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...

Nonlinear

...

Theory

...

The

...

nonlinear

...

chemical

...

doser

...

is

...

part

...

of

...

the

...

evolution

...

of

...

alum

...

dosing

...

techniques

...

in

...

an

...

effort

...

to

...

increase

...

the

...

maximum

...

flow

...

rate

...

capacity

...

of

...

AguaClara

...

plants

...

as

...

well

...

robustness

...

of

...

design

...

and

...

implementation.

...

A

...

frequent

...

problem

...

in

...

AguaClara

...

plants

...

was

...

the

...

occurrence

...

of

...

foam

...

in

...

the

...

flocculator

...

after

...

the

...

addition

...

of

...

alum

...

to

...

the

...

entrance

...

tank.

...

It

...

was

...

postulated

...

that

...

the

...

previous

...

free

...

fall

...

method

...

of

...

dosing

...

alum

...

was

...

contributing

...

to

...

the

...

occurrence

...

of

...

foam.

...

Any

...

new

...

dosing

...

mechanism

...

would

...

need

...

to

...

have

...

alum

...

dosed

...

below

...

the

...

level

...

of

...

water

...

in

...

the

...

entrance

...

tank.

...

Previous

...

linear

...

chemical

...

dosers

...

have

...

relied

...

upon

...

the

...

major

...

head

...

losses

...

caused

...

by

...

the

...

friction

...

of

...

the

...

dosing

...

tube

...

to

...

control

...

the

...

flow

...

of

...

alum.

...

The

...

important

...

point

...

in

...

distinguishing

...

the

...

need

...

to

...

move

...

to

...

nonlinear

...

flow

...

is

...

recognizing

...

the

...

relationship

...

between

...

head

...

loss

...

and

...

the

...

flow

...

rate

...

of

...

alum

...

in

...

the

...

two

...

dosing

...

methods.

...

In

...

previous

...

linear

...

dosing

...

designs,

...

the

...

flow

...

of

...

alum

...

is

...

proportional

...

to

...

major

...

head

...

losses

...

in

...

the

...

dosing

...

tube.

...

If

...

the

...

flow

...

rate

...

of

...

alum

...

were

...

to

...

increase

...

into

...

the

...

turbulent

...

range

...

then

...

the

...

head

...

loss

...

is

...

proportional

...

to

...

the

...

flow

...

rate

...

squared,

...

as

...

shown

...

in

...

the

...

figure

...

below.

...

Since

...

the

...

relationship

...

between

...

flow

...

rate

...

and

...

head

...

loss

...

is

...

not

...

linear

...

for

...

turbulent,

...

significant

...

dosing

...

errors

...

would

...

result

...

in

...

the

...

linear

...

dosing

...

scheme.

...

In

...

order

...

to

...

allow

...

there

...

to

...

be

...

turbulent

...

alum

...

flow

...

in

...

the

...

dosing

...

tube,

...

an

...

orifice

...

controlled

...

nonlinear

...

doser

...

is

...

now

...

being

...

used.

...

In

...

the

...

nonlinear

...

orifice

...

controlled

...

doser,

...

the

...

majority

...

of

...

the

...

head

...

losses

...

is

...

due

...

to

...

the

...

minor

...

losses

...

caused

...

by

...

the

...

orifice.

...

In

...

the

...

nonlinear

...

system

...

the

...

flow

...

rate

...

of

...

alum

...

is

...

proportional

...

to

...

the

...

square

...

root

...

of

...

h

...

in

...

both

...

laminar

...

and

...

turbulent

...

ranges.

...

This

...

homogeneity

...

in

...

relationships

...

allows

...

there

...

to

...

be

...

reliable

...

dosing

...

even

...

in

...

turbulent

...

ranges.

...

 

Laminar

Turbulent

Linear doser

Latex
$$Q\alpha h$$

...

 Latex
$$Q^2 \alpha h$$
...
Orifice doser 
 Latex
$$
Q\alpha \sqrt h 
$$
...
 Latex
$$
Q\alpha \sqrt h 
$$
...
where: 
...
 Latex
...
$$\alpha $$
...
 = 
...
 proportional 
...
 to
...
The 
...
 connection 
...
 between 
...
 the 
...
 flow 
...
 control 
...
 and 
...
 the 
...
 flow 
...
 measurement 
...
 aspect 
...
 of 
...
 the 
...
 dosing 
...
 mechanism 
...
 is 
...
 important 
...
 to 
...
 understand 
...
 the 
...
 evolution 
...
 of 
...
 dosing 
...
 mechanisms 
...
 in 
...
 AguaClara 
...
 plants. 
...

 Since 
...
 the 
...
 nonlinear 
...
 chemical 
...
 doser 
...
 has 
...
 the 
...
 same 
...
 relationship 
...
 between 
...
 flow 
...
 and 
...
 head 
...
 at 
...
 the 
...
 turbulent 
...
 ranges; 
...
 AguaClara 
...
 plants 
...
 can 
...
 be 
...
 scaled 
...
 up 
...
 to 
...
 much 
...
 higher 
...
 flow 
...
 rates 
...
 without 
...
 being 
...
 limited 
...
 by 
...
 the 
...
 turbulence 
...
 in 
...
 the 
...
 dosing 
...
 tube. 
...
 This 
...
 is 
...
 a 
...
 huge 
...
 advantage 
...
 of 
...
 the 
...
 nonlinear 
...
 system 
...
 because 
...
 it 
...
 expands 
...
 the 
...
 AguaClara 
...
 plants 
...
 capabilities 
...
 to 
...
 serve 
...
 much 
...
 larger 
...
 communities. 
...
 The 
...
 size 
...
 of 
...
 AguaClara 
...
 plants 
...
 is 
...
 no 
...
 longer 
...
 limited 
...
 to 
...
 the 
...
 flow 
...
 limitations 
...
 in 
...
 the 
...
 dosing 
...
 tube.
...
 As 
...
 mentioned, 
...
 the 
...
 nonlinear 
...
 doser 
...
 uses 
...
 the 
...
 minor 
...
 losses 
...
 caused 
...
 by 
...
 the 
...
 orifice 
...
 instead 
...
 of 
...
 a 
...
 dosing 
...
 tube 
...
 (major 
...
 losses) 
...
 to 
...
 control 
...
 the 
...
 relationship 
...
 between 
...
 changing 
...
 plant 
...
 flow 
...
 rates 
...
 and 
...
 chemical 
...
 dose. 
...
 The 
...
 flow 
...
 rate 
...
 through 
...
 the 
...
 Chemical 
...
 Dose 
...
 Controller 
...
 (CDC) 
...
 is 
...
 related 
...
 to 
...
 the 
...
 available 
...
 head 
...
 by 
...
 the 
...
 equation:

{
 Latex
}$$Q_{Cdc}  = K_{orifice}\sqrt {2gh_{Cdc} } $${latex}
where
* {latex}
where
  •  Latex
    $$Q_{Cdc} $$
...
  •  is 
...
  •  the 
...
  •  chemical 
...
  •  flow 
...
  •  rate
...
  •  Latex
    $$ K_{orifice} $$
...
  •  is 
...
  •  the 
...
  •  orifice 
...
  •  coefficient
...
  • h 
...
  •  is 
...
  •  the 
...
  •  available 
...
  •  head
...
The 
...
 desired 
...
 chemical 
...
 dose 
...
 to 
...
 the 
...
 plant 
...
 can 
...
 be 
...
 determined 
...
 by 
...
 a 
...
 mass 
...
 balance:

{
 Latex
}$$C_p  = {{C_c Q_{Cdc} } \over {Q_{Plant} }}$${latex}
where
* C ~c~ is the chemical stock concentration
* C ~p~ is the chemical dose

The influent raw water leaves the entrance tank through the Rapid Mix Tube which is the entry point for the dosing of alum.  In the Rapid mix tube, an orifice is located in the tube to generate macro and micro-scale mixing. The entry point for the dosing of alum with the raw water has been redesigned to be submerged in the entrance tank in order to possibly reduce the occurrence of foam in the flocculator.  After conversations with Dan Smith, the AguaClara engineer in Honduras, there doesn't appear to be foam forming in the Agalteca plant, where this dosing system has been implemented.
The flow of water through an AguaClara plant can be modeled as minor losses due to flow expansions.  The relationship between plant flow rate and head loss through the plant is governed by the minor loss equation shown below. 
{latex
where
  • C c is the chemical stock concentration
  • C p is the chemical dose
The influent raw water leaves the entrance tank through the Rapid Mix Tube which is the entry point for the dosing of alum. In the Rapid mix tube, an orifice is located in the tube to generate macro and micro-scale mixing. The entry point for the dosing of alum with the raw water has been redesigned to be submerged in the entrance tank in order to possibly reduce the occurrence of foam in the flocculator. After conversations with Dan Smith, the AguaClara engineer in Honduras, there doesn't appear to be foam forming in the Agalteca plant, where this dosing system has been implemented.
 The flow of water through an AguaClara plant can be modeled as minor losses due to flow expansions. The relationship between plant flow rate and head loss through the plant is governed by the minor loss equation shown below. 
 Latex
}$$ h{}_{plant} = K_{plant} {{Q_{plant} ^2 } \over {2g}}$${latex}

where:
...

 Latex
...
$$ K_{plant} $$
...
 = 
...
 plant 
...
 minor 
...
 loss 
...
 coefficient
...
The 
...
 equation 
...
 above 
...
 is 
...
 important 
...
 since 
...
 it 
...
 defines 
...
 the 
...
 relationship 
...
 between 
...
 plant 
...
 flow 
...
 and 
...
 head 
...
 loss 
...
 through 
...
 the 
...
 plant, 
...
 which 
...
 is 
...
 important 
...
 as 
...
 well 
...
 for 
...
 our 
...
 dosing 
...
 apparatus. 
...
 The 
...
 CDC 
...
 uses 
...
 a 
...
 lever 
...
 arm 
...
 with 
...
 a 
...
 float 
...
 and 
...
 counterweight 
...
 to 
...
 relate 
...
 the 
...
 dosing 
...
 to 
...
 the 
...
 changes 
...
 in 
...
 the 
...
 entrance 
...
 tank 
...
 water 
...
 level, 
...
 which 
...
 is 
...
 a 
...
 function 
...
 of 
...
 the 
...
 influent 
...
 flow 
...
 rate. 
...
 An 
...
 increase 
...
 in 
...
 head 
...
 loss 
...
 links 
...
 the 
...
 chemical 
...
 flow 
...
 rate 
...
 to 
...
 the 
...
 plant 
...
 flow 
...
 rate 
...
 and 
...
 the 
...
 chemical 
...
 dose 
...
 (mg/L) 
...
 will 
...
 be 
...
 constant 
...
 as 
...
 plant 
...
 flow 
...
 varies.As 
...
 can 
...
 be 
...
 seen 
...
 above, 
...
 the 
...
 equation 
...
 which 
...
 describes 
...
 the 
...
 flow 
...
 of 
...
 water 
...
 through 
...
 plant 
...
 and 
...
 the 
...
 orifice 
...
 equation 
...
 which 
...
 relates 
...
 flow 
...
 of 
...
 alum 
...
 in 
...
 the 
...
 dosing 
...
 system 
...
 both 
...
 have 
...
 the 
...
 same 
...
 relationship 
...
 between 
...
 flow 
...
 rate 
...
 and 
...
 head 
...
 loss. 
...
 Since 
...
 the 
...
 flow 
...
 rate 
...
 for 
...
 both 
...
 the 
...
 plant 
...
 and 
...
 alum 
...
 flow 
...
 are 
...
 each 
...
 related 
...
 by 
...
 the 
...
 square 
...
 root 
...
 of 
...
 the 
...
 head 
...
 loss, 
...
 the 
...
 two 
...
 flows 
...
 can 
...
 be 
...
 linked 
...
 through 
...
 the 
...
 float 
...
 in 
...
 the 
...
 entrance 
...
 tank. 
...
 Any 
...
 rise
...

 For 
...
 a 
...
 detailed 
...
 step 
...
 by 
...
 step 
...
 description 
...
 of 
...
 the 
...
 steps 
...
 involved 
...
 with 
...
 measuring 
...
 the 
...
 plant 
...
 flow 
...
 rate 
...
 please 
...
 see 
...
 flow 
...
 measurement 
...
 section 
...
 of 
...
 the 
...
 Chemical 
...
 Dose 
...
 Controller 
...
 Manual. 
...

 The 
...
 dosing 
...
 tube 
...
 must 
...
 be 
...
 designed 
...
 to 
...
 minimize 
...
 major 
...
 losses 
...
 so 
...
 that 
...
 minor 
...
 losses 
...
 dominate 
...
 head 
...
 loss. 
...
 The 
...
 amount 
...
 of 
...
 major 
...
 losses 
...
 due 
...
 to 
...
 the 
...
 friction 
...
 in 
...
 the 
...
 dosing 
...
 tube 
...
 will 
...
 vary 
...
 based 
...
 on 
...
 the 
...
 flow 
...
 of 
...
 alum 
...
 going 
...
 through 
...
 the 
...
 dosing 
...
 tube, 
...
 with 
...
 higher 
...
 head 
...
 loss 
...
 occurring 
...
 at 
...
 higher 
...
 alum 
...
 flow 
...
 rates. 
...
 As 
...
 a 
...
 result, 
...
 it 
...
 is 
...
 not 
...
 desirable 
...
 to 
...
 set 
...
 the 
...
 size 
...
 of 
...
 the 
...
 orifice 
...
 based 
...
 on 
...
 the 
...
 varying 
...
 major 
...
 losses 
...
 through 
...
 the 
...
 dosing 
...
 tube, 
...
 so 
...
 the 
...
 head 
...
 loss 
...
 caused 
...
 by 
...
 the 
...
 orifice 
...
 needs 
...
 far 
...
 outweigh 
...
 major 
...
 losses 
...
 so 
...
 the 
...
 friction 
...
 in 
...
 the 
...
 tube 
...
 can 
...
 be 
...
 ignored. 
...
 As 
...
 a 
...
 result, 
...
 the 
...
 dosing 
...
 tube 
...
 needs 
...
 to 
...
 have 
...
 a 
...
 large 
...
 enough 
...
 diameter 
...
 so 
...
 that 
...
 the 
...
 effects 
...
 of 
...
 major 
...
 losses 
...
 are 
...
 not 
...
 significant 
...
 when 
...
 compared 
...
 to 
...
 the 
...
 minor 
...
 losses 
...
 causes 
...
 by 
...
 the 
...
 orifice. 
...
 The 
...
 graph 
...
 below 
...
 illustrates 
...
 the 
...
 relative 
...
 contributions 
...
 of 
...
 head 
...
 loss 
...
 caused 
...
 by 
...
 a 
...
 large 
...
 2 
...
 mm 
...
 orifice 
...
 cap 
...
 as 
...
 opposed 
...
 to 
...
 the 
...
 major 
...
 losses 
...
 through 
...
 the 
...
 dosing 
...
tube. 
Image Added
 Center
classmyclass
Figure 1: Major loss contribution to the total head loss
As can be seen in the graph above, the relative contribution of major losses to the total head loss in the dosing system at a maximum, using a 2 mm diameter orifice (the large orifice), is minor, approximately 2%. With the contribution of the major losses less than 2%of the total losses, so it can effectively be ignored in the sizing of the dosing orifice.