## AH yAHTpTbi AHvap Tbj

This results in a rough estimate of the relative volatility aij, based on the boiling points only:

eWbi-Wb where J =

### AHp RTb

This gives us the Antoine coefficients: a-,. = (AHVap/R) (1/Tbi) # ln Pref, fe = -AHVap/R, c, = 0. In most cases Pref = 1 atm. For an ideal mixture that satisfies Raoult's law we have aij = po(T)/po(T) and we derive:

If we do not know AHap, a typical value p be used for many cases.

13 can

We see that the temperature dependency of the relative volatility arises from different specific heats of vaporization. For similar values (AHap + AH"ap), the expression simplifies to:

Example For methanol (L) and w-propanol (H), we have TBL = 337.8 K and TBH = 370.4 K and the heats of vaporization at their boiling points are 35.3 and 41.8kJmol~1 respectively. Thus Tb = ^337.8 • 370.4 = 354K and AHp = ^35.3 • 41.8 =38.4. This gives p = AHp/RTb = 38.4/(8.83 • 354) = 13.1 and a + e131 '326/354 + 3.34 which is a bit lower than the experimental value.

Material Balance on a Distillation Stage

Based on the equilibrium-stage concept, a distillation column section is modelled as shown in Figure 3. Figure 3 Distillation column section modelled as a set of connected equilibrium stages.

liquid composition on the stage above (xn

Figure 3 Distillation column section modelled as a set of connected equilibrium stages.

Note that we choose to number the stages starting from the bottom of the column. We denote Ln and Vn as the total liquid and vapour molar flow rates leaving stage n (and entering stages n — 1 and n + 1, respectively). We assume perfect mixing in both phases inside a stage. The mole fraction of species i in the vapour leaving the stage with Vn is yi,n, and the mole fraction in Ln is xi,n.

The material balance for component i at stage n then becomes (in mol s"1):

where Ni,n is the number of moles of component i on stage n. In the following we will consider steady-state operation, i.e: dNi>n/dt = 0.

It is convenient to define the net material flow (w) of component i upwards from stage n to n + 1 (mol s"1):

At steady state, this net flow has to be the same through all stages in a column section, i.e. wi>n = wi,n + 1 = wi.

The material flow equation is usually rewritten to relate the vapour composition (yn) on one stage to the

The resulting curve is known as the operating line. Combined with the VLE relationship (equilibrium line), this enables us to compute all the stage compositions when we known the flows in the system. This is illustrated in Figure 4, and forms the basis of McCabe-Thiele approach.

### Assumption about Constant Molar Flows

In a column section, we may very often use the assumption about constant molar flows. That is, we assume Ln = Ln+1 = L (mols"1) and Vn-1 = Vn = V(mols"1). This assumption is reasonable for ideal mixtures when the components have similar molar heats of vaporization. An important implication is that the operating line is then a straight line for a given section, i.e. y1>n = (L/V)x1>n +1 + w/V. This makes computations much simpler since the internal flows (L and V) do not depend on compositions. ## Solar Panel Basics

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