An FMC de-aerator 2.97 m high and 96.5 cm in diameter is rated to de-aerate from 4.2 to 8.4 kg

step 1 Hi. So for this problem, let us review some concepts first. When a gas is absorbed into a liquid

An FMC de-aerator 2.97 m high and 96.5 cm in diameter is...

Key Concepts

Geometric Considerations in Process Design

The design and operation of process equipment such as de-aerators are heavily influenced by geometric parameters like height and diameter. These dimensions dictate the available area for fluid flow, influence the calculation of film thickness and velocity, and thereby affect the overall performance of the separation process. Understanding how geometry interacts with fluid dynamics is crucial for optimizing equipment performance.

Non-Newtonian Fluid Rheology

Non-Newtonian fluids do not follow Newton’s law of viscosity, meaning their viscosity can change with the rate of shear. This behavior is often modeled using the power law, which introduces a flow behavior index and a consistency index to quantify how the fluid’s resistance to flow changes. These parameters are essential in calculating flow characteristics like film thickness and velocity for fluids with such complex behavior in engineering applications.

Residence Time Estimation

Residence time in the context of fluid films is the time that any element of the fluid remains within a designated region, which impacts processes like gas bubble separation. Calculating the residence time involves using the film length (or the characteristic distance, such as half the de-aerator height) and the average velocity of the fluid film, offering insights into how long gas bubbles have to migrate from the liquid phase into the vapor space.

Mass Flow Rate and Continuity

This concept involves using the relationship between mass flow rate, density, cross?sectional area, and velocity to determine the parameters of fluid flow. In calculating film thickness and film velocity, engineers rely on the mass conservation principle which ensures that the mass entering a system is equal to the mass exiting it, allowing for the determination of velocity if one knows the mass flow rate and the area over which the fluid flows.

Thin Film Flow Dynamics

Thin film flow refers to the behavior of a fluid flowing over a surface in a layer whose thickness is small compared to its other dimensions. In process equipment such as de?aerators, understanding thin film flow is critical since it affects heat and mass transfer, residence time of the fluid, and the opportunity for phase separation such as gas bubble removal. Key parameters derived from such dynamics include film thickness and velocity.

Transcript

0:00 Hi.

00:02 So for this problem, let us review some concepts first.

00:06 When a gas is absorbed into a liquid, the transfer of molecules from the gas phase to the liquid phase is controlled by mass transfer.

00:14 So in the liquid, this transfer is often limited by diffusion through a thin region, this is called the film, near the interface where decomposition changes from the interfacial value to the bulk value.

00:26 So different mass transfer theories try to describe how this region behaves.

00:31 So we have the film theory.

00:33 This assumes a stagnant film of thickness next to the interface with steady state diffusion across it.

00:40 And then we have the penetration theory which assumes fluid elements are brought to the interface for a finite contact time and then then swept away.

00:50 So diffusion is unsteady within each element.

00:55 And then we have the surface renewal theory which assumes that the surface is continually renewed at every rate s and each element has a distribution of residence time before it is replaced.

01:07 So all three theories that i've mentioned connect the observable total flux to diffusion coefficient and to mass transfer coefficients.

01:17 So in this problem, we have carbon dioxide that is absorbed from air into water in a packed column.

01:24 So assuming that only liquid -based resistance matters and that the liquid is essentially pure water, we need to determine the liquid phase mass transfer coefficient and the film thickness, the contact time for the penetration theory, and the average at the residence time, and the probability distribution for the surface renewal theory.

01:43 So i'll start with a.

01:48 Let's find the concentration of the pure water using its density in the molar mass.

01:54 This is the formula that we will use, density over molar mass.

01:59 Density of water is one gram.

02:06 Let me just just rewrite that one one gram per um l and then we have the molar mass that's 18 and this would give us 0 .0556 that's moles per cubic cm and next uh we'll find the mole fraction of co2 in the interface using hendrous law so the partial pressure of the gas at the interface is given we have 150 p .s.

02:49 And 10th constant for the system at the given temperature h is 9 ,000.

03:01 So we have this formula that the interface.

03:17 So we have 150, small fraction of co2 at the interface multiplied by 9 ,000.

03:32 So 150 over 9 ,000 is 0 .067.

03:37 Now at the bulk liquid, the concentration of carbon carbon dioxide is negligible.

03:43 So we have mole fraction at the bulk liquid b, b0.

03:52 So using film theory with negligible bulk concentration of the gas, the total flux is calculated as kc multiplied by the concentration, and then the mole fractions interface minus mole fraction, um, mole fraction of, co2 at the bulk liquid.

04:20 So i'll just rearrange this because we need to find the mass transfer coefficient kc.

04:39 Now the absorption rate is 0 .017.

04:45 Let's write that here.

04:51 This is in pound mole square feet hour and we can convert this to moles per square cm second this equivalent to 2 .31 times 10 to the negative 6.

05:06 This is the unit now just erase this this is most then log in the values here 2 .31 times 10 to the negative 6 over the total concentration c 0 .0556 multiplied by 0 .167 0 .0167 minus 0 .k so kc is 2 .49 times 10 to the negative three is one of our answers for letter a.

06:02 And according to the film theory, the ratio of the diffusion coefficient and the mass transfer coefficient is equivalent to the film thickness.

06:12 So this is formula.

06:20 And we're given with the diffusivity.

06:24 We have diffusivity of the o2 in water, two times 10 to the negative 5 square cm per second.

06:33 So let's plug in the values here.

06:40 Kc is 2 .49 times 10 to the negative 3.

06:47 So the film thickness is 0 .00803...

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