Review: Axial Motion of Material in Rotary Kilns

Abstract

The article examines the parameters of axial motion of bulk material in rotary kilns, including bed height, axial velocity, and mean residence time. The review includes summary tables of experiments from the scientific literature, detailing the conditions and ranges of operating parameter variations. Mathematical models from the literature are presented for each of the parameters discussed. The materials of the article cover studies from 1927 to 2025, including analysis of numerous works that were not published in international sources. Based on the review, the necessity of studying the impact of coating formation on the axial motion parameters is highlighted, along with the need for experiments on real facilities and pilot plants.

1. Introduction

Rotary kilns are integral to the cement, metallurgical, and chemical industries. Yet most automation systems struggle to keep these units in a continuously optimal state because kilns exhibit high thermal inertia, strong multivariable coupling, extreme temperatures, and concurrent chemical–physical transformations. To address these challenges, researchers are developing comprehensive models that act as operator advisors, optimize process parameters [1,2,3], and boost energy efficiency in mineral-processing facilities [4,5,6]. Such models can enhance kiln performance while reducing direct human intervention [7,8] and are now being explored for other pyrometallurgical units as well [9,10,11].

Integrated kiln models typically comprise three coupled sub-models: material flow, heat transfer, and chemical reactions. Material flow is analyzed in two orthogonal planes. Transverse flow governs mixing and therefore influences both heat transfer and reaction progress, whereas axial flow determines residence time, axial velocity, bed height profiles, and other key variables along the kiln length.

Early research on the axial flow of granular solids in rotary kilns began in 1927 [12]. Since then, many mathematical models and hundreds of rolling-regime experiments have been reported [13]. Despite this progress, no generalized model yet captures the realities of industrial kilns—namely, internal heat-exchange devices, property changes driven by high-temperature reactions, and evolving internal geometry caused by coating build-up.

The structure of the paper is as follows:

• Key Concepts—defines the terminology and core phenomena needed for the rest of the review;
• Experimental Data—surveys laboratory and pilot-scale studies on axial transport in rotating cylinders;
• Mathematical Models—compares analytical, empirical, and numerical approaches for predicting axial flow parameters;
• Conclusions—summarizes the main findings and proposes priority directions for future research.

2. Key Concepts

2.1. Axial Motion

Axial motion denotes the transport of particles along the kiln axis, from the feed end to the discharge end. It is characterized by the mean residence time (MRT), residence time distribution (RTD), axial velocity, local bed height, and fill degree. Because bed height and fill degree vary with position, they are treated as continuous profiles rather than single values [14].

All figures and tables are cited sequentially in the text (e.g., Figure 1,

Table 1. Material motion regimes [compiled by the authors].

Slipping Motion
The regimes in this group are primarily characterized by the absence of material mixing, caused by extremely low wall roughness of the cylinder.
Sliding	Occurs in cylinders with smooth walls. It is characterized by the material resting at a certain angle and simply sliding along the rotating wall of the cylinder. Material mixing is practically absent.
Surging	Compared to the sliding regime, the material still does not mix. However, due to the less smooth wall surface, the material adheres to the wall and moves with it for some time before sliding back to its initial position after covering a certain distance.
Cascading motion
The regimes in this group are primarily characterized by good material mixing and occur when the cylinder wall has sufficiently high friction. The specific regime within this group depends on the following factors: degree of filling, rotational speed, and particle size.
Slumping	Unlike the surging regime, the material does not slide back after traveling a certain distance along the cylinder wall. Instead, it cascades in an avalanche-like manner, creating a sort of discrete process where the motion alternates between the entire mass moving with the wall and subsequently tumbling downward.
Rolling	This type of motion is considered the most desirable in rotary kilns, as it provides optimal conditions for heat transfer and chemical reactions. It can be described as follows: In the cross-section, the material is divided into two zones: a stagnant zone and an active zone. In the stagnant zone, mixing is virtually absent, while in the active zone, intensive mixing occurs. At the same time, the surface of the material layer usually remains smooth.
Cascading	In this regime, the layer “rolls over”, its height increases, and it ceases to remain smooth.
Cataracting motion
The regimes in this group are primarily characterized by the detachment of individual particles from the bulk material and their ejection into the kiln space. These regimes are achieved due to the high rotational speed of the drum.
Cataracting	This regime is characterized by the ejection of particles into the kiln space, which intensifies with an increase in rotational speed.
Centrifuging	This regime is achieved at an even higher rotational speed and is characterized by the adhesion of particles ejected from the bulk material to the drum’s surface.

).

Two complementary approaches are used to analyze residence time in rotary cylinders. Mean residence time (MRT) condenses the transit of an entire charge into a single scalar—the average time a particle needs to pass from feed to discharge. Residence-time distribution (RTD), by contrast, is a full probability function that reveals slow and fast pathways, dead zones, and mixing efficiency. MRT is sufficient for routine performance assessment and basic control, whereas RTD is indispensable for detailed diagnostics and optimization.

This review concentrates on MRT. A full RTD analysis requires tracking particle segregation and the evolution of size, shape, moisture, bulk density, and repose angles along the kiln—topics that merit a dedicated study. An exemplary methodology for determining RTD is given in Ref. [15].

2.2. Transverse Motion

Transverse motion concerns particle dynamics across the kiln cross-section. Classical literature recognizes seven distinct regimes of transverse flow. The original taxonomy was introduced in Ref. [16] and later refined in Ref. [13]. Each regime is defined by three dimensionless parameters: the Froude number, fill degree, and effective friction coefficient. Representative numerical ranges for these parameters—including transitional states—are tabulated in Ref. [13]. For convenience, Table 1 below condenses the defining features of all seven regimes.

Further discussion will focus on the rolling regime, as it is the most desirable for rotary kilns.

Rolling Regime

It is important to highlight the concepts characteristic of this regime; for this purpose, let us refer to Figure 2.

In the rolling regime, the granular bed splits into two zones: a plug-flow core and an overlying active layer, separated by a thin shear interface. Particles in the plug-flow core rotate almost rigidly with the shell, so transverse mixing there is negligible. Upon reaching the free surface, they cascade down the slope, forming the active layer where vigorous mixing and most axial transport occur. The shear interface itself is a narrow region of near-zero transverse velocity that governs momentum exchange between the two zones and, hence, the overall dynamics of material movement in the kiln.

2.3. Angle of Repose

2.3.1. Dynamic Angle of Repose

Figure 2 shows that the granular bed rests on a slope whose inclination—the dynamic angle of repose—depends on particle friction, size distribution, and bulk density [17]. Because these properties evolve during high-temperature reactions (e.g., sintering), the dynamic angle can change along the kiln axis [18]. When a single cross-section is analyzed, however, the angle may be regarded as constant.

2.3.2. Static Angle of Repose

The static angle of repose is the steepest slope at which a granular surface remains stable: particles do not slip under gravity. Like its dynamic counterpart, this angle reflects intrinsic material properties and can vary as physicochemical transformations alter particle size, shape, or surface chemistry.

2.4. Filling Degree

The filling degree represents the fraction of the cross-sectional area of the cylinder occupied by the material. It is determined using the sector fill angle according to the following formula [13]:

$$\mathrm{Z}={\displaystyle \frac{1}{\pi }}\left({\displaystyle \frac{\epsilon }{2}}-\mathrm{s}\mathrm{i}\mathrm{n}\left({\displaystyle \frac{\epsilon }{2}}\right)\mathrm{c}\mathrm{o}\mathrm{s}\left({\displaystyle \frac{\epsilon }{2}}\right)\right).$$

(1)

3. Experimental Data

Following the classification proposed in [19], research on axial transport in rotary kilns falls into the following two broad categories:

1. Laboratory studies. These experiments use small, room-temperature drums with constant material properties. Investigators vary rotation speed, kiln tilt, feed rate, and fill degree across wide ranges and test multiple bulk solids. Drums may differ in geometry and occasionally include internal fittings. To emulate industrial conditions, the rolling regime is strictly maintained.

2. Full-scale kiln studies. Measurements are taken on operating industrial kilns, allowing researchers to capture effects that laboratory rigs cannot reproduce—chain curtains, coating build-up, evolving particle properties, and so forth. The trade-off is limited freedom to change operating parameters because adjustments are costly and potentially disruptive.

A balanced understanding of axial transport therefore requires integrating experimental data and modeling insights from both groups [20,21,22].

Table 2. Summary of experiments from the literature on material bed height determination [compiled by the authors].

№	Material	Particle Size (mm)	Particle Density (kg/m³)	Bulk Density (kg/m³)	Static Angle of Repose (°)	Dynamic Angle of Repose (°)	Length (m)	Diameter (m)	Rotational Speed (rpm)	Inclination Angle (°)	Filling Degree	Flow Rate	Special Conditions
[12]	Sand	0.5	-	-	-	34.96	2.13	0.152	2.5	2	25	0.0144 m3/h	-
[23]	Cement raw mix	-	-	-	-	-	79.25	2.5	0.91	2.29	-	7.95 m3/h	REAL Chains
[23]	Cement raw mix	-	-	-	-	-	99.06	2.5	0.91	2.29	-	7.95 m3/h	REAL Chains
[23]	Cement raw mix	-	-	-	-	-	109.73	3.4	1	2.29	-	14.56 m3/h	REAL Chains
[23]	Cement raw mix	-	-	-	-	-	112.78	3.4	0.86	2.29	-	16.58 m3/h	REAL Chains
[24]	Clinker	3.5–5	-	1550	37	46.1	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Clinker	2–3.5	-	1518	38.4	47.5	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Clinker	0.6–2	-	1570	38.2	47.5	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Red brick	3.5–5	-	915	42.0	53.2	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Red brick	2–3.5	-	915	39.4	52	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Red brick	0.6–2	-	945	39.3	51.4	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Limestone	3.5–5	-	1340	39.4	44.5	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Limestone	2–3.5	-	1245	40	47.35	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Limestone	0.6–2	-	1230	40	48.5	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Marl	0.2–2	-	1265	45	50	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Shale	0.2–2	-	777	40	44.3	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Sand	0.2–2	-	2650	33.25	37.4	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Cement raw mix	-	-	1155	43.18	49.48	45	2.7	1.07	2.18	-	6.19 t/h	REAL
[25]	Cement raw mix	-	-	-	-	-	150	3	-	-	-	-	REAL Chains
[26]	Iron ore	3–6	-	-	-	-	2	0.3	0.3–0.7	1–3	-	30 kg/h	DH-
[26]	Iron ore	3–6	-	-	-	-	2	0.3	0.3–0.7	1–3	-	30 kg/h	DH
[27]	Iron ore	-	-	1600	35	-	2	0.2–0.6	0.6	1.5	-	0.25–1.25 kg/h	-
[27]	Iron ore	-	-	1600	35	-	2–4.8	0.3	0.6	1.5	-	0.25–1.25 kg/h	-
[27]	Iron ore	-	-	1600	35	-	2	0.2–0.6	0.6	1.5	-	0.25–1.25 kg/h	DH
[27]	Iron ore	-	-	1600	35	-	2–4.8	0.3	0.6	1.5	-	0.25–1.25 kg/h	DH
[14]	Iron ore	3–6	-	-	-	-	2	0.3	0.3–0.7	1.5	-	0.5 kg/h	CL
[14]	Iron ore	3–6	-	-	-	-	4.8	0.3	0.3–0.7	1.5	-	0.5 kg/h	CL
[28]	Ilmenite	0.1–0.3	4200	-	27.4	-	5.9	0.147	1–3	0.78–1.37	-	6.8–36 kg/h	DH
[29]	Coal	5–20	1240	750	47	37	6	0.6	2–4	1	8–23	0.424–0.835 m3/h	-
[29]	Coke	5–20	720	480	47	37	6	0.6	2–4	1	8–23	0.424–0.835 m3/h	-
[30]	Sand	0.49	-	1600	32	-	1	0.119	3–7.5	0–5	5–25	-	-
[30]	Sand	0.49	-	1600	32	-	1	0.0515	3–7.5	0–5	5–25	-	-
[31]	Sand	0.46	-	1600	32	-	1	0.0515	5.6–7.5	0–5	2–25	-	DH
[31]	Sand	0.46	-	1600	32	-	1	0.053	5.6–7.5	0–5	2–25	-	DH
[32]	Granulated concrete	0.40	-	1360	32	-	16	4.096	0.267–0.89	1–2.2	3.67–12.24	-	-
[32]	Granulated concrete	0.40	-	1360	32	-	16	4.096	0.267–0.89	1–2.2	3.67–12.24	-	DH
[33]	Quartz sand	0.1–0.4	-	1570	-	32	5	0.4	1–8	1–5	-	45–440 kg/h	DH
[33]	Clinker	1–12	-	1410	-	31	5	0.4	1–8	1–5	-	45–440 kg/h	DH
[33]	Glass beads	0.4–0.84	-	1560	-	21	5	0.4	1–8	1–5	-	45–440 kg/h	DH
[33]	Quartz sand	0.1–0.4	-	1570	-	32	6.7	0.25	3.5	1–2	-	25–90 kg/h	-
[33]	Clinker	1–12	-	1410	-	31	6.7	0.25	3.5	1–2	-	25–90 kg/h	-
[33]	Glass beads	0.4–0.84	-	1560	-	28	6.7	0.25	3.5	1–2	-	25–90 kg/h	-
[34]	Cement raw mix	-	948	-	35	-	55	2.3	0–3.57	1.72	-	0–24 t/h	REAL
[35]	Raw wood chips	5–15x 2–7x 1–3	-	280	-	42	4.2	0.21	2–4	1–2	5.2–18.1	4–8 kg/h	-

DH—the experiment assessed the impact of plugs, partitions, or barriers; CL—the experiment assessed the impact of geometry simulating coating or ring formation; REAL—the experiment was conducted under real operating conditions; Chains—a chain curtain was used in the cylinder.

, Table 3 and Table 4 summarize the key studies, organized by each axial-motion parameter.

3.1. Material Bed Height

3.1.1. Laboratory Studies Without Internal Structures

One of the first experimental investigations of bed height in rotary kilns was carried out in 1927 [12]. The resulting data have since been widely used to develop and verify mathematical models; a tabulated version appears in [36]. This seminal study therefore serves as a logical starting point for examining drums without internal fittings.

Extensive results for six different bulk materials were presented in [24]. That work systematically varied kiln inclination, rotation speed, particle size, and drum diameter, thereby spanning a broad range of material properties. A comparable sand experiment was reported in [30]; it considered three operating variables—rotation speed, fill angle, and fill degree—and, unlike most studies in Table 2, also captured transient responses to abrupt changes in feed rate, speed, and inclination.

The observed trends are consistent. Increasing kiln inclination lowers bed height at the inlet and raises it at the outlet. Higher rotation speed reduces bed height along the entire length. A sudden rise in feed rate first causes material accumulation at the inlet and then produces a deeper profile downstream. Reducing any of these parameters has the opposite effect in each case.

These findings are consistent with the results reported in [26,33,35]. In the coal- and coke-based experiments of [29], the authors observed that, once back spillage appeared, the bed height profile became independent of rotational speed.

Reference [27] explored geometric effects: the fill degree dropped sharply as cylinder diameter increased to 55%, but larger diameters had little additional impact, and extending kiln length produced no significant change in bed height.

A laboratory kiln of near-industrial size was investigated in [32]; the dataset was later expanded in [33] to include quartz sand, clinker, and glass beads. Iron-ore motion in an empty cylinder was examined in [26].

All of the studies above dealt with particles that are effectively spherical. To capture shape effects, non-spherical wood chips were studied in [35].

3.1.2. Laboratory Studies with Internal Structures

Studies on drum geometry have focused first on outlet plugs. Reference [26] examined plugs that blocked 0–50% of the kiln’s internal diameter. By jointly adjusting rotation speed, kiln tilt, and outlet diameter via plug size, the authors achieved an almost uniform bed height profile along the kiln. They also found that a taller plug raised the overall material level, yielding a flatter profile with a slight peak in the middle.

The dataset in [26] was later expanded with additional trials that used plugs of the same relative dimensions in cylinders of various lengths and diameters; the observations remained consistent. These findings agree with the results of [28], which supplies Equation (2) for selecting plug size to obtain a uniform bed height profile.

$$h_0=R\left(1-cos\left({\displaystyle \frac{\epsilon }{2}}\right)\right).$$

(2)

Reference [32] investigates the use of baffles in laboratory kilns whose dimensions approach those of industrial units and confirms the earlier trends. A complementary study, Ref. [31], evaluates baffles positioned not only at the outlet but also inside the kiln. Three internal-baffle geometries—thin, thick, and conical—were installed at the drum midpoint. At low rotation speeds, a thin baffle caused a sharp drop in bed height immediately downstream, effectively dividing the kiln into two zones with different loadings; higher speeds eliminated this drop and produced a uniform profile. Thick baffles reduced the drop at all speeds, whereas conical baffles provided the most even profile. The authors therefore regard the conical design as a promising option for bed height optimization. The influence of a conical baffle at the kiln exit is analyzed in Ref. [33].

Special attention is warranted for Ref. [14], which concludes a three-part series (see also [26,27]) by examining coating formation. Conical baffles, terminating 66% of the kiln length from the feed end, simulated the coating. Various cone sizes and angles were tested in two kilns of equal diameter (300 mm) but different lengths (2000 mm and 4800 mm). In their narrowest section, the baffles blocked 15%, 30%, 45%, 60%, or 75% of the internal diameter. When the simulated coating exceeded 15% blockage, the bed height profile not only increased but also changed shape, leading to material build-up in specific kiln regions.

3.1.3. Industrial Experiments

Several experimental campaigns have measured bed height profiles in industrial cement kilns. Study [25] reported tracer-atom tests at the Sebryakov plant but did not supply operating parameters or feed composition. More recent work [34] includes broader process data, yet still omits practical details such as chain-curtain geometry and raw-mix formulation. The most comprehensive dataset remains that of study [24] from the Leningrad Cement Plant; alongside bed height and axial velocity, it recorded the static and dynamic angles of repose and the bulk density of the feed as it progressed along the kiln.

Process disturbances have also been analyzed. Reference [23] shows that an isolated increase in fuel rate shifts the temperature zones upstream, raises exhaust gas temperature, produces drier, finer granules after the chain curtain, and temporarily increases bed height in that segment. Similar transient thickening occurs when draft or excess air is raised or when a drier feed is supplied. If these new conditions persist for several hours, the bed height in the burning zone gradually returns to its previous level. Lowering rotation speed causes a comparable sequence: clinker discharge falls, secondary air temperature changes, the hot zones move upstream, exhaust gas temperature rises, and bed height in the burning zone grows, followed by a period of lower loading after speed is restored.

Profiles in study [25] exhibit local “dams”—sharp, isolated rises in bed height. One dam appeared at the exit of the chain-curtain zone and another just downstream of it. The authors concluded that slurry thickening, rather than the chains themselves, is the principal cause.

3.2. Axial Velocity of Material

3.2.1. Laboratory Studies Without Internal Structures

Axial velocity was first quantified in Sullivan’s experiments [12]; the tabulated results are reproduced in [36]. Subsequent work [17] measured axial velocity in a horizontal drum for two bulk solids—sand and rye—marking the first use of nearly cylindrical particles. The material palette was later broadened in [24], which reported velocities for clinker, red brick, limestone, marl, shale, and sand under a wide range of operating conditions. Additional measurements were published in [29].

Table 3. Summary of experiments from the literature on determining the axial velocity of material [compiled by the authors].

№	Material	Particle Size (mm)	Particle Density (kg/m³)	Bulk Density (kg/m³)	Static Angle of Repose (°)	Dynamic Angle of Repose (°)	Length (m)	Diameter (m)	Rotational Speed (rpm)	Inclination Angle (°)	Filling Degree	Flow Rate	Special Conditions
[12]	Sand	0.5	-	-	-	34.95	2.13	0.152	2.5	2	25	0.0144 m3/h	-
[17]	Sand	0.2–1	-	1490	33	-	0.65	0.11	10–50	0	20–40	-	-
[17]	Rye	7×2	-	710	40	-	0.65	0.11	10–50	0	20–40	-	-
[23]	Cement raw mix	-	-	-	-	-	79.25	2.5	0.91	2.29	-	7.95 m3/h	REAL Chains
[23]	Cement raw mix	-	-	-	-	-	99.06	2.5	0.91	2.29	-	7.95 m3/h	REAL Chains
[23]	Cement raw mix	-	-	-	-	-	109.73	3.4	1	2.29	-	14.56 m3/h	REAL Chains
[23]	Cement raw mix	-	-	-	-	-	112.78	3.4	0.86	2.29	-	16.58 m3/h	REAL Chains
[24]	Clinker	3.5–5	-	1550	37	46.1	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Clinker	2–3.5	-	1518	38.4	47.5	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Clinker	0.6–2	-	1570	38.2	47.5	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Red brick	3.5–5	-	915	42.0	53.2	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Red brick	2–3.5	-	915	39.4	52	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Red brick	0.6–2	-	945	39.3	51.4	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Limestone	3.5–5	-	1340	39.4	44.5	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Limestone	2–3.5	-	1245	40	47.35	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Limestone	0.6–2	-	1230	40	48.5	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Marl	0.2–2	-	1265	45	50	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Shale	0.2–2	-	777	40	44.3	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Sand	0.2–2	-	2650	33.25	37.4	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Cement raw mix	-	-	1155	43.18	49.48	45	2.7	1.07	2.18	-	6.19 t/h	REAL
[25]	Cement raw mix	-	-	-	-	-	150	3	-	-	-	-	REAL Chains
[25]	Cement raw mix	-	-	-	-	-	150	4	-	-	-	-	REAL Chains
[37]	Cement raw mix	-	-	-	-	-	150	4	1.07	4	-	40.5 t/h	REAL
[37]	Cement raw mix	-	-	-	-	-	150	4	1.07	4	-	40.5 t/h	REAL R
[37]	Cement raw mix	-	-	-	-	-	50	3.2	0.79–1.17	5	-	-	REAL
[37]	Cement raw mix	-	-	-	-	-	50.4	3.1	0.8	3.5	-	-	REAL
[37]	Cement raw mix	-	-	-	-	-	85.5	2.9	-	3.5	-	-	REAL
[37]	Cement raw mix	-	-	-	-	-	85.5	2.9	-	3.5	-	-	REAL R
[38]	Sand	1–2	2660	1342	29.7	-	1.8	0.3	0.5–10	0–5	-	-	DH R
[38]	Wood chips Paper Rubber		777.6	225	48.5	-	1.8	0.3	0.5–10	0–5	-	-	DH R
[29]	Coal	5–20	1240	750	47	37	6	0.6	2–4	1	8–23	0.424–0.835 m3/h	-
[29]	Coke	5–20	720	480	47	37	6	0.6	2–4	1	8–23	0.424–0.835 m3/h	-
[30]	Sand	0.49	-	1600	32	-	1	0.0515	3–7.5	0–5	5–25	-	-
[30]	Sand	0.49	-	1600	32	-	1	0.119	3–7.5	0–5	5–25	-	-
[19]	Cement raw mix	-	-	-	-	-	185	5	0.93	-	-	-	REAL Chains
[19]	Cement raw mix	-	-	-	-	-	185	5	0.918	-	-	-	REAL Chains
[19]	Cement raw mix	-	-	-	-	-	185	5	1.2	-	-	-	REAL Chains
[39]	Coke	3.5	-	755	34	-	3	0.45	1–8	1.74–3.76	-	-	DH
[39]	Coke	3.5	-	755	34	-	3	0.554	1–8	1.74–3.76	-	-	DH
[32]	Granulated concrete	0.40	-	1360	32	-	16	4.096	0.267–0.89	1–2.2	3.67–12.24	-	-
[34]	Cement raw mix	-	948	-	35	-	55	2.3	0–3.57	1.72	-	0–24 t/h	REAL
[40]	Nepheline Limestone	0.5–10	-	-	-	-	23 out of 150	3.4	1.8	1°43′	-	44 m3/h	REAL Chains

DH—the experiment assessed the impact of plugs, partitions, or barriers; REAL—the experiment was conducted under real operating conditions; Chains—a chain curtain was used in the cylinder; R—the experiment assessed the impact of ribs.

Table 3. Summary of experiments from the literature on determining the axial velocity of material [compiled by the authors].

№	Material	Particle Size (mm)	Particle Density (kg/m³)	Bulk Density (kg/m³)	Static Angle of Repose (°)	Dynamic Angle of Repose (°)	Length (m)	Diameter (m)	Rotational Speed (rpm)	Inclination Angle (°)	Filling Degree	Flow Rate	Special Conditions
[12]	Sand	0.5	-	-	-	34.95	2.13	0.152	2.5	2	25	0.0144 m3/h	-
[17]	Sand	0.2–1	-	1490	33	-	0.65	0.11	10–50	0	20–40	-	-
[17]	Rye	7×2	-	710	40	-	0.65	0.11	10–50	0	20–40	-	-
[23]	Cement raw mix	-	-	-	-	-	79.25	2.5	0.91	2.29	-	7.95 m3/h	REAL Chains
[23]	Cement raw mix	-	-	-	-	-	99.06	2.5	0.91	2.29	-	7.95 m3/h	REAL Chains
[23]	Cement raw mix	-	-	-	-	-	109.73	3.4	1	2.29	-	14.56 m3/h	REAL Chains
[23]	Cement raw mix	-	-	-	-	-	112.78	3.4	0.86	2.29	-	16.58 m3/h	REAL Chains
[24]	Clinker	3.5–5	-	1550	37	46.1	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Clinker	2–3.5	-	1518	38.4	47.5	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Clinker	0.6–2	-	1570	38.2	47.5	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Red brick	3.5–5	-	915	42.0	53.2	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Red brick	2–3.5	-	915	39.4	52	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Red brick	0.6–2	-	945	39.3	51.4	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Limestone	3.5–5	-	1340	39.4	44.5	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Limestone	2–3.5	-	1245	40	47.35	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Limestone	0.6–2	-	1230	40	48.5	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Marl	0.2–2	-	1265	45	50	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Shale	0.2–2	-	777	40	44.3	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Sand	0.2–2	-	2650	33.25	37.4	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Cement raw mix	-	-	1155	43.18	49.48	45	2.7	1.07	2.18	-	6.19 t/h	REAL
[25]	Cement raw mix	-	-	-	-	-	150	3	-	-	-	-	REAL Chains
[25]	Cement raw mix	-	-	-	-	-	150	4	-	-	-	-	REAL Chains
[37]	Cement raw mix	-	-	-	-	-	150	4	1.07	4	-	40.5 t/h	REAL
[37]	Cement raw mix	-	-	-	-	-	150	4	1.07	4	-	40.5 t/h	REAL R
[37]	Cement raw mix	-	-	-	-	-	50	3.2	0.79–1.17	5	-	-	REAL
[37]	Cement raw mix	-	-	-	-	-	50.4	3.1	0.8	3.5	-	-	REAL
[37]	Cement raw mix	-	-	-	-	-	85.5	2.9	-	3.5	-	-	REAL
[37]	Cement raw mix	-	-	-	-	-	85.5	2.9	-	3.5	-	-	REAL R
[38]	Sand	1–2	2660	1342	29.7	-	1.8	0.3	0.5–10	0–5	-	-	DH R
[38]	Wood chips Paper Rubber		777.6	225	48.5	-	1.8	0.3	0.5–10	0–5	-	-	DH R
[29]	Coal	5–20	1240	750	47	37	6	0.6	2–4	1	8–23	0.424–0.835 m3/h	-
[29]	Coke	5–20	720	480	47	37	6	0.6	2–4	1	8–23	0.424–0.835 m3/h	-
[30]	Sand	0.49	-	1600	32	-	1	0.0515	3–7.5	0–5	5–25	-	-
[30]	Sand	0.49	-	1600	32	-	1	0.119	3–7.5	0–5	5–25	-	-
[19]	Cement raw mix	-	-	-	-	-	185	5	0.93	-	-	-	REAL Chains
[19]	Cement raw mix	-	-	-	-	-	185	5	0.918	-	-	-	REAL Chains
[19]	Cement raw mix	-	-	-	-	-	185	5	1.2	-	-	-	REAL Chains
[39]	Coke	3.5	-	755	34	-	3	0.45	1–8	1.74–3.76	-	-	DH
[39]	Coke	3.5	-	755	34	-	3	0.554	1–8	1.74–3.76	-	-	DH
[32]	Granulated concrete	0.40	-	1360	32	-	16	4.096	0.267–0.89	1–2.2	3.67–12.24	-	-
[34]	Cement raw mix	-	948	-	35	-	55	2.3	0–3.57	1.72	-	0–24 t/h	REAL
[40]	Nepheline Limestone	0.5–10	-	-	-	-	23 out of 150	3.4	1.8	1°43′	-	44 m3/h	REAL Chains

DH—the experiment assessed the impact of plugs, partitions, or barriers; REAL—the experiment was conducted under real operating conditions; Chains—a chain curtain was used in the cylinder; R—the experiment assessed the impact of ribs.

Across these studies, the effects of process variables are clear. Higher rotation speed and steeper kiln inclination both increase axial velocity, whereas a larger material load decreases it. Reference [29] explains the latter effect through redistribution: a thicker bed shifts particles from the active layer to the passive core, and, as noted in [18], axial transport is dominated by the active layer.

Nearly all cited experiments were performed under steady conditions. Reference [30] instead examined transitions triggered by step changes in feed rate, rotation speed, and tilt. Axial velocity measurements for a full-scale, unheated kiln are given in [32].

Rotation speed also influences the kiln’s thermal profile. An indirect assessment in [41] shows that faster rotation shortens the residence time in the passive layer, limiting conductive heating from the wall, while enhanced mixing in the active layer increases contact with hot gas. As speed rises, the temperature profile becomes more symmetric and the depth of heat penetration decreases, shifting the balance between surface and volumetric heating—an essential consideration in thermal-regime design.

3.2.2. Laboratory Studies with Internal Structures

Study [38] examines internal structures and their effect on axial velocity. Both longitudinal ribs and ring ribs reduce axial velocity, but through different mechanisms. Ring ribs form barriers that flatten the bed surface and deepen the material layer upstream, slowing the flow. Longitudinal ribs restrict transverse mixing, promote segregation, and thus lower axial velocity. The greatest reduction occurs when both rib types are used together.

The same study evaluates outlet baffles that block 10% and 16.7% of the kiln’s internal diameter; axial velocity drops because the material must climb over the exit obstruction. Study [39] reports a speed-dependent effect: at rotation speeds below 2 rpm baffles slow the flow, whereas above 2 rpm they accelerate axial motion.

3.2.3. Industrial Experiments

Reference [23] used the tracer-atom method to measure axial velocity in an operating kiln. Particles introduced at the same time traveled at different speeds: velocity rose from the chain-curtain zone, peaked in the calcination zone, and fell to a minimum in the burning zone. Marked velocity differences were observed from kiln to kiln, independent of rotation speed or shell diameter. The study attributes these differences to granulation—smaller particles move faster—and suggests that CO₂ released in the calcination zone creates an airflow effect that further accelerates the bed.

The same work shows that any change boosting exhaust gas temperature (for example, a higher fuel rate) increases velocity upstream of the burning zone; lower feed moisture has a similar effect. Uniform granulation beyond the chain curtain is therefore seen as essential for stable operation.

Reference [25] did not observe a velocity peak in the calcination zone and proposes that the discrepancy with [23] may arise from differences in feed composition.

Measurements on three kilns at the Topki plant, reported in [19], generally confirm the trend: velocity climbs from the cold end to the decarbonization zone and then declines toward the sintering zone. The absolute maximum, however, varied between runs, possibly because operating conditions—exhaust gas temperature ranged from 199 to 311 °C—fluctuated during firing. Reference [19] also supplies a correlation matrix linking zone-wise velocity to process variables such as exhaust gas temperature, rotation speed, draft, calcination zone temperature, productivity, clinker exit temperature, secondary air temperature, and chain surface area.

Experiments focused on the chain-curtain zone at the Pikalevo alumina plant (high-silica feed) are summarized in [40] and build on earlier work [42,43,44].

Reference [24] provides a detailed velocity profile, although zone boundaries are not marked, which complicates direct comparison with [19]; nevertheless, its curve (Figure 18 in [24]) shows a similar nonlinear pattern.

Study [37] again records the highest velocity in the calcination zone. Thresholds installed to slow the flow in that region successfully reduced velocity there while increasing it upstream. The same study estimates the enthalpy change in the feed along the kiln length and covers wet, dry, and semi-dry units.

3.3. Mean Residence Time

3.3.1. Laboratory Studies Without Internal Structures

As with axial velocity and bed height, the first laboratory data on mean residence time (MRT) come from Sullivan’s experiments [12]; the tabulated values also appear in [36]. Subsequent MRT measurements in horizontal cylinders with cylindrical and spherical particles were reported in [17], and similar work continued in [45]. Khodorov’s monograph [24] later extended the database to a range of materials—sand, clinker, red brick, limestone, shale, marl, and additional sand trials.

Study [45] showed that MRT decreases with higher feed rate, increases with greater fill degree, decreases as rotation speed falls, and increases with steeper cylinder inclination; these trends are consistent with [28,29]. Reference [39] adds that the drop in MRT becomes less pronounced above 4 rpm, and the increase due to cylinder tilt is almost linear.

Table 4. Summary of experiments from the literature on determining mean residence time [compiled by the authors].

№	Material	Particle Size (mm)	Particle Density (kg/m³)	Bulk Density (kg/m³)	Static Angle of Repose (°)	Dynamic Angle of Repose (°)	Length (m)	Diameter (m)	Rotational Speed (rpm)	Inclination Angle (°)	Filling Degree	Flow Rate	Special Conditions
[12]	Sand	0.5	-	-	-	34.95	2.13	0.152	2.5	2	25	0.0144 m3/h	-
[17]	Sand	0.2–1	-	1490	33	-	0.65	0.11	10–50	0	20–40	-	-
[17]	Rye	7×2	-	710	40	-	0.65	0.11	10–50	0	20–40	-	-
[23]	Cement raw mix	-	-	-	-	-	79.25	2.5	0.91	2.29	-	7.95 m3/h	REAL Chains
[23]	Cement raw mix	-	-	-	-	-	99.06	2.5	0.91	2.29	-	7.95 m3/h	REAL Chains
[23]	Cement raw mix	-	-	-	-	-	109.73	3.4	1	2.29	-	14.56 m3/h	REAL Chains
[23]	Cement raw mix	-	-	-	-	-	112.78	3.4	0.86	2.29	-	16.58 m3/h	REAL Chains
[24]	Clinker	3.5–5	-	1550	37	46.1	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Clinker	2–3.5	-	1518	38.4	47.5	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Clinker	0.6–2	-	1570	38.2	47.5	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Red brick	3.5–5	-	915	42.0	53.2	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Red brick	2–3.5	-	915	39.4	52	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Red brick	0.6–2	-	945	39.3	51.4	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Limestone	3.5–5	-	1340	39.4	44.5	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Limestone	2–3.5	-	1245	40	47.35	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Limestone	0.6–2	-	1230	40	48.5	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Marl	0.2–2	-	1265	45	50	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Shale	0.2–2	-	777	40	44.3	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Sand	0.2–2	-	2650	33.25	37.4	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Cement raw mix	-	-	1155	43.18	49.48	45	2.7	1.07	2.18	-	6.19 t/h	REAL
[25]	Cement raw mix	-	-	-	-	-	150	3	-	-	-	-	REAL Chains
[25]	Cement raw mix	-	-	-	-	-	150	4	-	-	-	-	REAL Chains
[45]	Dolomite	28×35, 35×48, 48×65–TS *	-	-	-	-	0.24	0.08	20–80	0–3	-	1.8–8.64 kg/h	-
[37]	Cement raw mix	-	-	-	-	-	150	4	1.07	4	-	40.5 t/h	REAL
[37]	Cement raw mix	-	-	-	-	-	150	4	1.07	4	-	40.5 t/h	REAL R
[37]	Cement raw mix	-	-	-	-	-	50	3.2	0.79–1.17	5	-	-	REAL
[37]	Cement raw mix	-	-	-	-	-	50.4	3.1	0.8	3.5	-	-	REAL
[37]	Cement raw mix	-	-	-	-	-	85.5	2.9	-	3.5	-	-	REAL
[37]	Cement raw mix	-	-	-	-	-	85.5	2.9	-	3.5	-	-	REAL R
[46]	Sodium carbonate	0.137	2530	-	-	-	0.6	0.25	1–10	0	16–40	0.0025–0.0102 m3/h	-
[26]	Iron ore	3–6	-	-	-	-	2	0.3	0.3–0.7	1–3	-	30 kg/h	DH-
[27]	Iron ore	-	-	1600	35	-	2	0.2–0.6	0.6	1.5	-	0.25–1.25 kg/h	-
[27]	Iron ore	-	-	1600	35	-	2–4.8	0.3	0.6	1.5	-	0.25–1.25 kg/h	-
[26]	Iron ore	3–6	-	-	-	-	2	0.3	0.3–0.7	1–3	-	30 kg/h	DH
[27]	Iron ore	-	-	1600	35	-	2	0.2–0.6	0.6	1.5	-	0.25–1.25 kg/h	DH
[27]	Iron ore	-	-	1600	35	-	2–4.8	0.3	0.6	1.5	-	0.25–1.25 kg/h	DH
[14]	Iron ore	3–6	-	-	-	-	2	0.3	0.3–0.7	1.5	-	0.5 kg/h	CL
[14]	Iron ore	3–6	-	-	-	-	4.8	0.3	0.3–0.7	1.5	-	0.5 kg/h	CL
[28]	Ilmenite	0.1–0.3	4200	-	27.4	-	5.9	0.147	1–3	0.78–1.37	-	-	DH
[28]	Ilmenite	0.1–0.3	4200	-	27.4	-	5.9	0.147	1–3	0.78–1.37	-	-	-
[47]	Sand	2.02	2630	1640	36	-	1.05	0.192	0–100	0	-	22.68–109.08 kg/h	DH LIF
[38]	Sand	1–2	2660	1342	29.7	-	1.8	0.3	0.5–10	0–5	-	-	DH R
[38]	Wood chips Paper Rubber		777.6	225	48.5	-	1.8	0.3	0.5–10	0–5	-	-	DH R
[29]	Coal	5–20	1240	750	47	37	6	0.6	2–4	1	8–23	0.424–0.835 m3/h	-
[29]	Coke	5–20	720	480	47	37	6	0.6	2–4	1	8–23	0.424–0.835 m3/h	-
[48]	Sand	0.366	-	1370	-	33	4.635	0.4	1–3	3	2–11	59.9–149.536 kg/h	DH
[39]	Coke	3.5	-	755	34	-	3	0.45	1–8	1.74–3.76	-	-	DH
[39]	Coke	3.5	-	755	34	-	3	0.554	1–8	1.74–3.76	-	-	DH
[32]	Granulated concrete	0.40	-	1360	32	-	16	4.096	0.267–0.89	1–2.2	3.67–12.24	-	-
[35]	Raw wood chips	5–15× 2–7× 1–3	-	280	-	42	4.2	0.21	2–4	1–2	5.2–18.1	4–8 kg/h	-
[49]	Sand	0.55		1422	39	-	1.95	0.101	2–12	2–5	-	0.68–2.5 kg/h	DH LIF
[49]	Rice	3.8×1.9	-	889	36	-	1.95	0.101	2–12	2–5	-	0.68–2.5 kg/h	DH LIF
[34]	Cement raw mix	-	948	-	35	-	55	2.3	0–3.57	1.72	-	0–24 t/h	REAL
[50]	Wood chips	5–17× 2–8× 1–4	1506	260	42	-	4.2	0.21	0.5–21	1–3	8–10	2.5–7.5 kg/h	R LIF
[51]	Quartz sand	1.25–2	2650	-	-	-	1.2	0.1	1–3	0.5–3	15	18–36 kg/h	-
[51]	Quartz sand	3–3.75	2650	-	-	-	1.2	0.1	1–3	0.5–3	15	18–36 kg/h	-
[52]	Coal	0.053–0.579	-	530	-	29.2	1	0.095	1.5–10	1.1–2	2.8–13	0.2–2.3 kg/h	-
[52]	Olive pits	1.144–3.624	-	790	-	38.0	1	0.095	1.5–10	1.1–2	2.8–13	0.2–2.3 kg/h	-
[52]	Sand	0.257–0.416	-	1470	-	34.0	1	0.095	1.5–10	1.1–2	2.8–13	0.2–2.3 kg/h	-
[53]	Aluminum hydroxide	-	-	-	-	-	51	2.56	1.98	1.43	2.1	2640 m3/s	REAL
[53]	Aluminum hydroxide	-	-	-	-	-	51	2.56	1.98	1.43	2	2640 m3/s	REAL
[53]	Aluminum hydroxide	-	-	-	-	-	60	2.84	1.98	1.43	2.4	3140 m3/s	REAL
[53]	Aluminum hydroxide	-	-	-	-	-	50	2.24	1.04	1.43	2.1	1830 m3/s	REAL
[53]	Aluminum hydroxide	-	-	-	-	-	50	2.42	1.35	1.43	1.5	1970 m3/s	REAL
[53]	Aluminum hydroxide	-	-	-	-	-	50	2.05	1.182	1.43	1.7	1750 m3/s	REAL
[53]	Aluminum hydroxide	-	-	-	-	-	60	2.84	1.8	1.15	2	3060 m3/s	REAL
[53]	Aluminum hydroxide	-	-	-	-	-	60	2.84	1.98	1.72	1.8	3400 m3/s	REAL
[53]	Aluminum hydroxide	-	-	-	-	-	60	2.584	1.8	1.72	1.8	3350 m3/s	REAL

* DH—the experiment assessed the impact of plugs, partitions, or barriers; REAL—the experiment was conducted under real operating conditions; Chains—a chain curtain was used in the cylinder; R—the experiment assessed the impact of ribs; CL—the experiment assessed the impact of geometry simulating coating or ring formation; LIF—the experiment assessed the impact of blades; TS—Taylor System.

Table 4. Summary of experiments from the literature on determining mean residence time [compiled by the authors].

№	Material	Particle Size (mm)	Particle Density (kg/m³)	Bulk Density (kg/m³)	Static Angle of Repose (°)	Dynamic Angle of Repose (°)	Length (m)	Diameter (m)	Rotational Speed (rpm)	Inclination Angle (°)	Filling Degree	Flow Rate	Special Conditions
[12]	Sand	0.5	-	-	-	34.95	2.13	0.152	2.5	2	25	0.0144 m3/h	-
[17]	Sand	0.2–1	-	1490	33	-	0.65	0.11	10–50	0	20–40	-	-
[17]	Rye	7×2	-	710	40	-	0.65	0.11	10–50	0	20–40	-	-
[23]	Cement raw mix	-	-	-	-	-	79.25	2.5	0.91	2.29	-	7.95 m3/h	REAL Chains
[23]	Cement raw mix	-	-	-	-	-	99.06	2.5	0.91	2.29	-	7.95 m3/h	REAL Chains
[23]	Cement raw mix	-	-	-	-	-	109.73	3.4	1	2.29	-	14.56 m3/h	REAL Chains
[23]	Cement raw mix	-	-	-	-	-	112.78	3.4	0.86	2.29	-	16.58 m3/h	REAL Chains
[24]	Clinker	3.5–5	-	1550	37	46.1	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Clinker	2–3.5	-	1518	38.4	47.5	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Clinker	0.6–2	-	1570	38.2	47.5	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Red brick	3.5–5	-	915	42.0	53.2	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Red brick	2–3.5	-	915	39.4	52	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Red brick	0.6–2	-	945	39.3	51.4	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Limestone	3.5–5	-	1340	39.4	44.5	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Limestone	2–3.5	-	1245	40	47.35	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Limestone	0.6–2	-	1230	40	48.5	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Marl	0.2–2	-	1265	45	50	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Shale	0.2–2	-	777	40	44.3	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Sand	0.2–2	-	2650	33.25	37.4	2.7	0.5–0.3	1.05–3.47	1–4.5	5–10	-
[24]	Cement raw mix	-	-	1155	43.18	49.48	45	2.7	1.07	2.18	-	6.19 t/h	REAL
[25]	Cement raw mix	-	-	-	-	-	150	3	-	-	-	-	REAL Chains
[25]	Cement raw mix	-	-	-	-	-	150	4	-	-	-	-	REAL Chains
[45]	Dolomite	28×35, 35×48, 48×65–TS *	-	-	-	-	0.24	0.08	20–80	0–3	-	1.8–8.64 kg/h	-
[37]	Cement raw mix	-	-	-	-	-	150	4	1.07	4	-	40.5 t/h	REAL
[37]	Cement raw mix	-	-	-	-	-	150	4	1.07	4	-	40.5 t/h	REAL R
[37]	Cement raw mix	-	-	-	-	-	50	3.2	0.79–1.17	5	-	-	REAL
[37]	Cement raw mix	-	-	-	-	-	50.4	3.1	0.8	3.5	-	-	REAL
[37]	Cement raw mix	-	-	-	-	-	85.5	2.9	-	3.5	-	-	REAL
[37]	Cement raw mix	-	-	-	-	-	85.5	2.9	-	3.5	-	-	REAL R
[46]	Sodium carbonate	0.137	2530	-	-	-	0.6	0.25	1–10	0	16–40	0.0025–0.0102 m3/h	-
[26]	Iron ore	3–6	-	-	-	-	2	0.3	0.3–0.7	1–3	-	30 kg/h	DH-
[27]	Iron ore	-	-	1600	35	-	2	0.2–0.6	0.6	1.5	-	0.25–1.25 kg/h	-
[27]	Iron ore	-	-	1600	35	-	2–4.8	0.3	0.6	1.5	-	0.25–1.25 kg/h	-
[26]	Iron ore	3–6	-	-	-	-	2	0.3	0.3–0.7	1–3	-	30 kg/h	DH
[27]	Iron ore	-	-	1600	35	-	2	0.2–0.6	0.6	1.5	-	0.25–1.25 kg/h	DH
[27]	Iron ore	-	-	1600	35	-	2–4.8	0.3	0.6	1.5	-	0.25–1.25 kg/h	DH
[14]	Iron ore	3–6	-	-	-	-	2	0.3	0.3–0.7	1.5	-	0.5 kg/h	CL
[14]	Iron ore	3–6	-	-	-	-	4.8	0.3	0.3–0.7	1.5	-	0.5 kg/h	CL
[28]	Ilmenite	0.1–0.3	4200	-	27.4	-	5.9	0.147	1–3	0.78–1.37	-	-	DH
[28]	Ilmenite	0.1–0.3	4200	-	27.4	-	5.9	0.147	1–3	0.78–1.37	-	-	-
[47]	Sand	2.02	2630	1640	36	-	1.05	0.192	0–100	0	-	22.68–109.08 kg/h	DH LIF
[38]	Sand	1–2	2660	1342	29.7	-	1.8	0.3	0.5–10	0–5	-	-	DH R
[38]	Wood chips Paper Rubber		777.6	225	48.5	-	1.8	0.3	0.5–10	0–5	-	-	DH R
[29]	Coal	5–20	1240	750	47	37	6	0.6	2–4	1	8–23	0.424–0.835 m3/h	-
[29]	Coke	5–20	720	480	47	37	6	0.6	2–4	1	8–23	0.424–0.835 m3/h	-
[48]	Sand	0.366	-	1370	-	33	4.635	0.4	1–3	3	2–11	59.9–149.536 kg/h	DH
[39]	Coke	3.5	-	755	34	-	3	0.45	1–8	1.74–3.76	-	-	DH
[39]	Coke	3.5	-	755	34	-	3	0.554	1–8	1.74–3.76	-	-	DH
[32]	Granulated concrete	0.40	-	1360	32	-	16	4.096	0.267–0.89	1–2.2	3.67–12.24	-	-
[35]	Raw wood chips	5–15× 2–7× 1–3	-	280	-	42	4.2	0.21	2–4	1–2	5.2–18.1	4–8 kg/h	-
[49]	Sand	0.55		1422	39	-	1.95	0.101	2–12	2–5	-	0.68–2.5 kg/h	DH LIF
[49]	Rice	3.8×1.9	-	889	36	-	1.95	0.101	2–12	2–5	-	0.68–2.5 kg/h	DH LIF
[34]	Cement raw mix	-	948	-	35	-	55	2.3	0–3.57	1.72	-	0–24 t/h	REAL
[50]	Wood chips	5–17× 2–8× 1–4	1506	260	42	-	4.2	0.21	0.5–21	1–3	8–10	2.5–7.5 kg/h	R LIF
[51]	Quartz sand	1.25–2	2650	-	-	-	1.2	0.1	1–3	0.5–3	15	18–36 kg/h	-
[51]	Quartz sand	3–3.75	2650	-	-	-	1.2	0.1	1–3	0.5–3	15	18–36 kg/h	-
[52]	Coal	0.053–0.579	-	530	-	29.2	1	0.095	1.5–10	1.1–2	2.8–13	0.2–2.3 kg/h	-
[52]	Olive pits	1.144–3.624	-	790	-	38.0	1	0.095	1.5–10	1.1–2	2.8–13	0.2–2.3 kg/h	-
[52]	Sand	0.257–0.416	-	1470	-	34.0	1	0.095	1.5–10	1.1–2	2.8–13	0.2–2.3 kg/h	-
[53]	Aluminum hydroxide	-	-	-	-	-	51	2.56	1.98	1.43	2.1	2640 m3/s	REAL
[53]	Aluminum hydroxide	-	-	-	-	-	51	2.56	1.98	1.43	2	2640 m3/s	REAL
[53]	Aluminum hydroxide	-	-	-	-	-	60	2.84	1.98	1.43	2.4	3140 m3/s	REAL
[53]	Aluminum hydroxide	-	-	-	-	-	50	2.24	1.04	1.43	2.1	1830 m3/s	REAL
[53]	Aluminum hydroxide	-	-	-	-	-	50	2.42	1.35	1.43	1.5	1970 m3/s	REAL
[53]	Aluminum hydroxide	-	-	-	-	-	50	2.05	1.182	1.43	1.7	1750 m3/s	REAL
[53]	Aluminum hydroxide	-	-	-	-	-	60	2.84	1.8	1.15	2	3060 m3/s	REAL
[53]	Aluminum hydroxide	-	-	-	-	-	60	2.84	1.98	1.72	1.8	3400 m3/s	REAL
[53]	Aluminum hydroxide	-	-	-	-	-	60	2.584	1.8	1.72	1.8	3350 m3/s	REAL

* DH—the experiment assessed the impact of plugs, partitions, or barriers; REAL—the experiment was conducted under real operating conditions; Chains—a chain curtain was used in the cylinder; R—the experiment assessed the impact of ribs; CL—the experiment assessed the impact of geometry simulating coating or ring formation; LIF—the experiment assessed the impact of blades; TS—Taylor System.

Industrial-scale MRT measurements were presented in [32]. Non-spherical particles were examined in [35] using raw wood chips. More recent work has refined MRT-parameter correlations over narrower ranges: study [51], for example, investigated small inclination angles (0.5–1°) and found little effect on MRT—except at high feed rates, where slight decreases and backflow can occur once throughput capacity is exceeded. Study [52] supplied MRT data for biochar, olive pits, and sand.

3.3.2. Laboratory Studies with Internal Structures

Revisiting the datasets of [12] and [26], Sullivan did not regard feed rate as a key variable for MRT. This discrepancy was analyzed in [48], which showed that rotation speed affects MRT far more than feed rate, especially at high speeds. The kiln in [48] included an outlet baffle—a configuration also examined in [27]. That study found only a minor feed-rate effect when no baffle was present, but a pronounced MRT decrease when the feed rate rose with a baffle installed.

Reference [27] further reported a diameter effect: without a baffle, MRT falls as cylinder diameter increases; with a baffle, the trend is reversed.

One of the first systematic investigations of baffle geometry was conducted in [26]. MRT increased when a baffle was installed and rose further as baffle height grew. Similar behavior has been observed under other materials and operating windows [28,49], although [39] notes that MRT begins to decline once rotation speed exceeds 2 rpm.

Study [38] considered axial and ring ribs and found a significant MRT increase only with sufficiently tall ring ribs. Blades were assessed in [50]; they lengthen MRT by promoting backflow.

A distinctive contribution is [14], which modeled coating build-up by installing truncated conical inserts of various diameters. The inserts increased MRT; a ring blocking more than 25–30% of the kiln diameter and located 60–70% of the length from the feed end made operation nearly impossible because of severe backflow and very long residence times. Smaller rings mitigated these effects if the rotation speed was raised. When such a restriction is present, overall MRT is governed almost entirely by residence time in the constricted zone. These findings are critical because coating and ring formation remain persistent challenges in industrial cement kilns, yet they are rarely addressed in the literature.

Infrared thermography has proven to be an effective non-invasive method for assessing the thickness of internal coatings (rings) in rotary kilns by analyzing temperature variations along the kiln shell. As demonstrated in multiple studies, including those by [54,55,56], regions with thicker coatings exhibit lower external shell temperatures due to their insulating effect, while thinner or absent coatings correspond to higher temperatures. This inverse relationship allows for indirect estimation of coating thickness using shell temperature profiles, which are routinely recorded in modern plants. Study [55] confirms that ash rings formed in the calcination zone lead to a measurable drop in shell temperature, while Wirtz et al. [54] employ a one-dimensional heat transfer model coupled with CFD simulations to iteratively reconstruct coating thickness from infrared data. Rippon et al. [56] further demonstrate how long-term thermographic monitoring can visualize ring formation at different timescales and locations. These findings establish infrared thermography as a reliable tool for detecting, localizing, and tracking the evolution of coatings in rotary kilns, supporting both maintenance planning and the development of digital process control systems.

3.3.3. Industrial Experiments

Reference [23] highlights the practical importance of residence-time distribution (RTD) in industrial kilns. In kiln No. 7, the mean residence time (MRT) was 250 min, with the fastest particles exiting in 180 min and the slowest in 300 min. Because no rings were present, the kiln behaved as a mixer.

Reference [25] shows that higher slurry moisture extends residence time: when moisture rose from 38% to 41%, MRT increased from 258 min to 272 min.

Reference [19] presents a correlation matrix that links MRT to several process variables, including exhaust gas temperature, rotation speed, draft, calcination zone temperature, kiln productivity, clinker exit temperature, secondary air temperature, and chain surface area.

Together, these studies underscore how MRT depends on both kiln design and operating conditions.

4. Mathematical Models

4.1. Material Bed Height

A bed height model should predict how the layer depth varies along the kiln axis. The next section reviews the mathematical approaches published in the scientific literature.

$${\displaystyle \frac{dh}{dx}}={\displaystyle \frac{3\tan \left(\theta \right)M}{4\pi n\rho }}{\left(R^2-(h-R^2\right)}^{-{\displaystyle \frac{3}{2}}}-{\displaystyle \frac{\tan \left(\beta \right)}{\cos \left(\theta \right)}}.$$

(3)

$$h\left(0\right)=h_0.$$

(4)

$$h\left(0\right)=d_p.$$

(5)

One of the best-known methods for predicting bed height in rotary kilns is the Saeman model [57], expressed by Equations (3)–(5). It treats particle motion geometrically and yields a first-order, nonlinear differential equation posed as a Cauchy problem. Along the axial coordinate x (0 = kiln outlet, x=L = feed point), the boundary condition sets the bed height at the outlet equal to the baffle height—or to the particle diameter if no baffle is installed.

The model has matched experimental data well under the conditions tested in [32,34], yet other studies report substantial deviations [50]. To improve agreement, Ref. [52] recommends alternative initial conditions and introduces a dimensionless parameter, the Bed Depth Number (BDN).

$$\mathrm{B}DN={\left({\displaystyle \frac{2arccos\left(1-\frac{h\left(L\right)}{R}\right)-sin\left(2\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\left(1-\frac{h\left(L\right)}{R}\right)\right)}{3.5\pi }}\right)}^2.$$

(6)

Study [33] likewise tackles the boundary-condition problem: it introduces a dimensionless criterion and derives the following expression to apply it.

$$BDN={\displaystyle \frac{0.75Mtan\left(\sigma \right)}{\pi n\rho R^3}}.$$

(7)

The literature contains many adaptations of the Saeman equation for specific kiln geometries. Study [31], for example, investigated how internal thresholds of various shapes influence the bed height profile and accordingly modified both the equation’s form and its boundary conditions to account for these structures.

$${\displaystyle \frac{dh}{dx}}={\displaystyle \frac{3\tan \left(\theta \right)Q}{4\pi n{\left(R_i^2-{\left(R-h\right)}^2\right)}^{3/2}}}-{\displaystyle \frac{\tan \left(\beta \right)}{\mathrm{c}\mathrm{o}\mathrm{s}\left(\theta \right)}}.$$

(8)

$$h\left(0\right)=R-R_{dam}+d_p.$$

(9)

Study [35] adapts the model for non-spherical wood-chip particles by introducing three power-law correction coefficients.

$${\displaystyle \frac{dh}{dx}}={\displaystyle \frac{3Q^{C_Q}\tan \left(\theta \right)}{4\pi w^{C_w}{R}^3}}{\left({\displaystyle \frac{2h}{R}}-{\displaystyle \frac{h^2}{R^2}}\right)}^{-{\displaystyle \frac{3}{2}}}-{\displaystyle \frac{{\mathit{tan}}^{C_{\beta }}\left(\beta \right)}{\cos \left(\theta \right)}}.$$

(10)

A fully analytical solution to the model was derived in study [58] and later examined in research [59].

$$\begin{array}{c}\left(\cos \left(\epsilon \right)-\cos \left({\epsilon }_0\right)\right)-{\displaystyle \frac{2H{r}_1}{\left(r_1-r_2\right)\left(r_1-r_3\right)}}{\displaystyle \frac{1}{\sqrt{r_1^2-1}}}\times \left({tan}^{-1}\left({\displaystyle \frac{1-r_1\tan \left(\frac{\epsilon }{2}\right)}{\sqrt{r_1^2-1}}}\right)-{tan}^{-1}\left({\displaystyle \frac{1-r_1\tan \left(\frac{{\epsilon }_0}{2}\right)}{\sqrt{r_1^2-1}}}\right)\right)-\\ {\displaystyle \frac{2H{r}_2}{\left(r_2-r_1\right)\left(r_2-r_3\right)}}{\displaystyle \frac{1}{\sqrt{r_2^2-1}}}\times \left({tan}^{-1}\left({\displaystyle \frac{1-r_2\tan \left(\frac{\epsilon }{2}\right)}{\sqrt{r_2^2-1}}}\right)-{tan}^{-1}\left({\displaystyle \frac{1-r_2\tan \left(\frac{{\epsilon }_0}{2}\right)}{\sqrt{r_2^2-1}}}\right)\right)-{\displaystyle \frac{2H{r}_3}{\left(r_3-r_1\right)\left(r_3-r_2\right)}}{\displaystyle \frac{1}{\sqrt{r_3^2-1}}}\times ({tan}^{-1}\left({\displaystyle \frac{1-r_3\tan \left(\frac{\epsilon }{2}\right)}{\sqrt{r_3^2-1}}}\right)-\\ {tan}^{-1}\left({\displaystyle \frac{1-r_3\tan \left(\frac{{\epsilon }_0}{2}\right)}{\sqrt{r_3^2-1}}}\right))=X.\end{array}$$

(11)

$$r_1={\left({\displaystyle \frac{a}{b}}\right)}^{{\displaystyle \frac{1}{3}}}.$$

(12)

$$r_2=-\left({\displaystyle \frac{1}{2}}+{\displaystyle \frac{\sqrt{3}}{2}}i\right){\left({\displaystyle \frac{a}{b}}\right)}^{{\displaystyle \frac{1}{3}}}.$$

(13)

$$r_3=\left(-{\displaystyle \frac{1}{2}}+{\displaystyle \frac{\sqrt{3}}{2}}i\right){\left({\displaystyle \frac{a}{b}}\right)}^{{\displaystyle \frac{1}{3}}}.$$

(14)

$$H={\displaystyle \frac{a}{b}}.$$

(15)

$$a={\displaystyle \frac{3\tan \left(\theta \right)LM}{4\pi n{R}^4\rho }}.$$

(16)

$$b={\displaystyle \frac{Ltan\left(\beta \right)}{Rcos\left(\theta \right)}}.$$

(17)

$$X={\displaystyle \frac{bx}{L}}.$$

(18)

A formula derived from the Saeman model and presented in study [60] is also frequently cited in the literature.

$${\displaystyle \frac{dh}{dx}}={\displaystyle \frac{\mathrm{t}\mathrm{a}\mathrm{n}\left(\delta \right)}{\mathrm{c}\mathrm{o}\mathrm{s}\left(\sigma \right)}}-{\displaystyle \frac{360Qtan\left(\sigma \right)}{\pi f{D}^3}}{\left({\displaystyle \frac{4h}{D}}-{\left({\displaystyle \frac{2h}{D}}\right)}^2\right)}^{-1.5}.$$

(19)

The calculation method and boundary conditions are the same as in the Saeman equation. A less common, yet still widely cited, alternative was introduced by Ronco [29,61].

$${\displaystyle \frac{dh}{dx}}=tan\left({\displaystyle \frac{4Qsin\left(\beta \right)}{2.623\times {10}^{-2}\pi f{D}^3{Z}^{0.75}}}-\delta \right).$$

(20)

Khodorov’s material–motion model deserves special attention. The author has devoted several monographs to the topic, and the most detailed treatment of axial transport appears in study [24]. That work reports an industrial experiment on successive sections of a wet-process cement kiln in which bulk density, static and dynamic angles of repose, axial velocity, and bed height were measured. These data enabled a comprehensive model for predicting axial transport parameters. Study [24] presents the full calculation algorithm for every kiln section and includes a worked example. The kiln is divided into ten sections, each further split into smaller segments to reflect changes in material properties and kiln geometry along the axis.

As in earlier models, the computation starts at the kiln outlet. The initial bed height h(0)—usually equal to the height of the retaining ring [62]—is prescribed, so the first calculation is made at the outlet cross-section. The opening step is to determine the particle acceleration in the active layer from a force balance.

$$m_{p}g(\sin \left(\theta \right)-tg\left(s\right)\cos \left(\mathsf{\theta }\right)={\mathrm{m}}_{p}j.$$

(21)

Once the acceleration is known, the coefficient k₁—which characterizes the dynamics of particle motion—can be calculated.

$$k_1=1+0.00257{\displaystyle \frac{60\epsilon }{w}}\sqrt{{\displaystyle \frac{j}{Dsin\left(\frac{\epsilon }{2}\right)}}}.$$

(22)

It is noteworthy that a later study [62] employs a different constant when calculating this coefficient.

$$k_1=1+0.0013{\displaystyle \frac{60\epsilon }{w}}\sqrt{{\displaystyle \frac{j}{Dsin\left(\frac{\epsilon }{2}\right)}}}.$$

(23)

The next step is to determine the dimensionless angular function z(β,μ,θ) from the material flow rate equation, calculated with the following expression:

$$Q={\displaystyle \frac{k_1\pi }{45}}wz{(2R-h)}^{3/2}{h}^{3/2}.$$

(24)

By calculating z and knowing the dynamic angle of repose θ and the kiln inclination angle β, μ can be determined. There are two methods for this. The first is using the following equation:

$$z=\mathrm{c}\mathrm{o}\mathrm{s}(\mu )\sqrt{{\displaystyle \frac{{sin}^2\left(\theta \right)({cos}^2\left(\mu \right)\left({sin}^2\left(\theta \right)+2\sin \left(\beta \right)\sin \left(\mu \right)\cos \left(\theta \right)\right)-C-{sin}^2\left(\mu \right)E}{{sin}^2\left(\theta \right)\left({sin}^2\left(\beta \right)-2C\right)+E^2}}-1}.$$

(25)

$$C={sin}^2\left(\beta \right){cos}^2\left(\mu \right)+{sin}^2\left(\mu \right){cos}^2\left(\beta \right).$$

(26)

$$E={sin}^2\left(\beta \right){cos}^2\left(\mu \right)-{sin}^2\left(\mu \right){cos}^2\left(\beta \right).$$

(27)

The second approach uses the nomograms provided in studies [24] and [62]. After μ is obtained at the initial cross-section, the calculation advances in small steps toward the kiln’s cold end; study [24] used an angular increment of 0°15′. With μ known for the next step, the updated value of z is determined from Equation (25). The corresponding axial distance Δx for that angular change is then calculated. If the segment contains no internal devices, k₂ is set to 1; if it has no conical section, y is set to 0. Study [24] also provides tabulated k₂ values for particular chain-curtain types and for segments equipped with cascading blades.

$$\begin{array}{c}\left(2R_{x_0}\pm \left(x_1+x_2+...+x_n\right)tg\left(y\right)-h_{x_0}\pm x_{1}tg\left({\mu }_{x_1}\right)\pm x_{1}tg\left({\mu }_{x_1}\right)\pm ...\pm x_{n}tg\left({\mu }_{x_n}\right)\right)·(h_{x_1}\pm \\ \left(x_1+x_2+...+x_n\right)tg\left(y\right)\mp x_{1}tg\left({\mu }_{x_1}\right)\mp x_{1}tg\left({\mu }_{x_1}\right)\mp ...\mp x_{n}tg\left({\mu }_{x_n}\right))=\sqrt[3]{{\left({\displaystyle \frac{45Q}{k_1{k}_2\pi w{z}_{x_n}}}\right)}^2}.\end{array}$$

(28)

Each term carrying the subscript x_i refers to the current axial step. The quantity tg (y) is positive when the conical section widens and negative when it narrows. When the bed height rises, the angle μ is positive; therefore, the terms with tg (μ) inside the first bracket are positive, while those in the second bracket are negative [24].

The influence of internal heat-exchange devices on axial-motion parameters will be revisited in the section devoted to axial-velocity calculations.

In the present equation, the only unknown is the axial increment x for the current step. Once that value is obtained, the change in bed height for the step is determined directly from x and μ.

$$h_{x_n}=h_{x_0}{\pm x}_{1}tg\left({\mu }_{x_1}\right)\pm ...\pm x_{n}tg\left({\mu }_{x_n}\right).$$

(29)

The remaining steps are handled in the same way, progressing through every section from the hot end to the cold end of the kiln. A full computer algorithm for the procedure is given in [63]. Khodorov and Aronzon later simplified the calculations with nomograms [64], and study [65] introduced an analytical technique for drums in the rolling regime that avoids experimental measurement of the dynamic angle of repose. The same basic approach appears in [62], although the specific formulas for mean velocity and volumetric flow rate are different.

$$\mathrm{Q}=h_{al}{l}_{al}z{u}_{al}.$$

(30)

All previously unmentioned parameters, h_al, l_al, and u_al, can be obtained by simultaneously solving the following equations:

$$u_{al}=\sqrt{{\displaystyle \frac{l_{al}j}{6}}}.$$

(31)

$$l_{al}={\left(D^2-{\left(2\left({\displaystyle \frac{D}{2}}-h\right)+h_{al}\right)}^2\right)}^{0.5}.$$

(32)

$$u_{pr}={\displaystyle \frac{\pi w}{60}}\left(D-h_{pr}\right).$$

(33)

$$h_{pr}=h-h_{al}.$$

(34)

$$h_{pr}{u}_{pr}=h_{al}{u}_{al}.$$

(35)

In all other respects, the calculation procedure is completely identical.

4.2. Axial Velocity of Material

Among the sources that compare mathematical models for axial material velocity, three stand out: study [36], the more recent study [19], and the comprehensive review in book [24]. The tables from these references are brought together here and extended with additional studies.

As with experiments, the first study in international literature dedicated to determining the axial velocity of material is study [12].

$$u=N{\displaystyle \frac{Dn\beta }{1.77\sqrt{\theta }}}.$$

(36)

However, as study [24] points out, this formula ignores both dynamic operating conditions and the effect of bed filling on velocity, and it does not describe material flow in horizontal drums. Its scope is further restricted by the empirical coefficient N, which must be chosen from three separate correlations.

If the thickness of the uniform material layer is less than the height of the retaining ring,

$$N=e^{\left(\left({\displaystyle \frac{0.12L}{D_{dam}}}-3.86\right)lg\left({\displaystyle \frac{2.5V}{wv}}\right)+e^{(2.3-{\displaystyle \frac{0.32L}{D}})}-1\right){\displaystyle \frac{h}{D-2h}}}.$$

(37)

If the thickness of the uniform material layer is greater than the height of the retaining ring and the ring is not installed at the discharge end of the drum,

$$N={\displaystyle \frac{1}{\left(0.34-\frac{0.64D}{D_{dam}}\right)\frac{2.5V}{wv}+1.16}}.$$

(38)

If the thickness of the uniform material layer is greater than the height of the retaining ring and the ring is not installed at the discharge end of the drum,

$$N={\displaystyle \frac{\frac{D}{D_{dam}}-(0.8-0.3\frac{L}{D})}{0.3\frac{L}{D}+0.195}}.$$

(39)

Among the earliest studies, [66] should also be noted, as referenced in [24] and [19]. In this study, the velocity is presented as a formula for the cross-sectional average material velocity for a uniform layer.

$$u={\displaystyle \frac{\pi }{45}}Dn{\displaystyle \frac{{sin}^3\left(\frac{\epsilon }{2}\right)}{\epsilon -\mathrm{s}\mathrm{i}\mathrm{n}\left(\epsilon \right)}}\times {\displaystyle \frac{\beta +\lambda \mathrm{c}\mathrm{o}\mathrm{s}\left(\alpha \right)}{\sqrt{{sin}^2\left(\theta \right)-{sin}^2\left(\beta \right)}}}.$$

(40)

Bayard [67] extended this topic by giving a theoretical rationale for the coefficient N, assuming that the height of the non-uniform layer varies linearly. Study [24], however, notes that this assumption is unrealistic and therefore supplies two alternative formulas for axial velocity.

Outside the area influenced by the retaining ring,

$$u={\displaystyle \frac{Dni}{0.308(24+\theta )}}.$$

(41)

Within the area influenced by the retaining ring,

$$u={\displaystyle \frac{\left(D-h-0.5h_{bf}\right)wi2h_{bf}}{0.308\left(24+\theta \right)\left(\sqrt{h_{bf}h+h}\right)}}.$$

(42)

Another early study is article [68]. In its original form, the formula is presented differently, as it is expressed through the particle path in the kiln bed. This representation of the formula was derived in study [24], enabling the author to directly compare their own Model (44) with Model (43) from article [68].

$$u={\displaystyle \frac{\pi }{45}}Dn{\displaystyle \frac{{sin}^3\left(\frac{\epsilon }{2}\right)}{\epsilon -\mathrm{s}\mathrm{i}\mathrm{n}\left(\epsilon \right)}}\times {\displaystyle \frac{\beta +\lambda \mathrm{c}\mathrm{o}\mathrm{s}\left(\alpha \right)}{\mathrm{s}\mathrm{i}\mathrm{n}\left(\alpha \right)}}.$$

(43)

$$u=k_1{k}_2{\displaystyle \frac{\pi }{45}}zDn\left({\displaystyle \frac{{sin}^3\left(\frac{\epsilon }{2}\right)}{\epsilon -\mathrm{s}\mathrm{i}\mathrm{n}\left(\epsilon \right)}}\right).$$

(44)

Directly comparing both equations, the author of [24] notes that they differ in the method of calculating z, as well as in the fact that Equation (43) does not include coefficients accounting for dynamics and the presence of internal heat exchange devices.

In the doctoral dissertation of V.D. Ryvkin [53], a formula was also proposed for calculating the axial velocity of particle motion (Equation (45)).

$$u=8.37{\displaystyle \frac{n}{\chi \phi }}Fsin\left(\kappa \right)tg(\delta ).$$

(45)

$$F=0.5Dsin({\displaystyle \frac{\phi }{2}}).$$

(46)

With regard to heat-exchange devices, the most comprehensive mathematical treatment of material motion in the chain zone is found in the dissertation of V. Y. Abramov [40]. The velocity equation was derived from experiments performed under industrial conditions, and both the derivation and the supporting data are documented in detail in that source. The final expression for the velocity of plastic feed in the chain-curtain zone is as follows:

$$u=k_{3}Dw{tg}^{0.7}(Y){\left({\displaystyle \frac{F_c}{F_{ca}}}\right)}^{-0.3}.$$

(47)

The parameter k₃ reflects the properties of the material. For nepheline-limestone feed, it was assigned a value of 0.155 in study [40].

Study [29] compares the mathematical model from [12] with three more recent formulas [60,61] and ref. [36] using a laboratory setup.

$$u=5.328Dftan(\sigma )\left({\displaystyle \frac{1}{\sin \left(\delta \right)}}-\tan (\lambda ){\displaystyle \frac{\cot \left(\delta \right)}{\tan \left(\sigma \right)}}\right){\left({\displaystyle \frac{H}{D}}-{\displaystyle \frac{H^2}{D^2}}\right)}^{1.5}\left({\displaystyle \frac{1}{\phi -\sin \left(\phi \right)}}\right).$$

(48)

$$u=0.251Df(\delta +\lambda ){\displaystyle \frac{1}{Z^{0.25}}}\left({\displaystyle \frac{1}{\sin \left(\sigma \right)}}\right).$$

(49)

$$u=KDf\delta {\left({\displaystyle \frac{D^2/4}{h^2\cos \left(\sigma \right)+2h{\left(Dh-h^2\right)}^{0.5}\sin (\sigma )}}\times {\displaystyle \frac{\phi +2\tan \left(\phi /2\right)}{1-\cos (\phi /2}}\right)}^e.$$

(50)

Based on the results obtained, it can be concluded that the best prediction of axial velocity was provided by the models [61] and [36].

Later, the equation from study [36] was modified in article [69] and took the following form:

$$u=\left({\displaystyle \frac{0.09R{Q}_h^{0.946}{\delta }^{0.027}{f}^{0.045}}{h^{0.032}}}\right)\left({\displaystyle \frac{R^2}{{\left(h^2\cos \left(\sigma \right)+2h(2hr-h^2\right)}^{0.5}\sin \left(\sigma \right)}}\right){\left({\displaystyle \frac{\epsilon +2tg\left(\frac{\epsilon }{2}\right)}{1-cos\left(\frac{\epsilon }{2}\right)}}\right)}^{0.8}.$$

(51)

In study [36], mathematical models of axial velocity published in international sources prior to 1990 were reviewed and unified. Many of these models had already been compared with those in study [24]. For instance, study [36] also examined the equations of Friedman [70] (Equation (50)), Perry [71] (Equation (51)), Zablotny [72] (Equation (54)), and Heiligenstaedt [73] (Equation (55)).

$$u=1.585R{f}^{0.9}\phi .$$

(52)

$$u=1.675Rftg\left(\phi \right).$$

(53)

$$u=0.735R{f}^{0.85}{\phi }^{0.85}{\displaystyle \frac{1}{{\sigma }^{0.85}}}.$$

(54)

$$u=Rfsin(\phi ){\displaystyle \frac{1}{\sin \left(\sigma \right)}}\sqrt{1-{\displaystyle \frac{{sin}^2\left(\phi \right)}{{sin}^2\left(\sigma \right)}}}.$$

(55)

Models (52)–(55) demonstrated worse results compared to [61] and [36].

Similar approaches to calculating the axial velocity of material in rotary kilns, as previously discussed, were proposed in books [74,75,76].

$$u=Dn{\displaystyle \frac{{sin}^3\left(\frac{\epsilon }{2}\right)\sin \left(\beta \right)}{\sin \left(\theta \right)90\left(\frac{\epsilon }{2\pi }-\frac{sin\left(\frac{\epsilon }{2}\right)}{2}\right)}}.$$

(56)

$$u={\displaystyle \frac{200Dni}{\theta +24}}.$$

(57)

$$u=0.061{\displaystyle \frac{Dw\beta }{\theta }}.$$

(58)

The equations discussed so far are largely founded on similar principles, yet the literature also contains several distinct approaches. One such example is article [37], which, using data from industrial kilns, establishes an empirical link between feed-material velocity and its enthalpy change. This correlation makes it possible to quantify how physicochemical transformations in the material influence its axial velocity.

$$u=2.3Dn{\left({\displaystyle \frac{\beta }{\theta }}\right)}^{0.85}\left({\displaystyle \frac{H_1}{H_2}}\right).$$

(59)

Equations in the form of a polynomial dependency for determining material velocity in various zones of a cement kiln were presented in study [77] and later in the book [19]. This approach makes it possible to evaluate the influence of technological parameters of the firing process on the material motion within the kiln.

$$U_{\mathrm{c}.z}=0.0058T_{e.g}+0.0062P+4.79w-5.13.$$

(60)

$$U_{d.z}=0.335T_{e.g}-0.006T_{c.z}+0.0287P+8.36w-0.165L_c-5.42.$$

(61)

$$U_{s.z}=38.7-0.02T_{e.g}-0.0045T_{c.z}-0.0156P-0.153G_c-0.0078T_m-0.0057T_a.$$

(62)

$$U_{max}=0.18T_{e.g}+0.196P-0.027S_c+0.368G_c-0.0363T_a+24.23.$$

(63)

4.3. Mean Residence Time

The mean residence time of material is inextricably linked to its axial velocity. To determine the residence time from axial velocity, it is sufficient to divide the distance traveled by the material by its axial velocity.

$$MRT={\displaystyle \frac{L}{u}}.$$

(64)

Or, in integral form,

$$\mathrm{M}RT={\int }_0^L{\displaystyle \frac{dx}{u\left(x\right)}}.$$

(65)

$$\mathrm{M}RT={\int }_0^L{\displaystyle \frac{S\left(x\right)}{Q\left(x\right)}}dx.$$

(66)

For example, the residence time of the material can be calculated using the formula from studies [12,68].

$$MRT=1.77{\displaystyle \frac{L}{D}}{\displaystyle \frac{\sqrt{\sigma }}{w\delta N}}.$$

(67)

$$MRT={\displaystyle \frac{Lsin\left(\theta \right)}{2Rwi}}.$$

(68)

In the model presented in [24], the velocity is computed separately for each kiln segment because material speed varies from zone to zone as a function of local geometry and the temperature-driven physicochemical changes in the feed. Consequently, the velocity equations discussed earlier are omitted from this section. Instead, we introduce the correlation proposed in book [78], which has not yet been considered.

$$MRT={\displaystyle \frac{0.23L}{\delta w^{0.9}D}}.$$

(69)

And later refined in study [50].

$$MRT={\displaystyle \frac{0.33L}{\delta w^{0.9}D}}.$$

(70)

In the doctoral dissertation of V.D. Ryvkin [53], a formula was also proposed for calculating the mean residence time of particles in the kiln (Equation (71)), along with a coefficient accounting for the material properties (Equation (72)).

$$MRT=k_{14}{\displaystyle \frac{\epsilon L}{\mathit{sin}\left(\frac{\phi }{2}\right)Dntg\left(\delta \right)}}.$$

(71)

$$k_{14}=0.239{\displaystyle \frac{\chi }{sin\left(\kappa \right)}}.$$

(72)

The constant in the equation (x4) was obtained experimentally on industrial alumina kilns [53]. In the same work, simplified relations were proposed for cases where it is not possible to determine the bed height, the filling degree, and, accordingly, the central angle.

$$MRT=860+5.5·{10}^{-5}{\displaystyle \frac{L}{D}}nQtg(\delta ).$$

(73)

The residence time for kilns with a small inclination angle (0.5–1°) is specifically investigated in study [51], as the examples from the literature reviewed did not provide sufficiently accurate results in this range. The study included a comparison with works [12].

$$MRT=1.53{\displaystyle \frac{L\sqrt{\theta }}{Dw}}.$$

(74)

The literature also presents numerous power-law dependencies for calculating the mean residence time (MRT) of material in the kiln. One of the earliest works in this area is [26].

$$MRT=K{\displaystyle \frac{L^3}{Q_m}}{\left({\displaystyle \frac{\theta }{\beta }}\right)}^{1.054}{\left({\displaystyle \frac{Q_m}{L^{3}w}}\right)}^{0.981}.$$

(75)

The author later extended this relationship to include kiln-geometry parameters in study [27]. That expanded form has since been widely discussed; for example, study [52] proposes revised coefficient values. Even in its updated version, however, the model does not account for temperature-dependent changes in the material.

$$MRT={\displaystyle \frac{0.1026L^3}{Q_m}}{\left({\displaystyle \frac{\sigma }{\delta }}\right)}^{1.054}{\left({\displaystyle \frac{Q_m}{L^{3}w}}\right)}^{0.981}{\left({\displaystyle \frac{L}{D}}\right)}^{1.1}.$$

(76)

$$MRT={\displaystyle \frac{0.38L^3}{Q_m}}{\left({\displaystyle \frac{\sigma }{\delta }}\right)}^{1.054}{\left({\displaystyle \frac{Q_m}{L^{3}w}}\right)}^{0.981}{\left({\displaystyle \frac{L}{D}}\right)}^{1.1}.$$

(77)

Similarly, a series of dependencies for inclined cylinders was proposed in studies [28,69,79], corresponding to Equations (73)–(75).

$$MRT=1.4215h_0^{0.241}/{\delta }^{0.986}{f}^{0.875}{Q}_h^{0.059}{d}_p^{0.05}.$$

(78)

$$MRT=1315.2h_0^{0.24}/{\beta }^{1.02}{w}^{0.88}{M}_h^{0.072}.$$

(79)

$$MRT=k_5{\displaystyle \frac{Lsin\left(\sigma \right)}{2\pi Rn\left(tg\left(\delta \right)+\cos \left(\sigma \right)tg\left(\alpha \right)\right){cos}^2\left(\alpha \right)}}.$$

(80)

For horizontal cylinders, the following dependency was presented in study [17]:

$$MRT=0.91{\left({\displaystyle \frac{L}{D}}\right)}^2{\left({\displaystyle \frac{D}{h\left(0\right)}}\right)}^{0.6}{\displaystyle \frac{tg\left(\theta \right)}{w}}.$$

(81)

Article [14] continues the work of study [27] by examining how internal constrictions—used to simulate coating build-up—affect material flow in a rotary kiln. It is the only publication in the surveyed literature that attempts to quantify the coating’s influence on axial transport. Although the paper offers no direct formula for axial velocity, it does introduce a method for calculating an effective angle of repose that accounts for the constriction.

Mean residence time in kilns equipped with lifters is treated separately in study [80].

$$MRT=k_6\sqrt{gL}{\left({\displaystyle \frac{w^{2}D}{g}}\right)}^{k_7}{\left({\displaystyle \frac{D_{ex}}{D}}\right)}^{k_8}{\left({\displaystyle \frac{\theta }{\beta }}\right)}^{k_9}{\left({\displaystyle \frac{M_h}{\rho D^2\sqrt{gL}}}\right)}^{k10}{\left({\displaystyle \frac{4S_{lift}}{\pi D^2}}\right)}^{k11}{\left({\displaystyle \frac{\rho }{{\rho }_{tapped}}}\right)}^{k12}{\left({\displaystyle \frac{L}{D}}\right)}^{k13}.$$

(82)

Studies [19,77] adopted a different strategy: they related the mean residence time in an operating cement kiln directly to the process variables, expressing this relationship as a polynomial function.

$$MRT=337.1-0.217T_{e.g}-0.706P-294w-1.92M_m-0.0363T_m-0.00358T_a+24.23.$$

(83)

4.4. Modern Approaches to the Modeling of Rotary Kilns

A vast number of studies have been devoted to the modeling of rotary kilns, covering various aspects of the process. This article focuses primarily on mathematical models of axial material movement, which are less closely related to widely used approaches such as DEM and CFD, commonly applied for describing other subprocesses within the kiln. However, a comprehensive picture of modern research would be incomplete without addressing numerical modeling methods and machine learning techniques, which are increasingly used to simulate rotary kiln processes.

4.4.1. Numerical Modeling Methods

Computational Fluid Dynamics (CFD) offers powerful tools for modeling heat transfer and gas flow in rotary kilns, but its use for predicting the axial transport of bulk solids is constrained by several factors. First, CFD was designed for continuous media; to capture granular flows, it must be coupled with additional models—typically DEM or two-phase Eulerian formulations. Second, three-dimensional, long-residence-time simulations are computationally intensive. Third, extensive experimental data are required for calibration: turbulence parameters, heat-transfer coefficients, and interphase drag are often set empirically. Most CFD studies also simplify the problem by assuming monodisperse particles, constant physical properties, and perfectly smooth walls, which limits the accurate treatment of polydispersity, segregation, and phase changes. Geometric details such as chain curtains, lifters, and internal baffles—features that strongly influence axial motion—are seldom represented in full. Consequently, CFD alone cannot yet provide reliable axial-transport predictions without supplementary models and validation experiments.

Even so, several investigations have demonstrated successful CFD applications. In [81], a 3-D two-fluid Eulerian model with the kinetic theory of granular flow (KTGF) reproduces both the active and passive regions of a bed in rolling motion and matches experimental velocity and density profiles while evaluating kiln inclination, rotation speed, and fill level. Study [82] integrates methane combustion, bed movement (via the Discrete Phase Model with a three-hour residence time), and CaCO₃ calcination in a 76 m lime kiln; the model, validated against industrial data, quantifies energy losses, secondary air leakage, and the influence of moisture, particle size, and air ratio. In [83], the gas phase (turbulent coal combustion) and the solid bed (treated as a viscous pseudo-fluid undergoing sintering reactions) are solved in separate, coupled domains; the approach reproduces industrial temperature fields and clinker-formation profiles and assesses how axial versus swirling air ratios affect efficiency.

The Discrete Element Method (DEM) provides detailed insight into particle micro-dynamics, yet full-length kiln simulations remain limited by computational cost. Industrial kilns contain millions of particles, and the collision-resolution time step is extremely small, restricting DEM runs to real-time intervals of only a few seconds—far shorter than the minutes-to-hours time scales of axial transport. Nevertheless, pioneering studies are emerging. In [84], an axial-dispersion model for biomass pyrolysis links the Peclet number (Pe) to residence-time distributions (RTDs) validated against DEM data; for Pe < 10, dispersion markedly alters temperature and conversion, whereas for Pe > 50, its effect is negligible. More than 13,000 simulations show that in 30% of cases, accounting for RTD changes product yields by over 2%, underscoring the need to include axial dispersion when Pe is low or when lifters and mixers are present.

Study [85] couples DEM with a population-balance model (PBM) to simulate layering granulation of iron-ore fines on coarse nuclei. A hybrid aggregation scheme, combining one-to-one and one-to-many coalescence, enlarges the permissible time step and lowers computing cost—an approach that could benefit axial-transport problems involving size and shape evolution. Two-way CFD–DEM couplings can also capture bed heat transfer. In [86], the authors resolve conduction through particles and the kiln wall by solving the 3-D heat equation and computing thermal fluxes from actual contact areas; validation against a heated rotating drum and a fixed packed bed confirms the method.

A particularly promising hierarchical framework appears in [87]. DEM is first used to extract rheological properties—viscosity, friction, and shear-rate dependence—which are then fed into a 3-D CFD model of the entire kiln. A DEM-calibrated heat-transfer sub-model accounts for conduction and radiation among particles and through the wall. Applied to a 15 m kiln with variable fill, speed, and an outlet dam ring, the framework accurately predicts axial profiles of velocity, temperature, and viscosity at reasonable computational cost, making it attractive for engineering optimization.

4.4.2. Neural Networks and Machine Learning Methods

Neural networks have recently become popular for modeling complex physical processes—including heat and mass transfer, granular-media dynamics, and reactive transformations. For rotary kilns, axial quantities such as mean residence time, temperature profiles, and conversion are governed by many coupled factors (geometry, rotation, feed composition, boundary conditions, and so on). Data-driven models can capture these nonlinear dependencies and predict axial profiles directly from experimental or numerically generated data without solving partial differential equations.

Study [59] developed four such models—support-vector regression (SVR), a general regression neural network (GRNN), a radial-basis-function network (RBFNN), and a multilayer perceptron (MLP)—to predict the axial bed-depth profile. Trained on an extensive experimental set (1106 points) and benchmarked against analytical solutions, SVR performed best, with a mean error of 1.72% and R² = 0.9981; GRNN was nearly as accurate and more resistant to outliers. The MLP and RBFNN, by contrast, suffered from overfitting and weak generalization. The authors conclude that machine-learning surrogates can replace time-consuming iterative calculations and deliver rapid, accurate predictions for kiln optimization and automatic control.

Neural models trained on large numerical datasets can also approximate simulation outputs and thus shorten computation times. In [88], three methods—an artificial neural network (ANN), extremely randomized trees (ERT), and a particle-swarm-optimized SVR (PSO-SVR)—were trained on 121 DEM simulations of rod-like particles, with four operating parameters varied. All three predicted the steady-state mixing index (SMI) with high accuracy (R² ≥ 0.94); PSO-SVR performed best for mixing time (R² = 0.90). A multisphere particle description and a sub-domain SMI allowed the models to account for particle shape, density, and size asymmetry. The study shows that neural and regression techniques can stand in for costly DEM runs when analyzing axial transport and mixing in complex drums.

5. Conclusions

This review brings together virtually all laboratory and industrial measurements that describe the axial movement of bulk solids in rotary and inclined cylinders of many different designs and evaluates the vast majority of mathematical models proposed to predict this process. It considers, in turn, the classical equations derived within the Saeman framework, their later modifications, and modern computational methods based on CFD, DEM, coupled DEM–PBM, and CFD–DEM schemes, as well as neural network surrogates. In the international literature, the formulations introduced by Kramers and Croockewit are now cited most frequently, yet the earlier Soviet work by Khodorov is no less valuable: it rests on carefully documented full-scale experiments, contains a section-by-section algorithm that mirrors the physical changes in the kiln, and provides a worked example for which no true equivalent yet exists. The approach proposed by Besediny et al. broadens the conceptual boundaries of cement kiln modeling and deserves closer attention.

Despite the extensive body of existing research, clear gaps remain. The first concerns rings and accretions. These deposits change the internal diameter, reorganize both gas and solid flows, and ultimately govern residence time and backflow, yet systematic investigations are rare. None of the mathematical models examined here explicitly accounts for ring growth, and the absence of such a framework prevents real-time infrared thermography—already capable of detecting rings and estimating their thickness—from being incorporated into transport calculations or control strategies.

The second limitation is the narrow industrial database. Most field studies have been carried out on cement kilns, whereas rotary units are equally indispensable in alumina calcination, vanadium production, direct-reduced iron manufacture, and many other processes. Without comparable data from these sectors, it is difficult to separate equipment-specific effects from truly universal transport behavior. Moreover, only a few studies document how high temperature and the accompanying physicochemical transformations alter velocities, bed depth and residence time.

Numerical modeling offers another path forward. At present, DEM is used only sparingly for axial transport simulations because a faithful computation must track every particle along the full kiln length for many revolutions, which is computationally intensive. As computing power increases and algorithms improve, this barrier will diminish, and coupled DEM–PBM schemes—already applied to bidisperse segregation—appear particularly promising for realistic industrial feedstocks.

Continued development of these experimental and numerical tools is vital for predictive control strategies and digital twin concepts for rotary tubular kilns. Until a model can adjust its parameters in real time, assimilate thermal imaging data on ring formation and reproduce axial profiles with sufficient accuracy, full model-based kiln automation will remain elusive. Bridging this gap—through the integration of refined transport equations, broader industrial measurements, and high-resolution simulations—stands out as one of the most pressing challenges in the field.