Applied Mathematics - Exercise Problems

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  1. Three tanks of 10,000 gal capacity are each arranged so 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.
  2. 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?
  3. 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 ?
  4. 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.
  5. 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?
  6. 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.
  7. 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?
  8. 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
  9. 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
    	0		0.1
    	10		1.1
    	20		2.1
    	30		3.1

    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 lb/min.

  10. 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 time?
  11. 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.
  12. 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.
  13. 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 temperature:
    qloss = k(Tinside - Toutside)
    where k = 740 kJ/( Repeat the calculation in (a), but include heat loss by conduction through the walls and roof.
  14. (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 m3

    Flow rate of river = Qriver = 12 x 106 m3/day
    Concentration of X in river = [X] = 6.5 x 10-6 kg/m3 water

    (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 included.

  15. The reaction of chemical P to chemical Q releases heat:

    P à 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.

  16. (b) Obtain mathematical expression for the temperature in the reactor as a function of [P].

    (c) The rate of the reaction P -> 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].

  17. 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 vary.
    (dV/dt = Fi - F)

  18. 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 conditions.

  19. 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 assumptions.

  20. 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.

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Last Modified on: 04-Feb-2022

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