ISSN 0018151X, High Temperature, 2011, Vol. 49, No. 4 pp. 580–587. © Pleiades Publishing, Ltd., 2011. Original Russian Text © V.A. Babkin, 2011, published in Teplofizika Vysokikh Temperatur, 2011, Vol. 49, No. 4, pp. 598–605.
HEAT AND MASS TRANSFER AND PHYSICAL GASDYNAMICS
Turbulent Flow through a Circular Pipe: Resistance and Heat Exchange at the Constant Boundary Temperature V. A. Babkin Petrozavodsk State University, pr. Lenina 33, Petrozavodsk, 185910 Russia email:
[email protected] Received February 19, 2010
Abstract—In this work, problems of the velocity profile, hydraulic resistance and heat exchange at constant equal temperature on the walls, steadystate turbulent flow, and established heat exchange in a straight chan nel limited by coaxial circular cylinders (a circular pipe) are solved. A moving incompressible fluid is consid ered as the medium, its viscous and heatconducting properties being defined not only by its physical prop erties, but also by stable vortex structures that are formed upon the turbulent flow and generate local anisot ropy of the medium. A vector called the director is a characteristic parameter of anisotropy. Director dynamics within the flow is assigned by a separate equation. The flow region consists of two nearwall subre gions, which are adjacent to solid flow boundaries. The boundary between the subregions is determined dur ing solving the problem. A closed set of equations is formulated for the desired values (velocity, temperature), and boundary conditions are laid. The velocity profile and temperature field in the flow were obtained in form of solutions to the corresponding boundary problems. The results of solution are compared with the experi mental data and empirical formulas. DOI: 10.1134/S0018151X11040031
incompressible fluid in a straight circular pipe are solved.
INTRODUCTION A number of theoretical [1–4] and experimental [5–8] studies have been devoted to turbulent flow in a circular pipe. However, the absence of a generally accepted turbulence model does not allow one to regard the research in this field as complete. Following the tradition of the preliminary analyti cal studies, in which different authors used various models, the problems set up in this work are studied within the framework of a model that has not been employed before. Over the past 20–30 years, research ers’ attention has been drawn to vortices stretched along the flow that are formed near the solid border of the turbulent flow of the system, which are called the horseshoe vortices (or Λvortices). They form a stable system and induce local anisotropy of flow, which determines the nature of resistance and heat exchange of the moving fluid. It appeared [9, 10] that the effect of anisotropy can be accounted for by introducing a parameter (the socalled director) that is typical of this type of anisotropy into the constitutive equations of the medium. The system of equations for the velocity and the director, together with the equations of conti nuity and energy and the constitutive equations of the medium, compose the system of equations for the model of anisotropic nearwall turbulence [9, 10]. Within the framework of this model, the problems of determining the profile of velocities and field of tem perature upon steadystate turbulent flow of an
MODEL EQUATIONS Model equations are considered in the Cartesian coordinate system xi. Velocity ui and unitary director ni are the local kinematic parameters of the medium in the model of anisotropic nearwall turbulence. All local values characterizing the state and motion of the system are by definition considered to be averaged over volume. For an incompressible fluid, the equations acquire the form [9, 10]
∂ui = 0, ∂xi
(1)
dui ∂piα = + ρ fi, dt ∂xα
(2)
⎛ dn ⎞ ∂β ρ d ⎜ I i ⎟ = ij + g i + ρFi, dt ⎝ dt ⎠ ∂ x j
(3)
ρ
∂q (4) ρ dU = pijeij + βij N ij − g iN i + Q − i , dt ∂x i where ρ is the fluid density, ui is the velocity, pij are the stresses, fi is the mass force density, U is the internal energy of a mass unit, Q is the intensity of a heat source, and qi is the heat flux. The quantities βij, gi, and Fi are called the generalized stress, generalized internal mass force, and generalized external mass force, respectively.
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TURBULENT FLOW THROUGH A CIRCULAR PIPE
σ ij = K nα ,i (n j,α − nα , j + n j nβnα ,β ) ,
(6)
τij = μ1nαnβeαβnin j + μ 0eij,
(7)
β ij = κ j ni + K ( ni, j − n j,i − n j nαni,α ) ,
(8)
g i = χ ni − ( κ βni ) ,β + K nαnβ ,αnβ ,i,
(9)
qi = − ( λ 0T,i + λ1ni n jT, j ) ,
x Fig. 1. The scheme of flow in the upper half of a horizontal circular pipe: u is the longitudinal velocity; n is the direc tor; R1 and R2 are the radii of the internal and the external walls, respectively; r0 is the radius of the cylinder of the maximum velocity; and x and r are coordinates.
ux = u(r),
⎛ ∂u ∂u ⎞ eij = 1 ⎜ i + j ⎟ , 2 ⎝ ∂ x j ∂ xi ⎠ d = ∂ + u ∂ , n = dni , j i dt ∂t xj dt N i = ni − ωiαnα, N ij = ni, j − ωiαnα, j , 2ωij = uij − u ji, ni, j =
n
∂ni , ∂x j
FORMULATION OF THE PROBLEM An incompressible fluid in the mode of steadystate turbulent flow moves within the region limited by two coaxial circular cylinders with radii R1 and R2 (R1 < R2), which are used to simulated smooth solid channel walls. Let us introduce the cylindrical coordinate sys tem with the x axis oriented along the cylinder axis toward the flow and the r axis oriented with respect to the cylinder radius away from their axis (Fig. 1). Thus, the flow takes place within the region R1 ≤ r ≤ R2 . Let us divide the solution to the problem into two stages. First, we assume that the flow is isothermal and find the velocity profile; next, the temperature distri bution will be found. We assume that the coefficients μ0, μ1, and K and the inertia parameter I are constant under the speci fied flow conditions. We neglect the external mass forces fi and Fi in Eqs. (2) and (3) and will seek ui and director ni in the following form: Vol. 49
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2011
ur = uφ = 0,
nx = cosθ(r ), nr = sinθ(r ), nφ = 0,
(10)
where, р is the pressure; T is temperature, λ0, λ1, μ0, μ1, and K are the model coefficients; δij is the Kronecker symbol; and χ and κi are arbitrary scalar and vector functions, respectively. Since the properties of the fluid near a solid wall are determined by the nearwall vortex flow structure, coefficients λ0, λ1, μ0, μ1, and K can be dependent on the parameters that provide a global characterization of the flow, e.g., the Reynolds number.
HIGH TEMPERATURE
R2
(5)
n
u
r0
pij = −pδij + σij + τij,
r
R1
The quantity I characterizes the averaged inertia of the structure upon rotation of elements of the vortex struc ture in the nearwall layer. It is assumed that, over repeated indices, summation is taken from 1 to 3. The constitutive equations take into account the specific character of the medium. For the turbulized fluid, they are [9, 10]
581
(11)
where θ is the angle between the director and the x axis. Substituting the desired functions (11) into Eqs. (1)– (3) and (5)–(9) and assuming arbitrary quantities χ and κi to be equal to zero, we obtain equations to determine u(r) and θ(r) in the following form (the con tinuity equations become an identity):
1 d(r τ xr ) = −P, r dr
(
∂p , ∂x
(12)
μ 0 du , 2 dr
(13)
P=−
τ xr = μ1 sin 2 θ cos 2 θ +
)
)(
)
2 ⎛ 2 ⎞ sin θ cos θ ⎜ d θ2 + 1 d θ⎟ − 2 − 3cos 2 θ d θ = 0. (14) r dr ⎠ dr ⎝ dr Integrating Eq. (12) under the condition that P = const provides
(
(15) τ xr = −P r + C , 2 r where C is the constant of integration. On the channel walls, r = R1 and r = R2 and the fluid velocity u(r) is taken as zero; therefore, point r = r0 exists in the interval R1 ≤ r ≤ R2 , where the velocity attains the maximum value u0. According to (13), τ xr = 0 . With this provision, ⎛ r 2⎞ (16) τ xr = − P ⎜ r − 0 ⎟ . 2⎝ r⎠ At boundaries r = R1 and r = R2, the absolute values of shearing stresses τ1 and τ2, respectively, are expressed by the following formulas:
582
BABKIN
⎛ ⎛ r2⎞ r2 ⎞ (17) τ1 = − P ⎜ R1 − 0 ⎟ , τ 2 = P ⎜ R2 − 0 ⎟ . 2⎝ R1 ⎠ 2⎝ R2 ⎠ Surface r = r0 with the velocities being maximum on it divides the flow region into two parts: (I) subregion R1 ≤ r ≤ r0 adjacent to the internal wall and (II) subre gion r0 ≤ r ≤ R2 adjacent to the external wall (Fig. 1). From physical standpoint, within the model accepted, this division of the flow attests to the existence of two nearwall current within the flow. Each subregion (I or II) has its own vortex structure with its own director field ni(r). Thus, the velocity profile over the region R1 ≤ r ≤ R2 has to be found separately in subregions I and II; then, the profile parts should be juxtaposed in point r = r0.
VELOCITY PROFILE Internal region (R1 ≤ r ≤ r0). In addition to the obvi ous solution θ = const, Eq. (14) has a first integral,
r sin θ cos θθ ' = B , 2
(18)
where B is the constant of integration; the stroke denotes the derivative with respect to r. For the sake of convenience, we take the constant B in form of B = b1R1, where b1 is the constant that needs to be deter mined. Equation (18) acquires the following form:
r sin θ cos 2 θθ ' = b1R1.
(19)
On a solid wall, the nearwall structure vortices are elongated along the flow; therefore, we set the bound ary condition for θ(r) as θ(R1) = 0 [9, 10]. By integrating Eq. (19) with this condition, we obtain
ξ cos θ = 1 − 3b1R1 ln , h 3
ξ= r , R2
R h = 1. R2
1/3
(20)
(21)
Formulas (13) and (16) together provide the equa tion for determination of the velocity profile in subre gion I:
(
μ11 sin 2 θ cos 2+
)
⎛ r 2⎞ μ 01 du = − P ⎜ r − 0 ⎟, 2 dr 2⎝ r⎠
⎡ k13 ⎛ 2 s − ⎜⎜ γ 1 − 1 arctg 2 2⎢ 2 6μ11b1 ⎣ 2γ 1 − 1 ⎝ γ1 − 1 1
G1(s) = +
γ1 γ1 − s ⎞ s 2 1 4 2 ln ⎟ + + ln| s − s − ε1| 2 γ1 + s ⎠ 2 4
2 2 2 2 γ1 − s ⎤ 3ξ 0 R2 2ε1 + 1 ln − [M1 ln| s − k1| ⎥ 4(2γ12 − 1) s 2 + γ12 − 1⎦ 2μ11 2M 2 2s + k1 + arcot 3k1 3k1
+
⎛ 2s + k1 ⎞ + M 3 ⎜ 1 ln(s 2 + k1s + k12 ) − 1 arcot ⎟ 3k1 ⎠ 3 ⎝2 + M 4 ln | s − γ1| M6 arcot s + M 5 ln(s + γ1) + 2 γ1 − 1 γ12 − 1
M7 ln(s 2 + γ12 − 1) ⎤ , ⎥⎦ 2 13 13 s = [1 − 3b1R2(ξ − h)] , k1 = (3b1R1 + 1) , μ ε1 = 01 , 2γ12 = 1 + 1 + 4ε1. 2μ11 Coefficients М1, М2, … М7 in (23) are determined by the equations 1 M1 = , 2 2 2 2 3 k1 − γ1 k1 + γ1 − 1
(
)( ) ( + 2) − γ ( γ − 1)⎤⎦ M = , 3 ( k + k γ + γ ) ⎡k − k ( γ − 1) + ( γ − 1) ⎤ ⎣⎢ ⎦⎥ k ( k − 1) + 2γ ( γ − 1) , M =− 3 ( k + k γ + γ ) ⎡⎣k − k (γ − 1) + (γ − 1) ⎤⎦ 2
k1 ⎡⎣k12
4 1
k12
2 2 1 1
3
4 1
2 1
4 1
2
2
(
(
M7
2 1
4 1
2 1
2 1
2 2γ 1
2 1
2 1
2 1
2 1
)(
(
2
)
γ1 , 3 3 − 1 k1 + γ1
(
)(
)
)
k1 γ1 − 1 2
) ⎣ + (γ − 1) ⎤⎦⎥ (γ − 1) = . ⎡ ⎤ (2γ − 1) ⎣⎢k + (γ − 1) ⎦⎥ 2γ12
2
γ1 , 3 3 − 1 k1 − γ1
2 2γ 1
3
M6 = −
2 1
2 1
4 1
M4 = − M5 =
4 1
2 1
2 2 1 1
(22)
where μ01, μ11 are the model coefficients μ0 and μ1 fit ted with an index “1” in order to denote their belong ing to the medium in region I. Let us substitute Eq. (21) into Eq. (22) and integrate it with a provision for the adhesion of fluid on the wall u(R1) = 0. By denoting the velocity in this region as u1, we obtain the velocity pro file in the following form:
(23)
+
2 1
If (ξ − h) h Ⰶ 1, then approximately ln (ξ h) = (ξ − h) h; formula (20) can be written as
cos θ = [1 − 3b1R2( ξ − h)] .
u1(s) = P ( G1(s) − G1(1)) ,
−1
⎡k 6 ⎢ 1
,
2
2 1
2 1
3
2 1
6 1
2 1
3
External region (r0 ≤ r ≤ R2). The profile is obtained in the same manner as in the previous case. We take the constant B in formula (18) as B = – b2R2, where b2 is the constant to be determined. Equation (18) acquires the following form: HIGH TEMPERATURE
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TURBULENT FLOW THROUGH A CIRCULAR PIPE
(24) r sin θ cos 2 θθ ' = −b2R2. By solving Eq. (24) with boundary condition θ(R2) = 0, we obtain (25) cos 3 θ = 1 + 3b2R2 ln ξ. In proximity of the cylinder r = R2, when (1– ξ) Ⰶ 1, approximately ln ξ = (ξ – 1). Then, Eq. (25) can be written as (26) cos θ = [1 − 3b2R2(1 − ξ)] . The equation for determining the velocity profile is similar to Eq. (22):
(
(
)
⎛ μ 02 du r 2⎞ (27) = − P ⎜ r − 0 ⎟. 2 dr 2⎝ r⎠ In order to emphasize that Eq. (27) refers to region II, the values are marked with index “2”. The substitution of formula (26) into Eq. (27) followed by its integra tion with a provision for the adhesion of fluid on the wall u(R2) = 0 yields the desired result in the following form (velocity u is redenoted as u2): u2(t) = P ( G2(t) − G2(1)) ,
G2(t) = +
1 6μ12b22
⎡ k23 ⎛ 2 t ⎜⎜ γ 2 − 1 arcot 2 ⎢ 2 ⎣2γ 2 − 1 ⎝ γ2 − 1
γ2 γ2 − t ⎞ t 2 1 4 2 ln ⎟ + + ln | t − t − ε 2 | 2 γ2 + t ⎠ 2 4
(
)
2
(28)
(
(
)
HIGH TEMPERATURE
)
Vol. 49
(
(
(
)
)
) (
No. 4
(
) (
)
(
N5 = −
2
)(
(
2 2γ 2
N6 =
N7
)
γ2 , 3 3 − 1 k2 − γ 2
(
)(
)
)
k2 γ 2 − 1 3
( 2γ
2 2
)
,
2
(
)
3 − 1 ⎡k26 + γ 22 − 1 ⎤ ⎢⎣ ⎥⎦
(γ − 1) = (2γ − 1) ⎡⎣⎢k + (γ
,
2
2 2
3 . −1 ⎤ ⎦⎥ The velocity profile over the entire flow region h ≤ ξ ≤ 1 is the combination of partial profiles (23) and (28): 2 2
6 2
2 2
)
⎧P [G1 ( s (r )) − G1 ( s (R1))], R1 ≤ r ≤ r0, (29) u( r ) = ⎨ ⎩P [G2 (t (r)) − G2 (t (R2))], r0 ≤ r ≤ R2. Similarly to [2], the boundary between subregions ξ = ξ0 is determined by the criterion of velocity conti nuity in this point, namely, by the equation (30)
)
2011
ANALYSIS OF THE SOLUTION. COMPARING IT WITH THE EXPERIMENTS
b1 = b2 = 4.80 м −1,
13
)
)
2
Let us discuss the solution for air flows under nor mal conditions: density ρ = 1.205 kg/m3 and kine matic viscosity ν = 1.50 × 10–5 m2/s. Based on com parison with the experimental data [5], we define the values of model parameters b1, b2, μ0, and μ1 for air as the equations
t = [1 − 3b2R2(1 − ξ)] , k2 = (3b2R2 − 1) , μ 2 ε 2 = 02 , 2γ 2 = 1 + 1 + 4ε 2. 2μ12 Coefficients N1, N2, … N7 in (28) are determined by formulas 1 N1 = , 2 2 2 2 3 ⎡⎣k2 k2 − 1 − γ 2 γ 2 − 1 ⎤⎦ k2 ⎡⎣k22 k22 + 2 − γ 22 γ 22 − 1 ⎤⎦ N2 = − , 2 4 2 2 4 4 2 2 2 3 k 2 + k 2 γ 2 + γ 2 ⎡k 2 − k 2 γ 2 − 1 + γ 2 − 1 ⎤ ⎢⎣ ⎥⎦
(
(
The assigned problem has been nominally solved. Next, the solution needs to be analyzed.
⎛ 2 t − k2 ⎞ 2 2 + N 3 ⎜ 1 ln| t − k2t + k2 | + 1 arcot ⎟ 3k2 ⎠ 3 ⎝2 N6 + N 4ln| t − γ 2| + N 5 ln(t + γ 2 ) + 2 γ2 − 1 N 2 2 × arcot t + 7 ln(t + γ 2 − 1)⎤ , 2 ⎥⎦ 2 γ2 − 1 13
)
2
G1( s ( ξ)) − G1(1) = G 2( t ( ξ)) − G 2(1).
−t ⎤ 2ε 2 + 1 + ln ⎥ 2 2 4 2γ 22 − 1 t + γ 2 − 1⎥⎦ 3ξ 2 R 2 ⎡ 2N 2 2 t − k2 − 0 2 ⎢N 1ln| t + k2 | + arcot 2μ12 ⎣ 3k2 3k2 γ 22
2
2 3 k24 + k22γ 22 + γ 42 ⎡k24 − k22 γ 22 − 1 + γ 22 − 1 ⎤ ⎢⎣ ⎥⎦ γ2 N4 = , 2 3 3 2 2γ 2 − 1 k2 + γ 2
13
μ12 sin 2 θ cos 2+
)
k2 k2 − 1 + 2γ 2 γ 2 − 1 2
N3 = −
(
583
μ 01 = μ 02 = 1.80 × 10 −6 Pa s, μ1i = 0.047u∗i Pa s,
(31)
u∗i = τ i ρ, i = 1,2, where τi are the absolute values of shearing stresses on the walls defined by formulas (17). The values bi, μ0i, and μ1i accepted in (31) only slightly differ from those obtained earlier upon similar air flows [11]. Since coefficients μ1i depend on differ ent dynamic velocities, the turbulized fluid in the lay ers adjacent to different walls should be considered as distinct.
584
BABKIN
ξ0 1.0
u, m/s 20
0.8
15
0.6
10
2
1 2
5
0.4 1 2 3
0.2
0
1
0.2
0.6
0.4
0 0.8
1.0 h
Fig. 2. Dimensionless radius ξ0 of the cylinder of maxi mum velocity as a function of the ratio between the wall radii h. The curve corresponds to the plot of formula (33); dots: (1) calculation using (30), (2) experimental data [5], and (3) experimental data [6].
The relationship between the dynamic velocities u∗i under different values of the ratio h follows from formulas (17): 2 2 2 τ1 u*1 ξ0 − h (32) . = 2 = τ2 u 2 h 1 − ξ20 * As can be seen from formulas (23) and (28), coor dinate ξ enters Eq. (31) in a complicated manner. Parameters εi and γi enter the rate u*i to be determined through μ1i. Therefore, Eq. (30) is solved numerically. It appears that the position of maximum of the rate ξ0 in the cross section of the flow can be considered dependent only on the ratio of radii h. The depen dence on other parameters is weak, if not absent. Experiments [5–7] and calculations [2] support this conclusion. Figure 2 shows solutions to Eq. (30) for different values of h, as well experimental points [5, 6] and the empirical curve [7]:
(
)
0.343
h+h ξ 0 = . (33) 0.343 1+h The calculated curve [2] was not placed into Fig. 2, since it merges with curve (33). It should be noted that formula ξ 0 = h + h1/3 1+ h was proposed instead of formula (33) in [2]; however, both formulas provide almost equal results. All results shown in Fig. 2, including the depen dence ξ0(h) that was obtained in this study, agree with each other both qualitatively and quantitatively. 1/3
0.2
0.4
0.6
0.8
1.0 η
Fig. 3. Velocity profiles upon air flow. The curves corre spond to the calculation using formula (29); dots, to the experimental data [5]: (1) h = 0.0625, (2) h = 0.562, umax = 18 m/s, η = (ξ – h)/(1 – h).
When determining the profile, it was assumed that the tilt angles of nearwall vortices with respect to the flow do not exceed 90° and are determined by formu las (21) and (26) in the corresponding regions. There fore, the flow parameters R2, h, b1, and b2 need to satisfy the system of inequalities ⎧0 ≤ 1 − 3b1R2(ξ − h) ≤ 1, ⎨ ⎩h ≤ ξ ≤ ξ 0; (34) ⎧0 ≤ 1 − 3b2R2(1 − ξ) ≤ 1, ⎨ ⎩ξ 0 ≤ ξ ≤ 1. The following double inequality follows from ine qualities (34):
1 ≤ξ ≤ h+ 1 . (35) 0 3b2R2 3b1R2 Under experimental conditions [5], R2 = 101.5 mm and b1 = b2 = 4.80 m–1 (see (31)). Then, it follows from inequality (35) that the minimum value of ξ0 at which the model can be used is equal to ξ0min = 0.316. According to formula (33), the minimum ratio between the radii hmin = 0.058 corresponds to it. It is necessary to use another model for h < hmin. In Fig. 3, velocity profiles obtained by calculating using formulas (23), (28), and (30) are compared with the experimental profiles [5]. Since the results are sim ilar for all profiles [5], two flows were selected for the comparison: with the largest and smallest gaps (h = 0.0625 and h = 0.562) and with equal maximum veloc ity of 18 m/s. The proximity of the calculated and experimental profiles is obvious. The friction coefficient f in a circular pipe is deter mined by the formula [12] 1−
f =
4P ( R2 − R1) , 2 ρw
HIGH TEMPERATURE
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(36) No. 4
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TURBULENT FLOW THROUGH A CIRCULAR PIPE
where w is the mean velocity over the gap section that is calculated using formula 1 ⎛ ξ0 ⎞ 2 ⎜ ⎟, (37) w= u ( s ( ξ )) ξ d ξ + u ( t ( ξ )) ξ d ξ 1 2 ⎟ 1 − h2 ⎜ ξ0 ⎝h ⎠ velocities u1 and u2 being assigned by formulas (23) and (28). Figure 4 shows the calculated and empiric depen dences of the coefficient f on the Reynolds number 2w ( R2 − R1) Re = . ν Let us obtain the calculated values f (points) as fol lows. First, we determine the boundary of regions ξ = ξ0 using the fixed h. Then, by assigning the absolute values of pressure gradient P at this value of h and inte grating profiles (23) and (28) with respect to the flow section, let us find the mean velocity w, the Reynolds number Re, and the coefficient f for each P value. The experimental data in Fig. 4 are represented by empirical curves f(Re). Curve 1 is the plot of formula [5]
∫
∫
f = 0.348 , Re1/4 curve 2 is the plot of formula
1.030 (39) , [1.82 log (Re 8)]2 which was selected according to the rule [8, 12] and the Filonenko formula [13] for the Reynolds number under consideration. It is evident that all calculated points are located near the empirical curves. HEAT EXCHANGE If we have expressions (23) and (28) for the turbu lent profile of velocities in the channel, the problem of heat can also be solved. Let us find the steadystate dis tribution of temperature in a semiinfinite pipe х ≥ 0 with constant equal wall temperatures Tw1 = Tw2 = Tw with provision that temperature T0 is constant in the inlet section x = 0. The temperature T in the region R1 ≤ r ≤ R2 in the adopted coordinate system r, ϕ, x is found as T = T ( r , x ) . Then, formulas (10) for heat flux can be written as q r = − λ 0 + λ1nr2 ∂T − λ1nr nx ∂T , ∂r ∂x (40) q φ = −λ 1nφ nr ∂ T + n x ∂ T , ∂r ∂x q x = −λ1nxnr ∂T − λ 0 + λ1nx2 ∂T . ∂r ∂x The equation of heat conduction [12], taking into account Eqs. (11) and (40) with the provision that
(
)
)
(
(
HIGH TEMPERATURE
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f 0.030 1 2
0.025
)
No. 4
2011
h = 0.2 h = 0.5 h = 0.8
0.020 0.015 0.010 4 10
105
106 Re
Fig. 4. The friction coefficient f as a function of the Rey nolds number. Curves: (1) diagram of formula (38), (2) formulas (39); dots correspond to the calculations for h = 0.2, 0.5, and 0.8.
thermal conductivity along the x axis can be neglected, coefficients λ0 and λ1 being constant, has the following form:
(λ
(38)
f =
585
0
2 + λ1 sin 2 θ ∂ T2 ∂r
)
⎛ λ + λ1 sin 2 θ ⎞ +⎜ 0 + λ1 sin 2θ θ ' ⎟ ∂T r ⎝ ⎠ ∂r = ρ c pu(r ) ∂T , ∂x
(41)
where cp is the specific heat capacity at constant pres sure, and u ( r ) is the velocity profile (29). Let us introduce dimensionless variables Θ=
T − Tw , T0 − Tw
ξ= r , R2
X = x R2
(42)
and substitute θ(ξ) in subregions I and II, respectively, with expressions (21) and (26). Then, the equation (41) acquires the following form: 2 ∂ Θ + Φ(ξ) ∂ Θ = Ψ(ξ) ∂ Θ ; 2 ∂ξ ∂X ∂ξ 2λ11b1R2 ⎧1 + , h < ξ < ξ0; ⎪ξ s(ξ) ⎡λ + λ 1 − s 2(ξ) ⎤ 01 11 ⎪ ⎣ ⎦ (43) Φ(ξ) = ⎨ b R λ 2 1 12 2 2 ⎪ − , ξ0 < ξ < 1; ⎪ξ t(ξ) ⎡λ 02 + λ12 1 − t 2(ξ) ⎤ ⎩ ⎣ ⎦ c u R ( ) ρ ξ ⎧ p 2 ⎪λ + λ 1 − s 2(ξ) , h < ξ < ξ0; 11 ⎪ 01 Ψ(ξ) = ⎨ ρc pu(ξ)R2 ⎪ , ξ0 < ξ < 1. ⎪ λ 02 + λ12 1 − t 2(ξ) ⎩
(
)
(
)
(
)
(
)
586
BABKIN
When passing on to formulas (47), it was taken into account that nr = sin θ and θ = 0 on the walls. On the walls of a circular pipe, the Nusselt numbers are determined by the equations [12]
Nu0 800 600 1
2
3
Nu1 =
400 h = 0.2 h = 0.5 h = 0.8
200
0
2
1
3
4 5 Re × 10–5
Fig. 5. The limiting Nusselt number as a function of the Reynolds number. The dots correspond to Nu0 values obtained from formula (50) using Nu values found by cal culation using formulas (46)–(49) at different h. Curves correspond to the plots of empirical formulas for deter mining Nu using (50): (1) formula (52), (2) formula (54), and (3) formula (53).
Here, indices “1” and “2” at coefficients λ0 and λ1 denote their correspondence to the nearwall subre gions I and II. The boundary conditions are as follows: Θ(h, X ) = Θ(1, X ) = 0,
Θ(ξ,0) = 1.
(44)
Based on the known empirical formulas for the Nusselt number in circular channels [12, 14–17] and the alreadyperformed solution to a similar problem in a circular pipe [11], let us define the model heat con duction coefficients λ0 and λ1 for air, taking into account the subregion they refer to, using formulas similar to formulas (31):
λ 0i = 0.28u*i R2(2.35 − 2.10h), λ1i = 46.5u*i (2.35 − 2.10h),
(45)
i = 1,2.
The approximate solution to Eq. (43) with bound ary conditions (44) is obtained using the Galerkin method in the following form:
∑ g (X)sin (kπ 1ξ −− hh), m
Θ(ξ, X) =
(46)
k
k =1
where gk (X) are functions determined during the solu tion process. In this problem, heat exchange occurs on both walls; therefore, heat transfer is characterized by Nus selt numbers corresponding to these walls. Since only the heat conductivity oriented along the normal toward the channel walls is considered to be signifi cant, it follows from formulas (40) that heat fluxes qwi on the walls acquire the following form [12]: q w1 = −λ 01 ∂ T ∂r
, r = R1
q w2 = λ 02 ∂ T ∂r
. r = R2
(47)
2 ( R2 − R1) qw1 , (Tw1 − T ) λ
Nu 2 =
2 ( R2 − R1) qw2 . (Tw2 − T ) λ
(48)
Here λ is the physical heat conduction coefficient of a fluid, Tw1 and Tw2 are the temperatures on the internal and external walls, and T is the bulk temper ature determined by the formula R2
2 T = Tu ( r ) rdr. wR22(1 − h 2)
∫
R1
In notations (42), formulas (48) are written as
Nu1 =
2λ 01(1 − h) ∂Θ , λΘ ∂ξ ξ=h
(49)
2λ (1 − h) ∂Θ Nu 2 = − 02 , λΘ ∂ξ ξ=1
where Θ = (T − Tw ) (T0 − Tw ) . In order to compare the results with the experi mental data, the calculations were carried out for the parameter values as follows: ρ = 1.205 kg/m3, λ = 2.57 × 10–2 W/(m K), сp = 1002 J/(kg K), Prandtl number Pr = 0.705, and internal diameter of the external pipe d2 = 200 mm, with the model parameters found using formulas (31) and (45). In solution (46), m = 25 and X = 100. The results of calculations of the Nusselt numbers for the entire pipe Nu = (Nu1 + Nu2) are given in Fig. 5 (points) as compared with the plots of empirical formulas of stabilized flows. For the total Nusselt number Nu, the formula follows from the experiments under the conditions of this problem [15]: Nu = ANu 0 .
(50)
Here, Nu0 is the Nusselt number for the flow under the same conditions in a circular pipe with diameter d0, being equal to the width of a gap in the circular pipe, d0 = d2 – d1; the coefficient A was determined by the formula
(
)
A = 1 + 0.86 h 0.84 − 0.14 h 0.6 (1 + h) −1.
(51)
The Nu values obtained by calculations using for mulas (46)–(49) were used to determine the corre sponding Nu0 in a circular pipe (dots), which are com pared with the values of empirical formulas for Nu0 in Fig. 5. Curve 1 [14] is the diagram for the formula
f RePr 8 , f 23 900 1+ + 12.7 Pr − 1 Re 8 curve 2 [16] for
Nu 0 =
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(
)
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(52)
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TURBULENT FLOW THROUGH A CIRCULAR PIPE
Nu 0 = 7.6 − 3.6 + 0.0096Re 0.87Pr 0.605, log Re and curve 3 [17] for
(53)
(54) Nu 0 = 0.0186Re 0.8 . Fairly good agreement between the calculated and experimental data can be clearly observed both for large (h = 0.2) and small (h = 0.8) gaps. CONCLUSIONS As can be seen from the solution, both hydrody namic analysis of the flow and analysis of the heat exchange in it can be appreciably adequately carried out within the framework of a unified turbulence model based on anisotropy of the nearwall vortex structure. The mathematical model consists of fluid mechanics equations together with the constitutive equations of turbulent medium. The viscosity and thermal conduc tivity are determined taking into account anisotropy, which emerges in the nearwall turbulent flow. The assigned problems were solved as if they were mathe matical problems: the solutions to equations satisfying to the boundary conditions were obtained. The experi mental data were used only for determining the medium parameters. For air vortices, the values of the defining parameters of the model appeared to depend on the ratio between the diameters of a circular pipe and the shearing stresses on the walls. The calculation results agree fairly well with the experimental data. REFERENCES 1. Michiyoshi, I. and Nakajima, T., J. Nucl. Sci. Technol., 1968, vol. 5, p. 354. 2. Gräber, H., Int. J. Heat Mass Transfer, 1970, vol. 13, p. 1645.
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3. Hanjalic, K. and Launder, B.E., J. Fluid Mech., 1972, vol. 50, p. 50. 4. Buleev, N.I., Teplofiz. Vys. Temp., 1978, vol. 16, no. 2, p. 339. 5. Brighton, J.A. and Jones, J.B., J. Basic Eng., 1964, vol. 86, p. 835. 6. Sparrow, E.M. and Lin, S.H., Int. J. Heat Mass Trans fer, 1963, vol. 6, p. 248. 7. Kays, W.M. and Leung, E.Y., Int. J. Heat Mass Trans fer, 1963, vol. 6, p. 537. 8. Eifler, W., Berechnung der turbulenten Geschwindigkeits verteilung und der Wandreibung in konzentrichen Ring spalten, Berlin: Wärme und Stoffübert, 1969, vol. 2, p. 36. 9. Babkin, V.A., Inzh.–Fiz. Zh., 2002, vol. 75, no. 5, p. 69. 10. Nikolaevskiy, V., Angular Momentum in Geophysical Turbulence: Continuum Spatial Averaging Method, Dor drecht: Kluwer Academic, 2003. 11. Babkin, V.A., Inzh.–Fiz. Zh., 2006, vol. 79, no. 1, p. 155. 12. Petukhov, B.S., Genin, L.G., Kovalev, S.A., and Solov’ev, S.L., Teploobmen v yadernykh energet icheskikh ustanovkakh (Heat Transfer in Nuclear Power Plants), Moscow: Moscow Power Engineering Institute (Technical University), 2003. 13. Filonenko, G.K., Teploenergetika, 1954, no. 4, p. 40. 14. Petukhov, B.S. and Polyakov, A.F., Teploobmen pri sme shannoi turbulentnoi konvektsii (Heat Transfer in Turbu lent Mixed Convection), Moscow: Nauka, 1986. 15. Heat Exchanger Design Handbook, New York: Hemi sphere, 1983, Vol. 1. 16. Ibragimov, M.Kh., Subbottin, V.N., Bobkov, V.P., Sabelev, G.N., and Taranov, G.S., Struktura turbulent nogo potoka i mekhanizm teploobmena v kanalakh (Structure of Turbulent Flow and Heat Transfer Mech anism in Channels), Moscow: Atomizdat, 1978, p. 296. 17. Petukhov, B.S. and Roizen, L.I., Teplofiz. Vys. Temp., 1964, vol. 2, no. 1, p. 78.
