Why isothermal compressor is desirable

Digitization of the polytechnical journals

From chief engineer R. von Stein, Karolinental.

Contents overview.

It is shown how, in an elementary way, while avoiding entropy diagrams, all dimensions of turbo fans and / or. Compressors can determine.

The methods communicated are explained and discussed on the basis of a turbo compressor.

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For the calculation of the steam turbines, insofar as this is done on a rational basis, the heat diagram is probably used exclusively today, be it as a T s diagram or as an is diagram (Mollier), and rightly so; because the complex processes in the multi-stage steam turbines would be difficult and imperfect to take into account in any other way.

The use of these diagrams, however, inevitably means that one has to work constantly with a concept which is difficult to perceive and thus difficult to understand, precisely that concept which Professor Peary aptly referred to as the one with which most people have so far probably not used anything Knowing to begin with what is right, rather accepting it as a purely mathematical concept without a sensible basis. If it is now difficult to avoid accepting this disadvantage with the turbines, as has already been emphasized, the situation is significantly different with a closely related group of rotating machines, namely the turbo blowers and compressors. Here the old p v diagram that every technician is familiar with, combined with a few simple constructions, is completely sufficient to determine all the quantities required for the construction of these machines in a simple and transparent manner. What has been said is based on the much simpler physical properties of air (generally the permanent gases) compared to those of superheated water vapor and the purpose of the following lines is to explain this with examples and thus to make the designer independent of the use of graphic boards that are not are always at hand and, despite the simplicity of the underlying principle, require a lot of practice in their use because of the confusing number of lines.

The compression lines of the permanent gases are almost exclusively the so-called. polytropic lines, which follow the law p vn = const., and of these again those two special cases, which with n = 1 and with the isothermal respectively. represent the adiabatic compression law. The actual compression lines of the turbo-machines are usually located between these two lines as borders; the true compression line rises above the adiabatic only when there is a complete lack of cooling, since the unavoidable loss of work inside the machine is added to the pumped liquid during its compression in the form of heat. The isotherm can be calculated as an equilateral hyperbola according to the generally known construction (Fig. 1), the adiabatic according to the equally often used method | 242 | by Professor Brauer with the easy to remember values ​​tg ∙ α = ⅓ and tg ∙ β = ½ (Fig. 2).

The latter construction, for all its simplicity and convenience, has the disadvantage that inaccuracies are propagated, since each subsequent point is found from the next preceding one, and further that the distances between the points determined grow rapidly, so that the associated volume cannot easily be obtained for a specific pressure and that finally for other values ​​of n the relationship between a and β is not so simple, but rather has to be determined by a logarithmic calculation according to the formula 1 + tg ∙ β = (1 + tg ∙ α) n.

The following construction given by the author is free of these inconveniences and should therefore not be without interest despite its somewhat greater inconvenience.

Instead of starting directly from the formula p vn = const., We choose the formula , which is obtained from the previous formula by connection with the Boyle-Gay-Lussac formula p v = R T.

If we denote the initial and final states with p1, v1, T1, and p2, v2, T2, furthermore with vi the isothermal final volume, then the following applies: and what used in above formula brings out. If we set vi = 1 for the sake of simplicity, we get and by taking the logarithm and finally: .

This expression can easily be determined with the aid of a logarithmic line which is easy to construct (FIGS. 3 and 3a). The latter line is obtained in the simplest way by starting from v1 and drawing horizontal ordinates at equal, incidentally arbitrary, vertical intervals, the following of which always represents the same aliquot part of the next preceding one. (In Fig. 3a, for example, each following ordinate was made = ¾ of the next preceding one.) The abscissas (calculated from bottom to top) then represent the logarithms of the associated ordinates on an incidentally indifferent scale. the latter the numbers for the logarithms represented by the abscissas.

The division of log. v1 in proportion The easiest way to do this is by means of a proportional angle, which can easily be determined for any n without a logarithmic calculation. In this way, the adiabats for air have been determined in FIG. 4 by means of the isotherm and the values ​​n = k = 1 ∙ 4. It is in this case to determine the proportional angle α. | 243 | Because the relationship at the same time represents the ratio of the absolute temperatures with polytropic and isothermal compression for the same final pressure, this construction gives the final temperature of the gas compressed to any final pressure after any given polytropic in the simplest way for a given initial temperature. Conversely, the course of the associated polytropes and the exponent n can be determined for any final temperature of the polytropically compressed gas.

You decide , determine with the help of v2 and vi the hatched characteristic triangle and with the help of the same tg ∙ α, then n - 1 = n ∙ tg ∙ α and .

How close one can get closer to the isotherm with the real compression line depends only on the recooling of the air during compression. In the past, intercoolers like multi-stage reciprocating compressors were used between every two groups into which the entire compressor was split up, but this method is currently being abandoned more and more by restricting oneself to intensive surface cooling. This has completely different successes than with reciprocating compressors, in that the much better distributed air comes into much closer contact with the cooled walls and sweeps past them at great speed.

You can at a final pressure of 7 at abs. reach a final temperature of 75 ° and even less with favorable cooling conditions, so that the use of particularly expensive intercoolers appears questionable. We will come back to this subject and hold on to the provisional result that we can determine the general course of the condensation line with a correspondingly assumed start and end temperature as a polytropic line by means of the specified drawing method. Deviations from the pure polytropic course will only be noticeable at the beginning, where the temperature difference between air and cooling water is still slight, so that the lower part of the condensation line will generally correspond more to the adiabatic (sometimes even rise above it), and then completely to gradually change into the polytropic. In any case, one can hardly proceed more arbitrarily with this type of determination of the compression line than with the use of entropy tables.

If there is no recooling at all, which occurs with fans with low pressure, in which the inconvenience and additional costs of cooling are not outweighed by the gain in work, or a high air temperature is desired, the compression line can be found as follows: Let's see from the Bearing friction decreases, the entire loss work inside the fan is converted into heat, which, apart from the low radiation, is only used to increase the air temperature. In the case of lossless compression, the same would run after the adiabatic, since heat is neither supplied nor discharged and the final temperature reached would be ta The so-called. is then

,

i.e. the ratio of the theoretically loss-free work done inside the fan to the work actually done there (based on 1 kg of air). This then differs from the overall efficiency η only in that with about 3 v. H. Estimated bearing friction. Let us take η = 72 v. H., as is indicated today for good designs, we find η ad = 0.72 + 0.03 = 0.75.

Now is:

vi: va = T1: Ta or vi: (va - vi) = T1: (Ta - T1)

and

vi: v2 = T1: T2 or vi: (v2 - vi) = T1: (T2 - T1)

From which follows:

and finally:

.

From this, since va and vi are known from the isotherms and adiabats, v2 can be calculated:

.

It can be seen from FIG. 5 that the adiabatic efficiency is shown in the p v diagram by the ratio of the double-hatched area to the entire hatched area.

By observing the temperatures t1 and t2, | 244 | it is easy to determine the adiabatic efficiency and thus also the overall efficiency of an existing fan after calculating ta, but it should be expressly emphasized that this type of calculation only applies to uncooled fans.1)

If the p v diagram corresponding to the circumstances has been determined according to the above, it is now a matter of subdividing the entire diagram area into the partial diagrams corresponding to the individual wheels or wheel groups. When discussing the wheels, we will derive the formula: measured as the pressure head for a pair of impellers and idlers in meters of water column.

Here ϕ means a constant factor dependent on the blades and the quality grade, u the peripheral speed of the impeller, g = 9.81 m / sec2. the acceleration of gravity, γ the spec. Weight of the pumped liquid and γ0 = 1000 kg / m3 the specific weight of the water. All of these quantities, with the exception of y, are constant, and if we still take place write, we get the relationship:

h ∙ v = const.

If we look for the meaning of this equation in FIG. 6, we can see that the overall diagram for the individual wheels or wheel groups is to be broken down into strips of equal area, which relationship we will briefly refer to as that in the following. In order to be able to carry out this division quickly and easily, we use a simple drawing process.

We first divide the height of the p v diagram into several, roughly four to five equal parts, and draw horizontal lines through the dividing points, which now divide the overall diagram into just as many unequal areas. We transform these patches of surface into rectangles, as indicated in FIG. 7 by the hatched compensating triangles. The mean heights of these rectangles of the same base or an aliquot part thereof (in Fig. 7½) give a measure of the content of the individual surface strips, and by successively adding them up on the successive horizontal sub-lines and connecting the endpoints thus obtained, the Integral curve of the diagram area, ie a curved line with the property that each (horizontal) ordinate of the same represents that area which lies between the base and this ordinate. The end ordinate therefore represents the entire diagram area F.

If we now want to break down the latter into three equal area diagrams, for example, we only need to divide the end ordinate of the integral curve into three equal parts, draw perpendicular lines through the dividing points up to the intersection with the integral curve, and then lay horizontal lines through the intersection points obtained in this way the overall diagram will be broken down into the required three area-equivalent partial diagrams. H1H2 and H3 (Fig. 7) then represent the pressure heights to be overcome in the three individual groups.

If the mean ordinates vm1, vm2, vm3 etc. are determined in FIG. 8 with the aid of the hatched compensation triangles, then these represent the mean air volumes for which the wheels of groups 1, 2, 3 etc. are to be calculated if the base v1 of the overall diagram means the volume of air to be sucked in at atmospheric tension. But if one understands vm1, vm2, etc. as specific volumes, then are , etc. = γ1, γ2 etc. the mean specific gravity in the individual groups, provided that represents the specific weight of the pumped liquid at atmospheric tension (generally at the suction tension).

If h is the pressure level that can be achieved with a pair of impellers and idlers, if the fluid is at atmospheric tension, then the mean pressure levels h1, h2, etc. that can be achieved in the individual wheel groups with one pair of wheels are. , etc. or also , etc. The number of pairs of wheels required for the individual groups is then given by: , etc.

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In reality, of course, the pressure levels generated by the successive pairs of wheels within a stage are not the same, but also follow the law of area, but this does not change the correctness of the above calculation method, in which it is tacitly assumed that the mean density of the conveyed liquid in a group is also the true density in the whole group, or in other words, that the conveying liquid within a group behaves like a drip liquid, ie is regarded as incompressible. Before we move on to determining the pressure height for a pair of wheels, the following investigation should be made:

We consider an ideal diagram (Fig. 9), the compression line of which therefore follows the isotherm.

According to what has been shown, the condition must be fulfilled: (pn + 1 - pn) - vn = k (law of area), while the condition of isothermal compression is achieved can be expressed, understood by k and k 'constants.

From the first equation it follows: what is inserted into the second equation

results from which equation follows:

This also gives: pn + 1 + pn = k "pn + 1 - k" pn or pn (k "+ 1) = pn + 1 (k" - 1) or . Let's sit now so we get:

pn + 1 = q ∙ pn,

i.e. the pressures in the successive stages increase according to a geometric series (with the quotient q).

This law, which we obtained by combining the generally applicable law of area with the isothermal compression law, is often used as a basis for the calculation of turbo compressors. However, as follows from what has been said, the same is only valid for isothermal compression, but by no means generally and can give rise to not inconsiderable errors in the case of larger deviations of the true compression line from the isothermal.

Determination of the wheels: We use the old central thread theory (one-dimensional), with the full awareness that it will correspond even less to the actual flow processes in compressible liquids than in drip liquids, but since the experiments available so far, even with the latter, better theoretical bases free from gaining, have not produced impeccable results in spite of great scientific effort, the old theory is all the more likely to be retained as it fits perfectly into the framework of the present treatise with its clarity and transparency. Incidentally, it is actually possible to obtain any desirable approximation to reality by introducing suitable coefficients.

(Sequel follows.)

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Regarding the, incidentally, insignificant corrections to be made, see: Z. d. V. d. I. 1910, No. 40.