Additional problems ebook

2

For isothermal compressible flow through tubes, the mass flow rate G has been shown to be : đ??ş = The 2đ??ş

đ?‘‘đ??ş đ?‘‘đ?‘?2

maxima =

in

this

function

đ?‘€ đ?‘?

2đ??żđ?‘“ 1 ��đ?‘™đ?‘› 1 + ďż˝.[−2đ?‘?2 ]âˆ’ďż˝đ?‘?12 −đ?‘?22 ďż˝(− )ďż˝ 2đ?‘…đ?‘‡ đ?‘?2 đ?‘‘ đ?‘?2 đ?‘?

2đ??żđ?‘“ ďż˝đ?‘™đ?‘› 1+ ďż˝ đ?‘?2 đ?‘‘

=0

is

obtained

by

differentiating

which on simplification gives : G2 = p 2 2.(M/RT) = V s2 . Ď 2 , and the tube is choked.

Can you do the same development for adiabatic reversible flow ?

đ?‘€ ďż˝đ?‘?2 −đ?‘?22 ďż˝ 2đ?‘…đ?‘‡ 1 đ?‘?1 2 đ??ż đ?‘“ ďż˝đ?‘™đ?‘› + ďż˝ đ?‘?2 đ?‘‘

w.r.t.

p2

:

EverNote: 9/6/2008 12:49 PM

EverNote: 9/6/2008 12:49 PM

EverNote: 9/6/2008 12:49 PM

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EverNote: 9/6/2008 12:49 PM

2

In a ladle holding liquid steel (density ρs), a bubble of argon of size d0 is introduced at the bottom. The temperatures of the steel and the gas is constant at T,K. The ladle is kept in a vacuum chamber where the pressure is pT. (a) Write an expression for the diameter of the bubble as a function of height h. [1] (b) As the bubble expands during its ascent, it does P-V work on the liquid. Calculate the PV work done by one such bubble during its entire travel through the bath. [2] (c) The bubble does additional work on the liquid, since bubble moves against the drag force exerted by the liquid. If one assumes the bubble travels at the terminal velocity throughout, the drag force is equal to the net buoyancy force. What is the work done against this drag force ? [3] (d) Show that the total work done on the liquid expressed as J/moles of Ar blown is [1]

2 ln ...

T T steel

p RT pρgH  +

where T is the temperature of the steel.

•(a) A nitrogen bubble moves up settling steadily in liquid steel ( near the surface) at the rate of 10 mm/s. What is its diameter? (Note: You need to test the regime of flow) (5 marks) 1 page (b) If this is the diameter near the surface, what is its diameter at the bottom of a 2m high bath ? What is the settling rate here ?

The ambient pressure above the steel surface is 100 Pa. (3 marks) 1 page.

For 0.1<Re<5

is a reasonable approximation.

Steel : density 7000 kg/m3; viscosity 0.006 Pa.s.

Capillary viscometer : A common type of viscometer for liquids consists of a relatively large reservoir with a very slender vertical outlet tube at the bottom, the rate of outflow being determined by timing the fall in the surface level. If oil of constant density flows out of the viscometer at the rate of 0.273x10-6m3/s, what is the kinematic viscosity of the fluid ?

Can you suggest an experimental method to determine the diameter of fine capillaries drawing inspiration from the above experiment ? End effects need to be eliminated.

1. (a) In a continuous slab caster casting at 1m/min, the wide face mold consists of a copper plate of 20 mm thickness and 1.25m wide(exposed surface), backed by a steel plate, with a slots for the water to flow as shown in the first figure. The typical heat flux profile in the mold is given in the second figure. Heat transfer coefficients for slots may be approximated by the use of correlation for tube heat transfers, with appropriate equivalent diameters ( hydraulic radius concept). For turbulent flow in pipes the following correlation may be used : Nu = 0.023.Re0.8.Pr0.4 Calculate the minimum velocity of water flow in the mold, so that at no place in the cold surface of copper plate, the temperature exceeds about 100 C, if the bulk water temperature is 20C. ( 5 marks) (b) If the height of the mold is 600 mm, what pressure drop would you expect ? Remember: for flow thru a tube one can show, Î&#x201D;P/Ď = (4L/D).(V2/2).f. Take a conservative friction factor f value of 0.05.

(2 marks)

Data : Water : Density = 950 kg/m3, Viscosity ~ 0.37e-3 Pa.s, Cp ~ 4195 J/kg, k ~ 0.67 W/m.K, Pr ~ 2.29.

EverNote: 8/13/2008 10:42 AM

EverNote: 8/13/2008 10:42 AM

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EverNote: 8/13/2008 10:42 AM

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3

A vacuum chamber, where an oxygen-evolving reaction is to be performed, is to maintained at 1 mbar pressure.

It is estimated that the total amount of gas to evacuated (which is the sum of gas evolution and leakage rates), is about 2x10-4 kg/s. The vacuum pump can only maintain a pressure of 0.4 mbar at its inlet, and the pipe connecting the vacuum chamber to the vacuum pump should at least be 3m long (assume friction factor f to be 0.015, as a conservative estimate). What should be the minimum diameter of the pipe ? Assume isothermal conditions at 300K. [ You need to confirm that the pipe is not choked].

A vacuum chamber, where an oxygen-evolving reaction is to be performed, is to maintained at 1 mbar pressure.

It is estimated that the total amount of gas to evacuated (which is the sum of gas evolution and leakage rates), is about 2x10-4 kg/s. The vacuum pump can only maintain a pressure of 0.4 mbar at its inlet, and the pipe connecting the vacuum chamber to the vacuum pump should at least be 3m long (assume friction factor f to be 0.015, as a conservative estimate). What should be the minimum diameter of the pipe ? Assume isothermal conditions at 300K. [ You need to confirm that the pipe is not choked].

1.

It is desired to roast 60 kg/min of copper pyrites particles in a fluidized bed . Pyrites is continuously added and withdrawn. Air at 1000 K is to be used for fluidization as well as roasting .

a.

Calculate the limits for the superficial velocity .

GivenD p = 10-4 m, density of copper pyrites = 4000 kg/m3 density of air at 1000 K = 0.43 kg/m3 viscosity of air at 1000 K = 30x 10-6 Pa.s, ε in packed bed = 0.4 .

SOLUTION : Minimum fluidization velocity : = (ρ p – ρ g ) (1-ε) g ( weight of bed of 1m2 area and 1 m height) 2531250V o + 70547 V o 2 = 23542 V o = 9.3x10-3 m/s.

Terminal velocity : Assuming Stokes regime : π/6 (d)3 (ρ p – ρ g ) g = 6 π . µ . d/2 . V t V t = d2 . (ρ p – ρ g ) g /(18 . µ ) = 0.726 m/s; Re = 1.0406 Stokes law not valid ?

Iterate : π/6 (d)3 (ρ p – ρ g ) g = πd2/4 . ρ g V t 2 .f 12.166/f = V t 2, V t = 0.78 The limits for superficial velocity are 9.3x10-3 and 0.78 m/s.

b.

V Re f 0.726 1.04 20 0.78 1.118 ~20

Show how a value for superficial velocity can be fixed if the temperature rise of the exit gases due to heat liberation in roasting is not to exceed a given value . Roasting is a process where Cu 2 S is oxidized to Cu 2 O.

Down a plane, which makes an angle of θ with the vertical, water flows as a film. The entire system is in still air. Perform a shell force balance in the fully developed flow regime, and obtain (a) an expression for the shear stress variation across the thickness of the water film (2 marks) (b) an expression for the velocity profile (2 marks) (c) list the assumptions made (1 marks) (d) What is the average velocity, to an accuracy of 0.1% ? The inclination of the tube is 60o to the vertical and the film thickness is 10 mm. (1 mark). (e) If the plate is moving up ( parallel to the plate) with a velocity Vo, what is the velocity profile?

(1 mark) (f) If the liquid flowing down the plane is non-Newtonian (power law), get the expression for the velocity profile (2 marks) ½ page

a. A cylindrical tank with the bottom side open and top side closed, weighing 100kg, is submerged in water as shown. The local barometer reading is 1.013x105 Pascals. The thickness of the tank may be neglected. What additional force need to be put on top of the tank to bring the top of the tank flushwith the water surface ?

3 marks

b. At this position the tank suddenly ruptures and a 10 mm diam. orifice is created at the top surface. What is the initial rate of air flow through the opening ?

3 marks.

3 -6 Air : density 1.18 kg/m ; Viscosity : 18x10 Pa.s.

Down a plane, which makes an angle of θ with the vertical, water flows as a film. The entire system is in still air. Perform a shell force balance in the fully developed flow regime, and obtain (a) an expression for the shear stress variation across the thickness of the water film (2 marks) (b) an expression for the velocity profile (2 marks) (c) list the assumptions made (1 marks) (d) What is the average velocity, to an accuracy of 0.1% ? The inclination of the tube is 60o to the vertical and the film thickness is 10 mm. (1 mark). (e) If the plate is moving up ( parallel to the plate) with a velocity Vo, what is the velocity profile?

(1 mark) (f) If the liquid flowing down the plane is non-Newtonian (power law), get the expression for the velocity profile (2 marks) ½ page

For the system shown in the figure calculate the velocity at sections 2 and 3, pressure at 4 and the wattage of the fan (efficiency 0.6). The pipes can be assumed to be hydraulically smooth. (6 marks) 2pages Water : density 1000 kg/m3; viscosity 0.001 Pa.s.

Air : density 1.2 kg/m3; Viscosity : 18x10-6 Pa.s.

friction factor in the turbulent region :hydraulically smooth pipes: the Blassius formula : f = 0.071/(Re)0.25.

A particle of density 2500 kg/m3 is settling steadily in water at the rate of 10 mm/s. What is its diameter? (4 marks)

Fanning friction factor( drag coefficient ) as function of Reynolds Number given in the figure. (Based on crosssectional area of the sphere). Re = ρV∞d/µ For 0.1<Re<2, C = (24/Re)[1+(3/(16)Re] is a good approximation.

1. (a) I am building a fluidized bed of area 0.1m2 and would like to use it for fluidizing 100 kg of some powder using air at room temperature (300K). Exit of the fluidized bed is open to atmospheric pressure. What is the pressure drop Î&#x201D;P fluid across the bed ? (b) For successful operation of the bed I have been told that I need a gas distributor at the bottom. This should offer a resistance such that the pressure drop in the distributor is at least 10% of the drop in the fluidized bed (0.1 Î&#x201D;P fluid ) . Since I do not have any specific material for this, I plan to build the distributor as a packed bed consisting of spherical particles of diameter 1 mm and pack them carefully, in a hexagonal close-packed arrangement (Îľ = 0.26). Calculate how high the bed should be if the superficial velocity desired is 0.50 m/s ? (c) What is the minimum density of the material for the spheres such that this bed itself does not show tendency for fluidization ? (d) The fluid bed is supplied air from a compressor with a supply tube of 100 mm diameter and 20m length ( assume hydraulically smooth). What is the pressure at the exit of the compressor ? Neglect losses in entrance/exit/pipe fittings.

2.

3.

4.

5.

4.8

• In a gas adsorption experiment a viscous fluid flows upward through a small circular tube and then downward as film on the outside. Set up a momentum balance over a shell of thickness Δr in the film as shown in the figure.

Show that the velocity distribution in the falling film (neglecting end effects) is : (4 marks) 2 page max

•Write an expression for the volume rate of flow in the film. You need not integrate the expression. (2 marks) ½ page •If the accuracy of the flow rate has to be better than 0.1%, how would you interpret ρ in the above expression ? (1 mark) few lines only

The desktop PC in my lab has several large IC chips. One of them has a top surface of 50x50mm. A heat sink made of aluminium is pasted on top. The heat sink consists of a 11x12 grid of 1x3 mm fins of 35mm height standing upright (schematic shown). I can let the IC temperature (assume same as heat sink base temperature) to be 50oC. The room temperature is about 27oC, and the heat transfer coefficient must be about 20W/m2.K. What is the temperature of the tip of the fins ? How much heat is being dissipated by the heat sink ?

You may NOT assume adiabatic condition at the tip of the fin, without serious loss of rigour. kAl = 200 W/m.K.

1.

(a) Water is flowing freely as a film down a wide inclined plane. The thickness of the water film is small compared to its width. Perform a shell balance for the fully developed flow regime of the falling film of water and get an expression for the flow rate as a function of the thickness and the slope. List your assumptions and discuss them. Neglect the effect of the side walls. [BSL 2.2] How would you define a friction factor in such a case, and what would be the expression for it ?

Discuss the boundary condition of zero shear stress at the free surface and its validity (b) In the above problem, the water is flowing in a slit like tunnel as shown in the figure, and the water fills half the slit. Can you repeat the shell balance without assuming that air offers negligible resistance at the surface to the flow of water. (Neglect entrance exit effects). When is the assumption made in the previous problem reasonably valid?

You need to find the velocity profile in the air layer too.

What is the rate at which air is delivered at the exit?

A ladle of molten steel has a refractory tube at the bottom through which the liquid flows out, as shown in the figure. (a) Calculate the initial rate of metal flow, kg/s.

[2] (b) How much time would be needed for the height of the metal to drop by 1 m?

[3] (c) Calculate the pressure at 2 just as the flow starts and determine whether there is a possibility of atmospheric air being sucked in at this point if the refractory is porous.[2] Ď steel = 7000 kg/m3, steel = 0.006 Pa.s Additional frictional effect at entrance to tube is equivalent to 5 x diam.

Tube roughness, _ = 0.125mm; For rough tubes , 1/_f = 4.0 log10 (d/ _) + 2.28

1.

I have just seeded a new 2400 m2 lawn and the gardner recommends that I sprinkle it with 8 L/m2/day of water to make it grow properly. To do this I bought a pump, 25m of 25mm-i.d. plastic hose (smooth inside), and a sprinkler, which then are all connected. The pump draws water from a stream whose level is 3 m below the pump, the water intake is a large polyethylene pipe (negligible resistance) and the lawn is 9m above the pump. The pump is run for about 7 hr/day, its outlet pressure gauge reads 5.5bar, my electricity receipts show that I pay â&#x201A;š5.00/kWh and I must sprinkle for 32 days to develop a healthy lawn, so says the gardner. If the daytime temperature of water (when I plan to sprinkle) is 27oC and the pump is 25% efficient, find the cost of electricity for this 32-day job. [Levenspiel]

A log of 0.3m diam. holds water and oil as shown. Determine its specific gravity. (4 marks) 1 page Ď oil = 800 kg/m3. Since the friction of the dam is very high, assume that the log does not rotate.

1. A liquid flows down an inclined plane as shown. The flow is steady and laminar. a. For the fully developed flow region, obtain the shear stress variation across the thickness. Clearly state your boundary condition and justify it. b. If the fluid is Newtonian, what is the velocity profile ? c. What should be the upward velocity of the plate if the net down flow of the fluid should be zero ? d. What is the velocity profile if the fluid is non-Newtonian, and follows the power law ( shear thinning)?. In this case what should be the upward velocity for net down flow to be zero ?

1.

Oxygen is being delivered to a steel melt shop (SMS) at the rate of 500 Nm3/min through a 500 m long, 100 mm piping from a reservoir at the oxygen plant. The piping has five 90o bends of standard radius.

In addition one gate valve and one globe valve are situated in the pipeline. If the delivery pressure at the SMS should be 106Pa, what should be the minimum pressure at the reservoir ? Pipes are made of GI. (Nm3 represents cu.m. at STP).

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In a pellet hardening unit the flow rate of air at 1000K is 5x10-3m3/s, and the size of the spherically shaped pellets is 4 mm. Pellets are filled to the depth of 250 mm . The cross section area is 0.185m2. If the void fraction is 0.4.

Density of air at 1000 K = 0.43 kg/m3. Viscosity of air at 1000 K = 30x 10-6 Pa.s, Calculate the pressure drop in meters of water . What pump of 0.6 efficiency is needed to supply air to this unit?

SOLUTION : Ergun Equation :

1.

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V o = 5x10-3/0.185 = 0.027 m/s ( extremely small). D p = 0.004, Îľ = 0.4.

= 42.71 + 1.286 = 44.00 ; Î&#x201D;P = 11.00 Pa. >> 1.1 mm water column.

Check = Re = 1.548 >> Laminar. Second term small.

-W s

= 0.43 . 5x10-3 . (0.0272/2 + 11.00/0.43)/0.6 = 9.2x10-3 W

1.

The village of Schaffzell high in the Swiss Alps operates its own modest hydroelectric plant which produces electricity continuously whether it is used or not. When not needed the 240 kW of electricity runs a motor-turbine at 75% efficiency which pumps 5oC water at 0.1 m3/s through an equivalent length of 780 m of pipe (spiral welded) to a little lake located 153 m uphill. When extra electricity is needed the flow is reversed, water runs downhill at the same flow rate, 0.1 m3/s, from the little lake through the turbine to generate the needed electricity, again at 75% efficiency. How much power can be generated with this down-flow from the little lake ?

What size of pipe was used by the village of Schaffzell in their system that pumps water up to the little lake ? [Levenspiel]

1.(a) Perform a shell momentum balance for laminar flow of a fluid flow through a `HORIZONTAL’ circular pipe and obtain an expression for the SHEAR STRESS distribution across the cross-section, with an appropriate boundary condition. Note that the pipe is horizontal: no extraneous terms please. [2] (b) List your assumption clearly, and discuss them [1] (c) The fluid is Non-Newtonian and follows the power law for one-dimensional flow τxz = -0n∂∂ν z x

Where n is a positive constant. Obtain an expression for the velocity distribution after applying the appropriate boundary condition. Obtain an expression for the average velocity [3] (d) If the fluid, on the other hand, is Bingham plastic: τxz ± τ0 = -0 _

_ _ __ _ ∂ ∂ x zν

; where + applies when τxz is negative and – applies when τxz is positive. τ0 is a positive constant.

Sketch the typical velocity

An electro-static dust precipitator consists of a pair of oppositely charged plates between which dust-laden gases flow. Dust particles are usually charged and therefore get accelerated by the electric field. It is desired to establish a criterion for the minimum length of the precipitator in terms of the charge of the particle e, the electric field strength E, the pressure difference (po-pL), the particle mass m and the gas viscosity Âľ. That is for what length L will the smallest particle present (mass m) reach the bottom of the plate just being swept out of the channel. Assume that the flow is laminar, and the particle velocity in the direction of flow ( z-direction) is the same as that of the gas. Assume further that the Stokes drag on the sphere as well as the gravity force acting on the particle as it is accelerated in the x-direction can be neglected. (BSL-see second edition)

A fan draws air at 1 atm. pressure and 300K from a room through a 0.3m diam tube and discharges it at 1 atm. pressure outside. Measurements made give the value as shown in the figure.

Friction losses are negligible. Calculate a) The average velocity inside the tube. b) The value of h c) The mass flow rate and the rating of the fan if the fan efficiency is 0.5. a) Now if there is an additional frictional loss of 8 J/kg/(m length of tube) what is the new value of h?

The density of air at 1 atm pressure and 300 K is 1.18 kg/m3.

If there is friction, then the velocity profile inside is not uniform, but parabolic : hence the pitot tube placed at the axis does not give the average velocity, but the maximum. What is now the value of h ?

The parabolic velocity profile for laminar flow (assumption) can be written as : Vz = Vz,max {1-(r/R)2} Please note that the static pressure measured still corresponds to the entire crosssection. Further, Bernoulli equation written for the entire tube still should use the average velocity and not point- to-point variable velocity.

Find the minimum value of h for which the gate shown will rotate counterclockwise if the gate cross section is triangular (inverted) with 1.2 m base and 1.2 m height as shown in the figure.

Neglect bearing friction.(WWWR)

a. A cylindrical tank with the bottom side open and top side closed, weighing 100kg, is submerged in water as shown. The local barometer reading is 1.013x105 Pascals. The thickness of the tank may be neglected. What additional force need to be put on top of the tank to bring the top of the tank flushwith the water surface ?

3 marks

b. At this position the tank suddenly ruptures and a 10 mm diam. orifice is created at the top surface. What is the initial rate of air flow through the opening ?

3 marks.

3 -6 Air : density 1.18 kg/m ; Viscosity : 18x10 Pa.s.

Tutorial 7 September 29, 2011 --------------------------------------------------------------------------------------------------------------------------------------1.

Find the upward velocity of air at 20oC which will just float a table tennis ball.

Data: The balls used at Beijing Olympics weighed 2.7 gram, and had a diameter of 40 mm. Density of air at STP is 1.28 kg/m3 and a viscosity of 20x10-6 Pa.s at 20oC.

Solution : Ball size large. Porbably in Newton’s law regime : f=0.45 Force balance : 0.0027 x 9.81 = π d2/4 . ρ air .V t 2. f V t = 5.83 m/s ; Re = 13906, f (from graph) ~ 0.45.

Agrees with the Newton’s regime assumption.

The free velocity of a spherical silver particle in a 20o C water is measured by a microscope and is found to me 1 mm/s. What is the size of the particle ?

A particle of density 2500 kg/m3 is settling steadily in water at the rate of 10 mm/s. What is its diameter? (4 marks)

Fanning friction factor( drag coefficient ) as function of Reynolds Number given in the figure. (Based on crosssectional area of the sphere). Re = ρV∞d/µ For 0.1<Re<2, C = (24/Re)[1+(3/(16)Re] is a good approximation.

1.

Every summer I carefully grow a giant tomato plant because I love the taste of its fresh-picked fruit.

Since these plants need 2 liters of water each day of the growing season to produce these delectable and irresistible fruit, how do I grow my plant next summer when I will be away for four weeks with no way to water it ?

One solution would be to connect a long plastic tube 0.4 mm-i.d. to the faucet at my home where the water pressure is 100kPa,g and lead it to the plant. Determine how long the tube would have to be to deliver 2 lit/day of water. Of course, everything is on the level. [Levenspiel]

1.

We have a 50mm water line which supplies water to our induction furnace. We would like to install a flow meter in the line to measure the flow rate of water. The water-over-mercury manometer that we would like to use should not show a height difference of more than 100 mm, since we have a height constraint. If the maximum flow rate that the line can deliver is not expected to exceed 100 lpm, what diameter of venturi would you suggest? Assume a discharge coefficient of 0.98.

1. [3 PAGE MAX]Two stationary parallel plates, infinite in extent, are kept vertically a distance 2B apart. A fluid of density Ď f is flowing between them, and a constant pressure gradient â&#x20AC;&#x201C;dp/dz is applied in the vertical (downward) direction. Gravity may not be neglected. We need to find expressions for the shear stress, velocity etc. in this flow system. a. List clearly the assumptions that you would like to make.

(1) b. Select an appropriate control volume and perform a mass and momentum balance for this flow, clearly writing the physical significance of each term. Convert this into a differential equation for the shear stress variation.

(1+0.5) c. At this stage apply a boundary condition and obtain an expression for the shear stress variation.

(0.5+0.5) d. Assume the fluid to be Newtonian. Write the differential equation for the velocity. Apply another boundary condition and obtain an expression for the velocity variation. What is the average velocity ?

(0.5+0.5+0.5+0.5) e. Define a Reynolds number and a friction factor for this flow situation. (0.5+1) f. *From the results obtained, obtain a relationship between the friction factor and the Reynolds number under the assumptions that you have made.

(1.5)

1. A water supply system consists of an overhead tank, supplying water to taps in various floors. Consider a simple case : the water level in the overhead tank (third floor terrace) is 15m above the tap in the ground floor. The supply line is a 50 mm downcomer of length 20m, with three 90o bends of short radius, and a 5m straight run of 25mm.

There is one gate valve at the top of the line and the tap is considered to be a globe valve. All pipes are made of GI (roughness = 0.2mm).

What is the flow rate in the tap when both valves are fully open ?

The equivalent length for the reducer is small : consider it to be 5d 2 , referred to smaller diameter. (6 marks) 1 page max

A water supply system consists of an overhead tank, supplying water to taps in various floors. Consider a simple case : the water level in the overhead tank (third floor terrace) is 15m above the tap in the ground floor. The supply line is a 50 mm downcomer of length 20m, with three 90o bends of short radius, and a 5m straight run of 25mm. There is one gate valve at the top of the line and the tap is considered to be a globe valve. All pipes are made of GI (roughness = 0.2mm). What is the flow rate in the tap when both valves are fully open ?

The equivalent length for the reducer is small : consider it to be 5D, referred to smaller diameter.