# Penetration Theory proposed by Higbie (1935)

• andreia3aral
In summary, the conversation revolved around the Penetration Theory proposed by Higbie (1935) for a bubble column filled with water saturated in oxygen. The speaker carried out laboratory experiments to determine the KL of the bubbles and had questions regarding the application of the theory in their specific case. They also questioned the validity of using the expression for instantaneous KL instead of the average KL over tc in their situation. Further discussions touched upon the calculation of tc and the slip velocity, as well as the original publication of the Penetration Theory. The expert summarizer notes that for accurate results, a more accurate estimate of KL should be used and suggests consulting additional references. Additionally, they clarify that even in cases where there is "negative" penetration, mass
andreia3aral
TL;DR Summary
Questions related to Penetration Theory proposed by Higbie (1935) for a bubble column filled with water saturated in oxygen.
Hello,

I have some questions related to the Penetration Theory proposed by Higbie (1935).

I carried out laboratory experiments in a bubble column of 1.3 m filled with water and saturated with oxygen. Air bubbles were rising and the liquid was stagnant (its motion was just due to the rise of the air bubbles). I would like to determine the KL of these bubbles.

Can I apply the Penetration Theory in this case? Even if the liquid is saturated with oxygen?, i.e., there is no oxygen transfer and the liquid elements have the same concentration as the bubble/liquid element interface?

Since liquid elements have the same concentration as the bubble/liquid element interface, so "penetration" does not take place, does it make sense that I use the expression for instantaneous KL (KL= sqrt (D/(pi*t)) instead of the expression for average KL over tc (KL= 2sqrt (D/(pi*tc)) to calculate KL in this situation?

I read that tc can be calculated based on the diameter of the bubble and its slip velocity, right? When does the slip velocity can be considered equal to the bubble rising velocity?
In the case of the laboratory experiments described, is it OK to consider the slip velocity is equal to the bubble rising velocity?
But now, imagine a tank where there are air bubbles rising from the bottom to the top but there is also a horizontal liquid flow, can I still consider that the slip velocity is equal to the bubble rising velocity? If not, how could I determine this value?

Do you know where I can find the original publication of the Penetration Theory?

Please let me know if something is not clear. Thank you very much in advance.

andreia3aral said:
Summary: Questions related to Penetration Theory proposed by Higbie (1935) for a bubble column filled with water saturated in oxygen.

Hello,

I have some questions related to the Penetration Theory proposed by Higbie (1935).

I carried out laboratory experiments in a bubble column of 1.3 m filled with water and saturated with oxygen. Air bubbles were rising and the liquid was stagnant (its motion was just due to the rise of the air bubbles). I would like to determine the KL of these bubbles.

Can I apply the Penetration Theory in this case?
In this case, Penetration Theory only provides a rough approximation to KL because of the more complicated flow.
Even if the liquid is saturated with oxygen?, i.e., there is no oxygen transfer and the liquid elements have the same concentration as the bubble/liquid element interface?
Who says there is no oxygen transfer? The partial pressure of the oxygen within the bubble is less than atmospheric, so there is a driving force for oxygen to transfer into the bubble.
Since liquid elements have the same concentration as the bubble/liquid element interface, so "penetration" does not take place, does it make sense that I use the expression for instantaneous KL (KL= sqrt (D/(pi*t)) instead of the expression for average KL over tc (KL= 2sqrt (D/(pi*tc)) to calculate KL in this situation?
In this case, there is "negative" penetration, in the sense that the liquid boundary layer is depleted of oxygen. There is also positive penetration of nitrogen into the liquid boundary layer, accompanying transfer of nitrogen out of the bubble.
I read that tc can be calculated based on the diameter of the bubble and its slip velocity, right? When does the slip velocity can be considered equal to the bubble rising velocity?
In the case of the laboratory experiments described, is it OK to consider the slip velocity is equal to the bubble rising velocity?
As I said, penetration theory only provides a rough approximation in this case.
But now, imagine a tank where there are air bubbles rising from the bottom to the top but there is also a horizontal liquid flow, can I still consider that the slip velocity is equal to the bubble rising velocity? If not, how could I determine this value?
If the shear rate of the horizontally flowing liquid is not too high, that would be fine.
Do you know where I can find the original publication of the Penetration Theory?

Please let me know if something is not clear. Thank you very much in advance.
For accurate results, you need to use a more accurate estimate of KL. In addition to the reference provided by Tom.G., see Transport Phenomena by Bird, Stewart, and Lightfoot and Mass Transfer Operations by Treybel.

Thank you very much for your replies.

I understand that there is still mass transfer happening due to hydrostatic pressure drop. However, how does the penetration theory deal with "negative" penetration?

Which more accurate estimation of KL can be use?

andreia3aral said:
Thank you very much for your replies.

I understand that there is still mass transfer happening due to hydrostatic pressure drop. However, how does the penetration theory deal with "negative" penetration?

Which more accurate estimation of KL can be use?
See the references that have already been provided. We require you to do the detailed research.

My point has nothing to do with hydrostatic pressure. The partial pressure of oxygen inside the bubble is only 20 % of the partial pressure of the oxygen in the water. So, oxygen will diffuse into the bubble from the surrounding water.

Thank you for helping me. However, I do not understand your point of view.

When water is saturated with oxygen, it means that the partial pressure of oxygen in the bubble is in equilibrium with the concentration of oxygen in the water. Thus, in this situation, there is no net diffusion of the oxygen between the bubble and water, ie, the movement of oxygen from the air bubble into the water is equal to the movement of oxygen from the water back into the air, right?!

Can you explain better your point of view?

andreia3aral said:
Thank you for helping me. However, I do not understand your point of view.

When water is saturated with oxygen, it means that the partial pressure of oxygen in the bubble is in equilibrium with the concentration of oxygen in the water. Thus, in this situation, there is no net diffusion of the oxygen between the bubble and water, ie, the movement of oxygen from the air bubble into the water is equal to the movement of oxygen from the water back into the air, right?!

Can you explain better your point of view?
I thought you said that the water was saturated with pure oxygen at 1 atm before the air was bubbled through it. Is that not the case?

A bubble column was filled with water. Air was bubbled in the bottom of the column. Dissolved oxygen concentration was measured over time. After a period of time, the water was saturated with oxygen. So, in this situation, there is no net diffusion of the oxygen between the bubble and water.

The penetration theory was developed considering a bubble of gas that rises through a liquid which absorbs the gas (unsteady mass transfer). Is it possible to apply the penetration theory for steady mass transfer, for example, when water is saturated with oxygen?

andreia3aral said:
A bubble column was filled with water. Air was bubbled in the bottom of the column. Dissolved oxygen concentration was measured over time. After a period of time, the water was saturated with oxygen. So, in this situation, there is no net diffusion of the oxygen between the bubble and water.

The penetration theory was developed considering a bubble of gas that rises through a liquid which absorbs the gas (unsteady mass transfer). Is it possible to apply the penetration theory for steady mass transfer, for example, when water is saturated with oxygen?
If the water is saturated with oxygen, then there will be no mass transfer of oxygen between the bubbles and the water. So it doesn't matter what the mass transfer coefficient is. But I would not be using penetration theory even if there were (except for back-of-the-envelope estimates). I would start by looking up information on the mass transfer coefficient for water flow past a sphere. This is in Transport Phenomena by Bird et al. They might even have a correlation for bubbles.

But pressure can play a role along the height of the column and, consequently, the concentration of oxygen dissolved on the water on the top of the column is slightly lower than on the bottom of the column, right? If this is considered, is mass transfer taking place? I need to understand better this situation, but for me, it's hard to find a clear answer to my question.

## 1. What is Penetration Theory proposed by Higbie (1935)?

Penetration Theory, also known as the Higbie's Penetration Theory, is a scientific theory proposed by Robert Higbie in 1935. It is used to describe the mass transfer process in porous media, such as in filtration, adsorption, and chromatography.

## 2. How does Penetration Theory explain mass transfer in porous media?

According to Higbie's Penetration Theory, the rate of mass transfer in porous media is directly proportional to the concentration gradient and the surface area available for mass transfer. It also takes into account the diffusivity and the thickness of the porous media.

## 3. What are the assumptions made in Penetration Theory?

Penetration Theory makes several assumptions, including:

• The concentration gradient is constant throughout the porous media.
• The diffusion coefficient is constant.
• The porous media is of uniform thickness.
• The mass transfer is a one-dimensional process.
• The surface area available for mass transfer is constant.

## 4. What are the applications of Penetration Theory?

Penetration Theory has many practical applications, including:

• Water and wastewater treatment processes.
• Food and beverage processing.
• Pharmaceutical production.
• Chemical and petrochemical industries.
• Environmental remediation.

## 5. How is Penetration Theory different from other mass transfer theories?

Penetration Theory is a simplified version of other mass transfer theories, such as the film theory and the surface renewal theory. It assumes a constant concentration gradient and does not take into account the effects of turbulence or surface renewal. However, it is still widely used in engineering applications due to its simplicity and practicality.

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