Pyrolysis Characteristics of Industrial Lignin for Use as a Reductant and an Energy Source for Future Iron Making.

The purpose of this study is to explore the possibility of using industrial lignin instead of pulverized coal as a reducing agent for the production of direct reduced iron (DRI), thus reducing CO2 emissions. The pyrolysis characteristics and kinetics of pulverized coal and industrial lignin were studied by nonisothermal thermogravimetry. In the three stages of pyrolysis, the weight loss rate of industrial lignin is higher than that of pulverized coal. The volatile matter of industrial lignin is easier to release than that of pulverized coal, but the coking process is longer than that of pulverized coal. The activation energies of pyrolysis of Lu'an anthracite (LA), Shen'mu bituminous coal (SM), alkali lignin (AL), and magnesium lignosulfonate (ML) were 71.10, 70.30, 55.20, and 37.34 kJ·mol-1 at the middle-temperature stage, and 133.64, 98.31, 57.78, and 46.68 kJ·mol-1 at the high-temperature stage, respectively. After pyrolysis, a few nanometer thick carbon film structure appears in alkali lignin coke, which is conducive to the reduction of iron ore powder.

Introduction

The current primary problems in the ironmaking industry are energy consumption and CO2 emissions. According to relevant studies, the CO2 emissions by the iron and steel industry accounts for 6.7% of the total global CO2 emissions and 16% of the total industrial CO2 emissions.[1−3] In 2018, an average of 1.85 tons of CO2 was emitted for every ton of steel produced.[4] CO2 is the main culprit causing the greenhouse effect, so reducing CO2 emission is an urgent task for the development of the iron and steel industry in the future. At present, the mainstream electric arc furnace (EAF) steelmaking in the world generally uses 50–70% scrap steel with 50–30% direct reduced iron (DRI). The world’s DRI production over the years is shown in Figure .[5] In 2018, the world’s DRI production was 100.49 million tons, exceeding the landmark of 100 million tons for the first time. Facing the increasing production capacity of EAF steel and the shortage of scrap resources in China, DRI has been paid more and more attention by the iron and steel industry and will usher in a good opportunity for development.

Figure 1

World DRI production over the years.

However, the essence of the coal-based direct reduction process is to use coal, a nonrenewable energy resource, which does not reduce CO2 emission in essence. Therefore, this study hopes to find biomass to replace coal as a reducing agent, thus reducing the use of coal. According to statistics, 200 million tons of black liquor produced by China’s paper industry every year contains about 20 million tons of industrial lignin.[6] Under the harsh process conditions in the paper industry, chemical treatment will lead to the chemical transformation of lignin’s natural structure, including structural depolymerization, redox, and condensation.[7,8] This transformation makes the structure of lignin highly complex and difficult to understand, which seriously hinders the exploration of its applicable and effective value-added methods.[9] The utilization rate of industrial lignin is extremely low, and it is generally burned as low-grade fuel or made into low-value-added products such as flocculants[10] and dispersants.[11] Lignin and lignosulfonate have been studied by many researchers, and the focus of these studies has been on their modification.[12−18] However, the research on using lignin or lignosulfonate as a reducing agent for producing DRI is still insufficient. If industrial lignin can be applied to the production of DRI, that is, iron ore powder can effectively promote the pyrolysis of industrial lignin, while the pyrolysis products can promote the reduction of iron ore powder. It can not only save coal resources and reduce CO2 emissions but also make effective use of the wastes generated by the pulp and paper industry, which has a very important practical significance for the development of environmental protection.

This work is mainly in the early exploration stage. Pyrolysis experiments are carried out on anthracite (LA), bituminous coal (SM), alkali lignin (AL), and magnesium lignosulfonate (ML) to explore the possibility of industrial lignin as a reducing agent to produce DRI. The main contents of this work are as follows: (1) The release characteristics of the volatile matter of four reducing agents are discussed. (2) The differences in the agravic section of the four reducing agents during pyrolysis are analyzed. (3) The activation energy of pyrolysis of four reducing agents in different temperature ranges is calculated, and the pyrolysis mechanism function is determined. (4) The surface morphology of the products after pyrolysis is observed, and it is found that industrial lignin has certain advantages as a reducing agent. We know that the process of lignin reducing iron ore powder is not lignin itself as a reducing agent, but reducing substances such as hydrogen, carbon monoxide, and coke produced by lignin pyrolysis play a reducing role. Therefore, the study of pyrolysis kinetics is of great significance for further understanding the pyrolysis mechanism of industrial lignin, predicting the reaction rate and reaction difficulty.

Industrial lignin is a very promising renewable and environmentally friendly reducing agent conducive to the green development of the iron and steel industry. The practical reduction application effect of industrial lignin on iron ore powder will be discussed in other articles.

Results and Discussion

Higher Heating Values (HHV) of Reducing Agents

It can be seen from Table that the volatile matter content of AL is 60.44%, and that of ML is 57.82%, which is much higher than that of pulverized coal (13.38% for LA and 28.28 for SM), but the fixed carbon content is much lower than that of pulverized coal. The content of oxygen and sulfur in industrial lignin is higher than that in pulverized coal. Most of the sulfur can be removed in ironmaking or steelmaking processes. According to the famous Mendeleev formula,[19] the higher heating value (HHV) of the four reducing agents is calculated, and the results are shown in Table .where HHV is the higher heating value, kJ·kg–1; C, H, and S are the mass fractions of carbon, hydrogen, and sulfur in combustible substances, respectively, %; and O is the total mass fraction of oxygen and nitrogen in combustible substances, %.

Table 5

Proximate and Ultimate Analysis of Different Reducing Agents

	proximate analysis (wt %, ad.)				ultimate analysis (wt %, ad.)
samples	F.C.a	ash	V.M.b	moisture	C	H	O	N	S
AL	26.35	8.31	60.44	4.89	59.56	6.82	29.67	0.20	1.82
ML	18.01	12.32	57.82	11.85	32.57	4.86	42.42	0.45	9.39
LA	75.90	10.01	13.38	0.71	82.00	3.41	8.50	1.79	0.20
SM	62.41	7.46	28.28	1.85	75.90	4.14	16.50	1.38	0.63

Fixed carbon.

Volatile matter.

Table 1

Higher Heating Value of Different Reducing Agents

samples	HHV (kJ kg–1)
AL	25 669.63
ML	13 483.38
LA	30 943.01
SM	29 014.94

It can be seen from Table that the HHV of pulverized coal is higher than that of industrial lignin, which indicates that pulverized coal will release more heat during combustion. Among the four reducing agents, LA has the highest heat output of 30 943.01 kJ·kg–1, and the lowest heat output of ML is 13 483.38 kJ·kg–1, which is mainly due to the high carbon content in LA and the low carbon content in ML.

Thermogravimetric Analysis

The pyrolysis process of pulverized coal and industrial lignin is very complicated, including a series of chemical reactions and physical changes. The thermogravimetric results of pulverized coal and industrial lignin under a nitrogen atmosphere are shown in Figure . It can be found that the pyrolysis process of pulverized coal and industrial lignin is basically similar, that is, they all go through three stages. The pyrolysis process can be divided into stage I, stage II, and stage III according to the initial temperature Ti and the ending temperature Te of volatile matter release. The temperature ranges of the three pyrolysis stages of different reducing agents are shown in Table and Figure .

Figure 2

Pyrolysis curve of different reducing agents: (a) LA, (b) SM, (c) AL, and (d) ML.

Table 2

Temperature Range of the Pyrolysis Process with Different Reducing Agents, K

sample name	stage I	stage II	stage III
LA	303–749	749–1006	1006–1273
SM	303–681	681–838	838–1273
AL	303–543	543–715	715–1273
ML	303–526	526–695	695–1273

Figure 3

Temperature range of the pyrolysis process with different reducing agents.

Stage I of pyrolysis, that is, when the pyrolysis temperature is low, is a drying and degassing process. The weight loss of pulverized coal in this stage is mainly due to the release of water, some gases (CH4, CO2, and N2) in the pores, and the thermal decomposition of some chemical bonds with weak bond energy. The weight loss of industrial lignin is due to the volatilization of free water and bound water, as well as the volatilization of a small amount of carboxylic acid during glass transition.[20] In this stage, the weight loss rate of ML is 21.2%, which is much higher than that of the other three reducing agents, mainly because of the highest moisture content in ML.

Stage II of pyrolysis of pulverized coal and industrial lignin is mainly the rapid release of volatile matter. The TG curve in Figure shows rapid weightlessness. At this time, the reaction mechanism and control links begin to change. The pyrolysis process of pulverized coal in this stage is mainly the escape of gas and tar, and the formation of semi-coke. The reason is that the side chains of aromatic fused ring compounds, the noncovalent molecules, and some oxygen-containing functional groups are broken and decomposed in pulverized coal, thus forming CO2, CO, CH4, H2O, H2, phenols, aliphatic hydrocarbons, and other substances. In industrial lignin, carbohydrate compounds are decomposed, ether bonds and part of carbon–carbon bonds are broken, resulting in the formation of low molecular phenols, alkanes, alcohols, and aldehydes, as well as most of CH4, CO2, and a small amount of CO.[20−22] The order of the weight loss rate in this stage is AL (33.78%) > ML (28.7%) > SM (14.24%) > LA (6.69%). There are two reasons for this phenomenon: (1) The volatile matter content of industrial lignin is much higher than that of pulverized coal. (2) Industrial lignin is mainly composed of the ether bond (R–O–R) with low bond energy (about 380–420 kJ·mol–1) and is easy to break during pyrolysis, while pulverized coal is mainly composed of polycyclic aromatic hydrocarbons connected by the carbon–carbon bond with large bond energy (about 1000 kJ·mol–1), which is not easy to break during heating.[23] We can also find that the weight loss rate of SM is higher than that of LA, which is related to the oxygen-containing functional groups in pulverized coal. SM has a low degree of coalification and high oxygen content, so the oxygen-containing functional groups with lower bond energy will also increase, eventually leading to a higher weight loss rate of SM.

Stage III of pyrolysis is carbonization, in which semi-coke is gradually transformed into coke. At this stage, pulverized coal mainly goes through the polycondensation reaction and produces more H2 and CO. There are two main sources of H2, one is the polycondensation dehydrogenation of aromatic and hydrogenated aromatic structures[24] and the other is the secondary pyrolysis reaction, such as C2H6 → C2H4 + H2, C2H4 → CH4 + C, and CH4 → 2C + 2H2.[25] There are also two sources of CO, one is due to the breakage of carbonyl and oxygen-containing heterocyclic rings and the other is the reaction between CO2 and semi-coke.[26] Industrial lignin mainly goes through the carbon-forming reaction and the decomposition or condensation reaction of the aromatic ring structure and generates CO2 and a large amount of CO.[27,28] The weight loss rate of industrial lignin in this stage is also higher than that of pulverized coal. The order of the weight loss rate from large to small is ML (17.1%) > AL (16.65%) > SM (9.69%) > LA (2.93%). This also fully shows that industrial lignin is easier to pyrolyze and will produce more gas at this stage, which is conducive to the reduction of iron ore powder.

In addition, it can be found from Figure that there will be a small weight loss peak at the high-temperature stage, mainly due to the following three possible reason: (1) decomposition of minerals (mainly carbonate);[29] (2) condensation of semi-coke into coke at high temperatures, resulting in the release of a small part of hydrogen; and (3) reaction of alkali metal salt with the carbon in semi-coke.[21,30]

It can be seen from Figure that a very interesting phenomenon appears in the pyrolysis temperature ranges of the four reducing agents. In stage I of pyrolysis, the order of temperature range is LA (719 K) > SM (651 K) > ML (513 K) > ML (496 K), while in stage III of pyrolysis, the temperature range shows a completely opposite phenomenon: LA (540 K) < SM (708 K) < ML (831 K) < ML (851 K). This shows that industrial lignin is better than pulverized coal in drying and degassing, while it takes longer than pulverized coal in the coking process. In this study, the heating temperature range is 303–1273 K, and a large coking temperature range means that the coking starting temperature is lower, that is, industrial lignin is easier to coke than pulverized coal.

It is well known that lignin is a kind of complex organic matter with many functional groups. The main structural units of lignin mainly include three kinds of phenylpropane monomers with different properties (coniferyl alcohol, coumaryl alcohol, and sinapyl alcohol types). The industrial lignin used in this study was obtained by the alkaline pulping process, which is closer to the characteristics of natural lignin. Recent studies have shown that pulverized coal is mainly composed of aromatic hydrocarbons, aliphatic hydrocarbons, and a small amount of oxygen-containing functional groups. Therefore, industrial lignin has a higher content of oxygen-containing functional groups than pulverized coal. Moreover, due to the low bond energy of oxygen-containing functional groups, the structure in industrial lignin is easy to break and generate gases such as CO and CO2.

The research shows that the CO and CO2 yield increases linearly with the increase in the O/C atomic ratio of coal.[31] In this study, the O/C atomic ratios of LA, SM, AL, and ML are 0.10, 0.22, 0.50, and 1.30, respectively. The O/C atomic ratio of industrial lignin is higher than that of pulverized coal, resulting in a higher content of CO and CO2, which is beneficial to the reduction of iron ore powder. Therefore, the oxygen-containing functional group is also one of the characteristics of the reducing agent, and detailed experimental data and analyses are our future work.

Pyrolysis Characteristics of the Reducing Agent

The pyrolysis characteristic parameters obtained from Figure are shown in Table . It can be seen from Table that the Ti of ML is only 526.3 K, and that of AL is 542.9 K, which is far lower than those of SM (681.2 K) and LA (749.1 K). In addition, the Te and Tmax of industrial lignin are also significantly lower than those of pulverized coal. The (dw/dt)mean values of AL, ML, SM, and LA are 2.05, 1.79, 0.93, and 0.27%·min–1, respectively, the (dw/dt)max values are 2.90, 2.68, 1.42, and 0.37%·min–1, and the P values are 4.52, 4.57, 0.66, and 0.01 10–7%2·K–3·min–2. This indicates that the volatile matter of industrial lignin is easier to release than that of pulverized coal, and the pyrolysis reaction is also easier.

Table 3

Pyrolysis Characteristic Parameters of Different Reducing Agents

	Ti (K)	Te (K)	Tmax (K)	Tmean (%·min–1)	(dw/dt)max (%·min–1)	p (10–8%2·K–3·min–2)
LA	749.10	1006.00	822.95	0.27	0.37	0.06
SM	681.20	837.80	728.86	0.93	1.42	2.53
AL	542.90	715.30	617.74	2.05	2.90	12.57
ML	526.30	695.10	563.70	1.79	2.68	11.31

The Ti, Te, and Tmax of ML are lower than those of AL, but the P values of the two are basically the same, which indicates that the difficulty of the release of the volatile matter of the two is basically the same.

Kinetic Analysis

As is known, the difficulty of the chemical reaction can be reflected by activation energy, that is to say, the reaction is difficult when the activation energy is large. On the contrary, the reaction is easier when the activation energy is small. The purpose of this work is to determine the degree of difficulty in the pyrolysis of pulverized coal and industrial lignin according to the activation energy.

According to the analysis and discussion on the pyrolysis process of pulverized coal and industrial lignin in Section , it can be seen that the control mechanism of pyrolysis is different at different temperature ranges. Because the single kinetic model cannot describe the whole pyrolysis process well, this study adopts the segmented method to fit the different stages of pyrolysis to deeply understand the control mechanism of pulverized coal and industrial lignin in the pyrolysis process. Since the pyrolysis process of pulverized coal and industrial lignin is mainly affected by moisture in stage I, it will not be discussed here. This study focuses on the pyrolysis mechanism of stages II and III. The reaction mechanism functions of the four reducing agents at stages II and III are fitted and calculated, and the results are shown in Figure and Table . The results show that stage II accords with the third-order chemical reaction model, and the mechanism function is G(α) = [(1 – α)−2 – 1]/2. The reaction mechanism of stage III accords with the fourth-order random nucleation and growth model and the mechanism function is G(α) = [−In(1 – α)]4.

Figure 4

Linear fitting of G(α) = [(1 – α)−2 – 1]/2 versus 1/T for stage II (a) and G(α) = [−In (1 – α)]4 versus 1/T for stage III (b).

Table 4

Kinetic Parameters Obtained by Means of the Coats–Redfern Method

sample name	temperature (K)	fitting equation	E (kJ·mol–1)	A (min–1)	R2
LA	749–1006	y = −8552.20x – 3.60	71.10	2.33 × 103	0.999
	1007–1273	y = −16074.58x + 3.26	133.64	4.19 × 105	0.996
SM	681–838	y = −8455.38x – 1.79	70.30	1.40 × 103	0.990
	839–1273	y = −11824.65x + 0.94	98.31	3.00 × 103	0.996
AL	543–715	y = −6639.86x – 2.06	55.20	8.46 × 102	0.996
	716–1273	y = −6949.51x – 2.67	57.78	4.81 × 102	0.934
ML	526–695	y = −4491.24x – 4.60	37.34	4.5 × 10	0.997
	696–1273	y = −5614.59x – 3.97	46.68	1.06 × 102	0.987

It can be seen from Figure and Table that the linear fitting results are good. In stage II of pyrolysis, the order of activation energy of the four reducing agents is LA (71.10 kJ·mol–1) > SM (70.30 kJ·mol–1) > AL (55.20 kJ·mol–1) > ML (37.34 kJ·mol–1). The order of activation energy in stage III is LA (133.64 kJ·mol–1) > SM (98.31 kJ·mol–1) > AL (57.78 kJ·mol–1) > ML (46.68 kJ·mol–1). It can be found that the activation energy of industrial lignin is lower than that of pulverized coal in stages II and III, which indicates that industrial lignin is easier to pyrolyze. This result is also consistent with the analysis in Section 2.2. For the same reducing agent, the activation energy of stage III is larger than that of stage II, which indicates that pyrolysis in stage III is more difficult. This is because the bond energy of the material structure in the high temperature is relatively large and is not easy to break.

Surface Morphology Analysis

The surface morphology of the four reducing agents after pyrolysis was analyzed by scanning electron microscopy (SEM), as shown in Figure . The morphology of industrial lignin after pyrolysis is significantly different from that of pulverized coal. The coal coke is in the form of particles with rough surfaces and different sizes. A few nanometer thick carbon film structure appears in AL coke. However, the ML coke has a larger and thicker sheet structure. AL coke has a large specific surface area, but ML coke does not seem to have an advantage over coal coke.

Figure 5

Surface morphology of the reducing agents after pyrolysis: (a) LA, (b) SM, (c) AL, and (d) ML.

Conclusions

Industrial lignin can replace pulverized coal as a reducing agent for the production of DRI. The pyrolysis of the four reducing agents all go through three stages: the low-temperature drying and degassing stage, the medium-temperature volatile matter rapid release stage, and the high-temperature carbonization stage. The volatile matter of industrial lignin is easier to release than that of pulverized coal during pyrolysis. The drying and degassing temperature range of industrial lignin is smaller than that of pulverized coal, while the coking temperature range is larger than that of pulverized coal. The pyrolysis activation energy of industrial lignin is lower than that of pulverized coal. A few nanometer thick carbon film structure appears in alkali lignin coke, which increases the specific surface area and is very beneficial to the reduction of iron ore powder.

Methods and Materials

Materials

The reducing agents used in the experiment include alkali lignin (AL), magnesium lignosulfonate (ML), Lu’an anthracite (LA), and Shenmu bituminous coal (SM). LA and SM are provided by an iron and steel company, while AL and ML are provided by Huawei Youbang Chemical Co., Ltd. Before the experiment, the samples were placed in an oven and dried at 378 K for 24 h.

Apparatus and Experimental Procedures

The pyrolysis characteristics of pulverized coal and industrial lignin under a nitrogen atmosphere were studied on a thermogravimetric analyzer (STA409CD, Netzsch Group, Germany). In this work, about 10 mg of the sample was placed in an alumina pan and heated at 10 K min–1 up to a temperature of 1273 K. The carrier gas (N2) flow rate was maintained at 30 mL min–1 for each experiment. A separate blank run was conducted using an empty pan under identical conditions and these data were used for baseline correction during the evaluation of the sample thermal gravimetric analysis (TGA) profile. To calculate the kinetic parameter for the sample pyrolysis work, we have taken the time–temperature data while heating up the samples from 303 to 1273 K.

According to the GB/T 212-2008 method, the proximate analysis of pulverized coal is carried out. The analysis of O, N, and H elements was determined by an ONH836 (Laboratory Equipment Corporation) and that of S and C elements was determined by an SC-144DR (Laboratory Equipment Corporation) analyzer. The proximate analysis and ultimate analysis of the reducing agents are shown in Table .

The particle size of the sample is determined by a particle size analyzer (Master sizer 3000, Malvern Instruments Ltd., U.K.). The particle size of the samples is more than 200 mesh, accounting for above 90%; the particle size distribution is shown in Figure .

Figure 6

Particle size distribution of the samples: (a) LA, (b) SM, (c) AL, and (d) ML.

The surface morphology of the sample after pyrolysis was observed by scanning electron microscopy (SEM) (JSM-6510LV, Japan).

Comprehensive Volatile Matter Release Characteristic Index

The comprehensive volatile matter release characteristic index P is used to characterize the difficulty of volatile release during pyrolysis.[32−34] The larger the value P, the easier the sample to decompose. The calculation formula is shown in eq .where P is the comprehensive volatile matter release characteristic index, %2·K–3·min–2. is the maximum value of the volatile matter release rate, %·min–1. is the average release rate of volatile matter, %·min–1. Ti is the initial release temperature of volatile matter, K. Tmax is the temperature corresponding to the maximum value of the volatile matter release rate, K. ΔT1/2 is the temperature range when the volatile matter release rate reaches half of the maximum value, that is, the half peak temperature width when , K. The parameter definition is shown in Figure . Te is defined as the end temperature of volatile matter release.

Figure 7

Pyrolysis curve and schematic diagram of parameter definition.

Kinetic Models

The change in the weight loss rate with temperature is a measurement method of reactivity, which can well understand the reaction kinetics. The nonisothermal thermogravimetric method given in Section has also been used by many authors for kinetic analysis.[35−39]

The pyrolysis process of pulverized coal and industrial lignin can be expressed by eq

The pyrolysis process of pulverized coal and industrial lignin is a complex multiphase chemical reaction. According to the law of mass action and the Arrhenius equation, many scholars have proposed different kinetic treatment methods. According to the form of the kinetic equation, it can be divided into a derivative method and an integral method. The pattern matching method is one of the most commonly used analysis methods in the field of kinetics of nonisothermal processes.[40] The integral method will not produce numerical errors due to derivative, and the data is not easy to be distorted in the process of calculation. Therefore, the Coats–Redfern[41] integral method is adopted in this paper.

The basic theory of classical chemical reaction kinetics is based on the isothermal process and the homogeneous reaction. Using the thinking method of infinitesimal differential, assuming that the change in temperature can be ignored at an infinitesimal interval time, then the nonisothermal process can be regarded as an isothermal process,[42] and the chemical reaction rate equation can be expressed by eq .where α is the conversion rate, %. t is the pyrolysis time, min. k is the reaction rate constant, min–1. f(α) is the chemical reaction mechanism that depends on the elementary reaction, and its functional form depends on the reaction type and reaction mechanism.

The expression of the conversion rate α is as followswhere w0, w, and w∞ are the initial mass of the sample, the mass at time t, and the mass at the end of the reaction, respectively, mg.

The reaction rate constant k is expressed by the classical Arrhenius equation[43,44]where A is the frequency factor, min–1, E is the activation energy, J·mol–1, R is the gas constant, 8.314 J·mol–1·k–1, and T is the absolute temperature, K.

The expression of the heating rate β is as followsFrom eqs to 7, we can get the following resultsThe integral equation can be obtained as follows

The Coats–Redfern integral method was used to deal with the exponential integral by the asymptotic series expansion, and the separation of variable integral could be obtainedwhere G(α) is the integral form of the kinetic mechanism function of the solid reaction.

For the zone of the general reaction temperature and activation energy, the expression is Therefore, eq can be simplified as followsWhen the correct reaction mechanism G(α) is determined, the relationship between and 1/T must be in a straight line. The activation energy can be obtained by the slope of the straight line, and the frequency factor can be obtained by the intercept.

The forms of commonly used solid reaction kinetic mechanism functions are shown in Table .[45−50]

Table 6

Expressions of G(α) and f(α) for the Kinetic Model Functions Usually Employed for the Solid-State Reaction

shorthand	reaction model	integral formula G(α)	differential formula f(α)
Nucleation Models
P1	Mampel power	α	1
P2	Mampel power	α1/2	2α1/2
P3	Mampel power	α1/3	3α2/3
P4	Mampel power	α1/4	4α3/4
A1	Avaramie Erofeev	[−ln(1 – α)]3/2	2/3(1−α)[−In(1−α)]1/2
A2	Avaramie Erofeev	[−ln(1 – α)] 2	1/2(1−α)[−In(1−α)]−1
A3	Avaramie Erofeev	[−ln(1 – α)]3	1/3(1−α)[−In(1−α)]−2
A4	Avaramie Erofeev	[−ln(1 – α)]4	1/4(1−α)[−In(1−α)]−3
Chemical Reaction
F1	first order	–ln(1 – α)	1 – α
F2	second order	(1 – α)−1 – 1	(1 – α)2
F3	third order	[(1 – α)−2 – 1]/2	(1 – α)3
Geometrical Contractions
R2	contracting area	1–(1 – α)1/2	2(1 – α)1/2
R3	contracting volume	1–(1 – α)1/3	3(1 – α)2/3
Jander Equation
D1	one-dimensional	α2	–1/2α–1
D2	two-dimensional	[1–(1 – α)1/2]1/2	4(1 – α)1/2[1 – (1 – α)1/2]1/2
D3	three-dimensional	[1–(1 – α)1/3]1/2	6(1 – α)2/3[1 – (1 – α)1/3]1/2