In the previous post, I discussed the too common error of ignoring the change in velocity and density due to molar change by reaction. One possible cause for this error is that reactor models are often written in dimensionless form with several dimensionless groups. It is tempting to assume that dimensionless groups have constant values in a reactor. One reason for this temptation is that this assumption is often made in introductory courses or texts in order to simplify a problem for illustration. Another reason is that physical properties are often assumed constant and since the dimensionless groups are composed of physical properties, then the groups can be also be assumed constant. Wrong!
In this post, let's examine the dimensionless groups that might appear in a reactor model and determine which may be assumed constant and which are variables.
As a guide to the dimensionless groups, I have used Table 5.7-1, pg 168 by Himmelblau and Bischoff (1968). That is an ingenious table that shows the terms in the balance equations that are used to develop each group.
I am going to assume that all physical properties except density are constant within the reactor. This is a very common and usually valid assumption. Also, from the continuity equation, we know that the mass flux (density times velocity = G) is constant. Thus, we are looking for groups that have either velocity or density alone in the definition.
The momentum balance may not be needed for some reactor models, but the Reynolds number, which comes from the momentum balance, often appears in correlations for other parameters needed in the model.
The Reynolds number is the only group in the momentum balance that can be assumed constant. The Froude number varies with V*V. The Bingham number varies inversely with V. The friction factor and the Weber number vary with V*G.
The mass Peclet number is the equivalent of the Reynolds number, except it varies with V. The Sherwood and the Damkohler I numbers are constant. The Damkohler II number varies inversely with density.
All of the groups from the energy balance are constant. The energy Peclet, the Damkohler III and the Stanton number all contain G.
Dealing with the variability
The usual method of dealing with this variation is to use extents of reaction and an expansion parameter that modifies the velocity and density. The method I use is to compute the changing molecular weight of the fluid. This in turn affects the fluid density if a gas. This method eliminates the need to introduce new variables.
In addition, I eliminate velocity completely from the balances. For example, the mass Peclet number is multiplied by density over density. The group now contains density and G. All of the underlined groups above can be modified similarly.
The density is expressed as a function of the state variables, i.e. mass fractions, temperature and pressure. This gets us back to the importance of using functions that I discussed in a previous post. With density as a function, there is no need to explicitly code the update calculation every time it is needed.
The need to assume constant physical properties may eventually be eliminated as computing speed increases. I already use variable heats of reaction mainly because I use enthalpy of formation at 298 K as the starting point. With some reactions, the heat of reaction at the reaction temperature can be significantly different from the standard state. Since I calculate the change from 298 K to the reactor inlet temperature, I just continue with the variable heat of reaction in the reactor. Of course, assuming constant physical properties may not be proper for some problems.
There is a broader message in addition to the warning about this particular error. When looking for a way to model a system, be careful about the examples you read. If they are an example for teaching, the author may be justified in simplifying the problem. However, the assumptions may not be appropriate for actual industrial application.
Himmelblau, D.M. and K.B. Bischoff, Process Analysis and Simulation: Deterministic Systems, Wiley (1968)