JP2002035588A - Nitrile production catalyst and nitrile production method using the same - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a high initial yield of acrylonitrile or methacrylonitrile and a long-term stability of the yield during operation in the production of acrylonitrile or methacrylonitrile in the ammoxidation reaction of propylene, isobutene or tertiary butanol. At the same time. SOLUTION: General formula Mo ₁₂ BiaAbFecXdY
eOf, α = b / (a + b), β = 1.5 (a + b) / (1.5c
+ D), 0 ≦ α <２/3, 0.03 ≦ β ≦
0.07 is simultaneously satisfied.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

The present invention relates to an oxide catalyst for ammoxidation, which is used in the production of acrylonitrile or methacrylonitrile by reacting propylene, isobutene or tertiary butanol with molecular oxygen and ammonia, and the use thereof. And a method for producing acrylonitrile or methacrylonitrile.

[0002]

BACKGROUND OF THE INVENTION The reaction of propylene, isobutene or tertiary butanol with molecular oxygen and ammonia.
A method for producing acrylonitrile or methacrylonitrile by a so-called ammoxidation reaction is well known, and many catalysts used for the ammoxidation reaction have been proposed. For example, Japanese Patent Publication No. 38-17967 proposes an oxide catalyst containing molybdenum, bismuth and iron, and Japanese Patent Publication No. 38-19111 proposes an oxide catalyst containing antimony and iron. Have been continuously improved from various viewpoints.

Among these improvements, Japanese Patent Application Laid-Open No. Sho 49-101 discloses
No. 331 and JP-A-57-180431 disclose molybdenum, bismuth, cerium and alkali metals,
An oxide catalyst containing an element such as thallium is disclosed in JP-B-61-4
Japanese Patent No. 3094 discloses an oxide catalyst comprising molybdenum, tungsten, bismuth and cerium. JP-B-58-38424 discloses an oxide catalyst containing molybdenum, tellurium and cerium, and one or more elements selected from iron, chromium, aluminum and bismuth, and JP-B-51-33888 discloses molybdenum. , Containing one or more elements selected from nickel, cobalt in addition to bismuth and iron, and further selected from alkali metals, rare earth elements, tantalum and niobium
Oxide catalysts composed of more than one kind of element are disclosed in JP-B-61-26.
No. 419 discloses that molybdenum, bismuth and iron are the basic components, one or more elements selected from cerium, lanthanum, neodymium, praseodymium, samarium, europium and gadolinium and one or more elements selected from potassium, rubidium and cesium Oxide catalysts containing molybdenum are disclosed in JP-A-59-204163.
2 selected from iron, cobalt, nickel, copper, zirconium and potassium in addition to bismuth, phosphorus and silicon
Oxide catalysts containing one or more elements and one or more elements selected from manganese, cerium, thorium, yttrium, lanthanum and thallium are further disclosed in Japanese Unexamined Patent Publication No. 7-328441. An oxide catalyst containing iron, nickel and nickel as essential components and containing at least one element selected from magnesium and zinc and further at least one element selected from potassium, rubidium and cesium is disclosed in JP-A-8-266899.
The publication contains molybdenum, bismuth, and iron as basic components,
At least one element selected from rare earth elements and yttrium and at least one element selected from nickel, cobalt, manganese, chromium, indium, magnesium, calcium, strontium, barium, zinc, sodium and phosphorus, and potassium, rubidium An oxide catalyst containing one or more elements selected from cesium and thallium is disclosed.

[0004] On the other hand, WO 99/54037 discloses that molybdenum, bismuth, iron, nickel, chromium and potassium are essential components and one or more elements selected from zirconium, lanthanum and cerium, magnesium, cobalt, manganese and zinc. An oxide catalyst comprising at least one selected element and further at least one element selected from vanadium, niobium, tantalum and tungsten and containing antimony as an optional component is disclosed.

Also, US Pat. Nos. 5,093,299, 5,175,334 and 5,212,13.
No. 7 discloses molybdenum, bismuth, iron, nickel,
An oxide catalyst containing magnesium, potassium and cesium as essential components, and further containing cobalt, manganese, chromium, phosphorus, antimony, tellurium, sodium, cerium and tungsten as optional components, and a method for producing acrylonitrile and methacrylonitrile using the same. Is disclosed.

[0006]

Although the catalysts disclosed above have greatly improved the high initial yield and the stability of the yield in operation, the catalysts which have not yet been sufficiently satisfied with the compatibility of these two performances have been obtained. is not. The present invention provides a method for simultaneously producing a high initial yield of acrylonitrile or methacrylonitrile and a long-term stability of the yield during operation in the production of acrylonitrile or methacrylonitrile in the ammoxidation reaction of propylene, isobutene or tertiary butanol. provide.

[0007]

Means for Solving the Problems The inventors of the present invention have made intensive studies to solve the above-mentioned problems, and as a result, as an oxide catalyst, a general formula Mo ₁₂ BiaAbFecNidXeYfOg was used.
(Wherein A is a rare earth element, X is one or more elements selected from magnesium, zinc and manganese, Y is one or more elements selected from sodium, potassium, rubidium, cesium and thallium, a, b, c , D, e, f and g represent the atomic ratios of bismuth, A, iron, nickel, X, Y and oxygen to 12 atoms of molybdenum, respectively, 0.05 ≦ a ≦ 1, 0 ≦ b ≦ 1, 1 ≦ c ≦ 5,
1.5 ≦ d ≦ 10, 0.2 ≦ e ≦ 5, 0.02 ≦ f ≦ 1
Where g is the number of oxygen atoms required to satisfy the valency requirements of the other elements present. ), Α = b / (a + b), β = 1.5 (a + b) / (1.5c
+ D + e), 0 ≦ α <2/3, 0.03 ≦ β ≦
0.07 is simultaneously satisfied. The oxidized product represented by) has a high acrylonitrile yield or methacrylonitrile yield when acrylonitrile or methacrylonitrile is produced by reacting propylene, isobutene or tertiary butanol with molecular oxygen and ammonia. And also found that the stability of the reaction was good even during long-time operation, and that the decrease in the yield of acrylonitrile or methacrylonitrile was small, and the present invention was completed.

That is, the present invention provides a high yield of acrylonitrile or methacrylonitrile at the initial stage, has good reaction stability even in a long-time operation, and reduces the yield of acrylonitrile or methacrylonitrile. An object of the present invention is to provide a small oxide catalyst for ammoxidation and a method for producing acrylonitrile or methacrylonitrile using the same.

In the above general formula, a more preferable composition is that A is yttrium, lanthanum, cerium,
At least one element selected from praseodymium, neodymium and samarium, and 0.1 ≦ a ≦ 0.5;
02 ≦ b ≦ 0.5, 1.5 ≦ c ≦ 3.5, 1.7 ≦ d ≦
9, 0.5 ≦ e ≦ 4.5, 0.05 ≦ f ≦ 0.5, 0.1 ≦ α ≦ 0.6, 0.035 ≦ β
≦ 0.065 at the same time, and more preferably, A is cerium, and 0.15 ≦ a ≦ 0.4,
0.05 ≦ b ≦ 0.3, 1.7 ≦ c ≦ 3, 2 ≦ d ≦ 8,
1.0 ≦ e ≦ 3.5, 0.05 ≦ f ≦ 0.3, 0.2 ≦ α ≦ 0.5, 0.04 ≦ β
≤ 0.06 at the same time.

When α is / or more, the initial yield of acrylonitrile or methacrylonitrile becomes low regardless of the value of β. Even when α is 0 or more and less than 2/3, if β is less than 0.03, the initial yield of acrylonitrile or methacrylonitrile is low, and if β is more than 0.07, acrylonitrile or methacrylonitrile is obtained. Although the initial yield of nitrile is good, the activity of the catalyst and the selectivity of acrylonitrile or methacrylonitrile decrease over time with the progress of operation, resulting in a decrease in the yield of acrylonitrile or methacrylonitrile.

When chromium or tellurium is contained as a catalyst component in the catalyst of the present invention, the initial yield of acrylonitrile or methacrylonitrile is good,
In the case of chromium, the activity of acrylonitrile or methacrylonitrile per catalyst decreases as the activity of the catalyst decreases over time, and in the case of tellurium, the selectivity of acrylonitrile or methacrylonitrile decreases with time. Therefore, the yield of acrylonitrile or methacrylonitrile decreases. Therefore, it is preferable that the catalyst of the present invention does not contain chromium and tellurium.

As the carrier of the oxide catalyst of the present invention, an oxide such as silica, alumina, titania, and zirconia is used, and silica is used as a preferred carrier. Silica itself is inert compared to other supports and has a good binding effect on the catalyst components without reducing the selectivity of the catalyst composition for the desired product. Further, the supported catalyst composition can impart abrasion resistance, and thus is preferable as the carrier of the present invention. The support is used in an amount of 30 to 70% by weight, preferably 4%, based on the total amount of the support and the catalyst composition.
It can be effectively used in the range of 0 to 60% by weight. The oxide catalyst of the present invention is a known method, for example, a first step of preparing a raw material slurry, a second step of spray-drying the raw material slurry, and a second step of calcining the dried product obtained in the second step. It can be obtained by a method comprising three steps.

Next, a preferred embodiment of the method for producing the catalyst composition of the present invention comprising the first to third steps will be described. No.
In step 1, a catalyst raw material is prepared to obtain a raw material slurry, which is used as an element source for molybdenum, bismuth, rare earth elements, iron, nickel, magnesium, zinc, manganese, sodium, potassium, rubidium, cesium, and thallium. Examples thereof include ammonium salts, nitrates, hydrochlorides, sulfates, and organic acid salts that are soluble in water or nitric acid. In particular, ammonium salts are preferable as molybdenum sources, and nitrates are preferable as element sources of bismuth, rare earth elements, iron, nickel, magnesium, zinc, manganese, sodium, potassium, rubidium, cesium and thallium.

As described above, an oxide such as silica, alumina, titania, or zirconia can be used as a carrier for the catalyst. However, silica is used as a suitable carrier, and silica sol is preferable as a silica source. Regarding impurities in the silica sol, a silica sol containing preferably 0.04 or less aluminum per 100 atoms of silicon is used, and more preferably a silica sol containing 0.1 to 100 atoms of silicon per 100 atoms of silicon.
A silica sol containing not more than 02 atoms of aluminum is used. The preparation of the raw material slurry is performed by adding an ammonium salt of molybdenum dissolved in water to a silica sol, and then adding bismuth, a rare earth element, iron, nickel, magnesium, zinc,
It can be carried out by adding a solution in which nitrate of an element source of each of manganese, sodium, potassium, rubidium, cesium and thallium is dissolved in water or an aqueous nitric acid solution. Thus, a raw material slurry can be prepared. At that time, the order of the above addition can be changed.

In a second step, the raw material slurry obtained in the first step is spray-dried to obtain substantially spherical particles.
The atomization of the raw material slurry can be performed by a method such as a centrifugal method, a two-fluid nozzle method, and a high-pressure nozzle method which are usually industrially performed. Next, the obtained particles are dried. As a drying heat source, it is preferable to use air heated by steam, an electric heater, or the like. The temperature at the dryer inlet is 100-400 ° C, preferably 150-300 ° C.

In the third step, the desired particles are obtained by calcining the dried particles obtained in the second step. The firing of the dried particles is performed, if necessary, at a temperature of 150 to 500 ° C, and then at a temperature of 500 to 700 ° C, preferably 550 to 700 ° C.
This is performed in a temperature range of 00 ° C for 1 to 20 hours. Firing in a rotary furnace,
It can be performed using a firing furnace such as a tunnel furnace or a muffle furnace. When supported on a carrier (preferably silica), the catalyst composition of the present invention has a particle size of 10%.
Preferably, it is distributed in the range of 150 to 150 μm.

Propylene using the oxide catalyst of the present invention,
The production of acrylonitrile or methacrylonitrile by the reaction of isobutene or tertiary butanol with molecular oxygen and ammonia can be carried out in either a fluidized bed reactor or a fixed bed reactor, but is preferably carried out in a fluidized bed reactor. The raw materials propylene, isobutene, tertiary butanol and ammonia do not necessarily need to be of high purity, and those of industrial grade can be used.
As the molecular oxygen source, it is usually preferable to use air, but it is also possible to use a gas having an increased oxygen concentration by mixing oxygen with air. As the composition of the raw material gas, the molar ratio of ammonia to air with respect to propylene, isobutene or tertiary butanol is (propylene, isobutene or tertiary butanol) / ammonia / air = 1.
/0.8-1.4/7-12, preferably 1 / 0.9-
It is in the range of 1.3 / 8 to 11. The reaction temperature is 35
The temperature is in the range of 0 to 550 ° C, preferably 400 to 500 ° C. The reaction pressure can be set in a range from slightly reduced pressure to 0.3 MPa. The contact time between the raw material gas and the catalyst is 0.5 to 20
(Sec · g / cc), preferably 1 to 10 (sec · g / cc).
g / cc).

[0018]

BEST MODE FOR CARRYING OUT THE INVENTION In the examples and comparative examples, the conversion, selectivity and yield used to express the reaction results are defined by the following equations. Conversion (%) = (moles of propylene reacted) / (moles of propylene supplied) * 100 Selectivity (%) = (moles of acrylonitrile produced)
/ (Moles of propylene reacted) * 100 Yield (%) = (moles of acrylonitrile generated) /
(Mole number of supplied propylene) * 100

The reaction apparatus used is a Pyrex (registered trademark) glass fluidized bed reaction tube having an outer diameter of 23 mm, the reaction pressure P is 0.15 Mpa, the charged catalyst amount W is 40 to 60 g,
The total supply gas amount F is 250 to 450 cc / sec (NTP
The reaction temperature T was 430 ° C. The contact time is defined by the following equation. Contact time (sec.g / cc) = (W / F) * 273 /
(273 + T) * P / 0.10 The composition of the supplied reaction gas was as follows. Propylene / ammonia / air = 1 / 1.25 / 8.0
１10.0 The yield after 24 hours from the start of the feed of the raw material was defined as the initial yield, and the stability of the yield was compared with the yield after 700 hours. The present invention will be described based on examples.

[0020]

Example 1 The composition was Mo.₁₂Bi_0.18Ce_0.08Fe_2.3N
i_5.5Mg_2.3K_0.09Rb_0.05Oxide represented by the formula (α =
0.3, β = 0.035) to 50% by weight of silica.
The supported catalyst was prepared as follows. 30% by weight of S
iO_Two1666.7 g of silica sol containing
To 413.3 g of 6% by weight nitric acid was added 17.6 g of bismuth nitrate.
[Bi (NO_Three)_Three・ 5H_TwoO], 7.01 g of nitric acid
Li (Ce (NO_Three)_Three・ 6H_TwoO], 187.5 g
Iron nitrate [Fe (NO_Three)_Three・ 9H_TwoO], 322.7 g
Nickel nitrate [Ni (NO_Three)_Two・ 6H_TwoO], 118.
9 g of magnesium nitrate [Mg (NO_Three)_Two・ 6H
_TwoO] 1.84 g of potassium nitrate [KNO_Three〕and
1.49 g of rubidium nitrate [RbNO_ThreeIs dissolved
And finally add 427.4 g of powder to 860.9 g of water.
Ammonium lamolybdate [(NH_Four)₆Mo ₇O_{twenty four}・
4H_TwoO] was dissolved therein. Hara obtained here
The feed mixture was sent to a co-current spray dryer, and the outlet temperature was about 20
Dried at 0 ° C. Atomization of the preparation is performed in the upper part of the dryer.
Using a nebulizer with a dish-shaped rotor installed in the center
went. The obtained powder was heated at 350 ° C. for 1 hour in an electric furnace.
After pre-calcination, the mixture was calcined at 670 ° C. for 2 hours to obtain a catalyst.

Using 50 g of the obtained catalyst, an ammoxidation reaction of propylene was carried out at a contact time of 5.0 (sec · g / cc), and the conversion after 24 hours from the start of the reaction was 99.7%. Acrylonitrile selectivity is 85.0%,
The acrylonitrile yield was 84.7%, the conversion after 700 hours was 99.6%, and the acrylonitrile selectivity was 84.8.
% And the acrylonitrile yield was 84.5%.

[0022]

Embodiment 2 The composition is Mo ₁₂ Bi _0.34 Ce _0.15 Fe _2.5 N
oxide represented by i _5.0 Mg _2.5 K _0.2 (α = 0.3, β
= 0.065) was prepared in the same manner as in Example 1 with a catalyst supported on 50% by weight of silica, and the results of ammoxidation of propylene are shown in Tables 1 and 2.

[0023]

Example 3 The composition was Mo ₁₂ Bi _0.23 Ce _0.15 Fe _2.1 N
oxide represented by i _5.9 Mg _2.4 Rb _0.14 (α = 0.4,
β = 0.050) was prepared in the same manner as in Example 1 by preparing a catalyst carrying 50% by weight of silica, and the results of ammoxidation of propylene are shown in Tables 1 and 2.

[0024]

Embodiment 4 The composition is Mo ₁₂ Bi _0.32 Fe _2.4 Ni _7.4 Mn
Oxide represented by _0.8 K _0.09 Cs _0.05 (α = 0, β =
0.041) was carried out in the same manner as in Example 1, except that manganese nitrate [Mn (NO ₃ ) ₂ .6H ₂ O] and cesium nitrate [CsNO ₃ ] were used as the catalyst supporting 50% by weight of silica. Tables 1 and 2 show the results of the preparation and the ammoxidation of propylene.

[0025]

Embodiment 5 The composition is Mo ₁₂ Bi _0.12 Ce _0.18 Fe _2.2 N
An oxide represented by i _5.5 Zn _2.5 K _0.09 Cs _0.05 (α =
0.6, β = 0.040) on 50% by weight of silica, except that zinc catalyst [Zn (NO ₃ ) ₂ .6H ₂ O] and cesium nitrate [CsNO ₃ ] were used as the catalyst. Tables 1 and 2 show the results of the preparation of the propylene ammoxidation reaction in the same manner as in Example 1.

[0026]

Comparative Example 1 The composition is Mo ₁₂ Bi _0.09 Ce _0.22 Fe _2.1 N
An oxide represented by i _5.4 Mg _3.0 K _0.09 Rb _0.05 (α =
0.7, β = 0.040) were prepared in the same manner as in Example 1 for a catalyst supporting 50% by weight of silica, and the results of ammoxidation of propylene are shown in Tables 1 and 2.

[0027]

Comparative Example 2 The composition is Mo ₁₂ Bi _0.11 Ce _0.05 Fe _2.3 N
An oxide represented by i _7.7 Mn _0.9 Rb _0.14 (α = 0.3,
β = 0.020) was prepared in the same manner as in Example 1 except that manganese nitrate [Mn (NO ₃ ) ₂ .6H ₂ O] was used as a catalyst supported on 50% by weight of silica. Tables 1 and 2 show the results of the ammoxidation reaction.

[0028]

Comparative Example 3 The composition is Mo ₁₂ Bi _0.43 Ce _0.18 Fe _2.5 N
oxide represented by i _5.0 Zn _2.6 K _0.2 (α = 0.3, β
= 0.081) was prepared in the same manner as in Example 1 except that zinc nitrate [Zn (NO ₃ ) ₂ .6H ₂ O] was used as a catalyst supported on 50% by weight of silica. Tables 1 and 2 show the results of the oxidation reaction.

[0029]

Comparative Example 4 The composition was Mo ₁₂ Bi _0.15 Ce _0.15 Fe _2.1 N
oxide represented by i _8.4 K _0.1 Rb _0.04 (α = 0.5, β
= 0.039) was prepared in the same manner as in Example 1 with a catalyst supported on 50% by weight of silica, and the results of ammoxidation of propylene are shown in Tables 1 and 2.

[0030]

Comparative Example 5 The composition was Mo ₁₂ Bi _0.45 Fe _1.8 Cr _0.45 N
_{i 5.0 Mg 2 Mn 0.45 K 0.09} Cs 0.07 in oxide represented (α = 0, β = 0.067 ) and catalyst chromium nitrate carried on 50 wt% silica [Cr (NO _{3) 3} · 9H
₂ O], manganese nitrate [Mn (NO ₃ ) ₂ .6H ₂ O] and cesium nitrate [CsNO ₃ ] were prepared in the same manner as in Example 1, and propylene ammoxidation was carried out. Are shown in Tables 1 and 2. In this comparative example, the conversion after 350 hours was 98.9%, the conversion after 700 hours was 97.5%, and the activity was decreasing with time.

[0031]

Comparative Example 6 The composition is Mo ₁₂ Bi _0.13 Ce _0.19 Fe _2.1 N
i _5.5 Mg _3.2 K _0.09 Cs _0.05 Te _0.1 An oxide (α = 0.6, β = 0.041) supported on 50% by weight of silica was treated with cesium nitrate [CsNO ₃ ] and telluric acid. Except that [H ₆ TeO ₆ ] was used, it was prepared in the same manner as in Example 1, and the results of ammoxidation of propylene are shown in Tables 1 and 2.

[0032]

[Table 1]

[0033]

[Table 2]

[0034]

The present invention provides a method for producing acrylonitrile or methacrylonitrile in the ammoxidation of propylene, isobutene or tertiary butanol, which has a high initial yield of acrylonitrile or methacrylonitrile and a long-term stability of the yield during operation. At the same time.

Continued on front page F term (reference) 4G069 AA02 AA03 AA08 BB06A BB06B CB53 DA05 FA01 FA02 FB14 FB19 4H006 AA02 AC54 BA02 BA06 BA07 BA08 BA13 BA14 BA15 BA16 BA19 BA30 BA55 BC10 BC11 BC19 BE14 BE30 4H039 CA70 CL50

Claims (4)

[Claims] 1. An oxide catalyst of the general formula Mo ₁₂ BiaA
bFecNidXeYfOg (where A is a rare earth element,
X is 1 selected from magnesium, zinc and manganese
At least one element selected from sodium, potassium, rubidium, cesium, and thallium; a, b, c, d, e, f, and g are bismuth, A, iron, and molybdenum, respectively; nickel,
Represents the atomic ratio of X, Y and oxygen, and 0.05 ≦ a ≦
1, 0 ≦ b ≦ 1, 1 ≦ c ≦ 5, 1.5 ≦ d ≦ 10, 0.
2 ≦ e ≦ 5, 0.02 ≦ f ≦ 1, and g is the number of oxygen atoms required to satisfy the valence requirement of the other element present. ), Α = b / (a + b), β = 1.5 (a + b) / (1.5c
+ D + e), 0 ≦ α <2/3, 0.03 ≦ β ≦
An oxide catalyst for ammoxidation for producing acrylonitrile or methacrylonitrile by reacting propylene, isobutene or tertiary butanol, ammonia and molecular oxygen, characterized by simultaneously satisfying 0.07. 2. A is at least one element selected from yttrium, lanthanum, cerium, praseodymium, neodymium and samarium, and 0.1 ≦ a ≦ 0.5,
0.02 ≦ b ≦ 0.5, 1.5 ≦ c ≦ 3.5, 1.7 ≦
The oxide catalyst for ammoxidation according to claim 1, wherein d≤9, 0.5≤e≤4.5, and 0.05≤f≤0.5. 3. A process of reacting propylene, isobutene or tertiary butanol with ammonia and molecular oxygen to produce acrylonitrile or methacrylonitrile, wherein an oxide catalyst represented by the general formula Mo ₁₂ BiaAbFe
cNidXeYfOg (where A is a rare earth element, X is one or more elements selected from magnesium, zinc and manganese, Y is one or more elements selected from sodium, potassium, rubidium, cesium and thallium, a,
b, c, d, e, f and g are each molybdenum 12
Represents the atomic ratio of bismuth, A, iron, nickel, X, Y and oxygen to the atoms, 0.05 ≦ a ≦ 1, 0 ≦ b ≦
1, 1 ≦ c ≦ 5, 1.5 ≦ d ≦ 10, 0.2 ≦ e ≦ 5,
0.02 ≦ f ≦ 1, and g is the number of oxygen atoms required to satisfy the valence requirements of the other elements present. )
Α = b / (a + b), β = 1.5 (a + b) / (1.5c
+ D + e), 0 ≦ α <2/3, 0.03 ≦ β ≦
0.07. A process for producing acrylonitrile or methacrylonitrile, characterized by using an oxide catalyst for ammoxidation, characterized by simultaneously satisfying 0.07. 4. The method according to claim 1, wherein A is at least one element selected from yttrium, lanthanum, cerium, praseodymium, neodymium and samarium, and 0.1 ≦ a ≦ 0.5;
0.02 ≦ b ≦ 0.5, 1.5 ≦ c ≦ 3.5, 1.7 ≦
4. The method for producing acrylonitrile or methacrylonitrile according to claim 3, wherein d ≦ 9, 0.5 ≦ e ≦ 4.5, and 0.05 ≦ f ≦ 0.5.