Dissolution of a metal sphere

In summary, the article discusses how to calculate the dissolution time of a sphere undergoing constant corrosion. It uses a series of graphs to illustrate the solution. The article gives an example of how to find the solution, and provides a step-by-step guide on how to do so.
  • #1
johnsonb.engr
12
0
Hello All

I could get some help I would greatly appreciate it.

I am trying to figure how to calculate the dissolution time of sphere undergoing constant corrosion at a rate
of corrosion.

Through a little google-fu, I found an article which gives me the solution ( http://arxiv.org/pdf/1208.5925.pdf )
but I am having trouble understanding the proof.

In this article there is a dissolving sphere

From this article we know that

the mass of the sphere is equal to dm/dt = -c*s(m) (1

and the general solution is

m(t)=mo - A*(mo^2/3)*t + (1/3)*(A^2)*(mo^(1/3))*(t^2) -1/27*(A^3)*(t^3)
or

m(t) = (a-k*t)^3
a = initial mass = (mo)^1/3
k = (A/3).

The article then gives an example where

mo = initial mass = 10 grams
p = density = 0.8 mg/mm^2
c = corrosion rate = -0.003 mg/(s*mm^2)

I am having difficulty understanding how to relate the rate of corrosion c to A.

I know that c = (dm/dt)/s(m)

The article shows a series of graphs for Mass, Radius, SA, and Volume vs time.

I copied these graphs into excel and add used excel to find a trendline.

m(t) = -2E-07*t^3 + 0.0002*t^2 - 0.079t+ 9.9879

By plugging values into the trend line above I was able to find a solution to

A = 0.0173.

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However, do not understand how to relate A to c(-0.003).

Some help would be greatly appreciated.
 

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  • #2
corrosion rate coefficient is given as a number c milligrams of mass dissolved per second from each square millimeter of surface area.
Therefore, an object with surface area A square millimeters will have an instantaneous rate of mass loss of ##\dot m =cA## milligrams per second.
You know how to find the surface area of a solid sphere from it's mass and density right?
 
Last edited:
  • #3
Hello Simon,

Thank you very much for your response.

Maybe I am not fully understanding your answer.

The A, I am trying to find is not the surface area of the sphere but rather a constant found in the solution to

dm/dt = -c*s(m) = -c*4πi*(3m/(4πi*ρ)^(2/3) (equation 1)

where the solution to the equation listed above is

m(t)=mo - A*(mo^2/3)*t + (1/3)*(A^2)*(mo^(1/3))*(t^2) -1/27*(A^3)*(t^3) (equation 2)

which simplifies to

m(t) = (mo^(1/3) - (A/3)t)^(1/3) (equation 3)

Where A is constant. Maybe I am wrong here, but I do not believe that A equation 2 and 3 is equal to the surfaces area of the sphere.

Am I wrong here?

If I am only given the density, rate of corrosion, and diameter of sphere, how would I go about finding equation 3.

Thank you very much for your help.
 
  • #4
Then you need to identify what physical property the article is using the letter A to stand for. So far you have A=3k ... but that begs the question: what is k? (Though it is possible that A is just the constant of integration...)
Either that - or discard the article and work out your own solution.
The second is probably faster... I've already done it: took me a couple of minutes but then, I've had practise.
 
Last edited:

1. What is the process of dissolution of a metal sphere?

The dissolution of a metal sphere refers to the gradual breakdown or disintegration of the metal sphere into smaller pieces due to the reaction with another substance, typically a liquid or gas.

2. What factors affect the dissolution rate of a metal sphere?

The dissolution rate of a metal sphere can be affected by various factors, including the type of metal, the temperature and pressure of the surrounding environment, the acidity or basicity of the solution, and the presence of other substances that may hinder or facilitate the reaction.

3. How does the dissolution of a metal sphere occur?

The dissolution of a metal sphere occurs through a chemical process known as corrosion, where the metal reacts with another substance and forms compounds that are more stable than the metal itself. This leads to the gradual breakdown of the metal into smaller pieces.

4. What are the potential consequences of the dissolution of a metal sphere?

The consequences of the dissolution of a metal sphere can vary depending on the type of metal and the surrounding environment. In some cases, it may lead to the degradation of the metal and compromise its structural integrity, while in others it may result in the release of toxic substances into the environment.

5. How can the dissolution of a metal sphere be prevented?

The dissolution of a metal sphere can be prevented by taking measures such as using protective coatings or materials that are resistant to corrosion, controlling the temperature and pH of the surrounding environment, and avoiding exposure to substances that may accelerate the dissolution process.

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