Applied Mathematics in Chemical Engineering
- Three tanks of 10,000 gal capacity are each arranged
that when water is fed into the first tank an equal quantity of solution overflows from the first tank to the second tank, likewise from the second to the third, and from third to some point out of the system. Agitators keep the contents of each tank uniform in concentration. To start, let each of the tanks be full of a solution of concentration Co lb/gal. Run water into the first tank at 50 gpm, and let the overflows function as above. Calculate the time required to reduce the concentration in the first tank to Co/10. Calculate the concentrations in the other two tanks at this time.
- A tank contains 100 ft3 of fresh water; 2 ft3
of brine, having a concentration of 1 % salt, is run into the tank per minute, and the mixture, kept uniform by mixing, runs out at the rate of 1 ft3/min. What will be the exit brine concentration when the tank contains 150 ft3 of brine?
- In an experimental study of saponification of methyl acetate by
sodium hydroxide, it is found that 25% of the ester is converted to alcohol in 12 min when the initial concentrations of both ester and caustic are 0.01 M. What conversion of ester would be obtained in 1 hr if the initial ester concentration were 0.025 M and the initial caustic concentration were 0.015 M ?
- In a chemical reaction involving two substances, the velocity of transformation dx/dt at any time
t is known to be equal to the product (0.9 - x) (0.4 - x). Express x interms of t, given that when t = 300, x = 0.3.
- A tank of volume 0.5 m3 is filled with brine containing 40 kg of dissolved salt.
Water runs into the tank at the rate of 1.4 X 10-4 m3/sec and the mixture, kept
uniform by stirring, runs out at the same rate. How much salt is in the tank
after two hours?
- The number N of bacteria in a culture grows at a rate proportional to N.
N = 100 at t = 0 and N = 332 at t = 1 hr. Find the value of N after 1.5 hrs.
- A compressed-air vessel has a volume of 10 ft3.
Cooling coils hold its temperature constant at 70oF. The
pressure now in the vessel is 100 psia. Air is flowing in at the rate of
10 lb/hr. How fast is the pressure increasing?
- A lake has a surface area of 100 km2. One river is
bringing water into the lake at a rate of 10,000 m3/s, while
another is taking water out at 8000m3/s. Evaporation and
seepage are negligible. How fast is the level of the lake rising or
falling? Answer: 72mm/h
- A vacuum chamber has a volume of 10 ft3. When the vacuum
pump is running, the steady-state pressure in the chamber is 0.1
lbf/in2. The pump is shut off, and the following pressure-time
data are observed:
Time after shutoff, min Pressure, psia
Calculate the rate of air leakage into the vacuum chamber when the pump
is running. Air may be assumed to be a perfect gas. The air temperature
may be assumed constant at 70oF. Answer: 0.0051
- The tank in fig1 is cylindrical and has a vertical axis. Its
horizontal cross-sectional area is 100 ft2. The hole in the
bottom has a cross-sectional area of 1 ft2. The interface
between the gasoline and the water remains perfectly horizontal at all
times. The interface is now 10 ft above the bottom. How soon will gasoline
start to flow out the bottom? Assume frictionless flow. Sp.gr of
gasoline:0.72 . Answer: 36.5 sec.
A tank has 1000 m3 of salt solution. The salt concentration is
10 kg/m3. At time zero, salt-free water starts to flow into the
tank at a rate of 10 m3/min. Simultaneously salt solution flows
out of the tank at 10 m3/min, so that the volume of the
solution in the tank is always 1000 m3. A mixer in the tank
keeps the concentration of of salt in the entire tank constant; the
concentration in the effluent is the same at the concentration in the
tank. What is the concentration in the effluent as a function of
- Repeat the above problem, with the change that there is a layer of
solid salt on the bottom of the tank, which is steadily dissolving into
the solution at a rate of 5 kg/min.
- Rework problem 11, with the change that the outflow is only 9
m3/min and the total volume of liquid contained in the tank is
thus increasing by 1 m3/min.
- The "heat capacity" of my house is 3300 kJ/oC. That is,
3300 kJ raises the temperature of my house 1oC. The heater in
my house can supply heat at a maximum rate of 5.2 x 104 kJ/hr.
(a) I return from vacation to a cold house. The inside temperature is
5oC and the outside temperature is -15oC (minus 15
deg C). I set the heater at its maximum rate at 8:00 pm. At what time will
the temperature in my house be 20oC?
(b) Heat escapes from my house through conduction through the walls and
roof. The rate of heat loss, qloss in kJ/hr, is proportional to
the difference between the inside temperature and the outside
qloss = k(Tinside - Toutside)
where k = 740 kJ/(oC.hr). Repeat the calculation in (a), but
include heat loss by conduction through the walls and roof.
- (a) Calculate the concentration of pollutant in the lake as a
function of time (in kg pollutant / m3 water).
Volume of water in lake = Vlake = 4 x 109
Flow rate of river = Qriver = 12 x 106
Concentration of X in river = [X] = 6.5 x 10-6
(b)The pollutant decomposes to inert substances at a rate proportional
to its concentration in water:
Rate of decomposition of pollutant = -k[X]
[X] has units of (kg X)/(m3 water) and k is a constant with
units (m3 water)/day.
Calculate the concentration of pollutant in the lake as a function of
time (in kg pollutant / m3 water) when decomposition is
- The reaction of chemical P to chemical Q releases heat:
Q + 770 kJ/mole.
Because pure P reacts explosively, the reaction is conducted in a
dilute water solution. Consider a batch reactor (no flow in or out)
initially charged with 1.0 kg of water and 0.12 mole of P (= 0.013 kg P)
at 50oC. Thus [P]o = 0.12 mole/kg water. The reactor
is thermally insulated.
(a) Calculate the temperature in the reactor after P has completely
reacted to form Q. You may assume that the heat capacity of the dilute
solution is the same as that of water.
(b) Obtain mathematical expression for the temperature in the reactor
as a function of [P].
(c) The rate of the reaction P à Q increases as the temperature
increases. Under the conditions here the rate is approximately
proportional to the temperature:
rate of reaction = d[P] / dt = - aT[P]
such that a is a
constant. Derive an expression for [P] as a function of time. Note that T
is a function of time. Note also that T is a function of [P].
Surge tanks are often used to smooth flow rate fluctuations in liquid
streams flowing between chemical processes. Consider a liquid surge tank
with one inlet (flowing from process I) and one outlet stream (flowing to
process II). Assume that the density is constant. Find how the volume of
the tank varies as a function of time, if the inlet and oulet flowrates
(dV/dt = Fi - F)
Surge drums are often used as intermediate storage capacity for gas
streams that are transferred between chemical process units. Consider a
drum, where qi is the inlet molar flow rate and q is the outlet
molar flow rate. Develop a model that describes the variation of pressure
in the tank with time. Assume that the tank is maintained in isothermal
Consider a perfectly mixed stirred-tank heater, with a single feed
stream and a single product stream. Assuming that the flowrate and
temperature of the inlet stream can vary, that the tank is perfectly
insulated, and the rate of heat added per unit time (Q) can vary, develop
a model to find the tank temperature as a function of time. State your
Assume that two chemical species, A and B, are in a solvent feed stream
entering a liquid-phase reactor that is maintained at a constant
temperature. Two species react inversibly to form a third species, P. Find
the reactor concentration of each species as a function of time.
© M.Subramanian, Lecturer, Chemical Engg, SVCE, Sriperumbudur - 602105,
Tamil Nadu, INDIA