Statistical Modeling of Head Loss Components in Water Distribution Within Buildings

8/12/2019 Statistical Modeling of Head Loss Components in Water Distribution Within Buildings

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International Journal of Science and Technology Volume 3 No. 2, February, 2014

IJST 2014IJST Publications UK. All rights reserved. 101

Statistical Modeling of Head Loss Components in Water Distribution within

Buildings

J. I. SodikiDepartment of Mechanical Engineering

Rivers State University of Science and Technology

P. M. B. 5080Port Harcourt, Nigeria

ABSTRACT

Fractions of the total head loss which constitute the loss through fittings were computed for varying lengths of first index pipe run,

total flow rate and number of sanitary appliances supplied, in a water distribution system within a building. The results were used in a

regression analysis to obtain second order equations which gave a general increase of the fraction from 0.263 to 0.420 for an increase

in length of first index pipe run of 28.3m to 140.3m; from 0.262 to 0.429 for an increase in the total flow rate of 0.6L/s to 4.4L/s; and

from 0.238 to 0.393 for an increase in the number of supplied sanitary appliances of 8 to 120. The correlation coefficients between theratio of loss through fittings to the total head loss and each of the variables of pipe length, flow rate and number of appliances were

0.97, 0.95 and 0.91, respectively; which were found to be acceptable for a 99% confidence level. The ratios are useful for quick estimate

of losses in water distribution systems in buildings, needed in determining the suitability of water reservoir elevations and lift pump

head requirements.

Keywords:Loss ratios, pipe fittings, water distribution within buildings, regression analysis.

1. INTRODUCTION

The available pressure at a given point in a building water

distribution system pipe work is progressively reduced away

from the pressure source (which is usually on elevatedstorage). This reduction is due to pipe frictional loss and loss

through pipe fittings such as elbows, tees, reducers and valves.

The latter loss is sometimes called separation loss.

Thus, extensive pipe runs would result in increased frictional

loss, while multiplicity of pipe fittings would be associated

with increased separation loss. However, the number and types

of fittings installed in a given pipe run are specified such as to

achieve system functionality; and it can be assumed that, for a

given system configuration, the ratio between the frictionalloss and the separation loss for an index pipe run may vary with

varying length of pipe run.

In an earlier paper (Sodiki and Orupabo, 2011) it has been

established that the fraction of the total loss which constitutes

the total separation loss increases as the first index pipe lengthincreases, in simple water distribution systems; an increase in

length of water distribution first index pipe usually implying

increases in the total design water flow rate and in the number

of sanitary appliances being supplied.

In this paper, the ratio of total separation loss to total head loss

is regressed, in turn, on the variables length of first index pipe

run, design flow rate, and number of sanitary appliances, to

obtain equations for predicting the relationship of each variable

with the ratio. Second order relationships are utilized in theregression analysis, as indicated by the slopes of the curvesobtained earlier (Sodiki and Orupabo, 2011) and the

coefficients of correlation are calculated to check the

usefulness of the resulting equations.

2. HEAD LOSS COMPONENTS

The frictional loss hfand the loss through pipe fittings hpare

analysed as follows:

2.1 Frictional Loss: This loss, for each section of the firstindex pipe run, has been derived for plastic pipe

material as (Sodiki, 2002)

hf = 1.1374 x 10-3 ld-4.867 q1.85

- - - - - (1)

where l = pipe length (m)

d = pipe diameter (m)

and q = flow rate (m3/s)

Thus, the head loss per metre pipe run is

hf/ l = 1.1374 x 10-3 d-4.867 q1.85

- - - - - (2)

A graphical representation of Eqn. 2 (Institute of

Plumbing, 1977) is utilized in the analysis.

2.2 Loss through Fittings: This has been derived for

each fitting as (Sodiki, 2003)

hp = 0.08256kd-4q2- - - - - (3)

where k is a head loss coefficient of the fitting which is taken

for this analysis as 0.75 for an elbow, 2.0 for a tee and 0.25 for


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International Journal of Science and Technology (IJST) Volume 3 No. 2, February, 2014


a gate valve (Giles, 1977). For reducers, kis expressed in terms

of the ratio of upstream diameter d1to downstream diameter d2

as in Table 1 (Giles, 1977); with the k values, so obtained,

being referred to the downstream diameter in Eqn. 3.

Table 1: Values of K for Reducers, in Terms of Ratio of Upstream Diameter (d1) to Downstream Diameter (d2)

(Giles, 1977)

Ratio d1/d2 k

1.2 0.08

1.4 0.17

1.6 0.26

1.8 0.34

2.0 0.37

2.5 0.41

3.0 0.43

4.0 0.45

5.0 0.46

3. THE DISTRIBUTION SYSTEMCONFIGURATION

The configuration utilized in the analysis is one in which water

is distributed by gravity from a high level reservoir to a range

of identical toilet rooms of a hotel building (Fig. 1). Each room

contains a water closet, a bath tub, a wash basin and a water

heater. The selected pipe work arrangement represents a

commonly occurring scenario in simple water distribution

systems.

3.1 Methods of Estimate

The analysis was first done for the pipe run from A to B and

up to the farthest fixture supplied by the branch from B

(considering the extension on the main distribution pipe from

B towards C as non-existent). In the next step, the analysis was

done for the pipe run from A to C and up to the farthest fixture

supplied by the branch from C (again, considering theextension from point C towards D as non-existent). Subsequent

steps were carried out in like manner for extended index pipe

runs up to the last step (for the index run from A to P up to the

farthest appliance outlet supplied from P). The progressive

extensions of the first index run provided a variation of the

complexity of pipe work.

The graphical method adopted for pipe sizing and estimationof frictional losses is illustrated using the pipe run from A to C

and up to pipe section 6 (which was the second step in the

analysis procedure described earlier).

This pipe run is shown as an isometric sketch in Fig. 2 in whichthe pipe sections are labeled using boxes. The number to theleft of the box is the pipe section number, the number to the top

right is the measured pipe length (in m), and that on the bottom

right is the flow rate in the section (in L/s).


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In the analysis, loading units, which account for the non-

simultaneous use of all the installed sanitary appliances, were

utilized to obtain the flow rates from the graph of loading units

versus flow rates (Fig. 3) (Institute of Plumbing, 1977). These

units were taken as 2 for a water closet cistern, 1.5 for a wash

basin, 10 for a bath tub and 2 for a water heater cylinder.

Cumulative units were, thus, utilized for each pipe section. For

loading units below 10 which are not presented in Fig. 3, linearextrapolations were made to obtain corresponding flow rates.

Now, for a reservoir height above point A, in Fig. 2, of 10m

and a height of the water heater in pipe section 6 (which is the

final section of this index run) above point A of 2.5m, the

pressure head H available in the first index run = 10m - 2.5m= 7.5m. The measured length of the index run is L = 36.3m.

Then, the rate of head loss per metre run (H/L) should not

exceed 7.5/36.3 = 0.207m/m run.

This H/L value and the sectional flow rates were used to select

pipe sizes from Fig. 4 (Institute of Plumbing, 1977). For

instance, for pipe section 2 which carries a flow rate of 0.6L/s,

a 25mm pipe was selected (at point A in Fig. 4). The actual

values of H/L were obtained at the intersection of the lines offlow rate and pipe diameter. For pipe section 2, as an example,

the actual H/L value (at point A in Fig. 4) is 0.085m/m run and

the measured pipe length is 11.0m. Thus, the head loss due to

friction for this pipe section is 0.085 x 11.0m = 0.935m

Table 2 gives a summary of the pipe sizing estimates and thecalculated head losses for this index run.


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Table 2: Parameters of Distribution System for 36.3m First Index Run, 16 Appliances, 0.95L/s Flow Rate

1 2 3 4 5 6 7 8 9 10 11

Pipe

section

No.

Loading

units

Design

flow (L/s)

Pipe

length (m)

Permissible

maximum

H/L

Dia (mm) Actual H/L Frictional head

loss, hf(m)

Fittings (other than

reducers)

Reducers

(mm x mm)

Loss thru

fittings, hp

(m)

1 62.0 0.95 20.0 0.207 32 0.07 1.400 3 elbows2 gate valves

1 tee

- 0.338

2 31.0 0.60 11.0 0.207 25 0.085 0.935 3 elbows

2 gate valves

1 tee

32 x 25 0.370

3 19.0 0.45 0.1 0.207 20 0.2 0.02 1 tee 25 x 20 0.220

4 7.0 0.24 2.5 0.207 20 0.065 0.163 1 tee - 0.059

5 3.5 0.12 0.2 0.207 15 0.15 0.03 1 tee 20 x 15 0.050

6 2.0 0.07 2.5 0.207 15 0.06 0.15 1 elbow

1 gate valve

- 0.008

36.3 2.698 1.045


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With the pipe sizes obtained, locations of reducers in thefirst index run were determined. Other types of fitting

(elbows, tees and valves) in the first index run werespecified in consideration of system functionality. In pipesection 6, for instance, there are one elbow and one gate

valve, such that hpfor this section (from Eqn. 3) = (0.75+ 0.25) x 0.08256 x 0.015-4x (0.07 x 10-3)2 = 0.008m,

where d = 0.015m and q = 0.07 x 10-3m3/s.

Similarly, in pipe section 5, there are one 20mm x 15mm

reducer (which has d1/d2= 1.33) and one tee k for thereducer (from Table 1) is 0.139. Then, from Eqn. 3, hpfor

pipe section 5= (0.139 + 2) x 0.08256 x 0.015-4 x(0.12x10-3)2= 0.050m

3.2 Resulting Parameters for the Different Index

Runs

In the manner illustrated in section 3.1, the other pipeconfigurations which number fourteen were analyzed.

However, in order to maintain brevity, only the isometricsketches of the shortest and longest first index pipe runs

are presented in Figs. 5a and 5b, respectively. Also,rather than listing all fourteen tables of calculationsummary corresponding to the fourteen other first index

pipe runs, only the two corresponding to the shortest andlongest runs are presented in Tables 3 and 4, respectively.

Table 5 summarizes the calculated total frictional and

separation losses, as well as the ratios of separation lossto total loss for the varying complexities of pipe work.The presentations in Excel graphics of Fig. 6 depict the

variation of the ratio of loss through fittings to total loss

with pipework complexity. Measures of pipeworkcomplexity are presented as length of first index pipe run,total flow rate from the reservoir and number of sanitaryappliances supplied from the reservoir.


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1 2 3 4 5 6 7 8 9 10 11

Pipe

section

No.

Loading

unitsDesign

flow

(L/s)

Pipe

length

(m)

Permissible

maximum

H/L

Dia (mm) Actual H/L Frictional headloss, hf(m)

Fittings (other

than reducers)Reducers

(mm x mm)Loss thru

fittings, hp

(m)

1 31.0 0.60 23.0 0.265 25 0.085 1.955 6 elbows3 gate valves

1 tee

- 0.552

2 19.0 0.45 0.1 0.265 20 0.200 0.020 1 tee 25 x 20 0.220

3 7.0 0.24 2.5 0.265 20 0.065 0.163 1 tee - 0.059

4 3.5 0.12 0.2 0.265 15 0.150 0.030 1 tee 20 x 15 0.050

5 2.0 0.07 2.5 0.265 15 0.060 0.150 1 elbow, 1gate valve

- 0.008

28.3 2.318 0.889


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1 2 3 4 5 6 7 8 9 10 11

PipesectionNo.

Loadingunits

Designflow (L/s)

Pipelength (m)

Permissiblemaximum

H/L

Dia (mm) Actual H/L Frictional headloss, hf(m)

Fittings (other thanreducers)

Reducers(mm x mm)

Loss thrufittings, hp

(m)

1. 475.0 4.40 20.0 0.053 65 0.027 0.540 3el, 2g.v 1 tee - 0.426

2. 434.0 4.00 8.0 0.053 65 0.024 0.192 1 tee - 0.148

3. 403.0 3.70 8.0 0.053 65 0.020 0.160 1 tee - 0.127

4. 372.0 3.50 8.0 0.053 65 0.019 0.152 1 tee - 0.1135. 341.0 3.20 8.0 0.053 50 0.050 0.400 1 tee 65 x 50 0.440

6. 310.0 2.95 8.0 0.053 50 0.045 0.360 1 tee - 0.230

7. 279.0 2.90 8.0 0.053 50 0.043 0.344 1 tee - 0.222

8. 248.0 2.70 8.0 0.053 50 0.040 0.320 1 tee - 0.193

9. 217.0 2.60 8.0 0.053 50 0.035 0.280 1 tee - 0.179

10. 186.0 2.20 8.0 0.053 50 0.026 0.208 1 tee - 0.163

11. 155.0 1.80 8.0 0.053 50 0.020 0.160 1 tee - 0.086

12. 124.0 1.55 8.0 0.053 50 0.016 0.128 1 tee - 0.059

13. 93.0 1.25 8.0 0.053 40 0.045 0.360 1 tee 50 x 40 0.152

14. 62.0 0.95 8.0 0.053 40 0.029 0.232 1 tee - 0.058

15. 31.0 0.60 11.0 0.053 32 0.030 0.330 3el, 2g.v 1 tee 40 x 32 0.138

16. 19.0 0.45 0.1 0.053 25 0.050 0.005 1 tee 32 x 25 0.091

17. 7.0 0.24 2.5 0.053 25 0.017 0.043 1 tee - 0.024

18. 3.5 0.12 0.2 0.053 20 0.020 0.004 1 tee 25 x 20 0.02219. 2.0 0.07 2.5 0.053 15 0.037 0.093 1 el, 1g.v 1 tee 20 x 15 0.009

140.3 4.311 2.880


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Table 5: Ratios of Loss through Fittings to Total Loss for Varying Pipe Work Complexities

1 2 3 4 5 6 7

Length of 1stindex

pipe run (m)

Total flow rate through

main distribution pipe

(L/s)

No. of appliances served by

main distribution pipe

Frictional loss in 1stindex

run (m)

Loss through fittings in 1st

index run (m)

Total loss in 1stindex

run (m)

Ratio of loss through

fittings to total loss

28.3 0.60 8 2.318 0.889 3.207 0.277

36.3 0.95 16 2.698 1.045 3.743 0.279

44.3 1.25 24 3.943 1.302 5.245 0.248

52.3 1.55 32 3.747 1.302 5.049 0.258

60.3 1.80 40 4.777 1.594 6.371 0.250

68.3 2.20 48 4.337 1.627 5.964 0.273

76.3 2.60 56 4.245 1.724 5.969 0.289

84.3 2.70 64 4.625 1.936 6.561 0.295

92.3 2.90 72 5.005 2.069 7.074 0.282

100.3 2.95 80 4.304 2.079 6.383 0.326

108.3 3.20 88 4.379 2.318 6.697 0.346

116.3 3.50 96 4.467 2.666 7.133 0.374

124.3 3.70 104 4.147 2.542 6.689 0.380

132.3 4.00 112 4.059 2.658 6.717 0.396

140.3 4.40 120 4.311 2.880 7.191 0.401


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Loss Through Fittings

Total Loss

0

0.1

0.2

0.3

0.4

0.5

0 10 20 30 40 50 60 70 80 90 100 110 120 130

No. of Appliances (X3)

Loss Through FittingsTotal Loss

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Design Fllow Rate (X2), L/s

Loss Through Fittings

Total loss

0

0.1

0.2

0.3

0.4

0.5

0 20 40 60 80 100 120 140 160

Index Pipe length (X1), m

Fig. 6 : Variation of Ratio of Loss Thru Fittings to Total Head Loss with Pipework Complexity


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4. STATISTICAL ANALYSIS OF ESTIMATES

The statistical tool of regression analysis was employed to

obtain equations which represent the variation of the ratio of

fitting loss to total loss with the different measures of pipe

work complexity. Thus, this ratio, denoted as y, was regressed

on the mentioned three measures of complexity, denoted as x1,

x2and x3,respectively.

An initial investigation had indicated a second order variation

of y with x (Sodiki and Orupabo, 2011) namely

y = a0 + a1 x + a2x2

- - - - - (4)

The effort here, therefore, would be to obtain the regression

parameters a0, a1 and a2 for each of the independent variables

x1, x2 and x3 by the solution of the set of analytical

simultaneous equations (Lipson and Seth, 1973)

y = na0 + a1 x+ a2x2

yx = a0 x + a1 x2+ a2x3

- - - - - (5)

yx2 = a0 x2+ a1 x3+ a2x4

where n = number of data points (in this case, equal to

15).In line with standard methods of statistics, Table 6 was set

up to summarize the computations which aid the analysis. In

Appendix 1, the relationship between y and x1, y and x2, and y

and x3are established, respectively, as

y =0.22801.246 x 10-3x1+ 1.558 x 10-5x12 - -- (6)

y =0.28163.538 x 10-2x2+ 1.565 x 10-2x22 --- (7)

and

y =0.2287 + 1.169 x 10-3x3+ 1.649 x 10-6x32 --- (8)


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Table 6: Compilation of Statistical Variables and Terms

IndexPipe

Length

X1(m)

DesignFlow

X2(l/s)

No. ofAppliances

X3

Frictional Loss

(m)

Lossthru

Fittings

(m)

TotalLoss (m)

Loss ThruFittings/ Total

Loss Y

yx1 x12 yx12 X13 x14 yx2 x22 yx22 x23 x24 yx3 x32 yx32 x33 x34

28.3 0.60 8 2.318 0.889 3.207 0.277 7.8391 800.89 221.842 22665.187 641424.792 0.1662 0.3600 0.09772 0.2160 0.1296 2.216 64 17.728 512 4096

36.3 0.95 16 2.698 1.045 3.743 0.279 10.1277 1317.69 367.636 47832.147 1736306.936 0.2651 0.9025 0.25180 0.8574 0.8450 4.464 256 71.424 5832 65536

44.3 1.25 24 3.943 1.302 5.245 0.248 10.9864 1962.49 486. 698 86938.307 3851367.000 0.3100 1.3625 0.38750 1.7531 2.4414 5.952 576 142.878 13824 331776

52.3 1.55 32 3.747 1.302 5.049 0.258 13.4934 2735.29 705.705 143055.667 7481811.384 0.3999 2.4025 0.61985 3.7239 5.7720 8.256 1024 264.192 32768 1048576

60.3 1.80 40 4.777 1.594 6.371 0.250 15.0750 3636.09 909.023 219256.227 13221150.49 0.4500 3.2400 0.81000 5.8320 10.4976 10.000 1600 400.000 64000 2560000

68.3 2.20 48 4.337 1.627 5.964 0.273 18.6459 4664.89 1273.515 318611.987 21761198.71 0.6006 4.8400 1.32132 10.6480 23.9256 13.104 2304 628.992 110592 5308416

76.3 2.60 56 4.245 1.724 5.969 0.289 22.0507 5821.69 1682.468 444194.947 33892074.46 0.7514 6.7600 1.95364 17.5760 45.6976 16.184 3136 906.304 175616 9834496

84.3 2.70 64 4.625 1.936 6.561 0.295 24.8685 7106.49 2096.415 599077.107 50502200.2 0.7965 7.2900 2.15055 19.6830 53.1441 18.880 4096 1208.320 262144 16777216

92.3 2.90 72 5.005 2.069 7.074 0.292 26.9516 8519.29 2487.633 786330.467 72578302.1 0.8468 8.4100 2.45572 24.3890 70.7281 21.024 5184 1513.728 373248 26873856

100.3 2.95 80 4.304 2.079 6.383 0.326 32.6978 10060.09 3279.589 1009027.027 101205410.8 0.9617 8.7025 2.83702 25.6724 75.7335 26.080 6400 2086.400 512000 40960000

108.3 3.20 88 4.379 2.318 6.697 0.346 37.4718 11728.89 4050.196 127023.787 137566860.6 1.1072 10.2400 3.54304 32.7680 104.8576 30.448 7744 2679.424 681772 5996536

116.3 3.50 96 4.467 2.666 7.133 0.374 43.4962 13525.69 5058.608 1573037.747 182944290.0 1.3090 12.2500 4.58150 42.8750 150.0625 35.904 9216 3446.784 884736 84934656

124.3 3.70 104 4.147 2.542 6.689 0.380 47.2340 15450.49 5871.186 1920495.907 238717641.2 1.4060 13.6900 5.20220 50.6530 187.4161 39.520 10816 4110.080 1124864 116985856

132.3 4.00 112 4.059 2.658 6.717 0.396 52.3903 17503.29 6931.303 2315685.267 306365160.8 1.5840 16.0000 6.33600 64.0080 256.0000 44.352 12544 4967.424 1404928 157351936

140.3 4.40 120 4.311 2.880 7.191 0.401 56.2603 19684.09 7893.320 2761677.827 387463399.1 1.7644 19.3600 7.76636 85.1840 374.8096 48.120 14400 5774.400 1728000 207360000

=1264.5

=38.3

= 960 = 4.684 =419.5892

=124517.35

=43315.142

=13518124.61

=1559928599.5

=12.7188

= 116.01 =40.31322

=286.0308

=1361.5298

=324.504

=79360

=28218.048

=7374536

=730365952


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Furthermore, the coefficient of correlation is given as (Lipson

and Seth, 1973)

r =

2

.

1

y

xy

S

S

( 9 )

where Sy.x =

3

2

1

n

yyn

i

ici

( 1 0 )= standard error of estimate,

yibeing the actual values

of y, yicbeing the values of

y computed from the

regression equation and n

the numbers of points

and Sy = 1

2

1

n

yy

n

i

i

(11)= sample standard deviation

of y

n3 is the number of degrees of freedom, as the number of

regression parameters to be estimated is three, a0, a1and a2

In Appendix II, the coefficients of correlation between y and

x1, y and x2, and y and x3are computed, respectively, as 0.97,

0.95 and 0.91.

5. DISCUSSION OF RESULTS

From statistical tables (Lipson and Seth, 1973), r required for

99% confidence is 0.661. Since all three computed values of r

exceed 0.661, there is 99% confidence that the variation of the

fraction of head loss through fittings is interdependent, in turn,

with variations of length of index pipe run, total flow rate and

number of appliances served. It further follows that estimates

of the fraction of head loss can be made using the derived

regression equations.

The fractions of head loss through pipe fittings calculated by

the regression equations in Appendix II show the following

general increases: from 0.263 to 0.420 for increase in first

index pipe length from 28.3m to 140.3m; from 0.262 to 0.429

for an increase in total flow rate from 0.6L/s to 4.4L/s; and

from 0.238 to 0.393 for an increase in number of sanitary

appliances from 8 to 120.

It is, thus, observed that the head loss fractions fall between

0.24 and 0.43 within the limits of system complexity utilized

in the analysis.

In fact, random computations and estimates done on other

water distribution configurations (different from the general

architectural arrangement of toilet rooms and sanitary

appliances employed in this analysis), but utilizing the same

available head of 7.5m and distributing to a single building

floor, as in this analysis, had given fractions of head loss in the

region of 0.2 to 0.4 for first index lengths of between 20m and

120m. Hence, needed estimates of fractions may be made by

interpolating between these limits and adding a safety margin.

Alternatively, the present results may be applied with

reasonable correctness.

6. CONCLUSIONS

Within the limits of system complexity of up to 140m index

pipe length, 4.4L/s flow rate and 120 sanitary appliances a ratio

of loss through fittings to total loss of 0.45 is not likely to be

exceeded for simple water distribution systems under an

available head of 7.5m.

The ratios of head loss obtained from the regression equations

are useful in the analysis of system losses. For instance, for a


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given first index pipe length, the total head loss can quickly be

estimated by adding the relevant fraction to the frictional loss.

The total head loss, so estimated, is useful in determining the

suitability of the given reservoir elevation and, hence, in

assessing the head requirements of the lift pump.

Other water distribution systems such as are utilized for groups

of buildings in housing estates and for distribution in town and

village mains could also be analyzed by the same procedure.

Furthermore, the relationship between the available head and

the computed ratios could be investigated by carrying out the

foregoing analysis for different commonly utilised reservoir

elevations

REFERENCES

[1].Barry, R. (1977). The Construction of Buildings, Vol. 5:

Supply and Discharge Services. Granada Publishing Ltd,

London.

[2].Giles, R. V. (1977), Fluid Mechanics and Hydraulics.

McGraw-Hill Book Co., New York

[3]. Institute of Plumbing (1977). Plumbing Services Design

Guide, Essex

[4].Lipson, C and Seth, N. J. (1973). Statistical Design and

Analysis of Engineering Experiments. McGraw-Hill

Book Co., New York

[5].Sodiki, J. I. (2002). A Representative Expression for

Swimming Pool Circulator Pump Selection. Nigerian

Journal of Engineering Research and Development. Vol.

1, No. 4, Pp. 24-35

[6].Sodiki, J. I. (2003). Design Analysis of Water Supply

and Distribution to a Multi-Storey Building Utilizing a

Borehole Source. Nigerian Journal of Industrial and

Systems Studies. Vol. 2, No. 2, Pp 16-32

[7].Sodiki, J. I. and Orupabo, S. (2011). Estimating Head

and Frictional Losses through Pipe Fittings in Building

Water Distribution Systems. Journal of Applied Science

and Technology. Vol. 16, Nos. 1 & 2, Pp 67-74


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APPENDIX I: REGRESSION EQUATIONS

(i) Regression of y on x1

Substitution of values from Table 6 into Eqn. 5 yields the simultaneous equation

4.684 = 15a0+1264.5a1+ 124517.35a2 ----- (a)

419.589 = 1264.5a0+ 124517.35a1 + 13518124.61a2 ---- (b)

43315.142 = 124517.35a0+ 13518124.61a1 +1559928599.5a2 --- (c)

Solving for a0, a1and a2yields the expression for Eqn. 4 as

y = 0.22801.246 x 10-3x1+ 1.558 x 10-5 x12

(ii) Regression of y on x2

Substitution of values from Table 6 into Eqn. 5 yields the simultaneous equations

4.684 = 15a0+38.3a1+ 116.01a2 ----- (d)

12.719 = 38.3a0+ 116.01a1 + 1386.03a2 ---- (e)

40.313 = 116.01a0+ 386.03a1 +1361.53a2 ---- (f)

Solving for a0,a1and a2 yields the expression for Eqn. 4 as

y = 0.28163.538 x 10-2 x2+ 1.565 x 10-2 x22

(iii) Regression of y on x3

Substitution of values from Table 6 into Eqn. 5 yield the simultaneous equations

4.684 = 15a0+960a1+ 79360a2 ----- (g)

324.508 = 960a0+ 79360a1 + 7374536a2 ---- (h)

28218.048 = 79360a0+ 7374536a1 +730365952a2 ---- (i)

Solving for a0,a1and a2 yields the expression for Eqn. 4 as

y = 0.2287 + 1.169 x 10-3 x3+ 1.649 x 10-6 x32

APPENDIX II: CALCULATION OF COEFFICIENTS

II(i) Coefficient of Correlation for x1

The regression equation is

yc = 0.28801.246 x 10-3x1 + 1.558 x 10-5 x12


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where, ycis the calculated value of y. In Table A1, A2 and A3 yiis the actual value of y obtained from the analysis of the losses of

pressure head, and y is the mean of the y values.

Table A1: Values for Calculating Correlation Coefficient for x1

i xi yi yi y (yi y )2x 104 yic yiyc (yiyic)2

x 104

1 28.3 0.277 -0.0236 5.5696 0.265 0.012 1.44

2 36.3 0.279 -0.0333 11.0889 0.263 0.016 2.56

3 44.3 0.248 -0.0526 27.6676 0.263 -0.015 2.25

4 52.3 0.258 -0.0426 18.1476 0.265 -0.007 0.49

5 60.3 0.250 -0.0506 25.6036 0.270 -0.020 4.00

6 68.3 0.273 -0.0276 7.6176 0.276 -0.003 0.09

7 76.3 0.289 -0.0116 1.3456 0.284 0.005 0.25

8 84.3 0.295 -0.0056 0.3136 0.294 0.001 0.01

9 92.3 0.292 -0.0086 0.7396 0.306 -0.014 1.96

10 100.3 0.326 0.0254 6.4516 0.320 0.006 0.36

11 108.3 0.346 0.0454 20.6116 0.336 0.010 1.00

12 116.3 0.374 0.0734 53.8756 0.354 0.020 4.00

13 124.3 0.380 0.0794 63.0436 0.374 0.006 0.36

14 132.3 0.396 0.0954 91.0116 0.396 0.000 0.00

15 140.3 0.401 0.1004 100.8016 0.420 -0.019 3.60

= 1264.5 = 433.8893 = 22.38

y = 0.3123

Sy.x = 01366.012

1038.22 4

Sy = 05567.014

108893.433 4

Coefficient of Correlation r = 97.005567.0

01366.01

2

II(ii): Coefficient of Correlation for x2


yc = 0.28163.538x 10-2x2+ 1.565x10-2 x22


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i xi yiyi y (yiy )2x 104

yic yiyc (yiyic)2 x

104

1 060 0.277 -0.0236 5.5696 0.266 0.011 1.21

2 0.95 0.279 -0.0333 11.0889 0.262 0.017 2.89

3 1.25 0.248 -0.0526 27.6676 0.262 -0.014 1.96

4 1.55 0.258 -0.0426 18.1476 0.264 -0.006 0.36

5 1.80 0.250 -0.0506 25.6036 0.269 -0.019 3.61

6 2.20 0.273 -0.0276 7.6176 0.280 -0.007 0.49

7 2.60 0.289 -0.0116 1.3456 0.295 -0.006 0.36

8 2.70 0.295 -0.0056 0.3136 0.300 -0.005 0.25

9 2.90 0.292 -0.0086 0.7396 0.311 -0.019 3.61

10 2.95 0.326 0.0254 6.4516 0.313 0.013 1.69

11 3.20 0.346 0.0454 20.6116 0.329 0.017 2.89

12 3.50 0.374 0.0734 53.8756 0.349 0.025 6.25

13 3.70 0.380 0.0794 63.0436 0.365 0.015 2.25

14 4.0 0.396 0.0954 91.0116 0.390 0.006 0.36

15 4.4 0.401 0.1004 100.8016 0.429 -0.028 7.84

= 36.02

Sy.x = 01733.012

1002.36 4

Sy = 0.05567 (as for x1)


01733.01

2

II(iii): Coefficient of Correlation for x3


yc = 0.2287 + 1.169 x 10-3x3 + 1.649 x 10-6x 32


i xi yiyi y (yi y )2x 104

yic yiyc (yiyic)2 x

104

1 8 0.277 -0.0236 5.5696 0.238 0.039 15.21

2 16 0.279 -0.0333 11.0889 0.248 0.029 8.41

3 24 0.248 -0.0526 27.6676 0.258 -0.070 1.00

4 32 0.258 -0.0426 18.1476 0.268 -0.010 1.00


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5 40 0.250 -0.0506 25.6036 0.278 -0.028 7.84

6 48 0.73 -0.0276 7.6176 0.289 -0.016 2.56

7 56 0.289 -0.0116 1.3456 0.299 -0.010 1.00

8 64 0.295 -0.0056 0.3136 0.310 -0.015 2.25

9 72 0.292 -0.0086 0.7396 0.321 -0.029 8.41

10 80 0.326 0.0254 6.4516 0.333 -0.007 0.49

11 88 0.346 0.0454 20.6116 0.344 0.002 0.04

12 96 0.374 0.0734 53.8756 0.356 0.018 3.24

13 104 0.380 0.0794 63.0436 0.368 0.012 1.44

14 112 0.396 0.0954 91.0116 0.380 0.016 2.56

15 120 0.401 0.1004 100.8016 0.393 0.008 6.40

= 61.85

Syx = 0227.012

1085.61 4

Sy = 0.05567 (as for x1)


0227.01

2

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