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MATHEMATICAL MODELING OF THE HEAT TRANSFER PROCESS IN A DENSE BLOWN LAYER OF GRANULAR MATERIAL
Irina Boshkova1, Igor Mukminov2
department of Oil and Gas Technologies, Engineering and Power Engineering, Odesa National University of Technology, Odessa, Ukraine ORCID: https://orcid.org/0000-0001-5989-9223
2Department of Oil and Gas Technologies, Engineering and Power Engineering, Odesa National University of Technology, Odessa, Ukraine ORCID: https://orcid.org/0000-0002-3674-9289
E] Corresponding author: Irina Boshkova, e-mail: boshkova.irina@gmail.com
ABSTRACT
The object of research: a mathematical model of unsteady heat transfer between particles forming a dense layer in a heat exchange channel and an air flow passing through the layer. Investigated problem: obtaining a mathematical model adequate to the physical phenomenon will significantly expand the field of research, reduce the time for determining the thermal characteristics of heat transfer and data processing.
The main scientific results: an explicit analytical dependence is obtained that allows calculating the local temperature of the layer in an arbitrary cross-section of the channel at selected moments of time. The presented analytical dependence was verified by analyzing
the calculated data on the material temperatures in the channel obtained by varying the main physical quantities: the effect of the heating duration, heat transfer coefficient, porosity, and density on the temperature distribution was studied. The temperatures along the length of the channel were calculated at different points in time with the following initial data: material - gravel: c=860, p=1500, X=0.6, s=0.4; aud=800. Based on the results of the calculations for the time t=60 s, t=900 s, t=1800 s, t=2400 s, t=1800 s, the graphs shown in the paper are constructed. It is shown that the mathematical model adequately describes the process of heating a layer of granular material in a heat exchange channel and allows obtaining satisfactory data on the temperature distribution along the length at different points in time. The calculated data correlate satisfactorily with the experimental results. Innovative technological product is an analytical dependence obtained for the first time, which is necessary for inclusion in the calculation method of regenerative heat exchangers intended for the utilization of low-potential heat flows.
The area of practical use of the research results is the heat engineering calculations of regenerative heat exchangers with a dense granular packing. The consumers of the developed regenerators are greenhouses and food industry. Such industries are characterized by a low-potential level of thermal emissions, the heat of which is difficult to utilize using existing heat exchange equipment.
Scope of the innovative product: design bureaus focused on the design of innovative heat exchangers for the utilization of low-potential heat flows, and research laboratories studying the processes of heat exchange between a dense layer of granular material and a through gas (air) flow.
© The Author(s) 2022. This is an open access article under the Creative Commons CC BY license
1. Introduction
1. 1. The object of research
The object of research is a mathematical model of the heat exchange process between a dense layer of particles and a through gas flow, which takes into account the conditions of heat exchange in the channel of a regenerative heat exchanger with a dense nozzle made of granular material intended for the utilization of low-potential heat.
1. 2. Problem description
Currently, research has intensified to develop heat exchangers designed to utilize low-potential heat flows, such as heat flows from exhaust ducts or air flows in greenhouses. One of the prom-
ARTICLE INFO
Article history: Received date 15.11.2022 Accepted date 22.12.2022 Published date 30.12.2022
Section:
Chemical industry
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KEYWORDS
unsteady heat transfer channel
method of separation of variables differential analytical dependence local temperature
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ising areas of designing such devices is the development of heat regenerators with a nozzle in the form of a dense layer of granular material designed to accumulate heat during the heating period and transfer it to the cooling period. Experimental data prove the feasibility of these developments, but conducting research in a wide range of operating parameters is difficult and time-consuming. Obtaining an adequate mathematical model will significantly expand the scope of research and reduce the time for determining the main thermal characteristics of the process. At present, there are no exact analytical solutions to mathematical models describing the unsteady heat transfer process in a dense layer of granular material. The ultimate goal of such a solution is to obtain an analytical dependence for calculating the temperature of the granular material at an arbitrary point along the length of the channel at a given time of the heating or cooling process. Considerable attention is paid to modeling the processes of heat transfer between gas and solid particles [1], due to the importance of knowledge about the influence of certain factors and conditions on the temperature distribution and heat transfer efficiency. To calculate the technologically important characteristics of the process and develop recommendations for its rational organization, it is necessary to maximize the approximation of the uniqueness conditions to the individual problem under consideration. In [2], heat transfer processes are modeled, and the solutions obtained allow determining the temperature and moisture content fields for devices with dense layered granular systems used in metallurgy, chemical A clear description of the processes of transfer in a dense, stationary layer of dispersed material is complicated by the fact that it is a two-component gas-solid particle system, with abrupt changes in physical properties and parameters at the interface between the components. When developing methods for calculating layered apparatus, various simplified models are used [3, 4], based on certain assumptions. One- and two-component models are quite reasonable. In one-component models, the layer is considered as a quasi-continuous medium with effective transfer coefficients. Such models are attractive because of their simplicity and can be used only in a limited area of parameter variation, when the temperatures of the gas and solid components are almost the same [5].
1. 3. Suggested solution to the problem
To solve the problem, namely, to obtain an analytical dependence for calculating the local temperature of the granular material under non-stationary heat transfer, it is necessary to develop a mathematical model. To solve the mathematical model of thermal conductivity in a differential form, taking into account the intercomponent heat transfer between the gas flow and the particles of the dense layer, it is advisable to use the method of separation of variables. The conformity of the mathematical model to the physical process is determined by its verification, namely, by analyzing the calculated data when the defining characteristics of the process change.
Aim of research was to develop a mathematical model and obtain a calculated dependence for the local temperature of a dense layer of bulk material when it is heated by a through gas flow.
2. Materials and Methods
The blown dense layer of granular material is considered as a system of interpenetrating components - gas and solid, in which heat transfer processes take place. The dense layer is considered as a quasi-homogeneous medium with effective transfer coefficients. The effective heat transfer coefficient takes into account the heat transfer in the particles (product), through the contacts and the gas gap between them. The intercomponent heat transfer in the layer is taken into account using the appropriate heat transfer coefficient, the value of which is determined experimentally [6, 7]. The scheme of the channel with a dense layer of granular material is shown in Fig. 1.
The model is based on the equation of thermal conductivity of a layer of granular material with a porosity e in a cylindrical channel, taking into account the heat exchange between particles and the gas flow, represented by dependence (1) [8]:
\ dtm (d2t 1 dt d2t 1 52t) t_ , / x
(1-e)-pm ■cm —m = Xm-I-+--+-+--| + a(x,z)-a (tg-tm ), (1)
v ' Sm m dx m ^dr2 r dr dz2 r2 d92) K ' Vg m'
where a - the specific surface area of the particles, t - time, tm - the temperature of the solid material, tg - the temperature of the gas flow, r - the radius of the channel, z - the longitudinal coordinate,
pm, cm - the density and heat capacity of the material, X*m - effective thermal conductivity of the solid material, 9 - the angle of rotation along the cross-section in the channel, a - the intercompo-nent heat transfer coefficient.
fa V0=tr-tm0
V=tr-t z=0
4-2-►
Fig. 1. Scheme of a channel with a dense layer of granular material
To solve Equation (1), it is advisable to apply the method of separating variables [9] with certain simplifications of the equation itself in accordance with the conditions of heat transfer in a channel with granular material. The solution to equation (2) should be obtained by using the method of separation of variables [9].
2. 1. Research procedure
The channel is assumed to be thermally insulated. This assumption is based on the actual conditions of use of the heat exchange section, for which it is important to minimize losses to the
d 2t
environment. Therefore, in equation (1) let's take —7 = 0. There is also no change in temperature
dr
1 d t
along the angle 9, so--- = 0. Also, instead of temperature, an excessive $ = t -tm.
r cfy
Then equation (1) becomes the following:
I1 -s)-Pm •cm 'fp^ + a( ^ T)"a •($) . (2)
The following conditions of unambiguity have been adopted:
1) geometric: channel length Z;
2) physical: A=const, c=const, a=const; tg = tg;
3) initial: at t=0, 3 = 30; = tg - tm;
4) boundary condition: t>0, z = 0: — = 0;
dz
5) is the boundary condition on the surface of contact between the solid particle layer and
the air: ) =a-(1
Boundary condition (4) shows that there is no heat transfer at the end of the channel. To simplify the derivation of the calculation formulas of the mathematical model, the coordinate z=0 is taken at the output, since in this case it is convenient to obtain constant integrations.
A common solution for $( z, t) = z), by multiplying the solutions of the two equa-
tions (6) and (7). Boundary and initial conditions are used to find the integration constants. Then let's obtain the following equation:
"T^f I-72- Aam • k 2 • Pm- cm + a acp ' Z,
C3 • ^ • Cr jam- k • Pm" Cm + a Ocp ' c°s(^-1 -)
J a • p • c
rv/\ Vmrmm
Z, T) = -1 -"- (3)
Ja • p • c
\ m r m m
where c = c j +c2
To determine the constant k, let's apply the boundary condition (5) on the contact surface of the solid particle layer with air.
To find k, let's use the method of [10] and obtain:
X •
Ja • k2 • p
Ai m ~ m
a
• (1 -*)■ J
a • p • c
m ~ m m
= ctg
Ja • k2 • p
y m ~ m
4
a • p • c
m ~ m m
(4)
Assuming that for a layer of granular material Bi =
a• Z !• (1 -e)
equation (4) takes the form:
kZ Bi
ctg (^ ).
(5)
From the characteristic equation (5), it is possible to obtain the value of k. Analysis of (11) shows that for each value of Bio there are many solutions. The simplest way to solve
( /-T2- ^
Jam ■ k ■ pm ■ cm +a • ar_ • z this equation is graphically [10]. Let's denote g (k) = ctg m m m '
f ( k ) =
X • Ja • k2 • p • c +a-a
m y m r m m q
4-
a • p • c
m ~ m m
and
and plot the corresponding graphs. The intersection of the
a (1 -S)" ' Pm ' Cm
cotangent g(k) with the curve f(k) gives the value of the root k of the characteristic equation (5). Fig. 2 shows an example of graphs obtained at Bio=10 (X=0.6 W/mK, s=0.4; Z=0.3 m, a=13 W/(m2K)): The exact values of the points of intersection of g(k) with f(k) were determined. The obtained values of k are summarized in Table 1 for different Bio.
Fig. 2. Graphical finding of the roots k of the characteristic equation (5) at Bio=10
Table 1
Values of the roots of the characteristic equation.
Bio=4
f(k)=g(k) 1.64 2.02 2.41 2.79 3.18 3.57 3.96
k 11.42 33.52 48.31 61.34 73.55 85.29 96.34
Bio=10
f(k)=g(k) 1.36 1.62 1.87 2.13 2.39 2.65 2.91
k 17.21 38.89 54.27 67.79 80.39 92.46 104.18
Bio=40
f(k)=g(k) 0.68 0.74 0.80 0.87 0.998 1.06 1.13
k 31.25 57.56 76.59 92.92 121.74 135.06 147.91
S = C
Ja-c•p-k2
+ a • a
Va-c-f
• cos
: • Va- c • p-k2
+a a
Va-c-f
(6)
There are many solutions to the roots of equation (6). Thus, each found value of the root k will correspond to a specific temperature distribution. The corresponding solutions obtained will satisfy the conditions of the differential equation for any values of the constants c1, c2, ..., cn, but none of these solutions will correspond to the actual temperature distribution at the initial time. However, by superimposing an infinite number of such distributions with an appropriate choice of cn values, it is possible to reproduce any real temperature dependence at the initial time. Based on the above, the general solution can be represented by the sum of an infinite series.
»=! Cn
Va- c-p-k2
+ a- a
Va-c-f
• cos
yja-c • p-k2
+ a • a
Va-c-(
(7)
The constant cn in equation (7) is found from the initial conditions and the methodology [10].
1
Cn = - [ J] ,
(8),
where
S =
Va- c-p-k2 +a-aud ■ Z . Va• c-p-k2 +a-aud ■ Z
Ja-c-(
V°-c- (
• y/a-c-p + yj a- c- p-k2 +a- aud ■ Z
(9)
2-*Ja- c- p
z
J = J F ( x )• M-
cos
yja • c • p • k2
Va-c • p
+a a
A
dz = • sin
Va-c•p-k
2 - A
+ a • ^ • Z
Va-c-f
. (10)
n=1
Scope of application of the obtained dependence (assumptions):
1) the gas flow temperature at the inlet is constant;
2) during heat exchange between the bed and the gas, the effective temperature is established in cross-sections along the length of the channel;
3) the thermal insulation of the outer surface of the duct is perfect, i.e. there are no heat losses to the environment;
4) the air flow rate determines the heat transfer coefficient (intercomponent heat transfer coefficient), which is calculated based on experimental data;
5) the heat transfer coefficient is assumed to be constant for the heating (cooling) period.
This assumption introduces an error in the calculated data, so if a more accurate calculation
is needed, the empirical dependence a=fx) obtained for a particular type of material should be used.
The obtained dependence was verified by analyzing the calculated data on the temperatures of the material in the channel obtained by varying the main physical quantities characteristic of the process under analysis: the effect of the heating duration, heat transfer coefficient, porosity, and density. The temperatures along the length of the channel at different points in time were calculated with the following initial data: material - gravel: c=860, p=1500, X=0.6, e=0.4; 0^=800. The length of the channel: 1 m. Based on the results of the calculations for the time t=60 s, t=900 s, t=1800 s, t=2400 s, t=1800 s, the graphs shown in Fig. 3 were constructed. The initial temperature of the material is 20 °C, the inlet air temperature is 40 °C.
Fig. 4 shows the results of calculating the temperature of the material along the length of the channel at different values of the heat transfer coefficient (which is associated with a change in air flow) and at different values of the heat transfer coefficient (which is associated with a change in air flow) at t=1800 s.
Fig. 3. Change of the layer temperature along the channel length at different time points
a=17 W/m2K, £=0.4
Fig. 4. Distribution of material temperature along the length of the channel at different values of a
The choice of a=17 W/m2K was determined by experimental data conducted under similar conditions [11].
Fig. 5 shows the results of calculating the temperature along the length of the channel at different values of the porosity e.
Fig. 6 shows the results of calculating the temperature of the layer along the length of the channel at different material densities.
Fig. 5. Effect of porosity e on the heating of the material layer at a=17 W/m 2K, t=1800 s
The calculations show that the mathematical model clearly reflects the physical picture of the influence of the main physical characteristics on heat transfer.
Fig. 6. Distribution of layer temperature along the channel length at different material densities
a=17 W/m2K, t=1800 s, £=0.4
4. Discussion
The analysis of the graph (Fig. 3) shows that the material temperature decreases along the length of the channel, and with an increase in the duration of heating the material by the air flow at a given heat transfer coefficient, the material temperature increases. When comparing the calculated data from the obtained mathematical model with the results of experiments, it turns out that after 3600 s, the average temperature of the material was 32 °C, which is 5.8 °C lower than the weighted average calculated one. This is due to the fact that in the experiment, which is conducted in the field, the inlet air temperature varied. In addition, the mathematical model does not take into account thermal and moisture processes that can affect the heating of the bed particles. The calculations of the material temperature along the length of the channel (Fig. 4) at different values of the heat transfer coefficient (due to changes in air flow) show that the mathematical model clearly reflects the physical picture of the effect of flow on heat transfer. An increase in the heat transfer coefficient leads to an intensification of the heat transfer process and, accordingly, an increase in the temperature growth rate. Thus, at a=4 W/m2K, the inlet temperature was t=27.8 °C, at the outlet t=24.8 °C, and at a=16 W/m2K, the inlet temperature was t=39.6 °C, at the outlet t=34.7 °C. Reducing the porosity leads to an increase in the temperature of the material, which is associated with a decrease in the proportion of solid particles and, accordingly, a decrease in the volume of the heated material. A comparative assessment of the values shows a significant effect of the porosity: for example, a decrease in £ by a factor of 1.17 (from £=0.42 to £=0.36) leads to a 1.42-fold increase in the temperature of the material at the inlet to the channel - from a temperature of t=24.3 °C to t=33.8 °C. The effect of material density on temperature was analyzed. The material with a density of 800 kg/m3 (expanded clay) heated up by 14.6 °C higher than the gravel fraction with a density of 1700 kg/m3 in 1800 s. Thus, the effect of density, determined by calculation, corresponds to the physical picture of heat transfer phenomena under the conditions under consideration.
The limitation of the study is that the models do not take into account heat losses from the channel surface. This assumption made it possible to obtain an exact analytical solution for calculating the temperature as a function of time and coordinate. If heat losses cannot be neglected, the problem becomes two-dimensional and calculations must be carried out by numerical methods. It is also assumed that the temperature of the gas entering the channel is constant. During the operation of real thermal regenerators, this assumption is not always fulfilled. With significant fluctuations in gas temperature, it is recommended to introduce a function into the mathematical model that takes these changes into account.
5. Conclusions
The obtained mathematical model adequately describes the process of heating a layer of granular material in a heat exchange channel and allows obtaining satisfactory data on the temperature distribution along the length at different points in time. The influence of density, heat transfer coefficient, duration of the heating process and porosity, determined by calculation, corresponds to the physical picture of heat transfer phenomena under the conditions under consideration.
When obtaining data using the presented temperature dependence, the assumptions made should be taken into account, in particular, the absence of heat loss to the environment, the constancy of air f low (heat transfer coefficient), and the fact that mass transfer processes are not taken into account, which can be significant at high humidity of the gas (air) flow or in the presence of condensation on the surface of particles in the material layer.
The calculated data correlate satisfactorily with the experimental data obtained under similar conditions. Thus, after 3600 s, the average temperature of the material in the experiment was 32 °C, which is 5.8 °C lower than the weighted average calculated temperature. To refine the calculation, it is necessary to have accurate data on the thermophysical characteristics of the material, its porosity, and the value of the specific surface area of the particles.
Conflict of interest
The authors declare that there is no conflict of interest in relation to this paper, as well as the published research results, including the financial aspects of conducting the research, obtaining and using its results, as well as any non-financial personal relationships.
Funding
The study was performed without financial support.
Data availability
Data will be made available on reasonable request.
Acknowledgments
ILB highly appreciates warm hospitality and generous contribution of Prof. Dr. Ulrich K. Deiters to mastering modern math packages during her research stay under the DAAD program at the University of Cologne.
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