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3-11 Chapter 3 The mass transfer rate, Amɺ, of species A across the interfacial area for mass transfer Ai is given by Amɺ = (Area for mass transfer)(mass transfer coefficient)(driving force) The driving force for mass transfer can be expressed in many different ways. It could be based on the mole or mass fraction in the gas, or liquid phase, or both. The mass transfer rate for mass transfer from the liquid to the gas phase can be written as Amɺ = Aiky(yAi − yA) = Aikx(xA − xAi) (3.2-5) In this expression, ky and kx are the individual mass transfer coefficients based on the gas and the liquid phase, respectively. The mole fractions yAi, yA, xA, and xAi are defined in Figure 3.2-1. The mass transfer rate for mass transfer from the gas to the liquid phase can be written as Amɺ = Aiky(yA − yAi) = Aikx(xAi − xA) (3.2-6) If we do not know the direction of mass transfer, we could use either equation (3.2-5) or equation (3.2-6). If the mass transfer rate calculated to be positive, our assumption of the mass transfer direction is correct. For example, if we use equation (3.2-5) and Amɺ is positive, then species A is being transferred from the liquid to the gas phase. Let a be the interfacial area per unit volume of packing (m2/m3). Multiplying both sides of equation (3.2-5) by a, we obtain Amɺa = Aiaky(yAi − yA) ⇒ aAmiA/ɺ = aky(yAi − yA) (3.2-7) Ai/a = (Interfacial area)/(Interfacial area/volume of packing) = Vpack = Volume of packing. Equation (3.2-7) can be written as ApackmVɺ = aky(yAi − yA) (3.2-8) Since, the interfacial concentrations are difficult to measure, the mass transfer rate is usually written in terms of the overall mass transfer coefficient that is based on the overall driving force for mass transfer (yA − yA*) or (xA* − xA) as shown in Figure 3.2-2.3-12 EquilibriumdataxyyAyAiy *AxAxAix *Am2m1 Figure 3.2-2 Concentration driving forces in interphase mass transfer. For mass transfer from the gas to the liquid phase (as shown in Figure 3.2-2) ApackmVɺ = aKy(yA − yA*) = aKx(xA* − xA) (3.2-9a) For mass transfer from the liquid to the gas phase ApackmVɺ = aKy(yA* − yA) = aKx(xA − xA*) (3.2-9b) The term yaK1 is the total resistance to mass transfer based on the gas phase, and xaK1the total resistance to mass transfer based on the liquid phase. We have neglected the resistance to mass transfer at the interface. In Figure 3.2-2, m1 is the average slope of the equilibrium curve between two points (xA, yA*) and (xAi, yAi), m2 the average slope of the equilibrium curve between two points (xAi, yAi) and (xA*, yA). The relation between the overall and the individual mass transfer coefficients can be derived as follows: yA − yA*= (yA − yAi) + (yAi − yA*) Since yA* = m xA and yAi = m xAi, we have yA − yA*= (yA − yAi) + m(xAi − xA) ⇒ AyNK = AyNk + mAxNk 1yK = 1yk + xmk (3.2-10a) Similar derivation gives3-13 zyA,outyA,inV|z+dzV|zL|z+dzL|zdz 1xK = 1ymk + 1xk (3.2-10b) We now want to determine the height of a packed bed required to change the concentration of the inlet gas from yA,in to yA,out in a distillation column. We will assume constant molar overflow so that the vapor molar flow rate, V, and the liquid molar flow rate, L, are constant over the height of the packed column. Let Ac be the cross-sectional area of the column, the material balance over the differential volume Acdz gives xAL|z+dz + yAV|z = xAL|z + yAV|z+dz Figure 3.2-3 Material balance over Acdz Rearranging the liquid and vapor flow rates gives xAL|z+dz − xAL|z = yAV|z+dz − yAV|z Dividing the equation by dz and letting the control volume Acdz approach zero, we have d(LxA) = d(VyA) For constant L and V Ld(xA) = Vd(yA) The molar flux of A across the interface is NA = Ky(yA − yA*) Multiplying the expression by aAcdz gives NAaAcdz = Ky(yA − yA*)aAcdz Since NAAiadz = ))(( timeareatransferMassvolumeareavolume = rate of A transfer from volume Acdz, we have Ky(yA − yA*)aAcdz = Vd(yA) Solving for dz gives dz = AAAydyyyaKV*)( −3-14 Integrating over the height of the packed bed we have h = ∫hdz0 = ∫−outAinAyyAAAcydyyyaAKV,,*)( (3.2-11) The height of an overall gas transfer unit, HOG, is defined as HOG = cyaAKV If HOG is a constant, equation (3.2-11) becomes h = cyaAKV∫−outAinAyyAAAyydy,,*)( (3.2-12) For distillation column where species A is transferred from the liquid to the gas phase, h is given by h = cyaAKV∫−outAinAyyAAAyydy,,)*( = HOGnOG (3.2-13) In this expression, nOG is the number of overall gas transfer unit. If the driving force is based on the driving force in the liquid phase NAaAcdz = Kx(xA − xA*)aAcdz = Ld(xA) (3.2-14) The height of the packed bed is then given by h = cxaAKL∫−outAinAxxAAAxxdx,,*)( = HOLnOL (3.2-15) HOL is the height of an overall liquid transfer unit and nOL the number of overall liquid transfer unit. We now need to evaluate the integral ∫−outAinAyyAAAyydy,,)*( or ∫−outAinAxxAAAxxdx,,*)(.3-15 zyA,outyA,inV LyAxAxA,inxA,outxAyAy *- yA AEquilibrium curveOperating lineyA,outyA,inxA Figure 3.2-4 Material balance over the lower section of the tower. Assuming L and V are constant and making an A balance over the lower section of the tower as shown in Figure 3.2-4 we have xA,outL + yAV = xAL + yA,inV Solving for yA we obtain an equation called the operating line yA = VLxA + yA,in − VLxA,out (3.2-16) At any location in the packed bed the bulk concentration in the vapor and liquid phase are yA and xA respectively. The point (xA, yA) is on the operating line shown in Figure 3.2-4. The number of transfer unit nOG or the integral ∫−outAinAyyAAAyydy,,)*(can be evaluated where (yA* − yA) is the difference between the concentration yA* that is in equilibrium with the liquid at xA to the vapor concentration at the point (xA, yA). If the operating line and equilibrium curves are straight, nOG can be evaluated analytically. This condition might occur in gas absorption with dilute solution where the equilibrium curve is straight and L and G are constant. We will drop the subscript A with the understanding that y and x are the mole fractions of the diffusing species in the gas and liquid phase, respectively. For gas absorption, we use G for the gas flow rate instead of V for the vapor flow used in distillation. Making a material


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Cal Poly Pomona CHE 313 - Chapter 3

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