*E-mail: vpugliese@uninorte.edu.co **E-mail: figueroao@uninorte.edu.co

Optimization of Power Dissipation through a hydraulic jump

Pugliese, Victor*; Figueroa, Oswaldo**

*Fundacin Universidad del Norte, Barranquilla, Colombia

**Fundacin Universidad del Norte, Barranquilla, Colombia - Assessor

ARTICLE INFO

Article history:

Elaborated 13 June 2015

ABSTRACT

Purpose: Optimize the efficiency of power dissipation during the hydraulic

jump.

Method: Using a statistical technique, known as a Response Surface

Methodology (RSM), in order to optimize the response variable.

Results: Efficient power dissipation is influenced by flow rate and flume

slope. The maximum efficiency reached was 35%.

Conclusions: The maximum efficiency of power dissipation was reach

when the open channel was horizontal. The flow rate was 62 lpm, the

width channel was 7,6 cm and flow depth 1,3 cm. In this condition, the

Froude number was 3

Key words

Hydraulic jump

Power dissipation

Subcritical flow

Supercritical flow

1. Introduction

1.1. Purpose of paper The condition of flow in an open channel can go

from a supercritical to subcritical flow due to

changes in the channel characteristics, boundary

conditions, or the presence of hydraulic

structures. This change occurs abruptly through a

hydraulic jump. A hydraulic jump is highly

turbulent, with complex internal flow patterns,

and is accompanied by a considerable loss of

energy. [1]

The amount of energy loss depends on several

variables involving flow conditions, geometric

conditions of the channel, presence of external

forces, among others.

One of the most widespread uses that have the

hydraulic jump is to dissipate the energy of water

flowing over dams, weirs and other hydraulic

structures, and thereby prevent scouring of the

structures downstream. It is also used for mixing

quickly additives in the water flow for water

treatment.

It is plan to use the concepts developed in the

Course of Design of Experiments to optimize the

efficiency of power dissipation during the

hydraulic jump, and the results and conditions

operation could be scaled to larger open channels.

1.2. Theory model Because the energy loss due to the hydraulic jump

is usually significant and unknown, we cannot use

the energy equation to relate the depths before

and after the phenomenon occurs.

2

Momentum is a quantity vector, and separate

equation are needed if there are flow components

in more than one direction. However, open

channel flow is usually treated as being one

dimensional, and the momentum equation is

written in the main flow direction. Consider a

volume element of an open channel between an

upstream section U and a downstream section D

as show in Figure 1.

Figure 1 Volume of control for conservation of momentum equation. [1]

The element has an average cross-sectional area

of , flow velocity , and length . The

momentum within this element is . The

momentum is transferred into the element at

section at a rate and out of the element

at section at rate . The external forces

acting on this element in the same direction as the

flow are the pressure force at section , =

and the weight component sin =

sin . The external forces acting opposite to

the flow direction are the pressure force at section

, = , friction force on the channel

bed, , and any other external force, , opposite

to the flow direction. [1] For stationary flow, the

conservation of momentum equation is,

(

2

+ )

+ 0

+

2= (

2

+

) (Eq. 1)

This equation relates the external forces (included

the component of weight) with the depths

upstream and downstream the hydraulic jump

(they are included in the cross-section areas).

However, the forces are hard to determine

experimentally. Using the energy equation,

including the loss of energy, we have,

( + +

2

2) = ( + +

2

2) +

(Eq. 2)

If the friction force is negligible and there are no

obstacles or contractions, is zero. So, the energy

loss is due to the hydraulic jump.

1.3. Response Surface method Response surface methodology (RSM) is a

collection of mathematical and statistical

techniques that are useful for modeling and

analysis of problems in which a response of

interest is influenced by several variables and the

objective is to optimize this response. In most RSM

problems, the form of the relationship between

the response and the independent variable is

unknown. Thus, the first step in RSM is to find a

suitable approximation for the true functional

relationship between the output variable and the

set of independent variables. A first-order model

is simple and efficient to move rapidly to a vicinity

of the optimum. Near the optimum condition, is

possible that the system has curvature; so a

second-order model bust be used to approximate

the process. [2]

2. Methodology

2.1. Material and equipment Flume

The practical work was development in the

installations of the Hydraulic Laboratory in

Universidad del Norte, Barranquilla. It was used a

flume to produce a controllable hydraulic jump.

The flume consists of a clear-sided rectangular

working section supported on a frame, with an

inlet tank at one end. The frame is supported on

pedestals, and a jack allows the flume to be tilted

3

(Figure 2). The flume is designed to be used with

an Arm-field F1-10 Hydraulic Bench, which

provides a re-circulating water supply and a

volumetric measuring facility. [3]

Figure 2 C4-MKII Multi-propuse Teaching Flume. [3]

Flowmeter

Provides a direct reading of the volume flowrate of

the water passing through the working section.

Overshot weir

The level in the working section of the flume may

be controlled by an overshot weir arrangement at

the exit consisting of stop logs in a slot. Stop logs

are simply added or taken away to provide the

required depth of water in the working section to

reach a subcritical flow and generate the hydraulic

jump. [3]

Pedestals and Jack

The flume is supported on a

pair of pedestals which are

bolted to the floor for

additional safety. The pedestal

at the inlet end of the working

section is fitted with a hand-

operated jack. This jacking

arrangement permits the slope

of the channel bed to be

manually adjusted. The jack is

operated by a handwheel and

the mechanism incorporates a

slope indicator calibrated

directly in units of % slope. [3]

Weir

The undershot weirs are adjustable allowing a

constant head of the upstream reservoir under

uniform flow conditions. During the experiments,

there were used two types of undershot weirs.

Figure 4 shows an undershot straight and curve

weir.

Figure 4 Undershot weir used.

According the equation (1), the flowrate, the

cross-sectional area before and after the hydraulic

jump, and the slope of the channel influence the

phenomenon. Table 1 shows the design factor and

their feasible operating region.

Table 1 Feasible Operating Region.

Design Factor Low Limit High Limit Flow Rate [LPM] 0 96

Overshot weir high [cm] 0 10

Flume slope [%] 0 3

Type of undershot weir Straight Curve

2.2. Detecting influential factors for power

dissipation The statistical design was a screening experiment

24. Table 2 reports the processing parameters

analyzed in the present paper. Initial values were

determined during a familiarization stage, where

trial and error experiments were conducted to

specify the experimentation window of the

process, and being sure that depths can be

measured and generate a hydraulic jump.

Figure 3 Jack and slope indicator.

4

Table 2 Experimental Operating Region.

Design Factor Low Limit High Limit Flow Rate [LPM] 40 50

Overshot weir high [cm] 3.5 6

Flume slope [%] 1 2

Type of undershot weir Straight Curve

Table 3 shows the runs used during the screening

stage. The high and low values are identified with

the symbols (+) and (-). The runs were

implemented randomly

Table 3 Factorial design 24.

Run Design Factors Flow rate (A)

Overshot weir

high (B)

Flume slope

(C)

Undershot weir (D)

1 - - - -

2 + - - -

3 - + - -

4 + + - -

5 - - + -

6 + - + -

7 - + + -

8 + + + -

9 - - - +

10 + - - +

11 - + - +

12 + + - +

13 - - + +

14 + - + +

15 - + + +

16 + + + +

2.3. The method of steepest ascent The method of steeps ascent is a procedure for

moving sequentially along the path of steepest

ascent, that is, in the direction of maximum

increase in the response. [2] By definition, the

direction of maximum increase in a point of a

function is the gradient of the function evaluated

in the point.

If the fitted first-order model is

= 0 + =1 (Eq. 3)

The gradient is defined by the coefficients .

2.4. Analysis of second order response

surface After obtaining an experimental region with the

presence of a local maximum, an model to

approximate the natural curvature of the process

is done. Regression analysis is performed and the

optimal combination of parameters is defined.

2.5. Measurement of the response variable The response variable was the efficiency of power

dissipation. This is the percentage of initial

(upstream the hydraulic jump) energy that is loss

when the supercritical flow becomes subcritical

flow. Equation (2) permits compute the energy

loss ( []) using the datum (), the flow depth

() and the flow velocity (), upstream and

downstream of hydraulic jump.

Figure 5 Measurement of response variable.

Figure 5 show two millimeter rulers that were

used in order to measure the flow depth.

Downstream ruler was positioned where visibly

the flow was stable. The position of each ruler, and

the bottom slope were used to calculate the

datum, taking as zero the level of the channel end.

Flow velocity was computed using the registered

flow and the cross-section area (flow depth times

channel width).

The efficiency of power dissipation was calculated

using,

[%] =

(Eq. 4)

Where, is left side of equation (2).

5

3. Results

3.1. Measurement results Table 4 reports the efficiency of power dissipation,

Reynold number and Froude number for each run

are displayed in Table 4. The Reynold number

confirms that flow conditions were turbulent in all

runs. The Froude number corresponds at flow

conditions before the hydraulic jump occurs, and

is necessary a supercritical flow ( > 1).

Table 4 Results for screening stage.

Run Measured Variables Efficient Power

Dissipation

Reynolds Number

Froude Number

1 0,17 28611 2,9 2 0,31 35461 4,1 3 0,21 27101 3,7 4 0,31 34894 4,0 5 0,01 25000 1,6 6 0,03 30494 1,7 7 0,06 26933 2,2 8 0,15 32867 2,6 9 0,06 26154 1,7

10 0,16 32075 1,9 11 0,19 28652 3,3 12 0,30 34722 3,5 13 0,17 27000 2,2 14 0,20 32244 2,1 15 0,09 27067 2,2 16 0,16 33467 2,7

3.2. Critical factors that affect Power

dissipation The effects of each factor on Table 2, and the

interactions between them, were computed and

displayed in a normal plot graphic, in order to

define possible factors that may have a significant

influence at the response variable. See Figure 6.

Figure 6 permits to lump interactions as

experimental error. Despite of undershot weir (D)

appears to have a little effect in the efficient

power dissipation, it was included in the ANOVA

analysis.

According to Table 5, the independent variables

that have a significant effect in the response

variable are flow rate and flume slope.

The approximation of the response surface is

= 0,162 + 0,041 0,053 Eq. (5)

Where,

1 =[] 45

5

3 = 1,5%

0,5%

Figure 6 Normal plot of effect in efficiency of power dissipation.

6

Table 5 ANOVA Analysis.

SoV SS DoF MS F A 0,0267 1 0,0267 5,3 4,8

B 0,0085 1 0,0085 1,7 4,8

C 0,0440 1 0,0440 8,7 4,8

D 0,0006 1 0,0006 0,1 4,8

Error 0,0555 11 0,0050

Total 0,1879 15

SoV: Source of variation. SS: Sum of squares. DoF: Degree of freedom. MS: Mean square. F: Statistic of Fisher

: Reference value.

For the following stages, we use only a straight

undershot weir.

3.3. Model adequacy checking The decomposition of the variability in the

observations through an analysis of variance

(Table 5) requires that certain assumptions be

satisfied.

The residuals are normal and independently

distributed with mean zero and constant variance.

A check of the normality assumption is construct a

normal probability plot of the residuals. Figure 7

shows that error distribution resemble a straight

line. This indicate the assumption is satisfied.

Figure 7 Normal probability plot for residuals.

Plotting the residuals in time order of data

collection permits detecting correlation between

the residuals. Figure 8 shows no tendency in the

residuals of the run. This implies that the

independence assumption on the errors has not

been violated.

Meanwhile, the homoscedasticity is verified for

the significant factors, the flow rate and flume

slope. Figure 9 and Figure 10 show that the

variance has a little change when the factors vary

from low to high values. Even if the assumption of

homogeneity of variances is violate, the test in

only slightly affected in the balanced design used.

[2]

Figure 8 Residuals in time variation.

-0,2

-0,1

0

0,1

0,2

0 5 10 15 20Res

idu

als

Time

Independence

7

Figure 9 Residuals versus values of flow rate.

Figure 10 Residuals versus values of flume slope.

3.4. Steepest ascent The direction of maximum increase is,

= 0,04 0,05 = 0,05(0,8 1)

This indicates that an decrement of one unit for

the coded variable 3, require a increment of 0,8

units. Table 6 reports the path of steepest ascent.

It shows the values of the design factors and the

corresponding efficiency of power dissipation.

Flow increases 5 lpm in each condition, while

bottom slope decrease 0.5%. It is important to

note the natural limit for the slope; in this limit,

the channel is horizontal.

Figure 11 shows graphically the result of each

experiment condition on Table 6. The presence of

a local maximum seem to be between conditions

4-7. In this runs, only flow rate was changing, so

the response is a one-variable function.

Table 6 Path of steepest ascent.

Run Flow rate

[lpm]

Overshot weir high [cm]

Flume slope [%]

Undershot weir

Effic.

1 45 3,5 1,5 Straight 0,08

2 49 3,5 1,0 Straight 0,25

3 53 3,5 0,5 Straight 0,27

4 57 3,5 0,0 Straight 0,34

5 61 3,5 0,0 Straight 0,35

6 65 3,5 0,0 Straight 0,34

7 69 3,5 0,0 Straight 0,32

Figure 11 Path of steepest ascent.

For the 4-7 experiment conditions, the response

variable can be approximated with the following

equation.

[%] = 3 + 0,1106[]

0,0009([])2 Eq. (6)

3.5. Location of maximum response Table 1 shows the feasible values for each design

factor. The type of undershot weir is not a

significant factor, we choose a straight weir

because is widely use as gate in different uses.

Meanwhile, the overshot weir high at the channel

end, is fixed at a value of 3,5 cm. In the

familiarization stage, was noted that higher weir

avoided the formation of hydraulic jump, a choke

was produced.

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

-2 -1 0 1 2Res

idu

als

Flow rate

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

-2 -1 0 1 2Res

idu

als

C

Flume slope

0,000,050,100,150,200,250,300,350,40

0 1 2 3 4 5 6 7 8Ef

fici

en

t

Condition

Ascent

8

In a negative slope, the hydraulic jump cannot

occurs. This fixed the value of bottom slope as 0%.

See Table 6.

Using the equation (6), the value of flow rate was

stablish in order to get a maximum. This

correspond a flow rate of 62 .

4. Conclusions The independent variables that have a significant

effect in the response variable are flow rate and

flume slope.

The maximum efficiency of power dissipation was

reach when the open channel was horizontal. The

flow rate was 62 lpm, the width channel was 7,6

cm and flow depth 1,3 cm. In this condition, the

Froude number was 3. We can use the equation

for Froude Number, for a fixed flow rate, to design

a channel to dissipate energy.

References

[1] A. Osman Akan, Open Channels Hydraulics,

Butterworth-Heinemann, 2006.

[2] D. Montgomery, Design and Analysis of

Experiments. 5th Edition, Arizona: John Wiley

& Sons, 2004.

[3] ARMFIELD, "Instruction Manual Multipropose

Teaching Flume," 2012.