 A sheet of 5.0-mm thick steel has gaseous atmospheres on both sides at a constant temperature and
is permitted to achieve a steady-state diffusion condition. The diffusion coefficient is 2.95 × 10-10 m2/s,
and the diffusion flux is found to be 3.0 × 10-7 kg/m2-sec. Also, it is known that the concentration of the
gas in the steel at the high-pressure surface is 4.0 kg/m3. How far into the sheet (in mm) from this high-
pressure side will the concentration be 0.5 kg/m3? Assume a linear concentration profile.


To solve this problem, we can use Fick's First Law of Diffusion, which applies under steady-state conditions and with a linear concentration profile. The law is given by:

\[
J = -D \frac{dC}{dx}
\]

Where:
- \( J \) is the diffusion flux (in \( \text{kg/m}^2\text{-s} \)),
- \( D \) is the diffusion coefficient (in \( \text{m}^2/\text{s} \)),
- \( \frac{dC}{dx} \) is the concentration gradient (in \( \text{kg/m}^4 \)).

Given dataTo solve this problem, we can use Fick's First Law of Diffusion, which applies under steady-state conditions and with a linear concentration profile. The law is given by:

\[
J = -D \frac{dC}{dx}
\]

Where:
- \( J \) is the diffusion flux (in \( \text{kg/m}^2\text{-s} \)),
- \( D \) is the diffusion coefficient (in \( \text{m}^2/\text{s} \)),
- \( \frac{dC}{dx} \) is the concentration gradient (in \( \text{kg/m}^4 \)).

Given 
- \( J = 3.0 \times 10^{-7} \text{ kg/m}^2\text{-s} \)
- \( D = 2.95 \times 10^{-10} \text{ m}^2/\text{s} \)
- \( C_1 = 4.0 \text{ kg/m}^3 \) (concentration at the high-pressure side)
- \( C_2 = 0.5 \text{ kg/m}^3 \) (concentration at the distance we want to find)
- Thickness of the steel sheet \( L = 5.0 \text{ mm} = 5.0 \times 10^{-3} \text{ m} \)

Assuming a linear concentration profile, the concentration gradient can be expressed as:

\[
\frac{dC}{dx} = \frac{C_1 - C_2}{L}
\]

Substituting the values into Fick's first law:

\[
J = -D \frac{C_1 - C_2}{L}
\]

Now, we solve for \( L \) (distance \( x \) from the high-pressure side to the point where concentration is \( C_2 \)):

\[
L = \frac{-D(C_1 - C_2)}{J}
\]

We want to find the distance \( x \) where the concentration drops from \( C_1 = 4.0 \text{ kg/m}^3 \) to \( C_2 = 0.5 \text{ kg/m}^3 \):

1. Calculate the concentration difference:
   \[
   C_1 - C_2 = 4.0 - 0.5 = 3.5 \text{ kg/m}^3
   \]

2. Use Fick's Law to find \( x \):
   \[
   J = D \times \frac{3.5}{x}
   \]
   \[
   x = \frac{D \times 3.5}{J}
   \]

Substitute the given values:

\[
x = \frac{(2.95 \times 10^{-10} \text{ m}^2/\text{s}) \times 3.5 \text{ kg/m}^3}{3.0 \times 10^{-7} \text{ kg/m}^2\text{-s}}
\]

\[
x = \frac{1.0325 \times 10^{-9} \text{ m}^3/\text{s}}{3.0 \times 10^{-7} \text{ kg/m}^2\text{-s}} 
\]

\[
x = 3.44 \times 10^{-3} \text{ m}
\]

Convert meters to millimeters:

\[
x = 3.44 \text{ mm}
\]

**Answer:**
The concentration will drop to \( 0.5 \text{ kg/m}^3 \) at approximately \( 3.44 \text{ mm} \) from the high-pressure side.

Would you like more details on the calculation, or do you have any questions? 

Here are some related questions you might find helpful:

1. What is Fick's Second Law, and how does it differ from the First Law?
2. How does temperature affect the diffusion coefficient?
3. What are common methods to measure diffusion coefficients in materials?
4. How does the assumption of a linear concentration profile affect the solution?
5. What would happen if the sheet thickness was doubled?

**Tip:** Always ensure that the units in your diffusion equations are consistent to avoid calculation errors.

A sheet of 5.0-mm thick steel has gaseous atmospheres on both sides at a constant temperature and is permitted to achieve a steady-state diffusion condition. The diffusion coefficient is 2.95 × 10-10 m2/s, and the diffusion flux is found to be 3.0 × 10-7 kg/m2-sec. Also, it is known that the concentration of the gas in the steel at the high-pressure surface is 4.0 kg/m3. How far into the sheet (in mm) from this high-pressure side will the concentration be 0.5 kg/m3? Assume a linear concentration profile.

Solve a diffusion problem involving a 5.0-mm thick steel sheet, using Fick's First Law. Given a diffusion coefficient of 2.95 × 10^-10 m^2/s and a diffusion flux of 3.0 × 10^-7 kg/m^2-sec, find the distance at which the gas concentration drops from 4.0 kg/m^3 to 0.5 kg/m^3. Learn about the application of Fick's Law in steady-state conditions and linear concentration profiles.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation