This is probably a foolish attempt, trying to discuss Piketty without knowing any traditional economics and I have read only the first seven chapters so far out of a total of sixteen. I will depend on Branko Milanovic

review for overall view of the book. But it seems better to spell out doubts so that somebody may clarify them so that eventually I can understand it better. So far, Piketty uses traditional economics at only one place in Chaper 5, to get an asymptotic (long run) estimate for the rate of return r. I will ignore this and just discuss some of the stuff I read so far in terms of identities and their relation to data.

β is C/I where C is capital and I is income. To consider the long term changes, I will denote these quantities at a given time by subscripts β_t etc and after an year by β_(t+1) and growth rate for that year by g_t etc.
So C_(t+1)=C_t+r_t*C_t and I_(t+1)=I_t+g_t*I_t (here * denotes multiplication which I will omit it later and just use juxtaposition); If the population is not stagnant, growth rate in the population is added in g_t. Dividing both sides by I_(t+1) and simplifying, we have
β_(t+1)/β_t=(1+r_t)/(1+g_t) (call this Equation A)
from which we can estimate β_(t+k)/β_0 for k __years as __ a product of successive ratios:
β_(t+k)/β_t=((1+r(t+k-1)/(1+g_(t+k-1)).....((1+r_0/(1+g_0)) (Equation B)

By rearranging we also get another identity comparing r_t ,g_t:

r_t-g_t=((β_(t+1)/β_t)-1)(1+g_t) (call this Equation C)

Piketty's observation is that this difference r-g has been generally positive except for a period 1913-1970 in the twentieth century. Generally β has been less than 8 or so. Assuming that growth rate is bigger than -1 during the period under consideration, we see from Equation C that for r_t-g_t to be positive we also should have (β_(t+1)/β_t)-1)>0 or β_(t+1)>β_t. That is, β_t increases during that period. If it does not go to infinity, we should have β_t converging to some β in the long run during the period when r_t>g_t. This convergence implies from Equation A that 1+r_t gets close to 1+g_t or that r_t and g_t are nearly equal. If we assume that g_t converges to some g, then r_t also converges to the same number g.
So the conclusion is that if β_t does not explode, whatever the growth rate is (as long as it is bigger than -1) r_t modifies to mimic the behaviour of g_t.

But what are the numbers involved? Milanovic says on page 8 of his review "... g cannot exceed 2.5% per year. If r remains, as Piketty thinks, at its historical rate of 4-5% p.a., all the negative developments from the 19th century will be repeated."
If the above 'identities' are correct. some of the current data can be checked using, for example, Identity B. It seems from this, r cannot be that high compared to g without β getting much larger. So, I am not sure what is happening.

Milanovic asks "But, the reader will ask, if the capital/output ratio increases so much,

would not the marginal return to capital diminish? Would not r go down? This is obviously a soft point of Piketty’s machinery." The above analysis says that if β is bounded above, then r has to go to the level of g. What is this bound?

Last word from Milanovic and Piketty "A remarkable graph, reproduced below, shows a huge positive gap between r and g from Antiquity to the early 20th century, its disappearance (or rather, the inversion, g>r) for the most of the 20th century, and then recent reemergence. Moreover, Piketty sees, interestingly, today’s processes of expanding financial sophistication and international competition for capital as helping keep r high. While many people question financial intermediation and blame it for the onset of the Great Recession, Piketty sees it as helping uncover new and more productive uses for financial capital and maintaining the rate of return high. But far from making this high r a good thing for the economy, he regards it, unless checked by higher taxation, as a portender of disaster..............The validity of Piketty’s “model” thus depends on the key proposition of relative stability of the rate of return on capital. But in a methodological approach that he both pioneered and clearly prefers, the answer to that question cannot be given in the abstract but only by the empirical evidence that is still in the future. In other words, we shall have to wait for the judgement of history. "
P.S. Jack Morava has kindly texed it and sent me the pdf file https://www.dropbox.com/s/m0yycewv9vlx18j/SwaruponPiketty.pdf
P.P.S. The above is wrong. Corrections here http://gaddeswarup.blogspot.com.au/2014/03/trying-again-to-understand-piketty.html

P.P.S.2

Calculations of this type I was attempting are in a paper of Thomas Piketty and Gabriel Zucman http://behl.berkeley.edu/files/2013/02/Piketty-Zucman_WP2013-10.pdf