SIMPLIFICATIONS BASED ON TIME SCALES - Scales for Unknown Functions

Scales for Unknown Functions

3.4 SIMPLIFICATIONS BASED ON TIME SCALES

If the temperature or concentration is suddenly perturbed at some location, a finite time is required for the temperature or concentration changes to be noticed a given distance away the original disturbance. In a stagnant medium the time involved is the characteristic time for conduction or diffusion. This characteristic time is a key factor in formulating conduction or models, in that it determines how fast a system can respond to changes imposed at a boundary. A fast response may justify the use of a steady-state or pseudo-steady-state model. A slow may allow one to model a region as infinite or semi-infinite, because the effects of one or more distant boundaries

never "felt" on the time scales of interest.

The two examples in this section involve transient diffusion across a membrane L, as shown in Fig. The basic model formulation to both is described first. The solute concentration and within the membrane are t) respectively. (To simplify the notation. the subscripts usually used to identify individual species are omitted here.) The external solutions are assumed to be perfectly with solute concentrations and respectively. Each solution has a volume and the exposed area of the membrane is A. No solute is present initially, and at

on Time Scales

Membrane

Membrane 0

Diffusion through a separating well-mixed solutions. (a) Overall view; (b) enlargement the area indicated by dashed rectangle in (a).

the concentration in one external solution is suddenly changed to The objective to how the various concentrations evolve over time.

We focus first on the membrane, which is modeled as a homogeneous material.

species conservation equation from Table 2-3 reduces to

The initial and boundary conditions for are 0) = 0,

Lumped models for the external compartments are derived from integral

volume) statements of solute conservation based on Eq. (2.2-1). Assuming that the stirred compartments are closed except for mass transfer to or from the membrane, we obtain

represents the solute flux in the x direction. Evaluating the solute fluxes just inside the membrane, we have

= -

ac

completes the basic problem statement. Equations are coupled the concentration and flux conditions at = and = L, making this a difficult

to solve in a completely general manner.

The behavior of the concentration profile in the membrane at short times is quite than that at long times, as shown qualitatively in Fig. For small t, the changes resulting from the step change at = spread over only a fraction

concentration will change from KC, to zero over the distance The order-of-magnitude estimates

3-5. Qualitative behavior of the solute concentration in a membrane after the solute is suddenly added to of the external solutions. Concentration shown for several distinct times, such that

of the membrane thickness, as indicated by the curve for t = in Fig. The characteristic length for small is therefore penetration depth, This time-dependent scale is the distance over which significant concentration changes have occurred at any instant. If the external concentrations were to remain at their initial values, then the steady state corresponding to the curve labeled eventually would be achieved.

This would occur after the changes had spread over the entire thickness L and after additional time had elapsed to allow the concentration profile to adjust to the presence of the right-hand boundary. If the external compartments are large enough that their concentrations change slowly, a curve very close to that for = will in fact be achieved. The second series of changes, depicted in Fig. will ordinarily occur over a much longer time scale than the first. During this second period the external concentrations gradually equilibrate, until the final state corresponding to is reached. The purpose of the following examples is to establish the time scales for the two kinds of behavior and to show how the diffusion problem can be simplified when certain criteria are met.

Example 3.4-1 Penetration Analysis for Short Times We begin with an approximation to the membrane diffusion problem which is valid for small t for As suggested above, suppose that V is large enough that remains very close to its initial value, for these short times. The original boundary conditions, (3.4-3) and (3.4-4), are then replaced by

Because is known constant, the membrane diffusion problem is now uncoupled from the mass balances, Eqs. (3.4-5) and (3.4-6). Applying the second boundary condition at x =

instead of = L assumes that as in Example 3.2-2. The new feature here is that increases so that this assumption eventually fails.

The dependence of the penetration depth on time is deduced now from order-of-magnitude of the in using (3.2-2) and (3.2-3) to estimate the derivatives.

representative position the concentration will change zero to a significant fraction over the time r that has elapsed since the step change at the boundary. At any time the

Substituting these estimates into (3.4-1) and solving for gives 6-

shows that the penetration depth increases as the square mot of time, a well-known feature (or conduction) problems. It also provides an order-of-magnitude estimate of the time for diffusion to occur over a specified distance. Setting = L and solving for yields characteristic time for diffusion over the distance L,

arguments apply to transient heat conduction, for which the characteristic time is

where a is the thermal diffusivity. These time scales for diffusion and conduction are insensitive to the problem geometry. Moreover, as evidenced by the cancelation of the factor KC,, they are not affected by the concentration or temperature scales. Thus, these results can be used to estimate diffusion or conduction times in a wide variety of situations.

Returning to the membrane diffusion example, we wish to estimate the time interval for which (3.4-9), the boundary condition applied at x = will be valid. To ensure that

it follows from (3.4-13) that we must have

Given that V is large, this condition will be violated long before changes appreciably from Thus, it is the failure of (3.4-9), rather than Eq. which most severely limits the same membrane diffusion problem which is valid for long times. If the external concentrations change slowly enough, then we expect the instantaneous profile in the membrane to resemble that

a steady state as if and were constant). Neglecting Be time derivative in Eq.

the differential equation for is

to that for a steady state. The solution to Eq. (3.4-16) for given external concentrations and is

AND APPROXIMATION

methods for solving partial differential equations which are discussed in Chapter 4, the similarity method may be used with nonlinear as well as linear problems. The technique is illustrated by an example in this section and by several other applications throughout the book.

A similarity solution is developed here for the short-time, transient diffusion problem described in Example 3.4-1, involving a step change in concentration at and

= Using the dimensionless concentration the governing equations are

One requirement of the similarity method is seen to be satisfied, in that this problem contains no fixed characteristic length. That is, there is not a boundary condition imposed at some finite value of such as = L.

Assume now that can be expressed as a function of a single independent variable where

and is a scale factor which is to be determined. The only difference between and the penetration depth in section 3.4 is that, whereas was used only to describe orders of magnitude, will have specific numerical values. Although our knowledge of indicates that we will show how to derive without such prior information.

The derivatives in (3.5-1) are expressed now in terms of In this introductory example, subscripts are used as reminders of which variable is being held constant in each partial derivative. The derivatives are given by

terms of the similarity variable, (3.5-1) becomes

assumption that = is correct, then neither t nor can appear separately in equation and other conditions for 8. Because g this will be true

for (3.5-9) only if the product is a constant. The exact value of the constant is immaterial, except that a positive constant is needed to have and The sim- plest equation for is obtained by setting

With this choice, Eq. (3.5-9) reduces to

. Integrating 1) once yields

where a is a constant. Notice that the particular constant chosen in Eq. (3.5-10) has simplified the argument of the exponential in Eq. (3.5-12).

Turning now to the initial and boundary conditions, it is evident that Eqs. (3.5-2) and (3.5-4) will be equivalent to one another if

That is, t = and = will both correspond to = if Eq. (3.5-1 3) is satisfied. Requir- ing this, the boundary conditions for Eq. (3.5-1 1) become

Thus, only two boundary conditions must be satisfied, which is the number that can be accommodated by a second-order, ordinary differential equation such as

The similarity transformation has evidently been successful.

Integrating Eq. (3.5-12) and applying the boundary conditions yields

where and are the and complementary respectively.

functions are plotted in Fig. As varies from to increases from to 1, whereas decreases from 1 to The properties of the error function and related functions are discussed in many mathematical handbooks, including

(1970). Values of the functions are tabulated in such books, and they are available also from a variety of software packages spreadsheet programs for per- sonal computers).

To complete the solution, we need to determine Equation (3.5-10) is nonlinear but noticing that = it is a linear differential equation for namely,

Integrating Eq. with (3.5-13) as the initial condition, we obtain

TECHNIQUES This first approximation is equivalent to setting in the original equation, (3.6-1).

three roots of (3.6-4) are = - 2, = 0, and = 2, so that the first approximations to the desired roots are

+

^(3.6-5)

where denotes the root of (3.6-1).

Improved approximations to the roots are obtained now by considering the in (3.6-3). Essentially, we adopt the view that the terms have already been satisfied, and that the terms are still negligible. The problem is

Because the three are already known. (3.6-8) is a linear equation which may be solved for the corresponding It is found that =

-

and = Accordingly, the roots of the original equation are now

Continuing this procedure to gives the problem,

Equation (3.6-12) is linear in a,, just as (3.6-8) was linear in a,. Solving for yields - 2 1

2 8 (3.6-13)

were A key feature of this example was that the governing equations for

Linear, even though the original equation was not. This linearity starting with which is a feature of regular perturbation problems, greatly facilitates continuing the procedure to the desired level of accuracy.

3.6-2 Heated Wire with Variable

The problem in Example 2.8-1 is modified now by assuming that the thermal conductivity and source in an electrically heated wire are both functions of temperature. The energy equation is

that some range of temperatures the conductivity can be expressed as a function of the temperature,

Regular Perturbation Analysis 97

(3.6-17) where is the thermal conductivity at the reference temperature (the ambient temperature) and a is a constant. The local rate of heat generation is given by where i is the current density and K is the electrical conductivity of the wire. From Ohm's law, i = K where V is the applied voltage (assumed to be independent of r) and L is the length of the wire. It follows that Because conduction of heat and electricity in metals is by the same mechanism, we assume that k. Thus, the source term in Eq. (3.6-16) is written as

where is a constant. To simplify the analysis, it is assumed that Bi = so that the surface temperature is A convenient set of dimensionless quantities is

where the temperature scale is A^T= With these definitions, Eq. (3.6-16) is

. with the boundary conditions

, Note that (3.6-20) is nonlinear, because the coefficient of depends on There appears to be no general solution which is valid for all

Suppose now that the temperature dependence of k and is relatively weak, such that Treating the temperature as a function of and expanding the solution as a power series, we obtain

Because depends on the radial position as well as on the parameter the coefficients in the expansion are not constants, as they were in the previous example. The problem now is to determine the coefficient functions, We will be satisfied with the first correction to the temperature profile caused by the variable thermal properties. In other words, we will compute only

Substituting (3.6-23) into (3.6-20) yields

The boundary conditions expanded in a similar manner to give

The problem which is the same as that obtained by setting = in the original equations, is

These are the equations for constant values of k and solved earlier. From (3.2-6), the solution is

The problem for is

Equation (3.6-30) is a linear differential equation for with nonhomogeneous terms arising from the solution, The solution is found to be

Thus, the first two of the expansion for are

The procedure followed in obtaining can be extended indefinitely (as time and pa- tience to derive successively better approximations to @.

It was noted in the previous example that the perturbation procedure always yields a se- quence of linear equations beginning at Although not true in general, in the present example the equation (that governing happened to be linear The fact that was a simple polynomial greatly facilitated the determination of As a rule, the success of the perturbation approach in solving a differential equation hinges on whether the first term in the perturbation expansion 8,) can be expressed in of elementary functions.

PERTURBATION

feature of regular perturbation solutions to differential equations is that expansions valid for all values of the independent variable. Thus. the in Example is a good approximation to the temperature profile throughout in the Singular perturbation methods are needed to treat problems valid solutions cannot be found. The key characteristic of such problems is that each of two or more regions requires a different approximation. Singular analysis is relatively new, having been employed first in the 1950s for the

boundary problems in fluid mechanics, Out of this research developed what is now called the method of asymptotic expansions [see Chapter 4 of Van Dyke Matched asymptotic expansions have been described in many texts on applied mathematics. For information beyond the introduction given here, see Bender

and Orszag and Cole Lin and Segel Nayfeh

or Van Dyke (1964).

As shown in the examples which follow, a feature of singular perturbation problems is that the small multiplies the highest-order term in the equation the highest derivative). Consequently, setting = reduces the order of the equation. This reduction of order is sufficient to invalidate a regular perturbation expansion. However, not all singular perturbation problems have this feature (see Sec- tion 7.7, for example).

' , Example 3.7-1 Solution of an Algebraic Equation As in the discussion of regular perturbation methods, it is instructive to begin with an equation. Consider the roots of

where I . We will proceed as if the roots of this quadratic equation could not be found exactly, and use the exact answer only to check the results of the perturbation analysis. In regular perturbation analysis, as discussed in Section 3.6, the first approximation to the solution is found by

, setting in the governing equation. Doing that in (3.7-1) gives

(3.7-2)

or However, there is an immediate difficulty: what happened to the second root? It was inadvertently lost because setting = reduced the equation second order to first order. As

, already mentioned, a reduction in the order of the governing equation for a hallmark of singular perturbation problems. Such problems are singular in the sense that the solution obtained

, , by setting = 0 is radically different than the asymptotic solution for Thus, the solution for not an acceptable starting point for constructing better approximations. In the present

, the singularity manifests itself in the loss of one root; with differential equations it usually leads to the inability to satisfy one or more boundary conditions for =

, , ,

The flaw in the reasoning leading to was the implicit assumption that both roots. This led to the expectation that Resolving the difficulty requires that we pay careful attention to the scaling. As will be shown, the two roots of (3.7-1) have very

scales.

To correct the scaling problem, a new variable is defined by setting

to the used to scale the coordinates in Examples 3.2-2 and The Constraints which determine a are that the governing equation be second order (allowing us to

both roots) and that at most, Using in ( yields

What is needed now is to identify the most important terms for what is called a dominant balance. Aside from which is mandatory, either or both of the other terms on the

side might be important. Let us assume that the dominant terms are those involving This implies that = I or a = 1, so that (3.7-4) becomes

The term that does not involve X is consistent with our assumption that the others are dominant. Equation provides the basis for the perturbation solution, as shown below.

What if we had guessed the wrong set of dominant terms? ^If and the term involving were to be dominant in we would have concluded that = 1 a =

This would have given

The remaining term is now which contradicts the assumed dominant balance. This illus- trates how trial and error is sometimes involved in determining scales.

Equation (3.7-5) is solved by representing each root as a power series expansion in

n = O

computing successive as in Example The problem is

so that the roots at leading order are = - and The problem is

which gives - and = The roots X,, accurate up to are then

In of the original variable x, the roots are

The naive approach of setting in Eq. (3.7-1) gave us an approximation to but missed entirely. As already mentioned. the difficulty came from assuming that for both mots. As seen now in Eqs. (3.7-12) and but =

The exact roots of (3.7-1) are, of course, - 1 Expanding the numerator for 1 gives

It be seen from (3.7-14) and (3.7-15) that the "minus" and "plus" roots correspond to and respectively. Thus, the singular perturbation analysis has provided the first in an expansion of the exact solution.

Diffusion in a Cylinder with a Homogeneous Reaction this example singular perturbation approach is applied to a boundary-value problem. Consider a cylindrical catalyst pellet in fast, first-order reaction occurs at sites which are uniformly distributed

the pellet. The pellet is modeled as a permeable medium with an effective

the reactant and an effective homogeneous rate constant Both the and the rate depend on the catalyst material; see (1975) for a discussion of these parameters. The

reactant concentration at steady state, is shown qualitatively in Fig. 3-7. The fast,

reaction leads to a sharp drop in concentration near the outer surface, as for the problem involving a planar liquid film discussed in Section 3.2. The region in which the concentration declines sharply is called the concentration boundary layer:

The dimensionless reactant concentration and radial coordinate are given by

C (3.7- 16)

where is the concentration at the outer surface and R is the pellet radius. Using cylindrical the species conservation equation is

Because the reaction is assumed to be fast, the small parameter is The boundary tions are

The fact that multiplies the second derivative in Eq. (3.7-17) is an immediate indication that the perturbation problem is singular. If we set = 0, the differential equation reduces to = 0.

Although accurate for the "core" region in the center of the pellet, this solution is obviously

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