Mass Transfer in a Porous Medium: Multicomponent and Multiphase Flows
The term “mass transfer” is used here in a specialized sense, namely the transport of a substance that is involved as a component (constituent, species) in a fluid mixture. An example is the transport of salt in saline water. As we shall see below, convec
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Mass Transfer in a Porous Medium: Multicomponent and Multiphase Flows
3.1
Multicomponent Flow: Basic Concepts
The term “mass transfer” is used here in a specialized sense, namely the transport of a substance that is involved as a component (constituent, species) in a fluid mixture. An example is the transport of salt in saline water. As we shall see below, convective mass transfer is analogous to convective heat transfer. Consider a batch of fluid mixture of volume V and mass m. Let the subscript i refer to the ith component (component i) of the mixture. The total mass is equal to the sum of the individual masses mi so m ¼ Smi. Hence if the concentration of component i is defined as Ci ¼
mi ; V
(3.1)
then the aggregate density r of the mixture must be the sum of all the individual concentrations, r¼
X
Ci :
(3.2)
Clearly the unit of concentration is kg m3. Instead of Ci the alternative notation ri is appropriate if one thinks of each component spread out over the total volume V. When chemical reactions are of interest, it is convenient to work in terms of an alternative description, one involving the concept of mole. By definition, a mole is the amount of substance that contains as many molecules as there are in 12 g of carbon 12. That number of entities is 6.022 1023 (Avogadro’s constant). The molar mass of a substance is the mass of one mole of that substance. Hence if there are n moles in a mixture of molar mass M and mass m, then n¼
m : M
D.A. Nield and A. Bejan, Convection in Porous Media, DOI 10.1007/978-1-4614-5541-7_3, # Springer Science+Business Media New York 2013
(3.3)
47
48
3 Mass Transfer in a Porous Medium: Multicomponent and Multiphase Flows
Similarly the number of moles ni of component i in a mixture is the mass of that component divided by its molar mass Mi, ni ¼
mi : Mi
(3.4)
mi m
(3.5)
The mass fraction of component i is Fi ¼
so clearly SFi ¼ 1. Similarly the mole fraction of component i is xi ¼
ni n
(3.6)
and Sxi ¼ 1. To summarize, we have three alternative ways to deal with composition: a dimensional concept (concentration) and two dimensionless ratios (mass fraction and mole fraction). These quantities are related by Ci ¼ rFi ¼ r
Mi xi ; M
(3.7)
where the equivalent molar mass (M) of the mixture is given by M¼
X
Mi xi ;:
(3.8)
If, for example, the mixture can be modeled as an ideal gas, then its equation of state is PV ¼ mRm T
or PV ¼ nRT;
(3.9)
where the gas constant of the mixture (Rm) and the universal gas constant (R) are related by Rm ¼
n R R¼ : m M
(3.10)
The partial pressure Pi of component i is the pressure one would measure if component i alone were to fill the mixture volume V at the same temperature T as the mixture. Thus P i V ¼ mi R m T
or Pi V ¼ ni RT:
(3.11)
3.2 Mass Conservation in a Mixture
49
Summing these equations over i, we obtain Dalton’s law, P¼
X
Pi ;
(3.12)
which states that the pressure of a mixture of gases at a specified volume and temperature is equal to the sum of the partial pressures of the components. Note tha
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