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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:
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{latex}$$Q_{Cdc} = K_{orifice}\sqrt {2gh_{Cdc} } $${latex} |
where
is the chemical flow rateLatex Wiki Markup {latex}$$Q_{Cdc} $${latex}
is the orifice coefficientLatex Wiki Markup {latex}$$ K_{orifice} $${latex}
- 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:
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{latex}$$ Q_{Plant} = K_{orifice} \sqrt {2gh_{EtOrifice} } $${latex} |
where
is the plant flow rateLatex Wiki Markup {latex}$$ Q_{Plant}$${latex}
is the height of water above the entrance tank orficeLatex Wiki Markup {latex}$$ h_{EtOrifice} $${latex}
The chemical dose to the plant can be determined by a simple mass balance:
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{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
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{latex}$$V = \sqrt {2gh}$${latex} |
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{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:
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{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:
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{latex}$$h_l = K_{Exit} {{V_{VenaContracta}^2 } \over {2g}}$${latex} |
where
- h l is the head loss
is the minor loss coefficient from an exit to the atmosphereLatex Wiki Markup {latex}$$ K_{Exit}$${latex}
- V VenaContracta is the velocity in the orifice vena contracta
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{latex}$$ K_{Exit}$${latex} |
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{latex}$$A_{DoseTube} V_{DoseTube} = K_{Orifice} A_{DoseOrifice} V_{DoseOrifice} $${latex} |
where
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{latex}$$ K_{Orifice}$${latex} |
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{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.
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{latex}$${{HL_{Cdc_{Min} } } \over {HL_{Cdc_{\max } } }} = \left( {{{Q_{Cdc_{Min} } } \over {Q_{Cdc_{\max } } }}} \right)^2 $${latex} |
For this reason it is recommended that the total head loss range be increased to approximately 50 cm for the nonlinear CDC. For a chemical dose that is 10% of the maximum chemical dose the head required by the dosing tube is 1% of the maximum head loss (0.5 cm). Maintaining a positioning error of less than 0.5 cm may be difficult and would require a large float to minimize errors due to changes in moment caused by the dosing tube. The CDC lever must also be accurately calibrated to minimize errors at low chemical dosing rates. Thus the maximum head loss for the CDC should be at least 50 cm.
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