AU2019445705B2 - Devices for ohmically heating a fluid - Google Patents

Abstract

A heater for heating a conductive liquid includes a two-dimensional array of rod-like electrodes (22, 122, 322, 422, 522) extending parallel to one another, an electrical power supply having a plurality of poles, and power switches to connect different ones of the electrodes to different poles so that current flows between the poles through the liquid. The array desirably includes outer electrodes defining the boundary (24, 424) of the array and inner electrodes disposed within this boundary. The array may have regular or irregular spacings between the electrodes. The array can provide numerous different connection schemes to vary the electrical resistance between the poles and thus vary the heating rate. The array can be arranged to provide substantially equal currents through three poles of a three-phase power supply.

Description

2019445705 29 Nov 2021

DEVICES FOR DEVICES FOR OHMICALLY OHMICALLY HEATING HEATING A FLUID A FLUID CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] The present application is a continuation-in-part of United States Patent Application No. 16/346,354, filed April 30, 2019, which is the national stage of PCT International Application No. PCT/US2017/060192, filed on November 6, 2017, which in turn 2019445705

claims the benefit of the filing date of United States Provisional Patent Application No. 62/458,201 filed on February 13, 2017 and claims the benefit of U.S. Provisional Application No. 62/418,493 filed on November 7, 2016, the disclosures of which are hereby incorporated by reference herein. BACKGROUND OF THE INVENTION

[0002] The discussion of the background to the invention that follows is intended to facilitate an understanding of the invention. However, it should be appreciated that the discussion is not an acknowledgement or admission that any aspect of the discussion was part of the common general knowledge as at the priority date of the application.

[0002a] The present disclosure relates to ohmic fluid heating devices, and methods of heating a fluid. An ohmic fluid heater can be used to heat an electrically conductive fluid as, for example, potable water. Such a heater typically includes plural electrodes spaced apart from one another. The electrodes are contacted with the fluid to be heated so that the fluid fills the spaces between neighboring electrodes. Two or more of the electrodes are connected to a power supply so that different electrical potentials are applied to different ones of the electrodes. For example, where an ohmic heater is operated using normal AC utility power such as that obtainable from a household electric plug, at least one of the electrodes is connected to one pole carrying an alternating potential, whereas at least one other electrode is connected to the opposite pole carrying a neutral or ground pole. Electricity passes between the electrodes through the fluid at least one space between the electrodes, and electrical energy is converted to heat by the electrical resistance of the fluid. resistance of the fluid.

[0003] It is desirable to control the rate at which electrical energy is converted to heat, (the “heating rate”), in such a heater to achieve the desired temperature of the heated fluid. It has been proposed to vary the heating rate by mechanically moving electrodes closer relative to one

2019445705 29 Nov 2021

another, thereby varying the electrical resistance between the electrodes. Such arrangements, however, require complex mechanical elements including moving parts exposed to the fluid. Moreover, it is difficult to make such mechanisms respond quickly to deal with rapidly changing conditions. For example, if an ohmic heater is used in an “instantaneous heating” arrangement to 2019445705

1a

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heat water supplied to a plumbing fixture such as a shower head, the water continually passes

through the heater directly to the fixture while the fixture is in use. If the user suddenly increases

the flow rate of the water, as by opening a valve on the fixture, the heater should react rapidly to

increase the heating rate SO as to maintain the water supplied to the fixture at a substantially

constant temperature.

[0004] It has also been proposed to provide an ohmic heater with a substantial number of

electrodes and with power switches to selectively connect different ones of the electrodes to the

poles of the power supply. For example, an array of electrodes may be disposed in a linear

arrangement with spaces between the electrodes. The array includes two electrodes at the

extremes of the array and numerous intermediate electrodes between the two extreme electrodes.

To provide a minimum heating rate, the extreme electrodes are connected to opposite poles of

the power supply, and the intermediate electrodes are isolated from the poles. The electric

current passes from one extreme electrode through the fluid in a first space to the nearest one of

the intermediate electrodes, then through fluid in the next space to the next isolated electrode and

SO on until it reaches the last intermediate electrode, and flows from the last intermediate

electrode to the other extreme electrode. Thus, the fluid within all of the spaces is electrically

connected in series between the two extreme electrodes. This connection scheme provides high

electrical resistance between the poles of the power supply and a low heating rate.

[0005] For a maximum heating rate, all of the electrodes are connected to the poles SO

that each electrode is connected to the opposite pole from its next nearest neighbors. Stated

another way, alternate ones of the electrodes are connected to the hot pole and to the neutral

pole. In this condition, the fluid in each space is directly connected between the poles of the

power supply, in parallel with the fluid in every other space. The connection scheme provides

minimum resistance between the poles. Intermediate heating rates may be achieved by

connecting various combinations of electrodes to the poles of the power supply. For example, in

one such connection scheme, two of the intermediate electrodes are connected to opposite poles

of the power supply, and the remaining electrodes are electrically isolated from the poles of the

power supply. The connected intermediate electrodes are separated from one another by a few

other intermediate electrodes and a few spaces, SO that fluid in only a few spaces is connected in

series between the poles. This connection scheme provides a resistance between the poles that is

higher than the resistance in the maximum heating rate scheme, but lower resistance than the resistance in the minimum heating rate scheme. With fluid having a given conductivity, different connection schemes will provide different resistances between the poles, and thus different heating rates. Because the resistance with a given connection scheme decreases as the conductivity increases, a parameter referred to herein as "specific resistance" is used in this disclosure to characterize a circuit or a part of a circuit having elements electrically connected by a fluid. The specific resistance is the ratio between the electrical resistance of the circuit or part of a circuit and the resistivity of the fluid in the circuit.

[0006] Typically, the switches are electrically controllable switches such as

semiconductor switching elements as, for example, thyristors. Ohmic heaters of this type can

switch rapidly between connection schemes and thus switch rapidly between heating rates. Such

heaters do not require any moving parts in contact with the fluid to control the heating rate.

However, ohmic heaters of this type can only select from among the set of the specific

resistances fixed by the physical configuration of the electrodes, and thus the heating rate, in

steps. Under certain conditions, the available heating rates may not match the heating rate which

produces the desired fluid temperature. This drawback can be more significant for those heaters

which are used in a range of different conditions such as fluids of widely differing conductivities,

different flow rates of fluid flowing through the heater at different rates; different fluid inlet

temperatures and different fluid outlet temperatures. For example, if the heater provides a set of

different specific resistances between a highest specific resistance usable to provide a low

heating rate with a fluid of relatively high conductivity and a lowest specific resistance usable to

provide a high heating rate with a fluid of low conductivity, only a small subset of the available

specific resistances will be within a range useful to regulate the temperature of a particular fluid.

Adding more electrodes increases the cost of and size of the heater. Moreover, additional

electrodes can produce redundant connection schemes such that different ones of the connection

schemes provide the same specific resistance between the poles of the power supply, in which

case the additional electrodes offer little benefit.

[0007] One solution to this problem is disclosed in United States Patents 7,817,906

and 8,861,943, the disclosures of which are hereby incorporated by reference herein. As

disclosed in these patents, providing electrodes in an arrangement with non-uniform specific

resistances between pairs of neighboring electrodes as, for example, providing electrodes at

non-uniform spacings can provide an ohmic heater suitable for operation under a wide range of conditions. Desirably, the specific resistances between pairs of neighboring electrodes are 17 Oct 2025 selected so that, for a fluid of a given conductivity, the power levels available using different connection schemes include a series of non-redundant specific resistances extending over a very wide range. For example, such a heater may provide 60 or more specific resistances in a substantially logarithmic series, i.e., a series of specific resistances such that a ratio between each specific resistance and the next lower specific resistance is substantially constant. Such an arrangement provides a useful solution which has been employed commercially in demanding 2019445705 applications as, for example, an instantaneous heater for domestic hot water.

[0008] However, still further improvement would be desirable. For example, the commercial implementations of heaters as disclosed in the aforementioned ‘706 and ‘943 Patents have used electrodes in the form of electrically-conductive plates which are disposed in a dielectric housing so that the plates subdivide the interior of the housing into channels. The housing includes passages which direct the fluid through these channels. While this arrangement works well for mass-produced heaters of modest size as, for example, domestic water heaters for private homes or individual apartments, it is not optimum for large-scale industrial and commercial heaters. Such heaters typically are built to order in a custom size to fit the application. The cost of designing and fabricating the complex dielectric housing to suit the particular arrangement of electrodes required for a customized arrangement can be significant. Moreover, the components can be damaged if subjected to conditions such as extreme pressures and temperatures which may be encountered in industrial and commercial heaters, and may be difficult to repair or replace.

[0008a] According to one form of the invention there is provided a liquid heater comprising: (a) a chamber; (b) a plurality of rod-like electrodes disposed within the chamber and extending substantially parallel to one another, the electrodes being disposed a two-dimensional array including outer electrodes cooperatively defining an outer boundary of the array and interior electrodes disposed within the boundary; (c) a three-phase electrical power supply having three poles, the power supply being operable to supply alternating potentials offset by 120 º in phase to the respective poles; (d) power switches electrically connected between at least some of the plurality of electrodes and the poles, the power switches being operable to selectively connect each of the electrodes to one or another the poles and to selectively disconnect each of the electrodes from the poles so as to form current paths extending through liquid disposed in the chamber between electrodes connected to different ones of the poles, 17 Oct 2025 wherein the array includes a plurality of rows of electrodes extending in a first direction with the electrodes within each row spaced apart from one another in the first direction and aligned with one another in a second direction orthogonal to the first direction, the rows being spaced apart from one another in the second direction; and (e) a control processing unit coupled to the power switches and configured to actuate the power switches according to switch setting data retrieved from a memory, the memory storing a plurality of predefined connection schemes, wherein the 2019445705 connection schemes include electrode groupings having three-fold symmetry about a central axis and define current paths between pairs of poles of the three-phase power supply with substantially equal specific resistance.

[0008b] According to another form of the invention there is provided a liquid heater comprising: (a) a chamber; (b) a plurality of rod-like electrodes disposed within the chamber and extending substantially parallel to one another, the electrodes being disposed a two-dimensional array including outer electrodes cooperatively defining an outer boundary of the array and interior electrodes disposed within the boundary; (c) a three-phase electrical power supply having three poles, the power supply being operable to supply alternating potentials offset by 120º in phase to the respective poles; (d) power switches electrically connected between at least some of the plurality of electrodes and the poles, the power switches being operable to selectively connect each of the electrodes to one or another the poles and to selectively disconnect each of the electrodes from the poles so as to form current paths extending through liquid disposed in the chamber between electrodes connected to different ones of the poles, wherein the outer electrodes are disposed along an outer circle around a central axis and extend parallel to the central axis; and (e) a control processing unit coupled to the power switches and configured to actuate the power switches based on switch setting data retrieved from a memory, the memory storing a plurality of predefined connection schemes, wherein the connection schemes include groupings of electrodes defining current paths having three-fold symmetry about a central axis and substantially equal specific resistances between the poles of the three- phase power supply.

[0008c] According to another form of the invention there is provided a liquid heater comprising: (a) a chamber; (b) a plurality of rod-like electrodes disposed within the chamber and extending substantially parallel to one another, the electrodes being disposed a two-dimensional

4a array including outer electrodes cooperatively defining an outer boundary of the array and 17 Oct 2025 interior electrodes disposed within the boundary; (c) a three-phase electrical power supply having three poles, the power supply being operable to supply alternating potentials offset by 120º in phase to the respective poles; (d) power switches electrically connected between at least some of the plurality of electrodes and the poles, the power switches being operable to selectively connect each of the electrodes to one or another the poles and to selectively disconnect each of the electrodes from the poles so as to form current paths extending through 2019445705 liquid disposed in the chamber between electrodes connected to different ones of the poles, wherein the array includes three groups of electrodes having N-fold symmetry about a central axis wherein N is 3 or a multiple of 3, and wherein the power supply is a three-phase power supply having three poles, the power switches being operable to select sets of connected electrodes so that the connected electrodes include corresponding electrodes from each group so that the connected electrodes define current paths having three-fold symmetry about the central axis; and (e) a control processing unit coupled to the power switches and configured to actuate the power switches based on switch setting data retrieved from a memory, the memory storing a plurality of predefined connection schemes, wherein the connection schemes include groupings of electrodes defining current paths having three-fold symmetry about a central axis and substantially equal specific resistances between the poles of the three-phase power supply.

[0008d] According to another form of the invention there is provided a liquid heater comprising: (a) a chamber; (b) a plurality of rod-like electrodes disposed within the chamber and extending substantially parallel to one another, the electrodes being disposed a two-dimensional array including outer electrodes cooperatively defining an outer boundary of the array and interior electrodes disposed within the boundary; (c) a three-phase electrical power supply having three poles, the power supply being operable to supply alternating potentials offset by 120º in phase to the respective poles; (d) power switches electrically connected between at least some of the plurality of electrodes and the poles, the power switches being operable to selectively connect each of the electrodes to one or another the poles and to selectively disconnect each of the electrodes from the poles so as to form current paths extending through liquid disposed in the chamber between electrodes connected to different ones of the poles, wherein the electrodes are arranged in a hexagonal array including outer electrodes defining an outer regular hexagon and inner electrodes defining a an inner hexagon, the array including rows

4b extending along three sets of row axes disposed at 60° angles to one another, the row axes in 17 Oct 2025 each set being uniformly spaced and the spacings between row axes of the three sets being equal to one another, the row axes intersecting one another to form a grid of equilateral triangles, the electrodes being disposed at the vertices of the equilateral triangles, the power supply being a three-phase power supply having three poles, the power switches being operable to connect sets of the electrodes to the poles so that the electrodes connected to the poles have three-fold symmetry around a central axis and define current paths between the poles having three-fold 2019445705 symmetry around the central axis; and (e) a control processing unit coupled to the power switches and configured to actuate the power switches based on switch setting data retrieved from a memory, the memory storing a plurality of predefined connection schemes, wherein the connection schemes include groupings of electrodes defining current paths having three-fold symmetry about a central axis and substantially equal specific resistances between the poles of the three-phase power supply.

[0008e] Where any or all of the terms "comprise", "comprises", "comprised" or "comprising" are used in this specification (including the claims) they are to be interpreted as specifying the presence of the stated features, integers, steps or components, but not precluding the presence of one or more other features, integers, steps or components.

BRIEF DESCRIPTION OF THE DRAWINGS

[0009] Fig. 1 is a diagrammatic sectional view of a heater according to one embodiment of the invention.

[0010] Fig. 2 is schematic view of an electrical circuit in the heater of Fig. 1.

[0011] Fig. 3. is a diagrammatic perspective view of a heater according to a further embodiment of the invention.

[0012] Fig. 4 is a fragmentary sectional view along line 4-4 in Fig. 3.

[0013] Fig. 5 is a diagrammatic sectional view of a heater according to another embodiment of the invention.

4c

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[0014] Fig. 6 is a diagrammatic sectional view of a heater according to yet another

embodiment of the invention.

[0015] Fig. 7 is a schematic view of an electrical circuit in another embodiment of the

invention.

[0016] Fig. 8 is a diagrammatic sectional view depicting a heater used with the electrical

circuit of Fig. 7.

[0017] Figs 9, 10, 11 and 12 are diagrammatic views depicting certain connection

patterns used in operation of the heater of Fig. 8.

[0018] Fig. 13 is a diagrammatic view of an electrode array in a heater according to a

further embodiment of the invention.

DETAILED DESCRIPTION

[0019] A heater according to one embodiment of the invention includes a housing 20 and

numerous rod-like electrodes extending within the housing in the plane of the drawing. These

electrodes are disposed in an irregular two-dimensional array. As depicted in Fig. 1, the

electrodes are circular cylinders and thus are circular as seen in cross-section in Fig. 1. In the

irregular array, each of the electrodes has multiple neighboring electrodes. For example,

electrodes 22a, 22b, 22c, and 22d are all neighbors of electrode 22e. Unless otherwise specified,

the location of a rod-like or cylindrical electrode as used herein refers to the location of the axis

of the electrode. Electrodes 22b, 22c, 22d, 22f and 22g are "outer electrodes" as referred to

herein in that they cooperatively define the outer boundary 24 of the array. As referred to in this

disclosure, the outer boundary 24 of the array is the polygon formed by the shortest possible

combination of imaginary straight lines extending in a plane perpendicular to the axes 26 of

some of the electrodes 22 between the axes such the axes of all of the electrodes are either within

or on the outer boundary. By contrast, electrodes 22e, 22a and 22h are inner electrodes as

referred to herein because their axes 26 lie within, but not on, the boundary 24.

[0020] In the irregular array of Fig. 1, the electrodes lie at numerous different distances

from one another. The array of Fig. 1 is irregular in two dimensions, in that the spacing between

axes of the electrodes in both of the directions perpendicular to the axes of the electrodes,

indicated by arrows X and Y in Fig. 1.

[0021] The heater includes an electrical circuit (Fig. 2). The circuit includes a power

supply 36 incorporating two poles in the form of conductors 38 and 40. These conductors are

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connected to source of electrical power such as utility power mains. The conductors are

arranged SO that in operation different electrical potentials are applied to poles 38 and 40. For

example, conductor 40 may be a neutral conductor which receives a neutral voltage, typically

close to ground voltage, whereas conductor 38 may be a "hot conductor" which will receive an

alternating voltage supplied by the utility power mains. This particular power supply is a single

phase power supply in that only one alternating voltage is employed. Power switches 48 are

connected between the electrodes 22 and power source 36. The power switches 48 are arranged

SO that each electrode may be connected to either one of poles 38 and 40 or may be left isolated

from the poles. As used in this disclosure, the term "switch" includes mechanical switches

which may be manually actuated or actuated by devices such as relays or the like, and also

includes solid state devices that can be actuated to switch between a non-conducting condition

with very high impedance and a conducting condition with very low impedance. Examples of

solid state switches include elements such as triacs, MOSFETs, thyristors, and IGBTs. In the

particular arrangement depicted, two individual single pole switches are associated with each

electrode 22, each switch being operable to connect the associated electrode with a different one

of the poles and the electrode is isolated from both poles when both switches are open.

However, this arrangement can be replaced by any other electrically equivalent switching

arrangement.

[0022] In operation, an electrically conductive fluid as, for example, a conductive liquid

such as potable water is passed through the housing 20 SO that the fluid fills the space within the

housing and contacts the surfaces of electrodes 22. One or more of the electrodes 22 are

connected to the hot pole 38 by power switches 48, whereas one or more of the electrodes 22 are

connected to the neutral pole 40 SO that current flows between the different poles through the

fluid contained in the housing. The current flow varies inversely with the resistance between the

poles. The resistance between the poles depends on the specific resistances of all of the current

paths through the fluid between pairs of the electrodes connected to different poles, conducting

in parallel with one another. Moreover, in this arrangement, there are conductive paths through

the fluid between a given one of the electrodes and every other one of the electrodes. For

example, if only electrodes 26c and 22g are connected to opposite poles, current will flow

between these electrodes. Because other electrodes, such as electrodes 22e and 22h are disposed

in the path of the flowing current, and these electrodes are electrically conductive, some of the

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current will pass through the these electrodes, and the specific resistance of the current path

between electrodes 22c and 22b will differ appreciably from a hypothetical system in which

electrodes 22e and 22h were absent. If only neighboring electrodes 22c and 22b are connected to

opposite poles, current will flow between these electrodes. The current flows through all of the

fluid in the chamber, but the predominant flow path of this flow lies near the straight line

connecting the two electrodes. Thus, the presence of other electrodes, such as electrode 26e, will

affect the current flow to some extent, but this effect is small in comparison to the effect of

electrodes 22e and 22h in the preceding example. Because the distances through the fluid

between different ones of the electrodes differ from one another, and because the effect of other

electrodes is different, the specific resistances between different pairs of two electrodes differ

from one another. In this regard, the interior electrodes help to provide a wide range of specific

resistances between poles 38 and 40 which can be formed by connecting different electrodes to

the poles, SO that the heater can provide a wide range of heating rates and a large number of

distinct heating rates within this range. This wide range of heating rates can be provided in a

compact unit. In particular, the assembly may be compact in the dimensions transverse to the

axes of the electrodes. This is particularly desirable where the liquid to be heated is under

pressure SO that the housing holding the electrodes must be a pressure vessel. The cost and

weight of the walls of a pressure vessel required to withstand a given pressure increase as the

cross-sectional dimensions of the vessel increase.

[0023] The heater discussed above further includes an optional control circuit 56 (Fig. 2).

Although a particular control circuit is shown and discussed herein, it should be understood that

the heater can be controlled by manually controlling the switches and the control circuit may be

omitted. The particular control circuit of 56 includes a control processing unit 58 and one or

more sensors for sensing the one or more operating parameters of the heater. In one example,

the one or more sensors may include only an outlet temperature sensor (not shown) which is

physically mounted in or near the outlet of housing 20 to detect the temperature of fluid

discharged from the heater. The temperature sensor may include conventional elements as, for

example, one or more thermocouples, thermistors and resistance elements having electrical

resistance which varies with temperature. The control processing unit 58 is linked to power

switches 48 SO that the control processing unit can actuate the switches to provide various

connection schemes as discussed. The control processing unit may include a memory 70 such as

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a non-volatile memory, random access memory or other conventional storage element. The

memory desirably stores data for least some of the various connection schemes attainable by

operation of the switches. The data in the table for each connection scheme may include the

settings for each of the power switches 48 to form a particular connection scheme, as well as data

specifying, either explicitly or implicitly, a ranking of the stored connection schemes in order of

their specific resistances. For example, the data for each connection scheme may include the

specific resistance between the poles for that connection scheme, or equivalent data such as

values of resistance or conductivity for the various connection schemes all measured or

calculated for the case where the spaces are filled with a fluid of a given conductivity.

Alternatively, the explicit data may be simply an ordinal number for each connection scheme. In

an example of an implicit ranking, the data specifying switch settings for each connection

scheme may be stored at addresses within the memory, such that the data at a lowest address

specifies the switch settings for a connection scheme with the lowest specific resistance, the data

at the next lowest address specifies the data for the connection scheme with the next lowest

specific resistance, and SO on.

[0024] Control processing unit 58 further includes a logic unit 72 connected to

memory 70. The logic unit has one or more outputs connected to the power switches 48 as, for

example, by conventional driver circuits (not shown) arranged to translate signals supplied by the

logic unit to appropriate voltages or currents to actuate the switches. The logic unit may include

a general-purpose processor programmed to perform the operations discussed herein, a hard-

wired logic circuit, a programmable gate array, or any other logic element capable of performing

the operations discussed herein. Although the term "unit" is used herein, this does not require

that the elements constituting the unit be disposed in a single location. For example, parts of the

control processing unit, or parts of the logic unit, may be disposed at physically separate

locations, and may be operatively connected to one another through any communications

medium.

[0025] In operation, the control unit may start the heater in operation by retrieving the

switch setting data for the connection scheme with the highest specific resistance (lowest heating

rate) and setting the switches accordingly, SO that this connection scheme is set as the first

connection scheme in use. After startup, the control unit periodically compares the outlet

temperature of the fluid, as determined by the outlet temperature sensor, with a setpoint

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temperature. If the outlet temperature is below a setpoint temperature by more than a

predetermined tolerance, the control unit retrieves the switch setting data for a connection

scheme having specific resistance one step lower than the connection scheme then in use to

provide a greater heating rate, and sets the switches accordingly. This process is repeated

cyclically until the outlet temperature reaches the setpoint. If the outlet temperature exceeds the

setpoint by more than the tolerance, the control unit selects a connection scheme with a specific

resistance one step higher on the next cycle SO as to reduce the heating rate. In this way, the

control circuit will ultimately settle at a heating rate which brings the fluid to the desired output

temperature. Desirably, the control system actuates the switches to change the control scheme at

times when the alternating voltage applied to the hot pole 38 of the power supply is at or near

zero. Such zero crossing times occur twice during each cycle of a conventional AC waveform.

This arrangement minimizes switching transients and electrical noise generation. In other

embodiments, the control logic may use measured current flow between the poles and measured

flow rate of the liquid to determine a predicted temperature rise within the heater, and add the

predicted temperature rise to a measured inlet temperature of liquid entering the heater to arrive

at a predicted outlet temperature. If the predicted outlet temperature is below the setpoint

temperature by more than the tolerance, the control logic switches to a connection scheme

having a lower specific resistance to increase the current flow. The control logic takes the

reverse action if the predicted outlet temperature is above the setpoint temperature.

[0026] The electrical circuit of the heater may optionally include one or more shunting

busses 52 and shunting switches 50 operable to connect each electrode to the shunting bus or

busses and to disconnect each electrode from the shunting bus or busses. Each shunting bus can

be used to establish a low resistance conductive path between any two electrodes which are not

connected to the poles. In the example above where only electrodes 22c and 22g are connected

to opposite poles of the power supply and the other electrodes are disconnected from the poles of

the power supply and also are disconnected from the shunting bus, the specific resistance of the

current path is relatively high. However, if electrodes 22h and 22e are both connected to the

shunting bus, the conductive path will be a composite of two paths in parallel, i.e., a first path

from electrode 22c directly to the electrode 22g as discussed above, and a second path from

electrode 22c to electrode 22e, through the shunting bus to electrode 22h and from electrode 22h

to electrode 22g. Because the shunting switches 50 and shunting bus 52 have very low

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impedance, the path through electrodes 22e and 22h and the shunting bus will predominate. In

this instance, the specific resistance between electrodes 22c and 22g will be much lower. Where

the shunting bus is included, it provides additional connection schemes having further different

specific resistances. These additional connection schemes in are included in the data specifying

the various connection schemes and the specific resistances of the various connection schemes

stored in the memory 70 of the control unit 56, and the control unit is linked to the shunting

switches 52 SO that the control unit can open and close the shunting switches as needed.

[0027] The rod-like electrodes greatly simplify construction of the heater. For example,

as seen in Fig. 3, the housing may be an elongated hollow body 102 having a pair of end

walls 104 and 106. Cylindrical electrodes 122 extend through holes 108 (Fig. 4) in end wall 104.

Although only two electrodes 122 are depicted in Fig. 3 for clarity of illustration, in practice the

electrode array desirably includes numerous electrodes extending parallel with one another and

parallel to the axis of elongation 110 of body 102. The electrodes may be positioned in any

desired array simply by forming the holes 108 in the desired configuration, which facilitates

customization of the heater for a particular application. There is no need for elaborate baffle

systems to route the fluid through separate passageways between the electrodes. The end walls

of the hollow body may be formed from a dielectric material such as a polymer, or else may be

formed from a conductive material such as a metal and equipped with dielectric sleeves (not

shown) within holes 108. The exposed ends 125 of the electrodes can be readily connected to

the electrical circuit. The individual electrodes passing through the end wall can be secured in

place and sealed to the end wall by any of the well-known techniques commonly used to secure

elements such as tubes passing through a wall. For example, a seal may be formed by an O-

ring 126 seated in a groove 128 on the electrode, and may be secured in place by screw threads

(not shown) on the electrode engaged with corresponding screw threads (not shown) in

holes 108. The electrodes can be readily removed and serviced or replaced as needed. An inlet

(now shown) and an outlet (not shown) are provided in opposite ends walls 104 and 106 to pass

fluid through the interior of the heater.

[0028] A heater according to a further embodiment of the invention also includes an

array of rod-like electrodes 322 extending parallel to one another, in the directions into and out

of the plane of the drawing as seen in Fig. 5. In this embodiment, the array is partially regular

and partially irregular. The electrodes are disposed in columns 301 extending in the direction

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denoted by arrow "Y" and in rows extending in the direction denoted by arrow "X",

perpendicular to direction Y, both of these directions being perpendicular to the axes of the

electrodes. For example, electrodes 322aa, 322ab, and 322ac constitute column 301a, whereas

electrodes 322aa, 322ba, 322ca, 322da and 322ea constitute row 303a. The electrodes within

each row are disposed at the same location in the Y direction. The electrodes are regularly

spaced from one another in the Y direction, SO that the distances in the Y direction between

adjacent rows 303 are equal. The electrodes within each column are disposed at the same

location in the X direction. However, the distances C between mutually-adjacent columns are

unequal to one another, SO that the columns are irregularly spaced from one another in the X

direction. For example, distance Cab between columns 301a and 301b is larger than distance Cbc

between columns 301b and 301c. In the array of Fig. 5, those electrodes which are disposed in

the outer columns 301a and 301e, and those electrodes disposed in outer rows 301a and 301c

constitute the outer electrodes and define the boundary of the array, whereas those electrodes

which are disposed neither in an outer row nor in an outer column (electrodes 322bb, 322cb

and 322 db) constitute the inner electrodes disposed within the boundary.

[0029] The array of Fig. 5 is disposed in a housing 320 having dielectric walls. In the

particular embodiment depicted, the housing is arranged SO that liquid passing through the heater

flows predominantly in a direction transverse to the axes of the electrodes, in this case the X

direction, through the array from inlet 307 to an outlet 309. Here again, no complex structure is

required to route the fluid through individual spaces of the array. Alternatively, the flow may be

directed generally in directions parallel to the axes of the electrodes.

[0030] The electrodes 322 an connected to a power supply similar to that discussed

above, SO that each electrode can be connected to one or the other pole of the power supply, or

may be left disconnected. Where the power supply includes a shunting bus as discussed above,

the power supply can connect two or more of the electrodes which are disconnected from the

poles to the shunting bus as discussed above. An array of this type can provide numerous

combinations of current paths which provide numerous different specific resistances between the

poles of the power supply.

[0031] In a variant of the array discussed above with reference to Fig. 5, some or even all

of the spacings between columns may be equal to one another. In the extreme case where all of

the spacings C are identical, the array is a completely regular array. However, even in this case a

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substantial number of conduction schemes having different specific resistances can be provided.

As discussed above in connection with the heater of Fig. 1, the specific resistance between a

given pair of electrodes connected to different poles of the power supply will be affected by

other electrodes, and this effect varies with the location of the other electrodes relative to the pair

of connected electrodes. This effect increases the number of different conduction schemes which

can be provided by the array. For example, in the array of Fig. 5, the specific resistance between

electrodes 322aa and 322ab will differ from the specific resistance between electrodes 322ba

and 322bb. The latter electrode pair (322ba and 322bb) has four other electrodes in close

proximity, whereas the former electrode pair (322aa and 322bb) has only two other electrodes in

close proximity. In general, the specific resistance between a pair of outer electrodes disposed at

a given distance from one another will differ from the specific resistance between a pair of

electrodes disposed at the same distance from one another which pair includes one or more inner

electrodes. This effect is greater in a compact array with relatively small distances between

electrodes. One measure of compactness is the mean distance between neighboring electrodes.

For example the mean distance may be less than five times the mean diameter of the individual

electrodes, more desirably less than 3 times the mean diameter of the individual electrodes, and

still more desirably less than 2 times the mean diameter of the individual electrodes.

[0032] A heater according to a further embodiment of the invention (Fig. 6) includes a

housing 420 and an array of rod-like electrodes 422 having outer electrodes disposed at locations

on an outer circle 401 having a radius Ro around a central axis 410. Although circle 410 is

shown in solid line for clarity of illustration, it is a locus of the electrodes, not a physical

structure. Twelve outer electrodes are provided, and these are spaced at regular circumferential

intervals a, where a=30°. The outer electrodes thus define the boundary of the array as a regular

12-sided polygon 424, only a portion of which is depicted in Fig. 6. The array further includes

six inner rod-like electrodes 423 disposed at regular circumferential intervals of 2a on an inner

circle 403, concentric with the outer circle and with central axis 410. The radius R1 of the inner

circle is smaller than Ro. A first one 423a of the inner electrodes is offset in the circumferential

direction around central axis 410 from a first one 422a of the outer electrodes by a/2 degrees.

Thus, every other one of the inner electrodes 423 also lies at a circumferential location midway

between the circumferential locations of the two closest outer electrodes. All of the electrodes

extend parallel to one another and parallel to the central axis 410. In this embodiment as well,

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each of the electrodes can be connected to either pole of the power supply, or left disconnected

from the power supply. Where a shunting bus is provided, electrodes which are disconnected

from the power supply can be connected to the shunting bus. Here again, although this array of

electrodes has some degree of regularity, it provides a substantial number of unique specific

resistances between various combinations of electrodes. In a variant of this arrangement, the

circumferential spacings between inner electrodes 423, the circumferential spacings between

outer electrodes 442, or both may be wholly or partially irregular. In a further variant, further

electrodes may be added within the array, and those electrodes may be disposed at locations on

further circles. In yet another variant, the inner electrodes may be disposed on an inner circle

which is not concentric with the outer circle.

[0033] Three-phase electrical power is commonly used to power large-scale industrial

and commercial electrical equipment which consumes power on the order of a kilowatt or more.

A power supply 536 for use with three-phase power includes three poles 540, 542 and 546 which

are connectable to a three-phase utility circuit (not shown) to receive alternating potentials of

equal magnitude offset by 120° in phase from one another, i.e., at phase angles of 0°, 120°,

and 240°. Here again, power switches 548 are provided for selectively connecting each of the

electrodes to one of the poles. Only two of the electrodes 522 are depicted in Fig. 7 for clarity of

illustration; the same arrangement of power switches 548 typically is provided for every one of

the electrodes. The power switches depicted as including three switches associated with each

electrode, SO that any electrode can be connected to any pole. However, in many cases, it is

unnecessary to include all of these switches. For example, the power switches may include only

a single switch for each electrode, SO that a given electrode may be connected to one of the poles

or left disconnected. In this case, the power switches associated with different electrodes are

arranged to connect different ones of the electrodes to different poles. Optionally, one or more

shunting busses 552 and shunting switches 550 also may be provided. In three-phase operation,

current flows through the electrodes and through current paths in the fluid between each pair of

poles, i.e., between poles 540 and 542; between pole 540 and 544; and between poles 542 and

544. In operation of three-phase power supply as shown in Fig. 7, it is highly desirable to

maintain these three current flows equal to one another. Stated another way, the electrical

resistances between each pair of poles desirably is equal to the electrical resistance between each

other pair of poles. Assuming that the electrical resistivity of the fluid in contact with the

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electrodes is the same along each current path in the fluid, the specific resistance between each

pair of poles should be equal.

[0034] One heater used which can be used with the power supply of Fig. 7 is shown in

Fig. 8. The heater includes electrodes 522 disposed in a dielectric housing 520. The electrodes

desirably are arranged at locations of a hexagonal lattice as depicted in Fig. 8. Here again, all of

the electrodes 522 are rod-like and extend parallel to one another, into and out of the plane of the

drawing. In the hexagonal lattice, the electrodes are disposed in rows having row directions at

angles of 60° from one another, with the centers of the electrodes in each row disposed on row

axes (denoted A, B and C in Fig. 9) extending in the row directions. Although only a few row

axes are depicted in Fig. 9 for clarity of illustration, there are similar row axes extending along

the other rows of electrodes. The row axes parallel to one another are disposed at identical

spacings from one another SO as to define intersections at vertices of numerous equilateral

triangles, and the axes of the array are disposed at least some of the vertices. Here again, the

array includes outer electrodes which in this embodiment define an outer regular hexagon, and

inner electrodes which define an inner hexagon. The row axes also define a central vertex 510.

A central axis extends parallel to the axes of the electrodes through the central vertex, and the

array has six-fold symmetry about the central axis.

[0035] The same array can also be described as an arrangement of electrodes disposed on

concentric circles, where all of the electrodes disposed on the inner hexagon lie on an inner circle

(not shown) of radius R around the central vertex; the electrodes disposed at the corners of the

outer hexagon lying on an outermost circle (not shown) of radius Ro concentric with the inner

circle and central vertex 510; and the electrodes on the sides of the outer hexagon, shown shaded

in Fig. 9, are disposed on an intermediate circle (not shown) of radius RINT concentric with the

inner and outer circles, where R1 < RINT < Ro.

[0036] The power supply is arranged to connect at least some of the electrodes to the

poles of the power supply in connection schemes such that the connected electrodes include three

sets of electrodes connected to different ones of the poles 540, 542 and 544 of the power supply

(Fig. 7). One such of connection scheme is depicted in Fig. 9, with the electrodes 522a of a first

set, 522b of a second set and 522c of a third set shown with different cross-hatchings. In this

particular scheme, all of the electrodes 522 of the array are connected to the poles to provide low

specific resistances between the poles. In another pattern, (Fig. 10) each set of electrodes 522a,

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522b, 522c includes only one electrode, and these electrodes are outer electrodes at the corners

of the hexagonal array to provide very high specific resistance between the poles, and thus

provide the minimum heating rate. In this pattern, the remaining electrodes are disconnected

from the poles of the power supply. Numerous intermediate schemes to provide numerous

different specific resistances between the poles can be formed; one such intermediate scheme is

shown in Fig. 11. In each of the connection schemes discussed above, each connected electrode

and the corresponding electrodes of the other two sets are disposed at vertices of an equilateral

triangle having its center at the central axis of the array. For example, in Fig. 11

electrodes 522a1, 522b1 and 522c1 are disposed at the vertices of one equilateral triangle,

whereas electrodes 522a2, 522b2 and 522c2 are disposed at the vertices of another equilateral

triangle. Stated another way, in each of the connection schemes discussed above, the connected

electrodes of each are disposed at locations of the connected electrodes of the other sets rotated

120° about the central axis 510 from the locations of another set. In these connection schemes,

the electrode sets have three-fold symmetry about the central axis. These electrode sets thus

provide substantially congruent current paths between all three pairs of poles of the power

supply, and the effects of neighboring electrodes on the current paths will be identical. These

connection schemes therefore provide substantially equal specific resistances between the poles.

In other connection schemes, such as that depicted in Fig. 12, the connected electrodes 522a,

522b and 522c are disposed at vertices of an equilateral triangle which is not centered at central

axis 512. The conduction paths are of equal lengths and will have equal specific resistances

apart from any differences which may be caused by differences in the effects of neighboring

electrodes.

[0037] Although it is desirable to provide equal specific resistances between the poles,

perfect equality is not required. Thus, the connection schemes can include one or more

electrodes connected to one or two of the poles in such a way as to cause inequality. However, it

is desirable to select the electrodes SO that at least a substantial part of the current, and desirably

at least a majority of the current, flows through current paths having equal specific resistances.

This can provide additional heating rates different from those achievable with perfect equality,

while introducing only a limited amount of imbalance in the currents of the different phases. In a

variant of this scheme, electrodes which cause unequal current flows can be connected

cyclically. In each cycle, an electrode which causes unequal current flows with a greatest current

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through one pole is connected for a period and then disconnected and replaced by a second

electrode which causes a corresponding unequal flow with a maximum current directed through

a second pole, and the second electrode is then disconnected and replaced by a third electrode

which causes a corresponding unequal current flow with the maximum current through the third

pole. The third electrode is disconnected and replaced by the first electrode at the beginning of

the next cycle. In this manner, the unequal current flows rotate among the poles, which distribute

the effects of the excess current among the phases.

[0038] Arrays other than the regular hexagonal array can provide current paths with the

three-fold symmetry discussed above. For example, the array of electrodes shown in Fig. 13

includes three groups of electrodes 622a, 622b and 622c. The electrodes within each group are

shown with the same shading. The electrodes within each group are disposed at irregular radii

from a central axis 610, and at irregular intervals in the circumferential direction around the axis.

However, the groups are congruent with one another but each group is rotated 120° from the

position of another group. The power supply is arranged to connect sets of electrodes SO that the

connected electrodes include corresponding electrodes from all three groups. Here again, the

array has three-fold symmetry about the central axis, and thus the electrodes lie on vertices of

equilateral triangles having centers at the central axis 610.

[0039] Where shunting busses are used with an array having three-fold symmetry about

an axis, three shunting busses may be used SO that the set of electrodes connected to one another

by each bus each bus is congruent with the set of electrodes connected by another bus, but is

rotated 120° from the position of such other set.

[0040] In the discussion above, it is assumed that the liquid passing through the heater

has uniform resistivity. However, the resistivity of most liquids varies with the temperature of

the liquid. Where the flow of liquid is predominantly parallel to the axes of the electrodes, this

effect tends to affect all of the current paths equally. If the hotter liquid has lower resistivity, that

portion of each current path nearer the downstream ends of the electrodes will carry a greater

current than the portion of the same path nearer the upstream ends, but the relationship between

the currents carried by the various paths will be unaffected. However, if the electrodes and the

direction of fluid flow extend horizontally, convection may cause hotter liquid to flow

preferentially through those current paths disposed near the top of the array. Where the array is

connected to a three-phase power source, this may lead to asymmetric current flows between the

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poles. To suppress this effect, vanes (not shown) may be provided within the housing to induce

rotational flow around the axis of the housing, SO that the liquid follows a generally helical path.

The same effect may be achieved by configuring the inlet, outlet or both SO that the flow of fluid

into housing, out of the housing or both will induce rotational flow around the axis of the

housing.

[0041] In each of the embodiments discussed above, the rod-like electrodes are in the

form of right circular cylinders. However, other elongated rod-like elements may be employed.

For example, the rod-like elements may by tapered. In still other arrangements, the rod-like

electrodes may have non-circular cross-sectional shapes in the regions of the electrodes which

are exposed to the liquid. These electrodes may be generally cylindrical or conical to provide a

circular cross-sectional shape in the regions of the electrodes which penetrate the walls of the

housing.

[0042] In the embodiments discussed above, the electrodes are of equal diameter.

However, the diameters of the electrodes may be unequal. Also, the lattice arrangement as

depicted in Fig. 8 has all of the electrodes disposed at equal spacings. This arrangement may be

varied somewhat. For example, the diameter of the intermediate circle may be increased slightly

to move the electrodes on the intermediate circle away from the central axis. In this instance, the

smaller triangular sets of three connected electrodes as depicted in Fig. 12 will not have equal

distances between the connected electrodes, and therefore may induce some phase inequality, but

there will be additional unique specific resistances. The array of Fig. 8 is just one example of an

array where the entire array has N-fold symmetry about a central axis where N is 3 or a multiple

of 3. Moreover, the array of Fig. 8 includes subgroups of electrodes having N-fold symmetry

about other axes. Other arrays having either or both of these properties can be used to provide

three-phase balance.

[0043] In all of the arrangements discussed above, the number of electrodes can be varied

as needed.

[0044] As these and other variations and combinations of the features discussed above

can be employed, the foregoing description should be taken by way of illustration, rather than as

limiting the invention.

Claims (14)

THE CLAIMS DEFINING THE INVENTION ARE AS FOLLOWS: 17 Oct 2025

1. A liquid heater comprising: (a) a chamber; (b) a plurality of rod-like electrodes disposed within the chamber and extending substantially parallel to one another, the electrodes being disposed a two-dimensional array including outer electrodes cooperatively defining an outer boundary of the array and interior 2019445705

electrodes disposed within the boundary; (c) a three-phase electrical power supply having three poles, the power supply being operable to supply alternating potentials offset by 120º in phase to the respective poles; (d) power switches electrically connected between at least some of the plurality of electrodes and the poles, the power switches being operable to selectively connect each of the electrodes to one or another the poles and to selectively disconnect each of the electrodes from the poles so as to form current paths extending through liquid disposed in the chamber between electrodes connected to different ones of the poles, wherein the array includes a plurality of rows of electrodes extending in a first direction with the electrodes within each row spaced apart from one another in the first direction and aligned with one another in a second direction orthogonal to the first direction, the rows being spaced apart from one another in the second direction; and (e) a control processing unit coupled to the power switches and configured to actuate the power switches according to switch setting data retrieved from a memory, the memory storing a plurality of predefined connection schemes, wherein the connection schemes include electrode groupings having three-fold symmetry about a central axis and define current paths between pairs of poles of the three-phase power supply with substantially equal specific resistance.

2. A heater as claimed in claim 1 wherein the electrodes within at least one of the rows are disposed at unequal intervals in the first direction.

3. A heater as claimed in claim 2 wherein the electrodes are disposed in columns extending in the second direction, the electrodes disposed in each column being aligned with one another in the second direction, the columns being spaced apart from one another at unequal intervals in the 17 Oct 2025 first direction.

4. A liquid heater comprising: (a) a chamber; (b) a plurality of rod-like electrodes disposed within the chamber and extending substantially parallel to one another, the electrodes being disposed a two-dimensional array 2019445705

including outer electrodes cooperatively defining an outer boundary of the array and inner electrodes disposed within the boundary; (c) a three-phase electrical power supply having three poles, the power supply being operable to supply alternating potentials offset by 120º in phase to the respective poles; (d) power switches electrically connected between at least some of the plurality of electrodes and the poles, the power switches being operable to selectively connect each of the electrodes to one or another the poles and to selectively disconnect each of the electrodes from the poles so as to form current paths extending through liquid disposed in the chamber between electrodes connected to different ones of the poles, wherein the outer electrodes are disposed along an outer circle around a central axis and extend parallel to the central axis; and (e) a control processing unit coupled to the power switches and configured to actuate the power switches based on switch setting data retrieved from a memory, the memory storing a plurality of predefined connection schemes, wherein the connection schemes include groupings of electrodes defining current paths having three-fold symmetry about a central axis and substantially equal specific resistances between the poles of the three-phase power supply.

5. A heater as claimed in claim 4 wherein the inner electrodes are disposed along one or more inner circles coaxial with the outer circle.

6. A heater as claimed in claim 4 wherein the inner electrodes are disposed along a single inner circle coaxial with the outer circle.

7. A heater as claimed in claim 6 wherein M inner electrodes are disposed along the single 17 Oct 2025

inner circle and N outer electrodes are disposed along the single outer circle, and N>M.

8. A heater as claimed in claim 7 wherein the outer electrodes are disposed at equal circumferential spacings and the inner electrodes are disposed at equal circumferential spacings.

9. A heater as claimed in claim 8 wherein N=2M, and wherein each inner electrode is 2019445705

disposed at a circumferential location midway between the circumferential locations of two of the outer electrodes.

10. A heater as claimed in claim 5 wherein the inner electrodes include intermediate electrodes disposed on an intermediate circle of smaller diameter than the outer circle and innermost electrodes disposed on an inner circle of smaller diameter than the intermediate circle.

11. A liquid heater comprising: (a) a chamber; (b) a plurality of rod-like electrodes disposed within the chamber and extending substantially parallel to one another, the electrodes being disposed a two-dimensional array including outer electrodes cooperatively defining an outer boundary of the array and interior electrodes disposed within the boundary; (c) a three-phase electrical power supply having three poles, the power supply being operable to supply alternating potentials offset by 120º in phase to the respective poles; (d) power switches electrically connected between at least some of the plurality of electrodes and the poles, the power switches being operable to selectively connect each of the electrodes to one or another the poles and to selectively disconnect each of the electrodes from the poles so as to form current paths extending through liquid disposed in the chamber between electrodes connected to different ones of the poles, wherein the array includes three groups of electrodes having N-fold symmetry about a central axis wherein N is 3 or a multiple of 3, and wherein the power supply is a three-phase power supply having three poles, the power switches being operable to select sets of connected electrodes so that the connected electrodes include corresponding electrodes from each group so that the connected 17 Oct 2025 electrodes define current paths having three-fold symmetry about the central axis; and (e) a control processing unit coupled to the power switches and configured to actuate the power switches based on switch setting data retrieved from a memory, the memory storing a plurality of predefined connection schemes, wherein the connection schemes include groupings of electrodes defining current paths having three-fold symmetry about a central axis and substantially equal specific resistances between the poles of the three-phase power supply. 2019445705

12. A liquid heater as claimed in claim 11 wherein the electrodes of each group are disposed at irregular radii from the central axis, and at irregular intervals in the circumferential direction around the axis.

13. A liquid heater as claimed in claim 11 wherein the power switches are operable to select sets of connected electrodes so that the connected electrodes include corresponding electrodes from each group so that the connected electrodes define current paths having three-fold symmetry about the central axis and thereby provide current paths having equal current flows between the poles of the power supply, and so that the connected electrodes include other electrodes which provide current paths having unequal current flows between the poles of the power supply, and so that the paths of equal current flow carry at least the majority of the current flowing between the poles.

14. A liquid heater comprising: (a) a chamber; (b) a plurality of rod-like electrodes disposed within the chamber and extending substantially parallel to one another, the electrodes being disposed a two-dimensional array including outer electrodes cooperatively defining an outer boundary of the array and interior electrodes disposed within the boundary; (c) a three-phase electrical power supply having three poles, the power supply being operable to supply alternating potentials offset by 120º in phase to the respective poles; (d) power switches electrically connected between at least some of the plurality of electrodes and the poles, the power switches being operable to selectively connect each of the electrodes to one or another the poles and to selectively disconnect each of the electrodes from the 17 Oct 2025 poles so as to form current paths extending through liquid disposed in the chamber between electrodes connected to different ones of the poles, wherein the electrodes are arranged in a hexagonal array including outer electrodes defining an outer regular hexagon and inner electrodes defining a an inner hexagon, the array including rows extending along three sets of row axes disposed at 60° angles to one another, the row axes in each set being uniformly spaced and the spacings between row axes of the three sets being equal to one 2019445705 another, the row axes intersecting one another to form a grid of equilateral triangles, the electrodes being disposed at the vertices of the equilateral triangles, the power supply being a three-phase power supply having three poles, the power switches being operable to connect sets of the electrodes to the poles so that the electrodes connected to the poles have three-fold symmetry around a central axis and define current paths between the poles having three-fold symmetry around the central axis; and (e) a control processing unit coupled to the power switches and configured to actuate the power switches based on switch setting data retrieved from a memory, the memory storing a plurality of predefined connection schemes, wherein the connection schemes include groupings of electrodes defining current paths having three-fold symmetry about a central axis and substantially equal specific resistances between the poles of the three-phase power supply.