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Chapter 14: Problem 81

Define the penetration depth for mass transfer, and explain how it can be determined at a specified time when the diffusion coefficient is known.

Short Answer

Expert verified
Answer: In mass transfer, penetration depth (δ) is the distance it takes for the concentration of a substance being transferred to drop to about 37% (1/e) of its starting value. To determine the penetration depth when the diffusion coefficient (D) is known at a specific time (t), the following equation is used: \[ \delta = \sqrt{2 \cdot D \cdot t} \] By substituting the values of D and t into the equation and computing the final value, the penetration depth can be determined.

Step by step solution

01

Define penetration depth for mass transfer

Penetration depth, often denoted by the symbol δ, is a parameter used in the field of mass transfer to describe the distance in which the concentration of a diffusing solute (like a gas or a solute in a liquid) reaches 1/e (approximately 37%) of its initial concentration. Simply put, it is the distance it takes for the concentration of a substance being transferred to drop to about 37% of its starting value. It is often used to quantify the effectiveness of convective mass transport over diffusive mass transport in systems like boundary layers.
02

Understand the importance of the diffusion coefficient

The diffusion coefficient, denoted by the symbol D, is an essential parameter in determining the penetration depth. It represents the ease with which the solute particles can diffuse through the surrounding medium and is dependent on factors like the temperature and properties of the solute and the medium. A higher diffusion coefficient means that the solute can move more freely, and vice versa.
03

Introduce the equation for penetration depth

The formula used to calculate the penetration depth (δ) when the diffusion coefficient (D) and the specified time (t) are known is as follows: \[ \delta = \sqrt{2 \cdot D \cdot t} \] It represents the relationship between the penetration depth, the diffusion coefficient, and the time taken for the concentration to reach 1/e of its initial value.
04

Apply the formula to calculate penetration depth

To determine the penetration depth at a given time (t) when the diffusion coefficient (D) is known, plug the values of D and t into the equation: \[ \delta = \sqrt{2 \cdot D \cdot t} \] Then, proceed to compute the final value of δ, which will indicate the penetration depth for mass transfer at the specified time. In summary, understanding the penetration depth concept in mass transfer and its relationship with the diffusion coefficient is essential. By using the given equation and substituting the known values, it is possible to determine the penetration depth at a specified time.

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Most popular questions from this chapter

What is the physical significance of the Sherwood number? How is it defined? To what dimensionless number does it correspond in heat transfer? What does a Sherwood number of 1 indicate for a plane fluid layer?

The diffusion of water vapor through plaster boards and its condensation in the wall insulation in cold weather are of concern since they reduce the effectiveness of insulation. Consider a house that is maintained at \(20^{\circ} \mathrm{C}\) and 60 percent relative humidity at a location where the atmospheric pressure is \(97 \mathrm{kPa}\). The inside of the walls is finished with \(9.5\)-mm-thick gypsum wallboard. Taking the vapor pressure at the outer side of the wallboard to be zero, determine the maximum amount of water vapor that will diffuse through a \(3-\mathrm{m} \times 8-\mathrm{m}\) section of a wall during a 24-h period. The permeance of the \(9.5\)-mm-thick gypsum wallboard to water vapor is \(2.86 \times 10^{-9} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}^{2} \cdot \mathrm{Pa}\).

Using Henry's law, show that the dissolved gases in a liquid can be driven off by heating the liquid.

Benzene \((M=78.11 \mathrm{~kg} / \mathrm{kmol})\) is a carcinogen, and exposure to benzene increases the risk of cancer and other illnesses in humans. A truck transporting liquid benzene was involved in an accident that spilled the liquid on a flat highway. The liquid benzene forms a pool of approximately \(10 \mathrm{~m}\) in diameter on the highway. In this particular windy day at \(25^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) with an average wind velocity of \(10 \mathrm{~m} / \mathrm{s}\), the liquid benzene surface is experiencing mass transfer to air by convection. Nearby at the downstream of the wind is a residential area that could be affected by the benzene vapor. Local health officials have assessed that if the benzene level in the air reaches \(500 \mathrm{~kg}\) within the hour of the spillage, residents should be evacuated from the area. If the benzene vapor pressure is \(10 \mathrm{kPa}\), estimate the mass transfer rate of benzene being convected to the air, and determine whether the residents should be evacuated or not.

For the absorption of a gas (like carbon dioxide) into a liquid (like water) Henry's law states that partial pressure of the gas is proportional to the mole fraction of the gas in the liquid-gas solution with the constant of proportionality being Henry's constant. A bottle of soda pop \(\left(\mathrm{CO}_{2}-\mathrm{H}_{2} \mathrm{O}\right)\) at room temperature has a Henry's constant of \(17,100 \mathrm{kPa}\). If the pressure in this bottle is \(120 \mathrm{kPa}\) and the partial pressure of the water vapor in the gas volume at the top of the bottle is neglected, the concentration of the \(\mathrm{CO}_{2}\) in the liquid \(\mathrm{H}_{2} \mathrm{O}\) is (a) \(0.003 \mathrm{~mol}-\mathrm{CO}_{2} / \mathrm{mol}\) (b) \(0.007 \mathrm{~mol}-\mathrm{CO}_{2} / \mathrm{mol}\) (c) \(0.013 \mathrm{~mol}-\mathrm{CO}_{2} / \mathrm{mol}\) (d) \(0.022 \mathrm{~mol}-\mathrm{CO}_{2} / \mathrm{mol}\) (e) \(0.047 \mathrm{~mol}-\mathrm{CO}_{2} / \mathrm{mol}\)
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