Hunter’s Curve in the 21st Century

ACEEE Hot Water Forum

Steven Buchberger

November 4, 2013

What is Hunter’s Curve?

2

Hunter’s Curve Predicts Peak Flow

Fixture Units

GPM

3

4

Life in 1940

Population = 2.3B Gas = $0.2/gal5

Life in 2013

Population = 7.0B Gas = $4/gal6

Hour

Flo

w (

L/m

in) Friday, May 16

0 4 8 12 16 20 24

0

20

40

60

80

100

End User Demand (21 units)

7

t

T

q

One Fixture is a Bernoulli Trial

p = t/T = Average duration of flow

Avg time btn consecutive uses8

t

T

q

Three Key Parameters…..

Fixture Characteristics

Human Behaviorp = t/T

9

Many Fixtures Exist

1 2 3 n• • • k • • •

1

2

3

10

Many Fixtures Exist

1 2 3 n• • • k • • •

1

2

3

11Overlapping pulses

Design Problem

“Assuming that there are n (identical ) fixtures in a system, each operated once in Tseconds on the average, and that each operation is of t seconds average duration, what is the probability that k fixtures will be found operating simultaneously at any arbitrarily chosen instant of observation”?

(Roy Hunter, 1940)

12

Many Fixtures are Binomial

1 2 3 n• • • k • • •

Pr 1

0,1,...,

k n kexactly k busy fixtures np p

out of n total fixtures k

twhere p k n

T

13

Binomial Distribution Example

p=0.20; n=7

Binomial Distribution for Flush Tanks

Number of Busy Fixtures

pro

bability

0 2 4 6 8

0

0.1

0.2

0.3

0.4

77

Pr 0.2 0.8 0,1,...,7k kexactly k

kbusy fixtures k

14

p=0.20; n=7

Binomial Distribution for Flush Tanks

Number of Busy Fixtures

pro

bability

0 2 4 6 8

0

0.1

0.2

0.3

0.4

Design Condition is 99th Percentile

1% chance

15

99% chance

Design Flow, One Fixture Group

n

Fixture Group A

0 10 20 30 40

0

10

20

30

40

Q(0.99)

p=t/T=0.2

(gpm)n=7; Q=16 gpm

0.99

47 4 16

gpmQ n m q fixtures gpm

fixture

16

Design Flows, Two Fixture Groups

n

Fixture Group A

Fixture Group B

Q(0.99)

0 10 20 30 40

0

10

20

30

40

p=t/T=0.20

p=t/T=0.05(gpm)

17

Common Currency One Curve

Fixture Units

GPM

18

Hunter’s curve has withstood the test of time and is the basis for plumbing codes around the globe today.

Hunter’s curve went viral long before U-tube arrived; not surprising, it is clever, convenient, correct.

However, today Hunter’s curve is often faulted for giving overly conservative design….why?

Hunter’s Track Record

19

[1] Simplicity is seductive. Hunter’s curve has been applied to many situations for which it was not intended.

[2] Times have changed. Water use fixtures (hot and cold) have become much more efficient since Hunter’s pioneering work.

Two Main Issues

20

Hunter’s Curve in 1940

Fixture Units

GPM

21

Hunter’s Curve in 2013

GPMLEED, NZE, HE fixtures = lower q

uncongested use = lower n, p

22

National effort in US to update Hunter’s curve for peak water demands.

Driven by professional societies, not the US Gov’t (not Nat’l Bureau Standards).

Prevailing sentiment is to simply revise the fixture units in the code.

What would Roy Hunter do?

Old Habits Die Hard

23

IAPMO Sub-Task Group Orders

“….work singularly to develop the probability model to predict peak residential demands based on the number of plumbing fixtures of different kinds installed in one system.”

24

Aquacraft Data Sets

• 2011 California Single Family Home Water Use Efficiency Study (n=750)

• 2011 Albuquerque Retrofit Study

o Pre-retrofit (n=240)

o Post-retrofit (n=29)

• 2010 EPA Standard New Homes (n=302)

• 2010 EPA High Efficiency New Homes (n=25)

[1,346 homes ….. >15,000 home days]

25

Data Base Queries

1

2

3

4

5

6

7

8

home unique ID

range of home IDs

Aquacraft data set(s)

age of home

retrofit status of home (Y/N)

geographic location of home

fixture performance (NLF, LF, ULF )

fixture group

26

Data Base Queries

1

2

3

4

5

6

7

8

home unique ID

range of home IDs

Aquacraft data set(s)

age of home

retrofit status of home (Y/N)

geographic location of home

fixture performance (NLF, LF, ULF )

fixture group

9

10

11

12

13

14

15

16

indoor water use

outdoor water use

weekday water use

weekend water use

AM or PM use

hot or cold water use *

per capita daily water use

total annual household water use

27

Data Base Queries

1

2

3

4

5

6

7

8

home unique ID

range of home IDs

Aquacraft data set(s)

age of home

retrofit status of home (Y/N)

geographic location of home

fixture performance (NLF, LF, ULF )

fixture group

9

10

11

12

13

14

15

16

indoor water use

outdoor water use

weekday water use

weekend water use

AM or PM use

hot or cold water use

per capita daily water use

total annual household water use

17

18

19

20

21

22

23

24

home square footage

yard square footage

number of bedrooms

number of bathrooms

number of occupants

age of occupants

water meter size

? _____________

28

29

Six Types of Residential Fixtures

[1] Toilets (3 efficiency levels)

[2] Showers

[3] Bathtubs

[4] Faucets (all sinks)

[5] Dishwasher (energy star ratings)

[6] Clothes Washer (energy star ratings)

30

Three Characteristics of Fixtures

[1] Pulse Intensity (q)

[2] Pulse Duration (t)

[3] Pulse Frequency (T)

t

q

T

Water Pulse Characteristics

Fixture Group

No of Fixtures

Typical MinimumWater Pulse

Average (Nominal) Water Pulse

Typical MaximumWater Pulse

Standard Deviation Water Pulse

Sample Size

Terms and Units Water Pulse

n q t v=qt q t v=qt q t v=qt q t v N q t v

FG 1 100 1.00 1.50 1.50 1.50 2.00 3.00 2.00 2.50 5.00 0.25 0.25 1.00 774

FG 2 100 1.50 3.50 5.25 3.00 8.00 24.00 3.50 10.00 35.00 0.50 1.50 6.00 191 gpm min gal

FG 4 50 1.00 0.50 0.50 1.00 0.50 0.50 1.00 0.50 0.50 0.00 0.00 0.00 1040.3

Average (Nominal) Water Pulse

q t v=qt1.50 2.00 3.003.00 8.00 24.001.00 0.50 0.50

(gpm) (min) (gal)

Fixture Group

FG 1FG 2FG 4

(example, N=50 homes)

31

Peak Flow (99th percentile)

Fixture Group

FG 1, n=100FG 2, n=100FG 4, n=50

7 am 8 am0.026 0.0410.103 0.051

0.028 0.019

Probability of Fixture Use

p=t/T

(example, N=50 homes)

Hour ending 7 amFixtures Flow (gpm)

mean var mean var2.6 2.5 3.9 5.7

10.3 9.2 30.9 83.21.4 1.4 1.4 1.4

14.3 13.1 36.2 90.3

58.3 gpmQ(0.99)=

per Wistort 199432

Normal approximation (Wistort, 1994)

Computer simulation: SIMDEUM or PRPsym

Full enumeration of CDF (WDSA 2012)

Merge w/ Bldg Information Modeling (BIM)

“There’s an app for that!”

Tantalizing Possibilities

+ =+ +33

Steven.Buchberger@uc.edu

University of Cincinnati

Questions?

34

Phones to Faucets Analogy

“Arrival Rates”

Poisson Model Erlang 1918

“Time Between Uses”

Binomial Model Hunter 1940

35

Shutterstock.com

End User Examples - 1

Schools

Hospitals

36

Shutterstock.com

End User Examples - 2

Opera Houses

Bus/Rail Stations

37

Shutterstock.com

End User Examples – 3…

Hotels, CBD

Sports Stadiums

38