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The nonlinear doser uses a dosing orifice (minor losses) instead of a dosing tube (major losses) to control the relationship between changing plant flow rates and chemical dose. The flow rate through the CDC is related to the available head by the equation:

{latex}$$Q_{Cdc}  = K_{orifice}\sqrt {2gh_{Cdc} } $${latex}

where

• Latex
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  {latex}$$Q_{Cdc} $${latex}

  is the chemical flow rate

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  {latex}$$ K_{orifice} $${latex}

  is the orifice coefficient
• h is the available head

The entrance to the rapid mix tank is a rectangular orifice. The relationship between flow rate and head is governed by the equation:

{latex}$$ Q_{Plant}  = K_{orifice} \sqrt {2gh_{EtOrifice} } $${latex}

where

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  {latex}$$ Q_{Plant}$${latex}

  is the plant flow rate

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  {latex}$$ h_{EtOrifice} $${latex}

  is the height of water above the entrance tank orfice

The chemical dose to the plant can be determined by a simple 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

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The dosing tube must be designed to minimize major losses so that the major losses that deviate from the

{latex}$$V = \sqrt {2gh}$${latex}

{latex}$$V = \sqrt {2gh}$${latex}

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The dosing orifice is designed to produce the difference in head loss between the maximum CDC head loss and the actual head loss in the flexible dosing tube:

{latex}$$
h_l  = K_{DoseOrifice} {{V_{DoseTube}^2 } \over {2g}}
$${latex}

where

• h _l the difference in head loss between the maximum CDC head loss and the actual head loss in the flexible dosing tube
• K _DoseOrifice is the required minor loss coefficient through the orifice
• V _DoseTube is the velocity in the dosing tube

This head loss is equal to head loss in the vena contracta. Head loss in the vena contracta is modeled by the equation:

{latex}$$h_l  = K_{Exit} {{V_{VenaContracta}^2 } \over {2g}}$${latex}

where

• h _l is the head loss

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  {latex}$$ K_{Exit}$${latex}

  is the minor loss coefficient from an exit to the atmosphere
• V _{VenaContracta} is the velocity in the orifice vena contracta

{latex}$$ K_{Exit}$${latex}

{latex}$$A_{DoseTube} V_{DoseTube}  = K_{Orifice} A_{DoseOrifice} V_{DoseOrifice} $${latex}

where

{latex}$$ K_{Orifice}$${latex}

{latex}$$
K_{DoseOrifice}  = \left[ {{1 \over {K_{Orifice} }}\left( {{{V_{DoseTube} } \over {V_{DoseOrifice} }}} \right)^2 } \right]^2 
$${latex}

The chemical dose flow rate is not exactly proportional to the plant flow rate due to different exponential relationships between flow and head loss for the major losses. These differences arise from two sources. The most significant difference occurs when the flow in the dosing tube is laminar and thus the flow rate is proportional to the major head loss rather than to the square root of the head loss. The other source of error occurs for transitional flow where the friction factor decreases with increasing Reynolds number.

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The nonlinear relationship between flow and head loss makes it more difficult to accurately control the chemical dose when the dose is low.

{latex}$${{HL_{Cdc_{Min} } } \over {HL_{Cdc_{\max } } }} = \left( {{{Q_{Cdc_{Min} } } \over {Q_{Cdc_{\max } } }}} \right)^2 $${latex}

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