EXAMPLE 1 – Diluting a Toxic Water Supply (Elementary) A continuous stirred tank (CSTR) has a total volume of 5 liters. It contains 0.5 liters of “dirty” water with an initial concentration, 𝑐 0 =100 𝑔/𝐿, of some toxic substance represented by variable 𝑋. At some moment, 2.5 liters of clean water is added. Draw the process diagram. Does this process seem like batch or continuous? Calculate 𝑋 after adding the clean, fresh water. In order to be able to clean this water using micro-organisms, the concentration of 𝑋 needs to be lower than 0.1 𝑔/𝐿. Is it possible to add a sufficient amount of water to the tank such that the concentration of 𝑋 is below this limit?

EXAMPLE 2 – Diluting within a Stirred Tank Reactor (Medium) A continuous stirred tank is filled with water that contains kitchen salt at a concentration of 30 𝑘𝑔/ 𝑚 3 . The water volume is 100 liters. From time 𝑡=0 onward, fresh water flows in at a flow rate of 5 𝐿/𝑠. The flow rate out of the tank is the same. Draw the process diagram. What is the salt concentration if time 𝑡 goes to infinity? Setup the salt component mass balance for the stirred tank. Solve this balance and find what the concentration in the salt is at time 𝑡=20 𝑠.

EXAMPLE 3 – Antibiotics in a Body (Advanced) The human kidney removes all kinds of molecules from our bodies. This includes medicine, that you might receive in your blood during a serious illness. This could be an antibiotic, e.g. Gentamicin. In this exercise, we are going to investigate how long it takes to remove 50% of the used Gentamicin. We assume that the intake is instantaneous and that your blood has a concentration of 𝑐 𝑡=0 = 𝑐 0 of Gentamicin. Our kidneys do not process our entire blood stream during one cycle. They roughly process 20% of the total blood stream. They don’t remove all Gentamicin that flows through them; actually only a small fraction. We can model this by stating that a blood flow of 0.5 𝑚𝐿/𝑠 is going through our kidneys and is completely cleaned from Gentamicin. We can treat our blood system as a CSTR and our heart as the pump (of negligible volume) that pumps the blood around. For the removal of the Gentamicin, 0.5 𝑚𝐿/𝑠 of blood flows through our kidneys and gets cleaned. For the modeling of this system, treat the blood system as a 5 liter CSTR. Draw the process diagram. Set up a mass balance for the blood system, treating it as a CSTR. The mass to be considered is that of Gentamicin. At time 𝑡=0, the concentration of Gentamicin is 𝑐 0 . Solve this mass balance and compute how long it takes before the concentration of Gentamicin is half of 𝑐 0 .

EXAMPLE 4 – Unsteady State Mass Balance (Medium) Let’s consider a vessel where a certain mass flow rate of an aqueous salt solution ( 𝑐 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 =1.0 𝑔/𝐿) enters a well-defined volume and the same amount leaves the volume. Initially, the volume was filled with a salt solution with a concentration of 5 𝑔/𝐿 water. The vessel is well-mixed by a stirrer. Draw the process diagram. What is the concentration after 20 seconds in the vessel if the volume flow rate is equal to 10 𝐿/𝑠 and the volume of the vessel is 100 liters?

EXAMPLE 5 – Balance on a Mixing Unit (Elementary) An aqueous solution of sodium hydroxide contains 20% NaOH by mass. It is desired to produce an 8% NaOH solution by diluting a stream of the 20% solution with a stream of pure water. Draw the process diagram. Calculate the ratios (𝑔 𝐻 2 𝑂/𝑔 of feed solution) and (𝑔 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛/𝑔 feed solution). Determine the feed rates of 20% solution and diluting water needed to produce 2310 𝑙 𝑏 𝑚 /𝑚𝑖𝑛 of the 8% solution.

EXAMPLE 6 – Balance on a Distillation Column (Medium) A mixture containing 45% benzene (B) and 55% toluene (T) by mass is fed to a distillation column. An overhead stream of 95wt% B is produced, and 8% of the benzene fed to the column leaves in the bottom stream. The feed rate is 2000 kg/h. Determine the overhead flow rate and the mass flow rates of B and T in the bottom stream.

EXAMPLE 7 – Two Unit Distillation Process (Advanced) A labeled flowchart of a continuous steady-state two-unit distillation process is shown below. Each stream contains two components, A and B, in different proportions. Three streams whose flow rates and/or compositions are not known are labeled 1, 2, and 3. Calculate the unknown flow rates and compositions of streams 1, 2, and 3.