Recently, I was asked to suggest ways to dehydrate an agricultural product and so took the occasion to apply a modeling approach and specific models found in the book Food Process Design by Zacharias B. Maroulis and George D. Saravacos (Marcel Dekker Inc., 2003). The experience not only refreshed my understanding of dehydration but also taught some useful lessons about modeling and about relying on published examples.
Maroulis and Saravacos advocate an approach to modeling that uses the power of computerized spreadsheets, specifically Excel, to avoid having to program simulation models from scratch, along with the attendant headaches, familiar to those of us experienced enough to remember FORTRAN and other challenges. Most readers are somewhat familiar with spreadsheets, but may not realize how useful they can be.
The steps in process modeling, according to the authors, are: (1) process model formulation; (2) degrees of freedom analysis; (3) alternative problem formulations; (4) problem solution algorithm; (5) cost estimation and project evaluation analysis; and (6) process optimization.
As the authors say, "The equations of a model describe the physical laws which are valid for the process. They are derived from material and energy balances, thermodynamics equilibria, transport phenomena, geometry, equipment characteristics, and so forth" (p. 73). In the specific case of dehydration, some examples of such equations are psychrometric relationships of air and water, heat and mass transfer rates during drying, material and energy balances, physical dimensions of equipment, and cost relationships.
Drying rates, in particular, are difficult to predict from theory, though theory does suggest the form of successful correlating equations. This means that some experimental data is almost always required. For the purpose of illustration, the authors provide some material properties and correlating equations for some vegetables on p. 266 of their text. Fortunately, the material in which I was interested was similar enough that, in the absence of anything else to use, I used the properties of one of the vegetables. The examples that are presented in the text are based on the properties for carrots.
It is critical to base modeling on an accurate flow sheet. An example for a rotary dryer is Figure 1, courtesy of Maroulis. Using the flow sheet, one writes all of the governing equations and identifies all of the variables. As done by the authors, there are 25 equations with 41 variables (pp. 291–293).The difference gives the degrees of freedom—16 in the case of rotary drying. Common sense and experience suggest that certain variables are set by circumstances and by choice. These are such variables as raw material feed rate, moisture content, and size; dry product moisture content; ambient temperature, pressure, and relative humidity; dryer wall thickness; and available heating medium temperature (steam in the example given). Specifying these nine variables, called process specifications, leaves seven others that are considered design variables.
To solve the equations, the design variables must also be specified, but there usually is a set of these variables that is "best" in some sense. A common definition of "best" is that set that gives the lowest annual cost. In optimization terms, total annual cost is the objective function, and the design variables are the independent variables that are adjusted to determine the optimum. There exist add-in programs, prepared by third parties, which will "drive" an Excel-based model by adjusting design variables systematically until an optimum is located. Alternatively, a user can plot, using the chart tool in Excel, the effect of individual variables on total annual cost. An example of such a chart from the text is shown in Figure 2, again courtesy of Maroulis.
Having written the equations is not enough to solve them efficiently. The authors present the Lee-Christensen-Rudd Algorithm for equation ordering from Strategy of Process Engineering by D.F. Rudd and C.C. Watson (Wiley, 1968). This algorithm uses a structural array of the set of equations. A structural array of equations associates each equation with a row of a matrix and each variable with a column. A "1" is placed wherever a variable appears in an equation. A column in which there is only one "1" is located and the row and column eliminated from the matrix, keeping track of which equation is involved. The new matrix is examined and the process repeated. When all the equations have been removed, the remaining variables are the design variables. The equations are solved in the reverse order in which they were eliminated. If all the equations cannot be eliminated, there exists a recycle loop, which implies that some equations must be solved simultaneously. This can be done manually by guessing a value for one of the variables and then solving for it, substituting the new value for a second iteration. A stable system will converge quickly. It is preferred to avoid this situation if possible by proper choice of variables and equations.
To create a model, one first lists the "givens"—the process specifications and then the design variables. It is helpful to use the Excel tool "Naming" in which variables can be referred to by the names or symbols assigned to them instead of the cell address, which is the heart of spreadsheet logic. Then the equations are written in cells in the order determined previously so as to calculate the values of remaining variables. An important point is to keep track of units. As I tried to transfer the example model in the text to a spreadsheet, I was cruelly reminded of my inexperience with the System Internationale and the need to carefully observe whether rates were in seconds or hours, energy was in Joules or watt-hours, etc.
The Hazards of Taking Texts at Face Value
Typographical errors appear in every text, no matter how carefully proofed. As I tried to apply the example models in the text to my specific, practical case, I had some additional challenges, after I thought I had solved the units issues. After some correspondence with one of the authors, who generously sent the code he had used for the book, we determined that there was a transposition of a few constants in one of the equations. We also modified the model to resolve an apparent problem, namely that under some combinations of variables, flows seemed physically unrealistic. The change was to substitute the fraction of recycled air as a design variable for the specification of air velocity and then to calculate the velocity from the resulting air volume. (The velocity is needed in the equation for the drying rate constant.) As corrected and modified, the current model gives results reasonably close to those published in the book.
There are some useful lessons from this exercise. Hard as authors try, errors still make it into publications. Someone trying to use a published model or program must check it by hand and thoroughly understand every step. Even the underlying logic should be challenged.
This actually is one of the best ways to learn and understand a given process or operation. If you have worked through all the model equations and assured yourself that the physics, units, and values are accurate, then you understand that process.
Educators might think about assigning this sort of exercise to students. I know that I, at my age, learned a lot by trying to create a useful model for a practical problem from a published example.
One needs to check every value, every equation and every published result. It is encouraging to reproduce published results, but if other research has developed conflicting values, then one must apply some judgment. It is not a reflection upon authors that errors or challenges exist in their work; they are human, and errors creep in. Those I know have told me they value corrections brought to their attention, and I, as an author, feel the same way.
Saravacos is an emeritus professor and Maroulis is professor in food science at the National Technical University of Athens, one of the largest food science departments in Europe. They are hosting the International Conference on Engineering in Food (ICEF) in 2011.
It may be an artifact of the values used, but it seems from the model results that the optimum total annual cost comes with no recycle of air. Most people experienced with drying would wonder about this result.
On the other hand, for the conditions given, the temperature of drying air is relatively low, so discharging it is not that wasteful. Many dryers operate at much higher temperatures, where recycling air is much more favorable.
With permission from the authors and me, our current version of the process model is available by request. One must recognize that the built-in properties are for carrots and that we are still debating the right approach to modeling. So, whatever you get from us is just the starting point, but it may put you ahead of where you would be starting from scratch.
by J. Peter Clark,
Contributing Editor, Consultant to the Process Industries, Oak Park, Ill.