Fluid Thermodynamic Effects
For an ideal fluid, which is as approximated by cold water, hydrocarbon and amine fuels, and other low-vapor-pressure fluids, the limitation on suction performance is always leading-edge cavitation in the inducer. With certain fluids there is observed a thermodynamic suppression head (TSH) that acts to decrease the critical NPSH requirements of the inducer (refs. 1, 6-25). Among the fluids known to exhibit this effect are liquid hydrogen, liquid oxygen, storable oxidizers such as N2O4, and hot (over 200° F) water. For liquid hydrogen this effect may be so strong that the swallowing capacity is limited only by cavitation in the inlet duct; i.e., by c„,2/2g = (NPSH)tank, where c,„ is the meridional velocity, calculated for single-phase flow and (NPSH)ta„k is the minimum available net positive suction head in the tank at operating conditions.
Thermodynamic suppression head is an effect brought about by the decrease in fluid vapor pressure and additionally, in the case of two-phase flow, by the decrease in fluid density. The phenomenon of TSH is best defined and understood in the following mathematical formulations.
The basic condition for pump suction performance is that
(NPSHJrequired^ (NPSH) available (9)
The (NPSH)re(llUr(,d is determined by the characteristic pump suction specific speed S'*, which is obtained from pump performance with an ideal fluid (approximated in practice by cold water). From equations (1) and (4), with Q' as defined in equation (24)
assuming that the pump characteristic suction performance is independent of the pump fluid.
By definition, the net positive suction head is the excess of fluid total pressure ptotal above vapor pressure pv divided by the fluid density pF at the prevailing local conditions. It follows then that the pump sees the value
pf at inducer leading edge where ptoU1, pv, and pF are the local values measured at the inducer inlet; they are different from the values at the tank and depend on the flow conditions.
For the hypothetical ideal fluid, the NPSH value is of constant magnitude (except for line friction loss) throughout the inlet system from tank to inducer inlet under all flow conditions; i.e.,
PF tank where ptotaI, pv, and pF are the values in the tank at its outlet, and where Hloss is the line head loss due to friction.
By definition, the thermodynamic suppression head is determined by the following equation:
(NPSH) available = (NPSH) ideal fluid + TSH, Or TSH = (NPSH)a - (NPSH)if (13)
In practice, (NPSH)avallabIe has never been measured directly, but its value has been inferred from the measured pump suction performance with various liquids by assuming that the equal sign applies in equation (9) when head breakdown occurs.
Various semiempirical correlations of TSH with fluid properties and pump parameters have been attempted (refs. 24 through 27). These correlations are based on a relationship between the thermal cavitation parameter a, the thermal diffusivity of the liquid Kl, and the size and speed of the pump.
The thermal cavitation parameter and diffusivity are functions of the fluid properties only; i.e.,
PL CL
where
J = energy conversion factor, 778.2 ft-lb/Btu L = latent heat, Btu/lb c= specific heat of liquid, Btu/lb-°R T = fluid bulk temperature, °R PL — liquid density, lb/ft3 pv = vapor density, lb/ft3
kL = thermal conductivity of liquid, Btu/(sec-ft-°R) K/, = thermal diffusivity of liquid, ft2/sec a — thermal cavitation parameter, ft hL
Holl (ref. 22) combined the parameters a and KL to form the thermal factor
By hypothesis, TSH is a function of the fluid thermodynamic cavitation properties, the fluid velocity Uc on the cavity boundary, and the length Lc of the cavity. The variables TSH, a, Kl, Uc, and Lc form a set of three independent, physically significant, dimensionless groups through which the functional relationship is expressible. One set of three basic groups includes (TSH/a), (Lc/a), and (UCLJKJ, from which other sets are formed by combination, e.g., (TSH/LC), (LJa), and (f$2UJLc). The application of dimensionless groups to the analysis of pump performance studies requires a relation between a set of basic groups, e.g.,
where the constant C and the exponents ml, m2, and m3 are determined by tests on similar pumps, S is the blade spacing, and Uc is a function of the blade tip speed u. So far, no such relationship with well-established values for the constants and exponents has been found.
A cavitation number Kc, based on the cavity pressure instead of the liquid bulk vapor pressure, has been defined (refs. 24 through 27):
PF wV2 g where p8 — fluid static pressure, lb/ft2
pc = fluid vapor pressure in cavity at leading edge, lb/ft2 pF = fluid density, lb/ft3
w = fluid velocity relative to blade at tip, ft/sec g = gravitational constant, 32.174 ft/sec2
Venturi cavitation studies show that Kc is approximately constant while the conventional cavitation number K varies when both are measured over a large range of liquids, temperatures, velocities, and venturi sizes, provided the geometric similarity of the cavitated region is maintained (i.e., the ratio of cavity length to diameter, LJDC, is constant). The studies on venturi cavitation have produced information useful in understanding the problem of thermal suppression head in pumps. On the basis of these studies, attempts have been made to predict actual values for TSH for various fluids used in pumps (ref. 25). The correlations obtained, however, do not allow successful prediction of pump performance without the availability of reference data, i.e., data on the actual performance of the pump with a liquid having TSH effects.
Fluid thermodynamic effects on suction performance are considered in the design phase by a correction on the available NPSH value. An empirical allowance for TSH is added to the tank NPSH value (less the inlet line head loss). The assumed TSH value is based on previous experience with the fluid. No theoretical prediction is attempted at present. Presently established empirical values for the Mark 10 (F-l engine) liquid-oxygen pump (inducer tip speed: 300 ft/sec) and the Mark 15 (J-2 engine) liquid-hydrogen pump (inducer tip speed: 900 ft/sec) are as follows:
Mark 10-0: TSH = 11 ft, at 163° R Mark 15-F: TSH = 250 ft, at 38° R
These values are used with considerable reservation, however, when applied to other pumps, because the occurrence of thermodynamic suppression head is not well understood and the effect of a change in the characteristic parameters is not known. The TSH value of a fluid increases with temperature almost as a linear function of vapor pressure (ref. 20). Tests also indicate the existence of a speed and fluid velocity effect increasing the TSH with speed (rpm) at fixed flow coefficients (refs. 28 and 29).
Another common practice to allow for the fluid thermodynamic effect empirically in the design phase is to assume a value (based on experience) for an NPSH factor Z, defined by
which corresponds to a TSH correction of
2 g giving the total required NPSH value
2g where
for small <Povt.
Present liquid-hydrogen pumps are able to pump two-phase hydrogen at pump-inlet vapor volume fractions up to 20 percent at design liquid flow coefficient. The basic limit to pumping two-phase hydrogen occurs when, at high flow coefficients, the flow area within the inducer blade passages becomes less than upstream flow area. When this occurs, both two-phase flow and pure saturated liquid flow will choke (ref. 30). Further experimental investigations aimed at establishing proper criteria for two-phase flow are in progress.
2.1.5 Blade Profile
In a well-designed inducer cascade, the blade profile does not interfere with the free-streamline boundary of the cavitating flow at the blade leading edge.
If so-called real fluid effects due to viscosity of the fluid and surface roughness of the blade are neglected, the flow in cavitating inducers may be adequately described by potential flow models with a simplified geometry. These models all are based on the assumption of a two-dimensional, irrotational, steady flow of an incompressible, inviscid fluid through a two-dimensional cascade of blades. The cascade and flow conditions represent those of the actual inducer at some fixed radial station.
A set of physically significant, characteristic parameters relating to the state of the fluid, the entering and leaving flows, and the geometry of the cascade is illustrated in figure 8.
Figure 8.—Cascade and flow parameters.
The cascade geometry parameters are blade angle p, blade camber A¡3, chord length C, and cascade spacing S. Subscripts 1 and 2 or, occasionally and more distinctly, LE and TE denote the leading and trailing edges, respectively. The entering and leaving flows are characterized by the relative velocities wi and W2 and the angles yi and 72 of these flows with the cascade axis. The velocity components normal and parallel to the cascade axis are wm (or w„) and w„, respectively. These two components commonly are designated meridional and tangential components, referring to the equivalent usage for inducer flow. The meridional flow may or may not be axial but, by common usage, is always referred to as meridional because the cascade represents the meridional flow picture of the inducer. The cascade velocity wi is the vector sum of the two components represented by the blade velocity u and the fluid meridional velocity cm at the inducer inlet. The inlet velocity cm is assumed uniform over the inlet area; hence
irD2
where Q' is a corrected flowrate expressed by
giving the equivalent swallowing capacity of a hubless inducer.
Various models for the flow in flat-plate cascades with cavitating flow have been proposed and studied. These models differ essentially in the manner of cavity closure. There is no unique solution for a constant-pressure cavity of finite length, because the cavity can be terminated in a variety of ways. Among these cavity models are a reentrant jet, an image plate on which the free streamline collapses, and the free-streamline wake model where the flow gradually recovers pressure on a solid boundary that resembles a wake. Experimental and visual observations indicate that, of all these models, the free-streamline wake model (wake model, for short) simulates to some extent the actual wake downstream of the cavity terminus, where intense mixing may be seen. The wake model of the flat-plate cascade with semi-infinite blades yields the simplest possible simulation of the important flow features of a cavitating inducer with partial cavitation. The theory is described and derived in detail in reference 31. For supercavitating flow with an infinite cavity, existing solutions (refs. 32 and 33) treat cascades with a finite chord length and arbitrary camber.
The wake model gives a good approximation to the cavitating flow in the inducer and is a useful tool for the inducer designer in calculating the cavity boundary. The main difficulty lies in the evaluation of the free-streamline theory as a function of cavitation number and angle of incidence of the inducer flow; the analysis involves the numerical evaluation of some complex variable relationships for the cavity shape. A computer program written to accomplish these objectives for any given inducer blade is available (ref. 34).
The cavity velocity wc follows from Bernoulli's law and the definition of the cavitation number, giving
Similarly, combining this with the stagnation point condition for the velocity ratios results in wc — wi V 1 + K
where F stands for the expression
2 sin (/? - a) The wake or cavity height hc is found from he
which determines the maximum blade thickness that may be contained in the cavity with proper shaping of the leading edge (fig. 9).
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Blade Figure 9.—Blade in cavity. For the extreme case of supercavitation, the free streamline approaches a wedge shape at the leading edge with an angle equal to the angle of incidence a. The cavitation number K attains its minimum value at supercavitation: When a — 0 or a = P, then Kmln = 0; but these values are not realistic, because in the first case there is no deflection of the flow and in the second case there is no throughflow in the cascade. The equation does bring out clearly that, to obtain small cavitation numbers, the blade angle should be small. which also shows the need for small blade angles /? to get small values of K. Because of blade thickness and boundary-layer blockage, in actual operation with a real fluid the attained values of K are approximately two to three times greater than the maximum values of Kmln. | |
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