Head loss through porous dikes

SUBHASH C. JAIN, FORREST M. HOLLY, JR., A N D TIM H. LEE Department of Civil and Etzviror~mentnl Engineering and Iorvn It~stit~cte of Hydraulic Research, Universib of Iowa,

Iowa City. IA 52242, U.S.A.

Received August 6 , 1987

Revised manuscript accepted April 25, 1988

Porous dikes have been proposed for use in blocking access of fish to cooling water intakes in power plants using large cooling ponds for heat dissipation. Flow through such dikes is neither of the Darcy type nor quadratic, the friction factor depending on both the Reynolds number and material properties. Full-scale tests of the dike material proposed for the LaSalle County power plant confirmed the material-property and Reynolds-number dependencies reported in the literature and permitted calibration of the head-loss parameters for the prototype material under two placement configurations. Limited tests on dike clogging by surface debris permitted quantification of the additional head loss which clogging could cause.

Key words: porous media, cooling ponds, dikes, scale model tests.

Des digues poreuses ont CtC proposCes pour bloquCr l'accks des poissons aux prises d'eau de refroidissement des centrales Clectriques qui utilisent des Ctangs de grande superficie pour dissiper la chaleur. L'Ccoulement au travers de ces digues n'est ni du type Darcy, ni quadratique, le coefficient de friction Ctant fonction du nombre de Reynolds et des propriCtCs du matkriau. Des essais pleine grandeur du matCriau de digue proposC pour la centrale Clectrique de LaSalle ~ o u n t y o n t confirm6 cette dCpendance signalCe dans la documentation, et permis 1'Ctalonnage des paramktres de perte de charge du materiel d'essai selon deux configurations d'emplacement. Des essais limitCs de colmatage des digues par des debris de surface ont permis la quantification de la perte de charge additionnelle qu'entrainerait cette obstruction.

Mots clis : mmCdiu poreux, Ctangs de refroidissement, digues. essais de modkle i l'kchelle. [Traduit par la revue]

Can. J. Civ. Eng. 15,766-775 (1988)

Introduction The design of cooling-pond systems for nuclear and fossil-

fuel power plants must ensure that resident fish' populations cannot be drawn into, and thus clog, cooling-water intakes. This imperative has led to the development and use of various types of movable mechanical screening systems. The reliability and performance of mechanical screening systems must be guaranteed during emergency cooling situations. An alternative is the much simpler porous dike, constructed around the cooling-water intakes, and designed to act as a "filter" allowing the passage of large volumes of water while effectively blocking the passage of fish. It is not necessary for the voids of the porous dike to be smaller than the smallest resident fish; schooling fish tend to be deterred from passage by the mere perception of a solid structure.

Figure 1 shows the general layout of the Commonwealth Edison Company's LaSalle County Cooling Pond, Illinois (Tatinclaux et al. 1973), and the proposed porous dike that would be constructed around the entrance to the intake channel. The 1400 m long porous dike would be constructed of 30 cm nominal diameter riprap stone, with an average water depth of about 3 m at the upstream end of the dike and a flow rate of about 0.017 m3/(s.m) of dike length.

The design of such a dike must be based on an adequate evaluation of its structural integrity, biological efficiency, hydraulic performance, and maintenance requirements. The purpose of the study reported herein was to address only one of these aspects, namely hydraulic performance.

Review of previous studies Any porous dike design must be based on an improved

understanding of the mechanics and prediction of non-Darcy flow through porous media. Water passing through a porous

NOTE: Written discussion of this paper is welcomed and will be received by the Editor until February 28, 1989 (address inside front cover).

medium progressively loses energy due to viscous dissipation. In practical terms, for a porous dike this energy loss is the difference in water surface elevation between the upstream and the downstream face of the dike. Thus one of the hydraulic imperatives in dike design is to be sure that the available head (water surface elevation difference) is adequate to deliver the desired flowrate through the dike.

The flow regime through a porous dike lies somewhere between two well-understood but quite different flow regimes. At one extreme there is laminar flow through fine-grained material, where the relation between the bulk velocity (volu- metric discharge per unit area of dike perpendicular to the flow) and the hydraulic gradient (head loss per unit length parallel to the flow) is a linear one. This relation, known as Darcy's Law, is expressed as

where i is the hydraulic gradient, k is the coefficient of permeability, Q is the discharge, and A is the area perpendicular to the flow.

At the other extreme there is fully turbulent flow through large voids, such as through large pipes or dikes constructed of extremely large stones. For this case the linear law of [ I ] is replaced by the familiar quadratic law of turbulent head loss in conduits,

in which f is the (constant) Fanning friction factor, equal to '/4 of the Darcy-Weisbach friction factor (see for example, Chere- misinoff and Gupta 1983), D is the effective void (pipe) diameter, and g is acceleration due to gravity.

In between these two extremes is a zone in which the flow is neither fully turbulent nor fully laminar, the hydraulic gradient being neither linearly nor quadratically related to the bulk

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JAIN ET AL. 7 67

0 200 400 600 800 1000

SCALE (metres)

FIG. 1. Layout of the proposed porous dike.

velocity. If one retains the general quadratic law of [2], then the friction factor f is found to be related to the velocity through dependence on the Reynold's number R , defined as

where v is the kinematic viscosity. Figure 2 (Roberge et al. 1970) shows the dependence off on R as observed in several laboratory experiments for a variety of idealized and actual dike material sizes and shapes. From this figure the friction factor's inverse dependence on R can be clearly seen for R < 10, and its approach to R independence can be seen for R > lo4. Of particular note in Fig. 2 is the fact that at a given Reynold's number, the friction factor can vary over roughly an order of 10 for different types of dike material. Since the hydraulic gradient, and thus the total head drop across the dike, is directly dependent on the friction factor as seen in [2], this uncertainty in the friction factor must be resolved before dike head losses can be confidently predicted.

This friction-factor uncertainty can be somewhat reduced by adopting a general empirical head loss relationship identified by Wilkins (1956) and developed by Parkin et al. (1966), Olivier (1967), Leps (1973), and Linford (1967). The relationship is based on empirical observations which Wilkins fitted to a power-law relationship of the form

in which n and b are dimensionless empirical constants, W is a dimensional empirical parameter involving particle shape and water viscosity, m is the mean hydraulic radius of rock voids (void volume divided by particle surface area), and V, is the void velocity, also called seepage velocity. Vv is related to the bulk velocity by

in which e is the void ratio. Equation [5] implicitly assumes that the voids are saturated with moving water.

From his laboratory tests on material passing 19.0,76.l , and 203.2 mm sieves with hydraulic gradients ranging from 0.1 to 0.9, Wilkins (1956) found b = 0.5 and n = 1.86. Later, Parkin (1963) verified Wilkins' equation and the values of b and n through laboratory tests on rather fine (19.0 and 9.5mm) clean, angular gravel.

The utility of [4] depends on being able to evaluate Wand m for the particular material being studied. Wilkins (1956) and Parkin (1963) found that W ranges from about 3.6 mIJ2/s for crushed gravel, to about 4.4 rn1l2/s for polished marbles. While there is some uncertainty about the value of W for a given material, the rather limited range cited above (reported by Leps (1973)) limits the error in design which would result from that uncertainty.

The mean hydraulic radius m can be expressed as (Olivier 1967):

in which sv is the surface area per unit volume of stones. According to Olivier (1967), for stones with a particular shape, sv is inversely proportional to the spherical-equivalent stone diameter D,

with K = 6.0 for spheres; K = 7.45 for cubes; K = 7.81 for hexahedrons; K = 8.29 for crushed dolomite granite.

It is useful to combine these various relations into a working equation in which the uncertain parameters are clearly iden- tified. Since the total discharge per unit length of dike, q , is a convenient and appropriate design quantity, it can be used in conjunction with the local depth of flow within the dike, h (height of free surface above impervious dike base) to eliminate Vv in [4]. The resultant working equation, obtained through algebraic manipulation of [5]-[7] and adoption of b = 0.5, is

in which K1I2/ W = 0 . Thus, since the material properties e and D, can in principle be estimated, all uncertainty about the effects of rock shape and placement on head loss are lumped in the parameter 0 and exponent n.

Experimental apparatus and procedure The hydraulic tests on a 1 : 1 scale model of a porous dike were

conducted in the Environmental Flow Facility of the Iowa Institute of Hydraulic Research. A schematic of the experi- mental set up is shown in Fig. 3 and an overall view of the dike model is presented in Fig. 4. The Environmental Flow Facility is a recirculating flume with a test section 20 m long, 3 m wide, and 2.3 m deep. The working section has a 10 m long glass wall

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768 CAN. J. CIV. ENG. VOL. 15, 1988

REYNOLDS NUMBER IR

FIG. 2. Comparison of Fanning friction factor (from Roberge et al. 1976)

to facilitate flow visualization. The discharge to the flume is supplied by two 91.5-cm pumps driven by variable speed motors.

The model dike was 120 cm high with a 90 cm wide crest and 2: 1 side slopes and spanned the entire 3 m width of the flume. A 120 cm high dike was considered to be sufficient to produce flow conditions both with and without a downstream "seepage face." The dike material (Fig. 5) consisted of 30-cm nominal diameter riprap stone supplied by the River Products Company, Iowa City, IA.

Twelve pressure taps were installed at the bottom of the flume zlong its centerline. Ten pressure taps were mounted under the dike to measure piezometric head distribution. The other two pressure taps, one upstream and one downstream of the dike, were installed for the upstream and downstream flow-depth measurements. The pressure taps were connected to two banks of manometers, each with six glass tubes and a vernier scale to read piezometric heads with an accuracy of 0.3 mm. A full-width, 1.4 m high sharp-crested rectangular weir was installed at 4.6 m upstream froni the toe of the dike for the measurement of discharge in the flume. A baffle screen with a horsehair mat for stabilizing the water-level fluctuations was installed 1.2 m downstream of the weir.

At the beginning of each run, the speed of the pumps and the water level in the flume were adjusted to obtain the desired flow rate and the downstream flow depth. Once steady flow condi- tions were achieved, the piezometer and the point gage readings were recorded. In tests with a "seepage face," the flow depth at the intersection of the free surface with the downstream face of the dike also was measured. The occurrence of the seepage face was identified by visually observing the flow through the dike along one wall of the flume.

To evaluate the effect of rock placement on the head loss, the riprap stones were placed in two different manners during the dike construction in the flume. The first dike was constructed by randomly dumping the test stones. This arrangement is referred to hereinafter as the loose dike. The second dike was built by placing the stones in a compact form and is referred to hereinafter as the compact dike.

The void ratio of the two dikes was determined by placing the test stones in the manners described above in a 2-m diameter circular tank, and measuring the volume of water required to fill the voids. The void ratio was 0.89 for the loose dike and 0.67 for the compact dike.

Tests were conducted for a range of discharges and upstream and downstream flow depths. The ratio of the unit discharge to

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FIG. 4. Model dike.

FIG. 5. 12-inch riprap stone.

the average flow depth (i.e., q/h) for the proposed dike for the LaSalle County cooling pond is about 1.83 c m / s The value of [9] i = q/h in the present study was varied from about 0.61 to 9.15 cm/s. in which a, the loss coefficient, is given by

A total of 60 test runs were performed, for Reynold's numbers ranging from 1572 to 29 317. For each run, the [ lo] cw = (1 + e ) ,

~ , " 2 e3 /2 discharge per unit width was fixed, and the upstream and downstream flow depths, as well as the free-surface profile within the dike as determined from the pressure taps, were with b = 0.5. The relation between the Fanning friction factor

recorded. and loss coefficient can be derived as

Analysis of experimental results [ l l ] f=- The data were analyzed to determine the energy-loss expo-

nent n of [4], the ~ a n n i n ~ friction factor f ofVi2], and the There are several ways to evaluate the coefficient cw and dike-material parameter 0 of [8]. The analysis was most exponent n from the experimental data. The simplest method conveniently accomplished by rewriting [8] as (Nece et al. 1970) would be to assume a constant hydraulic

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gradient over the dike wetted area based on average flow velocity and flow distance. At the other extreme, the most elaborate method would require expressing the flow velocity in [9] in vector form, substituting it into the differential continuity equation and solving the resulting partial differential equation with appropriate boundary conditions. Such a rigorous method was not considered justified in the present studyin view of the uncertainties in quantifying several rock parameters. Instead, an ad hoc approach, referred to as the "variable gradient method," was developed to determine the loss coefficient and exponent n. The hydraulic gradient in this method varies across the dike wetted area which is divided, using an ad hoc procedure, into a number of subareas. The head loss in each subarea is com- puted from [9]. The variable gradient method is described in more detail below.

The wetted dike area is divided into several "flow channels" by the idealized flowlines, as shown in Fig. 6. The lines across the flow lines, referred to as "head lines, are straight lines which join points of equal piezometric head at the free surface and the dike bottom. The intersection of a head line with a flow line, or node, is identified by (i,j), where i varies from i = 1 at the downstream dike-face to i = P + 1 at the upstream dike-face, and ( P + 1) is the number of head lines. The change in the piezometric head at the free surface between successive head lines is held constant. The head lines are divided into m segments. The ends of the segments are numbered from j = 1 at the free surface to i = M + 1 at the dike bottom. The idealized flow lines are delineated by connecting, with straight lines, the ends of the head-line segments. The head loss, Ahu, from node (i,j) to node (i+ 1 ,j) along the jth flow line is determined from [9] applied locally,

in which vu = horizontal velocity component at node (i , j) and Al, = length of the flow line between nodes (i,j) and (i+ 1 ,j). It is assumed that v, along a flow line varies according to the continuity equation

in which hu equals the flow depth along the vertical line through node (i,j), and node (1 ,j) is located at the downstream dike face along the jth flow line. The computation procedure to determine a and n from the measured data is as follows:

(i) The first step in computations involves the determination of the free surface for the assumed values of a and n and for the measured values of the upstream flow depth, h,, and the downstream flow depth, hd (see Fig. 6) . Starting from the known coordinates of node (1, l ) , the coordinates of other nodes at the free surface are determined in terms of an assumed value of v,,. The computations are repeated for different values of v,, until the computed coordinates of the most upstream node ( P + 1, 1) agree with its measured coordinates.

(ii) The computations in the next step are carried out along the bottom flow line. The computation procedure is the same as outlined above for the free surface. A value of v~.,,~ which yields the computed coordinates of node (P+ 1, M+ 1) equal to its measured coordinates is determined. Note that vl,,+, < vl ,.

(iii) Once the coordinates of the nodes along the bottom flow lines are determined, the head lines are drawn by joining, with straight lines, the nodes at the free surface and the dike bottom. Each head line is divided into m equals segments and the flow lines are drawn as explained above. The coordinates of all

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CAN. J. CIV. ENG. VOL. 15, 1988

nodes, the length of flow lines between nodes, Ale, and the flow depth along the vertical line through each node, hU, are computed.

(iv) The velocities along intermediate flow lines are deter- mined next. The computations are started from node (l,j), j = 2,3, . . . , M, for an assumed value of vlj (note that v, > vU > vi,,+,) and the head loss along the jth flow line is computed using [12] and [13]. The computations are repeated for different values of vlj until the computed total head loss along the jth flow line is equal to the measured head loss. These computations yield the velocity, vij, at each node.

(v) The total flow rate through the dike isobtained from the computed velocity distribution along the upstream dike face. If the computed and the measured flow rates are the same, the assumed value of the loss coefficient, a, is correct, otherwise the entire computation is repeated for a new value of a.

Computations for several values of n were made; the values of m and p in the variableigradient method were 10 and 50, respectively. For each value of 1 2 , the value of a was plotted against Reynolds number and a straight regression line, a = constant, was fitted to the data by the least-squares method. The value of n which resulted in the least sum of squares of differences between observed values of a and the values given by the fitted regression line was adopted as the optimum for the dike material. The Fanning friction factor was then determined from [ l l ] .

A few of the computed flow and head lines for a typical run are shown in Fig. 7, which also includes the computed velocity distribution at the upstream dike face.

Test results The-optimum value of n for both the loose and compact dikes

was almost the same, varying between about 1.70 and 1.74. A value of n equal to 1.72 was adopted for the dike material tested in the present study. The computed results are compared with the measured data in Figs. 8 and 9. The agreement between the theory and experimental results is considered good except for very low and very high Reynolds numbers. The discrepancy at low Reynolds numbers is due to the inaccuracy in the measure- ments of head loss, which was less than about 3 mm across the dike. At high Reynolds numbers, a seepage face developed at the downstream dike face and the intersection of the free surface with the dike face could not be identified accurately.

The loss coefficient for the compact dike is about 12% higher than that for the loose dike. Since the head loss is proportional to an, the head loss through the compact dike is about 23% higher than that through the loose dike for the same flow conditions. To further evaluate the accuracy of the method, the computed and measured piezometric heads along the dike bottom were compared, and the agreement between them was very good, which suggests that the method is an accurate one for estimation of the head loss.

Using the void ratio and the loss coefficient determined from the data, the parameter 0, for the two dikes was determined from [lo] and found to be 0.47 s/m'I2 for the loose dike, and 0.38 s/m1I2 for the compact dike. The different values of 0 for the two dikes confirm the observation made by the earlier investigators that 0 depends on the stone placement.

The results of the present study are compared with the experimental results of Roberge et al. (1976) in Fig. 2 which includes results from other investigators. The friction factor for the dike material tested in the present study is about the same as

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1 I I I I I l l [ I I I 1 I T 1 I

1 0' REYNOLDS NUMBER

FIG. 8. LOSS coefficient.

REYNOLDS NUMBER

FIG. 9. Fanning friction factor.

for their 7.6- and 20.3-cm stone material and is about one third plastic sheet the top portion of the upstream face of the model of that for their quarry stone material. Such a high value of the dike. The percentage clogged area of the upstream face was friction factor for the quarry stone material is probably due to varied by conducting tests with different upstream flow depths. some fine material resulting in small size pores. The effect of flow distribution of the clogged dike face is to

increase the flow velocity near the clogged region. The Clogged dike additional head loss due to local increase in velocity is treated

The effect of possible clogging of the upstream face of the similar to a head due to expansion, i.e., dike by floating debris was examined for the compact dike only. The clogging of the dike face was simulated by covering with a [14] Ah, = P [(qlh,)' - (q/h,)2]/2g

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power plant. Iowa Institute of Hydraulic Research, Report No. 152. The University of Iowa, Iowa City, IA.

WARD, J. C. 1964. Turbulent flow in porous media. ASCE Journal of the Hydraulics Division, 90(HY9).

WILKINS, J. K. 1956. Flow of water through rockfill and its application to the design of dams. Proceedings of the 2nd Australia - New Zealand Soils Conference.

List of symbols A b D Ds e f g h he hu hd Ah,

area exponent void size nominal stone diameter void ratio Fanning friction factor acceleration due to gravity flow depth unclogged upstream flow depth upstream flow depth downstream flow depth head loss between nodes (i,j) and (i+ 1 ,j)

head loss due to clogging hydraulic gradient shape factor coefficient of permeability length of flowline between nodes (i,j) and ( i f 1 ,j) hydraulic radius of rock voids number of head-line segments exponent number of flow lines discharge unit discharge Reynolds number surface area per unit volume of stones seepage velocity horizontal velocity component at node (i,j) dimensional empirical parameter loss coefficient loss coefficient due to clogging K'I~I w kinematic viscosity

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