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Question

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VIPSolutions · Accepted Answer

# Step-by-Step Solution for the First Problem

## Problem Statement
For a laminar flow reactor exhibiting non-ideal residence time distribution, the convective-diffusion equation governs the tracer concentration profile. Establish the analytical solution for the E-curve $ E(\theta) $ in terms of the vessel Peclet number $ Pe_v $, assuming the open-open boundary conditions are satisfied at the inlet and outlet planes.

## Step 1: Write the Dimensionless Convective-Diffusion Equation

$$
\frac{\partial C}{\partial \theta} = \frac{1}{Pe_v} \frac{\partial^2 C}{\partial x^2} - \frac{\partial C}{\partial x}
$$

where:
- $ C $ is the tracer concentration,
- $ \theta $ is the dimensionless time ($ \theta = t/\tau $, with $ \tau $ as mean residence time),
- $ x $ is the dimensionless axial coordinate ( at inlet, 1 at outlet),
- $ Pe_v $ is the vessel Peclet number ($ Pe_v = uL/D $), with $ u $ as velocity, $ L $ length, and $ D $ axial dispersion coefficient.

## Step 2: Apply Initial and Boundary Conditions

- **Initial Condition:** $ C(x,) =  $ (no tracer initially)
- **Inlet Boundary (Open):** $ C(,\theta) =  $
- **Outlet Boundary (Open):** $ C(1,\theta) =  $
- **Pulse Input:** At $ \theta =  $, a pulse of tracer is added at $ x= $

## Step 3: Solution for the E-Curve

The E-curve (exit age distribution) is the response at the outlet ($ x=1 $) to a pulse at the inlet. The solution for this boundary value problem is:

$$
E(\theta) = \frac{Pe_v}{4\pi \theta^3}^{1/2} \exp\left[ -\frac{Pe_v (1-\theta)^2}{4\theta} \right] \exp\left( -\frac{Pe_v}{4\theta} \right)
$$

However, more commonly for the open-open boundary conditions (Danckwerts boundary conditions for a vessel of length 1, Pe_v = $ uL/D $), the analytical solution is

$$
E(\theta) = \frac{Pe_v}{4\pi \theta^3}^{1/2} \exp\left[ -\frac{Pe_v (1-\theta)^2}{4\theta} \right]
$$

But, for the normalized solution for a pulse input in a vessel of dimensionless length 1, the standard solution is:

$$
E(\theta) = \frac{Pe_v}{4\pi \theta^3}^{1/2} \exp\left[ -\frac{(1 - \theta Pe_v)^2}{4\theta/Pe_v} \right]
$$

Or, simplifying,

$$
E(\theta) = \frac{Pe_v}{4\pi \theta^3}^{1/2} \exp\left[ -\frac{Pe_v (1 - \theta)^2}{4\theta} \right]
$$

## **Final Answer**

The analytical solution for the E-curve in terms of the vessel Peclet number $ Pe_v $ is:

$$
\boxed{
E(\theta) = \frac{Pe_v}{4\pi \theta^3}^{1/2} \exp\left[ -\frac{Pe_v (1 - \theta)^2}{4\theta} \right]
}
$$

where $ \theta $ is the dimensionless time and $ Pe_v $ is the vessel Peclet number. This describes the residence time distribution for a laminar flow reactor with axial dispersion and open-open (Danckwerts) boundary conditions.

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