Scales for Unknown Functions
AND APPROXIMATION TECHNIQUES
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 vol- ume 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,
, . .
, .
, , =
=
where
is the partition coefficient.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
AND APPROXIMATION TECHNIQUES
The is
= Z - A sinh (3.3-3 1)
A fin that is dynamically or functionally "short" is one where or That is, the geometric length is much less than the characteristic length. Conversely, a dynamically "long" fin is one where In neither case is the aspect ratio, the controlling parame- ter. The limiting solutions for short and long fins are
The short fin is essentially whereas the temperature in the long fin reaches the ambient value well before the tip. These results resemble those for slow and fast homogeneous reactions, respectively, in a liquid film (see Section 3.2).
For and but with more general boundary conditions, the approach used here will yield a profile which is a good approximation in most of the fin, but near the base the base temperature is not constant if the imposed values of
may make it impossible to neglect temperature variations in the x direction in the vicinity of the base. In effect, the specified function introduces an additional length scale which may
nullify the use of the low-Biot-number approximation near the base, Because the base temperature is constant in the present problem, (3.3-31) is a good approximation throughout the fin. The
more general situation is discussed in Example 3.7-4, using the method of matched asymptotic The basic issue addressed in Example 3.7-4 is how to combine a two-dimensional
solution in one region with a one-dimensional solution in an adjacent region. As with the elemen-
tary of a fin presented here, that analysis serves as a prototype for other conduction and
diffusion problems,
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 vol- ume 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,
, . .
, .
, , =
=
where
is the partition coefficient.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
concentration will change from KC, to zero over the distance The order-of-magnitude estimates
the derivatives are then
ac
KC,(3.4- 10) at r
(3.4-1
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 charac- teristic 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 applicability of the penetration analysis.
The penetration problem defined by (3.4-2), and is much sim- pler than that stated originally. The solution for is completed in section 3.5 using the
method of similarity.
Pseudosteady Analysis for Times We now seek an approximation for 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
concentration will change from KC, to zero over the distance The order-of-magnitude estimates
the derivatives are then
ac
KC,(3.4- 10) at r
(3.4-1
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 charac- teristic 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 applicability of the penetration analysis.
The penetration problem defined by (3.4-2), and is much sim- pler than that stated originally. The solution for is completed in section 3.5 using the
method of similarity.
Pseudosteady Analysis for Times We now seek an approximation for 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