptions made in the derivation of dynamic equation for gradually varied flow. Starting from the first principle show that for a wide rectangular channel the slope of water surface can be expressed as :

3

0 3

1

1

n

c

y

ydyS

dx y

y

− = =

−

(b) A trapezoidal channel (z = 2) expands uniformly from a bottom width of 12.0 m at its upstream end to a width of 21.0 m at its downstream end over a distance of 1.5 km. The depth of flow at the down stream end is 4.8 m when the channel carries Q = 85 m3/s. The average slope of the river in this reach is 10– 3 with n = 0.019. Calculate the depth of flow at distances : 1.5, 1.0, 0.75, 0.6, 0.3 and 0.2 km from the downstream end of the reach towards the upstream. Use standard step method.

Q.8 (a) Show that hydraulic exponent (M) for critical flow conditions can be expressed as :

3y A dT

M TA T dy

= −

(b) Prove that the value of hydraulic exponent (M) is 3 and 5 for rectangular and triangular channel respectively.

Q.9 (a) Using the basic differential equation of Gradually Varied Flow (GVF), show that dy

dx is

positive for S1, M3 and S3 profiles.

(b) Sketch the GVF profiles produced on the upstream and downstream of a sluice gate introduced in a (i) steep slope, (ii) mild slope, and (iii) horizontal bed channel.

(c) Recognize the flow profiles for the following cases of flow situations :

(i) a mild channel ending with a drop,

(ii) an adverse channel discharging into a mild channel,

(iii) a mild channel discharging into a reservoir, and

(iv) a critical channel breaking into a mild channel.

36

Q.10 (a) In a very long wide rectangular channel the discharge intensity is 3.0 m3/s/meter width. The bed slope of the channel is 0.004 and the Manning’s n = 0.015. At a certain section in this channel the depth of flow is observed to be 0.90 m. What type of GVF profile occurs in the neighbourhood of this section?

(b) A trapezoidal channel having B = 6.0 m, m = 2.0, So = 0.0016 and n = 0.02 carries a discharge of 12 m3/s. Calculate and plot the back water produced due to the operation of a sluice gate at a downstream section which backs up the water to a depth of 4.0 m immediately behind it. Use varied-flow functions.

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