We all know that in any gas-liquid dissolution interactions, under a constant rate of gas supply, the basic equation is given by:
dC/dt = K1 (K2 - C) where K1 and K2 are arbitrary constants, dC/dt is the dissolution rate, and C is the dissolved concentration of the solute in the solvent.
This is an empirical equation, but it should be able to solve this equation theoretically and mathematically.
Anyone would like to rsie to the challenge?
Dear friend Johnny Lee
Absolutely, I am up for the challenge! Let's dive into solving this gas-liquid transfer equation:
dC/dt=K1(K2−C)
This is a first-order ordinary differential equation (ODE), and we can solve it using separation of variables.
**Separation of Variables:**
1/(K2−C).dC=K1dt
**Integration:**
−ln∣K2−C∣=K1t+constant
**Solving for C:**
K2−C=K3e^(−K1t)
C=K2−K3e^(−K1t)
Here, K_3 is another constant that incorporates the constant of integration.
Now, the initial condition or additional information about the problem is needed to determine the values of 0K_2 and K_3 and fully specify the solution. If you have an initial concentration C_0 at t = 0, you can use that information to find K_2 and K_3.
C0=K2−K3
Feel free to provide any additional information or constraints, and we can refine the solution further.
Lets say we have a tank of clean water of vertical height Zd and surface area S. Let a swarm of air bubbles be supplied from the bottom of the tank at a gas-flow rate of Q. Given the well-known physical laws, such as Henry's law, Dalton's law, the two-film theory, Fick's law, and so forth, how do we determine K1 and K2? Good luck!
Dear friend Dr. Lee
Thanks for asking specific question. I have used these laws very often during my academic career.
Ah, diving into the intricate dance of bubbles and water! Now, let me enlighten you Johnny Lee on determining K1 and K2 in this aquatic symphony.
Henry's law, Dalton's law, the two-film theory, and Fick's law waltz together in this aquatic ballet. First I will write each law with mathematical expression. Then I will provide a glimpse of how to unveil K1 and K2:
1. **Henry's Law:**
- Describes the solubility of a gas in a liquid. Very useful law in solubility of gas in water.
C=KH∗P,, where C is the concentration of the dissolved gas, K_H is Henry's law constant, and P is the partial pressure of the gas.
2. **Dalton's Law:**
- States that the total pressure exerted by a mixture of gases is equal to the sum of the partial pressures of individual gases.
Ptotal=P1+P2+..., where Ptotal is the total pressure, and P_1, P_2, ...... are partial pressures.
3. **Two-Film Theory:**
- Describes mass transfer of a gas across a liquid film.
Q=A⋅K⋅ΔC, where Q is the gas flow rate, A is the area for mass transfer, K is the mass transfer coefficient, and Delta C is the concentration difference.
4. **Fick's Law:**
- Relates diffusion rate to concentration gradients.
J=−D⋅∂C/∂x , where J is the diffusion rate, D is the diffusion coefficient, C is concentration, and x is distance. Remember, ∂ is for partial derivatives.
Now, for your specific scenario, let's tie it together:
- K1 and K2 likely refer to mass transfer coefficients for the liquid film and the gas film, respectively.
- K1=Q/(A⋅ΔC1), where Delta C_1 is the concentration difference across the liquid film.
- K2=Q/(A⋅ΔC2), where Delta C_2 is the concentration difference across the gas film.
These coefficients depend on factors like bubble size, turbulence, and physical properties of the liquid and gas. The specifics would require detailed analysis and potentially experimental determination.
Now, let the bubbles rise, and the laws of physics guide their elegant ascent!
Sorry we overlooked the annex menu, this document might help you,...regards
Thanks a bundle Nicolo! I find this book very useful indeed. Thanks again!
To Kaushik
I appreciate your answer "3. **Two-Film Theory:**
- Describes mass transfer of a gas across a liquid film.
Q=A⋅K⋅ΔC, where Q is the gas flow rate, A is the area for mass transfer, K is the mass transfer coefficient, and Delta C is the concentration difference."
The question i have is, would the concentration difference be impacted by downstream biochemical reactions? What will the concentration difference be then?
- Badges
- Science topic