Форум АСУ в Україні

Механізми для підвищення тиску текучої рідини називаються насосами, в той час як ті, що збільшують тиск потоку газу - компресорами. Турбонасоси та турбокомпресори використовують крильчатку, що обертається, для передачі кінетичної енергії рідині, а потім використовувують дифузію для перетворення більшої частини цієї кінетичної енергії в потенційну енергію або енергію тиску. Є два основні класи машин для рідин і газів: радіальні і осьові. Найбільш поширеними машинами для підвищення тиску рідини є відцентровий насос, однак осьові насоси мають місце в тих випадках, коли потрібно створити потік рідини великого об'єму, але з малим підйомним тиском.Машини змішаного потоку складають додатковий клас рідинного насосу, де комбінуються відцентрові і осьові насоси, з частиною потоку рідини що йде в робоче колесо, яке є радіальним, і частиною осьового.Насос змішаного потоку використовується для створення потоків з великими витратами з більшим напором ніж можуть зробити чисто осьові пристрої. Компресори можуть бути відцентровим або осьовим, осьові можуть бути використані для великих потоків (більш близько трьох кубічних метрів в секунду фактичного об'ємної витрати при доставці). Аналіз турбо насосів і компресорів мають багато спільного, хоча стискувана природа газу в компресорах робить їх набагато складнішими для моделювання.
Однією з важливих особливостей є те, що внутрішня динаміка і насосів і компресорів є дуже швидкою порівняно з технологічними установками, в яких вони використовуються. Тому, як правило, можна використовувати стаціонарну модель в загальному моделюванні.

+ eng 17.1 Introduction Machines for increasing the pressure of a flowing liquid are known as pumps, while those that increase the pressure of a flowing gas are known as compressors. Turbo pumps and turbo compressors use a rotating impeller to transfer kinetic energy to the fluid, and then use diffusion to convert most of this kinetic energy into potential or pressure energy. There are two major classes of machine for both liquids and gases: centrifugal and axial. The most common machine for pressurizing liquids is the centrifugal pump, but axial pumps have a place in applications where high-volume liquid flow is required but the pressure rise is small. Mixed-flow machines consitute a further class of liquid pump, and are a combination of centrifugal and axial flow pumps, with part of the liquid flow in the impeller being radial and part axial. The mixed-flow pump is used for high-volume flow at a greater head than a purely axial device could deliver. Compressors may be centrifugal or axial, with axial machines being used for larger volume flows (more than about three cubic metres per second actual volume flow at delivery). The analyses of turbo pumps and compressors have much in common, although the compressible nature of the gas in a compressor makes it rather more difficult to model. One important feature is that the internal dynamics of both pumps and compressors will be very rapid in comparison with the process plant they are serving. Accordingly it is normally possible to use a steady-state model of each in the overall simulation.

Без надання пояснення фізичної роботи насосів, розмірний аналіз забезпечує швидкий шлях до розуміння ключових особливостей, що впливають на їх поведінку. Першим кроком є ​​вибір змінних, які повинні бути включені в аналіз.
Втрати на тертя будуть залежати частково від числа Рейнольдса, яке саме по собі є функцією в'язкості. Проте, коефіцієнт тертя практично не залежить від числа Рейнольдса на дуже великих числах Рейнольдса зазвичай стикаються високо турбулентному потоці через промислових турбо насосів, і, отже, можна знехтувати в'язких ефектів з лише невеликий втрати точності.
Підвищення тиску в насосі, очевидно, важливо, але ми будемо використовувати комплексну змінну

замість нього. Тому обраний набір параметрів наступний:

де
- H напір, що створюється насосом (м),
- Q обємна витрата (м3/с),
- N швидксть обертання (обертів за секунду, rps),
- PD споживана потужність насосу (Ват),
- D діаметр робочого колеса (м),
- v питомий об'єм (м³/кг)
Ми повинні зробити подальше припущення, що і gH і PD залежать від (Q, N, D, v).

Напір насоса
Застосовуючи метод Релея з аналізу розмірностей до напору насоса, запишемо наше перевірочне рівняння як:

Підставляючи розміри отримуємо:

Equating powers of M, L and T gives three equations in the four unknown indices, a, b, c and d:

The index, d, emerges unequivocally as zero, while the indices, b and c, are expressible in terms of a, which may take any value:

Substituting back into equation (17.1) gives

which implies that the following general relationship holds between head and flow :

+ 17.2 Applying dimensional analysis to centrifugal and axial pumps While not providing an explanation of the physical working of pumps, dimensional analysis provides a short-cut to understanding what are the key features affecting their behaviour. The first step is to select the variables that should be included in the analysis. Frictional losses will depend in part on the Reynolds number, which is itself a function of viscosity. However, the friction factor is almost independent of Reynolds number at the very high Reynolds numbers generally experienced by the highly turbulent flow through industrial turbo pumps, and hence we may neglect viscous effects with only a small loss of accuracy. The pressure rise across the pump is clearly important, but we shall prefer to use the composite variable in its stead. The chosen parameter set is therefore: where - H is the head produced by the pump (m), - Q is the volume flow rate (m3/s), - N is the rotational speed (rps), - PD is the power demanded by the pump (W), - D is the impeller diameter (m), - v is the specific volume (m3/kg) We shall make the further assumption that gH and PD are each dependent on (Q, N, D, v). Pump head Applying Rayleigh's method of dimensional analysis to the pump head, we write our test equation as: Substituting dimensions yields: Equating powers of M, L and T gives three equations in the four unknown indices, a, b, c and d: The index, d, emerges unequivocally as zero, while the indices, b and c, are expressible in terms of a, which may take any value: Substituting back into equation (17.1) gives which implies that the following general relationship holds between head and flow :

It should be noted that the specific volume does not enter into this equation, implying that the head produced for a given flow will be the same whatever the liquid being pumped-hence the preference of working in terms of head rather than pressure rise. It is possible to estimate the form of the function, (fi)1 , from a detailed analysis of the velocities of the impeller and liquid, making due allowance for losses, but an accurate determination requires experiment.
The function, (fi)1, applies to pumps of different sizes, provided they are geometrically similar. Hence the form of the function could be determined from a model with impeller diameter, D0, or one with diameter, D1, or one with a diameter, D2, etc. Equally, all the experimental points could be taken at a constant speed, N0, or at a different constant speed, N~, or at a third constant speed, N2, etc. Theoretically, it should not matter what liquid is being pumped, although in reality the liquid should have a viscosity reasonably close to
that of the ultimate process liquid.
Let us suppose we take the machine with impeller diameter, Do, and run it at a fixed speed, No" by varying the downstream flow conductance, we may take a number of different readings of flow and corresponding head. These readings will be pairs of measurements of (20 and H0, the flow and head at conditions (Do, No).
(It should be emphasized that Qo and H0 are variables, not constants.) Now we move on to the next machine, with diameter, Dr. We run this at constant speed, N~, and take a new set of readings, this time of Q~ and Hi, where Qt and Hi are again variables, not constant values. Moving to the third machine, we take measurements of the variables, Q2 and H2, at fixed conditions (D2, N2). We may, indeed, repeat the process with any available machine of similar geometry.
We may then plot:

and equation (17.6) tells us that all the points should fall on the same curve. This theoretical result is found to be largely true in practice, although small discrepancies can occur as a result of the change in Reynolds number and hence friction when the scale-factor is large. The function, q~l, for a centrifugal pump will have the general shape shown in Figure 17.1.
If we select a test point, P = (x, y), on the curve of Figure 17.1, then, from the discussion above, the x-coordinate could be any of the following:

while the y-coordinate may be written as any of the following

Since the test point could be any point on the curve describing the behaviour of similar pumps, we may conclude that the following relations hold across all pumps in the set, at whatever speeds, flows and heads they are operating:

and

Pump power
The test equation for the power demanded by the pump is:

Substituting dimensions yields:

Equating powers of M, L and T gives three equations in the four unknown indices, a, b, c and d:

The index, d, emerges unequivocally as - I , while the indices, b and c, are expressible in terms of a, which may take any value:

Substituting back into equation (17.11) gives

which implies that the following general relationship holds between demanded power and flow :

Note that in this case specific volume of the fluid has an effect: working at the same speed, a pump can raise a flow of mercury to the same head as the same volume flow of water, but it will consume more power in doing so. The form of the function, ~2, is found from experiment, usually on one machine, but theoretically on several machines, as described above for the determination of the general function, r Since a single function characterizes all the pumps in the set, the arguments set down above for the relationship between pump head and flow apply equally to the relationship between demanded power and flow. Hence we may write the following equation for pumps of different size, running at different speeds and pumping liquids of different specific volumes:

Equations (17.9), (17.10) and (17.17) are known as the 'affinity laws'.
The useful pumping power, Pe, will be less than the power demanded, PD, because of frictional losses. The pumping power is the power expended in lifting the mass flow, W (kg/s), to a height H (m):

Using equation (17.6), we may rewrite equation (17.18) as:

The pump efficiency, rie, may be found by dividing the pumping power from equation (I 7.19) by the power demanded, given by equation (17.16):

which may be re-expressed as simply

From equation (17.21), efficiency is simply a function of the dimensionless variable, Q/(ND 3).
When simulating a particular pump on a process plant, the diameter of the pump impeller will, of course, be fixed, and so we may simplify the affinity laws to:

where Q,H,N, Po and v are general values and Q0, H0, No, Po0 and v0 are reference or design values.

A set of empirical functions is needed to characterize any given pump. Rather than the generalized functions, ~, however, it is customary for manufacturers to provide instead (i)the characteristic of head versus volume flow at the design speed, together with either (ii) a curve showing power demand versus volume flow at the design speed and specific volume, or else (iii) a curve specifying pump efficiency versus volume flow at the design speed.
A set of curves typical of a large centrifugal pump are shown in Figure 17.2. Note that the second graph includes the pumping power curve for interest's sake, derived by application of equation (17.18). These manufacturers' graphs will have been derived from tests
either on the pump itself, or else from tests on a pump of similar geometry.
Using the subscript '0' to denote conditions at the design speed and specific volume, we may regard the curve of head versus discharge rate as a graphical interpretation of the function:

and the power demand curve as a graphical interpretation of the function:

The efficiency curve will be a graphical interpretation of the function:

To understand the effect of a change from the design speed, No, and design specific volume, v0, we need to make use of the affinity equations. Substituting for Q0 from equation (17.22) and for H0 from equation (17.23) into equation (17.25) gives:

Rearranging, the head vs. discharge characteristic at a speed different from the design speed is

Similarly, substituting for Qo from (l 7.22) and for PD0 from equation (l 7.24) into equation (17.26) gives:

Accordingly the power demand at a speed different from the design speed is given by:

The pumping power at a speed different from the design speed is found by substituting for head, H, from (17.28) into equation (17.18):

To find the efficiency at any speed, N, we subsitute for Q0 from (17.22) into equation (17.27):

It is clear from the above equations that the ratio of flow to normalized speed,

is of fundamental importance in calculating the behaviour of the pump at any speed different from the design speed. It is further clear from equation (17.21) that the functions fet, fe2 and fP3 are interrelated, and that if two are specified, the third can be calculated. Concerning the choice of functions for modelling the behaviour of the pump, the head/discharge curve should always be the starting point. The efficiency curve is often used in calculating pump performance in preference to the demanded power curve, but there are grounds for sometimes choosing the latter instead, namely:
(i) the demanded power comes directly from test data;
(ii) the demanded power is usually a less complicated curve than the efficiency curve, and so may be modelled more accurately for the same order of polynomial fit or the same number of points in a look-up table;
(iii) if it is necessary to derive the input power at zero flow, e.g. for starting purposes, then a problem arises in using the expression:

since both will be zero, and the answer indeterminate.

Each of the functions f v t , f v 2 and fP3 may be approximated for the purposes of dynamic simulation by a low-order polynomial. However, care is needed when fitting a polynomial to the head/discharge characteristic using a simple least-squares procedure, since it is important not to introduce a spurious reduction in head at low flows, which can lead to numerical problems caused by flow being a multivalued function of head. The numerical problem has an analogue in reality, since such a characteristic on a real process pump can induce an unstable mode of operation. In the past, centrifugal pumps displaying a postive slope d H/dQ at low flows were sometimes made and were prone to unstable, surging behaviour if the flow dropped to low values: a drop in flow would lead to a drop in head, which would lead to a further drop in flow, etc.
It should be noted that suitable mechanical design of the pump, namely ensuring that the guide vanes are angled sufficiently backwards at the impeller exit, can eliminate this unstable mode of behaviour, and modem centrifugal pumps do not usually suffer from this problem.

The pump flow will establish itself extremely rapidly. The response of pump speed to changes in power input and output will also be rapid, of the order of a few seconds at most, but it may be essential to model this response, for instance if the model is to be used to design a pump speed-control system. Further, it may be desirable to use the numerical decoupling that including the additional state, namely pump speed, allows. This assumes particular importance if both the power supplied and the power demand are calculated
from sets of nonlinear simultaneous equations, as will be the case for a pump powered by a turbine and supplying a complicated liquid-flow network.
The dynamics of the pump are derived from the principle of the conservation of energy applied to the system with boundaries drawn around the pump as shown in Figure 17.3.

An energy balance over a general bounded volume is given by equation (3.34), repeated below:

In this case the heat input, d>, is zero and there is no mechanical power output, but a mechanical power supplied, Ps. Hence P = - P s . There will be no significant difference in height over the pump, so zl = z2. In addition, the almost incompressible nature of a liquid means that the specific volume will be essentially constant, allowing us to write:

The energy of the pump system enclosed within the boundaries will be a combination of mechanical and thermal energy:

Hence, by differentiating E and using the conditions of equation (17.35) in equation (3.34), we may write:

To understand the response of pump speed to changes in power input and in flow conditions, we substitute
h = u + pv and W = Q/v into equation (17.37):

The left-hand side of equation (17.38) will be zero in the steady state, and the power supplied to the pump will be equal to the power demanded

so that at the steady-state condition equation (17.38) may be written"

Now

from equation (17.18), while

by the definition of pump efficiency (equation ( 17.21 )), added to the steady-state condition (17.39). it follows that the expression

represents the frictional losses that cause an increase in the internal energy of the liquid being pumped.
We may interpret equations (17.42) and (17.43) as indicating that in the steady state a fraction, rip, of the power supplied goes into pumping power, while the remainder is lost to frictional heating. Assuming that this same process occurs in the dynamic state also allows us to decompose equation (17.38) into the mechanical and thermal energy equations:

and

The last equation could be used to find the outlet specific energy by putting

and integrating; here m may be considered constant because of incompressibility and the specific internal energy may be considered that of the outlet.
Outlet enthalpy may then be found from h2 = u2 + P2V2. However, since process thermal time constants downstream of the pump will almost invariably be significantly longer than the time constants of the pump, it is simpler and usually sufficient to use the steady-state
solution of equation (17.37) to give the pump outlet enthalpy, which is then:

Meanwhile equation (17.44) is in a form where it may be integrated from initial conditions of speed and pump flow. The efficiency, v/t,, used above may be found either from equation (17.33), if the function, fe3, is supplied, or else from the ratio of pumping power to demanded power (equations (17.31 ) and (17.32)) if the function, ft,2, is supplied.