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So, from that, we can actually derive this without much difficulty. The numerator actually becomes C p times U into C p times C w delta C w. And then, delta C w we express in terms of the angles and so on. So, it is a very simple two to three step derivation from which we can derive this long, rather longest expression, which we see here for total-to-static efficiency. Now, in this expression, we still have an unknown that is T 3 by T 2. T 2 we have already calculated, but we do not know the value of T 3, which also can be. In fact calculate, provided we know the exit stagnation temperature and the exit absolute velocity. So, if that is known, we can actually, we should be able to calculate because in this case, beta 3 is given and from the velocity triangle, since beta 3 is given, we can. And, U 3 is known, we can calculate C 3. What about T naught 3? For T naught 3, the power output is given, inlet stagnation temperature is given. So, we can calculate T naught 3 from there. T naught 3 minus C 3 square by 2 C p will give us T 3. And then, we can take ratio T 3 by T 2. That would be deriving the whole thing from the first principle or the whole thing can be expressed in a single definition term, which also I think I had mentioned in the last class. Which is basically, in terms of some of these parameters, which we know and T 3 by T 2 is actually defined terms of the velocities and loss coefficients. 1 minus U 2 square by 2 C p, T 2 into 1 plus r 3 by r 2 whole square multiplied by 1 plus zeta r cosecant square beta 3 minus 1 into cot square alpha 2. And, so if you substitute for these values here, the only unknown is zeta r. So, we should get 0.9396 minus 0.02187 zeta r. So, this, if you substitute in this expression where zeta n is also known, you get a quadratic equation and you can solve that zeta r, which is the rotor loss coefficient can be calculated as 0.62. So, this is one way of calculating zeta r. The other way, of course is to calculate T 3 by T 2 using what I had mentioned. T 2, we have already calculated. We know what the value of T 2 is. For calculating T 3, it involves two to three steps. One is to calculate the stagnation temperature at exit T naught 3, which can be calculated from the power expression. Power is equal to mass flow rate into C p into delta T. T naught 1 minus T naught 3. So, all the three parameters are there. All the parameters are known except T 0 3 we can calculate stagnation temperature. Then, to calculate T 3, we also need to knowPDF Image | Turbomachinery Aerodyanmics
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