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19PHYSICS AND MATHEMATICS
ТЕПЛОПЕРЕНОС В МНОГОСЛОЙНОМ ТОЛСТОСТЕННОМ ПОЛОМ ЦИЛИНДРЕ
Лавров Ю.А.
профессор, доктор физико-математических наук Санкт-Петербургский политехнический университет Петра Великого
HEAT TRANSFER IN MULTILAYERED THICK-WALLED HOLLOW CYLINDER
Lavrov Yu.
Professor Saint-Petersburg politechnical university
Аннотация
Рассмотрена задача управляемого переноса теплоты через многослойную цилиндрическую стену. Стена состоит из произвольного количества коаксиальных цилиндрических слоев с различными физическими параметрами. На каждой границе между соседними слоями обеспечен идеальный тепловой контакт. На внутренней и внешней поверхностях стены ставятся условия конвективного теплообмена. Температура сред у внутренней и внешней цилиндрических поверхностей имеет заданную кусочно-линейную зависимость от времени. Построено точное аналитическое решение задачи. Полученные на его основе численные результаты сравнены с результатами, полученными методом конечных разностей.
Abstract
The problem on controlled heat transfer through multilayered circular cylindrical tube is considered. A tube wall is a junction of an arbitrary number of coaxial layers with various physical parameters. Each boundary surface between neighbor layers provides an ideal thermal contact. On the inner and outer surfaces of the wall, convective heat transfer conditions are set. Environment temperatures at inner and outer wall surfaces have piecewise-linear time dependence. An analytic solution to the problem is obtained. The numerical results are verified by comparison with the results of using the finite difference method.
Ключевые слова: многослойный полый цилиндр, осевая симметрия, нестационарный процесс, точное аналитическое решение, метод конечных разностей, жилое здание.
Keywords: multilayered hollow cylinder, axe-symmetry, non-stationary processes, exact analytical solution, finite-difference scheme, residential building.
Introduction
The energy efficiency of a building largely depends on its shape. The spherical wall's shape is optimal, but not technologically advanced. The number of spherical shape buildings is small, that cannot be said about cylindrical buildings. The energy efficiency of a cylindrical building is higher compared to a parallelepiped-shaped building.
Multilayer building structures are widely considered in the scientific literature. Most attention is paid to flat structures. Thermal insulating barriers (walls) [1,2] as well as plates with internal heat sources (warm floor) [3] are considered. Knowledge of heat transfer by a flat wall cannot be transferred onto a cylindrical wall without amendment. The properties of cylindrical walls are also widely studied in the literature. Primarily, approximate analytical or numerical solutions [4,5,6,7] are offered. Some papers have analytical solutions for complicated cases: for thermoelasticity problem [8,9], for anisotropic [10] and orthotopic [11] materials, for generalized boundary conditions [12,13], for arbitrary heat generations within the layers [14], for heat stream across the tube walls from hot liquid [15,16], hot gas [17,18] or hot solid[19] inside, for finite hollow composite cylinders[20].
For accurate thermal calculation of the building, the external walls of which are cylindrical and multilayer, an explicit formulaic solution constructed in this paper would be useful.
The problem of fine temperature regime maintenance in a residential building assumes that the external temperature changes are unpredictable. Readout of external temperature usually acts from sensors at discrete points in time, usually with a constant readout step. It is logical to suppose, that the temperature has a linear dependence on time between readouts. A predictive temperature time dependence before next readout is ordinary linear too. Thus, long time temperature dependence may be considered as piecewise-linear. This dependence is approximate, but its error tends to zero when the readout step tends to zero.
A "Smart Home" concept assumes that the heating devices' power in the building automatically changes depending on the outer temperature. During waking hours, the inner temperature should be constant, regardless of outer temperature. At night, another conditions may be set, for example, the condition for the constancy of heating power. When the temperature reaches the lower threshold, the mode of maintaining the required constant low temperature may be activated.
The thermal calculation of the building by numerical methods demands significant computing expenses. In this sense, analytical methods take advantage over numerical.
The research aim of this paper is an investigation of unsteady heat transfer through a material of multi-
4 The scientific heritage No 45 (2020)
layered hollow cylindrical wall. The environment tem- designation = + + —+ is used, is m-
peratures at inner and outer wall surfaces have piece- th layer thickness. The processes dependence on axial
wise-linear dependence on time. (z) and angular ( <p) coordinates is not taken into ac-
The research task is to derive and numerically im- count. plement an exact analytical solution for corresponding The temperature within the cylinder material is
boundary-value problem of thermal conductivity. searched as piecewise homogeneous function
Statement of the Problem ©¿(r,x) = {©¿m(r,i), < r < flm}m=i,2.....M. It
Heat transfer through a hollow cylinder fl0 < r < satisfies the Fourier equation , —n < 9 < n, —ro < z < is considered. Coaxial surfaces r = flm (m = 1,2,... ,M — 1, M > 2) separate the layers of different material properties. The
920 jm(r,T) l88im(r,T^ 1 90im(r,x) dr2 r dr am 3x '
and non-homogeneous boundary conditions at inner and outer cylindrical surfaces,
(—¿+?HMr_fl =ifYo(-), (2)
1 r=«0 1
Here t is time, t¿-1 < t < tj, i is a number of time step, i = 1,2,..., /, t0 = 0, am = Am/(cmpm), Am, cm, pm are temperature conductivity, thermal conductivity, heat capacity and layer material density, respectively, m = 1,2,..., M; Yi7- (t) = k^ (t — Tj-1) + and a are environment temperature and heat exchange coefficient at inner (/ = 0) and outer (/ = 1) cylinder surface, respectively.
The ideal thermal contact conditions for of neighbor layers are
0im(fim — 0,t) — ©¿,m+i(fim + 0,t) = 0, (4)
Am--1 — Am+1-dr-1 = 0, (5)
m = 1,2,.,M — 1.
Initial cylinder temperature distribution for -th time step is
®i(r,T¿-1) = 0 ¡-1(r) , (6)
Oq (r) is the prescribed function, (r) is a result of ¿-th time step, i = 1,2,..., /.
An output result is the thermal field 0; (r, t), a radial heat flux density in arbitrary point of the wall G; (r, t) =
(r t)
{ Gjm(r,T), flm-1 < r < fim}m=1,2,.,M, Gim(r,T) = —^m—™ ' , and a heat flux density across the inner wall
surface q^M = £¿0(^0^). Analytical Solution
An exact analytical solution is presented as
0i(r,T) = £1=0 (k0 • ( U7-(r)(T — T-1) + V}(r)) + ^(r)) + W£(r,T). (8)
Coefficients A,m, B7-m, (/ = 0,1, m = 1,2, ...,M) forpiecewise-homogeneousfunction
W (r) = U-mln (f ) + B,-m, < r < flJ (9)
m=1.2.....M
m=l,2,...,M
are the solution of linear system of 2M equations. This system is obtained by substitution (9) into (2)-(5), with formal replacement (10),
Y o(t) = 1 — j, Yi(t)= j. (10)
Coefficients C,m, (/ = 0,1, m = 1,2, ...,M) for function
W = {¿^ (ln © — 1) + ^ + ^ln © + < r < *»}m=u.....M (11)
are the solution of linear system of 2M equations. This system is obtained by substitution (11) into (2)-(5), with formal replacement (10). Coefficients Ajm, in (11) are considered already found for the expression (9). A series
WKr,x) = ^^„S^e-^-^) (12)
submits to requirements (1)-(5) with formal replacement Yî0(t) = Yi1(x) = 0. It is expressed by a linear combination of functions
S„(r) = {£nm/o(Pn5mr) + ^nm^0(Pn5mr),^m-1 < r < ^m}m=1,2,...,M, (13)
7fc(y), ^c(y) are the first and the second kind Bessel functions, sm = l/^a^. The coefficients Wj„ (n = 0,1,2, ...) are arbitrary yet. The column of coefficients X = (£'n1,Fn1,£n2,Fn2, ...,£'nM,FnM)T (T is a transpose sign) is the solution of homogeneous system of linear equations
UpJ* = 0. (14)
The only elements of sparse matrix L(p) = ¿2mx2m(p) are distinct from zero:
¿11(P) = psj1(ps!flo) +ir/o(PS1flo), ¿12(P) = ps1^1(ps1flo)
Ài À1
^2m,2m-1(P) = /o(P5m^m), ^2m,2m = ^(^m^mX
^2m,2m+1(P) = —/o(P5m+1^m), ^2^2m+2 = —^0(P5m+1^m),
¿2m+1,2m-1(P) = -^mPsm/l(PsmRm), ¿2m+1,2m(P) = -mPSm^i (^m^mX
(P) = ^m+lPsm+Jl(Psm+1fim),
^2m+1,2m+2(P) = ^m+1P5m+1^1(P5m+1^m),
m = 1,2, ...,M — 1,
^2M,2M-1(P) = —P5^.^1(P5M^M) + ï /o(P5m^mX
ÀM
^2M,2M(P) = —P5M^1(PSM^M) + ^0(PSM^M).
A factor p„ (n = 0,1,2, ...,) is n-th positive root of equation
detL(p) = 0. (15)
Equation (15) means the non-zero solution existence condition for (14). The set of numbered factors p„ is the Sturm-Liouville eigenvalues problem solution for 5„(r) function.
At calculation of X column, for each p„, the appointment £n1 = 1 is recommended. Other X elements should be found by system (14) without one of its equations, for example, without the last one.
The pairwise orthogonal functions S„(r), n = 0,1,2,..., compose a basis on the interval [flo,flM]. Scalar product
(r), 4>2 (r)} = JflfloM y(r)% (r), ^2 (r)dr (16)
uses a weight function y(r) = {rA„/am, flm_1 < r < fim}m=1,2.....M.
Thus, the expression (8) submits to requirements (1)-(5). Scalar product (16) allows construct the following decompositions:
^_1(r) = £+ro^-1,„Sn(r), u,.(r) = ^^„(r), V}(r) = S+roV,„5n(r),
1 b = (îi-ii^M), u.n = (W^, v.n = (^nM), (17)
= (s„(r), 5„(r)}.
Expressions for coefficients u7-n, are finite and rapidly-calculated. For brevity, designations are introduced for infinite-dimensional vectors
$ = ($îo,$î1,$î2, .)T, = (Ujo,Uj1,Uj2, .)T, Vj = (vio,vi1,vi2,.)T, = (Wjo,Wj1,Wj2,.)T. Basis decomposition is unique, so, the last condition (6) of the problem statement means
Wj = $¿-1 — Z)=o(Kî7-V/ + (18)
Upon the time step termination the distribution of temperatures becomes
Oi(r) = 0j(r,x 0 =
= Z)=„ (k0 ( U7-(r)AT£ + V-(r)) + ^(r)) + £+r<> w^Me-"2^, (19)
At j = ti - t¿-:L. The heat flux density across the inner wall surface is
i
<7ioCO = -Ai ^ (k0 (u/(bo)(t - T-i) + VJ'(flo)) + -
7=0
-ao £+=o Win5n(fio)e-p22(T-Ti-l), (20)
= №) = C,i/fio - (^i - 2B,-i)flo/(4ai) .
An average heat flux density throw the inner wall surface for a time range t £ (t i-i, t¿) is
i
fco = ^ (k^^o) + + K^W^) -
7'=o
-fr^+=o^25n(fio)(l - e-^-^). (21)
ax i pn
Since the change in the power of heating devices occurs at discrete points of time (for example, at t = t j -i), an expression (21) should be used to select the power for the next time step. If the average heating power qio (or the instant heating power qio(T)) is above the maximum permissible value, or less than zero, the temperature regime Yio (t) (namely, the coefficient ko) should be adjusted, since it cannot be provided.
If the heating process starts at time t = To, the initial temperature distribution may be taken from the steady-state process, Oo00 = ^io^oOO + ^ii^i(r), cfo = ^io"o + ^ ^
If the heating process is periodic, and the period is T = t7 - To, then 0i(r,To) = 0,(7-,^), cf)o = cC, the equality (19) should be rewritten in the form
Ci = dr cfi-i + ) • xi + y;, (22)
d- = (e-PoA^e-p^g-piAT^ /) , = + 3?i=ÄTiZ}=oKi7-iii, 3 = g - ¿i£, g =
.T
~ - vi ''M/1) + ^-"i) J,i = ATiij=oKijl Xj is an arithmetic product of two vectors, Cj • Xj = (CjoXjo, CjiXji, Cj2Xj2, .■■ )T. Using (22) for i = 1,2,..., /, one can obtain
-i
C = Co = • <¿¿+2 • - • </) • (A • xXi + yO +
=i
+c7 • £7 + y7 + (di • <¿2 • - • d7) • cfo. (23)
All products of vectors in (23) are arithmetic. Solution of equation (23) for cfo is
Cfo = b • (£i-i(di+i • <¿¿+2 • - • • (Ci • x + y^ + c7 • x7 + y^ (24)
b = ((1- e^)-! (1 - e-^)-\ (1 - e^)^, -)T.
An analytical solution for the "reverse" geometric case, in which the outer surface of cylindrical wall is concave, may be constructed in a similar way. This case means = fio - -----ftm, m = 1,2, ...,M, hence,
^m < ^o.
Results and Discussion
The next parameters are accepted at calculations for periodic processes: M = 5, = = = 0.01 m, = 0.37, ao = 10 W/(m2 • °C). The layer's materials are: internal plaster, brickwork, heavy plaster, Styrofoam, light plaster, m = 1,2,3,4,5, respectively. Numerical experiments were carried out using the result of (8), (19)—(21), and verified using the slow finite difference method.
Dependence of environment temperatures on time is considered as daily-periodical,
Y O(T) = to, (25)
YI(T) * ticos(wr), (26)
t0 = 230C, ii = 100C, a round frequency is w = —, 7 = 86400 s (one day), t = 7 • i//, / = 288. Dependence (26) is approximated by piecewise-linear function Yj1(x) = k^Ct - t£-1) + where k^ = t1(cos(wxi) -cos(wTj-1))/ATj, ^¿1 = t1Cos(wTj-1).
An expressions Y (t) = u7e-1MT, 0 (r, t) = K(r)e-LMT, where i = V-1, used in [7] instead of (25)-(26). This approach gives a seemingly simple analytical solution for K(r). But this solution is not acceptable for multi-layered wall, for it generates a badly conditioned system of linear equations. An average heat flux density for all time period t £ (t0, t,) is
<?o =1^i=1ATi •<7io. (27)
The mathematical model (1)-(6) is linear,
12=1 J0T Yw (T)dT = t0, 1 Z{=1 J0r Yj1 (T)dT = 0
for (25)-(26), hence,
lim = q0 =-•(—+ —) 1, (28)
<70 is heat flux density throw a wall surface r = fi0 for steady-state processes in cylindrical wall [21]. Besides,
lim ¿?0 = <? = t0-(;1+^+:KUr), (29)
q is heat flux density throw a flat multilayered wall [21].
In some cases, the thermal calculation of the building with a cylindrical wall is the same as for a flat wall. This approach may result in a calculation error, for the ratio A = cf0/<7 may vary significantly from unity. The approximate formula
1VM hrn. 1 VM h vM h
A « A1 = 1 + — •2-Am a1 1-—-C-^ (30)
£^0 ao ai ira"1im
is effective for small e = — £m-1 Here, x = 1 for the convex external surface of the wall, x = —1 for concave surface.
Expressions (28) and (30) allow us to use an approximate formula
^0 ~ A1?. (31)
£^0
Figures 1, 2, 3 show a dependence of heat flux density on some geometrical and physical parameters. Figures' parts (a) and (b) are built for the convex (x = 1) and concave (x = —1) outer surface of the wall, respectively. For all the calculations, a relative error of formula (31) was less than 0.001.
Heat loss through the convex or concave cylindrical wall, respectively, is several percent higher or lower than through the flat wall. A cylindrical building is more energy efficient than a flat-rectangular building due to the lack of corners. The ring-shaped building, one of the "outer" walls of which is concave, is energy efficient, too.
Figures 1-3 show that heat loss through the multilayered wall significantly depends on the heat-insulating layer thickness ft4, while dependence on weather conditions (i.e. on a1) is rather weak. The designations ft41 = 0.05 m, ft42 = 0.1 m, ft43 = 0.15 m, ft44 = 0.2 m, a11 = 10 W/(m2 • °C), a12 = 20 W/(m2 • °C), a13 = 30 W/(m2 • °C), a14 = 40 W/(m2 • °C) are used.
(b)
Figure 1 Dependencies of the average heat flux density on the Styrofoam layer thickness. ai = aii, ai = ai4, lines 1, 2, respectively, flo = 5 m.
(a) (b)
Figure 2 Dependencies of the average heat flux density on the wall curvature radius.
lines 1, 2, 3, respectively, = 0.1 m. The curves of dependencies qo/ q and qo/for various ai merge into one line at the scale of thefigure.
(a) (b)
Figure 3 Dependencies of the average heat flux density on the wall curvature radius. h4 = h41, h4 = h42, h4 =
h43, h4 = h44, lines 1, 2, 3, 4, respectively, a1 = a11.
(a) (b)
Figure 4 Dependency of the temperature (a) and heat flux density (b) on radius r and time t. fl0 = 5 m, h4 = h42, = a11, r = fl0, fl2, fl3, fl4, R5, lines 0, 1, 2, 3, 4, 5, respectively, outer wall surface is convex. Line 7 is the crest one, 6 (r, t) = 1, line 6 is the lowest one, 6 (r, t) = 0.
Figure 4 presents 3D-illustration of thermal wave propagation through the wall material. The daily temperature range is very narrow at the inner wall surface, and is wide at the outer one. For visual range adjustment the special normalization of fields is applied:
0(r,x)- min 0(r,x)
8(r,x)=--
5(r,T) =
max 0(r,x)- min 0(r,x)' xo<x<x, xo<x<x,
G(r,x)- min G(r,x) v y xo<x<x, v y
max G(r,x)- min G(r,x)* xo<x<x, xo<x<x,
Figure 4 shows, that the heat wave propagates almost instantly through the heat-insulating material, while the wave is delayed by a significant period share over the brickwork material.
Conclusions
An exact solution to the problem of heat transfer through a multilayer cylindrical wall with piecewise
linear external environments' temperatures, is constructed. The expressions for the desired fields are built both for an arbitrary initial temperature distribution and for a periodic process.
It is numerically monitored that the temperature wave rapidly propagates across the insulating layer and propagates with a large delay across the brickwork layer.
Numerically revealed that heat loss through a multilayer wall strongly depends on the thickness of the insulating layer and weakly depends on heat exchange coefficient of the outer surface of the wall.
An approximate formula for the wall curvature correction factor for is proposed and numerically tested. Numerical experiments confirmed the high accuracy of the approximate formula.
References
1. Buratti C., Belloni E., Merli F., Zanella V., Robazza P., Cornaro C. An innovative multilayer wall composed of natural materials: experimental characterization of the thermal properties and comparison with other solutions. Energy Procedia. Vol.148. August 2018. Pp. 892-899.
2. Yufereva L.M., Lavrov Yu.A. Statsionarnoe teplovoe pole kusochno-odnorodnogo pryamou-golnogo brusa [Stationary thermal field of a piecewise-homogeneous rectangular bar]. Izvestiya RAN. Ener-getika. 2005. No. 2. Pp. 129-137. [Proceedings of Russian Academy of Sciences. Energetics. Issue 2. Pp. 129137].
3. Yufereva L.M., Lavrov Yu.A. O teplovom pole mnogosloinoi plastiny s vnutrennimi istochnikami [On thermal field of a multilayered plate with internal sources]. Izvestiya RAN. Energetika. 2009. No. 2. Pp. 77-82. [Proceedings of Russian Academy of Sciences. Energetics. 2009. Issue 2. Pp. 77-82].
4. Yufereva L.M., Lavrov Yu.A., Yuferev A.Yu. O teplovom pole mnogosloinogo pologo tsilindra [On thermal field of a multilayered hollow the cylinder]. Izvestiya RAN. Energetika. 2011. Vyp. 2. Pp. 122-128. [Proceedings of Russian Academy of Sciences. Energetics. 2011. Issue 2. Pp. 122-128].
5. Han, L.S., Periodic Heat Conduction Through Composite Panels. Journal of Thermophysics and Heat Transfer. 1986. Vol. 1. No. 2. Pp. 184-186.
6. Samarin, O.D. The temperature waves motion in hollow thick-walled cylinder. Magazine of Civil Engineering. 2018. 84(2). Pp. 161-168.
7. Samarin, O.D. The periodic temperature oscillations in a cylindrical profile with a large thickness. Magazine of Civil Engineering. 2019. 85(1). Pp. 5156.
8. Yufereva L.M., Lavrov Yu.A. Termome-hanicheskoe pole mnogosloinoi plastiny s vnutrennimi istochnikami [Thermomechanical field of a multi-layered plate with internal sources]. Vestnik Sankt-Pe-terburgskogo universiteta. Ser. 1. Issue. 2. Pp. 122-128 [Bulletin of St. Petersburg University. Maths. Mechanics. Astronomy. 2009. Issue. 2. Pp. 122-128].
9. Kalynyak B.M. Fredholm equations of the second kind for radial stresses aimed at the determination of the thermoelastic state of an inhomogeneous hollow long cylinder. Journal of Mathematical Sciences. 2015. Vol. 205. No. 5. Pp. 659-666.
10. Norouzi M., Biijandi A. K., Rahmani H., Joneidi A. A. A general exact analytical solution for an-isotropic nonaxisymetric heat conduction in composite cylindrical shells. International Journal of Heat and Mass Transfer. 2016. Vol. 93. Pp. 41-56.
11. Delouei A.A., Kayhani M.H., Norouzi M. Exact analytical solution of unsteady axi-symmetric con-
ductive heat transfer in cylindrical orthotropic composite laminates. International Journal of Heat and Mass Transfer. Vol. 55, Issues 15-16. July, 2012. Pp. 44274436.
12. Shevchuk V. A. Nonstationary one-dimensional problem of heat conduction for a cylinder with a thin multi-layer coating. Journal of Mathematical Sciences. 2012. Vol. 184. No. 2. Pp. 215-223.
13. Yang B., Liu S. Closed-form analytical solutions of transient heat conduction in hollow composite cylinders with any number of layers. International Journal of Heat and Mass Transfer. Vol. 108. Part A. May 2017. Pp. 907-917.
14. Fakoor-Pakdaman M., Ahmadi M., Bagheri F., Bahrami M. Dynamic Heat Transfer Inside Multi-layered Packages with Arbitrary Heat Generations. Journal of Thermophysics and Heat Transfer. 2014. Vol. 28. No. 4. Pp. 687-699.
15. Olek, S., Elias, E., Wacholder, E., and Kaizerman, S. Unsteady Conjugated Heat Transfer in Laminar Pipe Flow. International Journal of Heat and Mass Transfer. 1991. Vol. 34. No. 6. Pp. 1443-1450.
16. Yufereva L.M., Lavrov Yu.A., Yuferev A.Yu. Kraevaya zadacha teploprovodnosti dlya mnogosloinogo tsilindra, po kanalu kotorogo protekaet zhidkost. [Boundary value problem of thermal conductivity for a multilayer cylinder, through whose channel fluid flows]. Izvestiya Peterburgskogo universiteta putei soobsheniya. 2011. № 1 (26). Pp. 246-257. [Proceedings of St. Petersburg University of Railways. 2011. No. 1 (26). Pp. 246-257].
17. Noh J.-H., Cha K.-I., Yook S.-J. Three-dimensional inverse heat conduction modeling of a multi-layered hollow cylindrical tube using input estimation algorithm and thermal resistance network. International Journal of Heat and Mass Transfer. Vol. 106. March 2017. Pp. 793-803.
18. Noh J.-H., Cha K.-I., Ahn S.-T., Yook S.-J. Prediction of time-varying heat flux along a hollow cylindrical tube wall using recursive input estimation algorithm and thermal resistance network method. International Journal of Heat and Mass Transfer, Vol. 97. June 2016. Pp. 232-241.
19. Macher W., Komle N.I., Bentley M.S. The heated infinite cylinder with sheath and two thermal surface resistance layers. International Journal of Heat and Mass Transfer. Vol. 57, Issue 2. February 2013. Pp. 528-534.
20. Li M., Lai A.C.K. Analytical solution to heat conduction in finite hollow composite cylinders with a general boundary condition. International Journal of Heat and Mass Transfer. Vol. 60, May 2013. Pp. 549556.
21. Carslaw H.S., Jaeger J.C. Conduction of heat in solids. Oxford University Press, USA, 1986. 520 p.
