INDIAN INSTITUTE OF TECHNOLOGY, BOMBAY

Department of Metallurgical Engineering and Materials Science

MM 204: Transport Phenomena: 2014-2015

ASSIGNMENT 1

To be submitted by 19. 01.2015

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1. The figure below shows a schematic of a continuous stirred tank used for mixing in industries.

Let us take a simple case wherein one is interested in producing a salt solution (with water as

the solvent) of desired composition, C kgm-3

. Please note that the salt solution is dilute. The

desired production rate of salt solution is Q m3s

-1. To produce the salt solution, there are two

streams of input to the stirred tank, namely, water and salt through a powder feeder. Determine

the feed rate of water (in m3s

-1) and feed rate of salt (in kgs

-1).

To start the process, let us say, one needs to fill the stirred tank with pure water of volume V0by

closing the output valve. Once filled, both the salt feeder as well as the output valve are opened

simultaneously. If so, it would take some time before the salt solution of desired composition is

produced. Can you determine how the concentration of salt in the output stream will change as a

function of time from the time it was opened from the starting. Imagine that you have super

stirrer in the tank that it mixes instantaneously anything you put in! You can comment on the

assumption involved in making these calculations.

Figure 1: A schematic of a continuously stirred reactor

Now let us add more streams to the reactor as shown in the figure below. Instead of adding salt,

two separate streams, one for NaOH solution and another for HCl were added. Let us say the

inlet flow rate and concentration of NaOH stream be QNaOH and Cin

NaOH and that of HCL be QHCl

and Cin

HCl. The total output flow rate of the product solution is Qout

= Q + Qin

NaOH+ Qin

Hcl so that

the volume of the solution in the reactor does not change with QHCl time. In the reactor, not all

the NaOH and HCl react, it depends on the probability of collision between NaOH and HCl

molecule. Hence, let us say the amount of salt formed per second per unit volume of the tank is

given by

Gsalg= KCNaOHCHCl( in kg per second per unit volume of the reactor)

Perform mass balance for NaOH, HCl and Salt in the tank so that you can find the evolution of

their concentration with respect to time, starting with pure water in the tank.

Figure 2: A schematic of a continuously stirred reactor

2. In winter, most of us use a simple continuous water heater in our bath rooms. A schematic of

the water heater is shown in the figure below. We have a water heater with a power rating of

3kW. The reservoir in the heater holds 2 liters of water any time. During winter season, the

temperature of incoming water is approximately15oC. If I want the water to be heated upto 40

oC,

what should be the flow rate I should maintain? As per instructions, I should open the cold water

tap as soon as I switch the heater on.

Of course, then you will not start getting the hot water immediately after you switch on the

heater and the water tap to the heater. My 7 year old son, asked me one day “Pappa, why don’t

you wait for the water to heat up and then open the tap”. Can you tell me how long should I wait

before opening the tap? Remember, always there would be 2 liters of water in the reservoir at

15oC at the start. More often than not, I would either open the tap later than I planned or earlier!.

Can you draw a schematic of the evolution of temperature in the reservoir with time when I open

the tap as per planned, later than I planned and earlier than I planned? Please note that when I

open the tap, I would get the temperature of water same as that in the reservoir.

Figure 3: A schematic of a continuous water heater

3. What is the pressure at a point 10m below the free surface in a fluid that has a variable density,

ρ = 450 + ah (kg/m3), where a = 12 kg/m4 and h is the height in meters, measured from the free

surface ?

4. Derive equations that give the pressure and density at any elevation in a static ideal gas when

conditions are known at one elevation and the temperature gradient β is known. By use of the

above result determine the pressure and density at 3000m elevation. It is known that p=100 kPa,

abs. and temperature is 20oC, at an elevation of 300m for air, and β = -0.005

oC/m? (WWWR)

5. The tank is accelerated upwards at a uniform rate. Does the manometer level go up or down?

Give quantitative reasons. [psi stands for pounds per square inch (obsolete units still used in the

USA). The additional subscript g stands for gage pressure, which is that in excess of the

atmospheric pressure. Derived from the fact that pressure gages measure pressure with reference

to the atmospheric].

6. Find the minimum value of h for which the gate shown will rotate counterclockwise if the gate

cross section id (a) rectangular 1.2mx1.2m; (b) triangular (inverted) with 1.2 m base and 1.2 m

height. Neglect bearing friction. (6psig ~0.41 bar, above the atmospheric pressure).[WWWR]

7. A pressure die casting machine operates at a pressure of 106 Pa. Low melting metal like zinc

is taken into the cylinder and is forced to the die through a runner of 10 mm inner diameter. If

the stress in the runner is not to exceed 100 MPa, calculate the outer diameter of the runner.

{Stress is in the azimuthal direction. Slit the tube into two axially and find the pressure force/m

length acting on one of them. From this calculate stress in the shell}.

8. Beyond what height h will the beach ball shown be operative as a stopper? [WWWR]

9. A log of 0.3m diam. holds water and oil as shown. Determine

• The force per meter pushing it against the dam

• Log's specific gravity. ρoil = 800 kg/m3 . Since the friction of the dam is very high, assume

that the log does not rotate.

10. The Siemens process for production of pure silicon is a Chemical Vapor Deposition (CVD)

process. Distilled trichlorosilane (TCS or SiHCl3) is mixed in a reaction chamber with high-

purity hydrogen. The chamber contains a heated silicon rod upon which the reaction

production, silicon, deposits as polycrystalline solid. The rod grows by CVD. The rod is

heated to preferentially make the Si to form heterogeneously on the rod. The gas mixture has to

be maintained relatively at low temperature to avoid silicon formation on the reactor walls.

Consider a Siemens’ reactor shown in the figure below. The reaction can be written as follows

SiHCl3 + H2 = Si + 3 HCL

Consider a Siemens reactor with a molar flow rate of inlet feed, Fin (moles/s). The composition

of this inlet feed is: xSiHCl3, xH2 and the rest inert gas. Assuming that the reactor is continuously

stirred tank reactor wherein the gas composition in the reactor is uniform through the reactor and

the rate of formation of Si is given by

Rsi (in moles of Si /sec) = Arod * k * CTCS * CH2

Here, Arod is the Area of the Si rod, k is the reaction rate constant, C is the concentration.

Find out the expression to estimate the rate of Si growth for a known radius of the Si rod, R and

length L. Assume appropriate symbols for other variables to solve the problem.