# Simulation of Thermal Decomposition of Calcium Oxide in Water with Different Activation Energy and the High Reynolds Number

Authors: Abid A. Memon; Saqib Murtaza; M. Asif Memon; Kaleemullah Bhatti; Mojammel Haque; Mohamed R. Ali
Journal: Complexity
 (2022)

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## Abstract

In this article, we are going to suggest the parameters to control the thermal decomposition in a reactor that is continuously providing a cooling environment inside the tube. For this purpose, 5 governing partial differential equations gained through mass, momentum, and energy balance laws and one ordinary differential equation are used to simulate this chemical reaction with finite element package COMSOL Multiphysics 5.6. In the simulation, the thermal decomposition of calcium oxide in the water is controlled with the use of Reynolds numbers ranging from 100 to 1000, activation energy from 75,000 j/mol to 80,000 j/mol, and an initial concentration of 1% to 5%. The results are presented through the graphs and tables for conversion profile, temperature distribution, enthalpy change, diffusivity, the heat source of reaction, and Sherwood and Lewis number along the axial length of the reactor. Specifically, it was found a low-speed profile at the inlet will give a 100% conversion at Re = 100 for 3% to 5% of the initial concentration. The maximum temperature and enthalpy change in the reactor are decreasing increase in the Reynolds number. Also, the decrement of the Sherwood number along the length showed that the mass diffusion is always dominant over convection-diffusion for all cases of parameters. The validation is made by comparing the numerical results with the experimental correlations.


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## Body

## 1. Introduction

The applications of renewable energies are largely studied for many decades with the help of chemical reactions of calcium oxide [1–4]. It is defined to be one of the energy sources, which can facilitate light and electricity without demoralizing the surroundings. Different procedures are applied to get the energy stored, of which the thermal storage procedures are largely applied due to having the low cost. The thermal storage procedures are often followed by the use of exothermic chemical reactions because of providing energy density with a high amount. The reaction of calcium oxide CaO with water is one of the most promising exothermic reactions that will generate energy storage due to cost-efficiency of both reactants [5–8], and they also being nontoxic [9]. Because of possessing the high capacity for storing energy, the large applications of CaO can be seen along with the production of efficient heat pumps [10–12], capturing carbon and energy storage [13–15], and hydrogen generation with sorption enhancement [16, 17]. The mixing of calcium oxide in water was already described in various studies regarding the thermal equilibrium and enthalpy of the reaction [18, 19]. Later, several models due to the chemical reaction of CaO and water were proposed. Based on the first-order reaction, a model was proposed by [20] based on the partial pressure of steam in the range of 50–500 kPa with a particle size of 345.5 µm. The dehydration of calcium oxide was studied [21] with an apparatus producing high pressure with a range of 0.67–3.8 MPa and keeping the temperature of 1023 K. At this level, the activation energy was computed to be 8.4 kJ/mol. A parametric and numerical study was done due to the chemical kinetics of calcium oxide and steam with a high temperature in the range of 80 to 450°C and with different concentrations of the steam from 1.5% to 15% [22]. In a study [23], a model was settled for calcium oxide in a bed reactor to mix with the mixture containing the steam and nitrogen. The study was done to capture the mass and heat transport in the reactor, and finally, this model was compared with results from available literature [24].The objective of the present study is to focus on the steady-state decomposition of the CaO in water in a reactor whose curved wall is continuously providing the cooling environment inside to handle the explosion. In the paper, we are focusing on the distribution of temperature, the enthalpy changes for the chemical reaction, the diffusion coefficient of CaO, and the concentration of dissolved components along the length of the reactor with the variation of chosen parameters. These parameters are the Reynolds number, initial concentration, and activation energy. The results are conveyed and verified with the available correlation of the Sherwood number and the mesh independent study. Lewis’s number is also specially discussed at the end.


## 2. Methodology

### 2.1. Physical Construction and Details of Chemical Reaction

The chemical reactions are largely observed in the industrial area to observe the heat and mass flow applications. A variety of exothermic and endothermic reactions are used under applying certain boundary conditions to achieve the requirements in the form of energies. Various types of reactors are used to observe chemical reactions. The multicomponent tubular reactors are largely observed for exothermic reactions due to having a cooling jacket around them. The function of this cooling jacket is to supply constantly the inward heat flux inside the tube. The present problem is a two-dimensional axisymmetric problem where a multicomponent tubular reactor having radiusRa = 0.1 m and length L = 1 m is under observation (see Figure 1). The component contains a cooling jacket called a coolant, which is constantly trying to maintain the cooling environment inside the tube because of inward heat flux condition q0s. The fluid that enters from the entrance of the channel will face the temperature of Tin = 312 K in which we call inlet temperature. Initially, it assumes that the tubular reactor is a vacuum with a reference temperature of Tref = 293 K.Figure 1
A short view of the multicomponent tubular reactor.The present problem involves the thermal decomposition of the calcium oxide in water where the water is present in the axis. Therefore, the rate of reactionRr is of the first order and is only depending on the concentration of calcium oxide Cco.(1)Rr=kfCco.Here,kf is the forward rate constant and measured with the Arrhenius equation, which is given as(2)kf=Ae−ERT.To boost the chemical reaction of water and calcium oxide for the present problem, we are going to observe a parametric study for activation energy or threshold energyE in the range from 75,000 j/mol to 80,000 j/mol. Moreover, it should be noted that normally energy of 75,362 j/mol is required to let the chemical reaction happen between the calcium oxide and water. In Reference (2), factor A is known as the frequency factor and is fixed in the problem; for simplicity, we are testing the value 16.96 × 1012 1/h. We put normal enthalpy for this exothermic reaction ΔH=−64,300j/mol. From the inlet of the reactor tube, the initial concentration of the calcium oxide is taken from 1% to 5%. The velocity transport of the mixture is controlled with the use of the Reynolds number from 100 to 1000, which will yield a total flow rate vf from the inlet of the subjected reactor in the range from 1.57 × E−5 m3/s to 1.57 × E−4 m3/s. Let ui be the initial velocity of the fluid entering from the inlet region; then, the total flow rate and the inlet velocity can be calculated by(3)ui=vfπRa2.The Reynolds number is the nondimensional number, and by the definition, it is the ratio between inertial forces and viscous forces. The Reynolds number used here to predict the flow is calculated by using the physical properties of water because the amount of water is in access. Therefore, we calculate the Reynolds number with(4)Re=ρwateruiLcμwater.With the use of (3) in (4), the following equation can be written:(5)Re=ρwatervfLcμwaterπRa2.Further details of the parameters used for the fluid flow, heat, and mass transfer problem with the influence of chemical reaction are given as in Table1.Table 1
Summary of parameters.
ParametersValueDescriptionE75,000 j/mol–80,000 j/molActivation energyA16.96e12 1/hFrequency factorUk1300 W/m2/KOverall heat-transfer coefficientκ0.559 W/m/KThermal conductivity mixtureTin312 KInlet temperatureTc273 KThe inlet temperature of the coolantΔH−64,300 J/molHeat of reactionvf1.57 × E−5 m3/s−1.57 × E−4 m3/sTotal flow ratei.c1%–5%Percentage concentration of calcium oxidecwater43210mol/m3Water concentration, inletRa0.1 mReactor radiusL1 mReactor lengthmco56.0774 g/molMolar weight, calcium oxidemwater18 g/molMolar weight, watermcho74.093 g/molMolar weight, calciumρco3.34 g/cm3Density, calcium oxideρwater1000 kg/m3Density, waterρcho2.24 g/cm3Density, calcium hydroxideμref1e − 3 PasReference dynamic viscosity, waterTref293 KReference temperature viscositycp75.36 J/mol/KMolar heat capacity, waterRe100–1000Reynolds numberCThe concentration of the mixture inmol/m3vrVelocity along the radial (r) directionvzVelocity along the axial (z) directionTThe temperature in the reactor was measured in kelvin (K)


### 2.2. Mesh Independent Study for Local Nusselt Number

To get the numerical results with excellent accuracy, it is a crucial step to do the mesh independent study. The goal of the study is to find the mesh density where the numerical results are independent of mesh. The present problem of mass transfer via the chemical reaction is performed in a tubular reactor with finite element package COMSOL Multiphysics 5.6. The whole rectangular domain is decomposed by using irregular triangular elements. In Figure2, the zoom-in view of the meshing process is attached where it can be seen that on the right wall, a very fine mesh is used due to the inward heat flux condition on that boundary. Five different meshes with different configurations are used to obtain the numerical results where the number of elements is tried from 8489 to 184809. The targeted variable is the local Nusselt number computed at the lower wall of the channel for normal to extremely fine mesh (see Figures 3(a) and 3(b)). In Figure 3(a), the zoom-out graph is given, showing that there is the most negligible difference in the numerical results of the local Nusselt number due to the trial of different mesh densities. In Figure 3(b), the zoom-in graph is uploaded, showing that when to increase the density of meshes from normal to extremely fine meshes, the numerical results for the local Nusselt number are not much changing or stopped further improving. At this stage, the numerical scheme is achieved as a mesh independent solution. To get the most reliable results, extremely fine mesh is used for this whole problem of fluid, heat, and mass transfer.Figure 2
Meshing process of the geometry with the number of elements. (a) Normal mesh with 8489 elements; (b) fine mesh with 16442 elements; (c) fine mesh with 34206 elements; (d) extremely fine mesh with 71606 elements; (e) extremely fine mesh with 184809 elements.
(a)(b)(c)(d)(e)Figure 3
Numerical results gained with mesh independent study of the local Nusselt number. (a) Zoom-out; (b) zoom-in.
(a)(b)


### 2.3. Governing Equations and Computational Parameters

Testing exothermic reactions in different types of reactors are not easy due to taking care of the explosion in the reactor. Therefore, it is a need for such material or boundary around the surface of the reactors, which is capable to provide continuous cooling in the reactors. In the present problem, we consider a reactor that is folded by a cooling jacket and is capable to avoid an explosion in the reactor. That jacket is created by applying the inward heat flux boundary conditions. Moreover, the present problem is simulated using the finite element-based software COMSOL Multiphysics 5.6 where we are going to consider governing partial differential equations for momentum, mass, and energy balance. Also, one of the best things in the software is to handle the chemical reaction of calcium oxide with water with the use of a chemical engineering interface.We assume first that the diffusivity of the three species is taken identical throughout the simulation. About six constitutive ordinary and partial differential equations in the two-dimensional axisymmetric channel are taken part to observe the current simulation for fluid, heat, and mass transfer with the help of chemical kinetics. The simulation to visualize the fluid flow in the reactor will be handled by two momentum balance and one mass balance equation. One energy balance equation will be used to grip the heat flow in the core of the reactor, and one will get the hold to handle the mass diffusion of the chemical species in the reactor. One ordinary differential equation for the temperature of the cooling jacket will take a part in the simulation. All governing equations, boundary conditions, and computational parameters are given as follows:(6)1r∂rvr∂r+∂vz∂z=0,(7)1r∂∂rrvrvr+∂∂zrvrvz=μρ2r∂∂rr∂vr∂r+∂∂zr∂vr∂z−2vrr2+∂∂z∂vz∂r−1ρ∂p∂r,(8)1r∂∂rrvzvz+∂∂zrvzvz=μρ1r∂∂rr∂vz∂r+2∂∂z∂vz∂z+1r∂∂rr∂vr∂z−1ρ∂p∂z,(9)vr∂T∂r+vz∂T∂z=kρcp∂∂z∂T∂z+1r∂∂rr∂T∂r+−ΔHRiρcp,(10)vr∂C∂r+vz∂C∂z=kρcp∂∂z∂C∂z+1r∂∂rr∂C∂r−1ρcpRi,(11)−∂T∂r=UkkT−Tc.


#### 2.3.1. Boundary Conditions

At inlet of the reactorz = 0,(12)Cr,0=initialconcentration=c0,Tr,0=Tin,vz=inletvelocity=ui=fRe.At outlet of the reactorz = L,(13)∂C∂rr,L=0,−∂T∂rr,L=0,p=0.Around the wall of the reactorr = Ra,(14)∂C∂r=0,−∂T∂r=UkkT−T0.Equations (13) and (14) are applied [25].


#### 2.3.2. Computational Parameters

(15)Heatsourceofreaction=Qs=−RiΔH,molar concentration=Mco=numberofmolesofsolutelitersofsolution,(16)thermal diffusivity of calcium oxide=DCao=7.4×10−8TxMH2OηρCaO/MCaO0.6.Equation (16) is the correlation given by Wilke–Chang [26]. In the equation, x = 1 for nonassociated solvent and x = 2.26 is used specially for water. The other nondimensional numbers used in the problem are as follows:(17)lcal Nusselt numberNuz=convectiveheattransfercondectiveheattransfer=Ukzk,Sherwood numberSh=convectivemasstransferratediffusionrate=hD/L,Prandtl numberPr=momentumdiffusivitythermaldiffusivity=μcpk.And,(18)LewisnumberLe=thermaldiffusivitymassdiffusivity=αD.The governing equation from (6)–(11) corresponding the boundary conditions (12)–(14) will be solved by using the numerical method given in [27–29].The working wagon wheel is shown in Figure4 at which the emerging software COMSOL Multiphysics 5.6 is given.Figure 4
The working wagon wheel of COMSOL Multiphysics 5.6.


## 2.1. Physical Construction and Details of Chemical Reaction

The chemical reactions are largely observed in the industrial area to observe the heat and mass flow applications. A variety of exothermic and endothermic reactions are used under applying certain boundary conditions to achieve the requirements in the form of energies. Various types of reactors are used to observe chemical reactions. The multicomponent tubular reactors are largely observed for exothermic reactions due to having a cooling jacket around them. The function of this cooling jacket is to supply constantly the inward heat flux inside the tube. The present problem is a two-dimensional axisymmetric problem where a multicomponent tubular reactor having radiusRa = 0.1 m and length L = 1 m is under observation (see Figure 1). The component contains a cooling jacket called a coolant, which is constantly trying to maintain the cooling environment inside the tube because of inward heat flux condition q0s. The fluid that enters from the entrance of the channel will face the temperature of Tin = 312 K in which we call inlet temperature. Initially, it assumes that the tubular reactor is a vacuum with a reference temperature of Tref = 293 K.Figure 1
A short view of the multicomponent tubular reactor.The present problem involves the thermal decomposition of the calcium oxide in water where the water is present in the axis. Therefore, the rate of reactionRr is of the first order and is only depending on the concentration of calcium oxide Cco.(1)Rr=kfCco.Here,kf is the forward rate constant and measured with the Arrhenius equation, which is given as(2)kf=Ae−ERT.To boost the chemical reaction of water and calcium oxide for the present problem, we are going to observe a parametric study for activation energy or threshold energyE in the range from 75,000 j/mol to 80,000 j/mol. Moreover, it should be noted that normally energy of 75,362 j/mol is required to let the chemical reaction happen between the calcium oxide and water. In Reference (2), factor A is known as the frequency factor and is fixed in the problem; for simplicity, we are testing the value 16.96 × 1012 1/h. We put normal enthalpy for this exothermic reaction ΔH=−64,300j/mol. From the inlet of the reactor tube, the initial concentration of the calcium oxide is taken from 1% to 5%. The velocity transport of the mixture is controlled with the use of the Reynolds number from 100 to 1000, which will yield a total flow rate vf from the inlet of the subjected reactor in the range from 1.57 × E−5 m3/s to 1.57 × E−4 m3/s. Let ui be the initial velocity of the fluid entering from the inlet region; then, the total flow rate and the inlet velocity can be calculated by(3)ui=vfπRa2.The Reynolds number is the nondimensional number, and by the definition, it is the ratio between inertial forces and viscous forces. The Reynolds number used here to predict the flow is calculated by using the physical properties of water because the amount of water is in access. Therefore, we calculate the Reynolds number with(4)Re=ρwateruiLcμwater.With the use of (3) in (4), the following equation can be written:(5)Re=ρwatervfLcμwaterπRa2.Further details of the parameters used for the fluid flow, heat, and mass transfer problem with the influence of chemical reaction are given as in Table1.Table 1
Summary of parameters.
ParametersValueDescriptionE75,000 j/mol–80,000 j/molActivation energyA16.96e12 1/hFrequency factorUk1300 W/m2/KOverall heat-transfer coefficientκ0.559 W/m/KThermal conductivity mixtureTin312 KInlet temperatureTc273 KThe inlet temperature of the coolantΔH−64,300 J/molHeat of reactionvf1.57 × E−5 m3/s−1.57 × E−4 m3/sTotal flow ratei.c1%–5%Percentage concentration of calcium oxidecwater43210mol/m3Water concentration, inletRa0.1 mReactor radiusL1 mReactor lengthmco56.0774 g/molMolar weight, calcium oxidemwater18 g/molMolar weight, watermcho74.093 g/molMolar weight, calciumρco3.34 g/cm3Density, calcium oxideρwater1000 kg/m3Density, waterρcho2.24 g/cm3Density, calcium hydroxideμref1e − 3 PasReference dynamic viscosity, waterTref293 KReference temperature viscositycp75.36 J/mol/KMolar heat capacity, waterRe100–1000Reynolds numberCThe concentration of the mixture inmol/m3vrVelocity along the radial (r) directionvzVelocity along the axial (z) directionTThe temperature in the reactor was measured in kelvin (K)


## 2.2. Mesh Independent Study for Local Nusselt Number

To get the numerical results with excellent accuracy, it is a crucial step to do the mesh independent study. The goal of the study is to find the mesh density where the numerical results are independent of mesh. The present problem of mass transfer via the chemical reaction is performed in a tubular reactor with finite element package COMSOL Multiphysics 5.6. The whole rectangular domain is decomposed by using irregular triangular elements. In Figure2, the zoom-in view of the meshing process is attached where it can be seen that on the right wall, a very fine mesh is used due to the inward heat flux condition on that boundary. Five different meshes with different configurations are used to obtain the numerical results where the number of elements is tried from 8489 to 184809. The targeted variable is the local Nusselt number computed at the lower wall of the channel for normal to extremely fine mesh (see Figures 3(a) and 3(b)). In Figure 3(a), the zoom-out graph is given, showing that there is the most negligible difference in the numerical results of the local Nusselt number due to the trial of different mesh densities. In Figure 3(b), the zoom-in graph is uploaded, showing that when to increase the density of meshes from normal to extremely fine meshes, the numerical results for the local Nusselt number are not much changing or stopped further improving. At this stage, the numerical scheme is achieved as a mesh independent solution. To get the most reliable results, extremely fine mesh is used for this whole problem of fluid, heat, and mass transfer.Figure 2
Meshing process of the geometry with the number of elements. (a) Normal mesh with 8489 elements; (b) fine mesh with 16442 elements; (c) fine mesh with 34206 elements; (d) extremely fine mesh with 71606 elements; (e) extremely fine mesh with 184809 elements.
(a)(b)(c)(d)(e)Figure 3
Numerical results gained with mesh independent study of the local Nusselt number. (a) Zoom-out; (b) zoom-in.
(a)(b)


## 2.3. Governing Equations and Computational Parameters

Testing exothermic reactions in different types of reactors are not easy due to taking care of the explosion in the reactor. Therefore, it is a need for such material or boundary around the surface of the reactors, which is capable to provide continuous cooling in the reactors. In the present problem, we consider a reactor that is folded by a cooling jacket and is capable to avoid an explosion in the reactor. That jacket is created by applying the inward heat flux boundary conditions. Moreover, the present problem is simulated using the finite element-based software COMSOL Multiphysics 5.6 where we are going to consider governing partial differential equations for momentum, mass, and energy balance. Also, one of the best things in the software is to handle the chemical reaction of calcium oxide with water with the use of a chemical engineering interface.We assume first that the diffusivity of the three species is taken identical throughout the simulation. About six constitutive ordinary and partial differential equations in the two-dimensional axisymmetric channel are taken part to observe the current simulation for fluid, heat, and mass transfer with the help of chemical kinetics. The simulation to visualize the fluid flow in the reactor will be handled by two momentum balance and one mass balance equation. One energy balance equation will be used to grip the heat flow in the core of the reactor, and one will get the hold to handle the mass diffusion of the chemical species in the reactor. One ordinary differential equation for the temperature of the cooling jacket will take a part in the simulation. All governing equations, boundary conditions, and computational parameters are given as follows:(6)1r∂rvr∂r+∂vz∂z=0,(7)1r∂∂rrvrvr+∂∂zrvrvz=μρ2r∂∂rr∂vr∂r+∂∂zr∂vr∂z−2vrr2+∂∂z∂vz∂r−1ρ∂p∂r,(8)1r∂∂rrvzvz+∂∂zrvzvz=μρ1r∂∂rr∂vz∂r+2∂∂z∂vz∂z+1r∂∂rr∂vr∂z−1ρ∂p∂z,(9)vr∂T∂r+vz∂T∂z=kρcp∂∂z∂T∂z+1r∂∂rr∂T∂r+−ΔHRiρcp,(10)vr∂C∂r+vz∂C∂z=kρcp∂∂z∂C∂z+1r∂∂rr∂C∂r−1ρcpRi,(11)−∂T∂r=UkkT−Tc.


### 2.3.1. Boundary Conditions

At inlet of the reactorz = 0,(12)Cr,0=initialconcentration=c0,Tr,0=Tin,vz=inletvelocity=ui=fRe.At outlet of the reactorz = L,(13)∂C∂rr,L=0,−∂T∂rr,L=0,p=0.Around the wall of the reactorr = Ra,(14)∂C∂r=0,−∂T∂r=UkkT−T0.Equations (13) and (14) are applied [25].


### 2.3.2. Computational Parameters

(15)Heatsourceofreaction=Qs=−RiΔH,molar concentration=Mco=numberofmolesofsolutelitersofsolution,(16)thermal diffusivity of calcium oxide=DCao=7.4×10−8TxMH2OηρCaO/MCaO0.6.Equation (16) is the correlation given by Wilke–Chang [26]. In the equation, x = 1 for nonassociated solvent and x = 2.26 is used specially for water. The other nondimensional numbers used in the problem are as follows:(17)lcal Nusselt numberNuz=convectiveheattransfercondectiveheattransfer=Ukzk,Sherwood numberSh=convectivemasstransferratediffusionrate=hD/L,Prandtl numberPr=momentumdiffusivitythermaldiffusivity=μcpk.And,(18)LewisnumberLe=thermaldiffusivitymassdiffusivity=αD.The governing equation from (6)–(11) corresponding the boundary conditions (12)–(14) will be solved by using the numerical method given in [27–29].The working wagon wheel is shown in Figure4 at which the emerging software COMSOL Multiphysics 5.6 is given.Figure 4
The working wagon wheel of COMSOL Multiphysics 5.6.


## 2.3.1. Boundary Conditions

At inlet of the reactorz = 0,(12)Cr,0=initialconcentration=c0,Tr,0=Tin,vz=inletvelocity=ui=fRe.At outlet of the reactorz = L,(13)∂C∂rr,L=0,−∂T∂rr,L=0,p=0.Around the wall of the reactorr = Ra,(14)∂C∂r=0,−∂T∂r=UkkT−T0.Equations (13) and (14) are applied [25].


## 2.3.2. Computational Parameters

(15)Heatsourceofreaction=Qs=−RiΔH,molar concentration=Mco=numberofmolesofsolutelitersofsolution,(16)thermal diffusivity of calcium oxide=DCao=7.4×10−8TxMH2OηρCaO/MCaO0.6.Equation (16) is the correlation given by Wilke–Chang [26]. In the equation, x = 1 for nonassociated solvent and x = 2.26 is used specially for water. The other nondimensional numbers used in the problem are as follows:(17)lcal Nusselt numberNuz=convectiveheattransfercondectiveheattransfer=Ukzk,Sherwood numberSh=convectivemasstransferratediffusionrate=hD/L,Prandtl numberPr=momentumdiffusivitythermaldiffusivity=μcpk.And,(18)LewisnumberLe=thermaldiffusivitymassdiffusivity=αD.The governing equation from (6)–(11) corresponding the boundary conditions (12)–(14) will be solved by using the numerical method given in [27–29].The working wagon wheel is shown in Figure4 at which the emerging software COMSOL Multiphysics 5.6 is given.Figure 4
The working wagon wheel of COMSOL Multiphysics 5.6.


## 3. Validation of the Model

The problem with modeling and simulation of decomposition of calcium oxide in the reactor is verifying the correlation perceived by the Sherwood number and the Schmidt number [24]. The correlation is derived from the correspondence that the Prandtl number forecasts the same in heat transfer as the Schmidt number ensures in the problems of mass transfer. In Figure 1, the comparison is made by displaying the Sherwood number figured due to correlation and present work. In the first place, we come to recognize that our computed number is getting 90% accuracy due to the experimental cognitive of the correlation. In the second place, we can understand that the Sherwood number is decreasing, which directs mass diffusion rate is consistently increasing concerning thermal diffusion rate. The outcomes are computed for Re = 100, with 1–3% initial concentration of calcium oxide with the activation energy of 75,000 and 77,000 j/mol as shown in Figures 5(a)–5(f). The results display stability with the increasing initial concentration of the chemical species. However, the results for comparison can be found to weaken due to increasing activation energy and Reynolds number. The cause behind this is due to increasing the mass diffusion rate. The rate of decrease of the Sherwood number is quickly improved with an increase in the initial concentration of calcium oxide.Figure 5
Comparison between the computed Sherwood number and the experimental correlation: (a) I.C = 1%, Re = 100,E = 75,000; (b) I.C = 2%, Re = 100, E = 75,000; (c) I.C = 3%, Re = 100, E = 75,000; (d)I.C = 1%, Re = 100, E = 77,000; (e) I.C = 2%, Re = 100, E = 77,000; (f) I.C = 3%, Re = 100, E = 77,000.
(a)(b)(c)(d)(e)(f)


## 4. Results and Discussion

### 4.1. Conversion Profile of Calcium Oxide in Water

We present the above graph to show the conversion profile of calcium oxide since five different initial concentrations are tested and mixed with water. Figures6(a)–6(c) are the graphs where we present the results for initial parametric studies. In Figure 6(a), the conversion profile is shown in the variation of the Reynolds number, where we can see the conversion of calcium oxide is increasing along the length of the reactor. For a Re = 100, the compound is converted to about 90%. The maximum conversion can be seen at the outlet of the reactor, and for a fixed concentration and the activation energy, it is decreasing with the increase in the Reynolds number. In Figure 6(b), the conversion profile shows positivity with a decrease in activation energy. It indicates that for a good decomposition at the low initial concentration for the Reynolds number, the activation might be good at around 70,000 to 80,000 in j/mol. In Figure 6(c), it can be suggested that a 100% conversion is taking place for the lower Reynolds number in taking the concentration between 3% and 5% for the lower Reynolds number. For higher values of the Reynolds number and the activation energy in the current values, the conversion profile is found to linearly increase along the length. The conversion is only 30 percent at the outlet of the reactor (see Figure 7(a)). Figure 7(b) shows that even taking a high initial concentration, the conversion profile is largely affected by activation energy. Figure 7(c) is revealing that all the initial high Reynolds numbers and activation give about the same conversion of calcium oxide.Figure 6
Conversion profile of CaO against the length of the reactor: (a) I.C = 1%,E = 75,000 J/mol; (b) I.C = 1%, Re = 100; (c) Re = 100, E = 75,000 J/mol.
(a)(b)(c)Figure 7
Conversion profile of CaO against the length of the reactor: (a) I.C = 5%,E = 80,000 J/mol; (b) I.C = 5%, Re = 1000; (c) Re = 1000, E = 80,000 J/mol.
(a)(b)(c)In Table2, the percentage change in the conversion profile is described for the initial concentration of 1% to onward. We can conclude that for lower or moderate Reynolds numbers, the percentage change remains consistent. The maximum percentage change can be found in the case of Re = 400 and E = 75,000 j/mol. In short, the high Reynolds number, as well as the activation energy, will never give a favor to the conversion profile of calcium oxide. Finally, we ought to suggest to get the maximum amount of production keep the Reynolds number must be lower.Table 2
Percentage improved conversion profile from 1% of initial concentration to onward.
ReE (J/mol)χpc 1–2%χpc 1–3%χpc 1–4%χpc 1–5%100750008.58909811.6531911.936311.939811007700016.6006138.1403959.4910468.162261007900010.4422524.097742.9224670.89008100800007.40431616.3872827.5954942.12024007500013.2646331.4327357.332692.87567400770007.06255215.5515126.0041339.33254400790003.402767.12153711.2084915.72893400800002.3438234.8353457.49073910.32883800750007.91078517.6281329.9465846.23588800770003.8548118.11751712.8652818.20152800790001.8319573.7527875.7709327.894467800800001.2603162.5624723.9084245.3015711000750006.54109214.2906823.6664235.360861000770003.1513086.57028810.300914.394411000790001.4960423.0521964.6668926.3502531000800001.030972.0865193.1735314.287668


### 4.2. Temperature Distribution in the Reactor

As we are testing an exothermic reaction from the examples of chemical engineering, definitely the temperature along the selected domain is increasing along the length. However, we are interested to control the temperature distribution with the parameters used in the problem. The temperature distribution in the middle of the channel is presented in Figures8(a)–8(c) and 9(a)–9(c). Initially, the temperature of 312 K is imposed at the inlet of the channel. An increase in the temperature along the length can be seen in Figure 8(a) with the increase of the Reynolds number. The increment is nonlinear, and the maximum temperature at the outlet of the reactor is decreasing with the increase in the Reynolds number. In Figure 9(a), we also found an increment in the temperature along the length of the reactor for high initial concentration and the activation energy and found 4 K more temperature increment when compared with lower initial concentration and the activation energy. For the extreme case in Figure 8(b), the increment in temperature is found linearly. With fixing the Reynolds number and initial concentration, the maximum temperature at the outlet is decreasing with the increase in activation energy (see Figures 8(b) and 9(b)). It is also a clear message from these graphs that for the high Reynolds number as well as the initial concentration, the inclination in the temperature along the length is very high and linear. Moreover, we can understand from the graphs (Figures 8(c) and 9(c)) adding that more initial concentrations of calcium oxide will tend to increase the maximum temperature at the outlet. For the low Reynolds number as well as the activation energy, we found that an increment of the temperature at the outlet is high. Finally, we are presenting Table 3, where we are presenting the percentage increment in the maximum temperature at the outlet of the channel when the initial concentration is moved from 1% onward.Figure 8
Temperature distribution along the length of the channel: (a) I.C = 1%,E = 75,000 J/mol; (b) I.C = 1%, Re = 100; (c) Re = 100, E = 75,000 J/mol.
(a)(b)(c)Figure 9
Temperature distribution along the length of the channel: (a) I.C = 5%,E = 80,000 J/mol; (b) I.C = 5%, Re = 1000; (c) Re = 1000, E = 80,000 J/mol.
(a)(b)(c)Table 3
Percentage change of the temperature when initial concentration is changing from 1% to onward.
ReE (J/mol)Tpc 1–2%Tpc 1–3%Tpc 1–4%Tpc 1–5%100750002.187634.401486.508668.59267100770001.629693.868266.6599.23803100790000.776581.754323.051174.90507100800000.515411.120771.850642.76151400750001.124572.615344.71037.75102400770000.507641.095531.791272.63816400790000.226740.469610.730951.01363400800000.152340.3120.479690.65621800750000.598661.305722.163983.24566800770000.267560.557030.872081.21741800790000.120990.246520.376920.51257800800000.081710.16550.251480.339771000750000.484391.039531.687622.462211000770000.217170.44870.696520.962971000790000.098550.20010.304840.412961000800000.066630.134650.204110.27507From Table3, it can be understood that for the present problem, the percentage change in the temperature at the outlet is maximum for Re = 100 and E = 77,000 j/mol and minimum for Re = 1000 and E = 80,000. We can also conclude that an increase in the initial concentration of the compound always supports an increase in the temperature at the outlet of the reactor.


### 4.3. Enthalpy Change in the System

Enthalpy is known as the heat content in the system. When a chemical reaction is encountered by an exothermic reaction, the enthalpy change is increasing. It is also rigorously defined as the combination of internal energy and the product of pressure and volume. Involving so many components in determining the internal energy, the enthalpy in the system is almost impossible to determine. Because having large applications of enthalpy change are very important, it is easy to determine because of possessing the property of state function. The computation of enthalpy change is done here due to the variation of chosen parameters in Figures10(a)–10(c) and 11(a)–11(c) until we have determined that a low speed and activation energy give favor to complete the chemical reaction and about 90% conversion. Figure 10(a) shows that the enthalpy change is decreasing with an increase in the Reynolds number for a fixed concentration and activation energy. Figure 11(b) shows for higher concentration and fixed activation energy, the enthalpy change is decreasing at the outlet of the reactor with increasing Reynolds number. For a fixed initial concentration and the activation energy, an increase in the Reynolds number gives a sufficient decrease in the enthalpy change. In Figure 10(b), we see that for a fixed initial concentration and Reynolds number, the enthalpy change in the reactor is increasing along the length and maximum at the outlet of the reactor. With increasing activation energy, the maximum enthalpy is decreasing at the outlet of the reactor. We also test the case by increasing the Reynolds number and initial concentration, and the enthalpy change is increasing in a nonlinear way for a lesser Reynolds number. Figures 10(c) and 11(c) show that adding more initial concentration of the solid gives favor increasing the enthalpy change at the outlet of the reactor. It can be also seen that for a 5% concentration when the Reynolds number and the activation energy are taken from lower to upper-level cases, the enthalpy change decrease is about 60%. Finally, we attach Table 4, where we are going to present the benefit for enthalpy change when the percentage is changing from 1% to onward. For simplicity, we make the idea that the values will be presented in the percentage increment.Figure 10
Enthalpy change of the system: (a) I.C = 1%,E = 75,000 J/mol; (b) I.C = 1%, Re = 100; (c) Re = 100, E = 75,000 J/mol.
(a)(b)(c)Figure 11
Enthalpy change of the system: (a) I.C = 5%,E = 80,000 J/mol; (b) I.C = 5%, Re = 1000; (c) Re = 1000, E = 80,000 J/mol.
(a)(b)(c)Table 4
Percentage change in the maximum enthalpy of initial concentration from 1% to onward.
ReE (J/mol)ΔHp 1–2%ΔHp 1–3%ΔHp 1–4%ΔHp 1–5%1007500028.1465856.6304983.74196110.55541007700022.6949853.8692592.73312128.64861007900011.6917726.4121145.9368773.84811100800007.97617817.3443628.6393842.735414007500016.3517538.0283368.48996112.7035400770007.85764816.9575527.7267540.83564400790003.6328797.52415311.7113216.24047400800002.4656565.0497797.76378610.6208800750009.17081220.0021533.1496549.71962800770004.2637278.87655813.897219.40011800790001.9668754.00756.1273768.332631800800001.3358122.7057374.1114925.5549261000750007.5161316.1326.1862438.205431000770003.4837497.19791611.1734415.447791000790001.6072683.2634914.9716196.7348711000800001.0918162.2063243.344454.507178From Table4, it can be detected that the maximum percentage change at the outlet of the reactor was found in the case Re = 100 and E = 77,000 j/mol when the initial concentration is turned from 1% to 5%. The minimum percentage increment in the enthalpy change can be seen for the extreme case of Re = 1000 and activation energy E = 80,000 j/mol.


### 4.4. Diffusivity of the Calcium Oxide in Water

The rate of spread of molecules in the solvent is known as diffusivity. Mathematically, it is the proportionality constant between the molar flux and concentration gradient due to molecular diffusion. Calcium oxide is solid and solvable in water easily. It is known very well that the diffusion rate of diffusive components is naturally very low. Once a compound starts to dissolve in the solvent, the diffusivity begins to increase as the concentration of that compound decreases. Here, we intend to measure the diffusivity of calcium oxide against the length and its impact by the parametric selection, and we throw a light on it (see Figures12(a)–12(c) and 13(a)–13(c)). It is clear from all the graphs that the initial diffusivity of calcium oxide for the present problem is around 1.5 × E−7 m2/s. By fixing the initial concentration and the activation energy, the mass diffusivity of calcium oxide is decreasing with the increasing Reynolds number (see Figure 12(a)). At the very low Reynolds number, the mass diffusivity is increasing at a nonlinear rate. The formation of calcium hydroxide is taken place normally at E = 75,000 j/mol. The present problem has tested the diffusivity of calcium oxide for 75,000 < E < 80,000. In Figure 12(b), diffusivity is decreasing with the increasing activation energy, providing that the initial concentration and the Reynolds number are kept constant. Diffusivity along the whole length of the reactor is always increasing whatever we put the initial concentration. The maximum diffusivity at the outlet is increasing with the increasing initial concentration of calcium oxide (see Figure 12(c)). The extreme cases for the Reynolds number, initial concentration, and activation energy are displayed in Figures 13(a)–13(c). In Figure 13(a), the diffusivity is checked with the initial concentration of 5% with the activation energy of 80,000 j/mol. It can be seen that the diffusivity is increasing at a very high rate as compared to the high Reynolds number. The maximum diffusivity at the outlet is decreasing with the increasing Reynolds number. In Figure 13(b), the extreme case for Reynolds number and initial concentration is tested with increasing activation energy. The diffusivity is decreasing with the increasing activation energy. For the extreme case of activation energy as well as Reynolds number, the maximum diffusivity at the outlet is increasing with initial concentration. Also, diffusivity for this case is increasing linearly (see Figure 13(c)).Figure 12
Diffusivity of calcium oxide in water: (a) I.C = 1%,E = 75,000 J/mol; (b) I.C = 1%, Re = 100; (c) Re = 100, E = 75,000 J/mol.
(a)(b)(c)Figure 13
Diffusivity of calcium oxide in water: (a) I.C = 5%,E = 80,000 J/mol; (b) I.C = 5%, Re = 1000; (c) Re = 1000, E = 80,000 J/mol.
(a)(b)(c)


### 4.5. Heat Source of Reaction

The energy earned or lost for a chemical reaction at a uniform temperature to retain all components balanced is known as the heat source of the reaction. In the present simulation, we are involving chemical kinetics to observe the heat and mass transfer problem where the chemical action of calcium oxide and water is taking place to give a product of calcium hydroxide. The reaction is exothermal. We estimate these results because we want to know about that energy balance to get all the products in a certain instant. In Figure14(a), the heat source of reaction is indicated against the length of the tubular reactor for all Reynolds numbers with a 1% initial concentration with an activation energy of 75,000 j/mol. It can be perceived at this level at the inlet of the reactor a 36,000 w/m3 heat source of reaction is needed. That heat source of reaction is decreasing along the length for each Reynolds number from 100 to 1000. However, for the minimum Reynolds number of 100, the heat source of reaction is decreasing quickly as compared to the higher Reynolds number. In Figure 15(a), the heat source of the reaction is presented with a higher initial concentration and activation energy for all Reynolds numbers. In the graph, we perceive that the heat source of the reaction is increasing along the length of the tubular reactor. The heat source of reaction hits a limit of 48,000 w/m3 at the exit of the channel. In the graph, the rate of increase of heat source of reaction is decreasing with the increase in Reynolds number. The heat source of reaction holds a negative relationship with activation energy if we observe along the length of the channel by fixing Re and initial concentration (see Figure 14(b)). However, with settling the initial concentration of species as 5% with a high Reynolds number, the heat source of reaction begins to increase along the length for particular activation energy (see Figure 15(b)). The impact of the heat source of reaction is also been observed by fixing Re and Reynolds number (see Figures 14(c) and 15(c)). It seems from these graphs the heat source of reaction reaches zero when the initial concentration increases from 3 to 5%.Figure 14
Heat source of reaction: (a) I.C = 1%,E = 75,000 J/mol; (b) I.C = 1%, Re = 100; (c) Re = 100, E = 75,000 J/mol.
(a)(b)(c)Figure 15
Heat source of reaction: (a) I.C = 5%,E = 80,000 J/mol; (b) I.C = 5%, Re = 1000; (c) Re = 1000, E = 80,000 J/mol.
(a)(b)(c)


### 4.6. Lewis Number

The Lewis number is nondimensional and often used to measure the ratio of thermal diffusivity to mass diffusivity. Moreover, another intention to measure the Lewis number is the comparison between the thermal and mass diffusivity that whether it is thermal diffusivity that dominates over mass diffusivity or vice versa. In Figures16(a)–16(c) and 17(a)–17(c), we see that for all the cases, the Lewis number is decreasing along the length of the reactor. This means that the mass diffusivity is always higher than the thermal diffusivity in all cases. In Figures 16(a) and 17(a), the Lewis number is found to decrease along the length of the reactor whenever the initial concentration and activation energy is fixed. At the low Reynolds number even for low initial concentration and the activation energy, the mass diffusivity is comparatively higher than the thermal diffusivity. Also, increasing the activation energy from 75,000 j/mol to 80,000 j/mol, the Lewis number is decreasing (see Figures 16(b) and 17(b)). The Lewis number is always decreasing in a nonlinear way for Re = 100. For a high Reynolds number Re = 1000, the Lewis number is decreasing linearly and attempts the minimum value at the outlet, which is optimum in the case of a high Reynolds number. From Figures 16(c) and 17(c), it was also shown that adding large values of the initial concentration for a fixed Reynolds number and activation energy will tend to decrease the Lewis number. In general, we can conclude that for the present case of decomposition of calcium oxide in water, mass diffusion is found always to be dominant over thermal diffusion.Figure 16
Lewis number along the length of the channel: (a) I.C = 1%,E = 75,000 J/mol; (b) I.C = 1%, Re = 100; (c) Re = 100, E = 75,000 J/mol.
(a)(b)(c)Figure 17
Lewis number along the length of the channel: (a) I.C = 5%,E = 80,000 J/mol; (b) I.C = 5%, Re = 1000; (c) Re = 1000, E = 80,000 J/mol.
(a)(b)(c)


## 4.1. Conversion Profile of Calcium Oxide in Water

We present the above graph to show the conversion profile of calcium oxide since five different initial concentrations are tested and mixed with water. Figures6(a)–6(c) are the graphs where we present the results for initial parametric studies. In Figure 6(a), the conversion profile is shown in the variation of the Reynolds number, where we can see the conversion of calcium oxide is increasing along the length of the reactor. For a Re = 100, the compound is converted to about 90%. The maximum conversion can be seen at the outlet of the reactor, and for a fixed concentration and the activation energy, it is decreasing with the increase in the Reynolds number. In Figure 6(b), the conversion profile shows positivity with a decrease in activation energy. It indicates that for a good decomposition at the low initial concentration for the Reynolds number, the activation might be good at around 70,000 to 80,000 in j/mol. In Figure 6(c), it can be suggested that a 100% conversion is taking place for the lower Reynolds number in taking the concentration between 3% and 5% for the lower Reynolds number. For higher values of the Reynolds number and the activation energy in the current values, the conversion profile is found to linearly increase along the length. The conversion is only 30 percent at the outlet of the reactor (see Figure 7(a)). Figure 7(b) shows that even taking a high initial concentration, the conversion profile is largely affected by activation energy. Figure 7(c) is revealing that all the initial high Reynolds numbers and activation give about the same conversion of calcium oxide.Figure 6
Conversion profile of CaO against the length of the reactor: (a) I.C = 1%,E = 75,000 J/mol; (b) I.C = 1%, Re = 100; (c) Re = 100, E = 75,000 J/mol.
(a)(b)(c)Figure 7
Conversion profile of CaO against the length of the reactor: (a) I.C = 5%,E = 80,000 J/mol; (b) I.C = 5%, Re = 1000; (c) Re = 1000, E = 80,000 J/mol.
(a)(b)(c)In Table2, the percentage change in the conversion profile is described for the initial concentration of 1% to onward. We can conclude that for lower or moderate Reynolds numbers, the percentage change remains consistent. The maximum percentage change can be found in the case of Re = 400 and E = 75,000 j/mol. In short, the high Reynolds number, as well as the activation energy, will never give a favor to the conversion profile of calcium oxide. Finally, we ought to suggest to get the maximum amount of production keep the Reynolds number must be lower.Table 2
Percentage improved conversion profile from 1% of initial concentration to onward.
ReE (J/mol)χpc 1–2%χpc 1–3%χpc 1–4%χpc 1–5%100750008.58909811.6531911.936311.939811007700016.6006138.1403959.4910468.162261007900010.4422524.097742.9224670.89008100800007.40431616.3872827.5954942.12024007500013.2646331.4327357.332692.87567400770007.06255215.5515126.0041339.33254400790003.402767.12153711.2084915.72893400800002.3438234.8353457.49073910.32883800750007.91078517.6281329.9465846.23588800770003.8548118.11751712.8652818.20152800790001.8319573.7527875.7709327.894467800800001.2603162.5624723.9084245.3015711000750006.54109214.2906823.6664235.360861000770003.1513086.57028810.300914.394411000790001.4960423.0521964.6668926.3502531000800001.030972.0865193.1735314.287668


## 4.2. Temperature Distribution in the Reactor

As we are testing an exothermic reaction from the examples of chemical engineering, definitely the temperature along the selected domain is increasing along the length. However, we are interested to control the temperature distribution with the parameters used in the problem. The temperature distribution in the middle of the channel is presented in Figures8(a)–8(c) and 9(a)–9(c). Initially, the temperature of 312 K is imposed at the inlet of the channel. An increase in the temperature along the length can be seen in Figure 8(a) with the increase of the Reynolds number. The increment is nonlinear, and the maximum temperature at the outlet of the reactor is decreasing with the increase in the Reynolds number. In Figure 9(a), we also found an increment in the temperature along the length of the reactor for high initial concentration and the activation energy and found 4 K more temperature increment when compared with lower initial concentration and the activation energy. For the extreme case in Figure 8(b), the increment in temperature is found linearly. With fixing the Reynolds number and initial concentration, the maximum temperature at the outlet is decreasing with the increase in activation energy (see Figures 8(b) and 9(b)). It is also a clear message from these graphs that for the high Reynolds number as well as the initial concentration, the inclination in the temperature along the length is very high and linear. Moreover, we can understand from the graphs (Figures 8(c) and 9(c)) adding that more initial concentrations of calcium oxide will tend to increase the maximum temperature at the outlet. For the low Reynolds number as well as the activation energy, we found that an increment of the temperature at the outlet is high. Finally, we are presenting Table 3, where we are presenting the percentage increment in the maximum temperature at the outlet of the channel when the initial concentration is moved from 1% onward.Figure 8
Temperature distribution along the length of the channel: (a) I.C = 1%,E = 75,000 J/mol; (b) I.C = 1%, Re = 100; (c) Re = 100, E = 75,000 J/mol.
(a)(b)(c)Figure 9
Temperature distribution along the length of the channel: (a) I.C = 5%,E = 80,000 J/mol; (b) I.C = 5%, Re = 1000; (c) Re = 1000, E = 80,000 J/mol.
(a)(b)(c)Table 3
Percentage change of the temperature when initial concentration is changing from 1% to onward.
ReE (J/mol)Tpc 1–2%Tpc 1–3%Tpc 1–4%Tpc 1–5%100750002.187634.401486.508668.59267100770001.629693.868266.6599.23803100790000.776581.754323.051174.90507100800000.515411.120771.850642.76151400750001.124572.615344.71037.75102400770000.507641.095531.791272.63816400790000.226740.469610.730951.01363400800000.152340.3120.479690.65621800750000.598661.305722.163983.24566800770000.267560.557030.872081.21741800790000.120990.246520.376920.51257800800000.081710.16550.251480.339771000750000.484391.039531.687622.462211000770000.217170.44870.696520.962971000790000.098550.20010.304840.412961000800000.066630.134650.204110.27507From Table3, it can be understood that for the present problem, the percentage change in the temperature at the outlet is maximum for Re = 100 and E = 77,000 j/mol and minimum for Re = 1000 and E = 80,000. We can also conclude that an increase in the initial concentration of the compound always supports an increase in the temperature at the outlet of the reactor.


## 4.3. Enthalpy Change in the System

Enthalpy is known as the heat content in the system. When a chemical reaction is encountered by an exothermic reaction, the enthalpy change is increasing. It is also rigorously defined as the combination of internal energy and the product of pressure and volume. Involving so many components in determining the internal energy, the enthalpy in the system is almost impossible to determine. Because having large applications of enthalpy change are very important, it is easy to determine because of possessing the property of state function. The computation of enthalpy change is done here due to the variation of chosen parameters in Figures10(a)–10(c) and 11(a)–11(c) until we have determined that a low speed and activation energy give favor to complete the chemical reaction and about 90% conversion. Figure 10(a) shows that the enthalpy change is decreasing with an increase in the Reynolds number for a fixed concentration and activation energy. Figure 11(b) shows for higher concentration and fixed activation energy, the enthalpy change is decreasing at the outlet of the reactor with increasing Reynolds number. For a fixed initial concentration and the activation energy, an increase in the Reynolds number gives a sufficient decrease in the enthalpy change. In Figure 10(b), we see that for a fixed initial concentration and Reynolds number, the enthalpy change in the reactor is increasing along the length and maximum at the outlet of the reactor. With increasing activation energy, the maximum enthalpy is decreasing at the outlet of the reactor. We also test the case by increasing the Reynolds number and initial concentration, and the enthalpy change is increasing in a nonlinear way for a lesser Reynolds number. Figures 10(c) and 11(c) show that adding more initial concentration of the solid gives favor increasing the enthalpy change at the outlet of the reactor. It can be also seen that for a 5% concentration when the Reynolds number and the activation energy are taken from lower to upper-level cases, the enthalpy change decrease is about 60%. Finally, we attach Table 4, where we are going to present the benefit for enthalpy change when the percentage is changing from 1% to onward. For simplicity, we make the idea that the values will be presented in the percentage increment.Figure 10
Enthalpy change of the system: (a) I.C = 1%,E = 75,000 J/mol; (b) I.C = 1%, Re = 100; (c) Re = 100, E = 75,000 J/mol.
(a)(b)(c)Figure 11
Enthalpy change of the system: (a) I.C = 5%,E = 80,000 J/mol; (b) I.C = 5%, Re = 1000; (c) Re = 1000, E = 80,000 J/mol.
(a)(b)(c)Table 4
Percentage change in the maximum enthalpy of initial concentration from 1% to onward.
ReE (J/mol)ΔHp 1–2%ΔHp 1–3%ΔHp 1–4%ΔHp 1–5%1007500028.1465856.6304983.74196110.55541007700022.6949853.8692592.73312128.64861007900011.6917726.4121145.9368773.84811100800007.97617817.3443628.6393842.735414007500016.3517538.0283368.48996112.7035400770007.85764816.9575527.7267540.83564400790003.6328797.52415311.7113216.24047400800002.4656565.0497797.76378610.6208800750009.17081220.0021533.1496549.71962800770004.2637278.87655813.897219.40011800790001.9668754.00756.1273768.332631800800001.3358122.7057374.1114925.5549261000750007.5161316.1326.1862438.205431000770003.4837497.19791611.1734415.447791000790001.6072683.2634914.9716196.7348711000800001.0918162.2063243.344454.507178From Table4, it can be detected that the maximum percentage change at the outlet of the reactor was found in the case Re = 100 and E = 77,000 j/mol when the initial concentration is turned from 1% to 5%. The minimum percentage increment in the enthalpy change can be seen for the extreme case of Re = 1000 and activation energy E = 80,000 j/mol.


## 4.4. Diffusivity of the Calcium Oxide in Water

The rate of spread of molecules in the solvent is known as diffusivity. Mathematically, it is the proportionality constant between the molar flux and concentration gradient due to molecular diffusion. Calcium oxide is solid and solvable in water easily. It is known very well that the diffusion rate of diffusive components is naturally very low. Once a compound starts to dissolve in the solvent, the diffusivity begins to increase as the concentration of that compound decreases. Here, we intend to measure the diffusivity of calcium oxide against the length and its impact by the parametric selection, and we throw a light on it (see Figures12(a)–12(c) and 13(a)–13(c)). It is clear from all the graphs that the initial diffusivity of calcium oxide for the present problem is around 1.5 × E−7 m2/s. By fixing the initial concentration and the activation energy, the mass diffusivity of calcium oxide is decreasing with the increasing Reynolds number (see Figure 12(a)). At the very low Reynolds number, the mass diffusivity is increasing at a nonlinear rate. The formation of calcium hydroxide is taken place normally at E = 75,000 j/mol. The present problem has tested the diffusivity of calcium oxide for 75,000 < E < 80,000. In Figure 12(b), diffusivity is decreasing with the increasing activation energy, providing that the initial concentration and the Reynolds number are kept constant. Diffusivity along the whole length of the reactor is always increasing whatever we put the initial concentration. The maximum diffusivity at the outlet is increasing with the increasing initial concentration of calcium oxide (see Figure 12(c)). The extreme cases for the Reynolds number, initial concentration, and activation energy are displayed in Figures 13(a)–13(c). In Figure 13(a), the diffusivity is checked with the initial concentration of 5% with the activation energy of 80,000 j/mol. It can be seen that the diffusivity is increasing at a very high rate as compared to the high Reynolds number. The maximum diffusivity at the outlet is decreasing with the increasing Reynolds number. In Figure 13(b), the extreme case for Reynolds number and initial concentration is tested with increasing activation energy. The diffusivity is decreasing with the increasing activation energy. For the extreme case of activation energy as well as Reynolds number, the maximum diffusivity at the outlet is increasing with initial concentration. Also, diffusivity for this case is increasing linearly (see Figure 13(c)).Figure 12
Diffusivity of calcium oxide in water: (a) I.C = 1%,E = 75,000 J/mol; (b) I.C = 1%, Re = 100; (c) Re = 100, E = 75,000 J/mol.
(a)(b)(c)Figure 13
Diffusivity of calcium oxide in water: (a) I.C = 5%,E = 80,000 J/mol; (b) I.C = 5%, Re = 1000; (c) Re = 1000, E = 80,000 J/mol.
(a)(b)(c)


## 4.5. Heat Source of Reaction

The energy earned or lost for a chemical reaction at a uniform temperature to retain all components balanced is known as the heat source of the reaction. In the present simulation, we are involving chemical kinetics to observe the heat and mass transfer problem where the chemical action of calcium oxide and water is taking place to give a product of calcium hydroxide. The reaction is exothermal. We estimate these results because we want to know about that energy balance to get all the products in a certain instant. In Figure14(a), the heat source of reaction is indicated against the length of the tubular reactor for all Reynolds numbers with a 1% initial concentration with an activation energy of 75,000 j/mol. It can be perceived at this level at the inlet of the reactor a 36,000 w/m3 heat source of reaction is needed. That heat source of reaction is decreasing along the length for each Reynolds number from 100 to 1000. However, for the minimum Reynolds number of 100, the heat source of reaction is decreasing quickly as compared to the higher Reynolds number. In Figure 15(a), the heat source of the reaction is presented with a higher initial concentration and activation energy for all Reynolds numbers. In the graph, we perceive that the heat source of the reaction is increasing along the length of the tubular reactor. The heat source of reaction hits a limit of 48,000 w/m3 at the exit of the channel. In the graph, the rate of increase of heat source of reaction is decreasing with the increase in Reynolds number. The heat source of reaction holds a negative relationship with activation energy if we observe along the length of the channel by fixing Re and initial concentration (see Figure 14(b)). However, with settling the initial concentration of species as 5% with a high Reynolds number, the heat source of reaction begins to increase along the length for particular activation energy (see Figure 15(b)). The impact of the heat source of reaction is also been observed by fixing Re and Reynolds number (see Figures 14(c) and 15(c)). It seems from these graphs the heat source of reaction reaches zero when the initial concentration increases from 3 to 5%.Figure 14
Heat source of reaction: (a) I.C = 1%,E = 75,000 J/mol; (b) I.C = 1%, Re = 100; (c) Re = 100, E = 75,000 J/mol.
(a)(b)(c)Figure 15
Heat source of reaction: (a) I.C = 5%,E = 80,000 J/mol; (b) I.C = 5%, Re = 1000; (c) Re = 1000, E = 80,000 J/mol.
(a)(b)(c)


## 4.6. Lewis Number

The Lewis number is nondimensional and often used to measure the ratio of thermal diffusivity to mass diffusivity. Moreover, another intention to measure the Lewis number is the comparison between the thermal and mass diffusivity that whether it is thermal diffusivity that dominates over mass diffusivity or vice versa. In Figures16(a)–16(c) and 17(a)–17(c), we see that for all the cases, the Lewis number is decreasing along the length of the reactor. This means that the mass diffusivity is always higher than the thermal diffusivity in all cases. In Figures 16(a) and 17(a), the Lewis number is found to decrease along the length of the reactor whenever the initial concentration and activation energy is fixed. At the low Reynolds number even for low initial concentration and the activation energy, the mass diffusivity is comparatively higher than the thermal diffusivity. Also, increasing the activation energy from 75,000 j/mol to 80,000 j/mol, the Lewis number is decreasing (see Figures 16(b) and 17(b)). The Lewis number is always decreasing in a nonlinear way for Re = 100. For a high Reynolds number Re = 1000, the Lewis number is decreasing linearly and attempts the minimum value at the outlet, which is optimum in the case of a high Reynolds number. From Figures 16(c) and 17(c), it was also shown that adding large values of the initial concentration for a fixed Reynolds number and activation energy will tend to decrease the Lewis number. In general, we can conclude that for the present case of decomposition of calcium oxide in water, mass diffusion is found always to be dominant over thermal diffusion.Figure 16
Lewis number along the length of the channel: (a) I.C = 1%,E = 75,000 J/mol; (b) I.C = 1%, Re = 100; (c) Re = 100, E = 75,000 J/mol.
(a)(b)(c)Figure 17
Lewis number along the length of the channel: (a) I.C = 5%,E = 80,000 J/mol; (b) I.C = 5%, Re = 1000; (c) Re = 1000, E = 80,000 J/mol.
(a)(b)(c)


## 5. Conclusion

In the current fluid flow problems, we put a two-dimensional axisymmetric multicomponent tubular reactor under investigation of an exothermic reaction of calcium oxide and water with a standard enthalpy ofΔH=−64,800j/mol. The initial concentration of the calcium oxide was taken from 1% to 5% as soluble, and the water was considered as the solvent. Due to the presence of water in access, the total rate of reaction of this chemical reaction was considered as in first-order and depended upon the concentration of calcium oxide. The rate of reaction is given by the forward rate constant multiplied by the concentration of calcium oxide. The forward rate constant was calculated by the Arrhenius equation where the activation energy was tested on the trial bases in the range from 75,000 j/mol to 80,000 j/mol. About six different constitutive equations were observed to handle the fluid flow, heat, and mass transfer problem with the help of the finite element package of COMSOL Multiphysics 5.6. To handle an exothermic reaction in the tube, it was necessary to provide a cooling environment in the reactor with the application of a cooling jacket surrounding in the reactor. The pattern of the fluid flow was handled with the use of a nondimensional Reynolds number from 100 to 1000, which is specially calculated with the use of the total inflow rate.(i)
It was found that the low speed of the fluid of low Reynolds number will give the best conversion of calcium oxide in the reactor. Although we found that at Re = 100, the calcium oxide is converted about 90%, and when considering the initial concentration in the range from 3% to 5%, it will give a full 100% conversion for Re = 100 and with an activation energy of 75,000 j/mol. The best conversion percentage can be achieved from 1% to onward at Re = 400 andE = 75,000 j/mol.(ii)
It was also found that for a lower Reynolds number, the temperature distribution along the channel is increasing but in a nonlinear way, whereas for the high Reynolds number, a linear pattern of increase has been recognized. The maximum temperature at the outlet is decreasing with the increase in Reynolds number. For the present problem, decrement at the outlet is about 4 K for high Reynolds number and the fixed activation energy and initial concentration. It was also suggested that keeping a low Reynolds number in an exothermic reaction will yield an optimum temperature distribution.(iii)
It was found that for all the cases, the enthalpy change at the inlet of the channel is identical. The enthalpy change is increasing always along the length of the reactor, but the maximum enthalpy at the outlet of the reactor is decreasing with the increase in the Reynolds number and the activation energy. However, adding more initial concentrations of calcium oxide gives more favor to enthalpy change in the system. It can be suggested a full conversion profile of the compound will always give the optimum enthalpy at the outlet of the channel.(iv)
The rate of diffusion of calcium oxide was found in the current modeling and simulation. It was deducted that for maximum diffusivity, calcium oxide at the outlet is decreasing with the increasing Reynolds number, activation energy, and the initial concentration, providing that the remaining parameters are kept fixed or constant. No doubt increasing these parameters will give a little boost increase in the maximum diffusion rate at the outlet of the reactor.(v)
The parameters vary the heat source of reaction in balancing all components used in the chemical reaction. It is elaborated that the heat source of reaction is decreasing along the length of the reactor to fix low initial concentration at the activation energy and increasing for high initial concentration and the activation. Also, it was observed that for a fixed low Reynolds number and the low activation energy, the heat source of reaction reaches zero for high initial concentration.



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