An analytical solution for flow in a manifold

An analytical solution for flow in a manifold A. W. Warrick and M. Yitayew Department of Soil and Water Science and Department of Agricultural Engineering, University of Arizona, Tucson 85721, USA

An analytical solution is developed for flow in a manifold. The interest is primarily for trickle irrigation laterals, but the solution has broader applications including those for which pressure increases in the direction of flow and for intake manifolds. Both velocity head losses and variable discharge along the manifold are considered in the fundamental analysis. The appropriate second order, nonlinear equation is solved for two flow regimes, laminar and fully turbulent. Results indicate that for most trickle irrigation laterals the velocity head loss is negligible, but for an example from a chemical processing system the effect is important.

INTRODUCTION The theoretical solution of manifold problems has several applications in water resources including such diverse considerations as canal locks and irrigation systems t'2. Initially, our interest was primarily for trickle irrigation laterals of porous pipes or emitters spaced at short finite intervals and considered as having a continuous output along the lateral. A similar problem was also considered by Acrivos et al. 3 for chemical processing streams. The solution developed here is applicable for laminar and fully turbulent flow conditions. Two examples are for trickle laterals, but the solutions are valid for other systems, including intake manifolds. For previous studies the velocity head contribution and the variable discharge properties have been neglected in the basic flow equations 4-v. Velocity head considerations have been treated in an analytical solution for constant flow (pressure-compensating) emitters a. In a later study, the variable flow is taken into account, but the velocity head neglected9. The solution we propose here includes both of these factors at the expense of being more difficult to evaluate and limited to 2 flow regimes (laminar and turbulent). Additionally, the flow from the outlet is necessarily proportional to the square root of piezometric head.

with q as a continuous function of the lateral coordinate X.

If continuity is preserved, then along the lateral A dv/dx = - q

(3)

and from equation (2) and (3) H = (AEs2/c2)(dv/dx) 2

(4)

Watters and Keller 5 have shown that for small diameter smooth pipes used in trickle laterals, the DarcyWeisbach equation can give accurate predictions for frictional loss based on conservation of energy. The relationship was also used for stream flows in chemical processing studies 3. Using the Darcy-Weisbach formula and equating change in total head along the lateral to the head loss, leads to d dx (H + v2 /2g) + fvZ /(2gD)= 0

(5)

with D the lateral diameter and g the gravitational acceleration constant. The last two equations considered together result in

THEORY Consider outflow from an orifice related to an internal piezometric head by q i = c H °'5

(1)

where q i is outflow rate from an individual orifice, H the piezometric head in the lateral, and c the orifice coefficient that includes areal and discharge effects. The spacing between orifices (s), the orifice coefficient (c) and crosssectional area (A) of the lateral pipe are taken as constants. Contribution from the Dept. of Soil and Water Science and Dept. of Agricultural Engineering, the Univ. of Arizona, Tucson, Ag. Exp. Stat. Paper No. 4282. Support was from Western Regional Project W-128. Accepted November 1986. Discussion closes August 1987. 0309-1708/87/020058-0652.00 © 1987 Computational Mechanics Publications

58

If the orifices are sufficiently close, the lateral can be regarded as a homogeneous system of a main tube and a longitudinal slot. The outflow rate per unit length q can then be described by q = (c/s)n °"5 (2)

Adv. Water Resources, 1987, Volume 10, June

d [ (A:s2 /c2)(dv/dx) 2] + (v/g) dv/dx d--~ + (f/2gD)v 2 = 0

(6)

The friction factorfis a function of the Reynolds number. With little loss of generality, it can be taken of the form f = f o vm-2

(7)

Also, a dimensionless velocity V and length X may be defined as V= V/Vo (8) X = x/x o

(9)

Flow in a manifold: A. W. Warrick and M. Yitayew with v the inlet velocity, i.e., v = v0 at x = 0, and Xo is a characteristic length x 0 which will be defined momentarily. By equations (7), (8) and (9), the differential equation (5) reduces to

d dV 2 d - X I ( - ~ ) ] + a V ( d V / d X ) + V"=O

(10)

provided the characteristic length x o and dimensionless parameter 'a' are defined from

X3o= OzZsZgD5 /8c2fo)v2 - m

(11)

a = (2D/foXo)V 2 -m

(12)

(To verify equation (10), first substitute voVfor v and form- 2 for f in equation (6). Then divide by fov"~/290 and simplify.) The length Xo and dimensionless parameters are chosen to scale the velocity from V= 0 to 1 and reduce the number of parameters in the differential equation. For laminar flow

f = 64/R e = 64v/vD

Fortuitously, for m = 1, values of :t = 0 and fl = 1/a 2 satisfy equation (19) which is the laminar case of equation (11). Also for m = 2, ct = - 2 / a 3 and fl = 0 satisfy equation (19) which is of interest in that it is just above the exponent m-- 1.75 for the smooth pipe (Blasius) solution. We thus examine the solution for these 2 cases, m = 1 and m = 2. We will refer to the cases where m = 1 and m = 2 as the laminar and fully turbulent cases, respectively. Unfortunately, no other solutions are obvious.

Laminar case (m = 1) For r e = l , ct=0 and fl= 1/a 2, Kamke 1° (esp. p: 303) gives the parametric solution to equation (17) in terms of U defined by

p=a-2U

(20)

and

f

dU

u-O.5+ 1

_ _(a3/2)V 2

(21)

By equations (15), (20) and (21)

(13a)

d X = p-O.5 dV or

fo = 64v/D, For turbulent flow

we

m = 1,

(laminar)

X = a S U - ° 5 dV+Const.

(13b)

As p ° 5 = d V / d X < O (for the outflow case) then U °'5 < 0 and

use 6

f = 0.316(v/vD) °25

(14a)

for which

X = a f u -°5 d V+Const.

fo = 0.316(v/D) °'25,

m = 1.75,

(turbulent) (14b)

Thus, the problem as formulated is a second order, nonlinear, ordinary differential equation (equation (10)) subject to boundary conditions V= 1 at X = 0 and V= 0 at the blocked end (X = X0). Representative values of 'a' and Xo will be discussed later when actual examples are considered. For an intake manifold, the same equation applies, but the appropriate piezometric head, in equation (1) and (2) is negative (the outside pressure is greater) and d V/dX would be positive rather than negative in equation (11).

We proceed to evaluate the analytical solutions. Define p by p = (d V/dX)2 (15) By equation (15), equation

p°'5(dp/dV)+aVp°'5 + V " = 0

(16)

-

(18)

with a, fl to be chosen. Thus, we seek ~ and fl from

V 2"- : = a2fl - (aaa/2)V:

-a3V2/2=2f(w-2+w-1)dw a 3 V2/2 = f ( U ) - f ( U m i , )

f(U)=-U+2U°'5-21n(l+U

(23)

°5)

(19)

(24)

and Umi, corresponds to V2 = 0 and X = X o. To examine the behaviour of f(U), it is useful to expand the logarithmic term of equation (24) into a series, and rewrite f ( U ) as

(4/3)[U15[[1 + (3/4)[ U°SI + (3/5)U + . . . ] (25)

(17)

which may be compared directly to equation (1.55) of (Kamkel°), esp. p. 303 taking his f, g, h and n equal to our V", 0, - a V and - 0.5, respectively). The solutions are analytical and are given by Kamke provided V2"-2 -- a2[/~ -a= I VdV]

(22)

Iu-°51dV

Multiplication of numerator and denominator of the integrand of equation (22) by U °'5 and the substitution w = 1 + U °'5 leads to

f(U) =

Multiplying by p-O.5, we find

dp/dV= - V"p -°~ - a V

So-X=a

or

ANALYTICAL S O L U T I O N

and take p ° 5 = d V / d X < O . (10) becomes

The boundary conditions are V= 0 at X = X o and V= 1 at X = 0 which lead to

Observe from equation (24) that if f(U) is real, then necessarily l+UO.S>0 This and the fact that U ° 5 < 0 necessitates that U is always between 0 and 1. It will be a maximum at X = 0 and minimum (Umi,)at X = X o. The f(U) is positive and monotonically increasing. The correspondence of X, U, V and U °'5 is shown as Fig. 1.

Adv. Water Resources, 1987, Volume 10, June

59

Flow in a manifold: A. W Warrick and M. Yitayew x=0 V= 1

Inlet

X=Xo V=0

Outlet

As a trial, assume w---. - o o corresponds to V = 0 and X = Xo. The corresponding Xo is

Trickle Lateral

X o = 2 a -2

U = Umax > 0

U = Umi n > 0

uO.5= --IUmaxl °.5

U°.5= -- [Urninl°.5

Fig. 1.

f

'max

wW-ldw,

X=2a

X o - X =- (2a)-°'S I (Umi,, U)

(35)

and X is

Correspondence of X, U and V

An alternative form of equation (22) is by substitution of a a v d v = (df/dU) dU resulting in

Wmax~ WR"(0

~c

-2

f wwmax

w W -1 dw

(36)

The parametric relationship is completed by (a3/2) In V=

(26)

i

Wmax

w2W -1 dw

~dw

with

f/ IUI-°[df/dU]dV I(Vmn, V ) = ;mm

[ f ( U ) - f(Umi.)] °'5

or

(27)

v=

As U approaches U . . . . X approaches 0. Corresponding to equation (23) is another relationship 03/2 = f ( Umax) - f(Umin)

(28)

One scheme for assuring consistency, is to specify a Um~n, then use equation (28) to solve for Umax which is substituted for U in equation (26). (This gives a corresponding Xo as X = 0 when U = Umax).

where

f

w2W 1 d w =

We now perform numerical calculations. Three examples will be presented, one for the laminar case (m = 1) and two for the fully turbulent (m = 2). The first two examples are relevant for a trickle lateral for which we take

(30)

D = 0.014 m

(diameter)

s= 1m

(spacing)

v = (1.01)(10)- 6 m2s - 1 (kinematic viscosity of water at about 20°C)

(29)

(a3/2) ln V + C

The solution sought is V= v/v o as a function of X = X/Xo. Closely related is the relative discharge rate q/qavg which is easily shown by continuity to be

q/qavg = -- Xo(d V/dX) Wmbw 3 + w + I

(31)

b = 2/a 3

(32)

(In this form we substitute w for u °'5 of Kamke.) By equation (16), dX is p-O.5 dV or

X=faw-'V-' dV+C' By equation (30), V- 1d V--- - 2a last result becomes

3w2

W - 1 dw and the

(37)

D I S C U S S I O N AND E X A M P L E S

Fully turbulent case (m = 2) For m = 2 , a = - 2 / a 3 and f l = 0 , Kamke ~° gives the appropriate integrating factor and parametric solution p = ( V/a)2w 2

wflWJ-'

(38)

The pressure distribution follows directly from q by equation (2). The examples follow.

Example 1: Laminary flow (m = 1) Take a lateral length L = 150m, emitter discharge coefficient c=(2.07)(10) -v m 25 ms -1 and q,vg= (5.56)(10) -v m E S - 1 (corresponding to 2 lh-1 for the I m spacing). The corresponding value for flow into the lateral then is Vo=0.541 m s -1. [Note t h a t Vo--4qavgL/(ytD2).] For laminar flow the frictional term is f = f o v- ~ and by equation (14) fo = (2.59)(10)- 3

X = - 2 a -2 f w W - 1 d w + C '

(33)

The exact range of w, we do not immediately know, but since p ° 5 < 0 , we know w < 0 . However, we note the integral in equation (30) will diverge as W--, - oo and of equation (33) will not. Also, we know there is one real root wR for the cubic equation (31), namely

1

1

[ - ~-b,(I\ 4 ~ 60

1

~0"511/3

llO.,l,+ 2-~--~// j

Adv. Water Resources, 1987, Volume 10, June

resulting in a characteristic length Xo and dimensionless 'a' of Xo=261 m

a = 0.0126

The dimensionless total length X o then is

Xo = L/xo = 0.575 The algorithm for the solution may be divided into a preliminary Stage 1 for evaluation of Umax and Umi, followed by Stage 2 for the velocity and discharge profiles: STAGE 1. Evaluate Um~, and Ur~x

(34)

1.

Choose a trial value of less than 1.

Umi n

just greater than 0, but

Flow in a manifold: A. W. Warrick and M. Yitayew 2.

Calculate a corresponding value of Umax from equation (28) by bisection or other numerical procedure (we know 0 < Umi, < Umax< 1, cf. Fig. 1). Solve for/(Umin, Umax)by equation (27) (see also the Appendix). Compare (2a)-°'5I(Umi., Ur,ax) tO X o as by equation (26). If they closely agree (e.g., 3 significant numbers) go to Stage 2, otherwise return to Step 1.

3. 4.

STAGE 2. Velocity and discharge profile 5. 6.

1.0

7.

oli

n-

2.0

Repeat 5 and 6 until desired points are calculated.

The above algorithm was programmed in 'TurboPascal'. The integral of equation (29) was evaluated by a combination of equation (A.3) and a Simpson's algorithm 11. The computational time was negligible, less than a few minutes for the calculations, but more time for the interactive part (Stage 1). The resulting velocity profile is given as Fig. 2A. The profile is nearly a straight line (Exactly a straight line would result if total uniformity was attained). In fact, the Christiansen Uniformity and Lower Quarter Distribution Uniformity were 0.99. [The Christiansen Uniformity is 1 - (average absolute deviation)/mean for the water added; the Lower Quarter Uniformity is the (average over the lowest

I

I

'

I

'

I

>

I..=4

(39)

I

I-=I

Specify U between Umi, and U~ax. Calculate X from equation (26) and d V/dX from equations (10) and (20), i.e.,

d V/dX = -[U°Sl/a

i

I

0"/ > 0

O"

m

01[

o = 0.0072 I

O.

0.

I

100. DISTANCE

1.0 ~

'

~

"

I

I,,

200. (m)

Fig. 3. Relative velocity head and side discharge for Example 2 (m = 2)

LJ 0

_I l.d >

quarter/mean)13.] The values were conformed using a Runge-Kutta numerical solution directly on equation (10) as well as using the analytical solution for a = 0 (Refs 9 and 13). The results were indistinguishable. Discharge as a function of distance q/qavg is by continuity (or equation (3))

.5

W >

m

'~

o -

< J Wr r

-

1

O. 012B

O. O,

,

,

05 i

,

.

I.

q/qavg= - Xo d V/dX

This is plotted as a function of position in Fig. 2B and corroborates that the discharge is nearly uniform, varying from about 1.03 to 0.98 for the 150 m length of the lateral.

B

Example 2: Fully turbulent case (m = 2) For Example 2 we take the q=(1.11)(10)-Tm2s -~ (4 lh-~ for the 1 m spacing), lengthen the line to 250 m and use c = (3.58)(10)-7 m2.5S-I. The Vo for this case is 1.08 m s - a. For m--2, we approximate f by fo where

0

~ 1.00 O" o

•

g5~; 0.

O. 0126

-

I

(40)

fo = 0.316(v/Dvo( V) )°25 = 0.0355

I

50. 100. DISTANCE (m)

150.

Fi9. 2. Relative velocity head and side discharge for Example 1 (m= 1)

with ( V ) a mean velocity of 0.5. This results in a = 0.0072, Xo = 114 m and X o = 2.19. The procedure is analogous to that of the earlier example with the initial step to find wmax by equation (37). This is done by varying Wmax~
Adv. Water Resources, 1987, Volume 10, June

61

Flow in a manifold: A. W, Warrick and M , Yitayew shown in Fig. 2B, the q/qavg varies from 1.8 at the entry to 0.75 at the far end of the lateral. The results were checked against a numerical solution for m = 1.75 (for the smooth (Blasius) pipe flow). The results were nearly identical to those for the approximation m = 2. Also, the results were also shown equivalent to the a = 0 solution. For the first 2 examples, the effect of 'a' is negligible, in fact comparisons with the solution for a = 0 showed equivalent results. However, if 'a' is larger it will have an effect. In fact, an examination of equation (10) reveals that the rate of change of (d V / d X ) 2 with X will change from negative to positive if a is sufficiently large. By Fig. 4, this corresponds to a pressure increase (rather than decrease) along the lateral. Such tends to become the case when the ratio of kinetic to potential energy (v2/29 to H) becomes large. An example, for which the effect of 'a' becomes quite important was solved numerically by Acrivos et al. 3 for a chemical processing stream. They consider an equivalent form to equation (11) with m = 1.75 and values of F o = 0 , 0.2, 0.5, 1 and 1.5 for - d V / d y = 1 at y = 0 where X = (2Fo)°'33y

(42)

a = 2°33Fo 0.667

(43)

and

(see esp. their equation (1.16) and Fig. 6). We will compare with their middle F 0 value of 0.5. (Comparisons for F0 = 0.2, 1 and 1.5 are similar; for F o = 0, equation (5) is easily evaluated analytically by observing d V / d X = [ C - V2/c2] °5 resulting in X as an inverse sine function of V). Example 3: Larger 'a' value case We compare with the example of Acrivos et al. a. As our solution of equation (1 I) is valid for m of 1 and 2, we write the approximation with their y (see equation (1.16)) d2V d V ----+ dy 2 dy

y=(2125Fo)-°333X=O.944X

SUMMARY AND CONCLUSIONS Analytical solutions have been derived for a variable outflow manifold for a laminar and fully turbulent regimes. The analyses include both velocity head changes and the variable outflow. F o r the trickle irrigation examples, the effect of including the velocity head term was negligible. However, for the third example, the inclusion of the velocity head term led to a different shape of an outflow profile, namely the side flow and pressure increased rather than decreased along the lateral. These solutions to the nonlinear continuous manifold equation are relatively tedious to set up for numerical evaluation, but once p r o g r a m m e d are rapidly executed. In addition to being of direct use, they are applicable for testing algorithms based on purely numerical techniques.

REFERENCES 1

3

a = 2°33(2°25Fo)

II

-°'167

|

II

=

I

|

1.78

l

4

(m=2)

II'"

I

5 |

1.4

6 7

o

-

1.78

~

D

8

Q

,#m

9 "0

10 0

-

0

|

It

|

11 |

0.0

|

|

I

.5

•

|

12

1.0

13

Y Fig. 4. Slope o f dimensionless V as a function o f position for a = 0 and a = 1.78 (Example 3)

62

Adv. Water Resources, 1987, Volume 10, June

(m=2)

The resulting solution for d V/dy is shown as Fig. 3 and shows an increase in IdV/dy I along the line (corresponding to an increase in p as well). The m a x i m u m length X o at which V= 0 is at X = 0.906 or y = 0.855. This is in agreement with their results and was independently verified using a Runge-Kutta numerical solution t2 for both m = 1.75 and m = 2. The solution for a = 0 is given also in Fig. 3. The shape is drastically different as necessarily [dV/dy[ is forced to decrease (for decreasing V along y). This is opposite to Examples I and 2 for which a was demonstrated to be negligible.

2

V ( d V / d X ) + FoV2 /l~°av'g25 ~ 0

This reduces to our equation (11) provided

1.15

and

14

Allen,J. and Albinson, B. An invesigation of the manifold problem for incompressible fluids with special reference to the use of manifolds for canal locks, Proc. Inst. Civil. Eng., 1955, 4, 114-138 Christiansen, J. E. Hydraulics of sprinkling systems for irrigation, Trans. ASCE, 1941, 107. 221-250 Acrivos, A., Babcock, B. D. and Pigford, R. L. Flow distributions in manifolds, Chem. Enor. Sci., 1959, 10, 112-124 Bui, U. Hydraulics of trickle irrigation lines, MS Thesis, University of Hawaii, Honolulu, Hawaii, 1972, p. 63 Watters,G, Z. and Keller,J. Trickleirrigation tubing hydraulics, Paper No. 78-2015, presented at the Summer Meeting of ASAE, Utah State University, Logan, Utah, 1978 Wu, I. P., Howell, T. and Hiler, E. Hydraulic design of drip irrigation systems, Technical Bulletin No. 105, Hawaii Agricultural Experiment Station, University of Hawaii, 1974 Wu, 1-Pai and Gitlin, H. M. Hydraulics and uniformityfor drip irrigation, J. lrr. Dr. Div., ASCE, 1973, 99, 157-167 Yitayew,M. and Warrick, A. W. Velocity head considerations for trickle laterals, J. lrrig, and Dr. Engr., ASCE (Accepted), 1986a Warrick,A. W. and Yitayew, M. Trickle lateral hydraulics I: An analytical solution, J. lrri9. and Dr. Engr., ASCE (Submitted), 1987 Kamke, E. Differentialgleichungen losungsmethoden and losungen, AkademischeVerlagsgesellschaftGeest and Portig K.G., Leipzig, 1959 Miller,A. R. Pascal programs for scientists and engineers, Sybex, Berkeley, CA, 1981 Shoup,T. E. Numerical methods for the personal computer, Prentice-Hall, Inc. Englewood Cliffs, NJ, 1983 Yitayew,M. and Warrick, A. W. Trickle lateral hydraulics II: Design and examples,J. lrrig, and Dr. Enqr., ASCE (Submitted), 1987 Dwight,H. B. Tables of integrals and other mathematical data, 4th Edn, The Macmillan Co., New York, 1961

Flow in a manifold: A. W. Warrick and M. Yitayew APPENDIX

, it.

Evaluation of I(U=~, U,.t, + 6) The evaluation of equation (27) is complicated by the fact that the denominator approaches zero at the lower limit. To make matters worse, U.un itself can be close to zero. Thus, the expression for the second argument close to Um~nis needed in a useful form. For U=Urm,+6, we write a Taylor Series approximation f(Umi n + U ) ~ f(Umin) + 3 f ' + (62/2)f "

where the 1st and 2nd derivatives f ' and f" are evaluated at Umm. Thus, we can expand the denominator as [ f ( V ) - f(Vmin)]- 0.5 ~ (3f')- 0"5[1 -- 0.25(f'//f')6] The df/dU of the numerator (of equation (27)) can be expanded similarly as

df/dU ~ f ' + 6f" Thus, the integrand is approximately [U I- °5(A3- °'5 +860.5)

A = ( f ' ) °'s

(A.1)

B = (0.75)(f"/f')A

(A.2)

The integral then is approximately I~ A

f~

dU

0 U°'5 (Urnin +

U)O.5 t-B

;

U°'S dU

U)O.5

(Umin "+"

By equation (195.04) of Dwight ~4, I is approximated by

I = B6°'5(Umi. + 6)0.5 + (A - 0.5 UminB)

uo.s (Umi"+ U)o. 5

and by Dwight, equation (195.01)

1 = Bf°'5(Umin + 3)0.5 + 2(A - 0.5UmmB){ln[(Umin + 6) 0.5 + 6 °5] - In Um0i~} (A.3)

Adv. Water Resources, 1987, Volume 10, June

63