I recommend that you use HSC Chemistry software package for this purpose. You just need to enter the reactants (propylene oxide in your case) and the anticipated reaction products. HSC will then balance the reaction in terms of number of moles and out and will calculate the thermodynamics parameters of this reaction as well as the reaction rate as a function of temperature.

Hope this information helpful to you.

Professor Yehia Khalil

Yale University

USA

Kinetic equations have been solved for difunctional additions in which reaction rates are statistically about equal for all active hydroxyls. Some initiator like acidified water must be used to start the reaction. The ratio of water to propylene oxide determines the final average molecular weight, assuming one molecule of water promotes one molecule of glycols. Reactor needs to be fairly warm to get a reaction in weak acid.

The PO addition is highly exothermic and is likely run out of control making an explosion unless a solvent is used to dilute the chemical potential and help with cooling. Mistakes with propylene oxide might or might not give you a second chance, compared to ethylene oxide which probably doesn't give a second chance. You need to get expert safety advice before using Propylene Oxide, because of possible fire and explosion.

Let me try to remember the formula for ratios of different molecular weight products, from some years ago. Assume all the water and PO reacts, then solvent is removed by boiling, taking the heat with it.   Let f1, f2 and f3 be the mole fractions of mono. di, and tri propylene glycols in a series where f0 is water. Note that some larger molecules are likely if not enough water was used. The mole fractions follow closely to an exponential Maclaurin series expansion based on powers of [-Ln(f1)]. It is the solution of an infinite number of differential equations in series where the rate constants are equal per active hydroxyl group, but reactions are not usually reversible.

Notice that 

e[-Ln(f1)] = 1/f1             and

 ex = 1 + x + x2/2! + x3/3! + ... + xn/n! + ...               then

f2/f1 =  [-Ln(f1)]             and 

f3/f1 =  [-Ln(f1)]2 / 2!           the answers to your question

but there are higher molecular weights in the series, maybe significant amounts if your starting ratios are wrong, or mixing is not good.

The mole fractions add to one.

1 =  f1 + f2 + f3 ...+ fn + ....

1 = f1 + f1[Ln(f1)]  + f1[-Ln(f1)]2 / 2!   + f1[Ln(f1)](n-1)/(n-1)! + ...

Giving your answers from the exponential series.

Ratios of water and PO moles can also be given in series. Product yield is one mole of product per mole of water reacting, hence a refluxing solvent conserves water in the reactor. Moles of PO per mole of product are given.

p = f1 + 2f2 + 3f3 + ... + nfn + ...

p = f1 + 2f1[-Ln(f1)]  + 3f1[-Ln(f1)]2 / 2!   + nf1[-Ln(f1)](n-1)/(n-1)! + ...

As always the actual result differs a bit from the ideal especially in higher molecular weights, but the ideal gives a way to estimate results and select feed ratios.

Notice you can get exactly the same products in the same ratios from reacting acetone in the aldol reaction, but at more severe conditions of temperature and catalyst. Acetone also has potential for strong exothermic polymerization leading to explosion. With acetone there is a series of intermediates called polyacetyls which finally decompose or make glycols unless stabilized by a capping agent. 

If you choose acetone as your solvent then it is not inert, but probably does not permanently alter the product mix unless you have impurities that cap the acetyls.

 PO is added drop wise and only as quickly as its reaction heat Is carried away by reflux condenser.

In no case should PO be allowed to accumulate in the reactor. Also the reactor contents should be prevented from flowing backward  into a PO reservoir.

With these systems start in small scale and protected equipment. Verify the exotherm before proceeding. First get safety advice. Stop when in doubt. Protect people and property from damage.

If you want all mono glycol, water is your reflux solvent.

I pulled a book off the shelf and found additional information written in the margin of one page. If A is the molecular weight of water and B is the molecular weight of PO, the average molecular weight is given after all water and PO are reacted or removed.

W = (A+B) + B[-Ln(f1)]

These results and those in the previous answer only apply when each molecule has 2 and only 2 active hydroxyls per molecule. There is a different margin in the book for other cases.

PO groups per mole average are calculated.

(W-A) /B =  1+ [-Ln(f1)]

From equations you can see it is fairly easy to avoid high molecular weights.

Be careful and make preparations for operations and abnormal conditions.

Here is a clue to how the average molecular weight is derived from an infinite series.

W = (A+B)f1 + (A+2B)f2 + (A+3B)f3 +  ... + (A+nB)fn + ...

Make the rearrangement to get the exponential series again.

(W - A - B)/{ Bf1[-Ln(f1)] } = 1 + [-Ln(f1)] + [-Ln(f1)]2/2! + ... + [-Ln(f1)]n/n!  + ...  =  1/f1

Accuracies are usually good for smaller molecules but start to deviate from lab results for larger molecules where the hydroxyl groups may be hidden inside a folded molecule. The result is a bit less production of components 4, 5, and 6 than predicted by the infinite series solutions.

Before completion the reaction is approximately second order and not reversible at normal conditions. It is nearly first order because the number of hydroxyl groups is not changing much, especially when nearly all production is of the first few small molecules. Reaction normally requires a dilute mild catalyst, acid base or salt and does not proceed in a cold reactor. Reaction rate increases with temperature beginning when the reactor is just warm. Reaction is highly exothermic and can quickly get out of control.

All of my comments are for liquid reactions with heat carried away into a reflux condenser.

Notice that abnormal conditions can make solid products that plug equipment and might tempt you to bypass safety systems and procedures. Such mistakes always end in disaster with published histories.

You might think of vapor reaction on solid catalyst at mild conditions after which the products might condense, tending to favor the small molecules more than my calculations.

In that case you need different math series solutions and appropriate safety systems.

Possibly my final comment on this question, liquid reaction rate per volume tends to decrease in some cases as high molecular weight products are produced due to the reaction volume increasing and the number of active hydroxyls not increasing. 

Notice that some processes may allow propylene glycols to be used as initiators instead of water. A similar math occurs in series solutions but the constant A is changed to the average molecular weight of the starting glycols. A safety advantage may be gained since part of the heat of reaction has already been removed in a previous reaction to make the starting glycols. Also the mix of products can be adjusted somewhat to match market demands.

Larger molecules have linkages of ether or poly ether that can be hydrolyzed to make propylene glycol in a different reactor with hot water and catalyst

Three reactions, five reagents ... It's impossible.

Regards.

ASPEN PLUS can do this calculation. It needs the appropriate data sets. You might need to declare the reaction rate, or add the catalyst reactions to the chemistry section. Both the kinetic reactor module and the Gibbs equilibrium model are useful for different reasons of production and safety.

Originally I did the calculations by hand, but with a research laboratory, pilot plant, and supporting staff of specialist. Hand calculation helps validate the ASPEN model which might otherwise be wrong from an inhibitor, a missing parameter, or a hidden symmetry. The infinite series solutions make a hand calculation possible and more accurate than simply ignoring the larger molecules. The laboratory helps to modify the series solutions, but also might prevent publishing of a more detailed result.

Infinite series of differential equations might sound a bit scary. In fact you only have to work with one term at a time. Terms of an infinite series are often simpler than an equivalent finite function. Only a few hundred mathematical problems can be solved with finite functions. Infinite series can answer a great many more technical questions.

I do not understand the discussion and answers the question.

Apart from particular situations, such as for instance large differences in reaction rates, on the basis of data provided, it is impossible to determine the degree of conversion of propylene oxide and selectivities.

You can only determine the value of the molar flow rate of the substrates ("before") and the sum of the molar flow rates of the reactants (oxide, water, mono, di and tri) ("after") using the ideal gas law.

To calculate "conversion of propylene oxide to mono,di,tri propylene glycol"  must be determined experimentally molar flow rate of the three reactants ("after"), and then can be used:

FnOo - FnO = FnM + 2FnD

FnWo - FnW = FnM + FnD - FnT

Fn - molar flow rate, O - oxide, W - water, M - mono, D - di, T - tree, o - initial ("before")

That all.

Regards.

Ard, My answer was based on the data provided. At this stage it is only stoichiometry and therefore the reaction scheme, conversions, selectivities, stoichiometric invariants. But the lack of relevant data excludes solution.

I do have a problem with changing the total pressure. Use of the term "flow rate" suggests the flow system (fixed-bed, fluidized bed and so on). But in such system the total pressure does not change. Change of pressure is characteristic for batch system. It is also unclear whether the term flow rate refers to the volume or the number of moles. But this is only digressions - nothing more.:

Ard, Do sorry, I forgot to add "kind regards".  So now

Kind regards.