Evaluation of the eigenvalues for the graetz problem in slip-flow

Int. Comm. Heat Mass Transfer, Vol. 23, No. 4, pp. 563-574, 1996 Copyright © 1996 Elsevier Science Lid Printed in the USA. All rights reserved 0735-1933/96 $12.00 + .00

Pergamon

eII S0735-1933(96)00040-1

EVALUATION OF THE EIGENVALUES FOR THE GRAETZ PROBLEM IN SLIP-FLOW

Randall F. Barron, Xianming Wang, Robert O. Warrington, and Tim Ameel Mechanical & Industrial Engineering Department and Institute for Micromanufacturing Louisiana Tech University Ruston, Louisiana 71272 USA

(Communicated by J.P. Hartnett and W.J. Minkowycz)

ABSTRACT A technique was developed for evaluation of the eigenvalues for the Graetz problem extended to slip-flow. The first four eigenvalues for Knudsen numbers of 0.02, 0.04 ..... 0.12 were found. By using a least square curve-fit method, simplified relationships between the eigenvalues and Knudsen number were obtained. The efficient and accurate determination of the eigenvalues will lead to better predictions of heat transfer in rarefied gas flows and for gas flows in microtubes.

Introduction The Graetz Problem The Graetz problem is a simplified case of the problem of forced convection heat transfer in a circular tube in laminar flow.

With the assumptions of steady and incompressible flow,

constant fluid properties, no "swirl" component of velocity, fully developed velocity profile, and negligible energy dissipation effects, the problem was first solved over a century ago [1]. The solution by Graetz involved an infinite number of eigenvalues; however, only the first two eigenvalues were evaluated. Since the accuracy of the Graetz solution depends on the number of eigenvalues, it is extremely important to obtain more eigenvalues [2]. For seventy years the research for this 563

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R.F. Barron et al.

problem focused mainly on finding more eigenvalues.

Vol. 23, No. 4

The first significant advancement

employed a fairly rapidly converging series solution of the Graetz equation to determine the lowest five eigenvalues with much more accuracy [3].

The problem was then extended to

include a more effective approximation technique for evaluation of the eigenvalues in which any number of eigenvalues could be estimated [4]. The Graetz Problem in Slip-Flow Applications of microstructures such as micro heat exchangers have led to increased interest in convection heat transfer in micro geometries.

Some experimental work has been

reported, such as the experimental investigations in microtubes [5], in microchannels [6], and in micro heat pipes [7]. Appropriate models are needed to explain the significant departures in the microscale experimental results from the thermofluid correlations used for conventional-sized geometries. For example, the measured heat transfer coefficients in laminar flow in small tubes exhibited a Reynolds number dependence, in contrast to the conventional prediction for fully established laminar flow, in which the Nusselt number is constant [5]. Also, an experimental investigation of fluid flow in extremely small channels showed that there are deviations between the Navier-Stokes predictions and the experimental observations [6]. Therefore, some effects and conditions that are normally neglected when considering macro-scale flow must be taken into consideration in microscale convection. One of these conditions is slip-flow [8,9]. It has been found that the analytical model combined with slip flow conditions can fit the experimental data in microchannels with uniform cross-sectional area [10] and with non-uniform cross-sectional area [11]. Slip-flow occurs when gases are at low pressures or for flow in extremely small passages. At low pressures, with correspondingly low densities, the molecular mean free path becomes comparable with the body dimensions, and then the effect of molecular structure becomes a factor in flow and heat transfer mechanisms [12]. The relative importance of effects due to gas rarefaction can be indicated by the Knudsen number, a ratio of the magnitude of the mean free molecular path in the gas to the characteristic dimension in the flow field. The effects of rarefaction phenomena on flow and heat transfer becomes important when the Knudsen number can no longer be neglected.

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EIGENVALUES FOR THE GRAETZ PROBLEM

565

In defining when slip-flow occurs, a classification of four flow regimes for gases has been proposed [9] as follows: Continuum flow: Slip-flow:

K n < 10.3

10.3 <_K n < 0.1

Transition flow:

0.1 z K n < 10

Free molecular flow

I0 _
When slip-flow occurs, the gas adjacent to the surface, in contrast to its behavior in continuum flow, no longer reaches the velocity or temperature of the surface. The gas at the surface has a tangential velocity; the gas slips along the surface. The temperature of the gas at the surface is finitely different from the temperature of the surface.

Expressions for the

temperature jump condition and slip velocity at the surface are well known [12].

The slip

velocity as a function of the velocity gradient near the wall may be expressed as:

u,

-

\-~rj

,°

(I)

While an alternate form of Eq. (I) is available [12] which includes the specular reflectance F, this form was not used in this investigation, since for most engineering surfaces, F has values near unity, in which case the alternate form reduces to Eq. (I). The original solution by Graetz is valid for continuum flow; however, for gases at low pressures or in extremely small tubes, the flow may enter the slip-flow regime, in which case the velocity at the tube surface is not zero. In this case, the heat transfer coefficient depends not only on R e and P r , but also on Kn.

This additional independent parameter complicates the

determination of eigenvalues for the slip-flow problem. Therefore, a new technique is needed to evaluate these eigenvalues. Mathematical Statement The Graetz Problem in slip flow considers flow in a cylindrical tube with a fully developed velocity profile and a constant wall temperature. For T = T(r,x), the problem may be stated as

upcp ~-x = rO-rr\

(2)

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where the velocity profile for a slip flow condition is [4]:

U ~- U m

(3)

l+8Kn

The boundary conditions are

T(r,O) = To

x <_O;

for

T(ro, x) = Tw

for

x> 0

(4a, 4b)

where the temperature jump condition is [12]

vw(0)=ro

F v+ler--,0,

The nondimensional form of the problem in slip-flow is:

-

2(l+8Kn)#r + (1-r+2+4Kn)r + r+

--~-)

(6)

with the boundary conditions:

0(1, x ÷) = 0

for x+ > 0 ;

O(r +, O) = 1

for x ÷ _<0

(Ta, 7b)

In view of the linearity of Eq. (3), it is necessary to have only the fundamental solution, known as the Graetz solution, to construct all other desired solutions. The Graetz Solution A solution to Eq. (6) may be formed utilizing Eqs. (Ta) and (Tb) and may be expressed as

O(r+'x+ ) = E C,R,,(r+ ) e-~x+~

(8)

n=o

where the coefficients C, are determined from the relation [13]

c,, =

-2

¢a£)

"k O;t. ) +=l.~..~.

The eigenvalues ~n must satisfy the differential equation

(9)

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EIGENVALUES FOR THE GRAETZ PROBLEM

i 2 + r + R, ~w+R,+3, r (1 + 4 K n - r

+2

567

)R,=0

(10)

where the prime (') refers to differentiation with respect to r +. The required boundary conditions are derived from Eqs. (7a) and (7b): R(1)=0;

R(0)=l

(lla, llb)

The solution of Eq. (10) can be obtained by the method of Frobenius resulting in: R(r+) : X akr+:~

(12)

k=0

The first boundary condition is used to derive the eigenfunction for the desired eigenvalues:

R(1)

=

~a k

--

a0 +a I +a 2 + .... 0

(13)

k=0

An approximate method for calculation of eigenvalues for the classical Graetz Problem, that is Kn = 0, is available [4]; however, it is not suitable for the Graetz problem in slip-flow. One of the key features of this work was to find a technique to calculate the eigenvalues for the latter problem. Methodology Since the coefficients ak in Eq. (12) for the Graetz problem extended to slip flow have the recursion relationship ao = 1

a1= -

(1 + 4Kn)

a,=-

[( 1L.l+4X'a,_,-a,_2j )

Sork=ZS,4

(13)

....

the eigenfunction can be rewritten in terms of the eigenvalues 3. as follows: R()v) = ~ k=0

) J ' d , : 1 + L2d, + ~.'d 2 + ~,6d3 +

....

0

(14)

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R.F. Barron et al.

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Since the numbers related to the order of )2 in ak need to be summed, ak may be split into bt, k as the elements of matrices Bixk. Thus

0

0

0

0

0

b2,1

b2,2 ~2

0

0

0

0

0

b3.2~

b3,3~

0

0

0

0

b4,2

b~,J [32 b4.4~4

0

0

0

o

b,.3~

b,,4[¢ b,~,~'

0

0

b6,3

b6,4[~2 b6,s[34 b6,6[~6

b ~.if~

B,xk =

o

where k = 1,2,3 .... i = k + 1, k + 2, k + 3 ..... 2k, and [3 = ( l + 4 K n ) for the problem in slip-flow. From the matrix, a formulation may be obtained which can be used to directly calculate the coefficients related to the index instead of the expansion of each term ak. The formulation is [141 -- l) i

b,,

(15a)

22,(it.) 2

and for i > k

.

[E

i,,

""

~z~=A

.%

-1

S2 +1)

r 11]1 S~

.-.

(15b)

[_:q=l

where A = i - k. The coefficients in the eigenfunction can be found from 2k

dk = ~-" b;,,,,

(16)

i=k

where b,',,, = bi.k[3

(17)

The coefficients C~ are determined from the relation: -2

C,,

Z 2k)~2k d k k~l

08)

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EIGENVALUES F O R THE G R A E T Z PROBLEM

569

Results In the implementation o f the above formulation, some inherently large numbers must be computed that can cause an overflow on a computer. Three different means were utilized to deal with this problem. First, the 22i term in bi,~of Eq. (15b). Second, by letting X' = Mg and d k' =

g2kdk and allowing the scaling factor, g, to be any number greater than 1, results in

k=O

Third, the magnitude of

Ibi, il in Eq. (15a) was reduced for computational purposes by

taking the logarithm o f both sides and later reversing this computation using the inverse of the logarithm function. Using 25 terms in the summations o f the formulation, the eigenfunction for a non slipflow condition (Kn = 0) was computed, as shown in Fig. 1. This data indicated that the first four eigenvalues are correct, but the last two (5 and

g(k')

6)

1.0

are

somewhat

inaccurate

due

to

the

k=25 g=lO

truncation of the eigenfunction expression. Fig 1

also

shows

that

the

eigenfunction

is

.5

oscillating with decreasing magnitude as X'

.

0 increases. Figure 2 shows the behavior of the

,

8

16v

-.5 FIG. 1 Plot of the eigenfunction with 25 coefficient terms forKn = 0, k= 25, and g = 10

eigenfunction as a function of the number of coefficient terms. The curve numbers indicate the

number

of

dk terms

used

in

=

the

computations. At least six terms are needed to

=

.5

obtain the two lowest eigenvalues, shown as the curve with number 6. Obviously,

0 the

second

eigenvalue -.5

determined by curve 6 is less than the "accurate value", which is slightly greater than 6.6 (see

FIG. 2 Behavior of the eigenfunction vs. the number of coefficient terms for Kn = 0, k=-25, and g = 10

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R.F. Barron et al.

Vol. 23, No. 4

Table 2), shown as curves 7, 8 and 9 etc. in Fig. 2. The true second eigenvalue lies between those obtained by curves 7 and 8. It is clear that the second eigenvalue obtained initially by curve 6 is rather rough; with one more term, the value is larger than the first one as shown by curve 7, but closer to the "accurate value"; the second eigenvalue given by curve 8 is a little less than that given by curve 7 but is approaching closer to the "accurate value". The same holds true for the case with nine terms, ten terms, and so on. From this discussion it may be concluded that: (1) the eigenvalues initially obtained with the minimum number of coefficients is always rather rough, such as the second eigenvalue obtained by curve 6 and the third eigenvalue obtained by curve 9, (2) the next to the last available eigenvalue that can be determined for a certain number of coefficients is always correct and sufficiently accurate (for instance, for curve 7, the second eigenvalue can be assumed to be reasonably accurate and correct), and (3) the eigenvalues and the convergence of the eigenfunction are sensitive to the accuracy of the coefficients dk. Graetz [1] found the first two eigenvalues for the traditional problem, )~ = 2.7043 and )~2 = 6.50, from the following eigenfunction: 0 = 1 - 0.1875 )2+ 0.007921 ~4_ 0.00014404 )6+ 145.92x10-8 ~8_ 94.938x10-10) ~0+ .,.

(20)

The coefficients dk from Eq. (20) are compared with those computed using Eqs. (15a) to (17) in Table 1, Significant differences are seen to exist for terms higher than k = 4, which explains the inaccuracy in Graetz's second eigenvalue. TABLE 1 Comparison of Coefficients dk (g = 10) Simulation by Mathcad Version 5.0 Eq. (15) 1.00000

-18.7500E-02

Numeric equivalent 1.00000

-18.75000E-02

79.2101E-04 -144.043E-06 145.080E-08

79.210069E-04 -144.04297E-06 145.07980E-08

-92.6715E-10 40.8619E-12 -13.1812E-14 3.24941E-16

-92.67144E-10 40.861856E-12 -13.181156E-14 3.2451315E-16

-0.646315E-18

Symbolic solution 1

-3/16 73/9,216 -59/409,600 603,793/416,197,814,400 -555,379/59,929,893,273,600 4,266,870,481/104,421,846,039,920,640,000 -37,217,872,147/282,356,671,691,945,410,560,000 41,377,942,693,441/127,507,755,229,335,476,282,327,040,000

-0.62971547E- 18 -9,281,940,782,645,851/14,739,896,504,511~181,058,237,005,824E+6

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EIGENVALUES F O R THE G R A E T Z PROBLEM

571

Table 2 shows a comparison with previously known eigenvalues [4] for the classical Graetz Problem ([3 = 1 or Kn = 0).

The first four eigenvalues from this work are in excellent

agreement with those previously published.

Therefore, this technique may be applied with

confidence to the Graetz Problem in slip-flow for evaluation of the eigenvalues. TABLE 2 Com ~arison with previously known eigenvalues Jakob [4] Analog computer [4] Sellar et al. (1956)

This paper

2.667 6.667

+1.47989 -0.80345

2.705 6.66

+1.477 -0.81

2.71 6.69

+1.46 -0.809

2.704 6.679

+1.476510 -0.806124

10.667 14.667

+0.58732 -0.47499

10.3 14.67

+0.385 -0.479

10.62 14.58

+0.592 -0.51

10.670 14.761

+0.57934 -0.512304

18.667

+0.40445

17.255

+0.312379

Table 3 shows the first five eigenvalues for slip-flow with different Kn. Note that each eigenvalue decreases in value with Kn. TABLE 3 Eigenvalues for different Kn

Kn 0.00

~1 2.704

~2 6.679

~3 10.670

~4 14.761

~5 17.255

0.02 0.04 0.06 0.08 0.10

2.578 2.468 2.371 2.284 2.206

6.320 6.013 5.747 5.513 5.305

10.071 9.561 9.120 8.737 8.396

13.815 13.099 12.560 11.963 11.514

16.576 15.836 14.646 14.573 13.938

0.12

2.136

5.119

8.096

11.074

14.273

RO?) Fig. 3 shows the behavior of the eigenfunction

for

1

various Kn under

slip-flow conditions. Note again that the

0

eigenvalues decrease as Kn increases. For Kn > 0, the curves appear unstable

-1

after the fifth root so that only the first four values are reliable.

The possible

cause for the instability is that the

-2 |

0

.5

1.0

1.5

FIG. 3 The effect of Kn on the eigenfunction

~,'

572

R.F. Barron et al.

Vol. 23, No. 4

truncation errors are magnified by the factor (l+4Kn) ~ on bi, i in the modified matrix A.

15

The data shown in Fig. 4 10 indicate that eigenvalues are functions of Kn.

For practical purposes, a

simplified expression for calculation of

the

eigenvalues

would

~'2 5

be

beneficial. An exponential expression

0

0

0.02

0.04

0.06

The general form for this simplified

0.08

0.10

0.12

Kn

was found to yield the best results.

FIG. 4 The first four eigenvalues as a function of Knudsen number

expression is )~, = C 1 + C 2 K n exp(C 3 K n )

(21)

where the constants C~, C 2, and C3 and the correlation coefficient R 2 are listed in Table 4 for various Kn. TABLE 4 Coefficientsin Equation (21)asfunctions o f ~ Z. Ci C2 C3

R2

2.704 6.679

2.704 6.679

-6.6236 -18.9118

-2.8482 -3.2003

0.9997

10.670

10.670

-31.7454

-3.3293

14.761

14.761

-49.5056

-4.2066

0.9972 0.9637

0.9997

Conclusion A new computationally effective and efficient technique has been applied to determine the eigenvalues for the Graetz problem in slip-flow. The eigenvalues for this problem have been shown to decrease with increasing Kn. A simplified exponential relationship for the eigenvalues as functions of K n provides a reliable and convenient computation method. The evaluation of eigenvalues for large )~ (n > 5) has been found to be extremely time consuming; an improved method with increased calculation speed would be of future interest.

Vol. 23, No. 4

EIGENVALUES FOR THE GRAETZ PROBLEM

573

Nomenclature

ak

coefficient in Eq. (12)

U

velocity in x direction

bi,k

elements in matrix Bixk

Uj

Cn

coefficient in Eq. (8)

average streamwise velocity of the incident molecules

c

characteristic velocity

u,,

average fluid velocity in tube

cp

heat capacity at constant pressure

ur

average streamwise velocity of the reflected molecules

dk

coefficient in Eq. (14)

us

slip velocity

D

tube diameter

Uw

F

specular reflection coefficient

average streamwise velocity of the surface

(ur- u) / (u~ - u)

x

distance along tube

g

scaling factor in Eq. (19)

x÷

nondimensional axial length

Gz

Graetz number (RePr(D/L))

k

thermal conductivity; number of terms in Eq. (15)

Greek Symbols [3

parameter in Eq. (8) (l+4Kn)

Kn

Knudsen number (A/D)

y

ratio of specific heats

r

radius

)t

eigenvalue; mean free path of gas

ro

tube radius

~,'

eigenvalue divided by g

r+

nondimensional radius (r/ro)

R

R(r+), function in Eq. (8)

Re

Reynolds number

Pr

Prandtl number

T

T(x, r), temperature

Subscripts

To

temperature of gas at x = 0

w

(x/r o)(RePr)- I

dynamic viscosity p

density

0

nondimensional temperature

(T-Tw)/(To-Tw) wall

References L. Graetz, Uber die Warmeleitungsfahigheit von Flussingkeiten, part 1, Annalen der Physik und Chemie, vol. 18, pp. 79-94, (1883); part 2, vol. 25, pp. 337-357 (1885). 2.

M. Tribus and J. Klein, Forced Convection from Nonisothermal Surfaces, Heat Transfer." A

Symposium, pp. 211-235 (1953). 3.

M. Abramowitz, On Solution of Differential Equation Occurring in Problem of Heat Convection Laminar Flow in Tube, Journal of Mathematics and Physics, vol.32, pp. 184-187 (1953).

4.

J.R. Sellars, M. Tribus, and J.S. Klein, Heat Transfer to Laminar Flow in a Round Tube or Flat Conduit-The Graetz Problem Extended, Trans. ASME, vol. 78, pp. 441-448 (1956).

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Vol. 23, No. 4

5.

S.B. Choi, R.F. Barton and R.O. Warrington, Fluid Flow and Heat Transfer in Microtubes, Micro-mechanical Sensors, Actuators, and Systems, ASME, New York, NY, DSC-Vol. 32, pp. 123-134 (1991).

6.

J. Pfahler, J. Hanley, H. Bau, and H.N. Zemel, Gas and Liquid Flow in Small Channels, Micro-mechanical Sensors, Actuators, and Systems, ASME, New York, NY, DSC-Vol. 32, pp. 49-60 (1991).

7.

G.P. Peterson, A.B. Duncan and M.H. Weichold, Experimental Investigation of Micro Heat Pipes Fabricated in Silicon Wafer, ASMEJHeat Transfer, vol. 115, pp. 751-756 (1993).

8.

M.I. Flik, B.I. Choi and K.E. Goodson, Heat Transfer Regimes in Microstructures, Journal of Heat Transfer, vol.114, pp. 666-674 (1992).

9.

A. Beskok and G.E. Kamiadakis, Simulation of Slip-Flows in Complex Micro-Geomitries, ASME Proceedings, DSC vol. 40, pp. 355-370 (1992).

10. E.B. Arkilic, K.S. Breuer and M.A. Schmidt, Gaseous Flow in Microchannels, Application of Microfabrication to Fluid Mechanics, ASME FED-Vol. 197, pp. 57-66 (1994). 11. J.Q Liu, Y.C. Tai and C.M. Ho, MEMS for Pressure Distribution Studies of Gaseous Flows in Microchannels, Proceedings of IEEE Micro Electro Mechanical Systems, pp. 209-215 (1995). 12. E.R.G. Eckert and R.M. Drake, Analysis of Heat And Mass Transfer, McGraw-Hill Book Co., New York, pp. 486 (1972). 13. W.M. Kays and M.E. Crawford, Convective Heat And Mass Transfer, McGraw-Hill, Inc. (1993). 14. X.M. Wang, Evaluation of Eigenvalues for Graetz Problem in Slip Flow, Master Thesis, Louisiana Tech University (1995).

Received December 14, 1995