Field of the Invention The invention relates to filtering and separating a continuous phase fluid with low specific gravity from a discontinuous phase fluid with a higher specific gravity, for example separating water from diesel fuel. The invention relates also to a coalescer vessel for such use.

Background of the Invention

A considerably variety of applications for coalescers have been used to remove small amount of a first phase, the discontinuous phase, from a second phase, the continuous phase, in both gasses, fluids and a mixture of gasses and fluids. Fluids may constitute mixtures or suspensions of immiscible liquids such as water and oil from petroleum based fuels used as aviation kerosene, diesel fuel and gasoline, and in liquid hydrocarbons from base oil or synthetic oils used in hydraulics, transmissions, bearings and gears.

As an example of a coalescing process a liquid hydrocarbon such as diesel fuel contaminated with suspended water passes through a coalescing medium. As the discontinuous phase water is forced by the continuous phase fluid to pass through the coalescing medium the water attracts and collects, or "coalesces" on the downstream side of the coalescer medium as conglomerated small droplets forms into fewer larger droplets. The efficiency of the coalescer is heavily affected by the ability to form droplets of considerable sizes that will prevent re- entrainment of the droplets in the continuous phase fluid that emerge the coalescer vessel before a complete separation of the discontinuous phase fluid has taken place. As coalescer medium metals such as steel, stainless steel, and synthetics i.e. surface treated or no-surface treated polymeric materials i.e. thermoplastics have been used for water separation of hydrocarbon-based liquid fuels.

Considering the laws of Stokes it is readily seen that re-entrainment of relative small droplets may easily be aggravated by a relative high flow of the continuous phase fluid to the coalescer vessel outlet. Various mechanisms have been introduced to avoid entrainment of readily formed small droplets. Among these embodiments are large coalescer vessels providing low flow velocities supporting a natural settling of the discontinuous phase fluid. Other coalescer vessels are provided with separator means following the coalescer media. However, efficiency of the coalescer assembly may be affected if the droplets of the discontinuous fluid are smaller than the orifices in the separator allowing droplets to pass through the separator with the continuous phase fluid.

Coalescers generally suffers from the ability to handle fluids with particulates causing the coalescer medium to close for the passage of the fluid being processed due to particle contamination at the coalescer medium upstream side. Consequently, coalescers often have high demands to effective particulate filters to gain sufficient lifetime of the coalescer media. In the field of fuel preparation, the demands for efficient and compact water separators are still increasing. Compared to fuels in the past, modern low and ultra low sulfur diesel fuels, biodiesel, and modern technology additive packages all have a generally lover interfacial tension (ITF), consequently, water contamination in modern fuels have a more stable emulation due to the dispersed and suspended smaller droplet sizes. In these fuels with reduced interfacial tension and decreased droplet size water removal by natural settling or by coalescing is therefore much more complicated. Furthermore, modern high pressure common rail diesel fuel pumps sets new lower limits for acceptable water contents due to the higher pressure settings that worsen pump damages due to water content.

Examples of water separators are disclosed in International patent application

WO2006/032270, US patent No. US5938921, and British patent application GB2115305. Other water separators in the form of a coalescer with a filter medium are disclosed in US patent No. US7938963 and US patent application No. US2011/0168647.

Description of the Invention

The coalescer vessel embodying the present invention overcomes many of the limitations encountered in the prior art by several means. In addition several coalescer functional aspects are integrated in one embodiment.

The objective is achieved with a coalescer vessel for filtering and separating a continuous phase fluid with low specific gravity from a discontinuous phase fluid with a higher specific gravity characterized in that the coalescer vessel comprises:

- a coalescer vessel container comprising an upper part and lower part assembled and connected by assembling means, for example with a sealing ring and a ring spanner, the lower part having a bottom flange and a side wall extending upwards from the bottom flange;

- a fluid inlet, a continuous phase fluid outlet, and a discontinuous phase fluid drain outlet,

- a coalescer filter medium arranged in the upper part of the said vessel container,

- an internal filter medium container confining the said coalescer filter medium, - a upstream fluid rising tube, for example in combination with a center tube, in the said coalescer filter medium directing incoming fluid to the upmost part of the said coalescer filter medium,

- a bottom surface of the said coalescer filter medium, the bottom surface comprising an outlet of the said coalescer filter medium for the downstream fluid flow from the said coalescer filter medium into the lower part,

- a horizontal orientated disc assembly below the bottom surface, the disc assembly comprising an upper and lower surface, the upper surface directing the fluid flow from the outlet of the said coalescer filter medium in a predominantly horizontal direction,

- a plurality of baffle plates arranged on the upper surface of said horizontal disc assembly for applying forces to the fluid flow that diverts the fluid flow away from a radial direction towards the rim of the said horizontal disc assembly before the fluid diverge to below the said horizontal disc assembly,

- a demister means located under and near the rim of the said horizontal disc assembly, the demister means pointing downwards,

- a continuous fluid outlet aperture connected to the said fluid outlet and located closer to an underside of the said horizontal disc assembly than to the bottom flange and closer to an axial centerline of the said coalescer vessel than to the wall of the lower part,

- a drainage collecting space located in the lowermost part of the said coalescer vessel.

For example, the fluid is diesel fuel with slight amount of water in the diesel fuel, which is to be separated from the diesel fuel. Other liquid combinations, where the coalescer according to the invention is useful are hydraulic oils, turbine oils, heat transmission oils and lubrication oils, especially with low viscosity.

In accordance with one aspect, the present invention utilizes a coalescer filter medium. For example, the filter medium is based on natural hydrophobic fibers, for example wool, flax, or kapok or a mixture thereof. The present invention utilizes the ability of the medium to manage particulate filtration and accelerate the coalescing process by having a relative large volume. It is readily known, that coalescer efficiency is proportional to the thickness of the coalescer medium. In accordance with the present invention the flow direction of the fluid is directed in axial direction through the structure of the filter and coalescer medium and by this means the fluid travel is elongated compared to radial, horizontal flow direction filters/coalescers. The axial direction for the flow, typically, is vertical or only slightly slanted from vertical. For example, the filter least comprises two axially orientated medium materials with different densities.

In accordance with a second aspect, the present invention has a coalescer element with deep filtration ability directed to enhance the lifetime of the coalescer medium significantly.

Furthermore, the coalescer vessel can handle standard fluids avoiding specification requirement to fluid quality and thus avoiding or lower costs for pre-filtration of the fluid prior to the fluid enters the coalescer vessel. In accordance with a third aspect, the present invention has an axial orientated coalescer medium confined in a canister, for example cylindrical shaped canister, with openings for the upstream fluid and downstream fluid in one end, allowing the system to be serviced and maintained without oil spill and preventing trapped particulate contamination to enter the downstream side of the coalescer vessel.

In accordance with a fourth aspect, the present invention allows coalesced droplets from the discontinuous phase fluid to start settling immediately in the continuous phase fluid stream where it leaves the coalescer medium. This part of the coalescer vessel is shaped as a horizontal orientated disc assembly leading the fluid stream to the perimeter of the coalescer vessel. The disc assembly is offset at a distance to the coalescer medium downstream side confining a space with sufficient low flow rate of the continuous phase fluid to allow the coalesced discontinuous phase fluid drops to start settling onto the said plate. Settled drops conglomerates further into larger droplets on the horizontal orientated disc assembly. In accordance with a fifth aspect, the present invention contains baffles on the horizontal orientated disc confining the space at the coalescer medium downstream side. The baffles have two functions. First, the baffles support the pressure originating in the fluid back pressure of the coalescer medium, second, the baffles partition and divert the flow away from a radial direction to impose centripetal and tangential forces on the downstream flow of the fluid stream to further conglomerate collected droplets of the discontinuous phase fluid on top of the horizontal orientated disc assembly.

In accordance with a sixth aspect, the present invention diverts the downstream fluid flow from the rim of the said horizontal disc assembly to the underside of the said horizontal disc assembly where the downstream flow is directed horizontally towards the center portion of the vessel where the continuous phase fluid outlet is located. The horizontal orientation of the downstream flow assists further separation of droplets of the discontinuous fluid and hinders the risk of re-entrainment of droplets in the continuous phase fluid. In accordance with a seventh aspect, the present invention contains a demister means near the edge of the said horizontal disc assembly. Although, the demister can be oriented horizontally, it is preferred to orient it slanted or vertically for easy flow of fluid downwards from the demister. The demister is of a material, preferably stainless steel or synthetic polymeric materials that are permeable for the continuous phase fluid and demists droplets form the discontinuous phase fluid. It is configured for directing part of the flow through the material of said demister means. Thus, advantageously, the demister is porous; for example configured such that is not wetted by the discontinuous phase fluid. Optionally, it is made of polymeric materials and with hydrophobic properties. For example, the demister means comprises polypropylene, fiberglass, or Teflon. It may have a pleated surface structure. The said demister has two functions. First, the demister back pressure spread the fluid flow at the underside of the horizontal disc assembly into a larger area, decreasing the effective flow rate per square area, promoting separation of small droplets, thus minimizing re-entrainment of the discontinuous phase fluid to enter the continuous phase fluid outlet. Second, the demister separates droplets formed in the upward pointing compartment of the horizontal disc assembly and guides these to the bottom of the coalescer vessel to the discontinuous phase fluid reservoir and drain facility.

In accordance with an eighth aspect, the present invention contains a final thin porous layer classifier in the coalescer media downstream surface, for example made of polymeric material such as non treated or surface treated polypropylene, fiberglass or Teflon; advantageously polymeric materials with hydrophobic properties. For example, the classifier has a porous structure that is not wetted by the discontinuous phase fluid. The said final classifier has two functions. First, the classifier secures that non bound fibers from the coalescer medium do not enter the fluid flow to the outlet of the coalescer vessel, second, the final classifier supports further conglomeration of small droplets formed in the coalescer medium.

In accordance with a ninth aspect, the present invention is provided with a check valve in the coalescer fluid outlet. For example, the check valve is connected between the said fluid outlet aperture and the continuous phase fluid outlet with the said check valve flow direction pointing towards the continuous phase fluid outlet. The valve exercises an initial pressure inside the coalescer vessel in all operational applications. This initial pressure will enable drainage of the discontinuous phase fluid even if the coalescer vessel outlet port is connected to an application with zero or negative pressure. Furthermore, the check valve secures that no fluid will flow back from the application through the coalescer vessel outlet during disassembly of the coalescer vessel for service. For example, the check valve has a cranking pressure above 0.2 BAR.

In accordance with a tenth aspect, the present invention can be disassembled for service and maintenance by an outer vessel assembly connected by a ring spanner. In order not to apply unwanted fluid spill during service seals are installed to prevent the fluid to rise above the rim of the lower coalescer vessel part.

By utilizing the means described in the present invention for a coalescer vessel several advantages can be achieved. The main features and benefits are that the coalescer does not require pre-filtering of particulates of the upstream fluid to ensure significant lifetime of the coalescer medium, the materials used for filtering and coalescing are natural fibers that can be recycled without damage to the environment, separation of the discontinuous phase fluid is optimal arranged in a relative small confined space avoiding re-entrainment and re- contamination of the continuous phase fluid, drainage or accumulation of the discontinuous phase fluid is easily done in all applications of the coalescer vessel, service and maintenance are supported by easy accessibility to vital components without fluid spill, and the final advantages, the coalescer overall compact size and cost effective performance.

Application of the coalescer vessel described is primary for filtering and separating a continuous phase fluid with low specific gravity from a discontinuous phase fluid with a higher specific gravity. The two fluids could be water suspended in liquid hydrocarbons, for example diesel oil or low viscous hydraulic oil. Typically viscosities of the oils are between 4 and 100 cST.

Optionally, a discontinuous accumulator tank assembly is arranged between the said drainage collecting space and the discontinuous phase fluid drain outlet. Optionally, the discontinuous phase fluid drain outlet comprises a valve that is automatically operated.

The objective is also fulfilled by the following method for filtering and separating a continuous phase fluid with low specific gravity from a discontinuous phase fluid with a higher specific gravity with a coalescer vessel according to the above description. The method comprises the steps as follows: orienting the coalescer vessel with the upper part above the lower part and with the upper surface of the horizontal disc assembly being oriented horizontal; pumping fluid through the fluid inlet, the upstream fluid rising tube, optionally the center tube, and into the free space above the coalescer filter medium; causing the fluid to downwardly traverse the coalescer filter medium and the bottom surface and to flow onto the upper surface of the horizontal disc assembly; causing the fluid to flow towards the rim of the said horizontal disc assembly and onto the demister means before the fluid fills the drainage collecting space from which the continuous phase fluid is drained via the continuous fluid outlet aperture and the discontinuous fluid is drained via a discontinuous phase fluid drain outlet at the bottom flange.

The description and claims use the terminology of "upper", "lower", "axial" and "radial", as this is related to proper use of the coalescer, where the axial direction refers to a direction along the center tube, and the radial direction refers to a direction perpendicular thereto. Although the coalescer in principle can be oriented differently, for example during transport, the orientation during proper use, however, is clear to the skilled person, why the terminology is justified and used for simplicity.

Use of the vessel are specifically in diesel fuel supply for engines for ships but also in maintenance and reclamation of turbine oil for power generators, heat transmission oils used in transformers and electrical machines for cooling means, oils applied in hydraulic machinery and lubrication oils used in gearboxes and combustion engines.

Interrelated Aspects

In the following, a number of interrelated aspects are described.

Aspect 1. A coalescer vessel for filtering and separating a continuous phase fluid with low specific gravity from a discontinuous phase fluid with a higher specific gravity wherein the coalescer vessel comprises:

a) a two part coalescer vessel container consisting of an upper and lower vessel part assembled with a sealing ring and a ring spanner,

b) a coalescer vessel provided with a fluid inlet, a continuous phase fluid outlet, and a discontinuous phase fluid drain outlet,

c) a coalescer filter medium arranged in the upper part of the said vessel container,

d) an internal coalescer filter media confining the said coalescer filter medium, e) a upstream fluid rising tube and a center tube in the said coalescer filter medium directing incoming fluid to the up most part of the said coalescer filter medium, f) a downwards pointing axial surface of the said coalescer filter medium for the downstream fluid flow from the said coalescer filter medium,

g) a horizontal orientated disc assembly with an upper and lower surface directing the fluid flow from the outlet of the said coalescer filter medium in a predominantly horizontal direction,

h) a plurality of baffle plates arranged on the said horizontal disc assembly that diverts the fluid flow in a centripetal direction towards the rim of the said horizontal disc assembly applying centripetal forces to the fluid flow before the fluid diverge to the other side of the said horizontal disc assembly.

i) a demister means located near the rim of the said horizontal disc assembly pointing downwards preferably made of a hydrophobic material,

j) a fluid outlet aperture connected to the said fluid outlet located close to or at the underside of the said horizontal disc assembly and close to the axial centerline of the said coalescer vessel ,

k) a drainage collecting space located in the lowermost part of the said coalescer vessel,

I) an internal coalescer container confining the said coalescer filter medium.

Aspect 2. A coalescer vessel according to aspect 1, wherein the fluid flow is directed axial through the said coalescer filter medium.

Aspect 3. A coalescer vessel according to aspect 1, wherein the said coalescer filter medium is confined in a cylindrical shaped canister where openings for the up and

downstream fluid are both located in one axial end of the said cylindrical shaped canister. Aspect 4. A coalescer vessel according to aspect 1, wherein the said coalescer filter medium is based on non-treated or surface-treated natural hydrophobic fibers like wool, flax or kapok.

Aspect 5. A coalescer vessel according to aspect 1, wherein the said coalescer filter medium has a significant volume providing particulate deep filtration.

Aspect 6. A coalescer vessel according to aspect 1, wherein the said coalescer filter medium may consist of at least two axial orientated medium materials with different densities. Aspect 7. A coalescer vessel according to aspect 1, wherein the said coalescer filter medium has a final classifier assembled in close contact with the said coalescer filter medium downstream surface. Aspect 8. A coalescer vessel according to aspect 1, wherein the fluid stream is predominantly horizontal on either sides of the said horizontal disc assembly.

Aspect 9. A coalescer vessel according to aspect 1, wherein the said demister means is orientated in any angle to envelope the fluid flow in order to direct the full flow through the said demister means.

Aspect 10. A coalescer vessel according to aspect 1, wherein the said demister means is preferably orientated vertical to envelope the fluid flow in order to direct the full flow through the said demister means.

Aspect 11. A coalescer vessel according to aspect 1, wherein the said demister means is preferably located near the rim of the said horizontal orientated disc.

Aspect 12. A coalescer vessel according to aspect 1, wherein the said demister means is of a material, supported or non-supported, that are permeable for the continuous phase fluid and demists droplets form the discontinuous phase fluid.

Aspect 13. A coalescer vessel according to aspect 1, wherein the said demister means preferably is with porous structure of a material that is not wetted by the discontinuous phase fluid.

Aspect 14. A coalescer vessel according to aspect 1, wherein the said demister means is of polymeric materials either coated or non-coated and with hydrophobic properties.

Aspect 15. A coalescer vessel according to aspect 1, wherein the said demister means preferably is of a polymeric material such as non-treated or surface treated polypropylene, fiberglass or Teflon.

Aspect 16. A coalescer vessel according to aspect 1, wherein the said demister means has a pleated surface structure.

Aspect 17. A coalescer vessel according to aspect 1, wherein a check valve is connected between the said fluid outlet aperture and the said continuous phase fluid outlet with the said check valve flow direction pointing towards the continuous phase fluid outlet.

Aspect 18. A coalescer vessel according to aspect 1, wherein a discontinuous accumulator tank assembly can be arranged between the said drainage collecting space and the said discontinuous phase fluid drain outlet.

Aspect 19. A coalescer vessel according to aspect 1, wherein the said discontinuous phase fluid drain outlet can be manually or automatically operated.

Aspect 20. A coalescer vessel according to aspect 3, wherein the said coalescer filter medium is provided with an internal rising tube for the fluid flow from on axial end of the said coalescer filter medium to the other axial end of the said coalescer filter medium. Aspect 21. A coalescer vessel according to aspect 3, wherein the said cylindrical shaped canister is sealed by a sealing means towards the said lower vessel part.

Aspect 22. A coalescer vessel according to aspect 7, wherein the said final classifier preferably has porous structure and of a material that is not wetted by the discontinuous phase fluid.

Aspect 23. A coalescer vessel according to aspect 7, wherein the said final classifier is of polymeric materials either coated or non-coated and with hydrophobic properties.

Aspect 24. A coalescer vessel according to aspect 7, wherein the said final classifier preferably is of a polymeric material such as non-treated or surface treated polypropylene, fiberglass or Teflon.

Aspect 25. A coalescer vessel according to aspect 7, wherein the said final classifier preferably has a porous structure not permeable for the fiber diameter of the said coalescer filter media.

Aspect 26. A coalescer vessel according to aspect 17, wherein the said check valve preferably has a cranking pressure above 0.2 BAR.

Brief Description of the Drawings

The invention is described in details in the chapter following with reference to a preferred embodiment depicted in two drawings in which:

Figure 1 is a sectional drawing of the present invention.

Figure 2 is an exploded view drawing of the present invention with identical number identification as used in Figure 1. Not all numbers used in Figure 1 are visible in Figure 2.

Description of a Preferred Embodiment

According to the invention, Figure 1 shows a coalescer filter vessel for separating a discontinuous phase fluid from a continuous phase fluid which as an example could be water dissolved in liquid hydrocarbons, for example diesel oil. According to the drawing in Figure 1, the outer coalescer vessel is confined by a lower container part 1 with a bottom flange 5 and an upper container part 2. The two outer vessel container parts 1 and 2 are assembled and closed with a sealing ring 3 and a conical spanner mechanism 4.The upper container part 2 has a top wall 2" and a side wall 2', the side wall 2' extending downwards from the top wall 2" for being connected to a side wall of the lower container part 1, the side wall of the lower container part 1 extends upwards from the bottom flange 5.The side wall of the lower container part 2 comprises a cylindrical section29, which is connected to the cylindrical upper container part2, and a conical section 28that connects the cylindrical section29 with the bottom flange 5. The bottom flange 5 has an inlet opening 26'for the upstream fluid flow 26 to enter a rising pipe 6, an opening 27' for the downstream continuous phase fluid flow 27 leaving the coalescer vessel at a continuous phase fluid exit in connection fitting 19, and an opening 22' for drainage of the coalesced discontinuous fluid drainable at outlet 22 by opening the valve 21.

The fluid to be processed 26is feed to the rising pipe 6connecting the inlet opening 26' and the coalescer center tube 7directed upwards to the upper part of the free space 8 in the coalescer filter medium container 9. The coalescer filter medium 10, 11 is confined by the filter medium container 9 with fluid inlet from center tube 7 and fluid outlet at bottom surface 13. The axial bottom surface 13 is pointing downwards from the filter media container 9and hasmultible openings distributed over the main part of the downward pointing axial bottom surface 13,allowing coalesced fluid to pass freely into the chamber enclosed by the bottom surface 13 and the horizontal disc assembly 14. In accordance with the present invention, the flow direction of the fluid is directed in axial direction from the free space 8 through the structure of the filter and coalescer medium 10, 11 and 12. Preferably the coalescer filter medium 10 and 11 may be made from natural or surface-treated wool, flax, or kapok, or a mixture thereof, with coalescing properties for water. For example, the material chosen and densities required can be made from needle felt or from nonwovens, bound or non-bound by polymeric materials. In order to enhance the particulate filter performance, preferably, the first coalescer filter media 10 is of lower density than the second coalescer filter media llenabling the first filter media 10 to have a relative higher dirt holding capacity than the second coalescer filter media 11 with dense structure and relative better coalescer performance than the first filter media 10.

The coalescer filter media assembly may further comprise a final classifierl2 with a porous structure that assists in conglomeration of the discontinuous fluid droplets coalesced from the filter media 11. Furthermore, the final classifier 12 also acts as a safety filter for non bound fibers which may dislodged from the coalescer filter media 11. The classifier 12 is preferably of a material that is not wetted by the discontinuous phase fluid. If the discontinuous phase fluid is water, then, polymeric materials, either coated or non-coated but with hydrophobic properties, will be preferred for the classifier 12. Once, the coalesced treated fluid exits the coalescer medium container 9, the fluid flow is directed towards the rim of the horizontal orientated disc assembly 14 guided by the baffle plates 15 in a direction towards the rim of the disc assembly 14. The baffles are arranged slightly deviating from a radial direction such that they provide the fluid flow direction with a tangential component in addition to the radial flow component. The result is a flow direction along a spiral path from the center towards the rim of the disc assembly 14. The height of the compartment between the lower coalescer media bottom surface 13 and the upper surface of the horizontal orientated disc assembly 14 is chosen to promote a sufficiently low flow of the continuous phase fluid,facilitating separation of the discontinuous phase droplets that by gravity fall to the upper surface of the

horizontaldisc assemblyl4.

At the rim of the horizontal orientated disc assembly 14, the primary flow of the continuous phase fluid is diverted below the horizontal orientated disc assembly 14 through the demister means 16to the continuous phase fluid outlet 17. The demister back pressure promotes separation by spreading the flow at the area of the demister surface 16, consequently lowering the flow rate per square unit towards the continuous phase fluid outlet 17. The demisterl6is preferably made of a material that is not wetted by the discontinuous phase fluid so as to form a drainage barrier, conglomerating droplets from the discontinuous phase fluid that by gravity seek to the bottom of the drainage collecting space 20. If the discontinuous phase fluid is water, then, polymeric materials either coated or non-coated, and with hydrophobic properties, will be preferred for the demisterl6 material. Furthermore the demister

16,preferably, is made from a mesh or woven material, supported or self supported, allowing the continuous phase fluid to pass with minimum pressure loss. The continuous phase fluid outlet 17 protrudes close the lower surface of the horizontal oriented disc assembly 14 and close to the center of the horizontal orientated disc assembly 14 in order to, thus, maintaining a predominantly horizontal flow direction of the continuously phase fluid, supporting a final separation of droplets that by gravity seeks to the lower part of the drainage collecting space 20. The continuous fluid emerges from the coalescer vessel through the aperture of the continuous phase fluid outlet 17, a check valve 18, and a connection fitting 19 to the applied installation. The check valve 18cranking pressure provides an initial pressure in the coalescer vessel that, when the drain valve 21 is opened, secures sufficient pressure for effective drainage of the discontinuous phase fluid collected from the drainage collecting space 20. Furthermore, the check valve 18 prevents fluid from the applied installation to flood the lower part of the coalescer vessel when dismantled for service or maintenance. The coalescer vessel can be disassembled for service and maintenance by removing the upper vessel container 2 by means of a ring spanner 4. The coalescer medium contained can then be withdrawn from the vessel assembly in order to facilitate immediate access to all internal parts. In order to prevent fluid spill during disassembly a sealing ring 24 prevents fluid from entering the compartment 25 and thus no surfaces above the sealing ring 24 is wetted with fluid. Typical dimensions of the coalescer vessel are given in the following: