1-s2.0-S0895717798001654-main

Mathl. Comput. Vol. 28, 11, pp. 1998 @ Elsevier Science All rights

Printed in Britain

PII: SOS957177(98)00165-4 0895-7177/98 $19.00 + 0.00

Analytical Solutions for Diffusive Finite Reservoir Problems Using a

Modified Orthogonal Expansion Method

L. J. DE CHANT Applied Theoretical and Computational Physics Division

Mail Stop D-413, Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.A. [email protected]

(Received September 1997; revised and accepted February 1998)

Abstract-In this paper, we consider an eigenfunction expansion solution method useful for a family of l-d, unsteady diffusion dominated transport problems (mass, e.g., effluent leakage from a landfill; momentum, e.g., deceleration of shaft in a journal bearing, and energy; e.g., unsteady thermal loading of the skin of a m-entry vehicle) characterized by movement from a bounded, and therefore, time dependent, reservoir to an infinite reservoir. It is demonstrated that simple orthogonality is not applicable to this type of problem. To overcome this limitation, an extended eigenfunction expansion method is developed by modifying the weighting function within the classical orthogonality definition. Using this method, an analytical series solution is obtained. This series solution is also obtained using Laplace transform methods and analytical inversion. Comparison with numerical and approximate methods is good. The analytical solution developed here provides a convenient and physically insightful solution form useful in its own right and as a test problem for numerical implementations. @ 1998 Elsevier Science Ltd. All rights reserved.

Keywords-Modified orthogonal method, Eigenfunction expansion, Diffusion, Heat equation.

NOMENCLATURE

a

A

DAB

h

i

k

K, K

L, L,

n

9

S

s

T

temporal separation variable

reservoir geometry cross section

diffusion coefficient

water level

discrete spatial location

thermal, hydraulic, or generalized conductivity

generalized governing equation parameters

reservoir geometry parameter

summation counter, discrete time level

flux

Laplace transform parameter

hydraulic storage coefficient

temperature

t

u

uio

time

general independent, scalar variable, velocity

general independent, scalar variable initial condition

length coordinate

diffusivity

Dirac delta

eigenvalue

dynamic viscosity

kinematic viscosity

eigenfunction

density

density species a

orthogonal weighting function

Typeset by &S-W

73

74 L. J. DE CHANT

INTRODUCTION

In this analysis, we consider an eigenfunction expansion solution [1,2] method for a family of

l-d, unsteady diffusion dominated transport problems characterized by transport from a bounded,

and therefore, time dependent reservoir to an infinite reservoir. The term reservoir is used

loosely. Though this analysis does indeed provide solutions for the leakage of mass, heat, or

concentration leaking through a diffusive media from a bounded reservoir, the decelerating flat

plate for the unsteady Couette [3,4] problem (described subsequently) may be interpreted as a

bounded reservoir of linear momentum. Regardless of the physical problem, the mathematical

analysis will be essentially the same. In this paper, it will be shown that simple orthogonal-

ity is not applicable to the finite reservoir problem. Though numerical solutions are available

for these problems, they do not provide the convenience and insight afforded by an analytical

solution. Additionally, analytically based solutions provide a useful suite of test problems for

testing numerical implementations. Finally, a possible error in a standard reference is noted and

discussed.

For physical examples, we may draw upon a wide range of diffusive problems. Examples

and practical applications from fluid mechanics, heat transfer, flow in porous media, and mass

diffusion are described in Table 1. Schematic diagrams of these problems are presented in Figure 1.

Table 1. Examples of finite reservoir diffusion problems.

Diffusion Type

Momentum [3,4]

Governing Equations Finite Reservoir Problem

2 Newtons law of viscosity: g = ~2, Suddenly decelerated plate,

i.e., unsteady Couette

au(l) - _ /J

(_->

au(l) -. problem, e.g., deceleration of

at PPLP ax a shaft in a journal bearing, viscometry.

Energy [5-81

Fouriers law:

+_[$;;$.

A slab with one face in contact with a layer of perfect conductor or well stirred fluid, e.g., heating of the skin of a m-entry vehicle.

Mass [9,10] (flow in porous media)

Darcys law: ah kh,,, a2h -=7T$-$ at

Ground water flow through a porous dam from a bounded

ah(l) - _ k&v, ah(l) at (-1 L, xi--

reservoir, e.g., leakage of effluent from a sealed sanitary landfill.

Concentration [5,6]

ab a% Ficks law: - = DAB -, at as

apa pa apdu -=- - ( > at L, as

Mass transfer through a semipermeable membrane (cell wall), e.g., biological models and separation processes.

Among applications, particular attention is drawn to hydrological/petroleum problems. Leak-

age of contaminants from finite sources, such as land fills, sewage retention ponds, mine tailing

disposal lagoons, and ruptured gasoline storage tanks represent important examples of diffu-

sive flow from finite bodies to a larger reservoir. In petroleum reservoir engineering, leakage of

produced or stored hydrocarbons through planar surfaces caused by sealing faults is of great

interest. As a solution to one of the classical problems in mathematical physics, i.e., the l-d heat

equation, a wide range of applications exist.

ANALYSIS

The governing equation for the l-d, unsteady finite reservoir problem is a scalar form of the

heat equation, and as such, requires no further discussion. The boundary conditions, however,

Analytical Solutions

decelerated plate

x=%1 r=J$zYz7

hfinite, constaut potential, reselvolr poms dam

. h(x,t) I* . . . . . ___.._... ..+

___..,..... .d. /

finite water

.--.-.-.-.. . . . . ..__________...... _ . . . . . . . lk%tXVO~ X x=1

Figure 1. Schematic of finite reservoir diffusion problems.

do require further analysis. At z = 0, the primary variable of interest is specified, say $0, t) = 0. Note that by defining a suitable transformation v(z,t) = ~(2, t) + ~0, it is always possible to convert our problem to one with nonhomogeneous boundary conditions. Consider the more interesting boundary condition between the diffusive field and the finite reservoir, chosen to be z = 1 (see Figure 1). To develop the required boundary condition, we perform a global balance of the entire finite reservoir. If it is assumed that the flow is only possible through diiusive media, it is possible to equate the local flux to the time rate of change of conditions within the reservoir

CAL,% = Aq(l),

where c is a density x capacitance term, and A (cross-sectional area) and L, (finite reservoir length) are finite reservoir geometry parameters, e.g., L, = length of the landfill, diameter of the journal bearing, thickness of the underlying structure in a reentry vehicle, etc. The necessity of physically demanding a highly permeable reservoir for the ground water flow problem or a well stirred fluid in the mass and heat transfer cases is apparent here. Using the appropriate flux law

q(1) = -kF,

we may write

76 L. J.DE CHANT

Finally, initial conditions for this problem are a given constant primary variable value, say, uic, at t = 0. Hence, our governing system becomes

_=KaZU &J & 8x2

where K is a diifusivity (or kinematic viscosity as required conditions are generalized to, at z = 1:

(4)

by the problem) and the boundary

at _K, au(1% t,

ax (5)

and x = 0.0; and the initial condition

u(0, t> = 0, u(x,O) = l&J. (6)

The problem ss described by equations (4)-(6) is the basic differential equation describing the finite reservoir problem. An eigenfunction expansion solution is developed and compared to a simple approximate analytical solution, and a finite difference numerical solution.

ANALYTICAL SOLUTION METHOD

An exact solution to equations (4)-(7) may be developed by the classical separation of vari- ables method. These equations are complicated by the unsteady behavior of equation (5). The consequences of equation (5) will become apparent later. Beginning with the separation of vari- ables procedure, we propose the solution, ~(2, t) = a(t)+(x). Separating, we obtain the expected ordinary differential equations

4(X) + P+(z) = 0,

a(t) + KX2a(t) = 0.

Considering the eigenvalue problem, equation (7) first, we condition at x = 0, is trivial and yields d(O) = 0. The unsteady simplified via equations (4) and (7) evaluated at x = 1, to yield

f$(l) = X2$(l).

(7) (8)

obtain the eigenfunction. The boundary condition (5), may be

Equation (7) is solved to yield the eigenfunction

&(x) = sin&z, (10)

and the eigenvalues

&tan& = $. (11)

Although, at this point equations (7) and (9) appear to be a classical Sturm-Liouville problem, such is not the case. Equation (9) contains the unknown eigenvalue within it. Thus, the concept of classical orthogonality cannot be applied to this problem. To see this we apply Greens formula

substitution of equations (7) and (11) yields

(A: - A;) (/l &(x)$z(z)dx + $JW)W)) = 0. (13) 0

Analytical Solutions 77

Since, the eigenvalues are distinct, we must demand the second term in equation (13) is zero for

all values, thus, we define the weighting function

u(z) = 1+ $6(s -

and therefore,

s 0

119 (14

= 0. (15)

This development is analogous to one presented by Haberman [l] for the wave equation (the particular problem involved a vibrating string with an attached mass). Haberman also states that this type of problem may be solved by Laplace transforms. Carslaw and Jaeger [7], present a number of related heat conduction problem solutions for a slab with one face in contact with a layer of perfect conductor or well stirred fluid. They use Laplace transform methods to obtain similar solutions. In the following section and Appendix II, we will discuss the use of Laplace transforms to solve equations (4)-(6) in detail. As pointed out by Ozisik [8] and Haberman [l], the modified orthogonality method provides solutions more directly. Tittle [ll] has applied orthogonal methods to composite regions in heat conduction. Indeed, the analysis here can be shown to be a limiting case of their problem. The straightforward nature of the current development is shrouded by the complexity of Tittles composite region solution.

The concept of modified orthogonality may be used to satisfy the initial condition of our

family of problems, i.e., equation (7)

Application of the orthogonality relationship equation (15) yields

Computation of the required integrals (and application of the Dirac delta function) yields the

constant zlje [(l/X,)(1 - cos X,) + (K/K) sin X,]

, = (l/2) - (1/4X,) sin2X, + (K/K) (sin2 X,) (W

This relationship may be simplified via equation (11) an several trigonometric relationships to d yield

7~ = sin2X, [(K/K)2 + (X,)2 + (W/K)] (19)

The details of this derivation are presented in Appendix I. Finally, solving equation (8) to yield the expected exponentially decaying solution, we may write the series solution

21(2, t) = 2 bn sin X,ze-K(X=)at. (29) n=l

Equations (ll), (19), and (20) yield a formally exact solution. The behavior and adequacy of this solution is discussed further in the next section. Comparisons are made to simple approximate and numerical solutions.

78 L. J. DE CHANT

RESULTS

In this section, we will discuss equations (19) and (20), the solution obtained in this paper, in detail. As a starting point, equation (20) is compared to the solution obtained in [7], which was solved using Laplace transform methods. Referring to reference [7, p. 128, Section 3.131, equation (8), Carslaw and Jaeger obtain the solution

I1(x,t) = 2 2uie(K/K) sin(X,z)esKAZt

n=i siri& [(K//K) + (X,)2 + (K//K)] * (21)

Equation (21) does not agree with equations (19) and (20). It is the authors opinion that the solution presented in [7] is erroneous (typographical error). The basis for this claim is as follows.

1. Direct computation of equation (21) with t = 0 fails to recover the proper initial condition. As is shown subsequently, equation (19) does indeed yield the proper initial condition.

2. Appendix II presents a solution of equations (4)-(6) using Laplace transform methods and analytical inversion. This solution is shown to recover equations (19) and (20) precisely.

Accepting the solution presented here, we consider equations (19) and (20), in detail. Starting with a contour plot of the solution. The effect of the rapid drop for x = 0.0 and subsequent diffusion from the reservoir at x = 1.0 is apparent from Figure 2. Further, the treatment of the proper treatment of the initial condition by equation (19) is shown. We note, however, that for x = 0.0 that the independent variable u(x,t) is always zero. This ambiguity between the boundary condition for small time ~(0,t + 0) = 0.0, and the initial condition u(x ---) 0,O) = 1.0 is well known [l]. The relative error in the initial condition and the series representation is shown in Figure 3. It is worth noting that the fully numerical solution described subsequently, also performs poorly in the presence of this singularity.

0 0.6 1 1.6 2 2.6 3 3.6 4 4.6 Dimenrlonlesa Time

Figure 2. Contour plot of analytical solution equations (19) and (20), K = K = 1.

The variable of greatest interest is the primary independent variable, at the diffusive media- reservoir boundary condition, i.e., U(X = 1). We compare equation (20) to a classical approximate

solution based upon a lumped capacitance procedure

For clarity, a detailed development of equation (22) is provided in Appendix III. As a more accurate comparison to the analytically based solutions, a discretized numerical so-

lution was developed. The parabolic form of (19)-(22) makes the second-order time, second-order space [12,13] method a reasonable choice. Since this method is a relatively standard technique, a

Analytical Solutions 79

lhw Appmctm I .O for x-0.0.

0.01 0. IO Dimensionless Space

1.00

Figure 3. Initial condition relative error ju(x,O) - 11. Notice the increase in error near I = 0 caused by ambiguity between the initial and boundary conditions for small time and 2 = 0.

detailed development will not be presented, but for the sake of completeness, a summary deriva- tion is now provided.

Discretization of the time and space derivative terms of equation (4) yields

-ruy&r + (1 + 2r)uy+ - ruyZrr = ru;+r + (1 - 2r)$ + ru~_r, (23)

where AtK

T=%s* (24)

Equation (23) clearly, represents a sparse banded matrix which may be efficiently solved by decomposition [13]. Application of boundary conditions for this problem, equations (5) and (6) must be applied. Condition (6) is trivial and yields ur = 0, for t > 0. The time dependent condition (5) may be written

(26)

Equation (62) contains two fictitious nodes at i max +l. These unknown values may be elimi- nated by application of (60) at z = L, (i = i max). Simplification of this result yields

( 1 + 2r - 5) IL::& - 2ruy,+k _1 =

( 1 - 2r - 5) uTmax + 2ruym, _ r ( (27)

which is in the correct form for matrix inversion. Comparison of the numerical result to the analytical results is shown in Figure 4.

Referring to Figures 4a and 4b, it is seen that the modified orthogonal solution, equation (20); agrees well with the numerical solution at the finite reservoir/diffusive media interface, i.e.,

L. J. DE CHANT

- Mod. OrcbgOMl rol.: oq. (20): x-1.0

-- - - Numerical solution: x-1.0

.~~...~~~~~~~.. Appmximate solutiw; equ. (U): x-1 .O

.-.-.- Mod.o~~o~rol.;og.(20):~.1

...-..-..-.. Numsrid solution; x-O.1

t I I I I I I I I I

0.00 2.00 4.00 6.00 8.00 IO.00 Dimensionkss Time

(a) Comparison between solution methods for e general finite reservoir diffusion prob- lem, linear scale.

soh*oaMdilod

- Mod. c&o8amlSDl.; cqu. (20); x-1 .o

----. Nwwi~I~~Iu(ion;rl.O . * . ~Wlution;ogu.(22~x-1.0

-.- -.- Mod.olthogomlml.:cqu(202,x-o.1

._.____.__... NW,&&,,,.&-,.,

0.00 2.00 4.00 6.00 8.00 10.00 Dimawioniess Time

(b) Comparison between solution methods for a general finite reservoir diiueion problem, logarithmic scale.

Figure 4.

Analytical Solutions 81

5 = 1.0. The simple approximate solution, equation (21), also predicts the general trend ade- quately. The disagreement between the analytical solution and the numerical solution is partially attributable to the discretization error; A z, A y = 0(1/10)2, but is also due to errors caused by slight instability of the numerical method. This numerical instability is shown more dramatically near the P = 0 boundary as shown in Figure 2, for x = 0.1. Though it is beyond the scope of this discussion, we note that Ferziger [13] provides a relevant discussion of the causes numerically unstable behavior associated with the supposedly unconditionally stable Crank-Nicolson method. This behavior is clearly linked to rapid changes and high frequency errors being poorly damped by this method. The analytical solution does not exhibit such a limitation, though the analytical solution does suffers from a Gibbs [l] jump phenomenon near the discontinuity.

In Figure 5, the effect of a range of values for K/K on the drawdown, i.e., ~(1, t) is presented. Recalling that K cc K/L,, where L, is a finite reservoir size parameter (length of the landfill, diameter of the journal bearing, etc.), it is apparent that small values of K/K correspond to large reservoirs, while large values indicate a more limited finite reservoir. Given this physical interpretation, the slow drawdown for K/K < 1 (large reservoir) with proportional increase as K/K increases is physically plausible.

.. ..

\. \..

...

. . \

- KVK-O.001

----- KKQ.OI

.___ .._.. _... KIK-I,,,,,

-_ ---_,.

.-.. ..._,

--.

.-._

o.oow 1 I 1 I 1 I , I 1 I

0.00 I.00 2.00 3.00 4.00 5.00

Dimensionless Time

Figure 5. Finite reeervoir drawdown, u( 1, t), for a range of K/K values.

CONCLUSIONS

In this paper, we have considered an eigenfunction expansion solution method useful for a family of l-d, unsteady diffusion dominated transport problems characterized by transport from a bounded, and therefore, time dependent reservoir to an infinite reservoir. Example problems were introduced from: hydrology, e.g., effluent leakage from a landfill; mechanical engineering, e.g., deceleration of shaft in a journal bearing, and aerospace fields; e.g., unsteady thermal loading of the skin of a m-entry vehicle. The ubiquitous appearance of the heat equation in mathematical modeling highlights the potential uses of this analysis. It was seen that simple orthogonality was not applicable to the finite reservoir problem. To overcome this limitation, an extended eigenfunction expansion method was developed by modifying the weighting function within the orthogonality definition. Using this method, an analytical series solution was obtained.

a2 L. J. DE CHANT

Comparison with numerical and approximate methods is good. As an aid in validation and as a direct comparison of solution methods the series solution is also obtained using Laplace transform methods and analytical inversion. We note in closing, that this type of analytical solution provides a convenient physically plausible solution form useful in its own right and as a test problems for numerical implementations.

APPENDIX I

DERIVATION OF EQUATIONS (19) AND (20)

Previously, the concept of modified othogonality was developed as extension to classical eigen- function expansion solution methods for a family of scalar diffusion problems. A series solution for a canonical diffusive problem was derived. Here we provide details of the integration process used to obtain this solution.

The eigenfunction expansion of the initial condition starts with

U(Z, 0) = uie = 5 b,&(z). n=l

(1.1)

Application of the orthogonality relationship

with

u(x) = 1+ $S(z - l),

yields

b, = sd_ wohd44x> dx s, #,&> dx *

Substituting the eigenfunction, equation (10) into equation (1.4),

.I 1

49(z>42(z)dr> da: = 0, 0

uio b, = [

Ji sin X,x dx + (I-P/K) sin A,.,]

[ s, sin2 X,X dx + (W/K) sin2 x~]

(I-2)

(1.3)

(1.4)

(I-5)

Computation of the required integrals yields the constant

vi0 [(l/&J (1 - cos X,) + (K/K) sin X,]

bn = (l/2) - (1/4X,) sin2X, + (K/K) (sin2 X,) * (1.6)

This relationship may be simplified via equation (11) and the trigonometric relationships

sin2 X, + cos2 X, = 1 (1.7)

and i sin 2X, = co9 X, sin X,,

to yield equation (19)

bn = sin2X, [(K/K)2 + (X,)2 (K//K)] (1.9)

Solving equation (8) a(t) + K&(t) = 0 (1.10)

Analytical Solutions 83

is trivial since it is a first-order linear equation. This solution provides the expected exponentially decaying solution permitting us to write the series solution, equation (20)

u(z, i!) = 2 b, sin Xnze-K(Xn)2t, (1.11) n=l

thus completing the derivation. Before leaving the series solution, it is of use to note that computation of the eigenvalues X,,

from equation (11) can be somewhat tedious since by definition they are periodic. Here Newtons method was combined with a set of asymptotic solutions which were used to obtain an accurate first guess. The asymptotic solutions

sin(x,) x 0 ==+ X, M n7r, (1.12)

for n 2 2 and K//K small and

cos(X,) M 0 I A, a (1.13)

for n 2 2 and K'/K large.

APPENDIX II

SOLUTION OF THE FINITE RESERVOIR PROBLEM BY LAPLACE TRANSFORM

In this Appendix, the finite reservoir is solved using Laplace transform methods. Using the inversion integral [1,14] and residue theorem [15], we can analytically invert from the Laplace plane to the real plane. This method recovers the form of equations (19) and (20), previously derived by eigenfunction expansion, exactly. Indeed, the alternative solution method presented here provides confidence in the correctness of equations (19) and (20), and casts doubt upon equation (21).

The solution begins with equations (4)-(6), repeated here for convenience. Governing equation

au i3=u -=K-@ at (11.1)

and the boundary conditions

azl(l,t)= at

_K,au(l,t) ax (11.2)

u(0, t) = 0, u(x, 0) = 7&l). (11.3)

Transforming equations (11.1) and (11.2) yields

and

Kd2?i -@ = STi - uio

,W,t) ~?'i--u~~=--K- dx

where the transformed variable is defined

(11.4)

(11.5)

J m

TiE esdtu(x, t) dt. 0

(11.6)

84 L. J. DE CHANT

Solution of the nonhomogeneous equation (11.4) is trivial and yields s l/2

n(x, s) = A(s) cash i7 ( > s 112

z + B(s) sinh i7 ( > x+3!. S (11.7) Boundary condition u(O,t) = 0 implies A(s) = --zlto/s. Combining equations (11.5) and (11.7) permits us to obtain B(s):

(K/(sK)li2) sinh (s/K)~ + cash (s/K) 12]

B(s) = (s/K)~ [Kcosh (s/K)12 + K (s/K)12sinh (s/K)12] (11.8)

hence, we write

n(x,s) = -T cash ; ( >

l/2 z

+ uie [(K/(sK)i2) sinh(s/K)li2 + cosh(s/K)1/2] sinh(s/K)i2z + s (11.9)

(s/K)f2 [Kcosh(s/K)1/2 + K(s/K)r/2 sinh(s/K)r/s] s .

The only singularities of equations (11.8) or (11.9) are

Kcosh (G)12 + K (a)12sinh (s)l12 = 0. (11.10)

Letting (s/K)i2 = fX,i, applying the complex identities sinh(iz) = isin and cosh(iz) = cos(a) and rearranging, one obtains the transcendental eigenvalue relationship, equation (11):

X,kUlX, = g. (11.11)

Inversion of equation (11.9) to the real plane may be accomplished using the Fourier transform inversion pair specialized for Laplace transforms combined with the residue theorem. Due to the existence of simple poles, the inversion is particularly simple and takes the form

F(s) = s. n

(11.12)

Applying this method to equation (11.9)

u(z, t)

= O uic [(K/(sK)j2) sinh(s/K)1/2 + cosh(s/K)1/2] sinh(s/K)i2x

c n=l & [Kco~h(s/K)l/~ + K(s/K)li2 sinh(s/K)1/2]

@K(S/K)t) . (11-13)

Computing the necessary derivatives, simplifying through equation (11.10) and using (s/K)lj2 = fX,i, with the complex identities sinh(iz) = i sin(z) and cosh(iz) = cos(z), one obtains

O U(X, t) = 2uio c

(K + (K2/K) (l/Xi)) cos A, sin X,X~-~+

(K + (K2/K)Xi + K) sin X, * (11.14)

n=l

Once again applying equation (5) to simplify and the two trigonometric identities

sin2 X, + cos2 X, = 1

and 1 2

sin 2X, = cos X, sin X, ,

(11.15)

(11.16)

yields

21(x, t> = 2 4uto(K/K) sin X,,xesKXit

n=l sin2X, [(K//K) + (X,)2 + (K//K)] (11.17)

which is precisely equations (19) and (20). This completes the solution of the finite reservoir problem in terms of Laplace transforms.

Though precisely the same solution has been obtained, it is the authors opinion that this has been achieved at the expense of greater algebraic complexity. Additionally, the inversion has several rather subtle requirements, i.e., recognition of the simple singularities and complex eigenvalues (s/K)~ = fX,i. It seems clear, however, that operational methods are to be preferred for more complex multiple region problems, especially when combined with numerical inversion.

Analytical Solutions 85

APPENDIX III

APPROXIMATE ANALYTICAL SOLUTION METHOD

The most apparent complexity in equations (4),(5) lies in the time dependent term of the boundary condition (5) :

W1,t) _ at

_p4 t) 7

(III. 1)

If, it is possible to predict the behavior of ut(l, t); the current problem would form a well- posed homogeneous equation with nonhomogeneous boundary conditions. Elementary analytical techniques permit the solution of this type of problem. The basis of the solution method that we will now describe is to estimate this time behavior, by developing an approximate solution.

The computation of the approximate solution may be performed by developing a solution that is only required to satisfy the boundary condition (5). The governing equation itself will not be satisfied. This type of approximate solution, often termed a lumped capacity solution, is widely used in heat transfer applications [6]. The development of this solution is begun by proposing a solution of the form

n(z, t) = g(t)% (111.2)

Equation (111.1) obviously satisfies the homogeneous boundary condition. The unknown function, g(t), is computed by applying the time dependent boundary condition (111.1):

g(t) = -Kg(t). (111.3)

Solution of this initial value problem and application for the initial condition ~(1, t) = 1li0 com- pletes the approximate (or lumped capacity solution) which is written:

n(z, t) = u&-KtZ. (111.4)

This completes the derivation of the approximate solution equation (22). It is desirable to make comparison between the exact analytical solution, i.e., equation (20) and

the approximate analytical solution, equation (111.4). For simplicity, the time dependent portion of both solutions will be examined. Thus by equation (20), with n = 1, we write

j&&(t) = e-K(Al)zt. (111.5)

As a first order estimate, we will describe the approximate analytical solution by equation (111.4):

far&) = /t. (111.6)

The relationship bridging (111.5) and (111.6) is the eigenvalue relationship equation (11) for n = 1:

xi tanA = $. (111.7)

Expressing the trigonometric function in (111.7) in Taylor series

tanAl=Al+y+... , (111.8)

and then retaining the lowest order term only

0d2 ; R-.

Thus, equation (111.5) may be approximated:

(111.9)

(111.10) e-K(XIPt M e--K(K/Wt = ,-Kt

86 L. J. DE CHANT

Therefore, it is possible to conclude that the approximate solution, equation (22) closely matches the exact solution for small values of the eigenvalues (AIL).

We note as an aside, that using this approximation we could derived an improved approximate solution by modifying the boundary condition equation (111.1):

aa t) 1 Lz(1,t) -Kt -M---==ZLiOe . ax K at (111.11)

However, since an exact series solution is available, this approximation is of lessor interest and is thus quoted without proof:

00

u(x, t> = uiOxe-Kt + c a,(t) 1 sin n- - 7rx, n=l ( > 2

where the terms are defined:

and

2uio K A= (n_I,2)2n2sin

I- (n - :/2)7r

a,(t) = (a,PnK,) [ e-Kt - emant 1 + a(0)emamt.

(111.12)

(111.13)

(111.14)

(111.15)

(111.16)

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