Hydraulic Dredges

Hydraulic Dredges

Table of Contents

There are two general and distinct classes of hydraulic dredges, the River Type and the Sea-going Hopper Dredge. The first is the smaller machine, built for use in the calm waters of rivers and sheltered harbors, and is rarely self-propelling except among those designed by the Mississippi River Commission to meet the special and unique conditions obtaining upon that mutinous water-course. The second is the more imposing, ocean-going steamer, with moulded hull and, in the great majority of cases, with self-contained hoppers for receiving and transporting the pumpings. Although at times economically adaptable to ship-channel work in rivers and harbors, it is primarily intended for the removal of obstructive ocean bars.

The type of dredge developed by the Mississippi River Commission under the direction of the U. S. Government, embodies many distinctive features, by virtue of which it has been isolated in the foregoing classification and will be discussed subsequently in this chapter. The text immediately following refers only to the

Radial-Feeding Hydraulic Dredge with Spud Anchorage

General Description.—Briefly, it consists of a centrifugal pump directly connected to a Steam engine or a motor and mounted on a hull equipped with a hinged ladder, carrying both suction pipe and cutter-head, and with some means of position control and feeding movement.

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The dredge may be said to have three principal functions; first, the breaking up or cutting of the bottom material so that it readily may be drawn into the suction pipe; second, the horizontal and vertical movement of the suction pipe and cutter so that the material may be fed constantly to the suction; third, that of pumping the material through suction pipe, pump and discharge pipe into the spoil basin or the adjacent deep water. The first is accomplished by means of a rotating cutter at the suction end of the ladder, keyed to a shaft carried on the ladder and driven by an engine, called the cutter engine, located at the ladder hinge. The vertical movement of the second function is obtained by boom, “A” frame, back guys, fall and hoisting engine, by which the ladder is raised and lowered.

The horizontal movement, laterally, is effected in one of two ways; either by swinging the dredge and ladder as a whole about one of the two stern spuds as a pivot, or by swinging the ladder only, the dredge being held by three or four spuds. The forward movement or the “advance in cut” is obtained in the first instance by the alternate use of the two stern spuds and, in the second, by “walking” spuds. The third function is accomplished by the pump and its drive with the appurtenant suction pipe, floating discharge pipe or pontoon line, and the shore pipe-line.

All operations, with the possible exception of the spud hoists, are controlled from the pilot house or lever-room. The “runner” or “leverman” faces the suction end and is guided in his manipulation of the dredge by pressure, vacuum, steam and depth gages and by the behavior of the cutter engine.

The size of the pump varies from twelve to forty-eight inches, but the twenty inch dredge is generally conceded the most advantageous for all around work. Twelve and fifteen inch dredges are useful for dredging in confined areas and for pumping into basins of small capacity, or into filled piers or behind bulkheads. The larger sizes are well adapted to the removal of silt and fine sand and to the rehandling of material previously excavated by bucket dredges and dumped from scows to the hydraulic dredge to be pumped ashore.gold-dredge-round-nose-cutter

Cutter Head Dredge

There are two principal types of cutters, the open or cage type as shown in Figure 25a, Page 61, and the closed or round-nose cutter as illustrated in Figure 25b. The former consists of a set of knives or blades, straight or nearly so, mounted on a pair of annular frames of different diameters so that the blades converge outboard. The blades are set at an angle of from 15 to 25 degrees with the cutter head axis or shaft and protrude a short distance beyond the end frame. The great wear caused by the excessive abrasion makes it advisable that the blades be independent of the frames to facilitate renewals. In the closed or round-nose cutter, the blades are spiral shaped and converge to a hub at the nose or outboard end into which the end of the shaft is fitted. In both types, the cutter shaft is set immediately above the suction pipe, the end of which must, of course, lie within the surface described by the cutter blades. The large diameter of the cutter-head, therefore, will be somewhat greater than twice the outside diameter of the suction pipe.

The relative merit of the two cutters will vary with the character of the material and even more perhaps in the judgement of dredging men. Mr. Charles Evan Fowler, in his “Subaqueous Foundations” writes “The round-nose type is best suited to soft material, but usually the ordinary inward-delivery cage cutter is to be preferred especially for the compact and harder materials.” If working in clay, a cutter with a relatively large number of blades closely spaced has a tendency to retain the clay within itself, finally becoming entirely choked with the sticky mass.

To exclude stones and other obstructions, various straining devices are used. The cutter becomes a strainer when fitted with a series of bands or rings at the blades in planes perpendicular to the shaft, resulting in a cage-like structure, admitting only such objects as are smaller than the trapezoidal openings between the blades and the transverse rings. The material is excluded from the rear or large end of the cutter either by a transverse screen or by a solid disc, which completes the closure partially effected by suction pipe and ladder. Another scheme involves the use of a circular transverse screen or grating set in the big end of the cutter, perpendicular to the shaft and a short distance in front of the end of the suction pipe. It is attached to the cutter and rotates with it, resulting in the continual procession of the strainer openings across the mouth of the suction. It is apparent that any long and narrow obstruction, such as a stick of wood or a long stone, passing through one strainer opening and protruding into the suction pipe would be subjected to considerable

gold-dredge-ladder

shearing strain and would cause high stresses in the strainer, cutter and shaft. It is well, therefore, generously to proportion these parts that they may be capable of stalling the cutter engine without overloading themselves.

The cutter shaft is of relatively large section (about eight inches for a twenty-inch dredge) because of the torsional stresses set up in it by the maximum moment of which the cutter engine, through its gears, is capable. It is carried on pillow block journals set on the top face of the ladder.

The ladder may be a heavy steel box girder enclosing the suction pipe; or a pair of plate girders braced together; or a pair of lattice trusses. The last mentioned type offers less resistance to the current and is on that account better adapted for use in swift streams. In swinging dredges, the ladder’s perpendicularity to the bow is maintained by guy rods to the forward corners of the hull, or by spreading the two girders forming the ladder so that it is triangular in plan, or by a combination of these two methods. The stresses in the ladder are due to the pull of the swinging wires, the reaction of the rotating cutter, horizontal and vertical hull movement when the cutter head is in the mud and finally to the upward pull of the fall from the boom in raising the ladder and the dead weight of the ladder itself. The ladder is set in a well or recess in the center of the bow of the dredge. In the case of the swinging ladder dredge this joint must permit both horizontal and vertical motion of the ladder with respect to the dredge, but in the swinging dredge vertical motion only is required. If the outboard end of the latter is guyed to the hull corners, the points of attachment of guy rods must be in line transversely and vertically with the center line of the hinge.

The maximum depth below the water surface to which a hydraulic dredge can dig is determined by the length of the ladder and the hinge construction. Ladders 70 feet in length are not uncommon, permitting dredging to a depth of about 45 or 50 ft. below the water.

Feeding a hydraulic dredge

The arrangement of spuds depends upon the method of feeding. The case of the swinging dredge will be first considered.

It has but two spuds set in the same transverse plane at or near the stern of the dredge, about 10 or 12 feet apart and, if the discharge pipe is centered in the hull, symmetrical with respect to the dredge axis. If the discharge pipe leaves the hull at a point near a stern corner, the spuds are set off center in the direction of that same corner, in order that the point of attachment of pontoon line to dredge shall be as close as possible to the center of rotation to minimize the effect of the rotation upon the alignment of the pontoon line. When working, one spud is always up, clear of the bottom and the other down in the mud, forming the pivot about which the dredge rotates or swings. This radial motion is effected by two swinging wires, extending from two anchors, one on each side of the bow, to the winding drums on the dredge. The pull of the wires may be applied by means of sheaves to the outboard end of the ladder near the cutterhead, to the bow corners of the hull or to a kind of triangular horizontal apron extending forward from the bow.

gold-dredge-cutter-head

The first method has the advantage of keeping the swinging wires down on the bottom out of the way of passing vessels but presents also the disadvantage of the possibility of entanglement of the slack swinging wire about the cutter in the event of careless operation. When it becomes necessary to advance the dredge in order again to thrust the cutter into the bank, the movement is accomplished by dropping the high spud and raising the other when the dredge has swung to the limit of the cut. Thus, the new pivot is further ahead in the cut and the center of rotation therefore, advanced. In this manner, the dredge moves ahead, by the alternate use of the two spuds, successive positions of which, if plotted and joined by straight lines, would form a series of saw teeth.

The swinging-ladder dredge has four spuds, two forward and two aft, all of which are in the mud while pumping. Thus the hull is held, while the ladder is swung by means of swinging wires from cutter head to winding drums after passing through guide sheaves at the bow corners. At least three of the four spuds are set in slotted wells, permitting them to incline forward from the vertical and are called “walking spuds.” Two wires extend from the two forward spuds at or near their toes to sheaves on the stern corners of the hull, thence to drums of the winding engine. To move ahead, these wires are wound on the drums, pulling the dredge forward and inclining the spuds. When desirable, the walking spuds can be raised and plumbed, one at a time. Such dredges usually are equipped with a stern wire for hauling astern. Obviously, the swinging-ladder dredge makes a much narrower cut than the swinging-dredge and can, therefore, work in more confined spaces. Further more, the absence of side wires is often an advantage as they interfere with traffic in the vicinity of the dredge. On the other hand, the total time lost in moving and shifting the dredge is greater for the swinging-ladder type because it covers less area with one set up.

It is well to remember that the stress in one swinging wire is nearly always greater than in the other, due to the fact that the rotation and, therefore, the reaction of the cutter is always in the same direction, which fact at times becomes a factor in locating the dredge to the best advantage.

Boom “A” Frame and Back Legs.—Boom and “A” frame are set well forward, the former at a fixed angle of from 30 to 4.5 degrees with the horizontal, the latter generally vertical. The ladder is suspended from the boom head by a wire fall, which is purchase rigged to the ladder near the cutterhead and partially bridled to relieve the bending stresses in the ladder. The purchase retards and eases the vertical motion and at the same time economizes in engine capacity. The boom may be a single strut, or, in dredges that swing as a unit, it may be, in plan, an acute “A” frame to prevent lateral movement. The boom of the swinging-ladder dredge necessarily swings with the ladder, its heel, therefore, being mounted upon the same rotating bearing that carries the ladder trunnions. The “A” frame varies greatly in design from the ordinary timber type to steel frames of rectangular and even polygonal outline. The stresses in boom, “A” frame and back-legs or guys may be obtained graphically, knowing the maximum pull of the ladder hoisting engines, the purchase ratio and the angles of lead. The “A” frame and back- guy stresses will be greater in the swinging-ladder type than in the swinging dredge, becoming maximum for the same positions of the boom as outlined under grapples.

In some dredges, built for use in localities where head room is a consideration, such as in canals with limited clearance under bridges, the boom angle is quite flat and the topping-fall almost horizontal, and at about the same elevation as the roof of the house, which is but one story high. A house boat is generally an indispensable adjunct to such dredges as there is no space for living quarters on the dredge. A notable example of this is afforded by dredges used in the construction and maintenance of the New York State Barge Canal.

The Pipe Line.—The total length of discharge pipe line comprises three distinct parts; First, the portion on the dredge from the pump to the point of leaving the hull; second, the floating or pontoon line; and, third, the shore line. Since the first constitutes a permanent integral part of the dredge and also because any break therein might flood the hull and sink the ship, it is made of heavy cast iron pipe with flanged joints. For the usual central transverse position of the pump, there are three principal methods of discharge pipe arrangement resulting in three possible locations for the point of attachment of the pontoon line. Referring to Figure 28, they are No. 1, a direct lead from the pump to the side of the hull; No. 2, a stern discharge near one corner; and No. 3, the axial stern location. No. 1 discharge is limited to the swinging-ladder dredge, since in the swinging dredge type it is essential that the point of connection of pontoon line to dredge be close to the spud about which the dredge rotates. Location No. 3 presents the objection of loss of head due to reverse curvature. For general use, Discharge No. 2 appears to be preferable. Some swinging-ladder machines are equipped to discharge either at the stern or side, combining Nos. 1 and 2, or 1 and 3.

gold-dredge-discharge-pipe

The precaution is sometimes taken to enclose the pump in a watertight trough so that, in the event of a break in the pipe, the hull will not be flooded. Location No. 2 is particularly adapted to this safety measure in that the side of the hull and the adjacent longitudinal bulkhead form the two sides of such a trough and easy access may be had to the pipe so located by substituting removable hatches for the deck over the pipe.

The floating line from the hull to the shore pipe consists of a series of pontoons, each carrying a section of pipe from 20 to 50 feet long, so coupled to each other as to provide the necessary flexibility to allow the dredge to swing and advance or, in other words, to “wag her tail.” Both pontoon and shore pipe are lap or spiral riveted, usually the former, the sheets varying in thickness from No. 10 gauge to 3/8 inch. The shore pipe is built in sections of such length as to preclude a weight too great for handling by manual labor alone. They are seldom greater than

gold-dredge-pontoon-line

20 feet long and frequently only 16 feet. The pontoon pipe, however, is built in long lengths to reduce the number of couplings and pontoons and is usually of heavier metal. The sections of the floating line are coupled either by heavy, rubber sleeves or by ball-joint connections. Because of the short life and expense of the rubber sleeves (they are seldom less than 10 ply) the ball joint method is more economical. The shore pipe is laid with slip joints, the male end of each length protruding into the female end of the next in the direction of flow, and wired together, a pair of hooks being riveted to each end of each section for this purpose. Leaks in such unions are almost the rule rather than the exception and are plugged by wedges made from wooden shingles or equally convenient shapes. In tide water, the varying elevation of the pontoon line with respect to the shore line is permitted by a ball-joint pipe.

The most common fittings used in connection with dredge discharge pipe are bends, elbows, “Y”s and gate valves. The necessity often arises for sharper curvature in the pontoon line than is permitted by the couplings and it is met by the insertion of one or two elbows in the line. The shore pipe deviates from the straight line in circumventing obstructions and in continual shifting of direction in the basin to control the elevation of the fill. Although the slip joints allow a certain amount of curvature, bends of 15 degrees and up are often essential fittings. In order to facilitate the addition of pipe sections in the basin without stopping the pump, it is common practice to divide the line into two branches by means of a “Y” and two gate valves at a point on the fill near the discharge, with the idea that one leg of the fork can be lengthened while pumping continues uninterruptedly through the other. For this to be a true economic advantage, it is apparent that the loss of head due to the fittings must be fully offset by the prevention of lost pumping time.

There are several types of pontoon, which may be designated as the “scow” pontoon, the “catamaran” pontoon and the “cask” pontoon. The first is simply a single box of wood or steel, rectangular in plan and section. The second consists of two long narrow floats, either steel cylinders or wood boxes, with their longitudinal axis perpendicular to the pipe mounted upon them amidships. By struts from one to the other of each pair, they are maintained at such a distance apart as to place one near each end of the pipe section. In the “cask” pontoon, the pipe rests upon a horizontal frame to the under side of which a number of strong barrells or large kegs are fastened, providing

gold-dredge-pump-and-thrust-bearing

the requisite buoyancy. For the convenience and safety of the crew and pipe gang, it is desirable to provide a continuous walkway on the pontoons parallelling the pipe.

The Pump.—Centrifugal pumps, as used for dredging, have the following features: They are horizontal pumps, i.e., the shaft is horizontal. They are high-pressure pumps, i.e., the head for which they are designed is greater than 50 feet. They are single stage pumps, i.e., the head is generated by one impeller only. They are single inlet pumps, i.e., the water enters the impeller on one side only, necessitating the interposition of a marine thrust bearing between the pump and its drive. They are backward-discharge pumps, i.e., the impeller tips are bent backward and the angle θ, Figure 31, Page 72, between the absolute velocity of exit of the water, V1, and the peripheral velocity of the impeller, V2 is greater than 90 degrees. This type of pump is the most common even in water pumps and, because the head decreases as the quantity increases, it is better able to take care of a fluctuating quantity without overloading its driver. It is this feature principally which makes it the most desirable type for dredging purposes.

There are no discharge vanes in dredge pumps.

The impeller is of the enclosed type. The open type, due to rapid wear, loses in efficiency.

gold-dredge-pump-impeller

The openings between blades and between the periphery of the impeller and the shell are large to permit the passage of large objects such as stones, short timbers, etc. The pump must be designed to take anything that can pass through the cutterhead and into the suction. Chokes in the pump involve loss of time and possibly serious damage.

Both impeller and shell are heavily proportioned to withstand the abrasive action of the pumpings. It is good practice to use an unlined pump of very heavy proportions fitted with lined side discs.

Shell and impeller are cast iron or cast steel.

The problem is to design a pump to stand the unusually severe wear and tear, to pass fairly large bodies, and yet be of reasonable efficiency. It is apparent that the efficiency of a dredge pump cannot be as high as that of a water pump, and seldom exceeds 50 per cent.

The total head to be overcome by the pump is the sum of three components: first, the suction head; second, the height through which the discharged pumpings are raised, which may be termed the elevation head; and third, the friction head, or the frictional resistance offered by the pipe to the passage of the pumpings. The pump, therefore, must create a partial vacuum in the suction pipe in order to give to the entering water a high velocity, sufficient to draw in the material and to keep it moving and must also produce the pressure necessary to force the water and material through the discharge pipe against the resisting elevation and friction heads. Twenty-inch pumps have been designed capable of elevating the pumpings to a height of 40 feet through 3,000 feet of pipe line.

The suction head is the amount of vacuum in the suction pipe and is read on the vacuum gage in the lever room of the dredge. This gage is usually calibrated in inches of mercury, i.e., the barometric column. Its readings may be converted into feet of water by multiplying by the constant 1.13 which figure is the ratio between the weight of a column of mercury one inch high and a column of water one foot in height, since the specific gravity of mercury is 13.56.

The elevation head is the vertical height from the center of the pump to the highest point in the discharge pipe line. The sum of the friction and elevation heads is obtained from the discharge pressure gage in the lever-room. The reading is in pounds of pressure per square inch and may be converted into feet of head by multiplying by the constant 2.304. As this gage is usually some distance above the center of the pump, the head represented by the gage reading must be increased by this distance in order to get the actual discharge head on the pump. The friction head increases as the velocity of discharge increases and decreases as the pipe diameter increases. It is directly proportional to the length of pipe line. Curvature in the line, sleeve and ball joints in the pontoon line and gate valves develope a certain amount of friction head, but, for the average line of reasonable straightness, this is neglected and, in the determination of the value of the friction head, the pipe is regarded as straight. Hyraulic dredging is by no means an exact science, and the friction head in particular varies with the nature of the material dredged so that it would be inconsistent to attempt to evaluate such minor head losses as are caused by pipe fittings and moderate curvature.

The Table on Page 75 copied from the Morris Machine Works Catalogue, gives full data as to friction head velocity and discharge for water flowing through pipes. Since the frictional resistance is greater for hydraulic dredgings or pumpings than for water, the values of the head in this table must be increased for dredging computations by an amount depending upon the nature of the material being dredged. By multiplying the friction values for water by 1.35, very good results are obtained for average hydraulic material. The increase varies from 0.1 to 0.5.

The key to the investigation of dredge pump performance is the relation between the total head and the peripheral velocity of the impeller or runner. From any discussion of the theory of the centrifugal pump, it can readily be determined that the peripheral speed is directly proportional to the square root of the total head, or if

P. V. = the peripheral velocity of the impeller
H = the total head and
C = a constant
then P. V. = C√H

The value of the constant, C, varies somewhat with the capacity, being higher for higher capacity; also with the ratio between the outside diameter of the runner and that of the throat opening, being higher for smaller diameter runners with the same throat opening; and also with the angle of slope of the vanes at the periphery, being

gold-dredge-capacity-in-gallons

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smaller for larger angles. The determination of the constant is purely a matter of judgment with the above for a guide, but for average practice a value of 435 may be used with very good results. Therefore, when H is in feet and P. V. in feet per minute.

P. V. = 435√H………………………………………………………(1)

From this important equation, the number of revolutions per minute necessary for an impeller to turn that it may

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develop a given head can be determined; or, having the R. P. M. and total head, the proper diameter of impeller can be selected; or, knowing the R. P. M. and the impeller diameter of any machine, the head of which it is capable can be found.

In this connection, it is well to remember that the diameter of the impeller, for efficient operation, should never be less than 2.3 times the size of the pump; e.g., a 20-inch pump requires- a runner at least 46 inches in diameter.

Power.—The theoretical horse power developed by the pump is equal to the continued product of the discharge in gallons per minute by the weight of one gallon of water in pounds by the total head in feet divided by 33,000 foot pounds per minute, or if

Q = the discharge in gallons per minute
and H = the total head in feet, then

Theoretical H. P. = Q x 8.33 x H/33,000
or Theo. H. P. = Q x H/3960…………………………………………(2)

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The actual horse power required to drive the pump is about twice the above theoretical value since the efficiency of a dredge pump is generally about 50 per cent, or a trifle less.

For maximum economy of power, the velocity through the pipe should be no more than just enough to carry the material along because higher velocity means increased friction head and increased quantity, with both of which the horse power varies directly. For heavy material, such as coarse sand and gravel, however, the velocity must be higher than for silt, mud and fine sand to prevent the settlement of the material in the pipe and the consequent choking.

Long pipe lines require a large impeller to give the high peripheral velocity necessary to overcome the great head. However, should the same impeller be used on short lines, the engine speed would be so retarded as to prevent the engine from delivering its maximum power. The smaller impeller in the short line allows greater engine speed, maximum power and, therefore, greater quantity of material pumped. It is obviously advantageous, therefore, that a dredge intended for miscellaneous work have several impellers of different diameters, increasing the size with lengthening pipe line.

Hydraulic Dredge Design

 

Forward Feeding or Mississippi River Type

General Description.—The maintenance of low-water navigation in the shallow waters of alluvial rivers through the persistent and shifting sand-bar formations is a problem requiring special treatment. In the Mississippi River, these bars, during seasons of high water, assume extensive proportions, but as the river falls, they are cut out by the current erosion, and it is to assist and hasten this natural deepening tendency that the dredges are designed. To complicate matters, the usual dredging season, i.e., that portion of the year during which the state of low water exists; is only four months in length—from August 15 to December 15. (The river has a range in stage of upwards of 50 feet in places.) The amount of dredging to be done in any one season cannot be predicted, because of the impossibility of forecasting the stage or the rapidity of fluctuation. The only solution, therefore, appears to be simply that the dredges be available during the low-water season, to do whatever dredging may be required.

After repeated failures to make and maintain the depth necessary for low-stage navigation by means of current deflectors, water jets, stirring and scraping machines and other devices, the Mississippi River Commission, in 1892, built the 30 inch hydraulic dredge, ALPHA, for experimental purposes. The ALPHA proved so successful that eight additional machines were built by the Commission within the next decade. Various improvements and changes were made in the successive dredges, as experience dictated, until the type of to-day was developed and the Mississippi problem solved.

The typical Mississippi River Dredge differs from the radial feeding machine with spud anchorage, principally in:

  1. Greater capacity and lighter draft, both of which features are necessitated by local conditions.
  2. The Method of Feeding.—While some units of the Mississippi River plant are self propelling, being equipped with side paddle-wheels, many are pulled forward over the bar to be dredged by two cables extending from hauling engines on the forward deck to mooring piles driven in advance of the dredge. For holding the dredge in position while running lines or shifting the mooring piles, a single spud is set well forward.
  3. The Use of Water Jet Agitators.—On the Mississippi the mechanical cutter has been largely replaced by the water-jet agitator. The suction head, which is at the bow of the dredge—as is the pump also—is flattened down to a depth of about 8 inches and flared horizontally to a considerable width varying from about 20 to 40 feet. Beneath the suction head is a pressure chamber with a series of nozzles, through which water is pumped under pressure, constituting the jet agitator, the province of which is to loosen the sand or other material for entrance into the suction.
  4. Smaller Maximum Depth Capable of being Made. Mississippi River Dredges are generally equipped to lower their suctions to a depth not exceeding 20 feet, and the practice, apparently, is to dig always as deep as the suction head will permit.
  5. Use of the Double Suction.—Most of the dredges built by the Commission have a single dredge pump set vertically in the plane of the longitudinal axis of the hull, with a double suction leading into it on both the port and starboard side, and a single discharge pipe extending axially from the pump to the stern of the dredge, thence to the pontoon line. Each suction is, roughly, 24 inches in diameter, and the discharge pipe from 32 to 36 inches. Incidentally, the balanced suction obviates the necessity for a thrust bearing.

As in the other river types, the upper deck contains quarters for the captain and crew, numbering generally from 45 to 50 men.

The bars to be dredged form as a rule diagonally across the river at the point of change in the direction of curvature of the current, so that they lie between two pools of relatively deep water, located at the outer bank of two successive bends in the river. The axis of the dredged cut should coincide with the direction of the current, using a sufficient length of pontoon line to discharge into the deep water below the bar. Successive parallel cuts yield the desired channel width.

More detailed information regarding the Mississippi River Dredges and the performance tests made by the Commission is contained in a paper by F. B. Maltby, M. Am., Soc. C. E., published in Vol. LIV., of the Transactions.