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Chapter 14: Problem 27

What is an impermeable surface in mass transfer? How is it expressed mathematically (on a mass basis)? To what does it correspond in heat transfer?

Short Answer

Expert verified
The mathematical expression for an impermeable surface in mass transfer is 0 = -D (dC/dx), where J is the mass flux, D is the mass diffusivity, C is the concentration of the substance, and x is the distance across the boundary. The corresponding concept in heat transfer is an adiabatic surface, which prevents the passage of heat across the boundary.

Step by step solution

01

1. Defining an impermeable surface in mass transfer

An impermeable surface is a boundary that does not allow the passage of a specific substance across it. In the context of mass transfer, it means that the mass flux (rate of mass transfer per unit area) of this substance across the boundary is zero.
02

2. Mathematical expression of an impermeable surface on a mass basis

The mass flux, J, can be expressed mathematically as: J = -D \frac{dC}{dx}, where J is the mass flux (kg/m^2s), D is the mass diffusivity (m^2/s), C is the concentration of the substance (kg/m^3), and x is the distance across the boundary (m). For an impermeable surface, the mass flux, J, is equal to zero. Therefore, the mathematical expression for an impermeable surface in mass transfer is: 0 = -D \frac{dC}{dx}.
03

3. Identifying the corresponding concept in heat transfer

The corresponding concept in heat transfer is an adiabatic surface. An adiabatic surface is a boundary that does not allow the passage of heat across it, which means the heat flux (rate of heat transfer per unit area) at the boundary is zero.
04

4. Conclusion

An impermeable surface in mass transfer is a boundary that prevents the passage of a specific substance, and its mathematical expression on a mass basis is 0 = -D \frac{dC}{dx}. The corresponding concept in heat transfer is an adiabatic surface, which prevents the passage of heat across the boundary.

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Most popular questions from this chapter

Consider a thin layer of liquid water on a concrete surface. The surrounding air is dry with a convection heat transfer coefficient of \(50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The liquid water has an emissivity of \(0.95\), and the air and surrounding temperature is \(30^{\circ} \mathrm{C}\). If the layer of liquid water has a uniform temperature of \(20^{\circ} \mathrm{C}\), determine the conduction heat flux through the concrete.

A recent attempt to circumnavigate the world in a balloon used a helium-filled balloon whose volume was \(7240 \mathrm{~m}^{3}\) and surface area was \(1800 \mathrm{~m}^{2}\). The skin of this balloon is \(2 \mathrm{~mm}\) thick and is made of a material whose helium diffusion coefficient is \(1 \times 10^{-9} \mathrm{~m}^{2} / \mathrm{s}\). The molar concentration of the helium at the inner surface of the balloon skin is \(0.2 \mathrm{kmol} / \mathrm{m}^{3}\) and the molar concentration at the outer surface is extremely small. The rate at which helium is lost from this balloon is (a) \(0.26 \mathrm{~kg} / \mathrm{h}\) (b) \(1.5 \mathrm{~kg} / \mathrm{h}\) (c) \(2.6 \mathrm{~kg} / \mathrm{h}\) (d) \(3.8 \mathrm{~kg} / \mathrm{h}\) (e) \(5.2 \mathrm{~kg} / \mathrm{h}\)

A recent attempt to circumnavigate the world in a balloon used a helium-filled balloon whose volume was \(7240 \mathrm{~m}^{3}\) and surface area was \(1800 \mathrm{~m}^{2}\). The skin of this balloon is \(2 \mathrm{~mm}\) thick and is made of a material whose helium diffusion coefficient is \(1 \times 10^{-9} \mathrm{~m}^{2} / \mathrm{s}\). The molar concentration of the helium at the inner surface of the balloon skin is \(0.2 \mathrm{kmol} / \mathrm{m}^{3}\) and the molar concentration at the outer surface is extremely small. The rate at which helium is lost from this balloon is (a) \(0.26 \mathrm{~kg} / \mathrm{h}\) (b) \(1.5 \mathrm{~kg} / \mathrm{h}\) (c) \(2.6 \mathrm{~kg} / \mathrm{h}\) (d) \(3.8 \mathrm{~kg} / \mathrm{h}\) (e) \(5.2 \mathrm{~kg} / \mathrm{h}\)

A glass bottle washing facility uses a well agi(Es) tated hot water bath at \(50^{\circ} \mathrm{C}\) with an open top that is placed on the ground. The bathtub is \(1 \mathrm{~m}\) high, \(2 \mathrm{~m}\) wide, and \(4 \mathrm{~m}\) long and is made of sheet metal so that the outer side surfaces are also at about \(50^{\circ} \mathrm{C}\). The bottles enter at a rate of 800 per minute at ambient temperature and leave at the water temperature. Each bottle has a mass of \(150 \mathrm{~g}\) and removes \(0.6 \mathrm{~g}\) of water as it leaves the bath wet. Makeup water is supplied at \(15^{\circ} \mathrm{C}\). If the average conditions in the plant are \(1 \mathrm{~atm}, 25^{\circ} \mathrm{C}\), and 50 percent relative humidity, and the average temperature of the surrounding surfaces is \(15^{\circ} \mathrm{C}\), determine (a) the amount of heat and water removed by the bottles themselves per second, \((b)\) the rate of heat loss from the top surface of the water bath by radiation, natural convection, and evaporation, \((c)\) the rate of heat loss from the side surfaces by natural convection and radiation, and \((d)\) the rate at which heat and water must be supplied to maintain steady operating conditions. Disregard heat loss through the bottom surface of the bath and take the emissivities of sheet metal and water to be \(0.61\) and \(0.95\), respectively.

The average heat transfer coefficient for air flow over an odd-shaped body is to be determined by mass transfer measurements and using the Chilton-Colburn analogy between heat and mass transfer. The experiment is conducted by blowing dry air at \(1 \mathrm{~atm}\) at a free stream velocity of \(2 \mathrm{~m} / \mathrm{s}\) over a body covered with a layer of naphthalene. The surface area of the body is \(0.75 \mathrm{~m}^{2}\), and it is observed that \(100 \mathrm{~g}\) of naphthalene has sublimated in \(45 \mathrm{~min}\). During the experiment, both the body and the air were kept at \(25^{\circ} \mathrm{C}\), at which the vapor pressure and mass diffusivity of naphthalene are \(11 \mathrm{~Pa}\) and \(D_{A B}=0.61 \times\) \(10^{-5} \mathrm{~m}^{2} / \mathrm{s}\), respectively. Determine the heat transfer coefficient under the same flow conditions over the same geometry.
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