for Diffusion in Domains
202 METHODS FOR AND DIFFUSION PROBLEM Fluid Mixture
+
Figure Detector with membrane inlet.
4-18. Time Response of a Detector with a Membrane, Inlet
One strategy for monitoring the concentration of a gaseous species or a vola- tile solute in a Liquid is to equip a mass spectrometer (or other suitable detector) with a membrane inlet which is to the species of interest. When the inlet device contacts the mixture, a vacuum maintained in the instrument causes the species to diffuse across the membrane and enter the An example of such a system was described by Lewis et (1993). Ideally, the inlet is designed so that the rate of removal of the species is small enough to have a negligible effect
on being monitored.
A schematic of a membrane inlet is shown in Fig. P4-18. Consider what happens when the solute of interest is added suddenly to the fluid. Assume that the detector itself responds instanta- neously and that its signal is proportional to the solute throughput the flux leaving the membrane at Thus, the overall response is limited by mass transfer in the fluid and diffu- sion through the membrane.
(a) How must the dimensionless quantities (x, Bi) be defined? Pertinent dimensional quantities include the concentration in the fluid for the membrane thick- ness (L), the solute in the membrane (D), and the mass transfer coefficient in the fluid. The coefficient, or concentration ratio at equilibrium, is denoted as
Use the method to determine t).
Evaluate the solute flux leaving the membrane at =
Estimate the half-time of the response the time required to reach one-half of the final detector signal) for Bi = 1.
4-19. Flow in a Porous Sphere
As discussed in Problem 2-2, if law is used to the incompress- ible fluid in a porous material, the pressure field is governed by
it is desired to measure the permeability (K) in a small, sphere radius R using the arrangement shown in Fig. P4-19. The sphere will be held at the end of a circular tube by suction. A known volumetric flow rate will be imposed, and the pressures and will be measured. At the sphere surface, assumed that = for and
420. Inner Product and Eigenfunctions for Uniform Convection
*
A differential operator which arises in problems involving conduction or accompa- nied by a uniform bulk velocity is
where Pe is the number. Equation (2.8-49) provides an example of such an operator. An examination of the eigenvalue problems associated with is needed for solving certain problems involving uniform convection. An application of these results is in
4-21.
(a) Consider eigenvalue problems based on where
together with any combination of the three types of homogeneous conditions.
Show that a weighting function is needed in the inner product to make such a problem self-adjoint.
(b) A useful for solving differential equations like that in part (a) is to modify the dependent variable so as to eliminate the first derivative. Show that this is accom- plished by setting and that the general solution is
Determine the eigenvalues and eigenfunctions for the boundary conditions, = = What happens to the exponential factors?
problem was suggested by R. A. Brown.
421. in Solidification
*
A stagnant-film model for unidirectional solidification was presented in Example 2.8-5. The objective here is to extend that model to describe transients associated with the start-up of the process. Assume that the initial concentration of dopant is uniform at and that solidification at velocity suddenly at t = (see Fig. 2-1 0).
(a) Use the method to determine t) in the melt. For on the inner product and eigenfunctions, see Problem 4-20.
An important consequence of the time-dependent concentration in the melt is that the dopant concentration in the solid is not uniform; that is, (y, r). Determine
t), assuming that diffusion the solid is negligible. If the dopant level within
204 FOR CONDUCTION AND DIFFUSION PROBLEMS the solid must be within 5% of the steady-state value to give acceptable material prop- erties, how much of the product must be discarded?
problem was suggested by R. A. Brown.
4-22. Point Source Near a Boundary Maintained at Constant Concentration
Assume that a reactive solute is released at a steady rate at a point source located a distance L from a catalytic surface. The reaction rate is sufficiently fast that at the surface. Construct a solution for
Inner Product and for Material
When the heat capacity per unit volume thermal conductivity vary with posi- tion, the eigenvalue problems for conduction differ from those for constant properties. For tran- sient, one-dimensional conduction in rectangular with varying properties, the dimensionless energy equation as
Particular care is needed when the dimensionless properties and are discontinuous, as when the region of interest is a layered composite consisting of two or more different materials.
(a) If and are continuous functions for 1 and if the usual types of homo- geneous boundary conditons apply at = and = show that the differential equation and inner product which yield a self-adjoint eigenvalue problem are
(b) If and are both piecewise constant, such that
then the operator is given by
If the usual types of boundary conditions apply at x = and = show that a self- eigenvalue problem is obtained with the inner product
provided that
What is the basis for these conditions at The results given here apply to a composite material consisting of two layers; this approach may be extended to any number of layers, as described and Amundson
Material 1 Material 2
Figure P4-24. Conduction in a layered composite consisting of materials I and 2, which have different mophysical properties.
4-24. Conduction in a Layered Composite
Consider transient conduction in a composite solid consisting of two layers, as shown in Fig. P4-24. Using the on the inner product and eigenfunctions in Problem 4-23, deter- mine t). [Hint: The characteristic equation for the eigenvalues comes from the matching conditions at Also, note that the of the eigenfunctions changes at y. It may be
helpful to let = for and = for 1
4-25. Green's Functions for Diffusion in Dimensions
(a) For steady diffusion in two dimensions, show that the Green's function may be written as
Note that in this case it is not possible to satisfy the condition as where r =
+
(b) For transient diffusion in two dimensions, show that the Green's function is
for - t') for t
4-26. Transient Diffusion from a Solid to a Stirred Solution of Finite Volume
In Problem 4-6, which involves diffusion from a solid to a stirred liquid, it is assumed that the solute concentration in the liquid is zero at all times. The physical situation considered here is similar, except that the liquid volume is assumed to be finite. Thus, the solute concentration in the liquid increases as that in the solid decreases, until a steady state is achieved.
Assuming that the mass transfer resistance the liquid is negligible, the solute concentration in the solid, t), is governed by
where denotes the concentration in the liquid.
In this problem the concentration field consists of two parts, for the solid and
. for the Liquid. For such problems it is helpful to the overall solution as a vector,
206 FOR CONDUCTION AND D
where the second form is obtained by using the condition at x = A vector operator is defined so that the differential conservation equations for t) and are expressed as
The nth basis function is also written as a vector. so that the differential equation for the eigenvalue problem is expressed as
The expansion for t) is written as
It is seen that the inner product of the solution and basis function vectors is a scalar, which represents the time dependence common to the two parts of the solution. overall approach closely parallels the procedure, but (as given below) a special form of inner product is needed to make the approach work.
(a) Show that the liquid concentration is governed by
How must and be defined?
Based on the differential equations for and the differential operator with 0 is written as
and the corresponding eigenvalue problem is
Show that if the inner product of the vector functions and is defined as
then the eigenvalue problem is self-adjoint, Note that this inner product of vectors is similar to the dot product of vectors, in that both result in a sum of scalar
Use the first line of the eigenvalue problem to determine to within a multiplica- tive constant. (The normalization condition is applied later.)
Use the second line of the eigenvalue problem to find the characteristic equation gov- erning
(e) Use the normalization condition,
to complete the solution for Also, prove that the eigenfunctions are orthogonal in the inner product defined in part [Hint: Use the approach in Section
(f) Show that is governed by
where H is a constant. Evaluate H and solve for (g) Complete the solution for t) and
Problems of this general type, where a solid or fluid region with a distributed concentra- tion or is in contact with a stimd are discussed in and
and et (1979).
Rigid-Body
Rotation
Purely rotational motion, like that of a rigid solid, is considered first. In the situation illustrated in 5-1, the fluid is rotating counterclockwise about an axis which is perpendicular to the page, points 1 and 2 are material points on two of the circular The positions of these points relative to the origin are given by the vectors and whereas the vectors and are normal to the axis of rotation.
Thus, R, and are to the paper, but in general and are not. The angular velocity is denoted as on, where n is a unit vector parallel to the axis of rotation.
(Letting represent a vector rather than a tensor deviates from our usual convention.) The linear and angular velocities at any point are related by
(5.2-1) According to the right-hand rule for cross (vector) products, the counterclockwise rota- tion shown in Fig. 5-1 requires that be directed toward the reader. Letting Av
= -
Ar
= - r,, and AR = - the relative velocities and relative positions of the two points are related byThe second is a consequence of the fact that any components of and in the direction to or n do not contribute to the cross product.
The customary measure of rotational motion at any point in a fluid is the vorticity,
For the special case of rigid-body rotation, there is a simple proportionality between w and Taking the curl of both sides of (5.2-1) and using identity (6) of Table A-1, we find that
Because is a constant vector and R does not vary in the direction of n, all terms vanish except that containing Recognizing that R amounts to a two-dimensional position vector, we find that R = 2, by analogy with It follows that
Rewriting (5.2-2) in of w, it is found that for rigid-body rotation,
Figure 5-1. Motion of material points in a fluid undergoing rigid-body rotation. One point is
on each of two circular The coordinate origin is point
streamline
Although in rigid-body rotation the vorticity is a reflection merely of the (constant) angular velocity, in complex flows it provides a measure of the local rotational motion at any point in the fluid. Note that when we use (5.2-3) to compute t) from t ) , there is no need to specify an axis of rotation. If w = everywhere, the flow is said to be
The velocity gradient, Vv, is a dyad whose nine elements in rectangular coordi- nates consist of all possible first derivatives of the three velocity components with spect to the three coordinates. We now make use of the observation that any tensor can be written as the sum of a symmetric tensor and an tensor (see Section A.2). Thus, the velocity gradient can be expressed as
where is the tensor and is the vorticity The advantage of this decomposition of Vv is that, as their names suggest, the symmetric and
parts are uniquely associated with the local rates of deformation and rotation, respectively. [Some authors omit the factor from the definitions of the rate-of-strain and vorticity tensors, so that (5.2-7) would be written as =
+
The defini- tions used here are the ones more commonly employed.]To complete the description of rigid-body rotation, we focus first on the properties of Any tensor has only three independent, nonzero elements. Thus, as pointed out in (1962, pp. 24-25), an tensor can be constructed from the three components of any vector. The vorticity tensor is related to the vorticity vector w according to
where is the alternating unit tensor (see Section The matrix in (5.2-10) shows the relationship between the elements of and the rectangular components of w.
Using the identity
which is for any vector a, we find that for any pair of material points in a fluid undergoing rigid-body rotation the relative velocity is
This result is used below in interpreting the difference in velocity between neighboring material points in an arbitrary flow.
5.3 CONSERVATION
O F
MOMENTUMThe linear momentum of a solid body with mass and translational (center-of-mass) velocity v is a vector. For a body of constant mass, Newton's second law of motion is stated as
where F is the net force acting on the object. Thus, the rate of change of momentum equals the net force. The objective of this section is to rewrite Eq. (5.3-1) in integral differential forms applicable to fluids. The resulting equations describing the conser- vation of linear momentum are comparable in most respects to the conservation equa- tions for mass, energy, and chemical species derived in Chapter 2. The first task is to describe the rate of change momentum of a body of fluid, using local velocities instead of the center-of-mass velocity. There are two basic ways to do that, both of which are informative and will be discussed. The second task is to characterize the forces exerted on a body of fluid by its surroundings, leading to an expression for As in Chapter 2, the derivations begin with macroscopic (control volume) equations, which are reduced eventually to differential equations valid at any point in a fluid.