Critical Review of C_p Calculation Within the Fluidized Bed of Cement Rotary Kilns

Abstract

One thermodynamic parameter that is crucial to heat transport within the fluidized bed inside the rotary kiln, during clinker production, is the specific heat capacity. The particular parameter is often considered constant in the open literature, while, in reality, it strongly depends on the fluidized bed’s temperature and composition, considering that the temperature inside the kiln ranges from approx. 800 K up to 2000 K. For the current study, a mixing rule reported in the literature was applied in order to calculate the C_p of the fluidized bed, utilizing temperature and composition profiles available in the literature. An in-house code was developed for the comparison of the literature-reported C_ps and those resulting from the mixing rule. It was discovered that the C_p of the fluidized bed had a proportional increase with the increase in the temperature along the length of the kiln. The deviation between the two values (calculated and literature) is relatively small in some cases, whereas, in others, it is quite significant, ranging from 1.56% to 52.49%, thus making the adoption of the temperature-dependence of C_p necessary. Establishing a more accurate relation for the specific heat capacity leads to a better energy balance inside the kiln, which, along with other improvements, can lead to a decrease in the energy consumed and a significant reduction in greenhouse gas emissions.

1. Introduction

The cement industry is one of the largest contributors to greenhouse gas emissions, while simultaneously being a very energy-intensive industry. Currently, the average energy consumption in more advanced rotary kilns (depending on the kiln type) is 3.5 GJ per ton clinker produced [1], a number that has improved due to the upgrades in kiln design and operational protocols that have been made over the years. In order to improve the kiln operation, it is important to study different parameters that can enhance energy efficiency and simultaneously preserve/enhance the quality of the final product.

Out of the total energy consumption in the cement industry, thermal energy accounts for approximately 80% of the primary energy consumption, with the electrical energy making up the remaining 20%. Depending on the type of the kiln (i.e., wet, dry, semi-dry), the above numbers can vary significantly [2] due to different percentages of moisture of the raw meal that needs to be evaporated. Typically, in the wet process kiln, the raw meal contains 36% moisture, while in the corresponding dry type, that percentage is only 0.5% [3]. In particular, the clinker production requires approximately 4 GJ per ton of clinker produced; however, depending on the kiln type, that amount can be reduced to 3.4 GJ/ton for the dry kiln, while it can increase up to 5.29 GJ/ton for the wet type [1]. For each ton of cement produced, an equivalent amount of gas emissions is released into the atmosphere [4]. In some countries with advanced plants for cement production, the energy consumption drops to 2.95 GJ/ton, whereas, on the other hand, the consumption can increase up to 5 GJ/ton if the cement plant is not well equipped [5,6,7]. In the US, the energy intensity of the cement industry improved from 4.45 GJ/ton in 1999 to 3.65 GJ/ton in 2009. Similar trends are followed by the data from the Cement Association of Canada, with the respective numbers being 3.91 GJ/ton in 2002 and 3.48 GJ/ton in 2010 [8,9,10]. In the work of Oda et al. [1], based on the type of the kiln and also on the existence of preheaters and pre-calciners, it was observed that the consumed thermal energy is proportional to the lower heating value. The highest consumption was found for the case of the wet-type kiln, while the lowest consumption belonged to the dry-type kiln coupled with 4 to 6 cyclone preheaters and a pre-calciner.

Figure 1 shows the time evolution of the thermal intensity of the clinker production in Europe and in the US, along with future projections based on indicators in the Reference Technology Scenario (RTS) and the roadmap vision (2DS). Similar information is also provided in Table 1 and Table 2. It is evident that during recent years, a significant reduction has occurred, which is projected to be continued in the future as well.

The rotary kiln sector is considered to be the clinker production process’s most polluting sector [2,3], as the emissions are carbon dioxide, nitrogen oxide, chlorofluorocarbons and methane (greenhouse effect gases) [4,5]. A well-designed rotary kiln is essential as it can reduce emissions and improve energy efficiency while preserving the final product quality. The rotary kiln under investigation in the current study is of the dry type, so its energy consumption is already lower than that of the corresponding wet-type kiln. Nevertheless, kiln parameters, along with those in the energy balance equation, must be optimized in order to reduce greenhouse-gas emissions and energy consumption. The fluidized bed’s specific heat capacity, C_p, is a parameter that can have a significant effect on the total energy consumption in the rotary kiln. Also, another factor that can be improved is the rotational speed of the kiln, as previous studies have shown that it can increase the convection heat transfer coefficient [6,7,8].

Table 1. Energy consumption in different cement kilns in chronological order.

Year	Source	Country	Energy Per Type of Process [GJ/ton Clinker]
			Wet	Dry
1994	[9]	U.S.	366 PJ total	-
2002	[10,11,12]	Global	5.3–7.1	3.06
2006	[13]	Global	6.38	3.55
2006	[13,14]	Global	-	3.42
2007	[5]	U.S.	6.5	5.3
2011, 2017	[15,16]	Asia	5.68–6.28	4.6
2014	[17]	Global	-	3.5
2020	[3]	Iran	-	3.4
2022	[18]	Global	5.29	3.4

Table 1. Energy consumption in different cement kilns in chronological order.

Year	Source	Country	Energy Per Type of Process [GJ/ton Clinker]
			Wet	Dry
1994	[9]	U.S.	366 PJ total	-
2002	[10,11,12]	Global	5.3–7.1	3.06
2006	[13]	Global	6.38	3.55
2006	[13,14]	Global	-	3.42
2007	[5]	U.S.	6.5	5.3
2011, 2017	[15,16]	Asia	5.68–6.28	4.6
2014	[17]	Global	-	3.5
2020	[3]	Iran	-	3.4
2022	[18]	Global	5.29	3.4

Table 2. Specific thermal energy by technology [19] (European Commission). n.a.: not available.

Technology	Specific Thermal Energy Demand
	[MJ/ton Clinker]	Weighted Average [MJ/ton Clinker]
Dry kiln with preheater and pre-calciner	3000 < 4000	3515
Dry kiln with preheater without pre-calciner	3100 to 4200	3700
Semi-wet/semi-dry process	3300 to 5400	3918
Dry kiln without preheater (long dry kiln)	Up to 5000	3570
Wet process	5000 to 6400	5512
Shaft kilns	3100 to 6500 and higher	n.a.

Table 2. Specific thermal energy by technology [19] (European Commission). n.a.: not available.

Technology	Specific Thermal Energy Demand
	[MJ/ton Clinker]	Weighted Average [MJ/ton Clinker]
Dry kiln with preheater and pre-calciner	3000 < 4000	3515
Dry kiln with preheater without pre-calciner	3100 to 4200	3700
Semi-wet/semi-dry process	3300 to 5400	3918
Dry kiln without preheater (long dry kiln)	Up to 5000	3570
Wet process	5000 to 6400	5512
Shaft kilns	3100 to 6500 and higher	n.a.

Figure 1. Clinker thermal energy intensity in EU27 and at global level (MJ/ton clinker) [19] along with key indicators in the RTS (Reference Technology Scenario) and the roadmap vision (2DS).

Figure 1. Clinker thermal energy intensity in EU27 and at global level (MJ/ton clinker) [19] along with key indicators in the RTS (Reference Technology Scenario) and the roadmap vision (2DS).

During the current study, it was observed that a common assumption encountered in most published studies considering the modeling of rotary cement kilns was that the specific heat capacity of the fluidized bed was kept constant throughout the process. The particular assumption was ubiquitous, even though the temperatures inside the rotary kiln can range from approximately 800 K in the kiln inlet to almost 2000 K near the burner. Besides the chemical reactions and their corresponding kinetic parameters, the process is also affected by additional factors such as the heat transfer between all the phases (i.e., fluidized bed, gas, and solid kiln), the wall of the kiln, and the technical features of the kiln itself. While testing different case scenarios from the literature, via “in-house” codes, in order to find the optimal parameters, it was identified that the use of variable values of C_p has an impact on the concentration profiles of the components as a function of the kiln length. Additional research was conducted on this matter, as most of the literature sources suggested the use of a constant value for the bed heat capacity, thus ignoring the variability of C_p.

Therefore, the main objective/motivation of the current study is to examine the validity of the assumption of using constant values for C_p along the rotary kiln (i.e., independent of the temperature profile). To this purpose, the literature-reported temperature and compositional profiles along the length of the rotary kiln are utilized and are coupled with a mixing rule for C_p. Calculations (accounting for temperature and compositional variations along the kiln length) are performed in order to quantify the difference between variable and constant C_ps, and significant deviations are identified.

2. Methodology

The raw material in the kiln inlet consists of a mixture of clay and finely ground limestone that is processed inside the kiln to produce clinker. The main compounds of clinker minerals are the dicalcium silicate [2CaO∙SiO₂ (Belite): denoted as C₂S], tricalcium silicate [3CaO∙SiO₂ (Alite): denoted as C₃S], tricalcium aluminate [3CaO∙Al₂O₃ (Aluminate): denoted as C₃A], and tetracalicium aluminoferrite [4CaO∙Al₂O₃∙Fe₂O₃ (Ferrite): denoted as C₄AF], which are also the main products of the reactions taking place in the rotary kiln [20,21]. High temperatures are necessary for the initiation of those reactions. Certain assumptions are often made for problem simplification. Typical assumptions reported in the literature are summarized below:

• The system is assumed to be in a steady state with constant flow [22].
• The model is considered one-dimensional (1D).
• The fluidized bed height remains constant along the entire length of the kiln [23].
• The fluidized bed is assumed to move transversely within the cement rotary kiln, allowing it to be treated as a plug flow reactor (PFR) [23,24].
• The decomposition of calcium carbonate is assumed to take place primarily in the pre-calciner, so the CO₂ source term of the energy equation can be ignored for the case of the rotary kiln.
• The chemical reactions are restricted to five main ones (shown in Table 3) with corresponding kinetics [25,26,27,28] whose reaction rate is determined by an Arrhenius-type law with fixed values in the enthalpies of the reactions, the latent heat of the solid, activation energies and pre-exponential factors remain constant.
• The rotary kiln is a dry-type kiln; therefore, heat transfer from H₂O release due to low moisture content in the raw material is ignored [29,30].
• The emission factors are assumed to remain constant over the entire length of the rotary kiln for the gas phase, the fluidized bed and the inner wall.

A schematic of the cross-sectional area of the rotary kiln is shown in Figure 2.

In the current study, the reactions occurring in the fluidized bed were limited to 5 main reactions with their respective temperature range indicated in Table 3. In order to calculate the C_p of the fluidized bed, it is necessary (i) to find an expression that describes the dependence of C_p on temperature for each one of the nine primary chemical components that participate in the chemical reactions shown in Table 3, (ii) to obtain an expression (i.e., mixing rule) for the composite fluidized bed, and (iii) to take into account the concentration changes in each component along the kiln length.

Table 3. Main reactions of clinkerization inside the rotary kiln [31].

Reactions		Temperature Range [K]
1	CaCO3 $\to$ CaO + CO2	823–1233
2	2CaO + SiO2 $\to$ C2S	873–1573
3	C2S + CaO $\to$ C3S	1473–1553
4	3CaO + Al2O3 $\to$ C3A	1453–1553
5	4CaO + Al2O3 + Fe2O3 $\to$ C4AF	1453–1553
6	Clinker(sol) $\to$ Clinker(liq)	>1553

Table 3. Main reactions of clinkerization inside the rotary kiln [31].

Reactions		Temperature Range [K]
1	CaCO3 $\to$ CaO + CO2	823–1233
2	2CaO + SiO2 $\to$ C2S	873–1573
3	C2S + CaO $\to$ C3S	1473–1553
4	3CaO + Al2O3 $\to$ C3A	1453–1553
5	4CaO + Al2O3 + Fe2O3 $\to$ C4AF	1453–1553
6	Clinker(sol) $\to$ Clinker(liq)	>1553

Based on the chemical reactions scheme, shown in Table 3, there are nine solid components that participate as either reactants or products, given here for practical reasons, as two distinct groups:

• Group A: CaCO₃, CaO, SiO₂, Al₂O₃, and Fe₂O₃.
• Group B: C₂S, C₃S, C₃A, and C₄AF.

In the following sections, the equations that have been used in the current study for the calculation of C_p are provided.

2.1. Calculations of C_p for Components of Group A

For all the components belonging to Group A, the component-specific equations of C_p as a function of temperature were obtained from Perry’s Handbook [32]. Namely:

• CaCO₃

$$C_p=19.68+0.01189T-{\displaystyle \frac{\mathrm{307,600}}{T^2}} 273 \mathrm{K}<T<1033 \mathrm{K}$$

(1)

• CaO

$$C_p=10+0.00484T-{\displaystyle \frac{\mathrm{108,000}}{T^2}} 273 \mathrm{K}<T<1173 \mathrm{K}$$

(2)

• SiO₂

$$\begin{gathered} C_p=10.87+0.008712T-{\displaystyle \frac{\mathrm{241,200}}{T^2}} 273 \mathrm{K}<T<848 \mathrm{K} \mathrm{c},\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{z} \mathrm{a} \\ {C}_p=10.95+0.00550T 848 \mathrm{K}<T<1873 \mathrm{K} \mathrm{c}, \mathrm{quartz} \beta \end{gathered}$$

(3)

• Al₂O₃

$$C_p=22.08+0.008971T-{\displaystyle \frac{\mathrm{522,500}}{T^2}} 273 \mathrm{K}<T<1973 \mathrm{K}$$

(4)

• Fe₂O₃

$$C_p=24.72+0.01604T-{\displaystyle \frac{\mathrm{423,400}}{T^2}} 273 \mathrm{K}<T<1097 \mathrm{K}$$

(5)

Also, similar equations for the cases of CaO, Al₂O₃, and SiO₂ (i.e., components belonging to Group A) were obtained from Haas Jr et al. [33].

• CaO

$$\begin{gathered} C_p={\displaystyle \frac{-2.55577\times {10}^5}{T^2}}-{\displaystyle \frac{4.31990\times {10}^2}{T^{0.5}}}+71.6851-2\times 3.08248\times {10}^{-3}T+2.23862\times {10}^{-6}{T}^2 \\ \hspace{1em} 273 \mathrm{K}<T<1800 \mathrm{K} \end{gathered}$$

(6)

• Al₂O₃

$$\begin{gathered} C_p={\displaystyle \frac{-2.46518\times {10}^3}{T^{0.5}}}+2.33004\times {10}^2-2\times 1.95913\times {10}^{-2}T+9.44410\times {10}^{-6} T^2 \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} 273 \mathrm{K}<T<1800 \mathrm{K} \end{gathered}$$

(7)

• SiO₂, quartz a

$$C_p={\displaystyle \frac{-7.77338\times {10}^2}{T^2}}+83.2101+2\times 1.09962\times {10}^{-2}T 273 \mathrm{K}<T<844 \mathrm{K}$$

(8)

• SiO₂, quartz β

$$C_p=58.9107+2\times 5.0208\times {10}^{-3}T 844 \mathrm{K}<T<1800\mathrm{K}$$

(9)

The behavior of Equations (1)–(6) is demonstrated as a function of temperature using the equations from Perry’s handbook in the figures below. In particular, Figure 3a shows C_p within the temperature range where the equations are applicable, while Figure 3b shows C_p at temperature conditions that are extrapolated to those that the rotary cement kiln operates. Similarly, Figure 4a,b show the same results utilizing the equations from Haas Jr. et al. [33].

Although the temperature range of applicability of the equations provided by Haas Jr. et al. [33] was closer to the kiln operating conditions, the calculations in the current study were performed using the equations from Perry’s handbook, as the missing information for the nine components was less (i.e., for self-consistency reasons). In particular, Perry’s handbook provided information for all components (5/5) of Group A, while Haas Jr. et al. provided information for only 3 out of 5 components (3/5) of Group A. Neither option provided information for components of Group B.

2.2. Calculations of C_p for Components of Group B

On the other hand, for the case of C₂S, C₃S, C₃A and C₄AF (i.e., Group B components), there is no data (either experimental or computational) available for the temperature dependence of C_p. Members of Group B are intermediate chemical components, appearing at locations along the rotary kiln, which are hard to isolate and sample, and thus really challenging to measure experimentally their corresponding C_p’s. Instead, Mai et al. [34] provided information for the specific heat under constant volume (C_v) within the temperature range of 0 up to 800 K. The specific heat capacity under constant volume was calculated by Mai et al. [34] via the Density Functional Theory (DFT) method [35], which is a computational quantum mechanical modeling technique.

In order to proceed, we recall from Thermodynamics that the relation between C_p and C_v is given by

$$C_p-C_v={\displaystyle \frac{T{\alpha }^2}{\rho \beta }}$$

(10)

where ρ is the density of the mixture calculated with atomistic simulations, α is the thermal expansion coefficient, and β is the compressibility of the bulk modulus. Those coefficients are given, respectively, from the relations below [36]:

$$\alpha ={\displaystyle \frac{1}{V}}{\left({\displaystyle \frac{\partial V}{\partial T}}\right)}_P$$

(11)

$$\beta ={\displaystyle \frac{1}{V}}{\left({\displaystyle \frac{\partial V}{\partial P}}\right)}_T$$

(12)

For ideal gases, the relation between C_p and C_v is given as $C_p-C_v=R$ but, when it comes to solids or liquids, the value of $C_p-C_v$ is so small that it can be considered negligible for most cases [36], as solids are considered incompressible, while liquids have limited compressibility. Therefore, for the aforementioned components, it is assumed, in the current study, that the C_v value is equal to the C_p value.

Next, the specific heat capacity under constant volume (C_v), reported by Mai et al., for the products of the chemical reactions, is plotted in Figure 5 as a function of temperature. Furthermore, the calculated values are fitted (in the temperature range 200–800 K) using Equation (14), and the resulting curves are shown in Figure 6. Each curve fitting is described by the following equation:

$$y=a_1\ln \left(x\right)+a_0$$

(13)

where the calculated parameters a₀, a₁ are shown in

Table 4. Constants of the curve equations.

Components	a1	a0	R2
C2S	41.774	−115.4	0.9642
C3S	46.334	−81.538	0.9108
C3A	52.824	−80.033	0.9125
C4AF	113.6	−366.6	0.9546

, along with the coefficient of determination R², which indicates the goodness of the fit.

In Figure 5, below, it can be observed that the C_v of the four components belonging to Group B remain relatively constant at temperatures higher than approx. 400 K. Therefore, for engineering-type calculations, it can be assumed that for such temperatures, the C_v value of each component of Group B remains constant.

In conclusion, for the aforementioned four components of Group B, it is assumed, in the current study, that the C_v value is equal to the C_p value and remains constant for temperatures above 800 K, following the asymptotic value corresponding to this temperature. This assumption is critical because inside the rotary cement kiln, the temperature ranges between approx. 800 and 2000 K [37].

2.3. Calculations of C_p for the Mixed Fluidized Bed

Considering that energy and volume are extensive thermodynamic quantities, in a mixture, they can be expressed as a sum of the individual quantities. So, the specific heat capacity at constant pressure can be written as a linear combination of the individual phases [38]. Combining the heat capacities of pure components from both the equations and the diagrams with the compositions of the components along the kiln that were taken from the literature, it becomes possible to calculate the needed parameter.

The mixing rule for the calculation is based on the work of Abdolhosseini Qomi et al. [38], given as follows:

$$C_{p,mix}{\displaystyle = (1-n){\sum }_{i=1}^m{\Phi }_{Ri}{C}_{p,Ri}+n{\sum }_{i=1}^m{\Phi }_{Pi}{C}_{p,Pi}}$$

(14)

where m is the number of reactive agents, and n is the reaction degree. Φ_Ri, and Φ_Pi represent the mass fractions of the reactants (denoted with subscript Ri) and the products (denoted with subscript Pi), respectively, and C_pi are the heat capacities of the pure components. Applying all the known information (the heat capacities of pure components and the concentrations along the kiln) in Equation (14), the C_p of the solid mixture can be calculated. Given that there are five reactions in the system rather than just one, the rotary kiln can be divided into small, nearly equal sections where the temperature is kept constant, and the compositions are either reactants of the subsequent step or products of the previous step, simplifying the reaction degree to either 1 or 0 in each instance. Shown in Figure 7 is a schematic of the discretization along the length of the rotary kiln.

3. Results and Discussion

Table 5. Literature sources in chronological order reporting temperature and compositional profiles. The last column indicates the particular chemical species for which the compositional profile was not reported in the original studies. [-] indicates that the compositional profiles for all nine components are reported in the original studies.

Authors	Year	Ref. Source	Missing Component
Spang et al.	1972	[39]	Al2O3; Fe2O3
Mastorakos et al.	1999	[37]	[-]
Mujumdar et al.	2006	[22]	SiO2; Al2O3; Fe2O3
Mujumdar et al.	2006	[23]	SiO2; Al2O3; Fe2O3
Darabi et al.	2007	[24]	SiO2; Al2O3; Fe2O3
Csernyei et al.	2016	[31]	SiO2; Al2O3; Fe2O3
Colina-Morles et al.	2016	[25]	Fe2O3
Gonzales et al.	2016	[26]	[-]
Abdelwahab et al.	2017	[27]	[-]
Pieper et al.	2020	[28]	SiO2; Al2O3; Fe2O3
Mungyeko Bisulandu and Marias	2021	[21]	[-]

shows a list of computational studies (in chronological order) reporting temperature and compositional profiles along the rotary kiln length.

For all calculations performed in the current study, in-house codes have been utilized, and the obtained results are shown in the following figures. At first, calculations were performed using the units of the mixing rule equation (i.e., [J/(mol K)]); however, in order to be consistent with the literature values for C_p, the units were subsequently converted to [J/(kg K)].

Figure 8 shows the parameters of interest to the current study plotted in four sub-figures, resulting from the calculations based on the work of Mastorakos et al. [37]. In particular, Figure 8a shows the reported solid bed temperature profile along the dimensionless kiln length, while Figure 8b shows the corresponding solid bed composition, expressed as mass fractions, along the dimensionless kiln length. Figure 8c shows the current calculated C_p of the solid mixture of the kiln bed in units of [J/(mol K)]. The calculated C_p values are plotted along the dimensionless kiln length, taking into account the temperature profile as well. Finally, Figure 8d demonstrates the comparison of the current calculated (denoted with red circles) C_p (converted in units of [J/(kg K)]) and the values used by Mastorakos et al. [37]. Figure 8d also indicates, for a limited number of characteristic locations along the rotary kiln length, the % dev. between the calculated $C_p^{curent}$ in the current study and the $C_p^{lit.}$ value reported by the authors in their original publication. % dev. is defined as follows:$\% dev.=100\times \left({\displaystyle \frac{C_p^{curent}-C_p^{lit.}}{C_p^{curent}}}\right)$. Therefore, negative values for the % dev. indicate an under prediction of the current study, while positive values indicate an over prediction.

Figure 9, Figure 10 and Figure 11 show the corresponding results for the cases of the studies of González et al. [26] (Figure 9), Abdelwahab et al. [27] (Figure 10), and Mungyeko Bisulandu and Marias [21] (Figure 11). Results are presented only for these particular four cases among all cases included in Table 5. The four cases report info for the compositional evolution along the kiln for all nine components that are involved in the process.

An interesting observation from the current analysis is that every source in the open literature provides different composition and temperature profiles, and also different sets of operational rotary kiln parameters. Furthermore, as

Table 6. Literature-reported values for C_p.

Literature Source	Cp [J/(kg K)]
Pieper et al. [28]	Polynomial for SiO2
Mastorakos et al. [37]	1500
Mungyeko Bisulandu and Marias [21]	-
Colina-Morles et al. [25]	Cp solids relation [Cs = 1000 × (0.88 + 0.000293 Ts)]
Mujumdar et al. [22]	800
Csernyei et al. [31]	1088 (from Darabi [24])
Darabi [24]	1088
Gonzales et al. [26]	800 (from Mujumdar et al. [22])
Abdelwahab et al. [27]	1090
Georgallis [30]	1150

indicates, there is a wide range of C_p values that have been used in the open literature. On the other hand, in the current study, for each system, there is a proportional increase in C_p with increasing temperature compared to the literature values, which remain constant throughout the process. The particular observations indicate that there is a need for the systematic analysis of the effect of errors on the overall energy balance of the rotary kiln and the resulting temperature and compositional profiles. However, since no systematic error analysis is provided in the earlier studies, such an effort is far beyond the scope of the current study and is left for the future.

From Figure 8, Figure 9, Figure 10 and Figure 11, we observe that the variable C_p values calculated in the current study range between 1000 and 1300 [J/(kg K)]. Depending on the constant value for C_p used in the original studies, we estimated that the % dev. can range between 21 and 53% (underestimation) for the case of Mastorakos et al., 19–35% (overestimation) for the case of González et al., and 7% (underestimation) up to 15% (overestimation) for the case of Abdelwahab et al. It should be pointed out that for the case of Mungyeko Bisulandu and Marias (e.g., see Figure 11d), the C_p value used was not reported in that study. Therefore, in Figure 11d, the constant C_p values from various literature sources (i.e., five cases) are also indicated with the constant horizontal lines. As a result, the % dev. between the calculated $C_p^{curent}$ values of the current study and the $C_p^{lit.}$ value can be calculated for all five cases. Such results are shown in Figure 12.

4. Conclusions and Future Outlook

Energy consumption differs depending on the type of kiln, as a wet process kiln requires more thermal and electrical energy due to the higher amount of moisture in the raw meal in comparison with a dry process [8]. From the calculations reported in the current study, it can be concluded that the heat capacity of the fluidized bed varies along the kiln length as a result of changes in composition and temperature. The calculated values fluctuate within a range, as the extracted information for the reported concentration profiles exhibits inconsistencies, and several parameter values used were not fully aligned. Since there is a strong dependence on the temperature and the composition for the calculated C_p, according to the applied mixing rule, it is concluded that during the simulations, the value of C_p should not be considered constant along the kiln length. Instead, using a position-dependent (corresponding to different temperatures along the rotary kiln) C_p is recommended. Incorporating a more realistic value of C_p along the kiln influences the system behavior and, in particular, the composition profiles of clinker along the process. Establishing a more accurate description of the specific heat capacity improves the energy balance inside the kiln and, along with other enhancements, can lead to a decrease in the energy consumed and to a further reduction in greenhouse gases. Thus, for more reliable predictions of the compositions and the temperature inside a rotary kiln, a variable C_p should be used. Another observation is that, in all examined cases, there is a proportional increase in the values of C_p with the increase in temperature, in contrast to constant literature values. Finally, yet importantly, although the deviation between the literature and the calculated values is relatively small, in some instances it becomes quite significant, fluctuating from 1.56% up to 52.49%, clearly demonstrating the dependence on temperature.