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Study of the Fluid Flow in the Elliptical Duct Flow by the Method of Characteristics

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(1)

Yuan Mao Huang Professor.

C. H. Ho Research Assistant. Mechanical Engineering Department, National Taiwan University, Taiwan

Study of the Fluid Flow in the

Elliptical Duct by the Method of

Characteristics

This study develops a mathematical model to determine the properties of laminar flow in the elliptical duct. With some assumptions, the nonlinear governing equations

of the air in the elliptical duct are transformed into the hyperbolic type. The method of characteristics is then applied. Numerical results are obtained by using the finite difference method and the uniform interval scheme. The air properties in the elliptical duct are analyzed. The local Nusselt number and the heat transfer coefficient along the duct are studied. The numerical results are compared and show good agreement with the available data.

Introduction

The circumferential length of the elliptical duct is longer than that of the circular duct if the cross-sectional areas are the same. Hence, when the heat exchanger requires good cool effect and the space is an important factor in design, the el-liptical duct is used practically.

The governing equations for the air flow in the elliptical duct are nonlinear. Bodoia and Osterle (1961) transformed equations to get the appropriate solutions by numerical meth-ods. Lundgren et al. (1964) studied the pressure drop due to the entrance region in ducts with arbitrary cross sections. Igbal et al. (1972) used variational method for the steady fully de-veloped laminar flow. Dunwoody (1962) investigated the steady, viscous and fully developed laminar flow with constant wall temperature for the entrance region. Gilbert et al. (1973) used the slug flow model and ignored the influence of the viscosity for the forced laminar and steady flow to determine the heat transfer on the boundary. Abdel-Wahed et al. (1984) investigated experimentally the laminar developing and fully developed flow with the ratio of the major axis length to the minor axis length being 2.

The purpose of this study is to develop a mathematical model to analyze the air properties and to determine the local Nusselt number and the heat transfer coefficient of the air in the elliptical duct.

Mathematical Approach

The coordinate system for an elliptical duct is shown in Fig. 1. For a laminar flow field, the gradients of the velocity com-ponents in the y and z directions are negligible compared with the gradient of the velocity component in the x direction. The energy generated by the friction force is assumed much smaller than all other terms.

At the location far from the duct wall, the speed and the temperature gradients and the second-order differential of the

Contributed by the Pressure Vessels and Piping Division for publication in the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received by the PVP Division, July 31,1990; revised manuscript received September 4,1992. Associate Technical Editor: F. J. Moody.

heat transfer and the viscosity are very small; therefore, their effects are negligible. Large variation exists only at the location near the duct wall. The air temperature at the duct wall is assumed to be equal to the temperature of the duct wall.

The assumed dimensionless temperature is used to simplify the second-order terms in the differential equations. The ex-ponential form assumed for the temperature distribution is

T(x,y,z)-Tw(x)

T,(x)-Tw(x) = 1 - (1) where T, is the highest air temperature at the duct center and Tw is the air temperature at the duct wall. The exponential form assumed for the velocity distribution is

- + T (2)

where u, is the highest air velocity at the duct center in the x direction and um is the air mean velocity. The air is assumed to be an ideal gas.

The nondimensional variables used are

(2)

p*

=

£-

T

*

=

llzI°L

Pa' (Tin-Ta) .* * -^ ~(Dh/uiny Wh V* = JL *_'_L y D„' Z Dh Win V w = — ; c ' = - (3) Win CQ

Using Eqs. (1), (2) and (3), the column vector Lt of

nondi-mensional forms of the continuity, momentum and energy equations can be obtained as

p*8u p*d2i p*5u Ui 0 RTt p*R{Tin-Ta) P*U, 0 0 0 * * P U, o — «i/ " i n ^ i n RTt P*R(Tin-Ta) 0 - r J u T °2i ou Mi, . „ • RT P*R(Tin-Ta) , Win " in P*/}7- p*i?r P* i ? r — ^ r 01/ — z r °n _ ^ "a u Cvl a du dXj dv* dXj dw* dXj

K

dXj dT* dx. 0 [G]* 0 0 [F\* P*U, (4) where 5y = 0 if / ^ j , by = 1 if / = j nu, [G] = and |0fl"in-D/i - a 2( a2- l ) 2 / \ 2-al \b a4 + b4 -a2

avr

_2_ ^2_ 2 ' T 2 (5) rCT* k{T*-n){Tm-Ta) PacvTauiaDh

iHf

V 4z a ft - « [

s'-r

d , - i _2 ^

«

2

V

(6) The method of characteristics is applied by using the arbi-trary functions o>i, o>2, 013, 0)4 and o>5 to combine these

non-dimensional equations (Rudinger, 1969; Huang, 1992). Let the five coefficient vectors of the combined equation L\o)\ + L2oi2 + 1,3013 + Z,4o)4 + L5o)5 = 0 be related. Then, there exists a

common normal vector for these coefficient vectors as

P*8UG)l+p* 11,0)2 + — 51;0)5 Ni = 0 * * p RT p 52,-a>i + p Ujbi3 + — — 5I;a)5 N, = 0 Cvl a p 83,0)1+p U,<J04-\ — 81,0)5 N, = 0 RT U, o>i + — (81,0)3 + 82,o)5 + 83,0)4) Win (7) 7V, = 0 p*RT(Tm-Ta) (Si/0)2 + 82,0)3 + 83,0)4) + I ~ P t//W5 M = 0

If a four-dimensional normal vector and the velocity vector, respectively, are a b Bi c Cv Dh Gz h k N Nu Pr R Re T t = = = = = = = = = = = = = = = =

semi-major axis length of ellipse

semi-minor axis length of ellipse

Biot no. = ht„/k sonic speed

specific heat at constant volume

hydraulic diameter of duct Graetz no. = RePrDh/x

local heat transfer coefficient thermal conductivity

normal vector

Nusselt no. = hDh/k

Prandtl no. = Cpii./k

ideal gas constant Reynolds no. = puDh/ix

temperature time; thickness U u V w X y

z

y p M 0) velocity vector velocity component in x direction velocity component in y direction velocity component in z direction

distance along x coordinate distance along y coordinate distance along z coordinate ratio of specific heat at stant pressure to that at con-stant volume density viscosity nondimensional temperature distribution arbitrary function

a.\ = exponent of temperature

distri-bution

a2 = exponent of velocity

distribu-tion Subscripts

a = condition at room temperature

and standard pressure (303 K and 101325 Pa)

/ = component or vector in = condition at inlet of duct

m = mean value of cross section t = condition at center of duct

w = condition at wall Superscript

= nondimensional quantity

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Table 1 The data of the elliptical duct Table 2 The initial data and the input data of air properties

Semi-major axis length (a) Semi-minor axis length {b) Hydraulic diameter (Dh)

Ratio of a to b Length of the duct (/) Thickness of the duct (t„)

0.015469 m 0.0077349 m 0.020 m 2.0 3.0m t„ < 0.01 k/h (Bi <0.01) N<= (NUN2,N„NA) and C / , = ( H W , 1 )

the solution for co,- not all equal to zero is (t7*/V,)3 = 0 or U,N,= ± — "in (8) (9) (10) (11) Inlet velocity Inlet temperature Inlet pressure

Assumed outlet velocity Assumed outlet temperature Assumed outlet pressure Room temperature Viscosity

Thermal conductivity Original exponent of the

temperature distribution {a,) Original exponent of the

velocity distribution (a2) 2.0 m/s 383 K 101329 Pa 1.415 m/s 303 K 101325 Pa 303 K 2.002 x 10~6 kg/m-s 0.02709 w/m-k 4.0 6.0

Table 3 The size of the grid

Ax Ay Az At 0.15 m 0.003094 m 0.001547 m 0.0005 s Ax' Az' At' 7.50 0.1547 0.07735 0.0005 (12)

where the sonic speed c is

c=(yRT)0-5

Each Nj defines a vector space which is normal to the normal vector and is also called the characteristic vector space. Using Eq. (10), two independent equations for each TV,- obtained are

R[T*{Tin-Ta) + Ta] n CO) + " Cv* a o v = 0 and co2Ni + co3iV2 + CO4/V3 = 0 (14)

Since Eqs. (13) and (14) are independent to each other, the simplest method is to consider o>i, co5, and co2, co3, co4 separately.

When one of the two sets is considered, the other set is chosen to be zero as follows:

(a) co; = (o>i, 0, 0, 0, co5)

From Eq. (13), we obtain

'-R[T*(Tin-Ta) + Ta]

P Duv + j A>,P

+ p*i?(71 T„) D ,= Q

" i n '

If TV, = ( 1 , 0 , 0, -u*) is chosen and co,- = (0, 0, 0, 1, 0) is selected, the compatibility equation is

*n . ,R[T*(Tin-Ta) + Ta] „ . P DUW + j Ao2P

" i n

+ P*R{TifTa) D„2T* = 0 (20)

" i n

When w, in Eq. (16) is used, Nj = (1, 0, 0, - "* - c*) is chosen to satisfy the restriction of Eq. (11). Then Eq. (16) becomes

co,= (C*2, yc'^f, 0, 0, 1 (21)

,0,0,0,1 (15)

Ca 1 a

(b) co, = (0, co2, co3, co4, 0)

From Eq. (14), the three variables co2, co3 and co4 have only

one restriction. The vector of co2/ + co3/' + co4& is just

per-pendicular to the vector of Nxi + N2j + N3k. The characteristic

directions are arbitrarily selected as long as they are satisfied with the physical meaning and the simplification of the equa-tions to reduce the computer time.

Using Eq. (11), we can obtain

The compatibility equation is

yp c DLu +yp c ZDU v +yp c D„w

+ C DLp +p

"in

^ £ — ^ 1 *- DLT*

Ta J "in

= y£J!m [G]*+BE [F]* ( 2 2)

along the direction

W ; = C yNtc* KT * "in »T *

yN2c —, yNtf (16)

T 1 * ^in . * * ^in * "in *

L~\u — + c , v —, w — , 1

Cn Ca (-a

(23)

When co,- in Eq. (15) is used, only one compatibility equation which is independent of Nt is

If. —TS

* in -* n . - _ * DUP

When co, in Eq. (16) is used and N, = ( - 1, 0, 0, u - c ) is chosen, co,- becomes

(1-7)

r

Ta + 1

+p

%^)D

u

r = ^lFl* (17)

along the direction

*2 * "in rt A 1

«/= \c » -yc —, 0, 0, 1 The compatibility equation is

-yp c DLu +yp c *Daiv +yp c D^w + —c DLp +p — - — 1 —DLT

"in V T„ "in

(24)

(18) ^^IG]*+^IF\' (25)

When co, = (0, co2, co3, co4, 0) and the normal vector Nj =

( 1 , 0 , 0 , - u*) are selected to match the restriction of Eq. (10), co,- = (0, 0, 1,0, 0) is chosen to match the restriction of Eq. (14). The compatibility equation is

along the direction

r / / * "in * * "in * "in * \ /~,r\

L =\u — - c , v —, w —, II (26)

(4)

1.00r

( x / Dh) / ( R e . P r ) x 1 0

Fig. 2 Air mean density along the duct

0.8 |5 0.7 2 a t *~ 0.6 a. i E c £ >-- 0.5 " ' ji ~o « S t-a K 0.3 £ 0.2 <u £ ^ 0.1

°

z ol 2 ^ s N S v. i 5 10 V s. N , \ ^ I i 20 30 ( x / Dh N S X S " N i i i 40 50 60 ) / ( R e . P r ) x 1 03 x s v 70 8(

Fig. 3 Air mean temperature along the duct

n3

Dunwoody Present work

Dunwoody Present work Fig. 4 Air temperature along the duct

Results

The data used in the analysis are shown in Table 1. If the duct is very thin and Biot number is less than 0.01, the wall temperature can be assumed equal to the air temperature which does not generate error more than 5 percent (Days and Craw-ford, 1980). Using the method of characteristics, the boundary values and the initial conditions of the elliptical duct should be given. The temperature on the surface of the duct is assumed isothermal. The variations of initial air temperature, pressure and velocity are all assumed linearly decreased along the duct. The initial data and the input air properties are shown in Table

Fig. 5 Air velocity along the duct

2.0 1.0 0 ^_^__ — - ^ - — _ L 1 1 1 ' Y/a = 0.0 0 20.4 0.6 0.8 1 2 3 4 5 6 (x/Dh)/(Re.Pr)

Fig. 6 Air velocity at the major axis along the duct

' i — - — , 1 » ' Z/b=0.0 ' 0 2 0.4 0.6 0.8 0 1 2 3 4 5 6 (x/Dh)/(Re.Pr)

Fig. 7 Air velocity at the minor axis along the duct

2. The time unit is 5 x 10~4 s for each grid. The size of grid

and its dimensionless values are shown in Table 3.

The nondimensional assumed data and the result of the air mean density and air mean temperature along the duct are shown in Figs. 2 and 3, respectively. The air temperature dis-tribution at each cross section is shown in Fig. 4. The coor-dinates are expressed in the dimensionless reciprocal Graetz number Gz"1. The dotted lines in Fig. 4 are data obtained by

Dunwoody (1962) used for comparison. The nondimensional air velocity distribution along the duct is shown in Fig. 5. The velocities at the major axis and the minor axis along the duct are shown in Figs. 6 and 7, respectively.

Calculated local Nusselt numbers along the major axis and the minor axis for the isothermal elliptical duct are compared with the results of Dunwoody (1962) and Abdel-Wahed et al.

(5)

750

Abdel-Wahed et a l .

0.001 0.01 0 1

( x / Dh) / ( R e . P r )

Fig. 8 Local Nusseit number along the major axis for the isothermal elliptical duct £ a <U x 0.0 1.5 x (m)

Fig. 10 Heat transfer coefficient along the duct

Abdel-Wahed et a l .

Ji £

0.01

( x / Dh) / ( R e . P r )

Fig. 9 Local Nusseit number along the minor axis for the isothermal elliptical duct

(1984) as shown in Figs. 8 and 9, respectively. The heat transfer coefficient for the duct obtained from the gradient of the temperature distribution along the wall is shown in Fig. 10.

Discussion

The initial values of «i and a2 are chosen arbitrarily equal to 4 and 6, respectively, to reduce the computer time. These values are revised by the calculated results of the air temper-ature and velocity until the final results of nondimensional values are converged with the deviation less than 0.001.

The characteristic directions are not unique. Hence, the se-lection of the characteristic directions has significant effect on the accuracy and the convergent speed of the result. Two fac-tors are considered for the selection of the characteristic di-rections. One factor is to calculate data of locations easily from locations with known data along the characteristic di-rections. It is related to the consideration of the arrangement of the grid points. The other factor is that the coefficients and their difference in the compatibility equations should be in the same order of magnitude to avoid the error in solving the simultaneous equations. The characteristic directions should be spread for the proper selection.

The size of the grid and the error are related. The grid points in the y and z directions are relatively less. The speed of con-vergence is slower and the permissible range for concon-vergence is larger.

The calculated results of the local Nusseit numbers along the major axis and minor axis for the isothermal elliptical duct are comparable with the results obtained by Abdel-Wahed et al. (1984), Dunwoody (1962), and Gilbert et al. (1973), as shown in Figs. 8 and 9, respectively. The calculated results agree very well with those obtained by Dunwoody (1962) and Abdel-Wahed et al. (1984) if the reciprocal Graetz number is increased. Therefore, the assumptions, chosen characteristic directions, arrangement and the density of the grids used in this analysis provide very good calculated results.

Conclusion

The calculated results of the three-dimensional flow in the elliptical duct are compared and show good agreement with the available data. The governing equations of the air flow were derived in general form. It is believed that the mathe-matical model can be extended to analyze the ducts with ar-bitrary cross sections if the assumed distribution functions are reasonable.

Acknowledgment

The authors would like to express their thanks to National Science Council of the Republic of China for the Grant NSC77-0401-E002-24 to complete this study.

References

Abdel-Wahed, R. M., Attia, A. E., and Hifni, M. A., 1984, "Experiments on Laminar Flow and Heat Transfer in an Elliptical Duct,'' International Journal

of Heat and Mass Transfer, Vol. 27, pp. 2397-2413.

Bodoia, J. R., and Osterle, J. F., 1961, "Finite Difference Analysis of Plane Pouseuille and Couette Flow Development," Applied Scientific Research, Vol. A10, pp. 265-276.

Days, W. M., and Crawford, M. E., 1980, Convective Heat and Mass

Trans-fer, 2nd Edition, McGraw-Hill, New York.

Dunwoody, N . T . , 1962, "Thermal Results of Forced Heat Convection Through Elliptical Ducts," ASME Journal of Applied Mechanics, pp. 1965-1970.

Gilbert, D. E., Leay, R. W., and Barrow, H . , 1973, "Theoretical Analysis of Forced Laminar Convection Heat Transfer in the Entrance Region of an Elliptic Duct," International Journal of Heat and Mass Transfer, Vol. 16, pp. 1501-1503.

Huang, Yuan Mao, 1992, "Study of Unsteady Flow in a Heat Exchanger by the Method of Characteristics," ASME JOURNAL OF PRESSURE VESSEL TECH-NOLOGY, Vol. 114, pp. 459-463.

Igbal, M., Khatry, A. K., and Aggarwala, B. D., 1972, " O n the Second Fundamental Problem of Combined Free and Forced Convection Through Ver-tical Noncircular Ducts," Applied Scientific Research, Vol. 26, pp. 183-208.

Lundgren, T. S., Sparrow, E. M., and Starr, J. B., 1964, "Pressure Drop Due to the Entrance Region in Ducts of Arbitrary Cross Section,'' ASME Journal

of Basic Engineering, Vol. 86, pp. 620-626.

Rudinger, G., 1969, Nonsteady Duct Flow: Wave Diagram Analysis, Dover Publications, Inc., New York.

數據

Fig. 1 Coordinates of the elliptical duct
Table 3 The size of the grid  Ax  Ay  Az  At  0.15 m  0.003094 m 0.001547 m 0.0005 s  Ax' Az' At'  7.50  0.1547  0.07735 0.0005  (12) where the sonic speed c is
Fig. 3 Air mean temperature along the duct
Fig. 9 Local Nusseit number along the minor axis for the isothermal  elliptical duct

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