Simplified DO sag models for rapid BOD removal

Abstract

Predictive models for dissolved oxygen deficit have been developed based on an exponential form of expression for the equivalent linear removal of the settleable component of biochemical oxygen demand. These models are more convenient to use than other models because they are applicable for any distance from the outfall and for point and non-point wastewater discharge conditions.

Appendix—list of equations

$$ S = S_{{{\text{o-}}x}} \left[ {1 - (t/T)} \right] + S_{{{\text{o-}}y}} \exp ( - kt) $$

(1)

$$ S = S_{{{\text{o-}}x}} \exp ( - k_{1} t) + S_{{{\text{o-}}y}} \exp ( - k_{2} t) $$

(2)

$$ {\text{d}}D/{\text{d}}t = k^{\prime}S - k_{\text{r}} D = - {\text{dO}}_{2} /{\text{d}}t $$

(3)

$$ {\text{d}}D/{\text{d}}t = mS_{{{\text{o-}}x}} \exp ( - k_{1} t) + k_{2} S_{{{\text{o-}}y}} \exp ( - k_{2} t) - k_{\text{r}} D $$

(4)

$$ \begin{aligned} D = & [m/(k_{\text{r}} - k_{1} )]S_{{{\text{o-}}x}} [\exp ( - k_{1} t) - \exp ( - k_{\text{r}} t)] \\ & + [k_{2} /(k_{\text{r}} - k_{2} )]S_{{{\text{o-}}y}} [\exp ( - k_{2} t) - \exp ( - k_{\text{r}} t)] + D_{\text{o}} \exp ( - k_{\text{r}} t) \\ \end{aligned} $$

(5)

$$ \begin{aligned} D_{\text{o}} = & [m/(k_{\text{r}} - k_{1} )]S_{{{\text{o-}}x}} [1 - (k_{1} /k_{\text{r}} )\exp \{ (k_{\text{r}} - k_{1} )t_{\text{c}} \} ] \\ & + [k_{2} /(k_{\text{r}} - k_{2} )]S_{{{\text{o-}}y}} [1 - (k_{2} /k_{\text{r}} )\exp \{ (k_{\text{r}} - k_{2} )t_{\text{c}} \} ] \\ \end{aligned} $$

(6)

$$ D_{\text{c}} = (m/k_{\text{r}} )S_{{{\text{o-}}x}} \exp ( - k_{1} t_{\text{c}} ) + (k_{2} /k_{\text{r}} )S_{{{\text{o-}}y}} \exp ( - k_{2} t_{\text{c}} ) $$

(7)

$$ \begin{aligned} D_{\text{o}} = & [m/(k_{\text{r}} - k_{1} )]S_{{{\text{o-}}x}} \left[ {1 - \left( {k_{1}^{2} /k_{\text{r}}^{2} } \right)\exp \{ (k_{\text{r}} - k_{1} )t_{i} \} } \right] \\ & + [k_{2} /(k_{\text{r}} - k_{2} )]S_{{{\text{o-}}y}} \left[ {1 - \left( {k_{2}^{2} /k_{\text{r}}^{2} } \right)\exp \{ (k_{\text{r}} - k_{2} )t_{i} \} } \right] \\ \end{aligned} $$

(8)

$$ D_{i} = \left[ {(k_{1} + k_{\text{r}} )/k_{\text{r}}^{2} } \right]mS_{\text{o}} \exp ( - k_{1} t_{i} ) + \left[ {(k_{2} + k_{\text{r}} )/k_{\text{r}}^{2} } \right]k_{2} S_{{{\text{o-}}y}} \exp ( - k_{2} t_{i} ) $$

(9)

$$ \begin{aligned} S = & B_{\text{o}} \exp [ - k_{2} (t_{n - 1} + t)] + pB_{1} \exp [ - k_{1} (t_{n - 1} + t)] \\ & + (1 - p)B_{1} \exp [ - k_{2} (t_{n - 1} + t)] + pB_{2} \exp [ - k_{1} (t_{n - 1} - t_{1} + t)] \\ & + (1 - p)B_{2} \exp [ - k_{2} (t_{n - 1} - t_{1} + t)] + pB_{3} \exp [ - k_{1} (t_{n - 1} - t_{2} + t)] \\ & + (1 - p)B_{3} \exp [ - k_{2} (t_{n - 1} - t_{2} + t)] \\ & + \cdots + pB_{n} \exp ( - k_{1} t) + (1 - p)B_{n} \exp ( - k_{2} t) \\ \end{aligned} $$

(10)

$$ \begin{aligned} {\text{d}}D/{\text{d}}t = & mp[B_{1} \exp \{ - k_{1} (t_{n - 1} + t)\} + B_{2} \exp \{ - k_{1} (t_{n - 1} - t_{1} + t)\} \\ & + B_{3} \exp \{ - k_{1} (t_{n - 1} - t_{2} + t)\} + \cdots + B_{n} \exp ( - k_{1} t)] \\ & + k_{2} [B_{\text{o}} \exp \{ - k_{2} (t_{n - 1} + t)\} + (1 - p)\{ B_{1} \exp \{ - k_{2} (t_{n - 1} + t)\} \\ & + B_{2} \exp \{ - k_{2} (t_{n - 1} - t_{1} + t)\} + B_{3} \exp \{ - k_{2} (t_{n - 1} - t_{2} + t)\} + \cdots \\ & + B_{n} \exp ( - k_{2} t)\} ] - k_{\text{r}} D \\ \end{aligned} $$

(11)

$$ \begin{aligned} D = & [mp/(k_{\text{r}} - k_{1} )][B_{1} \exp ( - k_{1} t_{n - 1} ) + B_{2} \exp \{ - k_{1} (t_{n - 1} - t_{1} )\} \\ & + B_{3} \exp \{ - k_{1} (t_{n - 1} - t_{2} )\} + \cdots + B_{n} ][\exp ( - k_{1} t) - \exp ( - k_{\text{r}} t) \\ & + [k_{2} /(k_{\text{r}} - k_{2} )][B_{\text{o}} \exp ( - k_{2} t_{n - 1} ) + \{ B_{1} \exp ( - k_{2} t_{n - 1} ) \\ & + B_{2} \exp \{ - k_{2} (t_{n - 1} - t_{1} )\} + B_{3} \exp \{ - k_{2} (t_{n - 1} - t_{2} )\} + \cdots + B_{n} \} ][\exp ( - k_{2} t) - \exp ( - k_{\text{r}} t)] \\ & + D_{{{\text{o-}}n}} \exp ( - k_{\text{r}} t) \\ \end{aligned} $$

(12)

$$ \begin{aligned} D_{{{\text{o-}}n}} = & [mp/(k_{\text{r}} - k_{1} )][B_{1} \exp ( - k_{1} t_{n - 1} ) + B_{2} \exp \{ - k_{1} (t_{n - 1} - t_{1} )\} \\ & + B_{3} \exp \{ - k_{1} (t_{n - 1} - t_{2} )\} + \cdots + B_{n} ][1 - (k_{1} /k_{\text{r}} )\exp \{ (k_{\text{r}} - k_{1} )t_{\text{c}} \} ] \\ & + [k_{2} /(k_{\text{r}} - k_{2} )][B_{\text{o}} \exp ( - k_{2} t_{n - 1} ) + (1 - p)\{ B_{1} \exp (k_{2} t_{n - 1} ) \\ & + B_{2} \exp \{ - k_{2} (t_{n - 1} - t_{1} )\} + B_{3} \exp \{ - k_{2} (t_{n - 1} - t_{2} )\} + \cdots + B_{n} \} ][1 - (k_{2} /k_{\text{r}} )\exp \{ (k_{\text{r}} - k_{2} )t_{\text{c}} \} ] \\ \end{aligned} $$

(13)

$$ \begin{aligned} D_{\text{c}} = & (mp/k_{\text{r}} )[B_{1} \exp ( - k_{1} t_{n - 1} ) + B_{2} \exp \{ - k_{1} (t_{n - 1} - t_{1} )\} \\ & + B_{3} \exp \{ - k_{1} (t_{n - 1} - t_{2} )\} + \cdots + B_{n} ]\exp ( - k_{1} t_{\text{c}} ) \\ & + (k_{2} /k_{\text{r}} )[B_{\text{o}} \exp ( - k_{2} t_{n - 1} ) + (1 - p)\{ B_{1} \exp ( - k_{2} t_{n - 1} ) \\ & + B_{2} \exp \{ - k_{2} (t_{n - 1} - t_{1} )\} B_{3} \exp \{ - k_{2} (t_{n - 1} - t_{2} )\} + \cdots + B_{n} \} ]\exp \left( { - k_{2} t_{\text{c}} } \right) \\ \end{aligned} $$

(14)

$$ \begin{aligned} D_{{{\text{o-}}n }} = & [mp/(k_{\text{r}} - k_{1} )][B_{1} \exp ( - k_{1} t_{n - 1} ) + B_{2} \exp \{ - k_{1} (t_{n - 1} - t_{1} )\} \\ & + B_{3} \exp \{ - k_{1} (t_{n - 1} - t_{2} )\} + \cdots + B_{n} ] + \left[ {1 - \left( {k_{1}^{2} /k_{\text{r}}^{2} } \right)\exp \{ (k_{\text{r}} - k_{1} )t_{i} \} } \right] \\ & + [k_{2} /(k_{\text{r}} - k_{2} )][B_{\text{o}} \exp ( - k_{2} t_{n - 1} ) + (1 - p)\{ B_{1} \exp ( - k_{2} t_{n - 1} ) \\ & + B_{2} \exp \{ - k_{2} (t_{n - 1} - t_{1} )\} + B_{2} \exp \{ - k_{2} (t_{n - 1} - t_{1} )\} \\ & + B_{3} \exp \{ - k_{2} (t_{n - 1} - t_{2} )\} + \cdots + B_{n} \} ]\left[ {1 - \left( {k_{2}^{2} /k_{\text{r}}^{2} } \right)\exp (k_{\text{r}} - k_{2} )t_{i} } \right] \\ \end{aligned} $$

(15)

$$ \begin{aligned} D_{i} = & \left[ {mp(k_{1} + k_{\text{r}} )/k_{\text{r}}^{2} } \right][B_{1} \exp ( - k_{1} t_{n - 1} ) + B_{2} \exp \{ - k_{1} (t_{n - 1} - t_{1} )\} \\ & + B_{3} \exp \{ - k_{1} (t_{n - 1} - t_{2} )\} + \cdots + B_{n} ]\exp ( - k_{1} t_{i} ) \\ & + \left[ {k_{2} (k_{2} + k_{\text{r}} )/k_{\text{r}}^{2} } \right][B_{\text{o}} \exp ( - k_{2} t_{n - 1} ) + (1 - p)\{ B_{1} \exp ( - k_{2} t_{n - 1} ) \\ & + B_{2} \exp \{ - k_{2} (t_{n - 1} - t_{1} )\} + B_{3} \exp \{ - k_{2} (t_{n - 1} - t_{2} )\} + \cdots + B_{n} \} ]\exp ( - k_{2} t_{i} ) \\ \end{aligned} $$

(16)

$$ \begin{aligned} {\text{d}}D/{\text{d}}t = & mpB[\exp \{ - k_{1} (n - 1)t_{\text{o}} \} + \exp \{ - k_{1} (n - 2)t_{\text{o}} \} \\ & + \exp \{ - k_{1} (n - 3)t_{\text{o}} \} + \cdots + \exp \{ - k_{1} (n - n)t_{\text{o}} \} ]\exp ( - k_{1} t) \\ & + k_{2} [B_{\text{o}} \exp \{ - k_{2} (n - 1)t_{\text{o}} \} + (1 - p)B\{ \exp \{ - k_{2} (n - 1)t_{\text{o}} \} + \exp \{ - k_{2} (n - 2)t_{\text{o}} \} \\ & + \exp \{ - k_{2} (n - 3)t_{\text{o}} \} + \cdots + \exp \{ - k_{2} (n - n)t_{\text{o}} \} \exp ( - k_{2} t)-k_{\text{r}} D \\ \end{aligned} $$

(17)

$$ \begin{aligned} {\text{d}}D/{\text{d}}t = & mpB \left[ {\frac{{\exp \{ - k_{1} (n - 1)t_{\text{o}} \} [\exp (k_{1} nt_{\text{o}} ) - 1]}}{{\exp (k_{1} t_{\text{o}} ) - 1}}} \right]\exp ( - k_{1} t) \\ & + K_{2} \left[ {B_{\text{o}} \exp \{ - k_{2} (n - 1)t_{\text{o}} \} + (1 - p)B\left\{ {\frac{{\exp \{ - k_{2} (n - 1)t_{\text{o}} \} [\exp (k_{2} nt_{\text{o}} ) - 1]}}{{\exp (k_{2} t_{\text{o}} ) - 1}}} \right\}} \right]\exp ( - k_{2} t) - k_{\text{r}} D \\ \end{aligned} $$

(18)

$$ \begin{aligned} D = & [mpB/(k_{\text{r}} - k_{1} )]\left[ {\frac{{\exp \{ - k_{1} (n - 1)t_{\text{o}} \} [\exp (k_{1} nt_{\text{o}} ) - 1]}}{{\exp (k_{1} t_{\text{o}} ) - 1}}} \right][\exp ( - k_{1} t) - \exp ( - k_{\text{r}} t)] \\ & + [k_{2} /(k_{\text{r}} - k_{2} )]\left[ {\left[ {B_{\text{o}} \exp \{ - k_{2} (n - 1)t_{\text{o}} \} + (1 - p)B\left\{ {\frac{{\exp \{ - k_{2} (n - 1)t_{\text{o}} \} [\exp (k_{2} nt_{\text{o}} ) - 1]}}{{\exp (k_{2} t_{\text{o}} ) - 1}}} \right\}} \right][\exp ( - k_{2} t) - \exp (k_{\text{r}} t)]D_{{{\text{o-}}n}} \exp ( - k_{\text{r}} t)} \right] \\ \end{aligned} $$

(19)

$$ \begin{aligned} D_{{{\text{o-}}n}} = & [mpB/(k_{\text{r}} - k_{1} )]\left[ {\frac{{\exp \{ - k_{1} (n - 1)t_{\text{o}} \} [\exp (k_{1} nt_{\text{o}} ) - 1]}}{{\exp (k_{1} t_{\text{o}} ) - 1}}} \right][1 - (k_{1} /k_{\text{r}} )\exp \{ (k_{\text{r}} - k_{1} )t_{\text{c}} \} ] \\ & + [k_{2} /(k_{\text{r}} - k_{2} )]\left[ {B_{\text{o}} \exp \{ - k_{2} (n - 1)t_{\text{o}} \} + (1 - p)B\left\{ {\frac{{\exp \{ - k_{2} (n - 1)t_{\text{o}} \} [\exp (k_{2} nt_{\text{o}} ) - 1]}}{{\exp (k_{2} t_{\text{o}} ) - 1}}} \right\}} \right] \\ & \times [1 - (k_{2} /k_{\text{r}} )\exp \{ (k_{\text{r}} - k_{2} )t_{\text{c}} \} ] \\ \end{aligned} $$

(20)

$$ \begin{aligned} D_{c} = & [mpB/k_{\text{r}} ]\left[ {\frac{{\exp \{ - k_{1} (n - 1)t_{\text{o}} \} [\exp (k_{1} nt_{\text{o}} ) - 1]}}{{\exp (k_{1} {\text{t}}_{\text{o}} ) - 1}}} \right]\exp ( - k_{1} t_{\text{c}} ) \\ & + [k_{2} /k_{\text{r}} ]\left[ {\left[ {B_{\text{o}} \exp \{ - k_{2} (n - 1)t_{\text{o}} \} + (1 - p)B\left\{ {\frac{{\exp \{ - k_{2} (n - 1)t_{\text{o}} \} [\exp (k_{2} nt_{\text{o}} ) - 1]}}{{\exp (k_{2} {\text{t}}_{\text{o}} ) - 1}}} \right\}} \right]\exp ( - k_{2} t_{\text{c}} )} \right] \\ \end{aligned} $$

(21)

$$ \begin{aligned} D_{{{\text{o-}}n}} = & [mpB/(k_{\text{r}} - k_{1} )]\left[ {\frac{{\exp \{ - k_{1} (n - 1)t_{\text{o}} \} [\exp (k_{1} nt_{\text{o}} ) - 1]}}{{\exp (k_{1} t_{\text{o}} ) - 1}}} \right]\left[ {1 - \left( {k_{1}^{2} /k_{\text{r}}^{2} } \right)\exp \{ (k_{\text{r}} - k_{1} )t_{1} \} } \right] \\ & + [k_{2} /(k_{\text{r}} - k_{2} )]\left[ {B_{\text{o}} \exp \{ - k_{2} (n - 1)t_{\text{o}} \} + (1 - p)B\left\{ {\frac{{\exp \{ - k_{2} (n - 1)t_{\text{o}} \} [\exp (k_{2} nt_{\text{o}} ) - 1]}}{{\exp (k_{2} t_{\text{o}} ) - 1}}} \right\}} \right] \\ & \times \left[ {1 - \left( {k_{2}^{2} /k_{\text{r}}^{2} } \right)\exp \{ (k_{\text{r}} - k_{2} )t_{1} \} } \right] \\ \end{aligned} $$

(22)

$$ \begin{aligned} D_{i} = & \left[ {mpB(k_{1} + k_{\text{r}} )/K_{\text{r}}^{2} } \right]\left[ {\frac{{\exp \{ - k_{1} (n - 1)t_{\text{o}} \} [\exp (k_{1} nt_{\text{o}} ) - 1]}}{{\exp (k_{1} t_{\text{o}} ) - 1}}} \right]\exp ( - k_{1} t_{1} ) \\ & + \left[ {k_{2} (k_{2} + k_{\text{r}} )/k_{\text{r}}^{2} } \right]\left[ {\left[ {B_{\text{o}} \exp \{ - k_{2} (n - 1)t_{\text{o}} \} + (1 - p)B\left\{ {\frac{{\exp \{ - k_{2} (n - 1)t_{\text{o}} \} [\exp (k_{2} nt_{\text{o}} ) - 1]}}{{\exp (k_{2} t_{\text{o}} ) - 1}}} \right\}} \right]\exp ( - k_{2} t_{i} )} \right]. \\ \end{aligned} $$

(23)