DOI 10.1007/s10891-015-1179-5
Journal of Engineering Physics and Thermophysics, Vol. 88, No. 1, January, 2015
COMPUTATIONAL MODELING OF CONJUGATE HEAT TRANSFER IN A CLOSED RECTANGULAR DOMAIN UNDER THE CONDITIONS OF RADIANT HEAT SUPPLY TO THE HORIZONTAL AND VERTICAL SURFACES OF ENCLOSURE STRUCTURES G. V. Kuznetsov, T. A. Nagornova, and A. É. Ni
UDC 536.33:536.244
We have carried out computational modeling of nonstationary conductive-convective heat transfer in a closed rectangular domain in a conjugate formulation with a local heat source (a gas infrared radiator). Four variants of possible description of the radiant energy distribution over the inner surfaces of enclosures have been considered. As a result of the computational modeling, differential (temperature fields and stream functions) and integral (Nusselt numbers) heat transfer characteristics have been obtained. It has been shown that the radiant flux distribution influences the heat transfer intensity. Keywords: infrared radiator, radiative heating, conjugate heat transfer, computational modeling, natural convection, heat conduction. Introduction. The steadily increasing energy consumption, the problems with operating NPPs, and the limited capacities of renewable energy sources make the use of energy-saving technologies more and more topical. The traditional convective heating of large-size industrial premises is at present ineffective in many cases. The use of gas infrared radiators (GIR) can make it possible to reduce the energy consumption, according to the authors of [1], by more than 30% due to by local heat supply to the working places. Investigations of radiant heating systems are carried out mainly with the aim of choosing optimal characteristics of the GIR [2–4]. The obtained experimental data [5, 6] have shown that a microclimate favorable for personnel is attained in using GIRs at lower expenditures of fuel resources. It is not always possible to perform natural experiments under the conditions of large-size production shops with locally sitting operating equipment. In this case, it is expedient to use methods of mathematical modeling in choosing the basic parameters of GIR-based heating systems and test subsequently the results of the theoretical analysis under production conditions. It has been found [7, 8] that mathematical modeling of the processes of conductive-convective heat transfer in a conjugate formulation in solving analogous problems creates conditions for a much more detailed analysis of the temperature regimes of investigated objects of heat supply. Mathematical models based on the thermal balance of a room [9] do not completely take into account the spatial character of the heat transfer typical in practice. The known solutions of heat transfer problems during operation of radiant heating systems [10, 11] ignore the convection. The mathematical model of conductiveconvective nonstationary heat transfer under radiant heating of one of the internal boundaries of a closed rectangular domain in a conjugate formulation [12] was formulated under the assumption that the heat flux is distributed uniformly only over the horizontal surface of the lower base. It is interesting to analyze the thermal regimes of heated objects under radiant heating with account for the possible distribution of energy released by the GIR over the surfaces of vertical enclosure structures. The aim of the present work is to model computationally the nonstationary process of radiant heating of a rectangular domain in a conjugate formulation with account for the radiant energy distribution over the inner horizontal and vertical surfaces of enclosure. Formulation of the Problem and Solution Method. We consider a nonstationary boundary-value problem of conductive-convective heat transfer in a conjugate formulation. The geometrical model of the process is shown in Fig. 1. The solution domain consists of 5 rectangular subdomains. At its outer boundaries heat insulation conditions are assumed, and at the inner boundaries "air–enclosure" boundary conditions of the 4th kind are assumed. In the numerical realization, we made the following assumptions: the air flow conditions are laminar; the thermal properties of the air and the materials Tomsk National Research Polytechnical University, Energy Institute, 30 Lenin Ave., Tomsk, 634050, Russia; email:
[email protected]. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 88, No. 1, pp. 165–174, January–February, 2015. Original article submitted May 26, 2014. 168
0062-0125/15/8801-00168 ©2015 Springer Science+Business Media New York
Fig. 1. Problem solution domain: 1) air; 2) enclosure structures; 3) gas infrared radiator (conventional notation). of enclosures are temperature-independent; the air is a viscous incompressible Newtonian fluid satisfying the Boussinesq approximation; the air is a diathermally transparent medium. We assumed that the total radiant flux from the GIR can be given as a sum of heat fluxes q1–q6 whose values were determined by the zonal method [13] with account for the angular emissivity for bodies of finite sizes. The investigated process of heat transfer is described by the nonstationary Navier–Stokes and energy equations for the air and by the heat conductivity equation for the enclosures. In the variables "stream function ψ–velocity vector ω– temperature," they have the form ⎛ ∂ 2ω ∂ 2ω ⎞ ⎛ ∂ω ∂ω ∂ω ⎞ ∂T ρ1 ⎜ +u +v = μ + 2 ⎟⎟ + ρ1β g y 1 , 1 ⎜ ⎟ 2 ⎜ ∂x ∂y ⎠ ∂x ∂y ⎠ ⎝ ∂t ⎝ ∂x ∂ 2ψ ∂x 2
+
∂ 2ψ ∂y 2
(1)
= −2ω ,
(2)
⎛ ∂ 2T ∂T1 ∂T ∂T ∂ 2T1 ⎞ + u 1 + v 1 = a1 ⎜⎜ 21 + ⎟, ∂t ∂x ∂y ∂y 2 ⎟⎠ ⎝ ∂x
(3)
⎛ ∂ 2T ∂T2 ∂ 2T2 ⎞ = a2 ⎜⎜ 22 + ⎟. ∂t ∂y 2 ⎟⎠ ⎝ ∂x
(4)
Equations (1)–(4) were reduced to their dimensionless form. For the distance scale, the cross section of the gas cavity was chosen. To reduce the system of equations to its dimensionless form, we used the following relations: U = Ω =
u , Vnc
ω , ω0
X =
Θ =
x , ( L − 2l1)
T − T0 , Th − T0
V0 =
Y =
y , ( L − 2l1)
τ =
t , t0
g β (Th − T0 ) ( L − 2l1) ,
V =
v , Vnc
ψ 0 = Vnc L ,
Ψ =
ψ , ψ0
ω0 =
Vnc . L
The dimensionless Navier–Stokes and energy equations in the variables "curl of velocity Ω–stream function Ψ– temperature Θ" have the form [7, 12] ∂Ω ∂Ω ∂Ω +U +V = ∂τ ∂X ∂Y
1 Gr
∇ 2 Ψ = −2Ω ,
∇ 2Ω +
1 ∂Θ1 , 2 ∂X
(5)
(6)
169
TABLE 1. Comparison of Mean Nusselt Numbers Ra
103
104
105
Test values
1.089
2.205
4.442
[18]
1.117
2.238
4.471
[19]
1.108
2.201
4.430
1 ∂Θ1 ∂Θ1 ∂Θ1 +U +V = ∇ 2 Θ1 , ∂τ ∂X ∂Y Pr Gr
(7)
∂Θ 2 = ∇ 2Θ2 . ∂Fo 2
(8)
The initial conditions for Eqs. (5)–(8) are as follows: Ψ ( X , Y , 0) = 0 ,
Ω( X , Y , 0) = 0 ,
U ( X , Y , 0) = 0 ,
V ( X , Y , 0) = 0 ,
Θ1 ( X , Y , 0) = 0 ,
Θ 2 ( X , Y , 0) = 0 .
The boundary conditions for Eqs. (5)–(8) are as follows: at the outer boundaries of the solution domain X = 0,
X = 1,
Y = 0,
Y = 1,
0 < Y ≤1:
∂Θ 2 ( X , Y , τ) = 0; ∂X
0 < X ≤1:
∂Θ 2 ( X , Y , τ) = 0; ∂Y
at the inner boundaries "solid wall–air:" Y =
Ψ1 = 0,
Y =
X =
170
h1 , ( L − 2 l1)
∂Ψ1 = 0, ∂Y
l1 l4 : < X ≤ ( L − 2 l1) ( L − 2 l1)
⎧Θ 2 = Θ1 , ⎪ λ1 ∂Θ1 ⎨ ∂Θ 2 ⎪ ∂Y = λ ∂Y + Ki i , 2 ⎩
h4 l1 l4 , : Ψ1 = 0, < X ≤ ( L − 21) ( L − 2 l1) ( L − 2 l1)
l1 , ( L − 2 l1)
h1 h2 : Ψ1 = 0,
