CC BY
16
UDC 536.3:621.45
DOI: 10.18698/0236-3941-2019-4-44-57
CONJUGATE HEAT TRANSFER AND ESTIMATION OF THERMAL STATE OF ELEMENTS OF THE THERMAL PROTECTION SHIELD PACKAGE
V.A. Tovstonog tovstonog@bmstu.ru
Bauman Moscow State Technical University, Moscow, Russian Federation
Abstract
This article presents the physical and computational models of heat transfer and the high-temperature flow path thermal protection with the use of radiation shields package. The analysis of the thermal insulating ability and temperature state of a multielement radiation shields package was performed. It is shown that the distribution the temperature of the package elements is uneven, which can cause different thermal deformation of elements, the distortion of the shield, the possibility of mechanical contact of neighboring elements and the deterioration of the heat insulating ability
Keywords
Radiation heat transfer, radiation shield, emissivity, thermal protection
Received 30.01.2019 © Author(s), 2019
Introduction. Ensuring the temperature mode of the construction of high-temperature technological, energy and propulsion systems is an important practical task, the relevance and importance of which increases with operating temperatures growth, which determines the energy efficiency of the structure. For thermal insulation of high-temperature technological equipment working area, for example, vacuum furnaces, or gas-dynamic flow paths of propulsion systems, the use of radiation shields is effective [1-3]. The heat insulating capacity of such thermal protection is mainly determined by the laws of radiative heat transfer in a package of shields, and its numerical estimates are most often limited by the conditions for setting temperatures or heat flux at the boundaries of a package of shields [4, 5]. At the same time, in a number of practical applications, the assignment of specific values of these parameters is difficult and it is necessary to take into account the mutual influence of the thermal state of insulation and heat exchange parameters at the boundaries, and this connection is enhanced with the transition to higher temperatures. This article analyzes the temperature state of the system elements, which includes the internal wall of combustion chamber heated by a high-temperature gas flow, separated from the enclosing wall (housing) by a package of radiation shields.
Problem formulation. We consider the heat transfer in the system is stationary (Fig. 1), system contains the internal wall of combustion chamber 1, streamlined by high-temperature flow with Vp speed at Tp flow temperature. External (heated) and internal surfaces have emissivity s1 =s1(T1) and s12 = s12(T1). The wall is considered to be thermally thin, its temperature state
Fig. 1. Model of heat transfer in the flow path with a package of radiation shields:
1 is internal wall of combustion chamber; 2, 3 are outer shields of package; 4 is outer housing
is characterized by temperature T1; heat exchange between the gas flow and the internal wall surface facing it is due to convection and radiation and is characterized by an effective heat transfer coefficient ap = ap (Tp, T1). Between the internal wall and the casing wall there is a package of shields, the far left (2) and far right (3) shields have emissivity s21 = s21(T2) and £34 = s34(T3), and internal Ne shields have emissivity of each surface s*. We assume that the temperature state of the package is determined by the outer shields temperature T2 and T3, and in the internal gaps between the shields there is a heat exchange not only by radiation, but also by convection and due to the thermal conductivity of the gas, which will be taken into account by the generalized heat transfer coefficient of the package apn = apn (T2, T3). Such a generalization is due to the fact that the screens in the package are almost always separated by heat-conducting distance elements [6], and if the gaps are not evacuated, then heat transfer occurs due to convection and thermal conductivity of the gas, which can be characterized by a generalized heat transfer coefficient of gaps in the package a pn, *.
Total thermal resistance Rpn of package gaps with the number of intermediate gaps Ne is determined by the relation
Rpn — '
1
Ne + 1
■— S
1
^pn i = 1 &pn, i
and the generalized heat transfer coefficient in the gaps apn, i we define through the effective thermal conductivity coefficient of the gap ^
Ti+i - Ti
apn, i (Ti +1 Ti ) — hi
h
From this we get
then
a
pn, i
Xi h
a p
Ne + 1 h
— X ^
^pn i = 1 h
where hi is the size of the gaps in the package. Introducing the effective thermal conductivity coefficient of the package Xef as the relation
a pn T - T3)— X h
Ne+1 Ti+1 - Ti
we get
i — 1
a
hi
— À,
ef
'Ne +1
(T2 - T3)/ X hi i — 1
pn
— Xef E hi.
i — 1
If we assume
then
V —0,5|> (T2) + Л, (T3)],
À (T2) + Л, (T3)
Œpn — a pn (t2 > t3) — •
2(Ne +1) hi
The wall of the housing 4 is also considered thermally thin, is characterized by temperature T4,has an emissivity of the internal surface s43 = s43(T4), and external (facing the environment) has an s4 = s4(T4). The gas stream flows around the outer surface at a rate of Vh and temperature Th; convective heat exchange with the external environment is characterized by a heat transfer coefficient ah = ah (T4, Th).
Generally, we will assume that in the gap between the internal wall of combustion chamber and the package of shields (hi2) there is heat exchange
1
by radiation, and also heat transfer is allowed due to convection with a heat transfer coefficient a12 = a12(T1, T2) and thermal conductivity of gas with thermal conductivity coefficient ^12 = ^12(T1, T2). Similarly, in the gap
between the screen package and the housing (h34) there is heat exchange by radiation and also there can be heat transfer by convection and heat conduction with appropriate heat transfer coefficient a34 = a34(T3, T4) and thermal conductivity coefficient ^34 = ^34(T3, T4). We can note that in narrow gaps without forced flow of gas, the effect of convection may be insignificant, less than thermal conductivity, but the decisive role in the studied temperature range (T «1000 - 2000 K) belongs to radiation, and the indicated molecular mechanisms of heat transfer are included in the mentioned model in order of generalizing the numerical algorithm to the regimes in which the influence of these factors can be significant.
Further, in order to simplify mathematical expressions, unless otherwise stated, the functional dependence of the heat transfer parameters from temperature will not be given.
Mathematical model and solution method. Effective radiation fluxes in the gaps qij and in a package of shields qpn are determined by the relations that follow from the problem solution of radiation heat transfer in a system of flat surfaces for a diathermisches medium:
q12 = ^ G0T14 + ^ (1 - S12) 00T24; (1)
£21 s12
q21 = ^(1 -S21)g0T14 ^ (2)
£21 s12
= G0T34 +^(1 -s34) aoT4; (3)
S43 S34
= ^(1 -643) V0T34 + ^ aoT44; (4)
£43 £34
qpn = s pn^o ( t24 - t34 ), (5)
where
1
S pi ="1 1 ; S p2
-+--1
S12 S21
£ pn =
— + _ 1 + Ne f.2. - 1 .S21 £34 Vsi
are reduced emissivity of surface systems, in which radiation heat exchange takes place; Ne is number of intermediate shields of the package.
We write the heat balance equations for all boundary surfaces of the system:
f X ^
ap(Tp -Ti) + 812^21 -si^oT]4 -si2aoT14 - ai2 \(T -T2) = 0; (6)
h
12
£2^1 -S21a0T)4 +a12(T1 - T2) + -p2CT - T2) - qpn - apn (T2 - T3) = 0; (7)
h12
qpn +Upn(T2 -T3)-834q3 -S34Q0T34 -a34(T3 -T4) = 0; (8)
A,
e43q34 +a34(T? - T4) ^-34(T3 - T4) -S43G0T4 -S4G0T3 - ah(T4 - Th) = 0. (9)
h34
Taking into account expressions (1)-(5), after transformations, the system of equations (6)-(9) can be written as a system of nonlinear algebraic equations referred to temperatures T1, T2, T3, T4:
Z1 = b11T4 + buT24 + b13T34 + buT4 + d11T1 + dxlT2 + duT3 + "uTa + C1 = 0; (10)
Z2 = b21T4 + b22T24 + b23T34 + b24T44 + d21T1 + d22T2 + d23T3 + d24T4 + C2 = 0; (11) Z3 = b31T4 + b32T24 + b33T34 + b34T44 + d31T1 + d32T2 + d33T3 + d34T4 + C3 = 0; (12) z4 = b41T4 + b42T24 + b43T34 + b44T44 + d41T1 + d42T2 + d43T3 + d44T4 + c4 = 0. (13) Coefficients bij, dZj- are determined by the relations:
E1
bn=-, b12 = -1, b13 = 0, b14 = 0;
£ p1
a p + a12 a12 , n , n ap Tp
an =-, d12 =--, d13 = 0, d14 = 0; C1 =--;
s p1<50 8p100 8 p1<50
b21 = 1, b22 = —, b23 = —, b24=0;
sp1 ep1
i r\ i a12 + &pn j «pn j a a
d21 =0, d22 =--, "23 =-, d24 =0; C2 = 0;
b31=0, &32=Spn, ^33=^23, b34=-Sp2;
А Г\ A a Pn A a Pn + 034 A a34 n d3i=U, d32 =-, Ü33 =--, d34 =-; Сз=и;
ао ао ао
&41=0, &42=0, &43=6p2, b44 = E3; d.41 =0, d.42 =0, d.43 =034, d.44 = — (Й34 ); C4 = UfrTh;
ai2 = ai2 +А12/ hi2, a34 = ^34/ Й34;
where
Ei =(ei +s12)-
S12S pi(1 -S21)
621
621s p1(1 -s12) e21 =--(621 +6pn );
612
e23 =(6 pn +634)"
6346 p2(1 -643)
643
6p2&43(1 _ 634)
e3 =--(643 +64).
634
To solve the nonlinear system of equations (10)-(13), the iteration method was used.
Let us introduce a vector function
Z =
Z1
Z2 Z3 Z 4
(14)
and the vector of unknown temperatures
T =
T1 T2 T3 T4
(15)
Then the system of equations (10)-(13) can be written as
Z (T) = 0
and an iterative algorithm for solving it as
T(n+1) = t(") - W"1(t(n))Z(T(")),
where n is approximation order;
(16) (17)
W (T ) =
dz\ dz\ dz\
дГ1 T dT3 dT4
dz 2 dz 2 dz 2 &2
dTi dT2 dT3 dT4
&з dz3 dz3 dz 3
dT1 dT2 dT3 dT4
dz 4 dz 4 dz 4 dz 4
dT1 dT2 dT3 dT4
(18)
is Jacobi matrix; W 1 (T) is inverse matrix.
Termination condition of the iterative process has the form
=1 W"1(T(n))Z(T(n))|| < ôer,
(19)
where b(n) = T(n+1) - T(n) is residual vector; || b || is norm of the residual vector; 5er is the absolute allowable error of temperature calculation. In the implemented algorithm it is accepted that
II b ||=.
(n + 1) _ T(n) j2
T
(n + 1)
- T
(n)
T
(n + 1)
- T
(n)
T
(n + 1)
- T
(n)
T
(n + 1)
- T
(n)
A solution of an auxiliary system of equations describing heat transfer in a system with one equivalent intermediate shield [7] was used as an initial approximation to the equations system solution (16)
Z(T ) = 0,
(20)
where T is the solution to this system of equations — incomplete temperature vector of the initial approximation of the equations system (16) corresponding
to the temperature of the internal wall of combustion chamber Ti = T1(0), equivalent intermediate screen T2 = T2(0) with reduced emissivity
2
=-;-
(2Ne )/Si - (Ne - 1)
and casing temperature T3= T4(0). The initial approximation for the temperature of the right wall of the screen package T3(0) is calculated as
T
(0) _ 4
(rf )4 +(T<0) )'
Such an approach ensures fast convergence of the equations system solution (16).
Temperature of i-th (i = 1, 2,..., Ne) intermediate shield of package is determined by solving a nonlinear equation
0(T-) = Spn00 ( T4 - T34 ) + a pn (T2 - T3) -
-s pi 00 T4 - T4 ) - a pi (T2 - Ti) = 0, (21)
including the constancy of the resulting heat flux in the package of shields. Here
s pi _
—+-L -1+(i -1) fi. _ 1
s21 si
-1
is the reduced emissivity of the package part containing i intermediate shields;
a pi = a pn (Ne +1)/i
is generalized heat transfer coefficient of i gaps of the package.
The temperature mode of the structural elements of the flow path and thermal protection is determined by thermal and gas-dynamic flow parameters. The following are the calculation results for the flow of lithium hydride combustion products in the air flow at a stoichiometric ratio of components. The composition of the combustion products and thermophysical characteristics are taken appropriate to the temperature Tp = 2500 K and pressure p = 0,5 MPa. The heat transfer coefficient in the flow path is calculated according to the common practice criterion formula [8]
Nu _ 0,0296 Re0,8 Pr
,0,8 D_0,46
Tw
T
n0,39 f r
V rp J
(22)
where Nu, Re, Pr are Nusselt, Reynolds and Prandtl criteria; Tw is surface temperature; Tp is gas flow temperature; Te = Tp (l + ((k - 1)/2)r M2 ) is
recovery temperature, r = Pr" is recovery coefficient, n = 1/2 for laminar and n = 1/3 for the turbulent flow mode, k = Cp / cv is adiabatic constant, Cp and cv are heat capacity at constant pressure and volume respectively. The physical properties of the flow were calculated at the determining temperature Tdef = Tp + 0,5(Tw - Tp) + 0,22(Te - Tp). By definition, heat transfer coefficient
is ap = NuA,/ dx, where dx is characteristic size (in the examples was taken dx =0.1m). Ambient temperature and heat transfer coefficient were taken Th = 300 K and ah =20 W / (m2 • K).
The most important characteristic that determines the temperature mode of the radiation heat shield is the emissivity of its structural elements. For heat-resistant materials, this value is extremely uncertain and the data given in the literature (see, for example, [9]) have a large scatter. In this regard, in the main series of calculations, the emissivity values are taken from the characteristic range for heat-resistant materials in the high-temperature region: s1 = s12 = = 84=0,9; 821 = S34 = S43 = Si = 0, 3.
Fig. 2 shows the temperature dependences of the internal wall of combustion chamber T1 and the outer housing T4, and Fig. 3 shows the corresponding dependences of the temperature of the "hot" (facing the internal wall) surface of the radiation shields package T2 on the gas flow rate taking into account only heat transfer by radiation in the system under consideration (assessment of the role of heat transfer in the gaps between the internal wall of combustion chamber, radiation shield and the housing by molecular mechanisms is given in work [7]).
An increase in the number of shield package elements leads to an unambiguous decrease of the casing temperature T4,to an increase in the temperatures of the internal wall T1 and the "hot" surface of the shield package T2. Moreover, the degree of increase in the temperature of the shields package "hot" surface with an increase in the number of elements is significantly greater (by 3-3.5 times) than in the temperature of the internal wall surface.
Further we study the temperature state of the shields package. Figures 4 and 5 show the temperature distribution between the elements of the flow path and the radiation shields package for a number of emissivity values of the package elements (the value e = 0.05 should be considered as hypothetical).
The emissivity of package elements has a weak effect on the temperature of intermediate elements. Thus, for a seven-element shield package, a change of s t from 0.5 to 0.05 results in a change in the temperature of the leftmost shield (i.e., T2 ) from 2045 to 2069 K and from 1056 to 998 K for the right-most shield (i.e., T3). However, in this case, the difference in temperatures between adjacent elements of the package changes significantly. So, if for the "hot" side of the package it is ~ 80 K, then for the "cold" it increases to 350-400 K (Fig. 4b). It should be noted that with a decrease in the number of elements of the shields package (Fig. 5), the manifestation of this factor becomes more noticeable.
Zl,K
a
T4, К
Fig. 2. Dependence of the internal wall temperature T1 (a) and the housing temperature T4 (b) on the gas flow rate for different radiation shields:
1 is no shield; 2, 3, 4, 5, 6 are one-, two-, three-, five- and seven-element shield package
Nonuniform heating of the radiation shields package elements can lead to distination deformation of the neighboring elements of the package, their mechanical contact and additional heat transfer due to contact heat conduction, i.e., deterioration of the heat insulating properties.
Multi-element radiation shield is a package of thin reflective shields, separated by distance elements. Their role can be performed by pads, backfilling, riffing or corrugations of shield elements, rods or grids [3, 6], but in any case, due to the thermal conductivity of the spacer elements, the heat shield properties of the multiscreen insulation deteriorate. Fig. 6 shows the temperature values of the flow path elements with the emissivity s21 = s34 = s 43 = si = s = 0.3 and different thermal conductivities of the spacer
7-2, к
Fig. 3. Dependence of the internal wall "hot" surface temperature on the gas flow rate:
1,2, 3, 4, 5 are one-, two-, three-, five- and seven-element shield package
2000 1500 1000 500 0
T, К
400 350 300 250 200 150 100 50 0
Fig. 4. The temperature of the flow path elements (a) and the temperature difference
T -Ti +1; i = 1, 2,..., 7, between the elements of the seven-element radiation shield (b) with the emissivity of the package elements e21 = s34 = s43 = si = s and the
combustion products flow rate Vp = 700 m / s: 1 is internal wall of combustion chamber (T1); 2 is "hot" shield surface (T2); 3 is "cold" shield
surface (T3); 4 is housing (T4)
I s = 0.5 I e = 0.3 I £ = 0.15 ■ e = 0.05
12
23
34
45
56
67
llll
1 2 3 4 5 6 7 The number of the radiation screens package element a
8 = 0.5 _
е = 0.3
- 1 1 8 = 0.15
II e = 0.05 m шип III« llll II
12 23 34 45 56 67
The number of the inter-screen gap
T, К
400 350 300 250 200 150 100 50 0
1 2 3 4 5
The number of the radiation screens package element
a
T, К
18 = 0.5 18 = 0.3 18 = 0.15 ] 8 = 0.05
m Pi
12
23 34
The number of the inter-screen gap b
45
Fig. 5. The temperature of the flow path elements (a) and the temperature difference Ti -Ti+1, i = 1,2, ... ,5, between the elements of the five-element radiation shield (b) with the emissivity of the package elements e21 = s34 = s43 = si = s and the combustion products flow rate Vp = 700 m / s:
1 is internal wall of combustion chamber (Ti); 2 is "hot" shield surface (T2); 3 is "cold" shield
surface (T3); 4 is housing (T4)
elements with their thickness of 1 mm. With increasing thermal conductivity, the temperature difference between the shield elements decreases and the temperature of the protected surface increases, i.e., the heat insulating properties of the shield deteriorate. For example, in the example above, the use of a niobium grid, which occupies 5 % of the screen element area, leads to an increase in body temperature by ~ 400 K.
T, K
2000 1500 1000 500 0
Fig. 6. The temperature of the flow path elements with five-element radiation shield
with the emissivity of the package elements s21 = s34 = s43 = s, = s = 0.3 and the combustion products flow rate Vp = 700 m / s with different thermal conductivity of
1 mm thick spacer pads: 1 is internal wall of combustion chamber (T1); 2 is "hot" shield surface (72); 3 is "cold" shield surface (T3); 4 is housing (T4 ); red color is no thermal conductivity distination; blue color is X = 0,2W / (m • K); green color is niobium mesh
Conclusion. On the basis of the developed model, the analysis of the heat insulating ability and the temperature state of the multi-element package of radiation shields was carried out. Such multi-element radiation shields are effective heat-shielding means of high-temperature devices. It is shown that the temperature distribution over the shield elements is uneven, which can lead to different thermal deformations of the elements, causing distortion of the shield, the possibility of mechanical contact of neighboring elements and the deterioration of thermal insulating ability.
Translated by K. Zykova
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Tovstonog V.A. — Dr. Sc. (Eng.), Professor, Department of Spacecraft and Launch Vehicles, Bauman Moscow State Technical University (2-ya Baumanskaya ul. 5, str. 1, Moscow, 105005 Russian Federation).
Please cite this article as:
Tovstonog V.A. Conjugate heat transfer and estimation of thermal state of elements of the thermal protection shield package. Herald of the Bauman Moscow State Technical University, Series Mechanical Engineering, 2019, no. 4, pp. 44-57. DOI: 10.18698/0236-3941-2019-4-44-57
