Macroscopic balance equations and state equations

An important modeling tool is the construction of macroscopic balance equations. These derive from the accounting principles of the macroscopic method. Consider the system depicted in Fig. 2.2. A number of macroscopic extensive state variables specify its state.

Macroscopic accounting considers all the effects that cause a state variable to change. This leads to equations for the rate of change of extensive state variables. For the general case, macroscopic extensive quantities change due to two types of causes. Firstly, there are flows to and from the environment that result in such change. Secondly, processes in the system may

State Variables

Processes Exchange

Flows

Fig.2.2. System for macroscopic accounting.

Exchange Flows

17 result in a change of the amount of the extensive quantity, e.g. chemical conversions may lead to such change. We can simply state:

Increase of state variable = Net rate of production in system + Net rate of exchange (2.1) We promised to keep the mathematics to the bare minimum but here we introduce some mathematical shorthand notation. For each of the state variables under consideration eqn. 2.1 mathematically phrases as:

Y

dt Y

dY 3 ) (2.2)

In eqn. 2.2, the differential at the left hand side stands for the rate of change of state variable Y. The two terms at the right hand side stand for the rate of change by transformation processes and the rate of change due to exchange respectively. Both terms at the right hand side of eqn. 2.2 can be positive as well as negative.

Eqn. 2.2 is the so-called state equation for the state variable under consideration. If we avail of expressions for the rates of exchange flows and processes, we can determine the time evolution of the state variable. This evolution defines the trajectory of the state variable. In fact, if we perform an exercise of the type discussed for every extensive state variable deemed relevant in the modeling exercise, we completely specify the time evolution of the system from the macroscopic perspective. This statement is only valid if we know the initial values of the state variables; we have to know the initial conditions. Later on, we see that this is only limitedly possible due to inherent limitations of the macroscopic method. Notwithstanding this complication, we achieve complete modeling of the system in a macroscopic sense if we perform the accounting exercise for all the relevant state variables, formulate rate equations for each of the exchange flows and the processes and have full knowledge of the initial conditions.

The rate equations derive from the theories and the underlying laws that govern the rates of these processes. We provide some examples of rate equations for exchange flows and transformation processes in the next note. Again, the readers only interested in the main arguments developed in this work, can skip this note. This applies to all augmenting and clarifying notes in this work.

Note 2.2. Examples of rate equations.

An example of a rate equation for a chemical transformation is the law of mass action kinetics. We consider a process in which a chemical compound A converts into a compound B and vice versa. Mass action law kinetics leads to the following equation for the rate of change of the concentration of A, i.e. the amount of A per unit volume, indicatedC : _A

B

A kCA k C

dt dC

1

1

In this equation C is the concentration of B and the k´s are chemical rate constants. _B

In fact, this equation involves a dangerous simplifying assumption as we assume the system´s volume constant. For the general case, we cannot apply macroscopic accounting to concentrations, i.e. amount per unit system volume, but only to the amounts of chemical substances, being the product of concentrations and volume. The correct macroscopic balance equation reads:

18

V dt r

VC d

A A) (

In this equation r_Ais the net rate of production of A per unit time and volume. Mass action law kinetics leads to the following expression for the net rate of production of A per unit volume:

rA k1CAk1CB

Combination of the last two equation leads to the full formulation of the macroscopic balance equation. It shows that the first equation proposed in this note only follows if V does not change.

A rate equation for heat transport to a system derives from the Fourier law for heat conduction. It relates heat transport to the temperature difference between the system and its environment. If we indicate this difference by 'T the rate of exchange of heat follows as:

T

Q S '

) O

In this equation the term at the left hand side stands for the rate of transport of heat, Ois the coefficient of heat conductivity, S is the surface area of the system.

We conclude this section discussing a few special cases of the general balance equation introduced here. Firstly, we consider an isolated system, i.e. it exchanges nothing with the environment. In that case, eqn. 2.2 transforms to:

dt Y

dY 3 (2.3) Eqn. 2.3 shows that only transformation processes contribute to the change of a state variable in an isolated system. Processes continue to proceed until the state variables become time independent and the system reaches equilibrium. This does not imply that the approach towards equilibrium in isolated systems is straightforward and smooth. Often the dynamics of the approach to equilibrium involves a very complex and lengthy evolution. We highlight this discussing the evolution of the universe (Chapter 9), a prime example of an isolated system, as there is nothing outside the universe to engage in exchange processes.

Another special example concerns the distinction between conserved and non-conserved macroscopic quantities. A conserved quantity is not subject to net production in the processes in the system. In that case, it changes by exchange flows only and the balance equation becomes:

dt Y

dY ) (2.4)

The prime example of a conserved extensive quantity is total energy. The first law of thermodynamics, one of the cornerstones of thermodynamic theory, proclaims its conservation. We discuss this law in Chapter 3.

Finally, we introduce the concept of steady state, a state in which state variables have become

19 constant, just as in the case of equilibrium for isolated systems. In a steady state, the balance equation reduces to:

3_Y)_Y 0 (2.5) Steady states beyond equilibrium may occur in open systems if these exchange at least two different resources with the environment. There may be complex dynamics involved in the approach to steady states. In addition, steady states that conform to eqn. 2.5 may or may not be stable. We discuss these matters further in Chapter 6.