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The performance of electrogasdynamic expanders with slightly conducting walls
Energy Conversion.
Vol. 15, pp. 127-135.
Pergamon Press, 1976. Printed in Great Britain
The Pe ormance of Elect asdynamic Expenders with Slightly ConductingWalls D, WADLOW and P, J, MUSGROVEt (Received 16 January 1975) AlOft't--In electrogasdynamic (EGD) devices the radial movement of charge carders to insulating duct walls can produce high parasitic electric fields and greatly degrade the overall performance. In principle these parasitic fields may be reduced by constructing the EGD ducts from slightly conducting materials, but there is then a power loss due to current flow through the wall. This paper
examines the effect of particle deposition velocity,wall resistivity,load resistance and aspect ratio, as well as ttuid friction and gas density, on this resistive wall power loss and on the overall performance of EGD devices. Given suitable and realistic values of the relevant parameters, isentropic efficienciesin excess of 85 per cent are predicted at pressures of the order 10-40 atmospheres. Electrogasdynamics deposition
Energy conversion
Charged aerosols
1. Introduction Electrogasdynamic (EGD) energy conversion systems promise to provide a means of converting heat to electricity efficiently, and have the advantage that no mechanical moving parts are required. For example Musgrove [1] has predicted an overall efficiency of 42 per cent for EGD systems operating on an Ericsson cycle between a heat source at 650°C and a heat sink at 30°C provided that the power output is in excess of a few kilowatts. Such systems therefore merit consideration for a wide range of applications, and it has in the past been suggested that EGD systems be used in central power stations, operating at the gigawatt (109 W) power level. Although Nuclear/EGD or Fossil Fuel/EGD power stations may one day be built, we believe that EGD is better suited in the near future for applications at lower power levels, e.g. for power outputs in the range from a few kilowatts to a few megawatts. In particular EGD systems appear eminently suitable for use in Total Energy Schemes. In this role they would appear to have three principal attractions: (1) EGD systems require no mechanical moving parts and so, when fully developed, should be extremely reliable. (2) The energy input to EGD systems is from an external heat source, and not by internal combustion of fuel. They should therefore have the low noise and pollution levels that one associates with continuous, external combustion systems. (3) The efficiency of EGD systems is independent of the unit size and so efficiencies of the order of 40 per cent should be obtainable at kilowatt as well as megawatt power levels. The efficiency predictions, and consequently the overall prospects for EGD systems, rest on a fundamental assumption. This is that the two principal EGD compo~"Department of Engineering and Cybernetics, University of Reading, Whiteknights, Reading, RG6 2AY, Berkshire, England.
Direct conversion
Corona discharge
Particle
nents, i.e. the EGD expander stage and the EGD compressor stage, can be developed to operate at isentropic efficiencies of the order of 85 per cent. To date this has not been demonstrated experimentally, and until this is done the outlook for EGD energy conversion remains uncertain. Experiments at Reading University on a newly constructed high pressure EGD test rig should soon resolve this uncertainty; in the meantime this paper examines theoretically the prospects for operating EGD components at high isentropic efficiencies. The general characteristics of an EGD expander stage are shown in Fig. 1. A gas is used to propel unipolar charged particles against a strong electric field, so the ~, EARTHEDRING -Hi" ELECTROOE ,TTB~TOR NEEI1E""~1 UkCH~ffJ~o
e e
e
NEGATWEHIGH,,,.,j VOLTAC~TERNINAL 1 [ ®
®
e
o
o
AEROSOL
Fig. I. Sehemafie e l e e ~ r o ~ m d c
generator.
electrical potential energy of each particle increases, at the expense of the energy in the flowing gas. Typically current is supplied by the upstream corona discharge at a voltage measured in tens of kilovolts and this current is subsequently collected, via a self-induced corona discharge near the high voltage terminal, at a voltage of several hundred kilovolts. There is direct conversion of part of the power in the flowing gas to electricity and associated with this there is a small pressure drop along the length of the EGD duct--hence its description as an EGD expander. An EGD compressor stage appears almost identical to the expander shown in Fig. 1, but in this case the charged particles have the opposite polarity to that of the high voltage terminal. The electric field consequently propels the particles through the duct, and the energy imparted to them in this way is transmitted to the surrounding gas by viscous forces. Since the EGD compressor stage imparts energy to the gas flowing 127
128
D. WADLOW and P. J. MUSGROVE
through it, it requires an electrical power input. In a complete EGD thermodynamic cycle, as proposed in [1] part of the electrical power output from the EGD expander stages is used to supply the power required by the EGD compressor stages. In EGD devices a variety of charge carriers may be used, but experiments at Reading University have concentrated on the use of fine solid particles, typically silicon carbide particles a few microns in diameter. Fine particles such as these ensure a low mobility; hence slip between the charged particles and the carrier gas is small and so are the slip losses. Experiments, see Musgrove and Wilson [2], have shown that with the right choice of particle size and number density, losses associated with the particle charging process are small. However, fluid friction losses are not negligible, and since these power losses are proportional to the cube of the gas velocity, whereas the output power is only directly proportional to the velocity, efficient operation requires relatively low gas velocities, of the order 50 m/see. For velocities in this region, and at pressures of the order 10 atmospheres, previous calculations, e.g. Musgrove [1], have indicated that isentropic efficieneies over 85 per cent are feasible. But these calculations omit an important loss mechanism, which experiments have shown to be very significant. This loss results from the radial movement of charged particles to the walls of the EGD duct, due in part to turbulence in the flow and in part to the effect of radial electric fields. Charge builds up on the insulating duct walls and produces strong parasitic electric fields which greatly reduce the performance. Experiments at Reading have shown, for example, that these parasitic fields reduce the power output of an EGD expander by a factor of about four. There are several methods by which this problem may be avoided, or the consequences minimized. One possibility is to prevent the deposition of charge on insulating walls by surrounding the charge carrying gas by a gas stream which carries no charged particles, see for example Lawson and Von Ohain [3]. A second possibility is to generate a.c. not d.c., so that there is no accumulation of charge on the duct walls and parasitic electric fields are avoided. This may ultimately prove to be an attractive proposition, but little work has so far been done on a.c. EGD devices. The third possibility, and the one considered in detail in this paper, is to replace the insulating duct wall of the conventional EGD expander (or compressor) by a wall that is slightly conducting. The charge deposited on the wall by the radial drift of particles can then be allowed to leak away without giving rise to excessive fields within the duct. That this technique can be effective was demonstrated by Collier and Gourdine [4]. However, the use of a slightly conducting wall does introduce two power loss mechanisms. Firstly, there is the power lost resistively as the charge deposited on the walls leaks away to earth through the duct walls. Secondly, there is the power lost due to the fact that part of the gross power output from the EGD expander flows to earth through the duct wall
rather than through the output load. Although this second loss mechanism is potentially the more serious, it should be pointed out that the quest for maximum cycle efficiency will no doubt lead, in time, to EGD systems being operated at peak cycle temperatures limited only by the condition that the duct wall must not be too conducting. If for this reason EGD expanders are to have slightly conducting walls, then it seems reasonable to consider using this wall conductivity as a means of eliminating the parasitic fields produced by the radial movement of charged particles. This paper therefore examines the magnitude of the power losses associated with the use of a slightly conducting duct wall. The effect of resistive wall losses and fluid friction losses on the isentropic efficiency of an EGD expander is also calculated as a function of the particle deposition velocity, duct aspect ratio, wall resistance, etc. Although the calculations presented relate specifically to EGD expanders of circular cross-section, they may readily be extended to predict the performance characteristics for other geometries and for EGD compressors. 2. Nomenclature Ratio of length to diameter of duct, i.e. aspect ratio, L/D, gas density, duct diameter, electricfield strength, wall friction factor, electriccurrent, duct length, pressure,in atmospheres, power, resistanceper unit length of duct wall, load resistance, absolutetemperature, radial deposition velocity, mean axial gas velocity, voltage, distancealong duct from low-voltageterminal, ratio of deposition velocity to gas velocity,v~/vg, absolute permittivity, ,7 isentropicefficiency,power output/(power output + losses), ratio of total duct wall resistance to load resistance, rL/R, p chargedensity. Unless otherwise stated S.I. units are used throughout.
A d D E f t', I L p P r R T va va V x
3. Theoretical Model
3.1 Assumptions In the model which is developed subsequently a number of assumptions are implicit. Firstly, it is assumed that the charged particles which migrate to the wall surrender their charge to the wall. If the charge carrying particles were too insulating it would be possible for charge to build up on the inside surface of the duct, despite the wall conductivity. The use of a relatively conducting material such as silicon carbide prevents this. Also, when using particulate charge carriers, particles are constantly being deposited and re-entrained. Under steady-state conditions there is a thin layer of particles adhering to the wall at any instant in time. This deposit is about a monolayer thick and is assumed to have no significant effect on the overall wall resistivity. The duct wall is assumed to be uniformly resistive, which implies a constant duct wall cross-section and a uniform resistivity. This assumption simplifies
The Performance of Electrogasdynamic Expanders with Slightly Conducting Walls
the analysis, but optimum performance may well be associated with a controlled variation of the resistivity along the length of the duct. The model further assumes that triboelectric charging at the walls is insignificant compared with the charge given to particles as they pass through the corona discharge. Experience indicates that this assumption is a reasonable one. If it were not, it would imply a high value of the particle deposition velocity, whereas measurements by Lovett and Musgrove [5], have shown that this is only a small fraction of the axial gas velocity. No attempt has been made, in the analysis, to relate the deposition velocity to the particle size, the flow Reynolds number or the magnitude of the electric fields within the duct. Instead the variation of the EGD duct performance is examined for a range of typical values of the deposition velocity. For given operating conditions this deposition velocity is assumed to be substantially constant along the length of the duct. At the outlet of the EGD duct the charged particles are neutralized by a space-charge induced corona discharge. This is assumed to remove effectively all the charge carded by the particles, and failure to achieve this would lead to reduced performance. However, experimental evidence, in the form of current balance measurements, has shown that the discharge corona does remove almost all the charge on the particles. Finally, it is assumed that the particle mobility is small enough for slip losses to be negligible. The mobility depends on the particle size and the charge per particle and experiments at atmospheric pressure, see for example [5], have confirmed that the mobility of micron sized particles is low. However, care will need to be exercised when operating at high pressures so as to ensure that mobifity--and particle drift velocities--do not become excessive. This may well require the use, at high pressures, of particles somewhat smaller than the 3 t~m size which appears to be near the optimum at atmospheric pressure. 3.2 The axial charge density variation Due to the radial movement of charged particles the charge density p(x) decreases with distance x along which the conversion section. See Fig. 2. The current entering x=O
CO~ON,AI ~
NEEDt.Ed ~"~" I
HdGH VOLTAGE
ZERO POTENTIAL
I ~1
~-2.,-~',
~
~
~1~
~
~
~1#
~1
Since p(x -t- 3x) = p(x) q- 3x dp (x) dx one
obtains
4 p(x) =
where
~=
-
p ( x ) -6
/)g Vg
Integration then gives:
(1) 3.3 The current through the wall The current 3i which is transported to the wall along an element 3x, see Fig. 3, divides into two components x~O l - -
. . . . -"-i ~x 'l
'x+6x . . . . .
xI, L
Fig. 3. Component electric curreats within a longitudinal element of the duet wall.
8il and 8&. Since the wall resistance between the element and the low voltage end is rx, and the total duct wall resistance is rL, it follows that 8il -- r(L -- x) + R . 8i rL+R and 3i2-
rx
rL+R
. 8i.
At the point x, 8i----7rDvap(x)3x and the variation of p is given by Equation 1, so it is possible to evaluate the currents il = ]~ 8iz and i2 = ~ 8i2. The current il, which is the sum over x of all the components 8iz which flow in the negative x direction is consequently given by
i,=,rDv,f With the boundary condition that il = 0 at x = L integration yields
TER.INAt",,~
.'~ ,--%
(Resistcmce = r L.~.)
x=k
129
--'~J
.~d ,w~,j _ ~ NEEDLE
'~
RENSTANCE)
,~D%gp(O) {-- [rL -- 4BAR -- 4#Ar(L -- x)] i~ = 16[3A(rL q- R) " exp [--4fl 7 ]
+ [ r L - 4BAR] exp (--4flA)}. (2)
Fig. 2. The theoretical model of an EGD expander.
cylindrical element of length 8x must equal the current leaving it. With the assumption that axial slip between the gas and the charged particles is zero one therefore has: a
~rD2
vgp(x) =
zrD 2
vgp(x + 3x) + 7rD3xvap(x).
Similarly, the current is which is the sum over x of all the components is which flow in the positive x direction, is given by: is = ~rD2vgp(O)rL 16JSA(rL + R)
{1 [,+4,.;] ox.(
(3)
130
D. WADLOW and P. J. MUSGROVE
For i2 the boundary condition is that iz = 0 when x=0. The current collected from the gas-borne charged particles at x -----L is given by ,rD 2
ic =
~
It is important to know the magnitude of this power loss relative to the power output from the EGD expander. This power output P0 is simply given by
where iL is the current through the load. The load current has two components. Firstly, there is the proportion rL/(rL + R) of the current ie collected at the high voltage terminal, ie is given by Equation (4), and so this current component, ie~ is
(4)
vgp(L).
Part of this will flow through the load and part will flow in the negative x direction through the resistive duct wall to the low voltage end of the conversion section. If this wall current is denoted by ic~, then iea ---- ioR/(R + rL) and substitution for ie from Equation (4) and p(L) from Equation (1) leads to ~rD 2 R ica -- 4-- vg r ~
p(0) exp (-- 4/3A).
(8)
Po = i~Rz
icb --
~rD~ 4
rL va • (rL + R) " p(O) exp (-- 4flA).
(9)
The second component of the load current is part of the current deposited on the wall, i2(L), whose magnitude is given by the substitution of x = L in Equation (3). The total load current is therefore
(5)
The nett wall current, /, at any point along the conversion section is simply the algebraic sum of the component currents, i.e.
iL --
I : i2 -- il -- ica
IrD z
16/3A
. vg . p(0) .
rL
(rL 4- R)
[1 -- exp (-- 4fla)] (10)
and the power output P0 becomes
and substitution of the expressions for it, i2 and iea into this equation gives
[TrD2va rL P0 = / 16flA " p(0). R (rL~ +)
}2 . [1 -- exp (-- 4/3A)pR
(11)
~-O 2 p(0) 1--16flA.Vg.rL+--R.
{rL[1 -- exp (-- 4fla)]
From Equations (7) and (11) the ratio of the resistive wall power loss to the power output is given by (6)
--[rL + R]4flAexp(--4fl?)}.
PD _ 2tzfl A
P0 3.4 Wall resistive power loss
PD = r f : I2 dx.
Substitution from Equation (6) and integration then gives
[ zrD2vgp(O)- -I 2 "
coth 2flA --/~ -- 2,
(12)
where coth x = [1 + e x p ( - - 2 x ) ] / [ 1 - - e x p (--2x)] and tz is the ratio of the duct wall resistance to the load resistance. Measurements made by Lovett and Musgrove [5] indicated/7 ___ 0.004 and the aspect ratio A is typically about 5, so/3A is of the order 0.02. Figure 4 shows the variation of PD/Po with /z for a range of values of flA. Small values of/z correspond to a wall of low resistance and much of the current transported by the gas to the
Knowing the wall current as a function of x, the power dissipated resistively in the wall may readily be calculated. Denoting this power loss as PD one has
PD = [16BA(rL + R)J
1+
rL[l -- exp (-- 4/3A)]. {2[3A(rL + R)2[1 + exp (-- 4flA)] -- rL(rL + 2R)[1 -- exp (-- 4/3A)]}.
/
!,:°i ,oli i %, i ;°oi\
/
/ / ,,o.o,y / \\
. . .3 . . . . .
/ / /
/
~
. . .3O. . . . .
io
'
~o
/
,.ooo
......
~oo
#a
Fig. 4. Ratio of duct wall resistive power loss to output power, as a function of Iz.
(7)
The Performance of Electrogasdymunic Expanders with Slightly Conducting Walls
131
high voltage end of the duct then flows back to earth If r were varied along the length of the duct then the through the wall. This reduces the useful output power variations of Et with x, which result from variations and gives substantial wall losses, so the ratio PD/Po in L could be smoothed out. This would probably improve becomes large. At large values of ~ the load resistance the duct performance but subsequent calculations is relatively small. This means that the terminal voltage, assume, for simplicity, that r is constant along the duct. and hence the power output is reduced. Total wall losses are substantially unaltered and the ratio Po/Po 3.6 Output voltage is again large. An intermediate value of /z therefore The voltage along the duct wall is given by gives the minimum ratio of resistive wall loss to power output, and this is clearly indicated in Fig. 4. DifferentiaV(x) = -- r f I dx, tion of Equation (12) reveals that this minimum corand with the boundary condition V(0) = 0 this gives responds to a value , [ 2flAcoth 2flA ]1/2 /~D = [2flA coth~-~----- 1
(13)
V ( x ) - ~rD2vgp(O)rL 16flA(1 -t- /~) " {(/z + 1)[1 -- exp (-- 4flAx/L)] -/z[1 -- exp (-- 4~A)lx/L}.
and for small values of/3A this approximates to
, /z D
The output terminal voltage V(L) is then
~/3 ---~ 2flA"
V(L) -- zrD2vgp(O)rL . [1 -- exp (-- 4flA)] 16flA(1 + t~)
If flA = 0.02 this approximation and Equation (13) both indicate that tz~ = 43-3, and with these values for t~ and flA the resistive wall loss is only 4.7 per cent of the power output. Although this demonstrates that resistive wall losses may be kept small compared with the power output, the value of t ~ determined in this way does not give the maximum duct isentropic efficiency. This is because no allowance has yet been made for wall friction losses. The foregoing theory also imposes no restriction on the magnitude of the electric fields in the duct. These points are considered in the following sections.
3.5 Wall electric field The current in the wall is I and the electric field tangential to the wall is therefore Et -----rL Under normal operating conditions, with R =/: 0, I and hence Et are a maximum at x = 0. From Equation (6) one therefore finds that the maximum field is given by Et(O)- ¢rD2vgp(O)r 16flA(1 +/~) " {/z[1 -- exp (-- 4flA)] -- 4flA(lz + 1)}.
(14)
If breakdown along the duct wall is to be avoided this field must be kept below a limit which will depend on the working gas and its pressure and temperature. Air is assumed as the working fluid in this paper, and based on experience a surface breakdown strength of 1 MV/m at standard air density is assumed. It is further assumed that this breakdown strength is proportional to the gas density. One therefore has the condition _< 273 IEt(0)l --~ ~ - .
106P.
The maximum permissible value of the duct resistance (per unit length) is then
rm = 2"39 X 109
e#A(1 + ~) {4flA(t~ -k 1) • TD2vup(O) - - / ~ [ 1 -- exp (-- 4flA)]} -1.
(16)
(15)
(17)
and is seen to be directly proportional to rL. The output terminal voltage is not necessarily the highest voltage along the duet; under some load conditions a voltage maximum, corresponding to E ( x ) = 0 and hence to I(x) = 0, will occur between x = 0 and x = L. From Equation (6) this maximum is at
xm L
l {(/~ + 1) 4/3A' In /~
4flA } . [1 - - exp (-- 4 ~ ]
(18)
For/z large and/3A small this approximates to
L
4~A
+ 2~.4
which indicates that (xm/L) = 1 corresponds to/~ _~ ½flA. Larger values of tz give a potential peak at x < L and for very large /z, (Xra/L)~ 0"5. It will be shown later that optimum operating conditions usually correspond to /~ < (½/3A) and this gives a continuously increasing voltage along the duct. Since V(L) is proportional to r, the maximum output voltage is obtained when r = rra is substituted from Equation (15) into Equation (17). This gives 273P Vm(L) = ~ • 106 L × {4flA(/~
1 -- exp (-- 4flA)
(19)
where the first part of this expression, 273P 106]T may be recognized as the surface breakdown field strength.
3.7 Output current and power The cloud of charged particles within the conversion section produces an electric field distribution. Wilson [6] has calculated the maximum radial (Er) and axial (Ea) fields due to a cylindrical column of uniformly distributed charged particles, and obtained pD Er = Ea -- 4 ~
for A >~ 1.
(20)
132
D. W A D L O W and P. J. M U S G R O V E
For aspect ratios not large compared with unity, fields are less than Equation (20) indicates, e.g. for A = 1 the axial and radial electric fields are only 76 per cent of the value given by the equation. Wilson's calculations assume no constraint on the potentials along the walls of the conversion section, whereas with a resistive wall there will be an imposed voltage variation along the wall, as is indicated by Equation (16). The solution of Poisson's equation with this boundary constraint presents considerable difficulty analytically and the numerical solution will be the subject of a future paper. However, one may reasonably expect that the maximum field due to space charge Es is directly proportional to the charge density and for simplicity it will be assumed, following Equation (20) that E, oc pD/4 E. Rearranging this last equation indicates that p oc 4~Es/D. For air the breakdown field strength Eb at the density corresponding to standard temperature and pressure is 3 MV/m. In order to allow for uncertainties in the estimation of the space charge field in the duct it will be assumed that Es = Ed3. The maximum possible charge density pm is then given by pm --
4~ D
Eb 3"
(21)
Experimental evidence, though limited, suggests that this assumption errs on the side of pessimism. For pressures and temperatures differing from s.t.p., Equation (21) may be generalized (for air as the working fluid) to 4~ 273P p,n----~-. T " 10n (22) and substitution of this limiting value into Equation (10) giVeS iLm = 1"90
× 10_ 3 vgD
P
b*
" T" t * + l
• [I
--
exp (-- 4flA)]. (23)
The output power Pora corresponding to the maximum charge density pm and the maximum allowable duct wall resistance is given by Pore = iLmVm(L). From Equations (19) and (23) this is
Pore=5"19 × I05. ~ -
~]-
Power output ---- Power output + Wall losses + Friction losses (25) may be determined for a variety of operating conditions. The friction power loss PI = (friction pressure drop) × volume flow rate, i.e.
PI = 2fdLDvg.8
(26)
Since air is assumed to be the working fluid d = 1.293P (273/T) and the friction factor f is tabulated in many reference works, e.g. Massey [7], as a function of Reynolds number and relative surface roughness. Equations (24) and (26) reveal one of the most basic characteristics of EGD components, that the power output is proportional to the gas velocity vg, whereas friction power losses are proportional to the cube of the gas velocity. Hence the choice of relatively low operating velocities, of the order 50 m/see. Although the ultimate concern is with the isentropic efficiency it is of interest to consider the relative magnitube of the wall friction power loss and the power output• From Equations (24) and (26) this is given by T ( / z + 1) P~" = 1.07 × lO-SfflAv2a P Pom t* × {4flA(t, + 1) --/~[1 -- exp (-- 4flA)]} [1 -exp (-- 4flA)] 2
(27)
This may be minimized with respect to t*, as was done in section 3.4 for the ratio PD[Po. But since Pa' is independent of t*, minimizing Pf/Pom is equivalent to maximizing Pore, the output power. Differentiation of Equation (27) reveals that this condition corresponds to
,
[
] 1/2
t~J = _4flA -- 1 + exp (-- 4flA)J
"
(28)
flA is usually small and this condition then approximates to
.
[1 -- exp (-- 4/3A)]2 × {4/3A(t, + 1) -- t~[1 -- exp (-- 4/3A)]}"
that wall resistive power losses PD need only be a small fraction of the output power P0, and the condition minimizing Pn/Po, is given by Equation (13). But this condition does not give the highest duet efficiency since no allowance was made for wall friction losses. These losses will now be calculated, so that the overall duct isentropic efficiency,
(24)
3.8 Friction power loss The output power, given by Equation (24), is a function of several parameters. Some, such as /3, are largely outside the experimenter's control; others, such as/z, A, D and vg may be varied more readily. In some applications the objective may be to maximize the power density, i.e. the power output per unit volume (or unit weight). However, and as was stated earlier, an all-EGD-thermodynamic cycle demands high efficiency EGD components. In Section 3.4 it was shown
!
Comparing this with the condition ~D = ~/3/2/3A, which minimizes resistive wall losses, one sees that maximizing the power output--and hence minimizing P1/Pom--requires a much smaller value of ~, i.e. for flA = 0.02, ~I -----5.0 whereas ~ = 43.3. From Equation (24) and with the assumption that BA ~ 1, one finds that (Pora) _ f~ [1 + (2flA)l/2] 2 (29) (Pom)'f /* + 1 (1 + 2tAft) P
t
With flA = 0"02 and ~ = 43.3 this ratio is 0.52, i.e. the peak power output from the EGD duct is almost
The Performance of Electrogasdynamic Expanders with Slightly Conducting Walls
double the power output which corresponds to minimum resistive wall loss.
3.9 The isentropic e~ciency The expressions for the power output, the resistive wall losses and the wall friction losses can be now combined to give the overall isentropic efficiency, as defined by Equation (25). One finds that
~l=(tz+ l)-Z (2(l + l)flAcoth2flA-- l +
133
efficiency does vary with the gas velocity vg, decreasing quite rapidly as v0 increases, and Fig. 6 and subsequent figures assume that vg=50 m/see. To some extent this value is arbitrary, but a much higher velocity would preclude the attainment of high efficiency, and a much lower velocity would lead to reduced re-entrainment and particle accumulation on the duct walls. At high pressures, when the laminar sub-layer in the gas flow
[ (1)
1"07 X 10-3flafvzgT
4flA 1 + ~
P
]t1
(30)
4flA)]2P}-z/2.
(31)
+exp(--4flA)-- 1
[1 -- exp (-- 4flA)] ~'
Optimizing this with respect to/~ gives ftopt =
{2flA coth 2flA[1 exp (-- 4flA)]2P + 4"28 . lO-3fl2A2fv2oT}1/2 X {1-07. lO-3flATfv#[4flA+ exp (-- 4flA) -- 1] 4- [23A coth 2flA -- 1]. [1 -- exp (--
-
Wall friction losses are proportional to the gas density, whereas the power output is proportional to the square of the density. At high pressures friction losses consequently become less significant and tZopt tends to the value given by Equation (13), i.e. /z~. Conversely, at low pressures friction losses become dominant and tZopt tends to the limit tz~ given by Equation (28). The variation of/Zopt with pressure for several representative values of flA is shown in Fig. 5, and illustrates these tendencies. Substitution of this optimum value of t~ into Equation (30) then gives the isentropic efficiency as a function of A, f, vg, P and T. Figure 6 (a) and (b) indicate how the efficiency varies with pressure, at temperatures of 30 and 650°C respectively, for the same typical values of /3A, and for a gas velocity of 50 m/sec. It is seen that at high pressures (or more
90
80
= ~A
~
70
~ ~o GAS TEMPERATURE
= 300C
5 6 7 8 PRESSURE, A" MOSPHERES
9
MEAN AXIALGAS VELOCITY~50m/s
2O
10
1
2
3
4
10
(a)
60 95i 90
MEAN AXIAL GAS VELOCITY =5Orals
50
80
~A=0,002
BA=0"002
L0
?O 60
oE
z
2O 0.03
10
~o
GAS TEMPERATURE =650°C MEANAXIALGASVELOC]TY=50mls
0'075
.
2
.
.
3
.
.
.
~ 5 6 7 PRESSURE,ATMOSPHERES
~
'
N 3o
9
Fig. 5. Ol~lmum value of tt (duct wall resistunce/lond resistance), as a function of pressure. Gas temperature 30°C.
precisely, at high gas densities) isentropic efficiencies of over 85 per cent appear feasible.
20
10
P"ESSU~, ATMOS~RES (b)
Fig. 6 (a) Ismtropic efficiency as a fundion of lm~ure, at 30°C. 4.
Discussion
It is worth noting that the isentropic efficiency, as given by Equation (30), is not size dependent. This is an important characteristic of EGD components since the efficiency of most alternative energy conversion systems is sensitive to size and these alternative systems suffer from reduced efficiency at low power levels. The
(b) Isentropicefficiencyas a functionof presmre, at 650°C. is thinner, enhanced re-entrainment may allow lower carrier gas velocities and, consequently, higher efficiencies. The isentropic efficiency also depends on the friction factor f for the flow through the duct. A relative roughness of 0.0004 has been assumed for the internal surface
134
D. W A D L O W and P. J. MUSGROVE
of the duct, corresponding to a surface roughness of 10/zm on a 25 mm diameter duct. The value o f f corresponding to a given Reynolds number was then determined from [7]. Friction factor is not a strong function of surface roughness. In fact a 50 per cent variation in relative roughness produces a variation in the overall efficiency only of the order 1 per cent. There is some experimental evidence to indicate that a relatively rough duct wall facilitates re-entrainment. This would give a slightly higher value o f f , but might allow operation at a lower gas velocity and hence at a higher efficiency. There is also experimental evidence that the presence of micron-sized particles in a gas flow reduces f significantly, but no allowance has been made in these calculations for any such reduction. Figure 6(a) and (b) indicate the variation of isentropic efficiency with pressure for temperatures of 30°C and 650°C respectively, and show very similar trends. The 275
25C 225 ASPECT RA'RO= 7.5 ......
0 2 m ~ BEPOSITION 0 5m/sJ" VELOCITY MEAN AXIAL GAS VELOCffY : 50m/s
200 i /
175
ac 8f~2S 100 75 50 I ~"~ "~x~ '~\\~ ~ 1 , 1
REgSTIVE WALL POWER SSOL
x 100
25 .........................
~7-2%
major difference between these two figures is that, due to the effect of temperature on density, operation at high temperatures requires higher pressures for a given efficiency. The increased viscosity associated with high temperature operation also has an effect on the efficiency, but this is relatively slight. As stated in Section 3.9 the curves shown in Fig. 6 assume that /z = ftopt and indicate that for small values of/3A, efficiencies up to 90 per cent appear feasible. For the values of va = 0.2 m/see and vu = 50 m/see quoted in Ref. 5, flA = 0.002 would correspond to A = 0.5. Increasing the deposition velocity from 0.2 to 0.5 m/see, hence increasing flA from 0.002 to 0.005, is seen to have very little effect at this small aspect ratio. However if the aspect ratio is 7.5, v a - 0.2 m/see gives flA = 0.03 and this reduces the efficiency at elevated pressures by about 10 per cent. At this aspect ratio there is also increased sensitivity to the magnitude of the deposition velocity, as the curves for ~A = 0.075 (e.g. A = 7.5, va = 0.5 m/see, vu = 50 m/see) indicate. One can conclude from these figures that flA "~ 0.03 if isentropic efficiencies of about 85 per cent are to be obtained within the pressure range considered. It is of interest to know whether the inefficiencies are due primarily to friction losses or to resistive wall losses. Figure 7(a) and (b) show these losses separately as a function of pressure for aspect ratios of 0.5 and 7-5 respectively and for two values of yd. These and subsequent figures assume T = 30°C as well as vg = 50 m/see. The component losses are largest at low pressures, and at the larger values of va and A, as would be expected from Fig. 6. Throughout the range of pressures considered, the friction losses exceed the resistive wall losses, though at high pressures and large A these two losses do become of comparable magnitude. Figure 8 shows how the output current (Equation (23)) and output voltage (Equation (19)) vary with pressure for the four permutations of va and A previously considered. It is necessary to specify the duct diameter, and this is taken to be 25 ram. Increasing the aspect ratio on a duct of a given diameter implies increasing the duct
PRESSURE,ATNOSPNERES
(a)
1000 . . . . .
130 ASPECT RATIO= 0.5
120
.....
110 ~\
~
100 ~ \
o.2rn/s "/DEPOSITION
0 5m/s~ VELOCITY
J
/ /"
~ WALL FRICTION POWER LOSS X 100
|
"1
/ DEPOSITION VELOCITY(Vd) : 0,2mls
t\
/'/
-1400
""
MEAN AXIAL GAS VELOCITY = 50m/s
90 80
['~S00
75].ASPECT RATIO
CONTINOoSuJs LINES REPRESENT VOLTAGE WHEREAS INTERRUPTED LINES REPRESENT CURRENT MEAN AXIAL G~S VELOCITY = 50 m/s DUCT DIAMETER = 0,025m
!,oo
,"/'" .'" -~" ] |
<
,-%+
~ o. 5o 30 20 RESISTIVE WALL POWER LOSS X100" . . . . . . . . . . . . . . . . . . . 10 ~ t POWEROUTPUT . _ ---'~ - - - - - - -31 . . . . a. . . . . . . . t-----r . . . . r- . . . . 2 3 4 ~ 6 7 8 PRESSURE, ATMOSPHERES
~
g
__111,2°/o O : ~ 10
(b)
Fig. 7. (a) Power losses as a f,mction of pressure, 30°C. (b) Power losses as a function of pressure, 30°C.
2
3
4
5 6 7 8 Pressure (Atmospheres)
9
10
Fig. 8. Terminal voltage and load current as a function of pressure, at 30°C.
The Performance of Eiectrogasdynamic Expanders with Slightly Conducting Walls
length. The voltages corresponding to A = 7.5 are therefore much larger than those corresponding to A = 0"5. Since efficiency is independent of scaling, the duct diameter may be selected, in conjunction with the aspect ratio and operating pressure, to give any desired output within the practicable range 100-1000kV. Increased duct diameters will, from Equation (23), give increased currents and the power output (Equation (24)) increases as the square of the diameter. However the power density (i.e. the power per unit duct volume) is inversely proportional to the diameter, if A is maintained constant. For the 25 mm diameter duct the variation of power density with pressure is shown in Fig. 9. At 10 atmospheres, 30°C, the power density
....
4
7 : 5 ) Aspect Rotio
Meon Axial Gos Vetocity = 50mls
/
/
E Deposition
/ /
- ""
izl
g. 1 Pressure, Atmospheres
Fig. 9. Power density as a function of pressure, at 30°C.
is of the order 2-4 MW/m 8, the higher values being associated with the smaller aspect ratio and the lower deposition velocity. Figure 5 indicated how the optimum value of ~, 100 90 80
0
002
0005
- 70
135
varied with flA and pressure, and these values were used in subsequent calculations of isentropic efficiency etc. In practice it may be difficult, due to constraints on materials and due also to variations in the operating temperature, to maintain ~ at the optimum. Figure 10 therefore shows how the isentropic efficiency varies with ~ for one particular operating condition (10 atmospheres, 30°C). At low values of flA the isentropic efficiency is seen to be quite insensitive to the precise value of/~, but if EGD ducts need to be operated at higher values of flA, it is apparent that much closer control must be exercised on the value of/~. 5. Conclusions
It has been shown that the reduction in performance of EGD expanders, due to the radial drift of particulate charge carriers, may be minimized by the use of slightly conducting duct walls. The theory reveals the importance of operating at low values of the parameter flA, that is at low values of the product of duct aspect ratio and particle deposition velocity. Since the latter is, to a large extent, beyond the experimenter's control, this indicates the need for relatively low aspect ratio ducts. With the assumption that air is the working fluid, and that the mean axial velocity is 50 m/sec, the calculations indicate that one can expect isentropic efficiencies of the order 85 per cent provided that flA ~ 0.03 and that the gas density is at least ten times atmospheric. The power density is predicted to be of the order 2-4 MW/m a for a 25 mm diameter duct. Experiments currently in progress at Reading University will test these predictions. Acknowledgements This Research has been sponsored in part by the Air Force Office of Scientific Research (AFSC), U.S.A.F., through grant no. AFOSR 74-2647 and in part by the Science Research Council, U.K., through grant B/SR/9898.
References
m 60
~ so
~ 30 20
OAS PRESSURE = 10 ATM. GAS TEMPERATURE = 30*C i','EAN AXIAL GAS VELOCITY = 50mls
P
Fig. I0. Isentropie eWaeiencyas a function of Ix.
[1] P. J. Musgrove, Electronics & Power 19, 327 (1973). [2] P. J. Musgrove and A. D. Wilson, Energy Conversion 12, 21 (1972). [3] M. Lawson and H. Von Ohain, J. Engng Power p. 201 (April 1971). [4] E. L. Collier and M. C. Gourdine, AIAA J. 6, 2278 (1968). [5] C. D. Lover and P. J. Musgrove, Proc. 2nd Int. Conf. on Pneumatic Transport in Pipes--Paper D5 BHRA Engng, Cranfield (1973). [6] A. D. Wilson, Ph.D. Thesis, Reading University (1970). [7] B. S. Masscy, Mechanics of Fluids, Van Nostrand, New York (1968).
Vol. 15, pp. 127-135.
Pergamon Press, 1976. Printed in Great Britain
The Pe ormance of Elect asdynamic Expenders with Slightly ConductingWalls D, WADLOW and P, J, MUSGROVEt (Received 16 January 1975) AlOft't--In electrogasdynamic (EGD) devices the radial movement of charge carders to insulating duct walls can produce high parasitic electric fields and greatly degrade the overall performance. In principle these parasitic fields may be reduced by constructing the EGD ducts from slightly conducting materials, but there is then a power loss due to current flow through the wall. This paper
examines the effect of particle deposition velocity,wall resistivity,load resistance and aspect ratio, as well as ttuid friction and gas density, on this resistive wall power loss and on the overall performance of EGD devices. Given suitable and realistic values of the relevant parameters, isentropic efficienciesin excess of 85 per cent are predicted at pressures of the order 10-40 atmospheres. Electrogasdynamics deposition
Energy conversion
Charged aerosols
1. Introduction Electrogasdynamic (EGD) energy conversion systems promise to provide a means of converting heat to electricity efficiently, and have the advantage that no mechanical moving parts are required. For example Musgrove [1] has predicted an overall efficiency of 42 per cent for EGD systems operating on an Ericsson cycle between a heat source at 650°C and a heat sink at 30°C provided that the power output is in excess of a few kilowatts. Such systems therefore merit consideration for a wide range of applications, and it has in the past been suggested that EGD systems be used in central power stations, operating at the gigawatt (109 W) power level. Although Nuclear/EGD or Fossil Fuel/EGD power stations may one day be built, we believe that EGD is better suited in the near future for applications at lower power levels, e.g. for power outputs in the range from a few kilowatts to a few megawatts. In particular EGD systems appear eminently suitable for use in Total Energy Schemes. In this role they would appear to have three principal attractions: (1) EGD systems require no mechanical moving parts and so, when fully developed, should be extremely reliable. (2) The energy input to EGD systems is from an external heat source, and not by internal combustion of fuel. They should therefore have the low noise and pollution levels that one associates with continuous, external combustion systems. (3) The efficiency of EGD systems is independent of the unit size and so efficiencies of the order of 40 per cent should be obtainable at kilowatt as well as megawatt power levels. The efficiency predictions, and consequently the overall prospects for EGD systems, rest on a fundamental assumption. This is that the two principal EGD compo~"Department of Engineering and Cybernetics, University of Reading, Whiteknights, Reading, RG6 2AY, Berkshire, England.
Direct conversion
Corona discharge
Particle
nents, i.e. the EGD expander stage and the EGD compressor stage, can be developed to operate at isentropic efficiencies of the order of 85 per cent. To date this has not been demonstrated experimentally, and until this is done the outlook for EGD energy conversion remains uncertain. Experiments at Reading University on a newly constructed high pressure EGD test rig should soon resolve this uncertainty; in the meantime this paper examines theoretically the prospects for operating EGD components at high isentropic efficiencies. The general characteristics of an EGD expander stage are shown in Fig. 1. A gas is used to propel unipolar charged particles against a strong electric field, so the ~, EARTHEDRING -Hi" ELECTROOE ,TTB~TOR NEEI1E""~1 UkCH~ffJ~o
e e
e
NEGATWEHIGH,,,.,j VOLTAC~TERNINAL 1 [ ®
®
e
o
o
AEROSOL
Fig. I. Sehemafie e l e e ~ r o ~ m d c
generator.
electrical potential energy of each particle increases, at the expense of the energy in the flowing gas. Typically current is supplied by the upstream corona discharge at a voltage measured in tens of kilovolts and this current is subsequently collected, via a self-induced corona discharge near the high voltage terminal, at a voltage of several hundred kilovolts. There is direct conversion of part of the power in the flowing gas to electricity and associated with this there is a small pressure drop along the length of the EGD duct--hence its description as an EGD expander. An EGD compressor stage appears almost identical to the expander shown in Fig. 1, but in this case the charged particles have the opposite polarity to that of the high voltage terminal. The electric field consequently propels the particles through the duct, and the energy imparted to them in this way is transmitted to the surrounding gas by viscous forces. Since the EGD compressor stage imparts energy to the gas flowing 127
128
D. WADLOW and P. J. MUSGROVE
through it, it requires an electrical power input. In a complete EGD thermodynamic cycle, as proposed in [1] part of the electrical power output from the EGD expander stages is used to supply the power required by the EGD compressor stages. In EGD devices a variety of charge carriers may be used, but experiments at Reading University have concentrated on the use of fine solid particles, typically silicon carbide particles a few microns in diameter. Fine particles such as these ensure a low mobility; hence slip between the charged particles and the carrier gas is small and so are the slip losses. Experiments, see Musgrove and Wilson [2], have shown that with the right choice of particle size and number density, losses associated with the particle charging process are small. However, fluid friction losses are not negligible, and since these power losses are proportional to the cube of the gas velocity, whereas the output power is only directly proportional to the velocity, efficient operation requires relatively low gas velocities, of the order 50 m/see. For velocities in this region, and at pressures of the order 10 atmospheres, previous calculations, e.g. Musgrove [1], have indicated that isentropic efficieneies over 85 per cent are feasible. But these calculations omit an important loss mechanism, which experiments have shown to be very significant. This loss results from the radial movement of charged particles to the walls of the EGD duct, due in part to turbulence in the flow and in part to the effect of radial electric fields. Charge builds up on the insulating duct walls and produces strong parasitic electric fields which greatly reduce the performance. Experiments at Reading have shown, for example, that these parasitic fields reduce the power output of an EGD expander by a factor of about four. There are several methods by which this problem may be avoided, or the consequences minimized. One possibility is to prevent the deposition of charge on insulating walls by surrounding the charge carrying gas by a gas stream which carries no charged particles, see for example Lawson and Von Ohain [3]. A second possibility is to generate a.c. not d.c., so that there is no accumulation of charge on the duct walls and parasitic electric fields are avoided. This may ultimately prove to be an attractive proposition, but little work has so far been done on a.c. EGD devices. The third possibility, and the one considered in detail in this paper, is to replace the insulating duct wall of the conventional EGD expander (or compressor) by a wall that is slightly conducting. The charge deposited on the wall by the radial drift of particles can then be allowed to leak away without giving rise to excessive fields within the duct. That this technique can be effective was demonstrated by Collier and Gourdine [4]. However, the use of a slightly conducting wall does introduce two power loss mechanisms. Firstly, there is the power lost resistively as the charge deposited on the walls leaks away to earth through the duct walls. Secondly, there is the power lost due to the fact that part of the gross power output from the EGD expander flows to earth through the duct wall
rather than through the output load. Although this second loss mechanism is potentially the more serious, it should be pointed out that the quest for maximum cycle efficiency will no doubt lead, in time, to EGD systems being operated at peak cycle temperatures limited only by the condition that the duct wall must not be too conducting. If for this reason EGD expanders are to have slightly conducting walls, then it seems reasonable to consider using this wall conductivity as a means of eliminating the parasitic fields produced by the radial movement of charged particles. This paper therefore examines the magnitude of the power losses associated with the use of a slightly conducting duct wall. The effect of resistive wall losses and fluid friction losses on the isentropic efficiency of an EGD expander is also calculated as a function of the particle deposition velocity, duct aspect ratio, wall resistance, etc. Although the calculations presented relate specifically to EGD expanders of circular cross-section, they may readily be extended to predict the performance characteristics for other geometries and for EGD compressors. 2. Nomenclature Ratio of length to diameter of duct, i.e. aspect ratio, L/D, gas density, duct diameter, electricfield strength, wall friction factor, electriccurrent, duct length, pressure,in atmospheres, power, resistanceper unit length of duct wall, load resistance, absolutetemperature, radial deposition velocity, mean axial gas velocity, voltage, distancealong duct from low-voltageterminal, ratio of deposition velocity to gas velocity,v~/vg, absolute permittivity, ,7 isentropicefficiency,power output/(power output + losses), ratio of total duct wall resistance to load resistance, rL/R, p chargedensity. Unless otherwise stated S.I. units are used throughout.
A d D E f t', I L p P r R T va va V x
3. Theoretical Model
3.1 Assumptions In the model which is developed subsequently a number of assumptions are implicit. Firstly, it is assumed that the charged particles which migrate to the wall surrender their charge to the wall. If the charge carrying particles were too insulating it would be possible for charge to build up on the inside surface of the duct, despite the wall conductivity. The use of a relatively conducting material such as silicon carbide prevents this. Also, when using particulate charge carriers, particles are constantly being deposited and re-entrained. Under steady-state conditions there is a thin layer of particles adhering to the wall at any instant in time. This deposit is about a monolayer thick and is assumed to have no significant effect on the overall wall resistivity. The duct wall is assumed to be uniformly resistive, which implies a constant duct wall cross-section and a uniform resistivity. This assumption simplifies
The Performance of Electrogasdynamic Expanders with Slightly Conducting Walls
the analysis, but optimum performance may well be associated with a controlled variation of the resistivity along the length of the duct. The model further assumes that triboelectric charging at the walls is insignificant compared with the charge given to particles as they pass through the corona discharge. Experience indicates that this assumption is a reasonable one. If it were not, it would imply a high value of the particle deposition velocity, whereas measurements by Lovett and Musgrove [5], have shown that this is only a small fraction of the axial gas velocity. No attempt has been made, in the analysis, to relate the deposition velocity to the particle size, the flow Reynolds number or the magnitude of the electric fields within the duct. Instead the variation of the EGD duct performance is examined for a range of typical values of the deposition velocity. For given operating conditions this deposition velocity is assumed to be substantially constant along the length of the duct. At the outlet of the EGD duct the charged particles are neutralized by a space-charge induced corona discharge. This is assumed to remove effectively all the charge carded by the particles, and failure to achieve this would lead to reduced performance. However, experimental evidence, in the form of current balance measurements, has shown that the discharge corona does remove almost all the charge on the particles. Finally, it is assumed that the particle mobility is small enough for slip losses to be negligible. The mobility depends on the particle size and the charge per particle and experiments at atmospheric pressure, see for example [5], have confirmed that the mobility of micron sized particles is low. However, care will need to be exercised when operating at high pressures so as to ensure that mobifity--and particle drift velocities--do not become excessive. This may well require the use, at high pressures, of particles somewhat smaller than the 3 t~m size which appears to be near the optimum at atmospheric pressure. 3.2 The axial charge density variation Due to the radial movement of charged particles the charge density p(x) decreases with distance x along which the conversion section. See Fig. 2. The current entering x=O
CO~ON,AI ~
NEEDt.Ed ~"~" I
HdGH VOLTAGE
ZERO POTENTIAL
I ~1
~-2.,-~',
~
~
~1~
~
~
~1#
~1
Since p(x -t- 3x) = p(x) q- 3x dp (x) dx one
obtains
4 p(x) =
where
~=
-
p ( x ) -6
/)g Vg
Integration then gives:
(1) 3.3 The current through the wall The current 3i which is transported to the wall along an element 3x, see Fig. 3, divides into two components x~O l - -
. . . . -"-i ~x 'l
'x+6x . . . . .
xI, L
Fig. 3. Component electric curreats within a longitudinal element of the duet wall.
8il and 8&. Since the wall resistance between the element and the low voltage end is rx, and the total duct wall resistance is rL, it follows that 8il -- r(L -- x) + R . 8i rL+R and 3i2-
rx
rL+R
. 8i.
At the point x, 8i----7rDvap(x)3x and the variation of p is given by Equation 1, so it is possible to evaluate the currents il = ]~ 8iz and i2 = ~ 8i2. The current il, which is the sum over x of all the components 8iz which flow in the negative x direction is consequently given by
i,=,rDv,f With the boundary condition that il = 0 at x = L integration yields
TER.INAt",,~
.'~ ,--%
(Resistcmce = r L.~.)
x=k
129
--'~J
.~d ,w~,j _ ~ NEEDLE
'~
RENSTANCE)
,~D%gp(O) {-- [rL -- 4BAR -- 4#Ar(L -- x)] i~ = 16[3A(rL q- R) " exp [--4fl 7 ]
+ [ r L - 4BAR] exp (--4flA)}. (2)
Fig. 2. The theoretical model of an EGD expander.
cylindrical element of length 8x must equal the current leaving it. With the assumption that axial slip between the gas and the charged particles is zero one therefore has: a
~rD2
vgp(x) =
zrD 2
vgp(x + 3x) + 7rD3xvap(x).
Similarly, the current is which is the sum over x of all the components is which flow in the positive x direction, is given by: is = ~rD2vgp(O)rL 16JSA(rL + R)
{1 [,+4,.;] ox.(
(3)
130
D. WADLOW and P. J. MUSGROVE
For i2 the boundary condition is that iz = 0 when x=0. The current collected from the gas-borne charged particles at x -----L is given by ,rD 2
ic =
~
It is important to know the magnitude of this power loss relative to the power output from the EGD expander. This power output P0 is simply given by
where iL is the current through the load. The load current has two components. Firstly, there is the proportion rL/(rL + R) of the current ie collected at the high voltage terminal, ie is given by Equation (4), and so this current component, ie~ is
(4)
vgp(L).
Part of this will flow through the load and part will flow in the negative x direction through the resistive duct wall to the low voltage end of the conversion section. If this wall current is denoted by ic~, then iea ---- ioR/(R + rL) and substitution for ie from Equation (4) and p(L) from Equation (1) leads to ~rD 2 R ica -- 4-- vg r ~
p(0) exp (-- 4/3A).
(8)
Po = i~Rz
icb --
~rD~ 4
rL va • (rL + R) " p(O) exp (-- 4flA).
(9)
The second component of the load current is part of the current deposited on the wall, i2(L), whose magnitude is given by the substitution of x = L in Equation (3). The total load current is therefore
(5)
The nett wall current, /, at any point along the conversion section is simply the algebraic sum of the component currents, i.e.
iL --
I : i2 -- il -- ica
IrD z
16/3A
. vg . p(0) .
rL
(rL 4- R)
[1 -- exp (-- 4fla)] (10)
and the power output P0 becomes
and substitution of the expressions for it, i2 and iea into this equation gives
[TrD2va rL P0 = / 16flA " p(0). R (rL~ +)
}2 . [1 -- exp (-- 4/3A)pR
(11)
~-O 2 p(0) 1--16flA.Vg.rL+--R.
{rL[1 -- exp (-- 4fla)]
From Equations (7) and (11) the ratio of the resistive wall power loss to the power output is given by (6)
--[rL + R]4flAexp(--4fl?)}.
PD _ 2tzfl A
P0 3.4 Wall resistive power loss
PD = r f : I2 dx.
Substitution from Equation (6) and integration then gives
[ zrD2vgp(O)- -I 2 "
coth 2flA --/~ -- 2,
(12)
where coth x = [1 + e x p ( - - 2 x ) ] / [ 1 - - e x p (--2x)] and tz is the ratio of the duct wall resistance to the load resistance. Measurements made by Lovett and Musgrove [5] indicated/7 ___ 0.004 and the aspect ratio A is typically about 5, so/3A is of the order 0.02. Figure 4 shows the variation of PD/Po with /z for a range of values of flA. Small values of/z correspond to a wall of low resistance and much of the current transported by the gas to the
Knowing the wall current as a function of x, the power dissipated resistively in the wall may readily be calculated. Denoting this power loss as PD one has
PD = [16BA(rL + R)J
1+
rL[l -- exp (-- 4/3A)]. {2[3A(rL + R)2[1 + exp (-- 4flA)] -- rL(rL + 2R)[1 -- exp (-- 4/3A)]}.
/
!,:°i ,oli i %, i ;°oi\
/
/ / ,,o.o,y / \\
. . .3 . . . . .
/ / /
/
~
. . .3O. . . . .
io
'
~o
/
,.ooo
......
~oo
#a
Fig. 4. Ratio of duct wall resistive power loss to output power, as a function of Iz.
(7)
The Performance of Electrogasdymunic Expanders with Slightly Conducting Walls
131
high voltage end of the duct then flows back to earth If r were varied along the length of the duct then the through the wall. This reduces the useful output power variations of Et with x, which result from variations and gives substantial wall losses, so the ratio PD/Po in L could be smoothed out. This would probably improve becomes large. At large values of ~ the load resistance the duct performance but subsequent calculations is relatively small. This means that the terminal voltage, assume, for simplicity, that r is constant along the duct. and hence the power output is reduced. Total wall losses are substantially unaltered and the ratio Po/Po 3.6 Output voltage is again large. An intermediate value of /z therefore The voltage along the duct wall is given by gives the minimum ratio of resistive wall loss to power output, and this is clearly indicated in Fig. 4. DifferentiaV(x) = -- r f I dx, tion of Equation (12) reveals that this minimum corand with the boundary condition V(0) = 0 this gives responds to a value , [ 2flAcoth 2flA ]1/2 /~D = [2flA coth~-~----- 1
(13)
V ( x ) - ~rD2vgp(O)rL 16flA(1 -t- /~) " {(/z + 1)[1 -- exp (-- 4flAx/L)] -/z[1 -- exp (-- 4~A)lx/L}.
and for small values of/3A this approximates to
, /z D
The output terminal voltage V(L) is then
~/3 ---~ 2flA"
V(L) -- zrD2vgp(O)rL . [1 -- exp (-- 4flA)] 16flA(1 + t~)
If flA = 0.02 this approximation and Equation (13) both indicate that tz~ = 43-3, and with these values for t~ and flA the resistive wall loss is only 4.7 per cent of the power output. Although this demonstrates that resistive wall losses may be kept small compared with the power output, the value of t ~ determined in this way does not give the maximum duct isentropic efficiency. This is because no allowance has yet been made for wall friction losses. The foregoing theory also imposes no restriction on the magnitude of the electric fields in the duct. These points are considered in the following sections.
3.5 Wall electric field The current in the wall is I and the electric field tangential to the wall is therefore Et -----rL Under normal operating conditions, with R =/: 0, I and hence Et are a maximum at x = 0. From Equation (6) one therefore finds that the maximum field is given by Et(O)- ¢rD2vgp(O)r 16flA(1 +/~) " {/z[1 -- exp (-- 4flA)] -- 4flA(lz + 1)}.
(14)
If breakdown along the duct wall is to be avoided this field must be kept below a limit which will depend on the working gas and its pressure and temperature. Air is assumed as the working fluid in this paper, and based on experience a surface breakdown strength of 1 MV/m at standard air density is assumed. It is further assumed that this breakdown strength is proportional to the gas density. One therefore has the condition _< 273 IEt(0)l --~ ~ - .
106P.
The maximum permissible value of the duct resistance (per unit length) is then
rm = 2"39 X 109
e#A(1 + ~) {4flA(t~ -k 1) • TD2vup(O) - - / ~ [ 1 -- exp (-- 4flA)]} -1.
(16)
(15)
(17)
and is seen to be directly proportional to rL. The output terminal voltage is not necessarily the highest voltage along the duet; under some load conditions a voltage maximum, corresponding to E ( x ) = 0 and hence to I(x) = 0, will occur between x = 0 and x = L. From Equation (6) this maximum is at
xm L
l {(/~ + 1) 4/3A' In /~
4flA } . [1 - - exp (-- 4 ~ ]
(18)
For/z large and/3A small this approximates to
L
4~A
+ 2~.4
which indicates that (xm/L) = 1 corresponds to/~ _~ ½flA. Larger values of tz give a potential peak at x < L and for very large /z, (Xra/L)~ 0"5. It will be shown later that optimum operating conditions usually correspond to /~ < (½/3A) and this gives a continuously increasing voltage along the duct. Since V(L) is proportional to r, the maximum output voltage is obtained when r = rra is substituted from Equation (15) into Equation (17). This gives 273P Vm(L) = ~ • 106 L × {4flA(/~
1 -- exp (-- 4flA)
(19)
where the first part of this expression, 273P 106]T may be recognized as the surface breakdown field strength.
3.7 Output current and power The cloud of charged particles within the conversion section produces an electric field distribution. Wilson [6] has calculated the maximum radial (Er) and axial (Ea) fields due to a cylindrical column of uniformly distributed charged particles, and obtained pD Er = Ea -- 4 ~
for A >~ 1.
(20)
132
D. W A D L O W and P. J. M U S G R O V E
For aspect ratios not large compared with unity, fields are less than Equation (20) indicates, e.g. for A = 1 the axial and radial electric fields are only 76 per cent of the value given by the equation. Wilson's calculations assume no constraint on the potentials along the walls of the conversion section, whereas with a resistive wall there will be an imposed voltage variation along the wall, as is indicated by Equation (16). The solution of Poisson's equation with this boundary constraint presents considerable difficulty analytically and the numerical solution will be the subject of a future paper. However, one may reasonably expect that the maximum field due to space charge Es is directly proportional to the charge density and for simplicity it will be assumed, following Equation (20) that E, oc pD/4 E. Rearranging this last equation indicates that p oc 4~Es/D. For air the breakdown field strength Eb at the density corresponding to standard temperature and pressure is 3 MV/m. In order to allow for uncertainties in the estimation of the space charge field in the duct it will be assumed that Es = Ed3. The maximum possible charge density pm is then given by pm --
4~ D
Eb 3"
(21)
Experimental evidence, though limited, suggests that this assumption errs on the side of pessimism. For pressures and temperatures differing from s.t.p., Equation (21) may be generalized (for air as the working fluid) to 4~ 273P p,n----~-. T " 10n (22) and substitution of this limiting value into Equation (10) giVeS iLm = 1"90
× 10_ 3 vgD
P
b*
" T" t * + l
• [I
--
exp (-- 4flA)]. (23)
The output power Pora corresponding to the maximum charge density pm and the maximum allowable duct wall resistance is given by Pore = iLmVm(L). From Equations (19) and (23) this is
Pore=5"19 × I05. ~ -
~]-
Power output ---- Power output + Wall losses + Friction losses (25) may be determined for a variety of operating conditions. The friction power loss PI = (friction pressure drop) × volume flow rate, i.e.
PI = 2fdLDvg.8
(26)
Since air is assumed to be the working fluid d = 1.293P (273/T) and the friction factor f is tabulated in many reference works, e.g. Massey [7], as a function of Reynolds number and relative surface roughness. Equations (24) and (26) reveal one of the most basic characteristics of EGD components, that the power output is proportional to the gas velocity vg, whereas friction power losses are proportional to the cube of the gas velocity. Hence the choice of relatively low operating velocities, of the order 50 m/see. Although the ultimate concern is with the isentropic efficiency it is of interest to consider the relative magnitube of the wall friction power loss and the power output• From Equations (24) and (26) this is given by T ( / z + 1) P~" = 1.07 × lO-SfflAv2a P Pom t* × {4flA(t, + 1) --/~[1 -- exp (-- 4flA)]} [1 -exp (-- 4flA)] 2
(27)
This may be minimized with respect to t*, as was done in section 3.4 for the ratio PD[Po. But since Pa' is independent of t*, minimizing Pf/Pom is equivalent to maximizing Pore, the output power. Differentiation of Equation (27) reveals that this condition corresponds to
,
[
] 1/2
t~J = _4flA -- 1 + exp (-- 4flA)J
"
(28)
flA is usually small and this condition then approximates to
.
[1 -- exp (-- 4/3A)]2 × {4/3A(t, + 1) -- t~[1 -- exp (-- 4/3A)]}"
that wall resistive power losses PD need only be a small fraction of the output power P0, and the condition minimizing Pn/Po, is given by Equation (13). But this condition does not give the highest duet efficiency since no allowance was made for wall friction losses. These losses will now be calculated, so that the overall duct isentropic efficiency,
(24)
3.8 Friction power loss The output power, given by Equation (24), is a function of several parameters. Some, such as /3, are largely outside the experimenter's control; others, such as/z, A, D and vg may be varied more readily. In some applications the objective may be to maximize the power density, i.e. the power output per unit volume (or unit weight). However, and as was stated earlier, an all-EGD-thermodynamic cycle demands high efficiency EGD components. In Section 3.4 it was shown
!
Comparing this with the condition ~D = ~/3/2/3A, which minimizes resistive wall losses, one sees that maximizing the power output--and hence minimizing P1/Pom--requires a much smaller value of ~, i.e. for flA = 0.02, ~I -----5.0 whereas ~ = 43.3. From Equation (24) and with the assumption that BA ~ 1, one finds that (Pora) _ f~ [1 + (2flA)l/2] 2 (29) (Pom)'f /* + 1 (1 + 2tAft) P
t
With flA = 0"02 and ~ = 43.3 this ratio is 0.52, i.e. the peak power output from the EGD duct is almost
The Performance of Electrogasdynamic Expanders with Slightly Conducting Walls
double the power output which corresponds to minimum resistive wall loss.
3.9 The isentropic e~ciency The expressions for the power output, the resistive wall losses and the wall friction losses can be now combined to give the overall isentropic efficiency, as defined by Equation (25). One finds that
~l=(tz+ l)-Z (2(l + l)flAcoth2flA-- l +
133
efficiency does vary with the gas velocity vg, decreasing quite rapidly as v0 increases, and Fig. 6 and subsequent figures assume that vg=50 m/see. To some extent this value is arbitrary, but a much higher velocity would preclude the attainment of high efficiency, and a much lower velocity would lead to reduced re-entrainment and particle accumulation on the duct walls. At high pressures, when the laminar sub-layer in the gas flow
[ (1)
1"07 X 10-3flafvzgT
4flA 1 + ~
P
]t1
(30)
4flA)]2P}-z/2.
(31)
+exp(--4flA)-- 1
[1 -- exp (-- 4flA)] ~'
Optimizing this with respect to/~ gives ftopt =
{2flA coth 2flA[1 exp (-- 4flA)]2P + 4"28 . lO-3fl2A2fv2oT}1/2 X {1-07. lO-3flATfv#[4flA+ exp (-- 4flA) -- 1] 4- [23A coth 2flA -- 1]. [1 -- exp (--
-
Wall friction losses are proportional to the gas density, whereas the power output is proportional to the square of the density. At high pressures friction losses consequently become less significant and tZopt tends to the value given by Equation (13), i.e. /z~. Conversely, at low pressures friction losses become dominant and tZopt tends to the limit tz~ given by Equation (28). The variation of/Zopt with pressure for several representative values of flA is shown in Fig. 5, and illustrates these tendencies. Substitution of this optimum value of t~ into Equation (30) then gives the isentropic efficiency as a function of A, f, vg, P and T. Figure 6 (a) and (b) indicate how the efficiency varies with pressure, at temperatures of 30 and 650°C respectively, for the same typical values of /3A, and for a gas velocity of 50 m/sec. It is seen that at high pressures (or more
90
80
= ~A
~
70
~ ~o GAS TEMPERATURE
= 300C
5 6 7 8 PRESSURE, A" MOSPHERES
9
MEAN AXIALGAS VELOCITY~50m/s
2O
10
1
2
3
4
10
(a)
60 95i 90
MEAN AXIAL GAS VELOCITY =5Orals
50
80
~A=0,002
BA=0"002
L0
?O 60
oE
z
2O 0.03
10
~o
GAS TEMPERATURE =650°C MEANAXIALGASVELOC]TY=50mls
0'075
.
2
.
.
3
.
.
.
~ 5 6 7 PRESSURE,ATMOSPHERES
~
'
N 3o
9
Fig. 5. Ol~lmum value of tt (duct wall resistunce/lond resistance), as a function of pressure. Gas temperature 30°C.
precisely, at high gas densities) isentropic efficiencies of over 85 per cent appear feasible.
20
10
P"ESSU~, ATMOS~RES (b)
Fig. 6 (a) Ismtropic efficiency as a fundion of lm~ure, at 30°C. 4.
Discussion
It is worth noting that the isentropic efficiency, as given by Equation (30), is not size dependent. This is an important characteristic of EGD components since the efficiency of most alternative energy conversion systems is sensitive to size and these alternative systems suffer from reduced efficiency at low power levels. The
(b) Isentropicefficiencyas a functionof presmre, at 650°C. is thinner, enhanced re-entrainment may allow lower carrier gas velocities and, consequently, higher efficiencies. The isentropic efficiency also depends on the friction factor f for the flow through the duct. A relative roughness of 0.0004 has been assumed for the internal surface
134
D. W A D L O W and P. J. MUSGROVE
of the duct, corresponding to a surface roughness of 10/zm on a 25 mm diameter duct. The value o f f corresponding to a given Reynolds number was then determined from [7]. Friction factor is not a strong function of surface roughness. In fact a 50 per cent variation in relative roughness produces a variation in the overall efficiency only of the order 1 per cent. There is some experimental evidence to indicate that a relatively rough duct wall facilitates re-entrainment. This would give a slightly higher value o f f , but might allow operation at a lower gas velocity and hence at a higher efficiency. There is also experimental evidence that the presence of micron-sized particles in a gas flow reduces f significantly, but no allowance has been made in these calculations for any such reduction. Figure 6(a) and (b) indicate the variation of isentropic efficiency with pressure for temperatures of 30°C and 650°C respectively, and show very similar trends. The 275
25C 225 ASPECT RA'RO= 7.5 ......
0 2 m ~ BEPOSITION 0 5m/sJ" VELOCITY MEAN AXIAL GAS VELOCffY : 50m/s
200 i /
175
ac 8f~2S 100 75 50 I ~"~ "~x~ '~\\~ ~ 1 , 1
REgSTIVE WALL POWER SSOL
x 100
25 .........................
~7-2%
major difference between these two figures is that, due to the effect of temperature on density, operation at high temperatures requires higher pressures for a given efficiency. The increased viscosity associated with high temperature operation also has an effect on the efficiency, but this is relatively slight. As stated in Section 3.9 the curves shown in Fig. 6 assume that /z = ftopt and indicate that for small values of/3A, efficiencies up to 90 per cent appear feasible. For the values of va = 0.2 m/see and vu = 50 m/see quoted in Ref. 5, flA = 0.002 would correspond to A = 0.5. Increasing the deposition velocity from 0.2 to 0.5 m/see, hence increasing flA from 0.002 to 0.005, is seen to have very little effect at this small aspect ratio. However if the aspect ratio is 7.5, v a - 0.2 m/see gives flA = 0.03 and this reduces the efficiency at elevated pressures by about 10 per cent. At this aspect ratio there is also increased sensitivity to the magnitude of the deposition velocity, as the curves for ~A = 0.075 (e.g. A = 7.5, va = 0.5 m/see, vu = 50 m/see) indicate. One can conclude from these figures that flA "~ 0.03 if isentropic efficiencies of about 85 per cent are to be obtained within the pressure range considered. It is of interest to know whether the inefficiencies are due primarily to friction losses or to resistive wall losses. Figure 7(a) and (b) show these losses separately as a function of pressure for aspect ratios of 0.5 and 7-5 respectively and for two values of yd. These and subsequent figures assume T = 30°C as well as vg = 50 m/see. The component losses are largest at low pressures, and at the larger values of va and A, as would be expected from Fig. 6. Throughout the range of pressures considered, the friction losses exceed the resistive wall losses, though at high pressures and large A these two losses do become of comparable magnitude. Figure 8 shows how the output current (Equation (23)) and output voltage (Equation (19)) vary with pressure for the four permutations of va and A previously considered. It is necessary to specify the duct diameter, and this is taken to be 25 ram. Increasing the aspect ratio on a duct of a given diameter implies increasing the duct
PRESSURE,ATNOSPNERES
(a)
1000 . . . . .
130 ASPECT RATIO= 0.5
120
.....
110 ~\
~
100 ~ \
o.2rn/s "/DEPOSITION
0 5m/s~ VELOCITY
J
/ /"
~ WALL FRICTION POWER LOSS X 100
|
"1
/ DEPOSITION VELOCITY(Vd) : 0,2mls
t\
/'/
-1400
""
MEAN AXIAL GAS VELOCITY = 50m/s
90 80
['~S00
75].ASPECT RATIO
CONTINOoSuJs LINES REPRESENT VOLTAGE WHEREAS INTERRUPTED LINES REPRESENT CURRENT MEAN AXIAL G~S VELOCITY = 50 m/s DUCT DIAMETER = 0,025m
!,oo
,"/'" .'" -~" ] |
<
,-%+
~ o. 5o 30 20 RESISTIVE WALL POWER LOSS X100" . . . . . . . . . . . . . . . . . . . 10 ~ t POWEROUTPUT . _ ---'~ - - - - - - -31 . . . . a. . . . . . . . t-----r . . . . r- . . . . 2 3 4 ~ 6 7 8 PRESSURE, ATMOSPHERES
~
g
__111,2°/o O : ~ 10
(b)
Fig. 7. (a) Power losses as a f,mction of pressure, 30°C. (b) Power losses as a function of pressure, 30°C.
2
3
4
5 6 7 8 Pressure (Atmospheres)
9
10
Fig. 8. Terminal voltage and load current as a function of pressure, at 30°C.
The Performance of Eiectrogasdynamic Expanders with Slightly Conducting Walls
length. The voltages corresponding to A = 7.5 are therefore much larger than those corresponding to A = 0"5. Since efficiency is independent of scaling, the duct diameter may be selected, in conjunction with the aspect ratio and operating pressure, to give any desired output within the practicable range 100-1000kV. Increased duct diameters will, from Equation (23), give increased currents and the power output (Equation (24)) increases as the square of the diameter. However the power density (i.e. the power per unit duct volume) is inversely proportional to the diameter, if A is maintained constant. For the 25 mm diameter duct the variation of power density with pressure is shown in Fig. 9. At 10 atmospheres, 30°C, the power density
....
4
7 : 5 ) Aspect Rotio
Meon Axial Gos Vetocity = 50mls
/
/
E Deposition
/ /
- ""
izl
g. 1 Pressure, Atmospheres
Fig. 9. Power density as a function of pressure, at 30°C.
is of the order 2-4 MW/m 8, the higher values being associated with the smaller aspect ratio and the lower deposition velocity. Figure 5 indicated how the optimum value of ~, 100 90 80
0
002
0005
- 70
135
varied with flA and pressure, and these values were used in subsequent calculations of isentropic efficiency etc. In practice it may be difficult, due to constraints on materials and due also to variations in the operating temperature, to maintain ~ at the optimum. Figure 10 therefore shows how the isentropic efficiency varies with ~ for one particular operating condition (10 atmospheres, 30°C). At low values of flA the isentropic efficiency is seen to be quite insensitive to the precise value of/~, but if EGD ducts need to be operated at higher values of flA, it is apparent that much closer control must be exercised on the value of/~. 5. Conclusions
It has been shown that the reduction in performance of EGD expanders, due to the radial drift of particulate charge carriers, may be minimized by the use of slightly conducting duct walls. The theory reveals the importance of operating at low values of the parameter flA, that is at low values of the product of duct aspect ratio and particle deposition velocity. Since the latter is, to a large extent, beyond the experimenter's control, this indicates the need for relatively low aspect ratio ducts. With the assumption that air is the working fluid, and that the mean axial velocity is 50 m/sec, the calculations indicate that one can expect isentropic efficiencies of the order 85 per cent provided that flA ~ 0.03 and that the gas density is at least ten times atmospheric. The power density is predicted to be of the order 2-4 MW/m a for a 25 mm diameter duct. Experiments currently in progress at Reading University will test these predictions. Acknowledgements This Research has been sponsored in part by the Air Force Office of Scientific Research (AFSC), U.S.A.F., through grant no. AFOSR 74-2647 and in part by the Science Research Council, U.K., through grant B/SR/9898.
References
m 60
~ so
~ 30 20
OAS PRESSURE = 10 ATM. GAS TEMPERATURE = 30*C i','EAN AXIAL GAS VELOCITY = 50mls
P
Fig. I0. Isentropie eWaeiencyas a function of Ix.
[1] P. J. Musgrove, Electronics & Power 19, 327 (1973). [2] P. J. Musgrove and A. D. Wilson, Energy Conversion 12, 21 (1972). [3] M. Lawson and H. Von Ohain, J. Engng Power p. 201 (April 1971). [4] E. L. Collier and M. C. Gourdine, AIAA J. 6, 2278 (1968). [5] C. D. Lover and P. J. Musgrove, Proc. 2nd Int. Conf. on Pneumatic Transport in Pipes--Paper D5 BHRA Engng, Cranfield (1973). [6] A. D. Wilson, Ph.D. Thesis, Reading University (1970). [7] B. S. Masscy, Mechanics of Fluids, Van Nostrand, New York (1968).