The CRC Handbook of Mechanical Engineering. Chapter 4. Heat and Mass Transfer (776127), страница 51
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The number density of species i in a mixture or solution of n species is defined asNumber density of species i º Number of molecules of i per unit volume(4.7.1)º N i molecules m 3Alternatively, if the total number of molecules of all species per unit volume is denoted as N , then wedefine the number fraction of species i asni ºNi;NN =åNi(4.7.2)where the summation is over all species present, i = 1,2,…,n. Equations (4.7.1) and (4.7.2) describemicroscopic concepts and are used, for example, when the kinetic theory of gases is used to describetransfer processes.© 1999 by CRC Press LLC4-207Heat and Mass TransferWhenever possible, it is more convenient to treat matter as a continuum. Then the smallest volumeconsidered is sufficiently large for macroscopic properties such as pressure and temperature to have theirusual meanings. For this purpose we also require macroscopic definitions of concentration.
First, on amass basis,Mass concentration of species i º partial density of species iº ri kg m 3(4.7.3)The total mass concentration is the total mass per unit volume, that is, the density r = åri. The massfraction of species i is defined asrirmi =(4.7.4)Second, on a molar basis,Molar concentration of species i º number of moles of i per unit volumeº ci kmol m 3(4.7.5)If Mi (kg/kmol) is the molecular weight of species i, thenriMici =(4.7.6)The total molar concentration is the molar density c = åci. The mole fraction of species i is defined ascicxi º(4.7.7)A number of important relations follow directly from these definitions. The mean molecular weightof the mixture of solution is denoted M and may be expressed asM=r=cåx Mii(4.7.8a)or1=MmiåM(4.7.8b)iThere are summation rulesåm = 1(4.7.9a)åx =1(4.7.9b)ii© 1999 by CRC Press LLC4-208Section 4It is often necessary to have the mass fraction of species i expressed explicitly in terms of mole fractionsand molecular weights; this relation ismi =x i MiåxjMj= xiMiM(4.7.10a)and the corresponding relation for the mole fraction isxi =mi MiåmjMj= miMMi(4.7.10b)Dalton’s law of partial pressures for an ideal gas mixture states thatP=åP,iwhere Pi = ri Ri T(4.7.11)Dividing partial pressure by total pressure and substituting Ri = R/Mi givesPir RTcR TRT= i= ci= xi= xiP Mi PPP(4.7.12)Thus, for an ideal gas mixture, the mole fraction and partial pressure are equivalent measures ofconcentration (as also is the number fraction).A commonly used specification of the composition of dry air is 78.1% N2, 20.9% O2, and 0.9% Ar,by volume.
(The next largest component is CO2, at 0.3%.) Since equal volumes of gases contain thesame number of moles, specifying composition on a volume basis is equivalent to specifying molefractions, namely,x N 2 = 0.781;x O2 = 0.209;x Ar = 0.009The corresponding mass fractions are calculated to bemN 2 = 0.755;mO2 = 0.231;mAr = 0.014Concentrations at InterfacesAlthough temperature is continuous across a phase interface, concentrations are usually discontinuous.In order to define clearly concentrations at interfaces, we introduce imaginary surfaces, denoted u ands, on both sides of the real interface, each indefinitely close to the interface, as shown in Figure 4.7.1for water evaporating into an airstream.
Thus, the liquid-phase quantities at the interface are subscriptedu, and gas-phase quantities are subscripted s. If we ignore the small amount of air dissolved in the water,x H2O,u = 1. Notice that the subscript preceding the comma denotes the chemical species, and the subscriptfollowing the comma denotes location. To determine x H2O,s we make use of the fact that, except inextreme circumstances, the water vapor and air mixture at the s-surface must be in thermodynamicequilibrium with water at the u-surface.
Equilibrium data for this system are found in conventional steamtables: the saturation vapor pressure of steam at the water temperature, Ts, (Ts = Tu), is the required partialpressure PH2O,s . With the total pressure P known, x H2O,s is calculated as PH2O,s / P. If mH2O,s is required,Equation (4.7.10a) is used.© 1999 by CRC Press LLC4-209Heat and Mass TransferFIGURE 4.7.1 Concentrations at a water-air interface.For example, at Ts = 320 K, the saturation vapor pressure is obtained from steam tables as 0.10535´ 105 Pa.
If the total pressure is 1 atm = 1.0133 ´ 105,x H2O,s =mH2O,s =0.10535 ´ 10 5= 0.10401.0133 ´ 10 5(0.1040)(18)= 0.06720(0.1040)(18) + (1 - 0.1040)(29)For a gas or solid dissolving in a liquid, equilibrium data are often referred to simply as solubilitydata, found in chemistry handbooks. Many gases are only sparingly soluble, and for such dilute solutionssolubility data are conveniently represented by Henry’s law, which states that the mole fraction of thegas at the s-surface is proportional to its mole fraction in solution at the u-surface, the constant ofproportionality being the Henry number, He. For species i,xi,s = He i xi,u(4.7.13)The Henry number is inversely proportional to total pressure and is also a function of temperature.
Theproduct of Henry number and total pressure is the Henry constant, CHe, and for a given species is afunction of temperature only:He i P = CHei (T )(4.7.14)Solubility data are given in Table 4.7.1.TABLE 4.7.1 Henry Constants CHe for Dilute Aqueous Solutions at ModeratePressures (Pi,s /xi,u in atm, or in bar = 105 Pa, within the accuracy of the data).Solute290 K300 K310 K320 K330 K340 KH2SCO2O2H2COAirN24401,28038,00067,00051,00062,00016,0005601,71045,00072,00060,00074,00089,0007002,17052,00075,00067,00084,000101,0008302,72057,00076,00074,00092,000110,0009803,22061,00077,00080,00099,000118,0001,140—65,00076,00084,000104,000124,000© 1999 by CRC Press LLC4-210Section 4For example, consider absorption of carbon dioxide from a stream of pure CO2 at 2 bar pressure intowater at 310 K.
From Table 4.7.1, CHe = 2170 bar; thusHe CO =22170= 1085;2x CO2,u =1= 9.22 ´ 10 -41085Dissolution of gases into metals is characterized by varied and rather complex interface conditions.Provided temperatures are sufficiently high, hydrogen dissolution is reversible (similar to CO2 absorptioninto water); hence, for example, titanium-hydrogen solutions can exist only in contact with a gaseoushydrogen atmosphere. As a result of hydrogen going into solution in atomic form, there is a characteristicsquare root relationmH2 ,u aPH122,sThe constant of proportionality is strongly dependent on temperature, as well as on the particular titaniumalloy: for Ti-6Al-4V alloy it is twice that for pure titanium. In contrast to hydrogen, oxygen dissolutionin titanium is irreversible and is complicated by the simultaneous formation of a rutile (TiO2) scale onthe surface. Provided some oxygen is present in the gas phase, the titanium-oxygen phase diagram(found in a metallurgy handbook) shows that mO2 ,u in alpha-titanium is 0.143, a value essentiallyindependent of temperature and O2 partial pressure.
Dissolution of oxygen in zirconium alloys has similarcharacteristics to those discussed above for titanium.All the preceding examples of interface concentrations are situations where thermodynamic equilibrium can be assumed to exist at the interface. Sometimes thermodynamic equilibrium does not exist atan interface: a very common example is when a chemical reaction occurs at the interface, and temperatures are not high enough for equilibrium to be attained. Then the concentrations of the reactants andproducts at the s-surface are dependent both on the rate at which the reaction proceeds — that is, thechemical kinetics — as well as on mass transfer considerations.Definitions of Fluxes and VelocitiesThe mass (or molar) flux of species i is a vector quantity giving the mass (or moles) of species i thatpass per unit time through a unit area perpendicular to the vector (Figure 4.7.2). We denote the absolutemass and molar fluxes of species i, that is, relative to stationary coordinate axes, as ni (kg/m2sec) andNi (kmol/m2sec), respectively.
The absolute mass flux of the mixture (mass velocity) isn=ånand the local mass-average velocity isFIGURE 4.7.2 Flux vectors: (a) mass basis, (b) molar basis.© 1999 by CRC Press LLCi(4.7.15)4-211Heat and Mass Transferv=nm secr(4.7.16)The velocity n is the velocity that would be measure by a Pitot tube and corresponds to the velocityused in considering pure fluids. On a molar basis, the absolute molar flux of the mixture isåN(4.7.17)Nm secc(4.7.18)N=iand the local molar-average velocity isv* =The absolute fluxes at species i have two components.
On a mass basis we write(4.7.19)ni = riv + jiwhere ri n is transport of species i by bulk motion of the fluid at velocity n and is the convectivecomponent. Thus, ji is transport of species i relative to the mass average velocity; it is called the diffusivecomponent because most commonly it is due to ordinary (concentration) diffusion of the species. On amolar basis the corresponding relation isNi = ci v * + Ji*(4.7.20)Some important relations follow from these definitions:å j = åJ*iiNi =(4.7.21)=0niMi(4.7.22)ni = ri v + ji = miån + jNi = ci v * + Ji* = xiåN + Jii(4.7.23a)i*i(4.7.23b)Mechanisms of DiffusionOrdinary DiffusionFick’s law of ordinary diffusion is a linear relation between the rate of diffusion of a chemical speciesand the local concentration gradient of that species.
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