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FORMULATION OF PHYSICOCHEMICAL PROBLEMS

1.1 INTRODUCTION

Modern science and engineering require high levels of qualitative logic before the act of precise problem formulation can occur. Thus, much is known about a physicochemical problem beforehand, derived from experience or experiment (i.e., empiricism). Most often, a theory evolves only after detailed observation of an event. This first step usually involves drawing a picture of the system to be studied.

The second step is the bringing together of all applicable physical and chemical information, conservation laws, and rate expressions. At this point, the engineer must make a series of critical decisions about the conversion of mental images to symbols and, at the same time, how detailed the model of a system must be. Here, one must classify the real purposes of the modeling effort. Is the model to be used only for explaining trends in the operation of an existing piece of equipment? Is the model to be used for predictive or design purposes? Do we want steady‐state or transient response? The scope and depth of these early decisions will determine the ultimate complexity of the final mathematical description.

The third step requires the setting down of finite or differential volume elements, followed by writing the conservation laws. In the limit, as the differential elements shrink, then differential equations arise naturally. Next, the problem of boundary and initial conditions must be addressed, and this aspect must be treated with considerable circumspection.

When the problem is fully posed in quantitative terms, an appropriate mathematical solution method is sought out, which finally relates dependent (responding) variables to one or more independent (changing) variables. The final result may be an elementary mathematical formula or a numerical solution portrayed as an array of numbers.

1.2 ILLUSTRATION OF THE FORMULATION PROCESS (COOLING OF FLUIDS)

We illustrate the principles outlined above and the hierarchy of model building by way of a concrete example: the cooling of a fluid flowing in a circular pipe. We start with the simplest possible model, adding complexity as the demands for precision increase. Often, the simple model will suffice for rough, qualitative purposes. However, certain economic constraints weigh heavily against overdesign, so predictions and designs based on the model may need be more precise. This section also illustrates the “need to know” principle, which acts as a catalyst to stimulate the garnering together of mathematical techniques. The problem posed in this section will appear repeatedly throughout the book, as more sophisticated techniques are applied to its complete solution.

1.2.1 Model I: Plug Flow

As suggested in the beginning, we first formulate a mental picture and then draw a sketch of the system. We bring together our thoughts for a simple plug flow model in Figure 1.1a. One of the key assumptions here is plug flow, which means that the fluid velocity profile is plug‐shaped, in other words, uniform at all radial positions. This almost always implies turbulent fluid flow conditions, so that fluid elements are well mixed in the radial direction; hence, the fluid temperature is fairly uniform in a plane normal to the flow field (i.e., the radial direction).

FIGURE 1.1 (a) Sketch of plug flow model formulation. (b) Elemental or control volume for plug flow model. (c) Control volume for Model II.

If the tube is not too long or the temperature difference is not too severe, then the physical properties of the fluid will not change much, so our second step is to express this and other assumptions as a list:

1. A steady‐state solution is desired.
2. The physical properties (ρ, density; Cp, specific heat; k, thermal conductivity, etc.) of the fluid remain constant.
3. The wall temperature is constant and uniform (i.e., does not change in the z or r direction) at a value Tw.
4. The inlet temperature is constant and uniform (does not vary in r direction) at a value T0, where T0 > Tw.
5. The velocity profile is plug‐shaped or flat and constant at a value of υ0; hence, it is uniform with respect to (wrt) z or r.
6. The fluid is well mixed (highly turbulent), so the temperature is uniform in the radial direction.
7. Thermal conduction of heat along the axis is small relative to convection.
8. The heat transfer coefficient h at the wall is taken to be constant.

The third step is to sketch, and act upon, a differential volume element of the system (in this case, the flowing fluid) to be modeled. We illustrate this elemental volume in Figure 1.1b, which is sometimes called the “control volume,” which has a volume A (Δz), where A is tube cross‐sectional area.

We act upon this elemental volume, which spans the whole of the tube cross section, by writing the general conservation law

Since steady state is stipulated, the accumulation of heat is zero. Moreover, there are no chemical, nuclear, or electrical sources specified within the volume element, so heat generation is absent. The only way heat can be exchanged is through the perimeter of the element by way of the temperature difference between wall and fluid. The incremental rate of heat removal can be expressed as a positive quantity using Newton's law of cooling, that is,

As a convention, we shall express all such rate laws as positive quantities, invoking positive or negative signs as required when such expressions are introduced into the conservation law (Eq. 1.1). The contact area in this simple model is simply the perimeter of the element times its length.

The constant heat transfer coefficient is denoted by h. We have placed a bar over T to represent the average between T(z) and T(z + Δz)

In the limit, as Δz → 0, we see

Now, along the axis, heat can enter and leave the element only by convection (flow), so we can write the elemental form of Eq. 1.1 as

The first two terms are simply mass flow rate times local enthalpy, where the reference temperature for enthalpy is taken as zero. Had we used Cp (T − Tref) for enthalpy, the term Tref would be canceled in the elemental balance. The last step is to invoke the fundamental lemma of calculus, which defines the act of differentiation

We rearrange the conservation law into the form required for taking limits and then divide by Δz:

Taking limits, one at a time, then yields the sought after differential equation

where we have canceled the negative signs.

Before solving this equation, it is good practice to group parameters into a single term (lumping parameters). For such elementary problems, it is convenient to lump parameters with the lowest‐order term as follows:

where

It is clear that λ must take units of reciprocal length.

As it stands, the above equation is classified as a linear, inhomogeneous equation of first order, which in general must be solved using the so‐called integrating factor method, as we discuss later in Section 3.3.

Nonetheless, a little common sense will allow us to obtain a final solution without any new techniques. To do this, we remind ourselves that Tw is everywhere constant and that differentiation of a constant is always zero, so we can write

This suggests we define a new dependent variable, namely,

hence Eq. 1.9 now reads simply

This can be integrated directly by separation of variables, so we rearrange to get

Integrating term by term yields

where ln K is any (arbitrary) constant of integration. Using logarithm properties, we can solve directly for θ

It now becomes clear why we selected the form ln K as the arbitrary constant in Eq. 1.14.

All that remains is to find a suitable value for K. To do this, we recall the boundary condition denoted as T0 in Figure 1.1a, which in mathematical terms has the meaning

Thus, when z = 0, θ (0) must take a value T0 − Tw, so K must also take this value.

Our final result for computational purposes:

We note that all arguments of mathematical functions must be dimensionless, so the above result yields a dimensionless temperature

and a dimensionless length scale

Thus, a problem with six parameters, two external conditions...