Mean Residence Time where Dispersion Number is less than 0.01 Calculator

✖The Concentration of Solution is the quantity of a solute that is contained in a particular quantity of solvent or solution.ⓘ Concentration of Solution [c]			+10% -10%
✖Dispersion Coefficient at Dispersion Number < 0.01 is distinguished as Spreading of the Tracer in the reactor, that diffuses across a unit area in 1 s under the influence of a gradient of one unit.ⓘ Dispersion Coefficient at Dispersion Number < 0.01 [D_p]			+10% -10%
✖Velocity of Pulse for Dispersion Number <0.01 is the Velocity at which a Pulse of Material or Information travels through a Process or a System.ⓘ Velocity of Pulse for Dispersion Number <0.01 [u']			+10% -10%
✖The Length of Spread for Dispersion Number <0.01 of a Pulse provides Information about how far and how fast the Spread Propagates.ⓘ Length of Spread for Dispersion Number <0.01 [L']			+10% -10%

✖Mean Residence Time is the ratio of Time and Mean Pulse Curve.ⓘ Mean Residence Time where Dispersion Number is less than 0.01 [θ]

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Formula

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Mean Residence Time where Dispersion Number is less than 0.01

Formula

`"θ" = 1+sqrt((ln("c"*2*sqrt(pi*("D"_{"p"}/("u'"*"L'"))))*4*("D"_{"p"}/("u'"*"L'"))))`

Example

`"1.028239s"=1+sqrt((ln("44mol/m³"*2*sqrt(pi*("0.0085m²/s"/("40m/s"*"0.92m"))))*4*("0.0085m²/s"/("40m/s"*"0.92m"))))`

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Mean Residence Time where Dispersion Number is less than 0.01 Solution

STEP 0: Pre-Calculation Summary

Formula Used

Mean Residence Time = 1+sqrt((ln(Concentration of Solution*2*sqrt(pi*(Dispersion Coefficient at Dispersion Number < 0.01/(Velocity of Pulse for Dispersion Number <0.01*Length of Spread for Dispersion Number <0.01))))*4*(Dispersion Coefficient at Dispersion Number < 0.01/(Velocity of Pulse for Dispersion Number <0.01*Length of Spread for Dispersion Number <0.01))))
θ = 1+sqrt((ln(c*2*sqrt(pi*(D_p/(u'*L'))))*4*(D_p/(u'*L'))))
This formula uses 1 Constants, 2 Functions, 5 Variables

Constants Used

pi - Archimedes' constant Value Taken As 3.14159265358979323846264338327950288

Functions Used

ln - The natural logarithm, also known as the logarithm to the base e, is the inverse function of the natural exponential function., ln(Number)
sqrt - A square root function is a function that takes a non-negative number as an input and returns the square root of the given input number., sqrt(Number)

Variables Used

Mean Residence Time - (Measured in Second) - Mean Residence Time is the ratio of Time and Mean Pulse Curve.
Concentration of Solution - (Measured in Mole per Cubic Meter) - The Concentration of Solution is the quantity of a solute that is contained in a particular quantity of solvent or solution.
Dispersion Coefficient at Dispersion Number < 0.01 - (Measured in Square Meter Per Second) - Dispersion Coefficient at Dispersion Number < 0.01 is distinguished as Spreading of the Tracer in the reactor, that diffuses across a unit area in 1 s under the influence of a gradient of one unit.
Velocity of Pulse for Dispersion Number <0.01 - (Measured in Meter per Second) - Velocity of Pulse for Dispersion Number <0.01 is the Velocity at which a Pulse of Material or Information travels through a Process or a System.
Length of Spread for Dispersion Number <0.01 - (Measured in Meter) - The Length of Spread for Dispersion Number <0.01 of a Pulse provides Information about how far and how fast the Spread Propagates.

STEP 1: Convert Input(s) to Base Unit

Concentration of Solution: 44 Mole per Cubic Meter --> 44 Mole per Cubic Meter No Conversion Required
Dispersion Coefficient at Dispersion Number < 0.01: 0.0085 Square Meter Per Second --> 0.0085 Square Meter Per Second No Conversion Required
Velocity of Pulse for Dispersion Number <0.01: 40 Meter per Second --> 40 Meter per Second No Conversion Required
Length of Spread for Dispersion Number <0.01: 0.92 Meter --> 0.92 Meter No Conversion Required

STEP 2: Evaluate Formula

Substituting Input Values in Formula

θ = 1+sqrt((ln(c*2*sqrt(pi*(D_p/(u'*L'))))*4*(D_p/(u'*L')))) --> 1+sqrt((ln(44*2*sqrt(pi*(0.0085/(40*0.92))))*4*(0.0085/(40*0.92))))

Evaluating ... ...

θ = 1.02823892694706

STEP 3: Convert Result to Output's Unit

1.02823892694706 Second --> No Conversion Required

FINAL ANSWER

1.02823892694706 ≈ 1.028239 Second <-- Mean Residence Time

(Calculation completed in 00.004 seconds)

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Home » Engineering » Chemical Engineering » Chemical Reaction Engineering » Flow Pattern, Contacting and Non Ideal Flow » Dispersion Model » Mean Residence Time where Dispersion Number is less than 0.01

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Created by Pavan Kumar

Anurag Group of Institutions (AGI), Hyderabad

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< 9 Dispersion Model Calculators

Exit Age Distribution based on Dispersion Number

Go Exit Age Distribution = sqrt(Velocity of Pulse Measuring Variance^3/(4*pi*Dispersion Coefficient at Dispersion Number > 100*Length of Spread))*exp(-(Length of Spread-(Velocity of Pulse Measuring Variance*Time Required for Change in Concentration))^2/(4*(Dispersion Coefficient at Dispersion Number > 100*Length of Spread)/Velocity of Pulse Measuring Variance))

Mean Residence Time where Dispersion Number is less than 0.01

Go Mean Residence Time = 1+sqrt((ln(Concentration of Solution*2*sqrt(pi*(Dispersion Coefficient at Dispersion Number < 0.01/(Velocity of Pulse for Dispersion Number <0.01*Length of Spread for Dispersion Number <0.01))))*4*(Dispersion Coefficient at Dispersion Number < 0.01/(Velocity of Pulse for Dispersion Number <0.01*Length of Spread for Dispersion Number <0.01))))

Concentration using Dispersion where Dispersion Number less than 0.01

Go Concentration at Dispersion Number < 0.01 = 1/(2*sqrt(pi*(Dispersion Coefficient at Dispersion Number < 0.01/(Velocity of Pulse for Dispersion Number <0.01*Length of Spread for Dispersion Number <0.01))))*exp(-(1-Mean Residence Time)^2/(4*(Dispersion Coefficient at Dispersion Number < 0.01/(Velocity of Pulse for Dispersion Number <0.01*Length of Spread for Dispersion Number <0.01))))

Standard Deviation of Tracer based on Mean Residence Time for Large Deviations of Dispersion

Go Standard Deviation based on θ at Large Deviations = sqrt(2*(Dispersion Coefficient at Dispersion Number > 100/(Length of Spread*Velocity of Pulse))-2*((Dispersion Coefficient at Dispersion Number > 100/(Velocity of Pulse*Length of Spread))^2)*(1-exp(-(Velocity of Pulse*Length of Spread)/Dispersion Coefficient at Dispersion Number > 100)))

Standard Deviation of Spread based on Mean Residence Time for Small Extents of Dispersion

Go Standard Deviation based on θ at Small Extents = sqrt(2*(Dispersion Coefficient at Dispersion Number < 0.01/(Length of Spread for Dispersion Number <0.01*Velocity of Pulse for Dispersion Number <0.01)))

Variance of Spread of Tracer for Small Extents of Dispersion

Go Variance of Spread for Dispersion Number <0.01 = 2*(Dispersion Coefficient at Dispersion Number < 0.01*Length of Spread for Dispersion Number <0.01/Velocity of Pulse for Dispersion Number <0.01^3)

Dispersion Coefficient based on Variance of Spread of Tracer at Small Extents of Dispersion

Go Dispersion Coefficient at Dispersion Number < 0.01 = (Variance of Spread for Dispersion Number <0.01*Velocity of Pulse^3)/(2*(Length of Spread for Small Extents))

Length of Spread based on Variance of Spread of Tracer for Small Extents of Dispersion

Go Length of Spread for Small Extents = (Variance of Spread for Dispersion Number <0.01*(Velocity of Pulse^3))/(2*Dispersion Coefficient at Dispersion Number < 0.01)

Velocity of Pulse based on Variance of Spread of Tracer

Go Velocity of Pulse Measuring Variance = (2*((Dispersion Coefficient at Dispersion Number > 100*Length of Spread)/Variance of Spread))^(1/3)

Mean Residence Time where Dispersion Number is less than 0.01 Formula

What is Dispersion Number ?

Suppose an ideal pulse of tracer is introduced into the fluid entering a vessel. The pulse spreads as it passes through the vessel, and to characterize the spreading according to this model, we assume a diffusion-like process superimposed on plug flow. We call this dispersion or longitudinal dispersion to distinguish it from molecular diffusion. The dispersion coefficient D represents this spreading process.

What is θ ?

θ, Mean Residence Time, is the ratio between time and the mean of Cpulse curve.

How to Calculate Mean Residence Time where Dispersion Number is less than 0.01?

Mean Residence Time where Dispersion Number is less than 0.01 calculator uses Mean Residence Time = 1+sqrt((ln(Concentration of Solution*2*sqrt(pi*(Dispersion Coefficient at Dispersion Number < 0.01/(Velocity of Pulse for Dispersion Number <0.01*Length of Spread for Dispersion Number <0.01))))*4*(Dispersion Coefficient at Dispersion Number < 0.01/(Velocity of Pulse for Dispersion Number <0.01*Length of Spread for Dispersion Number <0.01)))) to calculate the Mean Residence Time, The Mean Residence Time where Dispersion Number is less than 0.01 formula is defined as Relationship between the Dispersion Number and the Concentration for small extent of Dispersion. Mean Residence Time is denoted by θ symbol.

How to calculate Mean Residence Time where Dispersion Number is less than 0.01 using this online calculator? To use this online calculator for Mean Residence Time where Dispersion Number is less than 0.01, enter Concentration of Solution (c), Dispersion Coefficient at Dispersion Number < 0.01 (D_p), Velocity of Pulse for Dispersion Number <0.01 (u') & Length of Spread for Dispersion Number <0.01 (L') and hit the calculate button. Here is how the Mean Residence Time where Dispersion Number is less than 0.01 calculation can be explained with given input values -> 1.028239 = 1+sqrt((ln(44*2*sqrt(pi*(0.0085/(40*0.92))))*4*(0.0085/(40*0.92)))).

FAQ

What is Mean Residence Time where Dispersion Number is less than 0.01?

The Mean Residence Time where Dispersion Number is less than 0.01 formula is defined as Relationship between the Dispersion Number and the Concentration for small extent of Dispersion and is represented as θ = 1+sqrt((ln(c*2*sqrt(pi*(D_p/(u'*L'))))*4*(D_p/(u'*L')))) or Mean Residence Time = 1+sqrt((ln(Concentration of Solution*2*sqrt(pi*(Dispersion Coefficient at Dispersion Number < 0.01/(Velocity of Pulse for Dispersion Number <0.01*Length of Spread for Dispersion Number <0.01))))*4*(Dispersion Coefficient at Dispersion Number < 0.01/(Velocity of Pulse for Dispersion Number <0.01*Length of Spread for Dispersion Number <0.01)))). The Concentration of Solution is the quantity of a solute that is contained in a particular quantity of solvent or solution, Dispersion Coefficient at Dispersion Number < 0.01 is distinguished as Spreading of the Tracer in the reactor, that diffuses across a unit area in 1 s under the influence of a gradient of one unit, Velocity of Pulse for Dispersion Number <0.01 is the Velocity at which a Pulse of Material or Information travels through a Process or a System & The Length of Spread for Dispersion Number <0.01 of a Pulse provides Information about how far and how fast the Spread Propagates.

How to calculate Mean Residence Time where Dispersion Number is less than 0.01?

The Mean Residence Time where Dispersion Number is less than 0.01 formula is defined as Relationship between the Dispersion Number and the Concentration for small extent of Dispersion is calculated using Mean Residence Time = 1+sqrt((ln(Concentration of Solution*2*sqrt(pi*(Dispersion Coefficient at Dispersion Number < 0.01/(Velocity of Pulse for Dispersion Number <0.01*Length of Spread for Dispersion Number <0.01))))*4*(Dispersion Coefficient at Dispersion Number < 0.01/(Velocity of Pulse for Dispersion Number <0.01*Length of Spread for Dispersion Number <0.01)))). To calculate Mean Residence Time where Dispersion Number is less than 0.01, you need Concentration of Solution (c), Dispersion Coefficient at Dispersion Number < 0.01 (D_p), Velocity of Pulse for Dispersion Number <0.01 (u') & Length of Spread for Dispersion Number <0.01 (L'). With our tool, you need to enter the respective value for Concentration of Solution, Dispersion Coefficient at Dispersion Number < 0.01, Velocity of Pulse for Dispersion Number <0.01 & Length of Spread for Dispersion Number <0.01 and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well.

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