Jet-pumps, shock waves and the exponential constant e -
a hidden secret of Euler’s equations
Brian Worth
Joint Research Centre, Ispra, Italy
e-mail:
Key words: Jet-pumps, gas dynamics, shock waves, Euler’s exponential constant
Abstract
This article presents a study of supersonic flow in a convergent-divergent nozzle with shock wave formations, in relation to the optimization of design parameters for supersonic ejectors and jet pumps. The study brings to light certain unique flow parameters related to the isentropic pressure drop in a supersonic nozzle and the non-isentropic static pressure rise across a shock wave, with regard to energy minimization principles. It is shown that an “optimized” Mach number exists which promotes the maximum pressure rise across a shock for the minimum expenditure in upstream energy. This fact allows a supersonic nozzle to be optimized for pulling maximum vacuum during the notoriously unreliable “start-up” phase of a steam-water ejector prior to full condensing flow. It also demonstrates a rare beneficial use for shock waves as a natural phenomenon for creating a strong vacuum. An unexpected result is the appearance of a fundamental constant as a limiting value in the flow equations. It shows that Euler’s exponential constant e =2.71828…, introduced more than 250 years ago, may be defined as a simple limit rather than an infinite series, and may crop up in the most unlikely of places.
Introduction
Euler’s flow equations, along with the Rankine-Hugoniot relationships, provide a well-established mathematical basis for studying shock waves and supersonic flow phenomena. Shock wave formation occurring in turbo-machinery such as compressors and jet engines are generally an undesirable but natural consequence of compressible flow – a phenomenon to be accomodated rather than utilized for good effect. To find a beneficial use for shock waves is therefore unusual. When designing a high-performance jet-pump or ejector, the optimization of design parameters for the primary supersonic nozzle can benefit from the formation of a compression shock at the nozzle exit plane. It is then possible to design the primary nozzle for maximum pumping efficiency, minimal steam consumption and optimal transfer of kinetic energy into pressure rise across the shock wave, by adopting certain idealized flow parameters.
This paper describes the development of these optimized flow parameters as derived from the general equations for the steady supersonic flow of a compressible fluid in a convergent-divergent nozzle. This scenario is common to many flow situations found in rocket nozzles, jet pumps and other gas dynamic situations.
An unexpected finding is the appearance of an alternative definition of Euler’s exponential constant e =2.71828…, appearing as a limit rather than an infinite series, and in the form of a basic thermodynamic ratio.
Preliminary considerations
We consider the steady supersonic flow of a gas or vapour through a quasi-one dimensional convergent-divergent nozzle. Under certain inlet and outlet conditions, a plane normal shock wave will form inside the nozzle, with supersonic flow (Mach number M>1) upstream of the shock expanding into subsonic flow (M<1) downstream of the shock. This has been analyzed many times, however, it is instructive to consider the conditions under which the pressure rise across the shock can be maximized with respect to a given isentropic pressure fall in the upstream flow, as it relates, for example, to the optimization of jet pump operating parameters. This is instructive for determining the ideal boundary conditions necessary for the improved performance of jet pumps and ejectors, and in providing some characterizing flow parameters for designing the supersonic nozzle, in particular the critical pressure and area ratios, and “optimal” Mach numbers upstream and downstream of the shock.
With this particular optimal Mach number upstream of the shock, to be derived later and given by , it can be shown that the Mach number downstream of the shock has a universally constant value of , independent of the isentropic exponent g (ratio of specific heats, cp/cv) of the working fluid itself. These considerations are based on an unusual design strategy which utilizes the pressure change across a shock wave to maximize the suction achievable in a high-lift jet pump at the moment of start-up. By putting the shock wave to good use, it is possible to optimize the pumping performance whilst minimizing the energy requirements for normal operation. The validity of this approach has been demonstrated many years ago in fully working supersonic high-lift ejectors used in the nuclear reprocessing industry for pumping highly corrosive, dense radioactive liquids [1].
A surprise outcome, first noticed in 1973 as a mathematical curiosity, is the existence of an alternative definition for Euler’s exponential constant e (2.71828…), based on a simple thermodynamic ratio. When the optimum upstream Mach number defined above is substituted into the standard expression for isentropic pressure ratio as obtained from Euler’s equations for compressible gas flow, the optimum pressure ratio becomes
.
Permissible (i.e. physical) values for g are in the range 1<g £1.67. If however g is set equal to 1.0 as an idealized lower limit, this expression, which tends to the undefined limit , has in fact the limiting value of 2.71828… . This immediately suggested a “new definition” of Euler’s constant, namely:
.
This expression can be written as, simplifying to (for n = 0), which can be yet further simplified to:. Thus Euler’s fundamental constant e, a bedrock of modern mathematics, can be defined as a simple limit rather than an infinite series, and the reasoning lies in considerations of theoretical fluid dynamics.
Fluid dynamics considerations
We start by considering singlephase, onedimensional compressible flow through a supersonic convergentdivergent nozzle in which a shock wave has formed in the divergent diffuser section, as shown in Fig.1.
Fig. 1: Schematic representation of convergent-divergent nozzle with typical pressure distribution and shock profile.
If isentropic expansion is assumed to proceed from positions 0 to 1, the upstream pressure ratio (p0/p1) is given by:
(1)
The pressure ratio (p2/p1) across a stationary shockwave which forms in the divergent section of the nozzle is given by the RankineHugoniot relationships:
(2)
Flow through a shock wave is non-isentropic, i.e. an increase in entropy always accompanies the increase in pressure as the flow decelerates from supersonic to subsonic conditions across the shock. A shock wave can thus be considered as a natural and very thin “diffuser” with a typical thickness on the order of 0.03 microns. Any change in Mach number across the shock wave, from M1 to M2’ can be obtained from the expression:
(3)
Equation (1) divided by equation (2) is continuous for all M1 ³ 1. The “overall pressure ratio”, given by (p0/p2), then becomes:
(4)
If we first invert (4), then differentiate with respect to M1 and equate to zero, we obtain the maximum or minimum value of p2/p0 for a given ‘optimum’ upstream Mach number M1 for which the ratio of shock pressure rise to upstream isentropic pressure fall assumes a limiting value. This follows more readily by taking the logarithm prior to differentiating, i.e.
for a minimum value of (p2/p0).
After some algebraic manipulation, one obtains an ‘optimum’ upstream Mach number, M1opt, given by:
. (5)
By taking the second derivative of p2/p0 with respect to M1, it can be easily shown that
< 0, for all .
Thus for a maximum value of p2/p0. (6)
is therefore the ‘ideal’ upstream Mach number to obtain the maximum possible pressure rise across a shock wave for the minimum isentropic pressure drop in the convergentdivergent nozzle. If the back pressure p2 is fixed (say atmospheric), then p1 takes a minimum value and p0 is the minimum required upstream stagnation pressure. The nozzle can then be designed so that the shock wave stands initially just inside the nozzle exit, having an optimum upstream pressure ratio (p0/p1)opt to be defined shortly for a particular nozzle flow rate.
As stated, this optimum overall pressure ratio (p2/p0)opt gives the maximum conversion of upstream kinetic energy in the gas phase into pressure rise (i.e. potential/strain energy change) across the shock wave, for the conditions of minimum gas (or vapour) pressure at the nozzle inlet. In this way, a normal shock wave can be considered as a naturally occurring mechanism for the efficient transformation of kinetic energy into pressure, achieved most efficiently using the above optimum Mach number conditions. Experiments on prototype high-lift ejectors [1] confirmed that the suction produced in the pumped liquid reaches a maximum when the primary nozzle exit velocity has a comparatively low Mach number (about 1.4) according to equation (6), and with a lowest possible inlet pressure. This vacuum is seen to deteriorate, contrary to ‘intuitive’ expectations based on momentum exchange arguments, with higher nozzle exit velocities. Flow velocities having a Mach number higher than offer no advantages and can exacerbate the start-up process by ‘choking’ the nozzle with a ‘captive’ shock wave, unable to escape into the downstream flow.
Substituting into equation (1) gives the optimum upstream pressure ratio (p0/p1)opt for the nozzle:
. (7)
Now, the critical pressure ratio for a nozzle is defined as that ratio of upstream to downstream pressure for which the mass flow rate is a maximum value. Choking occurs (M=1) at the throat section of a convergent-divergent nozzle. Assuming polytropic expansion in a one-dimensional nozzle, this pressure ratio is readily obtained from the general energy equation by differentiating the mass flow rate (as a function of upstream pressure and density, gas constant and downstream pressure) with respect to the critical downstream pressure pc and then equating to zero. This gives the standard text-book formula for the critical pressure ratio in a nozzle as being where n is the polytropic gas constant and pc is the pressure at the critical throat section. For the simplest case of steady frictionless adiabatic flow (no heat addition), n is then the isentropic exponent g and the gas expands according to Boyle’s law, p/rg = constant. In this case, the critical pressure ratio for flow in a nozzle (suffix c referring to sonic, or choked, conditions at the throat) is
(8)
from which it is noted that the optimum pressure ratio, (p0/p1)opt is equal to the square of the critical pressure ratio, (p0/pc)2, i.e..
or . (9)
By way of comparison, for steam with a typical value of g = 1.13, the optimum nozzle exit Mach number is relatively low at 1.437, whilst the optimum pressure ratio (p0/p1)opt is only about 3. This is shown schematically in Fig. 2.
Fig. 2 Pressure profile and position of shockwave in primary nozzle of a supersonic ejector for optimized startup
Under these optimum flow conditions, the initial pressure ratio across the normal shockwave then becomes (from (2))
. (10)
The overall pressure ratio (p0/p2) from combining (7) and (10) becomes
. (11)
This is shown plotted in Fig. 3 as p2/p0 against p0/p1, for various values of g.
Using equation (3) to calculate the Mach number M2 downstream of the shockwave, inserting from (6), gives
or (12)
On the assumption that g remains unchanged throughout the expansion and subsequent shock compression, the Mach number M2 just downstream of the shock has always the constant value 1/Ö2 (i.e. 0.7071…) when and is independent of the nature of the gas or vapour flowing in the nozzle.
Fig. 3. Overall nozzle pressure ratio (p2/p0) plotted against isentropic
pressure ratio (p0/p1) for various values of isentropic exponent g.
For upstream Mach numbers which are higher than the optimum value, the downstream Mach number will always be lower than 0.7071, and the initial entrainment rate will then be correspondingly less. It is desirable, therefore, to try to minimize the pressure p1 upstream of the shock (which also minimizes p0), for a given initial back pressure p2, in order to maximize the entrainment rate of secondary pumped fluid into the primary vapour during startup. This can be achieved most effectively by designing the nozzle such that a normal shock stands just at the nozzle exit plane at the first moment of gas or steam injection. Entrainment will then induce this shock wave to always move downstream as the back pressure p2 is reduced (through condensation in the mixing chamber), transforming into an expansion wave as fullydeveloped (condensing) pumped flow is established. To avoid ‘incipient choking’ of primary fluid (a common problem with jet-pumps), the optimum Mach numbers and pressure ratios for the supersonic primary nozzle as described above will ensure maximum chance of successful ‘startup’ followed by normal steady pumped flow. For vapour-liquid jet pumps, this implies fully condensing flow in the mixing chamber so achieving maximum suction, with stable static pressure recovery in the liquid diffuser of the jet-pump.
Primary nozzles designed for higher exit Mach numbers than the optimum value will suffer from shock wave formation within the nozzle diffuser and will be located upstream closer to the throat. Even though the potential for higher momentum exchange between primary and secondary fluid then exists, it becomes more difficult for this shock to travel the increased distance downstream to leave the nozzle exit plane. In practice, highlysupersonic nozzles have unreliable startup characteristics because the shock wave becomes trapped within the primary nozzle, intensifying as it moves downstream. Such ‘captive’ shock waves cannot emerge from the nozzle exit, thus inhibiting the transfer of primary momentum to the secondary fluid and failing to achieve fullycondensed pumped flow. The jet pump is then in a “stalled” subsonic condition, and varying the inlet pressure may be the only way to re-create conditions conducive to fully-condensing pumped flow. A variable inlet pressure however invariably complicates an otherwise very simple no-moving-part pump and may reduce overall system reliability. Moreover, in a correctly designed ejector, once fully-condensing flow has been established, the pressure downstream of the nozzle drops markedly allowing further continuous expansion of the steam downstream of the nozzle exit and thus increasing the momentum transfer to the secondary pumped liquid. Highly supersonic nozzles are therefore not an a priori requirement.
At the instant of ‘startup’, flow immediately downstream of the shock will be subsonic, with a constant Mach number, M2, for given optimum M1. This result could be obtained alternatively by substituting M22 in place of M12, from equation (3), directly into the inverted form of equation (4), and then differentiating (p2/p0) with respect to M2. Equating the result to zero again yields a particular value of M2 for which (p2/p0) is shown to be a maximum. The same value (M2=1/Ö2) results but the algebra is somewhat more tedious.