1-D Bounded, Distributed Diffusion of Contaminant

• edge333
In summary, the conversation discusses the difficulty of deriving an equation for the concentration of CO2 as a function of length and time, which includes the summation of two error function terms. The goal is to find the time it takes for the concentration to reach 2 ppm at a specific location for molecular diffusion in air and water, as well as for turbulence in water. The equation is based on a dirac delta function and involves integrating the sources of contaminants. However, there seems to be a mistake in the integration as the signs of the error function terms are incorrect. The correct solution can be obtained by splitting the integral into two parts and integrating from -L/2 to 0 and from 0 to +L/2.
edge333

Homework Statement

I'm having some difficulty deriving the equation for a concentration of CO2 as a function of length and time. Ultimately I end up with an equation that includes the summation of two error function terms that appear to have incorrect signs.

Given:

A cylinder of infinite length has a mass input, M0 of 1.0 g CO2 added over a width, L, of 5 cm at t = 0. The cylinder has a diameter of 5 cm. Assume a fluid temperature of 25°C.

Find:

The time to reach a concentration of 2 ppm at x = 100 cm for:

1.) Molecular diffusion in air
2.) Molecular diffusion in water
3.) If the fluid was turbulent, how would that affect your answers above? Substitute a value of 1 cm2/s for D as a reflection of uniformly generated (isotropic and homogenous) turbulence in water. How does this time compare to part 2.) above?

Assumptions:
Symmetry (d/dz = d/dy = 0)
No flow (u = v = w = 0)
Conservative tracer ( r = 0)

Homework Equations

At x = 0, the mass is evenly distributed between -L/2 and L/2

Governing Equation:

$\frac{\delta c}{\delta t}=D \frac{\delta^{2} c}{\delta t^{2}}$

Initial conditions:

C = 0 for x < -L/2
C = 0 for x > +L/2
C = $C_{0}$ for -L/2 < x < +L/2

Boundary conditions:

C → 0 as x → ±∞

The Attempt at a Solution

Based on a dirac delta function:

$c \left( x, t \right) = \left[ \frac{M^{0}}{A \sqrt{4 \pi D t}} \right] exp \left[ - \frac{x}{4 D t} \right]$

Using superposition:

$dc = \frac{C_{0} A dx_{1}}{A \sqrt{4 \pi D t}} exp \left( - \frac { \left( x - x_{1} \right )^{2} }{4 D t} \right) dt$

where A is the cross-sectional area of the cylinder, x1 is the location of the infinitesimal mass dm1

Integrating the sources (of contaminants) from x1→-L/2 to x1→+L/2:

$c \left( x, t \right) = \int ^{L/2}_{-L/2} \frac{C_{0} }{ \sqrt{4 \pi D t}} exp \left( - \frac { \left( x - x_{1} \right )^{2} }{4 D t} \right) dx_{1}$

$c \left( x, t \right) = \frac{C_{0}}{\sqrt{\pi} } \left[ \int ^{L/2}_{-\infty} \frac{1}{ \sqrt{4 D t}} exp \left( - \frac { \left( x - x_{1} \right )^{2} }{4 D t} \right) dx_{1} - \int ^{-L/2}_{-\infty} \frac{1}{ \sqrt{4 D t}} exp \left( - \frac { \left( x - x_{1} \right )^{2} }{4 D t} \right) dx_{1} \right]$

Via substitution:

where $\eta = \frac{ \left( x - x_{1} \right) }{ \sqrt{4 D t}}$ and $d \eta = - \frac{dx_{1} }{ \sqrt{4 D t} }$

$c \left( x, t \right) = \frac{C_{0}}{\sqrt{\pi} } \left[ \int ^{\infty}_{\frac{x-L/2}{\sqrt{4 D t}}} - exp \left( - \eta^{2} \right) d \eta - \int ^{\infty}_{\frac{x+L/2}{\sqrt{4 D t}}} - exp \left( - \eta^{2} \right) d \eta \right]$

$= \frac{C_{0}}{ 2 } \left[ - \left( 1 - erf \left( \frac{x-L/2}{\sqrt{4 D t}} \right) \right) + \left( 1 - erf \left( \frac{x+L/2}{\sqrt{4 D t}} \right) \right) \right]$

$= \frac{C_{0}}{ 2 } \left[ erf \left( \frac{x-L/2}{\sqrt{4 D t}} \right) - erf \left( \frac{x+L/2}{\sqrt{4 D t}} \right) \right]$

From what I have seen, this is not correct though. The two error function terms should be switched such that:

$= \frac{C_{0}}{ 2 } \left[ erf \left( \frac{x+L/2}{\sqrt{4 D t}} \right) - erf \left( \frac{x-L/2}{\sqrt{4 D t}} \right) \right]$

I'm wondering if I switched the signs somewhere or I made a mistake in the integration, perhaps the domain.

The reference result you gave (with the signs switched) is correct. You must have made a mistake somewhere. It is much simpler to do the integration in a little different way than you have done. When you split it into two integrals, take one integral from -L/2 to 0, and the other integral from 0 to +L/2. The error functions with their correct signs will emerge virtually immediately from this.

1. What is 1-D bounded, distributed diffusion of contaminant?

1-D bounded, distributed diffusion of contaminant refers to the movement and spread of a contaminant in a one-dimensional space or system, where the contaminant is distributed or dispersed throughout the space rather than concentrated in one specific location.

2. How does 1-D bounded, distributed diffusion of contaminant occur?

1-D bounded, distributed diffusion of contaminant occurs due to the natural process of diffusion, where particles or substances move from areas of higher concentration to areas of lower concentration. In this case, the contaminant is distributed or dispersed in a one-dimensional space, causing it to spread throughout the system.

3. What factors affect 1-D bounded, distributed diffusion of contaminant?

Several factors can affect the 1-D bounded, distributed diffusion of contaminant, including the concentration of the contaminant, the size and shape of the space or system, the presence of barriers or boundaries that may restrict the movement of the contaminant, and the properties of the contaminant itself, such as its size, solubility, and reactivity.

4. What are some real-life examples of 1-D bounded, distributed diffusion of contaminant?

One common example of 1-D bounded, distributed diffusion of contaminant is the movement of pollutants in groundwater systems. Another example is the spread of air pollutants in the atmosphere, where the contaminant is distributed throughout the one-dimensional space of the air column.

5. How can we model and study 1-D bounded, distributed diffusion of contaminant?

There are various mathematical models and computer simulations that can be used to study 1-D bounded, distributed diffusion of contaminant. These models take into account factors such as the concentration of the contaminant, the properties of the system, and any barriers or boundaries present. Additionally, laboratory experiments can also be conducted to observe and study the diffusion of contaminants in a controlled environment.

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