Method for producing a phosphorus-containing pig iron melt

Abstract

The invention relates to a method for producing a phosphorus-containing pig iron melt (11).

Description

[0001] The invention relates to a method for producing a phosphorus-containing pig iron melt.

[0002] The closure of phosphorus cycles is exemplified in the "Ordinance on the Reorganization of Sewage Sludge Utilization" (Sewage Sludge Ordinance). Instead of phosphorus recovery, the currently practiced land-based utilization will only be permitted from 2029 onwards for sewage sludge from wastewater treatment plants with a design capacity of up to 100,000 population equivalents, and from 2032 onwards only for sewage sludge from plants with a design capacity of up to 50,000 population equivalents. Furthermore, phosphorus must be materially utilized; see [reference to relevant section]. https: / / www.bmuv.de / gesetz / verordnung-zurneuordnung-der-klaerschlammverwertung The sewage sludge is, among other things, subjected to mono-incineration, whereby phosphorus and / or phosphates are extracted and recovered from the ash produced during combustion, which is complex and expensive.

[0003] State of the art also includes the production of phosphorus-containing (converter) slags by using phosphorus-rich ores and slagging them with limestone to form "Thomas slag", which is finely ground ("Thomas flour") and used as phosphate fertilizer.

[0004] Furthermore, DD 44 860 A5 discloses how the conversion of phosphorus-rich pig iron into steel and the conversion of Thomas iron into steel are carried out in a converter. DE 10 15 839 A1 also describes a process for producing carbon steels by refining phosphorus-rich pig iron in a converter vessel.

[0005] From the publication "Electric Arc Furnace Steelmaking" by Miroslaw Karbowniczek, 2021, available at http: / / doi.org / 10.1201 / 9781003130949, values ​​for carbon, phosphorus etc. and their temperature reduction coefficients are known in Table 10.16 on page 239, which, with the help of formula 10.26 on page 238, can be used to calculate the approximate melting point of a steel depending on the content of its constituents.

[0006] The task is to provide a process for producing a phosphorus-containing pig iron melt, which in particular proposes an alternative variant for closing phosphorus cycles and / or preferably can have a positive influence on reducing CO2 emissions in steel production.

[0007] The problem is solved by a method according to claim 1. Further advantageous embodiments are described in the dependent claims.

[0008] According to the invention, iron carriers and additives are melted in an electric melter to form a smelter slag covering the pig iron melt, wherein, to adjust the phosphorus content in the pig iron melt to between 0.150 and 5.0 wt.%, organic substances in the form of iron carriers and / or additives with a phosphorus and / or phosphate content of at least 1.0 wt.% are added and melted together.

[0009] In particular, taking into account Table 10.16 in conjunction with Formula 10.26 of Karbowniczek's publication, the invention thus combines the following advantageous features: Phosphorus in a pig iron melt increases the entropy of the system and can thus have a positive influence on lowering the melting point of a phosphorus-rich pig iron melt. This is advantageous when, after tapping the pig iron melt, transport for further processing, for example to a converter, and the associated logistics are required under specific conditions. Phosphorus can also have a positive influence by substituting carbon while essentially maintaining and / or adhering to a specified melting point. This reduces the amount of decarburization work required and increases the amount of dephosphorus removal work, for example through a blowing process in a converter, thus also reducing the amount of harmful CO2 produced.The use of phosphorus from organic materials, for example residual materials, is particularly advantageous for phosphorus recovery and / or for closing a phosphorus cycle. This can prove especially beneficial during further processing, preferably in a converter, as a phosphorus-containing converter slag can be produced through the blowing process. This slag can then be further processed into fertilizer, thus eliminating the need to deplete natural phosphorus resources for such production.

[0010] The phosphorus content in the pig iron melt can be, in particular, at least 0.250 wt.%, 0.350 wt.%, preferably at least 0.450 wt.%, 0.550 wt.%, more preferably at least 0.60 wt.%, 0.70 wt.%, and most preferably at least 0.750 wt.%, 0.850 wt.%, or 1.0 wt.%. The higher the phosphorus content, the more organic substances with a phosphorus and / or phosphate content of at least 1.0 wt.% can be added, and in particular, the carbon content in the pig iron melt can also be reduced, preferably while maintaining and / or adhering to a predetermined melting point.

[0011] The phosphorus and / or phosphate content of the organic substance can be, in particular, at least 1.2, 1.4, 1.6 wt.%, preferably at least 1.8, 2.0, 2.2 wt.%, preferably at least 2.4, 2.6, 2.8 wt.%. The phosphorus and / or phosphate content of the organic substance can be a maximum of 10.0 wt.%, in particular a maximum of 9.0, 8.0 wt.%, preferably a maximum of 7.0, 6.0, 5.0 wt.%.

[0012] The phosphorus content in the pig iron melt can be, in particular, a maximum of 4.0 wt.%, 3.50 wt.%, preferably a maximum of 3.0 wt.%, 12.50 wt.%, preferably a maximum of 2.0 wt.%, 1.80 wt.%, and especially preferably a maximum of 1.60 wt.%, 1.40 wt.%, 1.20 wt.%, in order to avoid, for example, excessively frequent slag removal (slag replacement).

[0013] In the production process, the molten pig iron can contain, in addition to phosphorus (P), the following components in wt.%: carbon (C): 1.2 to 5.0%, silicon (Si): 0.015 to 3.0%, manganese (Mn): 0.020 to 6.0%, balance: iron (Fe), and impurities, which together total 100 wt.%. The impurities can include one or more of the elements Al, Mg, Ti, S, N, and O, listed here by way of example (and this list is not exhaustive), with a total of up to 1.0 wt.% or less. In conventional iron production, P would also be considered an impurity; however, the essential advantages of P, as described, are utilized, among other things, in the present invention.

[0014] To produce pig iron molten metal in an electric melter, iron carriers are required. These include or consist of reduced iron, unreduced iron, ferrous scrap, ferrous residues and / or recycled materials.

[0015] Reduced iron is produced from iron ores containing iron oxides with gangue minerals such as silicon, aluminum, and magnesium in oxide form. The concentrations of these individual components can vary depending on the ore's origin. Using a known direct reduction process, iron ore is brought into contact with a hot reducing gas, which may contain or consist of CO and / or H₂, and metallized to form reduced iron. The cooled reduced iron can then be supplied as direct reduced iron (DRI), which can optionally be passivated to prevent reoxidation and loss of metallization. Alternatively, the reduced iron can be briquetted while still warm to form hot briquetted iron (HBI). Passivation is not necessary in this case.HBI can therefore also be provided as an iron beam.

[0016] The DRI and / or HBI has a metallization degree of at least 75%, in particular at least 80%, preferably at least 88%, and can ideally be up to 100%, in particular up to 99%. Depending on the reducing gas used, and in the case of DRI, when using a cooling gas with a high carbon content (see, for example, DE 10 2019 217 631 A1), the DRI and / or HBI can have a carbon content between 0 and 4.0 wt.%, in particular > 0.10 wt.%, preferably > 0.30 wt.%, preferably > 0.50 wt.%, and in particular a maximum of 3.50 wt.%, preferably a maximum of 3.0 wt.%, preferably a maximum of 2.0 wt.%.

[0017] The gangue structure during the metallization of iron ore to reduced iron by passing the iron ore through a reducing gas is not significantly affected or removed, so that it remains essentially unchanged.

[0018] Optionally, non-reduced iron, i.e., iron ore or only partially reduced iron, and thus in this case, iron oxide, can also be provided as the iron carrier. The feed rate can be up to 100 kg, in particular up to 90 kg, preferably up to 80 kg, preferably up to 70 kg per ton of pig iron melt produced, or even 0. Partially reduced iron, for example, has a metallization degree of at most 70%, in particular at most 50%, preferably at most 30%, and preferably at most 15%. In particular, sufficient reduction potential is present. Determining the degree of metallization of a substance is familiar to those skilled in the art, whereby the degree of metallization is defined as the quotient of the mass of elemental iron (Feelemental) and the total mass of iron (Fetotal).Iron oxide in the molten pig iron can have the advantage that the oxygen separates from the oxide iron and combines, for example, with dissolved silicon in the molten pig iron. The resulting silicon dioxide has a lower density compared to the surrounding molten pig iron and rises to the interface with the molten pig iron, thus being transferred into the slag above. Additional compounds with other components of the molten pig iron, especially impurities (Al, Mg, Ti), can also be transferred into the slag via the same mechanism.

[0019] Optionally, ferrous scrap can also be provided as an iron carrier. To improve the recycling rate in particular, up to 200 kg, especially up to 150 kg, preferably up to 100 kg, and preferably up to 50 kg of ferrous scrap can be added per ton of molten pig iron produced. The feed rate can also be zero.

[0020] Optionally, iron-containing residues and / or recycled materials can also be provided as iron carriers. Preferably, these iron-containing residues and / or recycled materials, which contain or consist of oxygen compounds, are in oxide form. Examples include iron-containing agglomerates or oxide recycled materials or metallurgical products generated in a smelting plant complex, particularly residues / dusts from a coking plant, sintering plant, and / or burden preparation facility. The advantage of these iron-containing residues and / or recycled materials, preferably in oxide form, is that they can be reduced to iron and thus substitute components such as silicon, as well as other elements, in the pig iron melt, thereby increasing the iron yield. In particular, sufficient reduction potential is present.

[0021] To produce molten pig iron in an electric melter, additives are required. These include or consist of at least one slag former and / or at least one reducing agent.

[0022] Slag formers comprise or consist of at least one component from the group consisting of SiO₂, CaO, MgO, and Al₂O₃. They are used to bind silicates and phosphates and to promote the slag flow of the smelter slag produced in the electric remelter. During the melting of DRI and / or HBI and optionally unreduced iron, the gangue leads to the formation of the smelter slag and contributes in particular to the composition of the smelter slag that forms on the molten pig iron during the melting process.

[0023] Alternatively, or preferably additionally, at least one reducing agent can be added as a further additive. The reducing agent comprises or consists of at least one carbon-containing substance in gaseous, liquid, and / or solid form with reducible free carbon, which can be introduced into an electric melter. Examples of solid substances include coke dust, coke slurry, coke grit, and / or (bio)coal particles. Examples of liquid substances include ethanol, methanol, and other (suitable) hydrocarbons. Examples of gaseous substances include carbon-containing gases such as carbon dioxide, methane (natural gas), carbon monoxide, propane, and butane.Preferably, reducing agents can be added in combination with an iron-containing residue and / or recycled material present in oxide form and / or unreduced iron, in order to reduce these to iron and, for example, to increase the iron yield.

[0024] The organic materials, which may be contained in the form of iron carriers and / or additives and enter the electric melter, may include or consist of components from sewage sludge, reduced and / or non-reduced iron, carbon-containing residues, such as briquetted dusts from metallurgical residues.

[0025] Sewage sludge is rich in nutrients, among other things, because microorganisms in the biological treatment stage use the wastewater constituents to build biomass, thus concentrating the nutrients contained in the wastewater. Nitrate, phosphate (especially with a P₂O₅ content of, for example, between 1.0 and 8.0 wt% based on dry matter), and other plant nutrients are of particular importance for agriculture. However, sewage sludge also contains microplastics, nanomaterials, and pharmaceutical residues, which preclude its direct agricultural use. Portions of the sewage sludge that are not directly used in agriculture can be effectively recycled using alternative technologies, thus recovering the phosphorus and / or phosphate content contained in the sludge and preventing it from ending up in landfills, for example.Sewage sludge, as an organic substance containing phosphorus and / or phosphate, could therefore be introduced in the form of an additive.

[0026] Depending on the ore mine, for example Kiruna in Sweden, but also at other ore mines such as Agbaja in Nigeria, iron ores can be contaminated with high levels of phosphorus and / or phosphate, for example, at least 0.8 wt%. Due to their high phosphorus content, these types of iron ores are rarely used for steel production and / or are mixed with ores containing less phosphorus. Therefore, alternatively or additionally, reduced and / or unreduced iron contaminated with phosphorus and / or phosphate can be added in the form of an iron carrier.

[0027] The additive can additionally or alternatively contain not only phosphorus and / or phosphate as an organic substance, but also free carbon, thus contributing to the enrichment of the pig iron melt to be produced in the form of a reducing agent containing phosphorus and / or phosphate.

[0028] The preferred use of reduced iron as an iron carrier, particularly as the main component of the iron carrier with at least 60%, more specifically at least 70%, preferably at least 80%, and, for example, a maximum possible 100%, also entails slag-forming components that are naturally present in the iron ore and cannot be driven off in a prior reduction process (gangue). If the gangue provided by the reduced iron is insufficient, further slag formers can be added as additional additives to produce a remelter slag suitable for further processing. Slag formers are preferably added such that a basicity B4 in the remelter slag is established between 0.7 and 1.8. B4 can be at least 0.8, preferably at least 0.9, and particularly at most 1.7, preferably at most 1.6.The basicity B4 corresponds to the ratio (CaO+MgO) to (SiO 2 +Al 2 O 3 ), whereby the determination of the characteristic quantities is generally known to those skilled in the art in solid slags.

[0029] To melt the iron carriers and additives, an electric melter has several electrodes that can be energized with electricity, thus providing the necessary energy to convert the materials into a liquid phase comprising molten pig iron and melting slag. Depending on the size / dimensions of the electric melter, three, four, five, six, or more than six electrodes can be used. The energy required for melting is preferably supplied, at least in part, from renewable energy sources (solar, wind, hydro, biomass). If the required energy can be supplied entirely by renewable energy sources, and if it is available in the necessary quantities, the electric melter can be operated in a climate-neutral (or more) manner.

[0030] The electric melter can preferably be an electric furnace of the OSBF (Open Slag Bath Furnace) type. This includes electric reduction furnaces, especially SAF (Submerged Electric Arc Furnace), which are melting furnaces with arc resistance heating that heat the solid and / or the melting slag by means of the Joule effect. In SAF furnaces, the electrodes are immersed in the charge and / or melting slag. Depending on the operating principle / mode, the electric reduction furnaces can be designed as alternating current arc reduction furnaces (SAFac) or direct current arc reduction furnaces (SAFdc).

[0031] Alternatively, electrically operated melters with direct arc ignition can also be used, which differ from the operating principle / mode described above. These are called EAFs (Electric Arc Furnaces) and form electric arcs between the electrodes and the solid. This includes the alternating current arc melting furnace (EAFac), the direct current arc melting furnace (EAFdc), and the ladle furnace (LF).

[0032] The advantage of electric reduction furnaces with arc resistance heating (SAF) is that they are operated with a reducing atmosphere, whereas melting furnaces with direct arc heating (EAF, LF) are operated with an oxidizing atmosphere.

[0033] The molten pig iron can have a temperature between 1300 °C and 1650 °C. Important states of matter can be derived from an iron-carbon diagram known to those skilled in the art. If the production of pig iron and its further processing into crude steel or steel must take place indirectly via suitable and known containers, for example, torpedo ladles or the like, a molten pig iron temperature with a certain buffer above the liquidus temperature is required to ensure that the molten pig iron does not (partially) solidify during transport in the suitable containers.The addition of phosphorus (P) can lower the liquidus temperature of a pig iron melt, independent of its carbon content. This can result in a lower temperature compared to conventional pig iron production, providing a sufficient buffer for transport. Therefore, the temperature of the pig iron melt when tapped from the electric melter can be a maximum of 1600 °C, preferably a maximum of 1550 °C, more preferably a maximum of 1500 °C, and more preferably a maximum of 1450 °C. Depending on the residence time of the tapped pig iron melt in the container, a buffer of at least 75 K, particularly at least 100 K, and preferably at least 125 K, can be provided, ensuring that the critical liquidus temperature is not reached during normal operation.Based on the lever law and the melting temperature of pure iron and the formation of mixed crystals, the liquidus temperature Tliq in °C can be determined to a good approximation based on fewer influencing elements using the following formula: . T liq ∘ C = 1625 ∘ C − 113 ⋅ C % + S i % 4,5 + P % 2 ∘ C % where the percentage of the elements C, Si and P is given in wt.%.

[0034] The invention also comprises a method for producing a phosphorus-containing converter slag comprising the steps of: - providing a phosphorus-containing pig iron melt produced according to the invention; - Treating the phosphorus-containing pig iron melt in a converter by adding at least one slag former and / or at least one reducing agent and by oxygen bubbling, wherein at the end of the treatment a steel melt comprising the following components in wt.%: C: 0.0010 to 0.80%, Si: 0.015 to 3.0%, Mn: 0.020 to 6.0%, P: up to 0.10%, balance Fe and impurities, which together amount to 100 wt.%, is tapped, wherein during the treatment and / or optionally at the end of the treatment a phosphorus-containing converter slag comprising or consisting of the following components in wt.%: FeO: up to 40%, MnO: up to 10%, SiO₂: 5 to 40%, CaO: 35 to 65%, MgO: up to 8%, Al₂O₃: up to 20% %, P 2 O 5 : 4 to 30 % and impurities totaling 100 wt.-%, which is produced and drawn off before tapping of the molten steel. The FeO content can be 0 or > 0, in particular > 1 wt.%, preferably > 2 wt.%. FeO can be a maximum of 35 wt.%, preferably a maximum of 30 wt.%, preferably a maximum of 25 wt.%. The MnO content can be 0 or > 0, in particular > 1 wt.%, preferably > 2 wt.%. MnO can be a maximum of 8 wt.%, preferably a maximum of 6 wt.%, preferably a maximum of 4 wt.%. The Al₂O₃ content can be 0 or > 0, in particular > 0.5 wt.%. Al₂O₃ can be a maximum of 15 wt.%, preferably a maximum of 12 wt.%, preferably a maximum of 10 wt.%.

[0035] The method of removing the converter slag can be customized depending on the converter's operation and / or design. For example, the converter slag generated in the converter by oxygen bubbles can be removed during the treatment process, i.e., after a predetermined treatment time, and optionally at the end of the treatment process, thus being removed at least twice. The first removal during treatment can yield converter slag containing the aforementioned components, while the second or, for example, final removal can yield converter slag whose composition differs from that of the first removal, at least in its phosphate content. This slag may contain up to 4.0 wt.% P₂O₅, more specifically up to 3.0 wt.%, and preferably up to 2.0 wt.% P₂O₅.

[0036] Alternatively, the converter slag can be removed only once, which means that, especially at the end of the treatment, a converter slag containing the aforementioned components can be provided.

[0037] The invention will now be explained in more detail using exemplary embodiments.

[0038] The Figure 1 Figure 1 shows a process flow diagram for producing a pig iron melt (11) with a phosphorus content between 0.150 and 5.0 wt.% and a smelting slag (12) covering the pig iron melt (11) in an electric melter (10). Iron carriers and additives are melted in the electric melter (10), whereby, to adjust the phosphorus content in the pig iron melt (11) between 0.150 and 5.0 wt.%, organic substances in the form of iron carriers and / or additives with a phosphorus and / or phosphate content of at least 1.0 wt.% are added and melted along with the smelting.

[0039] Furthermore, it shows Figure 1 also how a phosphorus-containing converter slag (22) is produced by treating the phosphorus-containing pig iron melt (11) by adding at least one slag former and / or at least one reducing agent and by oxygen bubbles in a converter (20). At the end of the treatment, a steel melt (21) comprising the following components in wt.%: C: 0.0010 to 0.80%, Si: 0.015 to 3.0%, Mn: 0.020 to 6.0%, P: up to 0.050%, balance Fe and impurities, totaling 100 wt.%, is tapped, whereby during the treatment and / or optionally at the end of the treatment, a phosphorus-containing converter slag (22) comprising or consisting of the following components in wt.%: FeO: up to 40%, MnO: up to 10%, SiO₂: 5 to 40%, CaO: 35 to 65%, MgO: up to 8%, Al₂O₃: up to 20%, P₂O₅: 4 to 30%, and impurities totaling 100 wt.%, is produced. is, which is withdrawn before tapping of the molten steel (21).

[0040] The withdrawn smelting slag (12) and converter slag (22) are fed into a further conventional route known to those skilled in the art, not shown here. In particular, they can optionally be subjected to granulation. The converter slag (22), which can optionally be granulated, can preferably be used for agricultural purposes, and more preferably further processed into a fertilizer.

[0041] The tapped steel melt (21) is fed to another conventional route known to those skilled in the art, not shown here.

[0042] On a laboratory scale, pig iron melt (11) with a phosphorus content of 0.65 wt.% and a carbon content of 4.1 wt.% was produced in an electric melter (10) of type SAF, wherein reduced iron with a metallization degree of 94% (87.1% of the impurity), which was reduced with 100% H₂, was used as an iron carrier with a total Fe content of 88.9% wt.%, 83.5% Fe metal, 0.050% S, 0.01% P, balance SiO₂, Al₂O₃, CaO, MgO, containing 80 wt.% C (4.0% of the impurity) as a reducing agent and additive, and sewage sludge (5.7% of the impurity) as an organic substance with a P content of 8.67 wt.% as a further additive. Although the sewage sludge contained CaO at 10 wt.% and MgO at 1.5 wt.%, an addition of additional slag formers (3.2% of the slag content) was required to melt a smelter slag (12) with a basicity B4 of 1.1.The temperature of the molten pig iron (11) was 1380 °C. The theoretical liquidus temperature Tliq would be 1110 °C according to the formula, resulting in a buffer of 270 K (temperature of the molten pig iron (11) - liquidus temperature). The resulting remelting slag (12) was removed, allowing the molten pig iron (11), containing 0.25 wt% Mn, 0.6 wt% Si, and 0.005 wt% S, with the remainder being Fe and unavoidable impurities, to be fed into a laboratory-scale converter (20). In the converter (20), the phosphorus-containing molten pig iron (11) from the electric remelter (10) was treated by adding at least one slag former and / or at least one reducing agent and by oxygen bubbling. At the end of the treatment, a steel melt (21) comprising the following components in wt.%: C: 0.05%, Mn: 0.06%, P: 0.052%, balance Fe and impurities, was produced and tapped.At the end of the treatment, a phosphorus-containing converter slag (22) comprising or consisting of the following components in wt.%: FeO: 20.7%, MnO: 2.3%, SiO2: 9.8%, CaO: 41.5%, MgO: 3.8%, Al2O3: 0.7%, P2O5: 21.2%, and impurities totaling 100 wt.%, was produced, which was drawn off before tapping of the steel melt (21).

Claims

1. Method for producing a pig iron melt (11) with a phosphorus content between 0.150 and 5.0 wt.%, characterized by the fact that Iron carriers and additives are melted in an electric melter (10) to the pig iron melt (11) and in the melter (10) to form a melter slag (12) covering the pig iron melt (11), wherein, to adjust the phosphorus content in the pig iron melt (11), between 0.150 and 5.0 wt.% organic substances in the form of iron carriers and / or additives with a phosphorus and / or phosphate content of at least 1.0 wt.% are added and melted together.

2. The method according to claim 1, wherein the iron supports comprise or consist of reduced iron, unreduced iron, ferrous scrap, ferrous residues and / or recycled materials.

3. A method according to any of the preceding claims, wherein the additives comprise or consist of at least one slag former and / or at least one reducing agent.

4. The method of claim 3, wherein the slag former comprises or consists of at least one component from the group consisting of SiO2, CaO, MgO, Al2O3.

5. Method according to claim 3, wherein the reducing agent comprises or consists of at least one carbon-containing substance in gaseous, liquid and / or solid form with reducible free carbon.

6. A method according to any of the preceding claims, wherein the organic substances comprise or consist of sewage sludge, reduced and / or unreduced iron, carbon-rich residues or mixtures thereof.

7. A method according to any of the preceding claims, wherein the pig iron melt (11) comprises or consists of the following components in wt.% in addition to P: C: 1.2 to 5.0%, Si: 0.015 to 3.0%, Mn: 0.020 to 6.0%, balance Fe and impurities, which together make up 100 wt.%.

8. Method according to one of the preceding claims, wherein the pig iron melt (11) has a temperature between 1300 °C and 1650 °C.

9. Method for producing a phosphorus-containing converter slag (22) comprising the steps of: - providing a phosphorus-containing pig iron melt (11) produced according to one of the preceding claims; - Treating the phosphorus-containing pig iron melt (11) in a converter (20) by adding at least one slag former and / or at least one reducing agent and by oxygen bubbling, wherein at the end of the treatment a steel melt (21) comprising the constituents in wt.%: C: 0.0010 to 0.80%, Si: 0.015 to 3.0%, Mn: 0.020 to 6.0%, P: up to 0.10%, balance Fe and impurities, which in total amount to 100 wt.%, is tapped, wherein during the treatment and / or optionally at the end of the treatment a phosphorus-containing converter slag (22) comprising or consisting of the constituents in wt.%-%: FeO: up to 40%, MnO: up to 10%, SiO2: 5 to 40%, CaO: 35 to 65%, MgO: up to 8%, Al2O3: up to 20%, P2O5: 4 to 30% and impurities totaling 100 wt.%, which is drawn off before tapping of the molten steel (21).

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