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

✖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%
✖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%
✖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%

✖Standard Deviation based on θ at Small Extents is Calculated using Mean of Pulse Curve and Dispersion Number, which is measure of Spread of Tracer.ⓘ Standard Deviation of Spread based on Mean Residence Time for Small Extents of Dispersion [S.D_S.E]

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

Formula

`"S.D"_{"S.E"} = sqrt(2*("D"_{"p"}/("L'"*"u'")))`

Example

`"0.021493"=sqrt(2*("0.0085m²/s"/("0.92m"*"40m/s")))`

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

STEP 0: Pre-Calculation Summary

Formula Used

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)))
S.D_S.E = sqrt(2*(D_p/(L'*u')))
This formula uses 1 Functions, 4 Variables

Functions Used

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

Standard Deviation based on θ at Small Extents - Standard Deviation based on θ at Small Extents is Calculated using Mean of Pulse Curve and Dispersion Number, which is measure of Spread of Tracer.
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.
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.
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.

STEP 1: Convert Input(s) to Base Unit

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

STEP 2: Evaluate Formula

Substituting Input Values in Formula

S.D_S.E = sqrt(2*(D_p/(L'*u'))) --> sqrt(2*(0.0085/(0.92*40)))

Evaluating ... ...

S.D_S.E = 0.0214931738405274

STEP 3: Convert Result to Output's Unit

0.0214931738405274 --> No Conversion Required

FINAL ANSWER

0.0214931738405274 ≈ 0.021493 <-- Standard Deviation based on θ at Small Extents

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

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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)

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

What is Standard Deviation ?

Standard Deviation is Square Root of the variance or the measure of the spread of the curve, which are directly linked to Dispersion Number.

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.

How to Calculate Standard Deviation of Spread based on Mean Residence Time for Small Extents of Dispersion?

Standard Deviation of Spread based on Mean Residence Time for Small Extents of Dispersion calculator uses 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))) to calculate the Standard Deviation based on θ at Small Extents, Standard Deviation of Spread based on Mean Residence Time for Small Extents of Dispersion formula is defined as Square Root of Measure of Spread of Tracer in the Reactor, based on Mean Residence Time. Standard Deviation based on θ at Small Extents is denoted by S.D_S.E symbol.

How to calculate Standard Deviation of Spread based on Mean Residence Time for Small Extents of Dispersion using this online calculator? To use this online calculator for Standard Deviation of Spread based on Mean Residence Time for Small Extents of Dispersion, enter Dispersion Coefficient at Dispersion Number < 0.01 (D_p), Length of Spread for Dispersion Number <0.01 (L') & Velocity of Pulse for Dispersion Number <0.01 (u') and hit the calculate button. Here is how the Standard Deviation of Spread based on Mean Residence Time for Small Extents of Dispersion calculation can be explained with given input values -> 0.021493 = sqrt(2*(0.0085/(0.92*40))).

FAQ

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

Standard Deviation of Spread based on Mean Residence Time for Small Extents of Dispersion formula is defined as Square Root of Measure of Spread of Tracer in the Reactor, based on Mean Residence Time and is represented as S.D_S.E = sqrt(2*(D_p/(L'*u'))) or 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))). 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, The Length of Spread for Dispersion Number <0.01 of a Pulse provides Information about how far and how fast the Spread Propagates & 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.

How to calculate Standard Deviation of Spread based on Mean Residence Time for Small Extents of Dispersion?

Standard Deviation of Spread based on Mean Residence Time for Small Extents of Dispersion formula is defined as Square Root of Measure of Spread of Tracer in the Reactor, based on Mean Residence Time is calculated using 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))). To calculate Standard Deviation of Spread based on Mean Residence Time for Small Extents of Dispersion, you need Dispersion Coefficient at Dispersion Number < 0.01 (D_p), Length of Spread for Dispersion Number <0.01 (L') & Velocity of Pulse for Dispersion Number <0.01 (u'). With our tool, you need to enter the respective value for Dispersion Coefficient at Dispersion Number < 0.01, Length of Spread for Dispersion Number <0.01 & Velocity of Pulse 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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