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

✖Dispersion Coefficient at Dispersion Number > 100 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 > 100 [D_p']			+10% -10%
✖The Length of Spread of a Pulse provides Information about how far and how fast the Spread Propagates.ⓘ Length of Spread [l]			+10% -10%
✖Velocity of Pulse is the Velocity at which a Pulse of Material or Information travels through a Process or a System.ⓘ Velocity of Pulse [u]			+10% -10%

✖Standard Deviation based on θ at Large Deviations is Calculated using Mean of Pulse Curve and Dispersion Number, which is measure of Spread of Tracer.ⓘ Standard Deviation of Tracer based on Mean Residence Time for Large Deviations of Dispersion [S.D_L.D]

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

STEP 0: Pre-Calculation Summary

Formula Used

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)))
S.D_L.D = sqrt(2*(D_p'/(l*u))-2*((D_p'/(u*l))^2)*(1-exp(-(u*l)/D_p')))
This formula uses 2 Functions, 4 Variables

Functions Used

exp - n an exponential function, the value of the function changes by a constant factor for every unit change in the independent variable., exp(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

Standard Deviation based on θ at Large Deviations - Standard Deviation based on θ at Large Deviations is Calculated using Mean of Pulse Curve and Dispersion Number, which is measure of Spread of Tracer.
Dispersion Coefficient at Dispersion Number > 100 - (Measured in Square Meter Per Second) - Dispersion Coefficient at Dispersion Number > 100 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 - (Measured in Meter) - The Length of Spread of a Pulse provides Information about how far and how fast the Spread Propagates.
Velocity of Pulse - (Measured in Meter per Second) - Velocity of Pulse 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 > 100: 410 Square Meter Per Second --> 410 Square Meter Per Second No Conversion Required
Length of Spread: 6.4 Meter --> 6.4 Meter No Conversion Required
Velocity of Pulse: 0.981 Meter per Second --> 0.981 Meter per Second No Conversion Required

STEP 2: Evaluate Formula

Substituting Input Values in Formula

S.D_L.D = sqrt(2*(D_p'/(l*u))-2*((D_p'/(u*l))^2)*(1-exp(-(u*l)/D_p'))) --> sqrt(2*(410/(6.4*0.981))-2*((410/(0.981*6.4))^2)*(1-exp(-(0.981*6.4)/410)))

Evaluating ... ...

S.D_L.D = 0.997454305299735

STEP 3: Convert Result to Output's Unit

0.997454305299735 --> No Conversion Required

FINAL ANSWER

0.997454305299735 ≈ 0.997454 <-- Standard Deviation based on θ at Large Deviations

(Calculation completed in 00.020 seconds)

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< Dispersion Model Calculators

Exit Age Distribution based on Dispersion Number

LaTeX 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))

Concentration using Dispersion where Dispersion Number less than 0.01

LaTeX 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

LaTeX 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)))

Variance of Spread of Tracer for Small Extents of Dispersion

LaTeX 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)

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

LaTeX Go

What is σ² ?

σ² is 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 Tracer based on Mean Residence Time for Large Deviations of Dispersion?

Standard Deviation of Tracer based on Mean Residence Time for Large Deviations of Dispersion calculator uses 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))) to calculate the Standard Deviation based on θ at Large Deviations, Standard Deviation of Tracer based on Mean Residence Time for Large Deviations of Dispersion formula is defined as Measure of how much the concentration profile of the tracer widens or spreads out over time and space. It's often characterized by a Dispersion Coefficient, which can be considered analogous to the Variance in Statistics. Standard Deviation based on θ at Large Deviations is denoted by S.D_L.D symbol.

How to calculate Standard Deviation of Tracer based on Mean Residence Time for Large Deviations of Dispersion using this online calculator? To use this online calculator for Standard Deviation of Tracer based on Mean Residence Time for Large Deviations of Dispersion, enter Dispersion Coefficient at Dispersion Number > 100 (D_p'), Length of Spread (l) & Velocity of Pulse (u) and hit the calculate button. Here is how the Standard Deviation of Tracer based on Mean Residence Time for Large Deviations of Dispersion calculation can be explained with given input values -> 0.905919 = sqrt(2*(410/(6.4*0.981))-2*((410/(0.981*6.4))^2)*(1-exp(-(0.981*6.4)/410))).

FAQ

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

Standard Deviation of Tracer based on Mean Residence Time for Large Deviations of Dispersion formula is defined as Measure of how much the concentration profile of the tracer widens or spreads out over time and space. It's often characterized by a Dispersion Coefficient, which can be considered analogous to the Variance in Statistics and is represented as S.D_L.D = sqrt(2*(D_p'/(l*u))-2*((D_p'/(u*l))^2)*(1-exp(-(u*l)/D_p'))) or 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))). Dispersion Coefficient at Dispersion Number > 100 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 of a Pulse provides Information about how far and how fast the Spread Propagates & Velocity of Pulse is the Velocity at which a Pulse of Material or Information travels through a Process or a System.

How to calculate Standard Deviation of Tracer based on Mean Residence Time for Large Deviations of Dispersion?

Standard Deviation of Tracer based on Mean Residence Time for Large Deviations of Dispersion formula is defined as Measure of how much the concentration profile of the tracer widens or spreads out over time and space. It's often characterized by a Dispersion Coefficient, which can be considered analogous to the Variance in Statistics is calculated using 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))). To calculate Standard Deviation of Tracer based on Mean Residence Time for Large Deviations of Dispersion, you need Dispersion Coefficient at Dispersion Number > 100 (D_p'), Length of Spread (l) & Velocity of Pulse (u). With our tool, you need to enter the respective value for Dispersion Coefficient at Dispersion Number > 100, Length of Spread & Velocity of Pulse 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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