Solving the Shrinking Core Model: Questions & Solutions

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In summary, the conversation discusses the solution d(ro*pi*D^3/6)/d(t)=-kr*Cao*pi*D^2 and the confusion over how to take the derivative of D. The solution uses the chain rule, where d(D3)/dt = 3D2 dD/dt, to arrive at ro/2* d(D)/d(t) = -kr Cao. The question remains on why d(D)/d(t) is still left behind when taking the derivative of D.
  • #1
cycling4life
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I have this solution that I don't quite understand

d(ro*pi*D^3/6)/d(t)=-kr*Cao*pi*D^2

from here I would have thought to separate the variables and integrate but the solution says

ro/2* d(D)/d(t) = -kr Cao

I guess my question is how are you able to do that? If you take the derivative of D, why is d(D)/d(t) still left behind?
 
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hi cycling4life! :smile:

(try using the X2 button just above the Reply box :wink:)
cycling4life said:
d(ro*pi*D^3/6)/d(t)
…
I guess my question is how are you able to do that? If you take the derivative of D, why is d(D)/d(t) still left behind?

chain rule … d(D3)/dt = 3D2 dD/dt :wink:
 

1. What is the Shrinking Core Model?

The Shrinking Core Model is a mathematical model used to describe the kinetics of solid-state reactions, specifically the dissolution or precipitation of a solid into a liquid or gas phase. It assumes that the solid is shrinking due to the reaction and that the reaction only occurs at the surface of the solid.

2. What are the main assumptions of the Shrinking Core Model?

The main assumptions of the Shrinking Core Model are:

  • The solid is spherical and is shrinking due to the reaction
  • The reaction occurs only at the surface of the solid
  • The reaction is first-order with respect to the concentration of the reactant in the liquid or gas phase
  • The reaction rate is controlled by mass transfer of the reactant from the bulk liquid or gas phase to the solid surface

3. How is the Shrinking Core Model solved?

The Shrinking Core Model is typically solved using a combination of analytical and numerical methods. The first step is to write and solve the differential equations that describe the change in concentration of the reactant in the solid and liquid or gas phases. This is done using the assumptions of the model and the appropriate boundary conditions. The resulting equations are then solved numerically using a computer program or by hand using mathematical techniques.

4. What are the limitations of the Shrinking Core Model?

Like all models, the Shrinking Core Model has its limitations. Some of the main limitations include:

  • The model assumes a spherical solid, which may not be applicable to all systems
  • The reaction is assumed to occur only at the surface, which may not be true for all systems
  • The model assumes first-order kinetics, which may not accurately describe the reaction
  • Mass transfer is assumed to be the rate-limiting step, but this may not be the case for all systems

5. When is the Shrinking Core Model used?

The Shrinking Core Model is commonly used in chemical engineering and materials science to model and understand solid-state reactions, such as the dissolution or precipitation of minerals or particles. It is also used in industries such as metallurgy, pharmaceuticals, and environmental engineering to design and optimize processes involving solid-state reactions.

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