pt flash handout 2010

CHEE 311 Flash Calculations Handout for non-ideal mixtures 1

VLE Flash Calculations for Non-Ideal Systems

We introduced flash calculations for ideal systems (Raoult’s Law) previously. With our revised models of chemical potential, we are now able to handle non-ideal systems quite accurately.

The basic P,T-flash problem:Given: P,T, z1,z2,…, zn Find: V,L, x1, x2,…xn, y1, y2,…, yn

Use a flash calculation whenever the overall composition of the system is known, but the composition of each phase is not.

Feedz1z2z3=1-z1-z2Tf, Pf

P,T

Vapoury1y2y3=1-y1-y2Liquidx1x2x3=1-x1-x2


VLE Flash Calculations from a Phase Diagram

To the right is the Txy diagram for thehighly non-ideal system of Ethanol(1)-Toluene(2) at P=1 atm.

1. Given a feed stream containing 25%ethanol, between what temperaturesdo we have two phases?

2. At 90°C, what are the compositionsof the liquid and vapour streams?

3. Under these conditions, what fractionof the system exists as a vapour?


VLE Flash EquationWhen a phase diagram is not at hand, a flash calculation using amodel for the phase behaviour is required.

Whenever confronted with a flash problem, apply one of the general flash equations:

(10.17)

or

(14.16)

This is the most versatile approach to solving flash problemsIn all but simple P,T flashes for binary systems, the general flash equation will produce the quickest answer.

1)1K(V1

Kzi i

ii =−+

∑

1)1K(V1

zi i

i =−+

∑


Partition Coefficient, Ki

The partition coefficient, Ki = yi / xi, is used to simplify the general flash equation.

It reflects the tendency of a component to vapourize. Those components with a large partition coefficient (Ki >1) concentrate in the vapour, while those with Ki <1 concentrate in the liquid phase.

The partition coefficient in a VLE system is provided by our phase equilibrium expression (derived from equivalence of chemical potential). Recall,

where,

Therefore,

Note that for a non-ideal system, Ki is a function of P,T and the compositions of both the liquid and vapour phase.

satiiiii PxPy γ=Φ

PP

xyK

i

satii

i

ii Φ

γ==

⎥⎦

⎤⎢⎣

⎡ −−φφ

=ΦRT

)PP(Vexpˆ sat

ili

sati

vi

i


Solving Non-Ideal Flash ProblemsThe “Classic” P,T-flash problems involves:Given: P,T, z1,z2,…, zn Find: V,L, x1, x2,…xn, y1, y2,…, yn

For a three component system, the VLE flash equation is:

14.16

Or, substituting for the partition coefficients:

The general solution involves:Find the vapour phase fraction (V≠0) that satisfies 14.16.Substitute V into:

and solve for yi using:

1)1

PP(V1

z

)1P

P(V1

z

)1P

P(V1

z

3

sat33

3

2

sat22

2

1

sat11

1 =−

Φγ

++

−Φγ

++

−Φγ

+

1)1K(V1

z)1K(V1

z)1K(V1

z3

3

2

2

1

1 =−+

+−+

+−+

)1K(V1zx

i

ii −+=

iii xKy =


Solving Non-Ideal Flash ProblemsThe non-ideal flash equation requires knowledge of the vapour and liquid compositions to evaluate Φi and γi, respectively.

These are the unknowns that we are attempting to calculateTherefore, flash calculations always require iteration.

Suppose you are given P,T, z1, z2, z3 and you are asked to find V, and the phase compositions.

1. Calculate the DEWP and BUBLP of the feed at the given temperature to ensure that two phases exist.

2. Use Raoult’s law to simplify the flash problem to the degree that it can (easily?) be solved.

This involves setting γi = 1 and Φi =1 for all componentsCalculate Pi

sat for each component at the given TSolve for V using the flash equation, 14.16Solve for xi, and yi

These are ESTIMATES that serve only to get us started!


Solving Non-Ideal Flash Problems3. Using the latest estimate of xi and yi, along with the P,T given, calculate:

γi for each component using an activity coefficient modelΦi for each component using an equation of state for the vapour.Calculate Ki = γiPi

sat / ΦiP for each component

4. Using the revised partition coefficients, calculate V through a trial and error procedure on the general flash calculation.

5. Recalculate xi,and yi for each component.

6. Repeat steps 3 through 5 until the solution converges.

1)1K(V1

zi i

i =−+

∑

)1K(V1zx

i

ii −+= iii xKy =


Non-Ideal VLE CalculationsThe Pxy diagram to the rightis for the non-ideal system ofchloroform-dioxane.

Note the P-x1 line representsa saturated liquid, and is commonly BUBL LINEreferred to as the bubble-line.

P-y1 represents a saturatedvapour, and is referred to as thedew line (the point where a liquid DEW LINEphase is incipient).

∑Φγ

=i i

satiii

BUBLPxP

∑ γΦ

=

isatii

iiDEW

Py

1P


What is Φi?

As mentioned previously:

What is and how can it be calculated?- This symbol stands for the “solution fugacity coefficient of component “i” in a non-ideal vapour mixture”- Solution fugacity coefficients can be calculated as shown in Lectures 11&12

How about ?- This is a pure-species property, called “the fugacity coefficient of i at saturation”; i.e., at P= Psat(T) for a given T- Examples illustrating its calculation were given in Lecture 9

As for the exponential term, it is called the “pointing factor” (see Lecture 10)

⎥⎦

⎤⎢⎣

⎡ −−φφ

=ΦRT

)PP(Vexpˆ sat

ili

sati

vi

i

viφ̂

satiφ


Non-Ideal BUBL P Calculations

The simplest VLE calculation of the five is the bubble-point pressure calculation.

Given: T, x1, x2,…, xn Calculate P, y1, y2,…, yn

To find P, we start with a material balance on the vapour phase:

Our equilibrium relationship provides:

(14.8)

which yields the Bubble Line equation when substituted into the material balance:

or(14.10)

∑ ==

=

ni

1ii 1y

PPxyi

satiii

i Φγ

=

∑Φγ

=i i

satiii PxP

∑Φγ

==∑i i

satiii

ii P

Px1y


Non-Ideal BUBL P Calculations

Non-ideal BUBL P calculations are complicated by the dependence of our coefficients on pressure and composition.

Given: T, x1, x2,…, xn Calculate P, y1, y2,…, yn

To apply the Bubble Line Equation:

requires:?√√

Therefore, the procedure is:calculate Pi

sat, and γi from the information providedassume Φi=1, calculate an approximate PBUBLuse this estimate to calculate an approximate Φirepeat PBUBL and Φi calculations until solution converges.

)T(PP

)x,...,x,x,T()y,...,y,y,P,T(

sati

sati

n21i

n21i

=

γ=γΦ=Φ

∑Φγ

=i i

satiii PxP


Non-Ideal Dew P Calculations

The dew point pressure of a vapour is that pressure which the mixture generates an infinitesimal amount of liquid. The basic calculation is:

Given: T, y1, y2,…, yn Calculate P, x1, x2,…, xn

To solve for P, we use a material balance on the liquid phase:

Our equilibrium relationship provides:

(14.9)

From which the Dew Line expression needed to calculate P is generated:

(14.11)

∑ ==

=

ni

1ii 1x

satii

iii P

PyxγΦ

=

∑γ

Φ=

isatii

iiP

y1P


Non-Ideal Dew P Calculations

In trying to solve this equation, we encounter difficulties in estimating thermodynamic parameters.

Given: T, y1, y2,…, yn Calculate P, x1, x2,…, xn

??√

While the vapour pressures can be calculated, the unknown pressure is required to calculate Φi, and the liquid composition is needed to determine γi

Assume both parameters equal one as a first estimate, calculate P and xiUsing these estimates, calculate ΦiRefine the estimate of xi and estimate γi ((12.10ab) Refine the estimate of PIterate until pressure and composition converges.

)T(PP

)x,...,x,x,T()y,...,y,y,P,T(

sati

sati

n21i

n21i

=

γ=γΦ=Φ

∑γ

Φ=

isatii

iiP

y1P


Non-Ideal Bubble and Dew T CalculationsThe Txy diagram to the rightis for the non-ideal system ofethanol(1)/toluene(2) at P =1atm.

Note the T-x1 line representsa saturated liquid, and is commonly DEW LINEreferred to as the bubble-line.

T-y1 represents a saturatedvapour, and is referred to as thedew line (the point where a liquidphase is incipient).

BUBL LINE

∑Φγ

=i i

satiii

BUBLPxP

∑ γΦ

=

isatii

iiDEW

Py

1P


Non-Ideal BUBL T Calculations

Bubble point temperature calculations are among the more complicated VLE problems:

Given: P, x1, x2,…, xn Calculate T, y1, y2,…, yn

To solve problems of this sort, we use the Bubble Line equation:

14.10

The difficulty in determining non-ideal bubble temperatures is in calculating the thermodynamic properties Pi

sat, Φi, and γi.

Since we have no knowledge of the temperature, none of these properties can be determined before seeking an iterative solution.

∑Φγ

=i i

satiii PxP

)T(PP

)x,...,x,x,T()y,...,y,y,P,T(

sati

sati

n21i

n21i

=

γ=γΦ=Φ


Non-Ideal BUBL T Calculations: Procedure

1. Estimate the BUBL T Use Antoine’s equation to calculate the saturation temperature (Ti

sat) for each component at the given pressure:

Use TBUBL = Σ xi Tisat as a starting point

2. Using this estimated temperature and xi’s calculate

Pisat from Antoine’s equation

Activity coefficients from an Excess Gibbs Energy Model (Margule’s, Wilson’s, NRTL)

Note that these values are approximate, as we are using a crude temperature estimate.

lnln

sat sati ii i i i

i i

B BP A T CT C A P

= − ⇒ = −+ −



3. Estimate Φi for each component.We now have estimates of T, Pi

sat and γi, but no knowledge of Φi. Assume that Φi=1 and calculate yi

’s using:

14.8

Plug P, T, and the estimates of yi’s into your fugacity

coefficient expression to estimate Φi.

Substitute theseΦi estimates into 14.9 to recalculate yi and continue this procedure until the problem converges.

Step 3 provides an estimate of Φi that is based on the best T, Pisat,

γi, and xi data that is available at this stage of the calculation.If you assume that the vapour phase is a perfect gas mixture, all Φi =1.

PPxyi

satiii

i Φγ

=



4. Our goal is to find the temperature that satisfies our bubble point equation:

(14.10)

Our estimates of T, Pisat, γi and Φi, are approximate since they are

based on a crude temperature estimate (T = Σ xi Tisat)

Calculate P using the Bubble Line equation (12.11)» If Pcalc < Pgiven then increase T» If Pcalc > Pgiven then decrease T» If Pcalc = Pgiven then T = TBUBL

The simplest method of finding TBUBL is a trial and error method using a spreadsheet.

Follow steps 1 to 4 to find Pcalc.Change T and repeat steps 2, 3, and 4 until Pcalc = Pgiven

∑Φγ

=i i

satiii PxP


Non-Ideal DEW T Calculations

The dew point temperature of a vapour is that which generates an infinitesimal amount of liquid.

Given: P, y1, y2,…, yn Calculate T, x1, x2,…, xn

To solve these problems, use the Dew Line equation:

14.11

Once again, we haven’t sufficient information to calculate the required thermodynamic parameters.

Without T and xi’s, we cannot determine Φi, γi or Pi

sat.

∑γ

Φ=

isatii

iiP

y1P

)T(PP

)x,...,x,x,T()y,...,y,y,P,T(

sati

sati

n21i

n21i

=

γ=γΦ=Φ


Non-Ideal DEW T Calculations: Procedure

1. Estimate the DEW T Using P, calculate Ti

sat from Antoine’s equation

Calculate T = Σ yi Tisat as a starting point

2. Using this temperature estimate and yi’s, calculate

Pisat from Antoine’s equation

Φi using the virial equation of state

Note that these values are approximate, as we are using a crude temperature estimate.

ii

isati C

PlnABT −−

=



3. Estimate γi, for each component Without liquid composition data, you cannot calculate activity coefficients using excess Gibbs energy models.A. Set γi=1

B. Calculate the Dew Pressure:

C. Calculate xi estimates from the equilibrium relationship:

D. Plug P,T, and these xi’s into your activity coefficient model to

estimate γi for each component.

E. Substitute these γi estimates back into 12.12 and repeat B through D until the problem converges.

satii

iii P

PyxγΦ

=

∑γ

Φ=

isatii

iiP

y1P



4. Our goal is to find the temperature that satisfies our Dew Line equation:

(14.11)

Our estimates of T, Pisat, γi and Φi, are based on an approximate

temperature (T = Σ xi Tisat) we know is incorrect.

Calculate P using the Bubble Line equation (14.10)» If Pcalc < Pgiven then increase T» If Pcalc > Pgiven then decrease T» If Pcalc = Pgiven then T = TDew

The simplest method of finding TDew is a trial and error method using a spreadsheet.

Follow steps 1 to 4 to find Pcalc.Change T and repeat steps 2, 3, and 4 until Pcalc = Pgiven

∑γ

Φ=

isatii

iiP

y1P


9.3 Modified Raoult’s LawAt low to moderate pressures, the vapour-liquid equilibrium equation can be simplified considerably.

Consider the vapour phase coefficient, Φi:

Taking the Poynting factor as one, this quantity is the ratio of two vapour phase properties:

Fugacity coefficient of species i in the mixture at T, PFugacity coefficient of pure species i at T, Pi

sat

If we assume the vapour phase is a perfect gas mixture, this ratio reduces to one, and our equilibrium expression becomes,

or

⎥⎦

⎤⎢⎣

⎡ −−φφ

=ΦRT

)PP(Vexpˆ sat

ili

sati

vi

i

satiiii

satiiiii

PxPy

PxPy

γ=

γ=Φ1


Modified Raoult’s LawUsing this approximation of the non-ideal VLE equation simplifies phase equilibrium calculations significantly.

Bubble Points:

Setting Φi =1makes BUBL P calculations very straightforward.

Dew Points:

∑γ

=

isatii

iP

y1P

∑Φγ

=i i

satiii PxP ∑ γ=

i

satiii PxP

∑γ

Φ=

isatii

iiP

y1P

pt flash handout 2010

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