Concentration using Dispersion where Dispersion Number less than 0.01 Calculator

Concentration Formula:

\[ C = \frac{1}{2\sqrt{\pi\left(\frac{D_p}{u'L'}\right)}} \cdot \exp\left(-\frac{(1-\theta)^2}{4\left(\frac{D_p}{u'L'}\right)}\right) \]

Dispersion Coefficient (D_p):

m²/s

Velocity of Pulse (u'):

m/s

Length of Spread (L'):

Mean Residence Time (θ):

Concentration (C):

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1. What is Concentration using Dispersion where Dispersion Number less than 0.01?

Definition: This calculator determines the concentration profile in a system where the dispersion number is less than 0.01, indicating minimal dispersion effects.

Purpose: It's used in chemical engineering and reactor design to predict concentration distributions in systems with low dispersion.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ C = \frac{1}{2\sqrt{\pi\left(\frac{D_p}{u'L'}\right)}} \cdot \exp\left(-\frac{(1-\theta)^2}{4\left(\frac{D_p}{u'L'}\right)}\right) \]

Where:

\( C \) — Concentration (mol/m³)
\( D_p \) — Dispersion coefficient (m²/s)
\( u' \) — Velocity of pulse (m/s)
\( L' \) — Length of spread (m)
\( \theta \) — Mean residence time (dimensionless)

Explanation: The formula describes how concentration changes with position and time in a system with minimal dispersion.

3. Importance of Concentration Calculation

Details: Accurate concentration prediction is crucial for reactor design, environmental engineering, and process optimization in chemical systems.

4. Using the Calculator

Tips: Enter the dispersion coefficient (default 0.0085 m²/s), velocity (default 40 m/s), length of spread (default 0.92 m), and mean residence time (default 0.98). All values except θ must be > 0.

5. Frequently Asked Questions (FAQ)

Q1: When is this formula applicable?
A: This formula is specifically for systems where the dispersion number (D/uL) is less than 0.01, indicating negligible dispersion effects.

Q2: What's a typical dispersion coefficient value?
A: In laminar flow, Dp might be 10^-5 to 10^-4 m²/s, while in turbulent flow it could be 10^-3 to 10^-1 m²/s.

Q3: What does θ = 1 represent?
A: θ = 1 represents the mean residence time where concentration peaks in the system.

Q4: How does velocity affect concentration?
A: Higher velocities generally lead to narrower concentration profiles (less spreading).

Q5: What are the units of concentration?
A: The concentration units depend on your input units, typically mol/m³ for chemical systems.