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Critical Time Formula

GoCritical Time when we have Critical Oxygen Deficit Formula

GoCritical Time given Self Purification Constant with Critical Oxygen Deficit Formula

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Critical Time refers to the minimum dissolved oxygen concentration occurs, found by differentiating the dissolved oxygen equation with respect to time. Check FAQs

t c = (\frac{1}{K R - K D}) \cdot \log 10 ((\frac{K D \cdot L t - K R \cdot D o + K D \cdot D o}{K D} \cdot L t) \cdot (\frac{K R}{K D}))

t_c - Critical Time?K_R - Reoxygenation Coefficient?K_D - Deoxygenation Constant?L_t - Oxygen Equivalent?D_o - Initial Oxygen Deficit?

Critical Time Example

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With units

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Here is how the Critical Time equation looks like with Values.

Here is how the Critical Time equation looks like with Units.

Here is how the Critical Time equation looks like.

697.8548 = (\frac{1}{0.22 - 0.23}) \cdot \log 10 ((\frac{0.23 \cdot 0.21 - 0.22 \cdot 7.2 + 0.23 \cdot 7.2}{0.23} \cdot 0.21) \cdot (\frac{0.22}{0.23}))

697.8548d = (\frac{1}{0.22d⁻¹ - 0.23d⁻¹}) \cdot \log 10 ((\frac{0.23d⁻¹ \cdot 0.21mg/L - 0.22d⁻¹ \cdot 7.2mg/L + 0.23d⁻¹ \cdot 7.2mg/L}{0.23d⁻¹} \cdot 0.21mg/L) \cdot (\frac{0.22d⁻¹}{0.23d⁻¹}))

697.8548 = (\frac{1}{0.22 - 0.23}) \cdot \log 10 ((\frac{0.23 \cdot 0.21 - 0.22 \cdot 7.2 + 0.23 \cdot 7.2}{0.23} \cdot 0.21) \cdot (\frac{0.22}{0.23}))

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Critical Time Solution

Follow our step by step solution on how to calculate Critical Time?

FIRST Step Consider the formula

t c = (\frac{1}{K R - K D}) \cdot \log 10 ((\frac{K D \cdot L t - K R \cdot D o + K D \cdot D o}{K D} \cdot L t) \cdot (\frac{K R}{K D}))

Next Step Substitute values of Variables

∴

t c = (\frac{1}{0.22d⁻¹ - 0.23d⁻¹}) \cdot \log 10 ((\frac{0.23d⁻¹ \cdot 0.21mg/L - 0.22d⁻¹ \cdot 7.2mg/L + 0.23d⁻¹ \cdot 7.2mg/L}{0.23d⁻¹} \cdot 0.21mg/L) \cdot (\frac{0.22d⁻¹}{0.23d⁻¹}))

Next Step Convert Units

∴

t c = (\frac{1}{2.5E-6s⁻¹ - 2.7E-6s⁻¹}) \cdot \log 10 ((\frac{2.7E-6s⁻¹ \cdot 0.0002kg/m³ - 2.5E-6s⁻¹ \cdot 0.0072kg/m³ + 2.7E-6s⁻¹ \cdot 0.0072kg/m³}{2.7E-6s⁻¹} \cdot 0.0002kg/m³) \cdot (\frac{2.5E-6s⁻¹}{2.7E-6s⁻¹}))

Next Step Prepare to Evaluate

∴

t c = (\frac{1}{2.5E-6 - 2.7E-6}) \cdot \log 10 ((\frac{2.7E-6 \cdot 0.0002 - 2.5E-6 \cdot 0.0072 + 2.7E-6 \cdot 0.0072}{2.7E-6} \cdot 0.0002) \cdot (\frac{2.5E-6}{2.7E-6}))

Next Step Evaluate

∴

t c = 60294655.3173632s

Next Step Convert to Output's Unit

∴

t c = 697.854806913926d

LAST Step Rounding Answer

∴

t c = 697.8548d

Critical Time Formula Elements

Variables

Functions

Critical Time

Critical Time refers to the minimum dissolved oxygen concentration occurs, found by differentiating the dissolved oxygen equation with respect to time.

Symbol: t_c

Measurement: TimeUnit: d

Note: Value should be greater than 0.

Formulas to find Critical Time

Reoxygenation Coefficient

Reoxygenation Coefficient is referred as the parameter used in water quality modeling to describe the rate at which oxygen is transferred from the atmosphere to the water body.

Symbol: K_R

Measurement: First Order Reaction Rate ConstantUnit: d⁻¹

Note: Value should be greater than 0.

Formulas to find Reoxygenation Coefficient

Deoxygenation Constant

Deoxygenation Constant is referred as the value obtained after decomposing of oxygen in sewage.

Symbol: K_D

Measurement: First Order Reaction Rate ConstantUnit: d⁻¹

Note: Value should be greater than 0.

Formulas to find Deoxygenation Constant

Oxygen Equivalent

The Oxygen Equivalent is referred as the oxidizable organic matter present in sewage.

Symbol: L_t

Measurement: DensityUnit: mg/L

Note: Value should be greater than 0.

Formulas to find Oxygen Equivalent

Initial Oxygen Deficit

Initial Oxygen Deficit is referred as the amount of oxygen required at initial levels.

Symbol: D_o

Measurement: DensityUnit: mg/L

Note: Value should be greater than 0.

Formulas to find Initial Oxygen Deficit

log10

The common logarithm, also known as the base-10 logarithm or the decimal logarithm, is a mathematical function that is the inverse of the exponential function.

Syntax: log10(Number)

Other Formulas to find Critical Time

Go Critical Time when we have Critical Oxygen Deficit

t c = \log 10 \frac{\frac{D c \cdot K R}{K D \cdot L t}}{K D}

Go Critical Time given Self Purification Constant with Critical Oxygen Deficit

t c = \log 10 \frac{D c \cdot \frac{f}{L t}}{K D}

Go Critical Time given Self Purification Factor

t c = - (\log 10 \frac{1 - (f - 1) \cdot (\frac{D c}{L t}) \cdot f}{K D \cdot (f - 1)})

How to Evaluate Critical Time?

Critical Time evaluator uses Critical Time = (1/(Reoxygenation Coefficient-Deoxygenation Constant))*log10(((Deoxygenation Constant*Oxygen Equivalent-Reoxygenation Coefficient*Initial Oxygen Deficit+Deoxygenation Constant*Initial Oxygen Deficit)/Deoxygenation Constant*Oxygen Equivalent)*(Reoxygenation Coefficient/Deoxygenation Constant)) to evaluate the Critical Time, The Critical Time formula is defined as the crucial time period in which certain tasks must be completed to avoid project delays. It often relates to the critical path method (CPM) where the longest stretch of dependent activities determines the shortest possible duration to complete a project. Critical Time is denoted by t_c symbol.

How to evaluate Critical Time using this online evaluator? To use this online evaluator for Critical Time, enter Reoxygenation Coefficient (K_R), Deoxygenation Constant (K_D), Oxygen Equivalent (L_t) & Initial Oxygen Deficit (D_o) and hit the calculate button.

FAQs on Critical Time

What is the formula to find Critical Time?

The formula of Critical Time is expressed as Critical Time = (1/(Reoxygenation Coefficient-Deoxygenation Constant))*log10(((Deoxygenation Constant*Oxygen Equivalent-Reoxygenation Coefficient*Initial Oxygen Deficit+Deoxygenation Constant*Initial Oxygen Deficit)/Deoxygenation Constant*Oxygen Equivalent)*(Reoxygenation Coefficient/Deoxygenation Constant)). Here is an example- 0.002576 = (1/(2.5462962962963E-06-2.66203703703704E-06))*log10(((2.66203703703704E-06*0.00021-2.5462962962963E-06*0.0072+2.66203703703704E-06*0.0072)/2.66203703703704E-06*0.00021)*(2.5462962962963E-06/2.66203703703704E-06)).

How to calculate Critical Time?

With Reoxygenation Coefficient (K_R), Deoxygenation Constant (K_D), Oxygen Equivalent (L_t) & Initial Oxygen Deficit (D_o) we can find Critical Time using the formula - Critical Time = (1/(Reoxygenation Coefficient-Deoxygenation Constant))*log10(((Deoxygenation Constant*Oxygen Equivalent-Reoxygenation Coefficient*Initial Oxygen Deficit+Deoxygenation Constant*Initial Oxygen Deficit)/Deoxygenation Constant*Oxygen Equivalent)*(Reoxygenation Coefficient/Deoxygenation Constant)). This formula also uses Common Logarithm (log10) function(s).

What are the other ways to Calculate Critical Time?

Here are the different ways to Calculate Critical Time-

Critical Time=log10((Critical Oxygen Deficit*Reoxygenation Coefficient)/(Deoxygenation Constant*Oxygen Equivalent))/Deoxygenation Constant
Critical Time=log10(Critical Oxygen Deficit*Self-Purification Constant/Oxygen Equivalent)/Deoxygenation Constant
Critical Time=-(log10(1-(Self-Purification Constant-1)*(Critical Oxygen Deficit/Oxygen Equivalent)*Self-Purification Constant)/(Deoxygenation Constant*(Self-Purification Constant-1)))

Can the Critical Time be negative?

No, the Critical Time, measured in Time cannot be negative.

Which unit is used to measure Critical Time?

Critical Time is usually measured using the Day[d] for Time. Second[d], Millisecond[d], Microsecond[d] are the few other units in which Critical Time can be measured.

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Critical Time Formula

​GoCritical Time when we have Critical Oxygen Deficit Formula

​GoCritical Time given Self Purification Constant with Critical Oxygen Deficit Formula

Critical Time Example

Critical Time Solution

Critical Time Formula Elements

Other Formulas to find Critical Time

How to Evaluate Critical Time?

FAQs on Critical Time

Variable Name

GoCritical Time when we have Critical Oxygen Deficit Formula

GoCritical Time given Self Purification Constant with Critical Oxygen Deficit Formula