The effect of swirl on MHD generator performance

Energy Conversion. Vol. 8 pp. 181-183. PergamonPress, 1968. Printedin Great Britain

The Effect of Swirl on MHD Generator Perfe nce JOHN B, HEYWOODt (Received 6 June 1968)

1. Ina'oduction

In the design of a coal-fired combustion chamber for an MHD generator, it is important to obtain good mixing between the coal and air, and to create and maintain a film of liquid slag on the combustion chamber walls. This insulating slag film with its high gas-side surface temperature (--~ 1800°C) considerably reduces the thermal radiation loss to the chamber walls. Since increasing the thickness of this layer reduces the combustion chamber heat loss, it is advantageous to deposit the slag droplets in the combustion zone onto the walls as quickly as possible. An attractive way of doing this is to introduce the preheated air into the combustion chamber with a tangential as well as axial component of velocity. The slag droplets are then thrown outwards by centrifugal forces. An example of a cyclone-type combustion chamber for an MHD power plant has been given by Young et al. [1]. The angle between the velocity vector and the duct axis is called the swirl angle, and in this paper a simplified model of the flow from the combustion chamber into the inlet of the generating duct is analysed so that the effect of this swirl on generator performance can be assessed. Numerical estimates are made of the magnitude of the swirl as the flow enters the generator, the length over which this swirl persists and the loss of stagnation pressure that results.

on the type of generator, but it will be assumed that most of the nozzle is of drcular cross-section. The magnetic field profile is also shown. The swirl at the inlet to the generator is related to the swirl in the combustion chamber by conservation of angular momentum. Thus, at any cross-section of the duct we can write R

f

COMBUSTIONCHAMBER NOZZLE

GENERATOR DUCT

where R is the duct radius, p the gas density, and v~ and vo are the axial and tangential'velocity components. Friction effects have been neglected. Since at any crosssection p and v~ are approximately constant, and since the mass flow rate (zrR2pvx for a circular duct) is conserved, Equation (1) reduces to R

(l/R2) / r2v° dr = constant. A mean swirl velocity ~0, where ts0 ---- (I/A) ( vo dA, ¢/

A

and A is the duct cross-sectional area, is now defined. Then for a circular duct, and provided the radial distribution of swirl does not change with axial distance along the duct [i.e. vo can be expressed in the form v o ( x ) f ( r / R ) ] Equation (2) can be shown to give ~oR = constant.

Thus the swirl angle at the inlet to the generator (which for a sonic inlet velocity will be the throat of the nozzle) is given by Vxt

SWIRL

I0 Tf

r

(3)

Vxt Rt

where subscripts t and c denote throat and combustion chamber values respectively, and a is the swirl angle. It is possible, for example, that swirl at the throat could be of the order of 20 ° for 45 ° swirl in the combustion chamber.

. B ve

(2)

0

tan at - - ~ot _ vxc Re tan ~c,

/

(1)

0

2. Magnitude of Swirl at Inlet to the Generator Figure 1 shows the model. It is assumed that the combustion chamber cross-section is circular. The generator duct may be rectangular or circular depending

2zrr2pvxvo dr = constant,

x L

Dig. L Mode! of eombek~don dmumber, nozzle and generator inlet, also showing m~mgnetJefield ~strJlmfion. f" Central Electricity Research Laboratories, Leatherhead, Surrey, England. Present address: Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.

3. Damping of the Swirl in the MHD Duct

It is well known that rotational motion of an electrically conducting material is resisted by induced currents, in a magnetic field not parallel to the axis of rotation (an example of the damping of the Karman vortex street is given by Heiser [2]). Therefore as the gas enters the 181

182

JOHN B. HEYWOOD

magnetic field, the swirl will be damped out by eddy currents which the swirl sets up in the gas. A qualitative examination of Ohm's law shows that these eddy currents flow in closed loops as indicated schematically in Fig. 2. The vo x B swift-induced electric fields act in opposite directions on each side of the duct and since at any given x and y, the potential on each side of the duct

axial force balance along a length L on each side of the duct to be of order

Ap__ 1 +flz'

which for f~o/Vz = 1/3 and other values as above, gives a pressure difference of the order of an atmosphere. This is of the same order as the transverse pressure difference set up by the axial current which may flow in a generator even when swirl is absent. The stability of a transverse pressure gradient has not yet been analysed in detail. Since the swirl-induced electric fields are shorted out as described above, the first order drop in generator efficiency is expected to be the stagnation pressure loss resulting from the increased dissipation. As the magnetic field and area variation along the duct, and radial distribution of swirl are not known in detail, we will estimate this loss on the assumption that the axial velocity and static pressure p remain unchanged (there are no net axial forces produced by the eddy currents Vx z set up by the swirl), while the tangential velocity is damped by these currents as the gas enters the magnetic ]Fig. 2. Schematic of emb'l-baleced cm'rem loops at inlet to the generator. field. The model is, of course, only approximate as the magnetic field does not have a sharply defined edge. must be approximately the same, a current loop flows This estimate would be expected to be reasonable when in the x - z plane which shorts out this induced field. the damping length L is short compared with the length This loop causes a Hall field in the y direction, again of the generator, but longer than the length over which acting in opposite directions on each side of the duct, the magnetic field rises to its steady value. In addition, and therefore shorted out by Hall current loops in the the change in stagnation enthalpy due to power extracy - z plane as shown. These current loops interact with tion and heat loss from the gas over the length L, will the magnetic field, and the current loop in the x - z plane also be neglected (again a reasonable approximation results in the torque which damps out the swirl; while when L is short compared to the generator length). From the Gibbs relation, Tds = dh -- dp/p, and with the Hall current loop in the y - z plane gives a torque about the y axis which sets up a pressure gradient across dhs = 0 and dp = 0, we can relate the drop in stagnation pressure with the rise in gas temperature T by integrating the duct. We can estimate the distance L required to damp out over the damping length L, and obtain the swirl by equating the retarding torque from the current loop in the x - z plane with the incoming angular (4) momentum:

NI #-:" '-f

f dp,= f dh

°roB2 R A L ~-- pv~A~oR, 1 +[33

where the subscript s denotes stagnation value. For a small pressure loss, Equation (4) can be written,

which gives

Aps _

L -----pv (1 + oB 2

f dT

p8

9/

AT

9/--I

T'

'

where o is the gas conductivity, B the magnetic field and fl the Hall parameter. With p = 3 × 10-1 kg/m s, vx = I0 a m/see,/g = 1, o = 5 mho/m and B = 6 Wb/mZ we find L "~ 3 m. Note that this characteristic length is independent of the swirl angle. The Hall current loop results in an axial body force in the positive x direction for z<0, and in the negative x direction for z>0. The axial pressure gradient for z<0 must therefore be decreased, and for z>0 increased from its mean value. Any non-uniformity in axial velocity would be resisted by the induced currents which would result. A pressure difference across the duct is therefore set up, and its magnitude can be estimated from an

where A represents the change over the length L due to swirl and 9/(the ratio of specific heats) = C~/C,j. To within the accuracy of the above assumptions, we can write C ~ A T = ¢0~/2, and hence the stagnation pressure drop is given by Ap, _ p,

9/

~

(9/-- 1) 2C~T

_

9/M2tan ~

2

'

(5)

where M is the Math number based on axial gas velocity, is given by Equation (3), and ~ and M are evaluated where the gas enter the magnetic field. For air combustion products, with 9/= 1.2, ~ = 20 ° and M = 1, Equation (5) gives A p d p , = - - 0 . 0 8 .

The Effect of Swirl on MHD Generator Performance 4. The Effect on Generator Performance

The major effects of the swirl on the M H D power station performance will be the increased duct inlet stagnation temperature due to decreased combustion chamber heat loss, and the increase in compressor power required to offset the stagnation pressure loss as the swirl is damped in the generator inlet. Obviously these two effects must be balanced against each other, and a net increase in station efficiency must result for the swirl to be worthwhile. The compressor power Pc required to compress the air before the preheater is given by Pc = maC~aTa(P°'zs6 -- 1)/~/c

where ma is the mass flow rate of air, Ta is air temperature at compressor inlet, C ~ is the specific heat, Pa the stagnation pressure at inlet to the air heater and ~7c the compressor efficiency. For a 2000 MW (thermal) station ma=663 kg/sec; and with Cva= 103 J/kg°K, Ta=300°K, ps = 8 atm, ~ = 0.87 and Apdps = - - 0 . 0 8 as calculated above, the additional compressor power required, APc, is 10 MW. The overall efficiency of an MHDsteam power station is affected by the compressor power (for a cycle with a directly fired air heater) through the term -

(100

-

,7,)t',/e~u

where ~a is the efficiency of the steam plant in per cent and P r H is the thermal input power. Thus, for the

183

example above, the overall station efficiency (as a percentage) will be reduced by about 0.3. Data on the effect of combustion chamber heat loss on station efficiency have been given by Wright [3]; and the linear relation A~s = -- 0.29 ALe, where ALc is the change in combustion chamber heat loss expressed as a percentage of the thermal input of the fuel, is a good fit to the data given. Thus an improvement of at least 1 per cent in Le is required to offset the increased compressor power for the example considered. The problem of the effect of swirl on heat transfer to the combustion chamber walls goes beyond the scope of the present paper. However, calculations by Hoy, Roberts and Wilkins [4] for a 60 MW coal-fired chamber show that swirl is required to obtain satisfactory distribution of coal in the chamber, and slag on the walls. Acknowledgement--This work was carried out at the Central

Electricity Research Laboratories as part of the British MHD Collaborative Research Programme. It is published by permission of the Central Electricity Generating Board. References

[1] W. E. Young, S. Way, T. C. Tsu, D. Q. Hoover and N. P. Cochran, Mech. Engng 89 (11), 49 (1967). [2] W. H. Heiser, AIAA Jnl 2, 2217 (1964). [3] J. K. Wright, Phil. Trans. R. Soc. A261, 347 (1967). [4] H. R. Hoy, A. G. Roberts and D. M. Wilkins, to be published in "Open Cycle MHD Power Generation" ed J. B. Heywood and G. J. Womack, Pergamon Press (1969).