Single-component systems are not adequate for realistic chemical engineering problems. It is rare to have a single component unless it is the product of many different unit operations. If chemical engineering is the science of chemical and physical change, then it is also a science of complexity. A major source of complexity comes as a result of having to deal with real systems that are composed of many interacting components. The objective of this chapter is to set up a strong foundation for the problem of multicomponent systems of all kinds.
4.1 The Concept of the Component Balance
The masses of components can be handled in much the same way that we have handled total mass. The total mass balance is simply the sum of each of the component balances. Imagine we are playing a game tossing black and gray balls into a box on a scale (see Figure 1). Each ball has the same mass. The player tossing the gray balls is more skillful than the one tossing the black ones, and as a result she is able to throw more gray balls into the box every minute than the fellow who is tossing black balls. The scale tells us how fast the total mass of balls, both black and gray, is changing. If we want to know how fast the mass of just black balls in the box is changing, then we need to know how many are being thrown per unit time over the period of the measurement and similarly for the gray balls. The sum of the arrival rates of the black and gray balls together is the rate of mass change in total within the box.
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Figure 1
The total material balance for this system is:
= mblack + ˚ dmtot[t]
˚ mgray
dt The component balances are:
dmblack[t]
= m˚black
dt dmgray[t]
dt = m˚gray
Restating the total material balance, we have:
dmtot[t] dmblack[t] dmgray[t]
= +
dt dt dt
Therefore, the sum of the component balances is the total material balance while the net rate of change of any component’s mass within the control volume is the sum of the rate of mass input of that component minus the rate of mass output; these can occur by any process, including chemical reaction. This last part of the dictum is important because, as we will see in Chapter 6, chemical reactions within a control volume do not create or destroy mass, they merely redistribute it among the components. In a real sense, chemical reactions can be viewed from this vantage as merely relabeling of the mass.
4.2 Concentration versus Density
1534.2 Concentration versus Density
To this point we have had to deal only with the mass per unit volume in the form of density, since we were concerned only with single-component systems. Multiple components share the volume and because of this we must use concentration as well as density. The density of a single component i is the mass of that component per unit volume:
massi
ρi = vol
For a multicomponent system the total density is the sum of the masses of the components per unit volume:
n i=1 massi ρtot =
vol
The concentration of any component i can be either a mass concentration or a molar concen
tration:
Although these definitions are straightforward, they do seem to cause problems more often than they should, especially for those who are just beginning to work with them in earnest.
The component material balance for a system with input and output, but no chemical reaction, is written as follows:
dmi
= m˚i,in − m˚i,out
dt
If the mass flows are those of liquids, then in terms of mass concentrations, this becomes:
V dmi V
The last statement is the typical form of a liquid-phase component mass balance. When this is divided through by the molecular weight of species i, this becomes a differential mole balance since the concentrations are expressed in molarity units:
1 d 1
MWi dt [CiV] = (Ci,in qin − Ci,out qout) MWi
d 1
[MiV] = (Mi,in qin − Mi,out qout) MWi dt
Typically, this last statement is written with the symbol C for molar concentration just as it is for mass concentration. Given that this is the case and it is not likely to change, the particular meaning of C must be understood from context. Fortunately, this is usually easy to do.
4.3 The Well-Mixed System
Once we move away from single component systems there is the real possibility that the components will partition themselves in different parts of the vessel due to different densities, solubilities, or miscibilities. Partitioned systems are also referred to as “distributed.” That means that the properties are not everywhere the same over macroscopic length scales. To handle distributed systems we typically have to choose a differential control volume, that is, an infinitesimal volume within the macroscopic system. We will see this when we consider plug flow down a tube.
Although partitioning is often encountered, and even though it may be advantageous in many cases, it is also true that many systems are either naturally homogeneous or are forced to be by the action of vigorous mixing. When a system is homogeneous, it means that the density and concentration are everywhere the same throughout the control volume.
This is referred to as the condition of being “well-mixed.” From the purely mathematical vantage, it refers to any system that can be described solely in terms of time as the independent variable. We turn now to problems of systems with multiple components and which are well mixed.