CC BY
10
MSC 76S05
DOI: 10.14529/ mmp 180410
MATHEMATICAL MODEL OF HEATING OF PLANE POROUS HEAT EXCHANGER OF HEAT SURFACE COOLING SYSTEM IN THE STARTING MODE
V.l. Ryazhskikh1, D.A. Konovalov1, S.V. Dakhin1, Yu.A. Bulygin1,
V.P. Shatskiy2
1
2
E-mails: ryazhskikh_vi@mail.ru, dmikonovalov@yandex.ru, secretary-d21203705@yandex.ru, bulyginu@gmail.com, sha.vladim@yandex.ru
Based on the conjugate Darcy-Brinkman-Forchheymer hydrodynamic model and Schumann thermal model with boundary conditions of the second kind, a model with lumped parameters was proposed by means of geometric 2D averaging to identify the integral kinetics of the temperature fields of a porous matrix and a Newtonian coolant without phase transitions. The model was adapted for a heat-stressed surface by means of a porous compact heat exchanger with uniform porosity and permeability, obeying the modified Kozeny-Carman relation, in the form of a Cauchy problem, the solution of which was obtained in the final analytical representation for the average volume temperatures of the coolant and the porous matrix. The possibility of harmonic damped oscillations of the temperature fields and the absence of coolant overheating in the starting condition of the cooling system were shown. For the dimensionless time of establishing the stationary functioning of the porous heat exchanger, an approximate estimate was obtained correlating with the known data of computational and full-scale experiments.
Keywords: flat porous heat exchanger; heat-stressed surface; boundary conditions of the second kind; time to settle a stationary warm regime.
Introduction
The classical use of porous materials for the intensification of single-phase heat-exchange processes [1] is greatly supplemented by new substantive applications, such as geothermal energy supply [2], technologies for obtaining structured polymeric materials [3], waste utilization in combustion in porous media [4], bioconvection in porous tissues of living organisms [5], etc. This requires consideration of transport phenomena in porous systems under nonstationary conditions [6].
The analysis of a wide range of phenomenological mathematical models of non-stationary hydrothermal fields in porous media [7] showed that the representation of the fundamental Darcy-Brinkman-Forchheymer and Schumann equations in the Hsu-Cheng form for the laminar flow regime of the Newtonian coolant [8]
V • v = 0,
(1)
Pf \dv (v • V) v
+
(2)
£ dr £
(3)
(1 - е) (рор)з7^ = V (AS • Vis) Т asf asf (is - if) (4)
dts
Or
is the most convincing when interpreting experimental data and meets the criteria of qualitative and quantitative adequacy, where t is time; (p,^,cv)f is density and dynamic viscosity of the liquid; e is porosity; (p, cp)s is density and heat capacity of the porous skeleton; v is a liquid velocity vector; g is a gravitational acceleration vector; p is pressure; K is a permeability of the medium; Xf, X^ are effective tensors of thermal conductivity coefficients of the liquid and the porous body skeleton material; tf, ts are temperature of the liquid and the skeleton of the porous body; asf is a coefficient of heat transfer between the liquid phase and the skeleton of the porous body; asf is a characteristic of surface area of the wetted surface in the porous medium.
For the first time the numerical analysis (1) - (4), under the assumption of homogeneity of the porous medium, constant thermophysical characteristics and local thermal equilibrium between the liquid phase and the porous skeleton, was given in [9] for the pulsational change in the liquid velocity at the entry to the porous layer with boundary thermal conditions of the second kind. The articles in [10, 11] are devoted to the experimental study of the settling of the temperature fields in porous media, in which the laws of the formation of the thermal initial region as a function of porosity were studied. However, the question related to the duration of the heating of the porous elements was not illuminated, in spite of the continuing interest in nonstationary processes in porous media [121.
1. Problem Statement and Synthesis of the Model
We consider a flat porous element in the 2 — D format in the Cartesian coordinate system with the origin at the edge of the lower bounding plane, which is heat-generating with the heat flux density q0 The longitudinal coordinate x is directed along the flow of the heat carrier (with known values of its velocity in the input section u0 = const and its temperature t0 = const), the transverse coordinate is perpendicular to the heat-generating plane. The porous medium is bounded by the heat-insulating surface located parallel to the heat-generating plane at the distance h (the surfaces bounding the porous medium
are impenetrable for the coolant). Under the same assumptions as in [9], but refusing to tf
as [131
ts
dU f dU ~d¥ + dX
„6U dP
f V— =--
dY dX
dU dV
dX + dY
1 ( d2 U d2 U
Re \dX2 ' dY2J VRe •Da ' v^
+
0,
1
+
B
d— ndV. vdd— = dP ^L (dV d2v\ ~de+ dX+ dY = -dY+Re IdX2 + dY2)
1
+
B
Re • Da v/Da
1 dTf dTf dTf f + + — f
е de
dX
• Pe ( dXf + dY0 + Pr PRe p ^ Tf)
dY Pr • Pe V dX2
(1 - е) Lu • Pr • Re
dTs d 2ТЯ + d2Ts
de
dX2 ' dY2 Nu p V Re
(Ö2 - Tf)
(5)
--Vu2 + v2^ u, (6)
VU2 + V^j V, (7)
(8)
(9)
f
where 9 = u0t/ (eh)] X = x/h; Y = y/h] U = u/u0; V = v/u0; u, v are components of the liquid velocity vector V; P = e2p/ (pf u^)] B = e2b] b is the Forchheymers factor; Tf = Af (tf — t0) / (q0h); Ts = (ts — t0) / (q0h); A = Af/A|; Re = pf u0h/ (if e2) is Reynolds number; Re p = pf u0dp/ [6 (1 — e) ¡if ] is pore Reynolds
number; Da = K/h2 is Darcy number; Pr = e (pcpf (Afe pf) is Prandtl number;
Nu p = asf dp[Af is pore I^^^^^^^mber; dp is a characteristic size of the extraporate
space; Lu
Afe! (pcp)f I \Ase/ (pcp)s] is modified Lykov criterion characterizing the diffusion of heat in the liquid with respect to the diffusion of heat in the skeleton of a porous medium.
The hypothesis of a unidirectional flow of a coolant in the laminar regime [13], taking into account the fact that the diffusion of heat in the transverse direction of Ohe porous layer is substantially greater than in the longitudinal one (3T2sfdY2 ^ dTf, jdX2)[14], allows to write the system (5) - (9) in the simplified form
1 dJl + dh = df , Nu pRe (AT _T ) nm
e d9 + dX Re • Pr dY2 + Re * • Pr ( s ±f) ' 1 j
dT d2T ( Re )2
(1—e) Lu Re •pr dS = dYi— Nu > (Re;) <ATs—T>) (11>
with the thermal boundary conditions corresponding to the formulation of the problem
Tf (X,Y, 0) = Ts (X,Y, 0) = 0, (12)
dTf (X, 0,9) dTs (X, 0,9)
dY dY
dTf (X, 1,0) dTs (X, 1,9)
-1, (13)
0. (14)
dY dY
When Da ^ 0, which corresponds to the maximum possible values of the heat transfer interphase [15], the liquid velocity profile in the porous layer is close to the ideal displacement regime, i.e. U ~ 1, which leads to the further simplification of system (10) -(14), since there is no need to solve the hydrodynamic problem.
The transition from the model with distributed parameters (10) - (14) to the model with lumped parameters is carried out according to the rule of geometric averaging
1 L
Tf,s (0) = L J J Tf s (X, Y, 9) dXdY, 0 0
where L = l/h, l is a length of the porous heat exchanger. As a result, we obtain the Cauchy problem for the system of ordinary differential equations of the first order
dT' e + sATs -(L + .T, (15)
d9 Pr Re Pr Re 2 \L PrRe 2
dTs =_1___Nu p Re AT | Nu p Re AT
d9 (1 — e) Lu Pr Re (1 — e)LuPrRe p s + (1 — e)LuPrRe p f' 1 J
Ts (0) = Tf (0) = 0. (17)
2. Closing Relations
The classical physical model of porous media, as a rule, is presented in the form of dense packing of spheres [16], the voids of which are interconnected and completely filled with liquid, with only two phases: a liquid and a porous non-deformable skeleton. As the thermophysical parameters in (15) - (17) are homogeneous in space coordinates and do not depend on the temperature, they can be calculated from the relations [17]:
sf = 6(1 - £) /dp
a
sf = xf
2 + 1,1Pr°1/3 (pfuodp/jjf )u'b /d
0,6
where Pr0 = // cpf/A/; A/ is heat conductivity of the liquid;
Xf
£ + (0,1 - 0,5) Pr°1/3 (pfUodp/Vf) xs = (1 - £) Xs,
Xf
(18)
(19)
(20) (21)
where As is heat conductivity of the skeleton of the porous medium; dp is number-average diameter of spherical particles in the porous layer. The dimensionless parameters in (15) - (17) on the basis of (18) - (21) cm be defined as follows:
Re = Re %2, Re p
Re [6(1 - £)], Pr = £ Pr0/(£ + 0, 3 Pr° Re J), Nu p
(2 + 1,1Pr°1/3ReJ/5) / (£ + 0, 3Pr°ReJ), where Re 0 = pfu°h/f Re J = pfu°dp/^f
a
3. Solutions
The change in the temperatures of the coolant and the skeleton of the porous medium from the dimensionless time is obtained by means of the one-sided integral Laplace transform [181
Tf (d) = £A\-
ш
[7AB + £ (L-1 + B)]2 < 4£7AB (L-1 + 2B),
2 ] °,5
(a + yAB)2 + ш2 , n. . / л ш
exp (ad) sin I шд + arctg
+
Y ЛВ
a2 + ш2
+ £Y ЛАВ
a2 + ш2 1
ш
- arctg - + a + 7ЛВ a
)
/1 \ °'5 Ш
I —--- ) exp (ad) sin ( шд — arctg — )
ш a2 + ш2 a
1
ш
a a2 + ш2
°,5
Ts (d) = yM i { ^'Jffl' exp (ad) sin (uti + arctg a+e[L-i+B) - arctg a) +
£ (L-1 + B) .
+ 2 . 2 > + £YAB
a2 + ш2
\ 1 ( 1 \ °'5 ш
) + £jAB — \ —-- I exp (ad) sin ( шд - arctg — )
ш a2 + ш2 a
ш\ 1
) + 2 1 2 a a2 + ш2
when [YЛB + £ (L-1 + B)]2 = 4£YЛB (L-1 + 2B),
Tf (d)
2ejAAB
Ts (d)
e7A(L-1+2B) (
so I
1+ 1+
-1 + s° (1 +
-1 + s° (1 +
so 27лВ
exp
so
e(L-1+2B)
exp
d
2
s
и
d
when
[7AB + e (L-1 + B)] 2 > 4s7Ab (L-1 + 2B) , (22)
= 2sy AAB eAsi + 2s7Aab eAs2 + 2s7Aab
Tf (0) =--1---r—exp (si9) +---T—exp(s20), (23)
S1S2 si (si - s2) s2 (s2 - si)
T 2ejA (L-1 + 2B) + 7Asi + s7a (L-1 + 2B) ( 0) + s (9) =-^-+-^-eXP (si9) +
+ 7As2 +s^-L-i1)+2B) exp(s29), (24)
where A = 1/(Pr • Re) B = Nu p Re / (Pr • Re p), 7 = 1/[(1 - e)Lu], u = ^4syAb (L-1 + 2B) - [7AB + e (L-1 + B)]2, so = - [7AB + e (L-1 + B)] /2,
si = so + ^s2 - 4s7AB (L-1 + 2B), s2 = so s2 - 4s7AB (L-1 + 2B). 4. Computing Experiment
Let's consider the real situation. Let the porous layer consists of dense packing of identical copper ball elements with the diameter dp = 0, 5• 10-3m, and e = 0,4; h = 0, 01m; l = 0, 02m. Thermophysical characteristics of the coolant are close to water, then p, = 1000 kg/m3; ps = 2700 kg/m3; = 5 • 10-4 Pa^s; cp, = 4190 J/feK); Cps = 880J/(kg^K); X, = 0, 68 W/(m^); Xs = 211 W/(m^K). The values of the determining parameters are given in Table.
Table
Reo Pr Rep Re Lu A Nup
20 0,930 0,278 125 0,004 0,007 2,719
100 0,245 1,389 625 0,015 0,027 1,235
200 0,128 2,778 1250 0,029 0,052 0,868
The dimensionless form of recording the equations of the mathematical model allows to abstract from the specific value of the heat flux released by the cooled heat-stressed surface. The results of the calculations show (Fig. 1) that with an increase in the Reynolds number, the dimensionless time of establishment of the stationary regime increases. However, in denominate quantity, it decreases, as value u0 grows faster than 9. This means that a larger flow of the coolant through a porous element leads to an early onset of a stationary regime, which is consistent with physical concepts of heat transfer in porous media.
The local maximum of the temperature of the porous skeleton is explained by the fact that the coolant is present in the region adjacent to the inlet for some time. At the same time, the heat comes from the heat-generating surface by the mechanism of thermal conductivity into the skeleton itself, which is still free from the coolant, which causes the increase in its temperature. With the complete passage of the coolant through the porous element, the temperature of the skeleton decreases and approaches the stationary value. For the coolant temperature, there is no such a regularity. Thus, the proposed model makes it possible to answer the question of the impossibility of the impact of overheating
Fig. 1. Dynamics of the temperatures of the coolant and the porous heat-exchange element when the heat-stressed surface is cooled for different Reynolds numbers of the inlet coolant flow: a - 20; b - 2; c - 5; 1 - Tf; 2 - Ts
of the porous skeleton in the starting condition on the creation of the conditions for the phase transition in the coolant.
The increase in porosity (Fig. 2) leads to the increase in the dimensionless and real transition time, since in this case, the Reynolds number for the interskeleton space decreases and in addition the transition temperature naturally increases.
0.002 0.004 0 006 OOOS 0 010
Fig. 2. Dynamics of Fig. 3. Dynamics of Fig. 4. Dynamics of
the temperatures of the the temperatures of the the temperatures of the
coolant and the porous coolant and the porous coolant and the porous
heat exchange element at heat exchange element heat exchange element
Re ° = 20 £ = 0, 5 Re
Tf Ts
dp Ts
°= 5
200 Re °
200 and
3 . 10-5: 1 - Tf, 2 - h = 0,02rn: 1 - Tf; 2 - Ts
Reducing the dispersion of the particles in the porous element shortens the transition time and reduces the temperature of the coolant and the porous skeleton by increasing the surface area of the heat transfer (Fig. 3).
The increase in the thickness of the porous layer leads to the decrease in the temperature of the coolant and the porous skeleton with the reduction in the transition time (Fig. 4).
Since, for the vast majority of practically important cases, condition (22) is satisfied and si is substantially smaller than s2, then, for example, from (24) we can find an approximate estimate of the dimensionless time for setting the stationary cooling regime from
Ts (Oc)Ts (œ) - 1 = S,
from where
Oc
iln
si
2S (si - S2) jAB
e
/
/
/
/
S2 (si + 2yAB)
where S is predetermined relative accuracy of determination OC (usually S = 0, 01).
Proceeding from the assumption that the thermal initial section in the flat porous layer in the hydrodynamic regime of the ideal displacement of the coolant with constant velocity is formed in time similarly to the heat transfer coefficient along the axial coordinate, comparison with the experimental data from [19] (Fig. 5) shows a satisfactory correlation with the calculation results according to the proposed formula (25).
(25)
/
/
/
100
Re
Fig. 5. Dependence of the determination of the dimesionless time on the Reynolds
number for the coolant:---is empirical
approximation [19]; • is calculation by (25)
Conclusion
The proposed model makes it possible to identify the duration of the transient thermal regime when cooling heat-stressed surface with a porous heat exchanger without complicated calculations and to not only select the rational macro- and micro-geometry of the porous skeleton, but also to evaluate the effect of thermophysical characteristics on the kinetics of the setting of the thermal stationary regime.
References
1. Kandlikar S.G., Garirriella S., Li D., King M.R. Heat Transfer and Fluid Flow in Minichannels and Microchannels. N.Y., Elsevier, 2014.
2. Hutter G.W. The Status of World Geotheririal Power Generation. Proceeding World Geotherrnal Congress, 2000, pp. 23-37.
3. Advani S.G., Sczer M. Process Modeling in Composite Manufacturing. N.Y., CRC Press, 2002.
4. Howell J.R., Hall M.J., Ellzey J.L. Combustion of Hydrocarbon Fuels Within Porous Inert Media. Progress in Energy and Combustion Science, 1996, vol. 22, pp. 121-145.
5. Bees M.A., Hill N.A. Wavelengths of Bioconvection Patterns. The Journal of Experimental Biology, 1997, vol. 200, pp. 1515-1526.
6. Nield D.A., Bejan A. Convection in Porous Media. N.Y., Springer, 1999.
7. Alazmi B., Vafai K. Analysis of Variants Within the Porous Media Transport Models. Journal of Heat Transfer, 2000, vol. 122, pp. 303-326.
8. Hsu C.T., Cheng P. Thermal Dispersion in Porous Medium. International Journal of Heat and Mass Transfer, 1990, vol. 33, no. 8, pp. 1587-1597.
9. Guo Z., Kim S.Y., Sung H.J. Pulsating Flow and Heat Transfer in a Pipe Partially Filled with a Porous Medium. International Journal of Heat and Mass Transfer, 1997, vol. 40, no. 17, pp. 4209-4218.
10. Vafai K., Alkire R.I., Tien C.L. An Experimental Investigation of Heat Transfer in Variable Porosity Media. International Journal of Heat and Mass Transfer, 1985, vol. 107, pp. 642-647.
11. Renken K.J., Poulikakos D. Experiment and Analysis of Forced Convection Heat Transfer in a Packed Led of Spheres. International Journal of Heat and Mass Transfer, 1988, vol. 31, pp. 1399-1408.
12. Teruel F.E. Validity of the Macroscopic Energy Equation Model for Laminar Flows Through Porous Media: Developing and Fully Developed Regions. International Journal of Thermal Sciences, 2017, vol. 112, pp. 439-449.
13. Ryazhskih V.I., Konovalov D.A., Slyusarev M.I., Drozdov I.G. [Analysis of Mathematical Model Heat Removal from the Flat Surface by the Laminar Moving Refrigerant through Conjugation Porous Medium]. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2016, vol. 9, no 3, pp. 68-81. (in Russian) DOI: 10.14529/mmpl60306
14. Ingham D.B. Governing Equations for Laminar Flows Through Porous Media. Emerging Technologies and Techniques in Porous Media, Dordrecht, Springer Science+Business Media, 2004, vol. 134, pp. 1-11. DOI: 10.1007/978-94-007-0971-3
15. Popov I.A. Gidrodinamika i teploobmen v poristyih teploobmennyi aelementah i apparatah [ Hydrodynamics and Heat Transfer in Porous Heat Exchange Elements and Apparatus]. Kazan, Center Informacionnyih Thenologii, 2007. (in Russian)
16. Bear J., Bachmat Y. Introduction to Modeling of Transport Phenomena in Porous Media. Dordrecht, Kluwer Academic Publishers, 1991. DOI: 10.1007/978-94-009-1926-6
17. Amiri A., Vafai K. Analysis of Dispersion Effects and Non-Thermal Equilibrium, Non-Darsian Variable Porosity Incompressible flow Through Porous Media. Journal Heat and Mass Transfer, 1994, vol. 37, no. 6, pp. 939-954.
18. Dech G. Rukovodstvo k prakticheskomu primeneniyu preobrazovaniya Laplasa i z-preobrazovaniya [A Guide to the Practical Application of the Laplace Transform and the z-Transform]. Moscow, Nauka, 1971. (in Russian)
19. Dehghan M., Valipour M.S., Saedodin S., Mahmoudi Y. Investigation of Forced Convection Through Entrance Region of a Porous-Filled MicroChannel: An Analytical Study Based on the Scale Analysis. Applied Thermal Engineering, 2016, vol. 99, pp. 446-454.
Received July 16, 2018
УДК 621.1.016.4(03) DOI: 10.14529/mmp 180410
МАТЕМАТИЧЕСКАЯ МОДЕЛЬ ПРОГРЕВА ПЛОСКОГО ПОРИСТОГО ТЕПЛООБМЕННИКА СИСТЕМЫ ОХЛАЖДЕНИЯ ТЕПЛОВЫДЕЛЯЮЩЕЙ ПОВЕРХНОСТИ В РЕЖИМЕ ПУСКА
В.И. Ряжских1,Д.А.Коновалов1,С.В. Дахин1, Ю.А. Булыгин1,
В.Л. Шацкий2
1
Российская Федерация
2
Российская Федерация
На основе сопряженных гидродинамической модели Дарси - Бринкмана - Фор-чхеймера и тепловой модели Шуманна с граничными условиями второго рода путем геометрического 2Б-осреднения предложена модель с сосредоточенными параметрами для идентификации интегральной кинетики температурных полей пористой матрицы и ньютоновского теплоносителя без фазовых переходов. Модель адаптирована для охлаждения теплонапряженной поверхности с помощью пористого теплообменника с однородной пористостью и проницаемостью, подчиняющейся модифицированному соотношению Козени - Кармана, в виде задачи Коши, решение которой получено в конечном аналитическом представлении для среднеобъемных температур теплоносителя и пористой матрицы. Показана возможность существования гармонического затухающего колебания полей температур и отсутствие перегрева теплоносителя в пусковом режиме системы охлаждения. Для безразмерного времени установления стационарного функционирования пористого теплообменника получена приближенная оценка, коррелирующая с известными данными вычислительного и натурного экспериментов Ключевые слова: плоский пористый теплообменник; теплонапряженная поверхность; граничные условия второго рода; пусковой режим; время установления стационарного теплового режима.
Литература
1. Kandlikar, S.G. Heat Transfer and Fluid Flow in Minichannels and Microchannels / S.G. Kandlikar, S. Garimella, D. Li, M.R. King. - Oxford: Elsevier, 2014.
2. Hutter, G.W. The Status of World Geothermal Power Generation // Proceeding World Geothermal Congress. - 2000. - P. 23-37.
3. Advani, S.G. Process Modeling in Composite Manufacturing / S.G. Advani, M. Sczer. - New York: Marcel Dekker, 2002.
4. Howell, J.R. Combustion of Hydrocarbon Fuels within Porous Inert Media / J.R. Howell, M.J. Hall, J.L. Ellzey // Progress in Energy and Combustion Science. - 1996. - V. 22. -P. 121-145.
5. Bees, M.A. Wavelengths of Bioconvection Patterns / M.A. Bees, N.A. Hill // The Journal of Experimental Biology. - 1997. - V. 200. - P. 1515-1526.
6. Nield, D.A. Convection in Porous Media / A. Bejan. - New York: Springer, 1999.
7. Alazmi, B. Analysis of Variants within the Porous Media Transport Models / B. Alazmi, K. Vafai // Journal of Heat Transfer. - 2000. - V. 122. - P. 303-326.
8. Hsu, C.T. Thermal Dispersion in Porous Medium / C.T. Hsu, P. Cheng // Journal of Heat and Mass Transfer. - 1990. - V. 33, № 8. - P. 1587-1597.
9. Guo, Z. Pulsating Flow and Heat Transfer in a Pipe Partially Filled with a Porous Medium / Z. Guo, S.Y. Kim, H.J. Sung // Journal of Heat and Mass Transfer. - 1997. - V. 40, № 17. -P. 4209-4218.
10. Vafai, К. An Experimental Investigation of Heat Transfer in Variable Porosity Media / K. Vafai, R.I. Alkire, C.L. Tien // Journal of Heat Transfer. - 1985. - V. 107. - P. 642-647.
11. Renken, K.J. Experiment and Analysis of Forced Convection Heat Transfer in a Packed Led of Spheres / K.J. Renken, D. Poulikakos // Journal of Heat and Mass Transfer. - 1988. -V. 31. - P. 1399-1408.
12. Teruel, F.E. Validity of the Macroscopic Energy Equation Model for Laminar Flows Through Porous Media: Developing and Fully Developed Regions // Journal of Thermal Sciences. -2017. - V. 112. - P. 439-449.
13. Рижских, В.И. Анализ матемаической модели теплосъема с плоской поврехности лами-иарио движущимся хладоагентом через сопряженную пористую среду / В.И. Ряжских, Д.А. Коновалов, М.И. Слюсарев, И.Г. Дроздов // Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. - 2016. - Т. 9, № 3. - С. 68-81.
14. Ingham, D.B. Governing Equations for Laminar Flows Through Porous Media / D.B. Ingham // Emerging Technologies and Techniques in Porous Media. - Dordrecht: Springer Science+Business Media, 2004. - V. 134. - P. 1-11.
15. Попов, И.А. Гидродинамика и теплообмен в пористых теплообменных элементах и аппаратах / И.А. Попов. - Казань: Центр инновационных технологий, 2007.
16. Bear, J. Introduction to Modeling of Transport Phenomena in Porous Media / J. Bear, Y. Bachmat. - Dordrecht: Kluwer Academic Publishers, 1991.
17. Amiri, A. Analysis of Dispersion Effects and Non-Thermal Equilibrium, Non-Darsian Variable Porosity Incompressible Flow Through Porous Media / A. Amiri, K. Vafai // Journal of Heat and Mass Transfer. - 1994. - V. 37, № 6. - P. 939-954.
18. Дёч, Г. Руководство к практическому применению преобразования Лапласа и z-преобразования / Г. Дёч. - М.: Наука, 1971.
19. Dehghan, М. Investigation of Forced Convection Through Entrance Region of a Porous-Filled MicroChannel: An Analytical Study Based on the Scale Analysis / M. Dehghan, M.S. Valipour, S. Saedodin, Y. Mahmoudi // Applied Thermal Engineering. - 2016. - V. 99. -P. 446-454.
Виктор Иванович Ряжских, доктор технических наук, профессор, кафедра «Прикладная математика и механика:», Воронежский государственный технический уни-вврситбт (г. Воронеж, Российская Федерация), ryazhskih_vi@mail.ru.
Дмитрий Альбертович Коновалов, кандидат технических наук, доцент, кафедра «Теоретическая и промышленная теплоэнергетика», Воронежский государственный технический университет (г. Воронеж, Российская Федерация), dmikonovalov@yandex.ru.
Сергей Викторович Дахин, ксшдидсхт твхничвеких hclvk. доцент, кафедра «Теоретическая и промышленная теплоэнергетика», Воронежский государственный технический университет (г. Воронеж, Российская Федерация), secretary-d21203705@yandex.ru.
Юрий Александрович Булыгин, доктор технических наук, профессор, кафед-
«»
технический университет (г. Воронеж, Российская Федерация), bulyginu@gmail.com.
Владимир Павлович Шацкий, доктор технических наук, профессор, кафед-
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(г. Воронеж, Российская Федерация), sha.vladim@yandex.ru.
Поступила в редакцию 16 июля 2018 г.
