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ADVANCES IN HEAT TRANSFER, VOLUME 21

Heat Transfer under Supercritical Pressures

A. F. POLYAKOV

Institute of High Temperatures U.S.S.R. Academy of Sciences

Moscow. U.S.S.R.

1. Introduction

The 1970 contribution of Petukhov [ 11 to Advances in Heat Transfer was devoted in part to heat transfer at supercritical pressures (SCP) of fluids. This problem is the most general and complicated case of heat transfer at varying physical properties of single-phase fluids. The term “under supercritical pressures” means the state of the medium with parameters near the thermo- dynamic critical point, but under pressures above the critical pressure, when the medium can be considered to be single phase. Since Petukhov’s paper [ 11 was published, investigations of heat transfer have advanced profoundly and versatilely, resulting in new data and generalizations of both theoretical and practical interest.

The results obtained before 1978 are reflected in the review paper of Hall and Jackson [2], showing an important role of buoyancy forces for development of heat transfer peculiarities in forced turbulent pipe flow under SCP and giving empirical estimations for boundaries of the beginning of the buoyancy effect. As is noted by those authors, reliable data on hydraulic resistance, velocity, and temperature fields for SCP fluid were lacking, physical models were not especially elaborated, and the scope of numerical modeling results was insufficient at that time.

This article presents an analysis both of the problems with the data in the mentioned publications and of the results of investigations mainly conducted during the past 10-12 years.

1 English translation copyright 01991 by Academic Press, Inc.

All rights of reproduction in any form reserved.

2 A. F. POLYAKOV

11. General Description of the Problem

A. THERMOPHYSICAL PROPERTIES OF FLUID

Specific features and peculiarities of heat transfer under SCP are determined by the character of the changes in the single-phase medium’s physical properties at subcritical state parameters. At present, many data on heat transfer to water, carbon dioxide, and helium are obtained; in addition, there are some results for nitrogen, hydrogen, freons, and hydrocarbonaceous combinations. Reliable data on thermophysical properties of fluid are needed to make analyses of and generalizations about experimental data, as well as to predict numerical solutions. However, such data are incomplete for a number of specific fluids, and work to verify data on thermophysical properties near the critical point continues for water, carbon dioxide, and helium. Naturally, the latter results will require further amendments of generalizations on heat transfer.

By way of example, Fig. 1 presents some data on the change of physical properties versus temperature for water (P, = 22.12 MPa, T, = 647.3 K) and for carbon dioxide (P, = 7.38 MPa, T, = 304.2 K) near the pseudocritical temperature T,,, i.e., the temperature corresponding to the maximum value C, at the given pressure. Figure 1 (from Ref. [3], with some modifications) illustrates the complicated character of a medium’s physical property variations in the region under consideration, as well as their amendment. In com-

T ( K ) T (K)

FIG. 1. Physical properties of (a) water (-, [4,8]; ---, [S]) and (b) carbon dioxide (-, c91; ---, csi).

HEAT TRANSFER UNDER SUPERCRITICAL PRESSURES 3

paring later data on viscosity [4] with previous data [ S ] , it can be seen that the difference can reach 20% while the character of the variation (T) remains the same. As is indicated by Altunin [6] and Neumann and Hahne [7], the presence of a local maximum in the distribution k(T) for a single-phase region, near the critical point, is well established. Previously, such particularity of thermal conductivity dependence on temperature has not been indicated, as is clear from comparison of water data presented in Fig. l a [5,8]. Similar data are shown in Fig. l b for carbon dioxide. The distribution k ( T ) with local maxima have been calculated by Popov and Yankov [3] using curves k ( p ) at constant temperatures near the critical point [9].

The above-mentioned amendments of thermophysical properties refer to a narrow range of parameters near the critical point. They are not presented in all reference books on thermophysical properties used in analytical works or generalizations of data on heat exchange. The presentation of heat transfer data is not significantly changed by such amendments in the case of turbulent flow, whereas in the case of laminar flow analytical data on heat transfer, taking into account the different character of the distribution k(T), can vary substantially. This will be illustrated by examples in Sections III,E, IV,A, and IV,B.

B. GENERAL PREMISES AND APPROACHES TO PROBLEM SOLUTION

Heat transfer in a single-phase flow near a wall, containing the very large changes in physical fluid properties versus temperature, is the subject of the following discussion. The principal focus is on medium motions that are stationary in the mean, i.e., quasistationary in turbulent fluctuation scales. For individual cases, experimental data on nonstationary and three-dimensional flows will be presented, whereas the basic material relates to momentum and heat transport, which is two-dimensional in the mean. The latter we shall interpret mathematically by means of the following system of equations written in the cylindrical coordinates as a boundary-layer approximation:

Equations (1)-(3) are written without taking into account physical property fluctuations, i.e., their variations are supposed to be in compliance with

4 A. F. POLYAKOV

changes of mean temperature (enthalpy), and their instantaneous variations caused by the fluctuating temperature are neglected.

The principal difficulty in solving Eqs. (1) and (2) consists in a search for the most reliable approximations for correlations characterizing turbulent heat (n) and momentum (m) transport.

In the study of heat transfer under SCP, the role of empirical data is decisive, and use of simpler and approved models as compared with developed models of high order for constant property fluid is justified in obtaining a numerical solution to this problem. The particular relations proposed for calculations in some cases of turbulent heat transfer under SCP, along with mathematical boundary conditions will be discussed later.

One of the most important factors affecting the development of heat transfer particularities concerns the very large changes in the density (illustrated in Fig. lb). In the first place, an occurrence of regimes with a sharp local wall temperature maximum (“peak”) may be considered as such a particularity; these regimes were conventionally referred to by Petukhov [ 11 as “degraded heat transfer regimes,” contrary to “normal regimes” without the “peak” in the wall temperature distribution. Others [2,10] relate the local deterioration of turbulent heat transfer to a free convection effect, when wall temperature peaks are experimentally obtained in vertical, heated pipe upflows, but they are absent in downflows under the same conditions, as well as to thermally induced acceleration, which arises from the sharp decrease in fluid density along the pipe, when wall temperature peaks are discovered in experiments independently of the flow direction. However, the mechanism of buoyancy and acceleration effects, as well as quantitative correlations between the development of these effects, and heat transfer changes were not explained. Polyakov [l 13 proposed, apparently for the first time, to take into account the influence of buoyancy and acceleration effects for the analysis of heat transfer particularities under SCP, connecting them with density fluctuations by way of a turbulent energy balance equation in the following form:

pu’d- + p’u’ + g + u- + E = 0 -au aY -( 3 (4)

I I1

Two terms of Eq. (4) take into account the density fluctuations. Term I is connected with the acceleration due to gravity (the positive sign refers to the upward flow in heated pipes, the negative sign refers to the downward flow); I1 is connected with the individual particle acceleration in averaged motion and is written in supposition of the presence of mean fluctuating mass flux only along the pipe. The first and the last terms of Eq. (4) are turbulence production due to mean velocity gradients and to the dissipation of turbulence, respectively.


Such an attempt provided a clear and plausible explanation of the heat transfer peculiarities mentioned above and it was used as a basis for the further development of analyses, generalizations, and numerical modeling.

Some more recent approaches have been made to use the “ K - E model” (the equations of balance for turbulence energy and for turbulence dissipation) for calculations of turbulent flow and heat transfer to SCP fluid. In particular, this model is used by Renz and Bellinghausen [ 121.

111. Heat Transfer at Forced Flow in Round Pipes

A. LAMINAR FLOW

A great importance is given to the study of heat transfer in turbulent flow, as it is necessary for practical purposes. However, for general analysis, it is interesting to consider the influence of variable physical properties, when only molecular momentum and heat transport determine heat transfer from the wall.

In Shenoy et al. [ 131, the numerical solution of Eqs. (1)-(3) for laminar flow (&’ = tlV) = 0) was obtained without taking into account buoyancy forces with the following boundary conditions on velocity and temperature:

u = v = O , T = T w f o r r = r ,

u = const., u = 0, T = T,, = const. ( 5 ) for x = 0, 0 I r l ro

(auldr) = 0, u = 0, (aT/dr) = 0 for x > 0, I = 0 The results of calculations for the hydrodynamic entry region, i.e., with-

out a preliminary developed velocity profile, on water heating conditions (Tw 2 Tpc > T,,) demonstrate a great increase in the heat transfer coefficient (Fig. 2a). Figure 2b shows dimensionless results of the prediction for carbon dioxide under other conditions, viz. for a thermal entry region (the parabolic velocity profile is set up at the start of heating) under boundary conditions of a second-order type (qw = const.). The other boundary conditions are in ac- cord with those listed in Eq. (5). Data presented in Fig. 2b were obtained at the Institute for High Temperatures of the Academy of Sciences in Moscow by Medvetskaya and were gently proposed by her for this review paper. This numerical solution was carried out for upward flow in vertical, heated pipes with taking into account buoyancy forces, i.e., the term pg in Eq. (2) presented in the dimensionless form as (Ga/Re2), (p/pin) = gdp;Jm2. By neglecting buoyancy effects (Ga = 0), it was found that the heat transfer rate decreases with the growth of values qw and under ti, < tpc. The comparison of this result with data shown in Fig. 2a denotes a different character of SCP heat transfer change for different hyrodynamic and heat boundary conditions

6 A. F. POLYAKOV

FIG. 2. Distribution of heat transfer along the pipe at Re,, = lo3; (l), constant physical properties.

and different temperature ranges, even in the simplest case of viscous flow. The case of mixed laminar convection is more complicate, with Archimedes force actions being interconnected with varying physical property actions. At low heat flux and under ti, .c tpc, these effects lead to the increase of heat transfer by 30-40% as compared with the case of constant physical properties. Whereas heat flux increases, buoyancy leads to the growth of heat transfer at constant properties (see, e.g., [14]); however, the effect of a significant decrease of fluid density and thermal conductivity in the wall region as compared with the flow core in this case dominates the first, manifesting itself in a reduction of the Nu number. As fluid heats at t , 2 tpc, medium physical properties vary with temperature similarly to gas properties. In this case the growth of thermal conductivity near the wall and the effect of Archimedes forces lead to increasing heat transport from the wall, the regions beyond the minimum value of Nu number on the curves Nu(X) corresponding to these phenomena.

B. TURBULENT FLOW WITHOUT SUBSTANTIAL INFLUENCE

OF GRAVITY FIELD

As is clear from the previous discussion, even in laminar flow a great change of all physical properties at subcritical fluid parameters results in a specific character of heat transfer and in a realization of a large variety of cases as compared with heat transfer for constant property fluids. However, to solve this problem only mathematical obstacles are to be overcome, viz. it is necessary to find a solution of three-dimensional nonlinear equations with sharply varying coefficients in the complete formulation.


In the case of turbulent flow, major difficulties are connected with the determination of objective physical regularities of turbulent momentum and heat transport.

The regimes with degraded local heat transfer cause a great anxiety in practice. As is indicated in Section I,B, at present it is known that in addition to physical property variability causing heat transfer decrease in some cases, buoyancy and thermal acceleration determine a high deterioration of heat transfer. All three effects are to be taken into account not only in Eqs. (1)-(3) for mean values, but also in the mathematical description of turbulent momentum and heat transport.

The manifestation of Archimedes forces and thermal acceleration is coupled with a density change that intensifies more and more with the growth of the thermal load, and it is naturally accompanied by the increase of other physical property variabilities. While presenting the data, however, we shall try to separate some conditions that allow the consecutive consideration of all three cases noted previously. Let begin the analysis of theoretical and experimental data by means of presentation of results for a rather low heat load corresponding to the small difference of temperatures (t, - tb), when the effect of thermal acceleration and buoyancy forces can be neglected. Analysis of the development of the substantial influence of the gravity field is presented later in a separate section.

The investigation of heat transfer in turbulent flow based on the system of Eqs. (1)-(3) demands the application of some dependences for correlations

and m. The traditional approximations by the Boussinesq relation

- au

aY / I u v = -vT-

and by a similar relation - ai

aY i’o’ = - vT Pr,-

are widely used for prediction of SCP heat transfer. Petukhov et al. [IS] demonstrated that the use of relations for turbulent

transport coefficients proposed for the forced flow with constant properties, without especially taking into account variable physical properties, Archi- medes forces, and thermal acceleration, does not allow for a correct description of the behavior of heat transfer to a single-phase fluid with parameters near the critical point.

Theoretical investigations are directed essentially to obtain some relations for turbulent transport coefficient fitted experimental data. The main volume of experimental material is formed from heat transfer data. Among the few experimental data on hydraulic resistance, velocity, and temperature fields, it is necessary to distinguish the complex results recently obtained for carbon

8 A. F. POLYAKOV

dioxide [16-181 as the most complete and reliable ones. Such experimental data provide many reasons to perfect the equivalent mathematical models.

A computational model accounting for all the indicated effects was suggested by Petukhov and Medvetskaya [19,20]. This model makes use of the simplified equation of turbulent kinetic energy balance similar to Eq. (4) to find a turbulent momentum transport coefficient; to obtain the coefficient Pr, v,, the simplified enthalpy balance equation is applied:

(7)

The second term in the second equation shows mean heat transport of fluctuating motion by means of convection in the mean flow.

By taking into account Eq. (6) and the following basic approximations borrowed from the general theory of turbulence (see, e.g., [Zl]),

the relations given by Eq. (7) were transformed in the expressions for coefficients of momentum and heat turbulent transport as follows:

(9) 5 V = {p); - (--) a,- C, p - Pr, 1 - ar ai[ +g + ( u- ;; + u- ;)]}1/2

c2 lZ

In Eqs. (9) and (10) the terms (vT/v)* and Pr,, are determined taking into account a variability of physical properties, but in the absence of density fluctuations ( p = 0) and therefore the influence of buoyancy forces and acceleration on turbulent transport. Here we shall not dwell in detail upon expressions for additional functions and constant values presented by Petukhov and Medvetskaya [20].

It is evident that the mathematical model under consideration possesses a lot of gross approximations. In particular, every approximation written in Eq. (8) may be criticized, and under the conditions of Archimedes force influence, the coefficient a, in the last approximation is dependent on the value


of the buoyancy parameter, as has been shown in experiments using air [ 141. Moreover, an objection of principle can be brought up relating to the possibility of using the gradient presentation of turbulent transport [Eq. (6)] under certain conditions; however, this concerns all the theoretical investigations on turbulent SCP heat transfer.

Together with the above-stated discussion, it is necessary to note that the described model has a series of specific and advantageous features as compared with the others: by introducing into the equation of turbulence kinetic energy balance the terms associated with gravity acceleration and with mean individual particle acceleration, the influence of density fluctuations on turbulent transfer is taken into account; by use of the equation for temperature fluctuation balance the influence of the same effects on turbulent heat transport (turbulent Prandtl number) is also taken into account. Adopted approximations and constant values were verified to be acceptable by correspondence of preliminary calculations of hydraulic resistance and local heat transfer to theexperimental data obtained for turbulent flow of water and air in pipes under significant influence of a gravity field.

Some results of calculations according to a described mathematical model [20] for the cases without buoyancy effects will be presented in the following discussion. For such classification it is necessary to know the limits of realization of the considered regimes.

Polyakov and Petukhov [11,14], on the basis of Eq. (4), obtained the following estimations of the boundaries, below which it is possible to neglect heat transfer change in vertical pipes by means of variations of turbulent momentum transport induced by effects of buoyancy and thermal acceleration:

IfGr, k J ( c 4 x 10-4Re2.8E = Bth (1 1)

where the acceleration parameter is

md Re = - fib

where 5 is evaluated at (Tw + &)/2.

10 A. F. POLYAKOV

The positive sign in front of the Grashof number Gr, in Eq. (1 1) is for upward (downward) flow in heated (cooled) vertical pipes, and the negative sign is for the opposite case; the positive sign in front of the parameter J is for the case of fluid heating (T, > Tb), and the negative sign is for the cooling

The analysis [22] of experimental data on heat transfer to water and carbon dioxide in a heated pipe upflow resulted in the following estimations of boundaries of the absence of Archimedes forces effects:

(Tw < Tb).

(GrJRe’) < 0.01 (14a)

(14b)

or

c1 - ( p ~ / p b ) l ( ~ ~ p / ~ ~ ’ ) < 0*01

where

Grp = g(pb - pw)d3/pbvt Transforming Eq. (1 1) to a form similar to Eq. (14) at J = 0, we can obtain

(Grp/Re2) < 4 x 10-4(Reo.*/Nu)(c/pb)h (15)

x 0.02(p”lpb) % 0.01 [1 + (pw/pb)] Thus Eq. (15), corresponding to Eq. (1 l), is similar to the Eq. (14) at different degrees of approximation. Protopopov [22] does not prefer Eqs. (14) over one another, and for his determination he uses both the data for vertical pipes and for horizontal ones.

Jackson and Hall [23], on the basis of a series of gross physical suppo- sitions, in particular, relationships between the thickness of the buoyancy, thermal, and hydrodynamic layers and the constancy and magnitude of the dimensionless thickness of specific hydrodynamic layers, suggested the following relation:

(GrJRe’.’) < (16) where

Grp = c1 - (p/pb)lgd3/vz

Thus, comparing Eqs. (1 l), (14), and (1 6), we shall use Eq. (1 1) for an esti- mation of the boundaries of the gravity field influence on heat transfer in vertical pipes. From Eq. (11) it is clear that the combined action of buoyancy and thermal acceleration at upward flow in vertical, heated pipes is the greatest (the signs in front of Gr, and J are the same) and leads to a decrease


of turbulent transport [see Eq. (9)], and thus to a decrease of heat transfer. Under high heat loads, when the temperature over the pipe arises considerably and ti, < t,, < tout, the influence of these effects can manifest itself at most in some logitudinal region of the pipe, resulting in local heat transfer deterioration. When the condition Gr, + J < &, is satisfied, the degraded local heat transfer is not observed, and such regimes are “normal regimes” according to conventional classification [ 11.

In Fig. 3 the results of calculations [20] are compared with experimental data [ 161 on the friction factor ( f ) and the complete hydraulic resistance factor (fd) as well as for distributions of wall temperature and bulk fluid temperature over the heated pipe (4, = const.) for upward carbon dioxide flow. In this regime t , < t,, < t,; however, the heads of temperature (to,, - ti,) and (t, - t b ) are relatively low, and despite the fact that (JIB,,) = 10-12 at (Grq/B,,) = 0.9-0.65, the local maximum t , is absent, and thus the local heat transfer decrease is also absent. The last is confirmed by the fact that in practice the value (J/&,)i, = 10 and (J/&)out = 12 is constant. Its magnitude re- veals a possible noticeable (and uniform over the pipe) decrease of turbulent transport, friction resistance, and heat transfer due to thermal acceleration, which is confirmed by the data on f/fo and the data on velocity distribution distortion shown in Fig. 3b. The absence of an Archimedes force effect [Grq/Blb) < 1) is confirmed by the comparison of experimental data on wall temperature values at upward and downward flows. According to classification [l], this regime is also referred to as a “normal regime” by the form of the t,(x) distribution. It is to be noted that in the considered case the Reynolds number changed over the range (3.4-5.6) x lo5.

With the increasing heat load, the character of heat transfer distribution changes over the pipe, as is illustrated in Fig. 3c. The relative parameter of thermal acceleration acquires the values (J/&)in = 280 and (J/Blh)oul = 78, which are considerably greater, changing largely along the pipe. The substantial decrease (JIB,,) toward the outlet leads to the growth of friction factor f, to the fall in t,, and thus to the increase of heat transfer. Thus the distribution t,(x) with a local maximum is formed. Such regime in Ref. [l] is classified as a “degraded heat transfer regime,” i.e., a regime with a local “peak” of wall temperature. Data on velocity profile change presented in Fig. 3d confirm an important role of thermal acceleration in that regime. In this experiment, Reynolds numbers were in a range of (0.41-1.36) x lo6 and the relative parameter of buoyancy had the values (Grq/Bth)in = 2.1 and (Grq/Blh)oul = 0.3, thereby confirming the impossibility of appreciable influence of the gravity field on heat transfer.

Cornelissen and Hoogendoorn [25] present a specific development in numerical investigations of turbulent flow and heat transfer for SCP fluid. Their work carried out a comparative calculation of heat transfer to helium

12 A. F. POLYAKOV

u 0.5 1 0

V

p -7.7 M Pa d-0.8cm

tb 40 ____ t m I;-

tin = i9.yc I

0 50 100 I

I l a 1.4 -

I I

- 50 100 150

P = 27MPa d=O.Bcm 200

C t,n'190C \

I 0 50 I00 150

I.. I I I

2.0 m =4034ug/m2

0 05 i

Y

- 1.01 I I I I I

50 100 150 x/d

0

FIG. 3. Comparison of predicted results (lines) [20] with experimental data (0, upward flow; 0,

downward flow) [16]. (c) For experimental downward flow rn = 4100 kg/m2 sec, and qw = 1150 kW/m2. (After Ref. [24].)

according to three models of turbulence: the model of the mixing length ( I model), the model using the equation for turbulent energy (K model), and the K--E model, The K--E model was found to give the worst results. The results of calculations according to two other models fit experimental data only for low heat flux. Known empirical relations for constant physical proper-


ties with consideration of local values of physical properties are used in the I model. Approximations for eddy viscosity and turbulence dissipation similar to those written for Eq. (8) supplemented by functions of damping near the wall are used in the K model. Gravity field influence is not considered. The results of the work allow the following conclusions: (1) The simple models of turbulence of the I-type model satisfactorily describe heat transfer to different SCP fluids under low heat loads, as was demonstrated by Petukhov and Hall and Jackson [1,2]. (2) The use of complex models, such as the K--E model, needs a special perfection of the approximations and constants for a prediction of the flow of media with very large changes in physical properties.

Special attention is merited by the complex theoretical and numerical investigations being developed by V. N. Popov in terms of a hypothesis of mixing length dependence on ow, p, p, y, and 6 for variable properties, identical to constant properties, and of preservation of the form of this dependence:

(17) 1 3 = Fi(y+, Y) 1 = F(ow, P, ~9 Y, 61,

where y+ = ( c ~ , p ) ' / ~ y / p , Y = y/6, and physical properties correspond to local values. As result, from Eq. (6) and the Prandtl formula vT = l 2 au/dy, taking into account Eq. (17) in Ref. [26], the formula for eddy viscosity at variable properties was obtained:

where 1o/0,1~ and ( V ~ / V ) ~ are determined by the relations for constant properties, but using their local values. The term with (a/aw) is interpreted as taking into account flow acceleration.

Equation (18), in conjunction with Eqs. (1)-(3), has been widely used for analyses of heat transfer and hydraulic resistance over a wide range of defining parameters and boundary conditions at turbulent motion of uniform media, with a different dependence of thermophysical properties on temperature (liquid and gas at pressures below critical; water, carbon dioxide, helium, and nitrogen at supercritical state parameters). Satisfactory results have been obtained in many cases. A series of these results will be described in the following sections.

The calculations using Eqs. (1)-(3) make it possible to follow local heat transfer development immediately from the start of heating. At the same time, in practice, the empirical formulas for heat transfer in the pipes do not take into account the dependence on x. Also, two questions occur: (1) Under which conditions is such an approach acceptable? (2) What is the degree of the effect of the flow's history on the heat transfer development along the pipe?

In Popov et al. [27], the effect of the flow's history was investigated using Eqs. (1)-(3) and (18). The graph calculated for single-phase helium near its

14 A. F. POLYAKOV

'0" 5 0 , roo t5o v'd

FIG. 4. Influence of upstream flow history on the heat transfer of helium at P = 0.29 MPa, d = 0.109 cm, rn = 178 kg/m2 sec, and qw = 8.75 kW/m2.

critical point (P, = 2.3 MPa, T, = 5.2 K) in Fig. 4 [27] shows the maximum possible difference between heat transfer for different fluid enthalpies at the start of heating. As can be seen from the figure, the greatest difference between the heat transfer coefficients h at small x/d was 50%. When x / d > 50, the value of h varies by _+ IS%, this variation decreasing with increasing x/d. Thus the heat transfer in a turbulent flow can be considered fully developed for x / d > 50, i.e., it does not depend on the thermal conditions at the start of heating or on the history of the flow.

Together with the development of theoretical and numerical investigations, the improvement of empirical formulas on heat transfer continues. The 18 empirical formulas presented in Ref. [23] are variations of power functions

n (19) N~ ~ ~ 0 . 8 f 0.1 ~ ~ 0 . 6 f 0 . 2 ~

or, in alternate form, NU - Nuo&

where F, and F,,, are different combinations of simplexes of physical properties.

This work [28] suggests another approach to the construction of an interpolation formula directed to describing SCP heat transfer far from the start of heating (x/d > 50). This formula is based on the relation for constant physical properties [29] , which may be presented as follows:

St = ( f / 8 ) / [ k , + k2(f/8)'/2(PrZi3 - l)] (20)

I I1 111

The denominator of this expression, known as the Reynolds analogy factor, may be interpreted as a sum of thermal resistances of the near-wall layer (term


11) and turbulent core (terms I and 111). The influence of variable physical properties, as well as the buoyancy effect and the thermal acceleration effect, is taken into consideration by way of corrections to corresponding terms of the Reynolds analogy factor. In order to predict SCP heat transfer, the Stanton number is determined by a difference of enthalpies, i.e., St = q,/rn(i, - ib), and the Prandtl number by some averaged effective value. Equation (20) does not predict heat transfer in the entry region and may be used at ( x / d ) > 50.

Equation (20) has the friction factor, which is to be determined taking into account the variability of fluid physical properties. Experimental data [ 171 on the friction factor for carbon dioxide were obtained and analyzed together with different analytical dependences. This analysis has shown that in the region of states near the region of the pseudophase transition under the conditions of an absence of buoyancy effect, the experimental data were described in the best manner by the following formula:

(2 1 a) Equation (21a) was established [30] as a result of generalization of calculated data. It is evident that the variability of molecular viscosity, taken into account in a number of empirical relations for fluids, together with density variations, affects the friction factor. As a result of a generalization of the great volume of calculated data for different media in a wide range of defining parameters, the authors of Ref. [27] have recently obtained an interpolation equation,

f / f O = (PW/P~)’ ‘~

which is written here with their amiable consent. The analysis [17] of experimental data of a number of authors on heat

transfer to water, carbon dioxide, and helium for “normal regimes” resulted in the suggestion of a relationship similar to Eq. (20):

St = (f/8)/[1 + 900/Re + 12.7(f/8)’/2(E2/3 - l)] (22) where

iw - ib p b Pr=-- tw - tb kb

and f is calculated from Eq. (21). All indicated data, as well as the relations obtained on the basis of such data,

strictly speaking, are related to constant (or weakly varying) wall heat flux under the conditions of heat supply to fluid. Most of the experimental data concerns heat transfer to water and carbon dioxide. Recent interest in the improvement of new engineering techniques has encouraged acquisition and development of additional data on heat transfer to other fluids.

16 A. F. POLYAKOV

0 40 80 120 160 (80

y d FIG. 5. Relative change in the heat transfer coefficient along a pipe for helium: P = 0.25 MPa,

qn = 4.25 K, and G = 0.25 kg/sec.

In Bogachev et al. [31], special attention is given to the conditions of heat transfer growth during a turbulent motion of helium, when a free convection effect can be neglected. The experiment was made with a vertical pipe of diameter 0.18 cm at qw = const. Figure 5 illustrates typical curves of relative heat transfer variation over the pipe. With the initial growth of q,, the ratio Nu/Nu, increases over the entire length of the pipe. The number Nu, was calculated using Eq. (22) for constant physical properties, the values of Re and = Pr being evaluated at the bulk temperature of the fluid in a given cross-section of the pipe. Beginning from some heat load in the distribution Nu/Nu,, a maximum appears at the coordinate after the pipe section where T, = Tpc (coordinates corresponding to this equality are illustrated by arrows directed downward). With the increase of q,, the maximum is displaced upstream in the flow; however, its value remains practically constant. In a regime with the highest heat load, a minimum of Nu/Nuo is observed for the coordinate corresponding to the condition Tb x Tpc. The decrease of heat transfer, especially at the values Nu/Nuo -= 1, seems to be connected with the above-mentioned effects. In the region Nu/Nu, > 1, the relative heat transfer distribution approximately follows the course of Change c p / c p b [ c p = (i, - ib)/(Tw - Tb)]. Arrows upward directed indicate the coordinates of maximum values of the simplex c p / c p b for some regimes. The values Nu/Nu, > 1 are described with an accuracy of about &20% by following the interpolation equation:

Nu/Nu0 = ( c p / c p b ) 0 ' 3 5 (23)

Equation (23) is analogous to well-known power expressions for the description of heat transfer to water and carbon dioxide [23]. The character of the heat transfer change shown in Fig. 5 in terms of quality is similar to the behavior of heat transfer to water and carbon dioxide.


Additional problems of SCP heat transfer may occur during the use of hydrocarbonaceous fluids, their specific feature being very strong viscosity dependence on temperature, changing by two to three orders of magnitude over the temperature range of interest. However, such investigations are only at the initial stage of development [32].

C. TURBULENT MIXED CONVECTION

In this section we shall turn to a consideration of the features of heat transfer under the essential influence of a gravity field. Under intensive thermal conditions at nearcritical parameters of a single-phase medium, the buoyancy effect manifests itself in one way or another for the majority of parameter combinations typically realized in practice. Naturally, in this case the character of the flow and heat transfer will depend on the orientation of flow direction with respect to the direction of gravity acceleration. At first we shall analyze the results of recent investigations for vertical pipes, then for horizontal pipes. The second case is much more difficult for mathematical modeling and generalization and is much less studied. To some extent it may be connected also with the fact that constructions with vertical channels are used more often, generally. All presented data are related to the case of fluid heating at qw = const.

In accordance with the fact that buoyancy effects (Archimedes forces) occurring with the differences of densities in nonisothermal fluids are considered in this discussion, we shall occasionally use the expressions “thermogravitational forces” and “thermogravity.”

1. Vertical Pipes

Analysis of experimental data [33] showed that at

Gr,/Re2 > 0.6 (24)

heat transfer is higher than that at constant physical properties, or (Nu/Nuo) > 1, and with the growth of heat load, or, more exactly, the value of the parameter Gr,/Re2, heat transfer increases at x/d > 50.

As is illustrated in Fig. 6, generalized results [ 141 for conventional fluids (air, water) at low nonisothermality obtained on the basis of experimental and theoretical research show the pattern of buoyancy influence on heat transfer.

The condition given by Eq. (24) relates to rising branches of curves after the values St = St, in Fig. 6 for the case of heated fluid upflow. The Stanton number St, corresponds to the limit Gr + 0 (horizontal dash-dot lines). As is seen from Eqs. (11) , (14), and (15), the parameter Gr,/ReZ determines a boundary of the beginning of thermogravity influence on heat transfer in

18 A. F. POLYAKOV

2

1 o-2

6

St 4

2

t 0-3

6

(Grq/Pr Re4)"2

FIG. 6. Stanton number versus buoyancy parameter for upward (solid line) or downward (hatched line) flow in heated vertical pipes at x/d > 50.

forced turbulent flow. However, there is no reason to believe all processes of mixed forced and free convection are determined by the only combination of Gr, Re, and Pr. Namely, data in Fig. 6 allow to such judgement. Petukhov and Polyakov [ 141 propose interpolational equations describing heat transfer over every range from the beginning of buoyancy influences on forced turbulent flow to their dominant manifestation in the limiting case for both upward and downward fluid motion. Here we shall confine ourselves to writing a relation only for the case of upward flow for heating and qw = const.,

-- (f/8) - (1 + 0.83e2)(1 + 0.042e2[E'/410g(Re/8)]-'}-' St

+ 12.7(f/8)1/2{Pr2/3[1 + 0.72e3(1 + 0.28e'/2)/(1 + 0.43e4)]

- (1 + 0.58e2)/(1 +0.83e2)} (25) where

e = 103Gr,/Pr Re2.75

E = Gr,/Pr Re4

Gr, = gpqwd4/v2

f = ((1 + 0.83eZ)/[1.8210g(Re/8) + 0.076e2E'/4]}2

For the case of the absence of a buoyancy effect, i.e., at Gr + 0 (E = 0, e = 0), Eqs. (25) and (26) reduce to equations for forced flow at constant physical properties Sto, fo. At the initial stage of manifestation of a buoyancy effect,


the heat transfer equation may be presented in the following relative form (see Ref. [14]):

St/St, = Nu/Nu, = 1 T 112 Gr,/Re2." Pr1.lS (27)

where the upper sign belongs to upward flow and the lower one, to downward flow.

In the limiting case Gr + co, i.e., for a free convection regime, Eq. (25) takes the form:

Stfree = (Gr,/Pr Re4)'/4[1.33 log(Re/8) + 3.3(Pr2I3 - 0.7)]-' (28)

For downward flow and Gr + co, i.e., for a regime of thermal generation of turbulence [14], the following formula is suggested:

Stgen = (Gr,/Pr Re4)1'40.5[Pr''4(1 + PI-'/^)]-' (29)

As is seen from Eqs. (25)-(29), the thermogravity effect over the whole range from Gr + 0 to Gr + 00 cannot be taken into account by only one of the combinations, Gr/Re"Pr", with constant exponents n and m. It is more justifiable to present the appreciation of the considerable influence of both variable physical properties and buoyancy forces on heat transfer in the form of relations such as Eq. (25), rather than in the form of an amendment to the ratio of the dimensionless heat transfer coefficients Nu/Nu, (or St/St,).

As follows from Eqs. (28) and (29), in the limiting case of a dominant effect of gravity field for both upflow and downflow, the relationship between Nusselt number and the Grashof number corresponds to free turbulent convection Nu - Gr:I4; however, dependences on Pr are different. Also, in the case of upward flow in Eq. (28) for a free convection regime, a weak dependence on Re (geometrical size) remains, which is connected with free convection development in a constricted volume of pipe, where boundary layers developed over the perimeter interact in the central region of flow.

Equations (25)-(29) were obtained for weakly varying physical properties of moving fluids. Let us first consider interpolation correlations for a calculation of SCP heat transfer. Then we shall analyze additional data on the physics of the process-experimental results on velocity and temperature fields, mathematical models, and their efficiency.

As follows from Eq. (1 l), the thermogravitational effect is substantial at

I( k Gr, + J)/Re2.' fil > 4 x Gr, 2 J (30)

(304

or, without taking J into consideration,

(Gr,/Re2.' E) > 4 x lop4

and there are regimes with degraded heat transfer in heated upflow, whereas there are no such regimes in downflow.

20 A. F. POLYAKOV

In the experiments carried out using pipes of relatively large diameter [33,34], the peaks of wall temperature, conditioned by thermogravity at upward flow were found to be placed near the start of heating, i.e., at x/d -= 50, when the condition given by Eq. (24) is satisfied.

For local heat transfer calculations [14] taking into account thermogravity and thermal acceleration for the regimes (including regimes with degraded local heat transfer) and corresponding to the conditions given by Eq. (30) and

(Gr,/ReZ) < 0.6 (31)

at x/d > 50, the following formula is suggested:

St = q,/m(i, - ib)

=b(l 8 +E+ Re 12.7($)1’2{[Prf(1 +i)]”’-O.1E2 - l})-’ (32)

where f, = 0.184/Ret2, Re, = md/p,, Pr = 2 Pr, Pr,/(Pr, + Pr2), and Re = md/pb, p,, pi, 8, and Pr, are evaluated at the temperature T, (enthalpy i,), which can be considered as the first approximation to the wall temperature. The value i , is calculated from i , = ib + 43.5 qw/m; Pr, and pz are evaluated at the ttmperature T, = + (il - ib)&/kb.

The parameter k is - k = lo3(J2/Pr: Re4)’I3 + B(Gr,/Ei Re4)’I3

where

984 d4 Gr, = +

Pr, = ~ -

kbVb

i Z - ib p b

TZ - Tb k b

-

The coefficient B = 800 is for an upward flow in a heated pipe and B = - 100 is for a downward flow;

/4p2/pb at Re I 2.5 x lo5

As distinct from all other formulas for calculation of SCP heat transfer, Eq. (32) does not need to be solved iteratively, i.e., at predetermined q,, none of the parameters of Eq. (32) requires a knowledge of T,, thereby complicating its evaluation but significantly simplifying calculations.


i b (K V 9 )

FIG. 7. Comparison of wall temperature distributions experimental data (points) [35].

calculated using Eq. (32) (lines) with

Equation (32) is obtained on the basis of a generalization of experimental data on water and carbon dioxide. It has as the limit the relation for St,, whereas in the absence of a buoyancy effect (Gr = 0) but at variable physical properties, i.e., for “normal regimes,” calculations using Eq. (32) may bring great deviations of the value Nu as compared with experimental data, especially at low differences (T, - Tb), as was shown by Petukhov et al. [17]. For these cases the use of Eq. (22) is recommended. An impressive example of the use of Eq. (32) is shown in Fig. 7.

In order to generalize experimental data on heat transfer to helium upflow in a vertical, heated pipe with a diameter of 0.18 cm, the above-mentioned relations along with other relations for x/d > 30 and for the relatively small change in physical properties (0.7 I p,/pb < 1; 0.7 I Cp/Cpb I 1.8; 0.7 I k,/kb I 1.6; 0.9 I pw/pb s 1.2) are analyzed [36]. To give a representa- tion of the heat transfer coefficient in the relative form Nu/Nu,, Bogachev et al. [36] have selected the complex Gr,/Pr Re3 as a buoyancy parameter, this selection being considered as giving the most successful generalization of experimental data and as being close to the parameter well-founded physically and included in Eqs. (25)-(32). The same parameter is used in Ref. [37] for a generalization of experimental data for helium downflow obtained with the same setup. The character of heat transfer change established [36,37] both for upward and downward flows corresponds to the behavior of curves shown in Fig. 6. The expression Gr,/PrRe3 = 4 x corresponding in

22 A. F. POLYAKOV

practice to Eq. (30) for values of the Reynolds numbers Re = lo4 that are specific to the work of Bogachev et al. [36], is recommended as the boundary of the beginning of the buoyancy influence on heat transfer. As is indicated in Ref. [36], Eq. (32) can be used to describe heat transfer in mixed convection of SCP helium in the region (Gr,/Pr Re3) c 0.25, which exceeds even the limit given by the Eq. (31).

The data examined above are obtained for pipes of relatively small diameter. Naturally, when the pipe diameter increases, the free convection influence is augmented more and more. As was noted previously, in the work of Petukhov et al. and Watts and Chou [33,34] (carbon dioxide, d = 2.9 cm; water, d = 2.5 and 3.22 cm, respectively), the local heat transfer deterioration was found to be disposed near the start of the heating, whereas intensive heat transfer, with the value for upward flow being close to that for downward flow, i.e., corresponding to regimes of free convection and thermal turbulence under the condition given by Eq. (24), was observed far from the start of the heating at x/d > 50. Different empirical equations generalizing their results are suggested in both papers. Nevertheless, it is notable that for very large Gr, the dependence of Nu on Gr, being similar to that for turbulent free convection, is always observed. In particular, it is necessary to distinguish the results of generalization for the minimum Nusselt number (Numi,), i.e., for the value of the wall temperature peak as determined by Watts and Chou [34]. The shape of the curves of dependence on the buoyancy parameter for upflow is quite similar to that shown in Fig. 6. A generalization for the averaged heat transfer downstream beyond the peak T, does not reveal any appreciable “hole,” i.e., values Nu/Nu, < 1. It is evident that empirical relationships from Watts and Chou [34] require a special verification to extend them over other liquids and ranges of parameters. In particular, the relation for Numi, in the entry region of the pipe does not include the influence of x/d and upstream flow history.

In the numerical analysis of mixed convection development in pipes [ 12,20,38-401, the use of different mathematical models of turbulence allows us to find the correlation between heat transfer, resistance, velocity, and temperature fields; to try to construct the most general method of calculations; and to give a more profound analysis of physics of the process.

With the purpose of explaining the experimentally observed velocity profile deformation, which results in the M-shaped distribution at upward fluid motion, Hauptmann and Malhotra [38] used the K--E model for closing Eqs. (1)-(3). In the equations for turbulent characteristics, the buoyancy effect was not taken into consideration. As a result, M-shaped velocity profiles are obtained, but data on heat transfer are qualitatively contrary to experimental data, i.e., heat transfer was higher in upflow as compared with that in downflow. Such a pattern of velocity and heat transfer changes is typical for laminar mixed convection (see, e.g., Ref. [ 141).


In Renz and Bellinghausen [12], the model used previously [38] was supplemented by some terms defining the buoyancy effect in the equations for turbulent kinetic energy and dissipation. A result was obtained that is qualitatively correct: the distribution of wall temperature along the pipe in downflow is monotonic, and the values of temperature are lower as compared with upward fluid motion; in the second case, in the distribution T,, zones with a sharp increase of T, were obtained, these zones displacing toward the inlet of the pipe when the heat load grew. At the same time there was no correspondence between analytical and experimental data, especially in the region of local heat transfer deterioration.

The work of Bellmore and Reid [39] merits great attention, wherein, in order to solve the problem of SCP heat transfer, density fluctuations are taken into account in the mean energy, momentum, and continuity equations. Density fluctuations are supposed to be induced only by fluctuations of enthalpy, whereas enthalpy and velocity component fluctuations are also supposed to be determined by gradients of mean values and corresponding scales, viz.:

p' x (ap/di) i ' = -p/?* i '

i' = -li(ai/dy)

u' = l,(du/dy) (33)

u' = -z,(du/ay)

The expressions for turbulent shear stress and for turbulent heat transport, accounting for density fluctuations, have been written as

Using Eqs. (33) and (34) and the approximation of Boussinesq [Eq. (6)], a relation for the coefficient of turbulent viscosity, allowing for density fluctuations, has been obtained:

By means of Eq. (35), Bellmore and Reid [39] carried out calculations of heat transfer at an upward flow of hydrogen (P, = 1.29 MPa, T, = 33.0 K ) in pipes with diameters from 0.56 to 1.29 cm. Heat transfer deterioration is connected with the generation of M-shaped profiles of velocity. Calculations presented for four regimes satisfactorily describe the experimentally obtained distribution of local heat transfer along the pipe.

24 A. F. POLYAKOV

The suggested method is not universal, evidently, and its improvement requires a wider verification for different conditions; however, the idea to present momentum and heat transport in the form of Eq. (34) and to allow for density fluctuations in the mean via Eqs. (1)-(3) appears fruitful and merits further development.

The most complete numerical modeling of flow and heat exchange in mixed convection in vertical pipes has been made by Popov and Valuyeva [40-431. Methods of prediction and results of calculations for water [40] and carbon dioxide [41] have been presented. Equations (1)-(3) are solved under boundary conditions [Eq. (5)] with two differences: x = 0 and uin = uo(r,Re) for a developed turbulent velocity profile; r = ro and (k/c,)(ai/i%) = qw for the boundary condition of a second-order type.

On the basis of a simplified balance equation of turbulent stresses [taking into account the buoyancy effect, a number of approximations, and sup- positions similar to those used to write Eqs. (17) and (18)], there have been found some relations for calculations of turbulent stress and turbulent viscosity:

vT., vT, = [2.24(/?gl4/C, ~r)lai/ar(] 1/2

is the turbulent viscosity in the absence of and in the dominant effect of the gravity field.

When the thermogravity effect is so low as to be neglected, Eq. (37) can be reduced to Eq. (18) written for vT = vT*. Equation (6) is used for turbulent heat transport, and the value of the turbulent Prandtl number is assumed to be one.

Before considering the results of mixed turbulent convection numerical modeling [40-431, for a more complete analysis of the heat transfer character in vertical pipes it seems justified to present Fig. 8 [14] and experimental data on the behavior of local heat transfer in air flowing upward through a heated pipe at practically constant physical properties. As can be seen from Fig. 8, the thermogravity effect in the thermal entry region (x /d < 50) is weaker than it is far from the start of heating. Data from Fig. 8 for x/d > 50 have been used to plot generalized graphs in Fig. 6. At the initial stage of the buoyancy effect, a sharp decrease in heat transfer takes place essentially over the whole length of the pipe, and is caused by a partial relaminarization of the entire flow. Some minimum level of heat transfer for the given values of Re and Pr is established over the whole length. When the heat load continues to augment, heat transfer


20

Nu

to

c4 P r = 0 7 R e = 5100 , . I -\\

FIG. 8. Experimental data for air at various values of buoyancy parameters Gr,/B,, = a and (Gr,/Pr Re4)’” x lo4 = b.

far from the start of heating begins to increase intensively, due to the beginning of the development of free turbulent convection (in Fig. 6 , the branches of the curves going upward beyond the minimum values). Near the start of heating (x /d < 30), heat transfer varies weakly even in the case when at x /d > 50 the values of Nu exceed Nu,.

In Fig. 9 results of calculations [40] are presented in comparison with experimental data on the distribution t , for water flow in pipes of relatively large diameters. A remarkable feature of Fig. 9a and b is the fact that wall temperature variations for single-phase flow have the same character at

50 0 50 0 30 60 30 60

350

300

2 50 - 2 0 0 . z 1.8 c.’

0 3

Z

2 1.0 L 0.6

x/ d x/ d x l d X / d

FIG. 9. Comparison of calculated results (solid lines) [40] with experimental data for upward water flow (points, dashed lines). Data for a and b from Ref. [44], d = 2 cm: (a) Re =

(1.2-4.7) x lo4, Gr,/Re2 = 0.10-0.25; (b) Re = (3.5-6.5) x lo4, Gr,/Re2 = 0.13-0.30. Data for c-e from Ref. [34], d = 3.22 cm, P = 25 MPa: (c) Re = (0.84-1.07) x lo5, Gr,/Re2 = 0.05 + 0.07; (d) Re = (5.2-7) x lo4, Gr,/ReZ = 0.20-0.38; (e) Re = (2.2-3.9) x lo4, Gr,/Re2 = 1.5-2.1.

26 A. F. POLYAKOV

subcritical and supercritical pressures. The equality of values of approximate buoyancy parameters Gr,/Re2 for given regimes can be used as a confirmation of the preceding statement. It follows from Eqs. (14) and (15) that the value Gr,/Re2 for these regimes is 5-10 times larger than the threshold value, i.e., the boundary of the beginning of the buoyancy influence on heat transfer. Data shown in Fig. 8 reveal the fact, that in this case, heat transfer in the entry thermal region is substantially lower than without the influence of body forces, and at x/d > 50 the values Nu failed to achieve the value Nu,. Such qualitative analysis is confirmed by distributions Nu/Nu, calculated by Popor and Valuyeva [40] shown in Fig. 9a and b.

Data in Fig. 9c-e illustrate heat transfer distributions along the pipe under different degrees of buoyancy force influences. In Fig. 9, the first case (c) is related to the initial stage of thermogravity (see the line for which a = 1 in Fig. 8). The increase of the buoyancy parameter by four to five times leads to a result that has been already shown in Fig. 7. Here it is necessary to em- phasize once more that the process at the initial stage is developing with an extremely large intensity-with the buoyancy parameter changing only by two-three times, heat transfer assumes a minimum value along the whole length of the pipe (see the line for which a = 2.4 in Fig. 8). In this connection there is evidence that it is unjustified to consider that heat transfer deterioration due to an Archimedes forces effect is localized only in the thermal entry region. Such conclusions are based on data obtained for the flow of SCP fluid in pipes of relatively large diameters, when, in practice, this narrow region of parameters is missed. The results shown in Fig. 7 are in favor of this opinion. With a further decrease of flow rate (in experiments by Watts and Chou [34], the change of degree of thermogravity influence is realized by the change of in under q , = const.), a substantial increase of buoyancy parameter results in heat transfer along the whole length of the pipe and surpasses the initial level, i.e., Nu/Nu, > 1. The last depends on the development of intensive free turbulent convection over the entire surface of heat transfer.

The results of calculations for helium [43] are compared in Fig. 10 with experimental data [45] obtained for a pipe having a 1.8-cm diameter. The character of the distribution, T, and Nu, is similar to that shown in Fig. 9b and d for water.

Kurganov et al. [18] obtained reliable experimental data on velocity and temperature (enthalpy) fields, on the local hydraulic resistance of the motion of carbon dioxide in a vertical pipe with a 2.27-cm diameter. Parts of these data shown in Fig. 11 are compared with the results of calculations [41,42]. The results presented in Fig. 11 have been obtained for very high values of Re w lo6. However, an appreciable influence of thermogravity is observed even for such great values of Re, its contribution being evaluated by comparison of t, calculations for both upward and downward flows. A local


0 20 4 0 0 20 40 0 X/d X j d

FIG. 10. Comparison of calculated results (lines) [43] with experimental data (points) [45]. (a) Re = (8.7-9.1) x lo4, Gr,/Re2 = 0.27-0.64; (b) Re = (9.1-9.7) x lo4, Gr,/Re2 = 0.30-0.95.

300

200 - 100 0 W

0 A y" 400 A x 300

.c_ 200

100

0

1.0

> 0.6

0.2

W n

.- I

<

co, 0 0 4

6.0

5.0

* * * * 4.0

3.0

2.0

/-. v 0,

c 3

FIG. 11. Comparison of predicted results (lines) [41,42] with experimental data for upward flow(points)[18]. Re = (8.7-18) x 105,Gr,/Re2 = 0.1-0.2(solidlines,upwardflow;dashedline, downward flow; hatched line, one-dimensional approach).

28 A. F. POLYAKOV

deterioration of heat transfer with wall temperature “peak” is realized near the region with t b x t,, in upflow, but is lacking in downflow. As can be seen, in this case the buoyancy effect is manifested in a local manner at tb x tpc, and, moreover, far from the start of heating. If we reanalyze Fig. 6, we shall state that the conditions of the downward branch of the beginning of heat transfer deterioration correspond to this regime, and after the region t b x t,, has been passed, we shall return to conditions without buoyancy influence. This case is close to results shown in Fig. 7 and differs from data presented in Figs. 9 and 10, illustrating the increasing development of the thermal convection effect.

The development of velocity profiles along the pipe takes place in such a manner that the profiles with the most strongly manifested M-shaped form are related to cross-sections near the highest buoyancy effect, degraded local heat transfer. Such a deformation of velocity profile is typical for mixed convection in upward flows of heated fluids in vertical pipes and under weakly varying physical properties (see, e.g., Ref. [14]). The M-shaped character of the velocity distribution in the pipe cross-section becomes more distinct under a strong influence of free convection. Distributions of enthalpy along the pipe for different radius coordinates are of special interest, demonstrating their difference from the distribution ib(X).

The regime presented in Fig. 1 1 is characterized by the great acceleration of flow, determined by large values of a local inertia factor:

f, = 4r0- Pb -[ 2j01 puz(r/ro)d(r/ro)] m dx

In addition, in this regime there are low values of the friction factor

f = - 8awpb/m2

and low values of the heat transfer coefficient along the whole pipe length. The hydraulic drag is equal to the sum of the inertia factor and the fric-

tion factor

For the regime under consideration

fd 2 fo x 0.316/Re1l4 x 0.184/Re0.’ (40)

In a one-dimensional approximation, the inertia factor can be presented as

The last equality has been obtained.for the condition qw = const.


The results of calculations of the inertia factor according to the one- dimensional model presented in Fig. 1 1 show that for the regimes with degraded heat transfer, one-dimensional approximations can lead to essential errors in calculations of hydraulic resistance. Such conclusions have been made as a result of both experimental investigations [17,18] and numerical predictions [20,27]. For regimes for which the influence of thermogravity and thermal acceleration is not important, the one-dimensional approximation is correct.

In fact, Eq. ( 1 1 ) finds its confirmation in the estimations of boundaries of the buoyancy effect and boundaries of local heat transfer deterioration due to thermal acceleration, as suggested by Petukhov et al. [17] on the basis of analysis of experimental data on components of hydraulic resistance. In particular, it is indicated that for values of mass velocities of carbon dioxide (m 2 1000 kg/m2), the regimes of “normal heat transfer” satisfy the condition of weak influence of buoyancy forces Gr,/B,, < 1. On a basis of analysis of experimental data [17,47] and data on heat transfer to gases, Petukhov et al. [ 171 have suggested the following condition of the existence of “normal heat transfer” under heating of SCP fluids and gases:

( f u / f d ) 5 - 1.3 (42)

Using Eq. (21) as recommended [17] for the value fd, Eq. (41) for f, = fu l ,

and Eq. (40) for fo, we can write Eq. (42) in the following form:

(f;/fd) = (44pqwd/Cpp Re0’8)(Pb/Pw) - 1.3 (43)

On the other hand, if the threshold acceleration parameter is written according to Eqs. ( 1 1)-( 13), we obtain

J/Bth = ( l O O p q w d / q p b Reo’g)(Pb/p) < (44)

Comparison of Eqs. (43) and (44) shows not only their qualitative correspondence, but also their quantitative proximity at the values of Re z lo5, typical for experimental data.

After the generalization of the considered data has been made, we can make the following presentation of the development of mixed turbulent convection. In the initial stage of gravity field influence on upward turbulent fluid motion in vertical, heated pipes, an intensive decrease of heat transfer takes place due to the suppression of turbulence. Such a heat transfer deterioration under SCP can have greater local manifestations connected with the very large change in the density near the pseudocritical temperature. As the thermogravity effect increases far from the start of heating, turbulent free convection begins to develop, thereby leading to the increase of heat transfer. Incidentally degraded heat transfer envelops some area of the thermal entry region, the

30 A. F. POLYAKOV

dimension of which becomes more and more shortened with the intensification of the buoyancy effect, approaching the start of heating. At very strong influences of free convection, “the entrance wall temperature peak” actually disappears, and in these regimes Nu/Nu, 2 1.

In addition to the considered picture of development of mixed turbulent convection, thermal acceleration (e.g., the parameter J) leading to the suppression of turbulence under heating, and hence to a decrease of heat transfer (see Fig. 3), as well as to a large change in density, heat capacity, viscosity, and other physical properties, affects the character of heat transfer in media under SCP. Various combinations of the indicated factors depending on the parameters of the regime can lead to the specific alternation of zones with “deteriorated” and “improved” heat transfer along the pipe. In particular, a realization of such a distribution of t , with two local maxima seems to be real in a long pipe: in the thermal entry region, at tb c t,, and t , I t,,, the distribution similar to that shown in Fig. 9b and d is set; at further advancement along the pipe, when tb x t,, is attained, a development of a second local deterioration of heat transfer is possible, and, consequently, a second wall temperature peak can appear. There are no peaks caused by a buoyancy effect in downflow.

It is necessary to note that the considered cases of relaminarization caused by suppression of turbulence by thermogravity and thermal acceleration refer to the conditions of nonperturbed fluid motion, i.e., experimental data have been obtained in the presence of hydrodynamic entry isothermal regions before the start of heating, and theoretical results have been obtained under the boundary condition for x = 0, given as the fully developed isothermal turbulent velocity profile. The distribution of local heat transfer appears to have qualitatively distinguishing features without the hydrodynamic entry isothermal region, viz. a local deterioration of heat transfer in the entry region at mixed convection is not realized.

2. Horizontal Pipes

The development of turbulent mixed convection in horizontal pipes has a three-dimensional pattern, thereby complicating theoretical investigations as well as generalizations of experimental data. Local heat transfer changes along the length, as well as around the perimeter, of the pipe. Hall and Jackson [2] presented experimental data showing the possibility for the wall temperature maximum to arise at coordinate x, when tb !z t,,, near the upper generatrix. In addition to that, the difference between the wall temperatures at the top and bottom generatrices can reach 200-250°C when water flows in pipes with diameters of 3.0 cm [46]. Even in pipes of small diameters the nonuniformity of temperature around the perimeter can have a substantial value, and has


attained 80°C in experiments [47] with water in a steel pipe with a diameter of 0.444 cm.

Petukhov and Polyakov [14] give a relation describing the boundaries of the beginning of the buoyancy effect on local heat transfer near the top (bottom) generatrix, far from the start of heating ( x / d > 50),

- Pr2I3 - 1

Gr, = Grth = 3 x Re2.” 1 + 2.4 ( Re’/8

where Gr,, E, and Re are defined by Eq. (13), in which i, and t , can be calculated from Eq. (22) without taking into account the influence of free convection.

When there is the condition J 5 Gr,, the experimental data on the difference between the temperatures at the top and bottom generatrices under 4, = const. and x /d > 50 can be generalized in the coordinates

where

%

pr2/3 - 1 cth = 3 x Re2.” 1 + 2.4 ( Re’’’ and

B = 281B2/(Pl + P 2 ) rv Pr = 2Pr1 Pr2/(Prl + Pr,)

FIG. 12. Generalization of experimental data on temperature differences between top and bottom generatrices of horizontal steel pipes at qu = const.

32 A. F. POLYAKOV

F ' ' 7 I 2.6 0 O n 0 2.2

FIG. 13. Comparison of calculated results using Eq. (32) at Gr -t 0 with experimental data on averages around the perimeter heat transfer coefficient for water flow [46].

Here 8, Pr,, b2, and Pr, are evaluated at the temperatures T, and T, defined in Eq. (32).

In Fig. 12 the experimental data [46,48,49] are plotted using the coordinates given by Eq. (46). There is a good deal of scatter. However, they suggest that there is a distinct relationship between the maximum wall temperature difference around the pipe circumference and the magnitude of the body forces. This relation could be used to obtain, without iteration, the estimations of this difference and the limits at which gravity begins to have an effect on local heat transfer.

Even though the difference between the wall temperatures at the top and bottom generatrices may become quite large, the average heat transfer around the perimeter remains practically unchanged. Calculations of St, using Eq. (32), neglecting body-force effects (Gr = 0), are compared in Fig. 13 with experimental values of 5 [48]. But the ratio of % to the values of the Stanton number 5, at constant physical properties gives deviations from experimental data of substantially greater values.

D. TURBULENT HEAT TRANSFER AT HEAT FLUX VARYING LENGTHWISE

Local heat transfer at upward flow of carbon dioxide (PIP, z 1.3) has been investigated [SO] experimentally under the conditions of linear growth or decrease of wall heat flux qw along a pipe with a diameter of 0.546 cm and with length 37d. Heat flux change over the length qvX/q:'" was from 6.6 to 9.4 at the values qy over the range 0.25-1.75 MW/m2. The Reynolds number and the temperature have been changed in experiments over the following ranges: Re = (0.57-3.7) x lo5, TWIT,,, = 0.94-2.7, and T,/T,, = 0.92-1.03.


It has been shown [ S O ] to be possible to use the Duhamel integral (superposition principle) for calculations of heat transfer in single-phase subcritical regions in the form

where Nu,, is the relation for heat transfer in SCP fluid at qw = const., qw(0) is heat flux at the start of heating, and el is a correction for heat transfer increase in the thermal entry region for qw = const. and constant physical properties.

(48)

Thus the superposition principle can be used as an empirical procedure to allow for the variability of qw, its influence on heat exchange being assumed to be the same at constant physical properties of the fluid. The influence of variability of physical properties and body and inertia forces is allowed for by the scale number Nu,, .

In Fig. 14 the results of calculations of wall temperature using Eqs. (47), (48), and (32) for Nu,, are presented. The combinations of experimental parameters satisfy the boundaries of validity of Eq. (32). The presented data

The following equation for E' has been suggested [SO]:

= 1 + 2.35 Pr- 0 . 4 ( ~ / d ) - 0 . 6 exp( - 0.39 Re-'.' x / d )

X / d x / d

FIG. 14. Wall temperature distribution and values of normalized Nusselt number at linear (a) growth of qw or (b) decrease of qw along the pipe on the basis of experimental data [SO].

34 A. F. POLYAKOV

correspond to the regimes with the greatest manifestation both of thermal acceleration effect and of thermogravity effect. Values of relative parameters JIB,,,, Grq/Bth are indicated for the coordinate x/d = 32 near experimental points for the Nusselt number. The curves [(Nu/Nu,), (xld)] for the growth and decrease of qw are calculated from Eq. (47) using Eq. (32) for Nu,, in which the parameters J and Gr are ignored. The experimental values Nu/Nu, for the conditions satisfying Eq. (1 1) are disposed near these predicted curves. Such points are not shown in Fig. 14. Heat flux variations over the length realized in experiments lead to some increase of heat transfer with the growth of 4,; on the contrary, with the fall of qw it has a more appreciable manifestation by diminishing heat transfer significantly. In the both cases the influence of acceleration and buoyancy substantially decreases the values of Nu/Nu,, as well as under boundary conditions qw = const. considered above.

As is seen from Fig. 14, the suggested approximative method correctly describes the distribution t , and consequently the local heat transfer at q,, which changes along the length of the pipe.

E. TURBULENT HEAT TRANSFER UNDER COOLING

It is much more difficult to obtain experimental data on local heat transfer under cooling conditions as compared with the case of heating. Experimental data have been published on local heat transfer in pipes under conditions involving cooling of water [Sl], carbon dioxide [52,53], and helium [54].

Systematic numerical investigations have been carried out [55,56] for carbon dioxide and water under cooling conditions, wherein the system of Eqs. (1)-(3), (6), and (18) is used with the boundary conditions of Eq. (5 ) , in which the following changes have been inserted: at x = 0 a developed velocity profile has been given for constant physical properties uin = u&); on the inside surface of the pipe the first-order boundary condition T,(x) or the second- order boundary condition qw(x) has been given. The range of calculation parameters for carbon dioxide corresponds to experimental regimes [52,53]. To obtain q,, the distributions T,(x) have been predetermined from experimental data, and to obtain T, the experimental distribution of heat flux along the channel has been specified. Physical properties were taken from data [6] allowing for the peak of thermal conductivity on the basis of other data [3]. Calculations were carried out without taking into account the influence of free convection, which is considered unimportant according to the obtained estimations.

The comparison of predicted data [values of qw(x) were taken from the experiment] [55] with experimental data on heat transfer [53] for the cases T, < Tpc -= Tb is presented in Fig. 15. Calculated data on the friction factor


3 0- A e

6 4

50 40 30

3 2

0.6

0 1.0 R 0 R 1.0 3 z

0 1.0 0 5 0

- Y

I- 4

x/d R

FIG. 15. Predicted results [55] and experimental data [53] for P = 8.0 MPa, d = 0.412 cm, rn = 2400 kg/m2 sec, and Re = (4.8-4.5) x 10’.

1; the inertia resistance coefficient ju, the total drag coefficient fd, and the velocity and temperature distributions are also shown in this figure.

While under heating conditions, the M-shaped velocity profiles are connected with the sharp decrease of shear stress and with the passage of its values through zero into the negative region at the coordinate of maximum velocity, according to Eq. (6); on the contrary, under cooling conditions, such peculiarities are not observed. The flow decelerates, velocity profiles are monotonic, and the distribution of shear stress is close to that at constant physical properties of the fluid.

Physical property variations under cooling conditions significantly increase friction resistance. Local inertia resistance is commensurable with local friction resistance, but they have different signs. At some combinations of parameters, this can lead to a sharp decrease of total hydraulic drag, which attains negative values, thereby resulting in the appearance of zones with pressure increasing along the pipe.

The results of numerical modeling demonstrated that with cooling, for all the investigated ranges of parameters, the inertia resistance coefficient can be calculated from the one-dimensional Eq. (41) with an error less or equal to f 10%.

The results of calculations for pressure that is closer to the critical value than was found in Baskov et al. [53] are compared in Fig. 16 with experimental data [52] for the heat transfer coefficient. For this case, predicted data, taking into account the heat conductivity “peak,” are compared with data that do not consider the peak (see Fig. 1 b). It can be seen that the results of calculations fit the experiment, and the consideration of the thermal conductivity peak

36 A. F. POLYAKOV

I

32 34 36 38 t b (“4

FIG. 16. Comparison of predicted results (lines) [ 5 5 ] with experimental data (points) [52]: (1) qw = 36 kW/m2, m = 990.5 kg/m2 sec; (2) qw = 14.4, m = 495.2 (solid lines, with peak of thermal conductivity; dashed lines, without peak of thermal conductivity).

leads to a change in the heat transfer coefficient by approximately 10%. Cal- culations for P = 8.0 MPa (with the thermal conductivity peak) plotted in the same graph illustrate a significant influence of pressure on heat transfer at the values close to P,. While moving from P, with the increase in pressure, the local maximum in the distribution k( T) loses itself rather quickly, as follows from Fig. lb. Calculations of turbulent heat transfer [ S S ] with different data on thermal conductivity resulted in concluding that taking into account the thermal conductivity peak leads to some variations of the Nusselt number under pressures close to the critical one, but at P/P, > 1.1, the difference in the data is not routinely observed.

Special multiparametric calculations for different boundary conditions (qw = const. and T, = const.) showed that, in fact, the type of boundary conditions has no influence on turbulent heat transfer and resistance.

The generalization [56] of calculated data on the friction factor for carbon dioxide, water, and helium suggests an interpolation equation,

where fUl is calculated from Eq. (41). The approach [28] based on a two-layer model has been used to suggest a

heat transfer formula. Analysis of the entire volume of calculated data has


convinced Petrov and Popov [56] that an equation such as Eq. (20) describes data more successfully than does an equation such as Eq. (19). From generalizations [56] of calculated data on coefficients of heat transfer for water, helium, and carbon dioxide, the following formula has been obtained:

Nu = (f/8)Re Pr/( 1.07 + 12.7({)1’2{A“3(2)1’2

x [ 1 - A , (‘7’)”’] - - [ 1 - A,(!$$-)”’]})

(50)

where for water A, = 1 . 1 , A, = 1.0; for helium A, = 0.8, A, = 0.5; and for carbon dioxide A , = 0.9, A, = 1.0. The value f is calculated from Eq. (49).

IV. Free Convection

A. VERTICAL SURFACES

Processes of heat transfer and motion in a free-convective boundary layer on the vertical plate are described by the system of Eqs. (1)-(3), which are written not for cylindrical coordinates (x, r), but for Cartesian coordinates (x, y) ; thus, naturally, the term in the momentum equation taking into account the change of static pressure at forced flow is eliminated. Then the problem is specified by the following boundary conditions: when y = 0

. . u = 0, u =0, I = t, or (a i ldy) , = -qwcpw/kw (51a)

when y + 00

. . u = o , t = t ,

1. Laminar Flow

Laminar heat transfer in SCP fluid, moving along a vertical surface, was studied experimentally and theoretically. Experimental investigations have been carried out for water [57], carbon dioxide [58-601, and helium [61] with vertical plates [57-59,611 and cylinders [60]. Some theoretical investiga- ations have been undertaken for these fluids. Among these studies, the work of Popov and Yankov [3] can be distinguished; in this work, experimental and predicted data from other studies were analyzed and generalized along with a great volume of Popov and Yankov’s calculations for water, carbon

38 A. F. POLYAKOV

(T,-T- ) ( K ) (T,-L)(k)

FIG. 17. Calculated results [3] for laminar free convection using physical property data [5,6] and comparison with results of calculations [62,63] and experiments [57]. [3]' shows calculated results [3] using values K from Refs. 4,8, and 9 (see Fig. 1).

dioxide, helium, and nitrogen. Results of these calculations [3] are shown in Fig. 17. The relations

q,x114 = Fl(T,,AT); hx'14 = F,(T,,AT) (52)

following from self-similarity of the solution under the boundary condition Tw = const., are justified by numerical calculations. In conjunction with this, data in Fig. 17 are plotted in the coordinates given by Eq. (52).

Calculations [3] carried out using different data on physical properties confirm that a difference in physical properties for laminar flows can lead to differences in heat transfer coefficients reaching 30-40%. Consequently, the coincidence of prediction and experiment concerning laminar free convection is determined not only by experimental precision, but also is greatly dependent on the accuracy of physical property data (primarily on transport properties).

In the relation h(AT), a maximum appears at the values of T, close to Tpc. The value of this maximum appreciably increases if calculations are made taking into account the thermal conductivity peak. The influence of boundary conditions (T, = const. and qw = const.) on heat transfer is illustrated in Fig. 17. When qw = const. and T, and AT = T, - T, have the same values that they have at T, = const., heat transfer is greater by approximately 15%. The calculations have not revealed any objective regularities for the ratio Nu,/Nu,, so, according to Popov and Yankov [3], this ratio is assumed to be constant and equal to some mean value:

Nu,/Nu, = 1.15 (53)

An interpolational equation for heat transfer coefficient at T, = const. has been proposed [3]:


hx NU = - = N U ~ ( C , / C , , ) ~ ~ ~ ~ ( ~ ~ / ~ , ) "

(54) k,

Nu, = 0.75[2E/(5 + 10Pr'" + 10fi;)]1/4Ra'/4 -

0.32, Cp/CPw I 1 n = { -

0.1, C,/C,, > 1

Ra = ( g p A T x 3 / v i ) Pr, - Pr = p w q / k w

= (i, - i,)/(T, - T,)

Equation (54) describes the entire set of experimental and predicted data on heat transfer to water, carbon dioxide, nitrogen, and helium better than other known empirical and interpolational equations. Reliable data on thermal conductivity close to Tpc is lacking, thus the generalization has been made using calculated data obtained both with and without taking into account the thermal conductivity peak.

2. Turbulent Flow

A number of experimental [60,61,64-661 and theoretical [67-691 investigations of heat transfer at turbulent free convection near vertical plates [60,6 I ,64,67-691 and cylinders [65,66] under heating [60,61,64,65,67-701 or cooling [66] conditions have been published recently.

In order to estimate boundaries of transition from a laminar to turbulent boundary layer, the following approximate equation has been proposed [70] :

( 5 5 )

Numerical modeling of turbulent heat transfer at free convection of carbon dioxide near a heated plate has been made [67,68] for regime parameters corresponding to experimental data [60,64].

Popov and Yankov [68] have used Eqs. (36) and (37), which were proposed by Popov and Valuyeva [40,41] for turbulent momentum transport. The influence of the thermal conductivity peak on the calculated heat transfer coefficient and a comparison with data [60,64] in the form h = h(AT) are depicted in Fig. 18. The lines show the results of calculations for the cross- section x = 9 cm corresponding to experimental points [64]. Taking account of peak "k" gives a noticeable increase (up to 30% at P = 7.5 MPa and up to 20% at P = 8.0 MPa) in the heat transfer coefficient in the region of its maximum values and is more in agreement with experimental data. Results of

Ra, = 5 x 10'o(~/Cp,)-'.66

40 A. F. POLYAKOV

A

3 1.5 < 2, m

1 1.0 0 JZ

0.5 4 6 8 10 12 14 16

FIG. 18. Comparison of calculated results (-, with peak of thermal conductivity; -.-, without peak of thermal conductivity [68]; --. [67]) with experimental data (A, x [64]; 0,. [60]).

calculations for P = 7.85 MPa and AT> 6 K appear to be essentially lower (by 30-50%) than experimental data [64]. It is important to pay attention to a certain disparity among experimental data. The points [64] for P = 7.85 MPa are close to data [60] for P = 7.5 MPa and greatly exceed the values h for P = 8.0 MPa. Probably, they should in fact be closer to the latter values.

In work aimed at finding turbulent viscosity (Seetharam and Sharma [67]), which is determined by Eqs. (6) and (8), Eq. (4) has been used, supplemented with convective and diffusive terms. Results of predictions are compared with experimental data and the empirical relation [60] for carbon dioxide:

NU = 0.135 Ra'/3(Cp/Cpm)0.75(pw/pm)0~4 (56) It is shown that heat transfer at turbulent free convection is not dependent on the type of boundary condition (T, = const. or qw = const.) in the limits of validity of the empirical Eq. (56).

In Fig. 18, the maximum value of the heat transfer coefficient can be revealed at T, % Tpc. It is shown distinctly in Fig. 19 by an example of the results of numerical modeling of turbulent free-convectional heat transfer to helium [69], which has been investigated experimentally [61]. Calculations [69] have been made using the same mathematical model applied in earlier work [68]. As can be seen from Fig. 19, the maximum h becomes less pro- nounced with an increase of PIP,.

Popov and Yankov [69] showed that Eq. (56) gives a bad description of experimental and calculated data for helium. They have suggested the following equation to describe heat transfer to the turbulent motion of helium along the vertical plate:

Nu = 0.12 Ra1~3(~/Cpm)n(pw/pm)o~15 (57)

n = 1.0

n = 0.5,

T, > Tpc

T, I Tpc


I 0.8 0.9 1.0 1 . 1

T . / Tpc

FIG. 19. Heat transfer coefficient at turbulent free convection of helium: experiment (points) [61]; theory (lines) [69].

Temperature distributions in the boundary layer (Fig. 20) have some particularities distinguishing them from monotonic distributions at constant physical properties. The shape of the curves of dimensionless temperature (T - T,)/(T, - T , ) varies substantially depending on the coordinate when turning from one regime to another. For regimes with T, > Tpc > T, there are some sections along the plate with negative values of the second-order derivative in cross-sections of the boundary layer, when local temperature is close to the pseudocritical value. Such particularity has been denoted in the case of laminar free convection of SCP helium [3].

As experimental data on carbon dioxide [66] showed, the character of the dependence h = h(T,) for turbulent free convection under cooling conditions is typical for heat transfer under SCP: near the pseudocritical temperature a heat transfer coefficient attains the maximum value, which increases when the pressure approaches the critical value. Maximum values of h versus T, correspond to the temperatures T,, which are slightly higher than Tpc. At the

lo-' 4 10" 4 10'

I

0

Nu b / X ) FIG. 20. Temperature distribution in turbulent free convection boundary layer of helium [69].

42 A. F. POLYAKOV

same time, experimental data on cooling are not described by the relation for the heat transfer coefficient at free convection in carbon dioxide under heating conditions [Eq. (56)]. The following empirical equation for the coefficient of local heat transfer to carbon dioxide under cooling conditions has been suggested [66]:

(58) hx

Nu, = - = N~ow(c,/c,w)"(P~/Pw) k W

where

NuOw = 0.135Ra:l3, Raw = [(p, - p,)/pw] -

( s x 3 / v 3 Prw, C, = (i, - iW) / (~ , - T,)

q / C p w < 1,

C,/Cpw > 1 ,

n = 0.4

n = 0.75 -

B. HORIZONTAL WIRES

In this section, in accordance with the results of the work of Neumann and Hahne [7], we shall discuss heat transfer from a horizontal heated thin wire under free convection in single-phase carbon dioxide with nearcritical parameters.

Typical results for the heat transfer coefficient from thin platinum wire are illustrated in Fig. 21. When the temperature of the medium far from the wire, t, , is appreciably lower than the pseudocritical value, the heat transfer coefficient sharply increases, when the wire temperature t , approaches t,, . At

i0 20 30 40 5060 70 R a: t w @)

FIG. 21. Heat transfer from horizontal platinum wires [7].


t , > t,, > t,, the coefficient h decreases, adopting always the same values, which are practically constant. When t, is close to the pseudocritical temperature, changes of h have a sharp character, and maximum values of the heat transfer coefficient become rather large at t , x tpc. For t , > tpc, heat transfer has a relatively low level and varies slightly. Thus at t , < tpc, the coefficient h is dependent both on t , and t , , while at t , > tPc , it remains dependent only on the temperature t , .

Special attention has been given [7] to the analysis of the contribution of the thermal conductivity anomaly (see Fig. 1) in heat transfer at free convection from horizontal wires. To distinguish a sharp increase in thermal conductivity near the pseudocritical temperature, the latter is presented in the form

(59)

where Ak is an anomalous increment of thermal conductivity and ko is an interpolation “without the peak.”

When the conditions t, < t , < (t,, - 2K) or (t, , + 2K) < t , < t , are satisfied and the variation of the medium physical properties is relatively small, experimental data are described with an accuracy of f 10% by the known relation of Van der Hegge Zijnen [71],

(60)

k = ko + Ak

Nu = 0.35 + 0.25 Ra,‘/* + 0.45 Ra!I4

which is valid for heat transfer from horizontal cylinders and wires over the range < Rai < lo9. Here

Rai = Gri Pri = ( 2, g d 3 ~ P,; ~ i ) ( z ~ i ii - ia)

“ i ki t , - t ,

(all the other values with subscript i are determined similarly). Equation (60) describes with the same accuracy, f lo%, the experimental

data [7] over the indicated range of temperatures, with even a simpler averaging of physical properties, when they are determined not as the integral mean values, but over the arithmetic mean temperatures ( t , + t,)/2.

The variability of physical properties has its most important manifestation over the range of temperatures (t,, - 2K) < t , < t , < ( t , , + 2K). To reveal the degree of influence of the thermal conductivity peak, the experimental data on heat transfer are presented [7] in modernized coordinates Nu:(Ral), which are calculated by use of “k” from Eq. (59), where Ak is ignored, i.e., the values k, are used in Nu: and R a l . The curves in Fig. 21 b are plotted over the data relating to this range of temperatures. The deviations from Eq. (60) using

44 A. F. POLYAKOV

the dimensionless numbers Nu? and Ra l , which have been observed, are found to begin when the wire temperature approaches the pseudocritical value (compare with Fig. 21a), and to attain the maximum value when the thermal layer around the wire is at a pseudocritical state. At the same time, when the diameter increases, the deviation begins at greater values of Ra l and has a lower value. With a growth of the intensity of free convection, the contribution of the thermal conductivity layer near the wall in heat transfer decreases. The dependence of the studied deviation on ( t , - tpc) , ( tpc - tm), Pis established in experiments, and in view of the generalizations of data, Neumann and Hahne [7] suggested the following modification of Eq. (60):

Nui = (0.35 + 0.25 Ra,"8 + 0.45 RafI4)(l + a,a,a,)-' (61)

where

a, = zexp(-z), z = 4.5[(Tw - Tpc)/TJ''2

a2 = tanh(30 Tpc T, - Tm ) a, = 1 - 0.3 tanh 1 5- ( * 'iCPC)

The empirical Eq. (61) describes all the experimental data [7] with an accuracy of f 10%. The comparison of Eq. (61) with experimental data is shown in Fig. 21a.

V. Special Problems

A. SOME DATA ON NONSTATIONARY HEAT TRANSFER

In this review we have considered the data generally related to stationary or quasistationary (i.e., stationary in the mean) conditions. The latter determine both developed turbulent flow, when the averaging interval corresponds to the scale of turbulence, and flows with fluctuations of hydrodynamic and thermal values, which are developed, for example, in some regimes of mixed convection, but when the conditions of quasistationary approximation for mean values are satisfied.

Some of the work devoted to studying nonstationary processes in SCP fluid is examined in the following discussions.

1. Nonstationarity Determined b y External Conditions

Smirnov and Krasnov [72] presented results of experimental investigations on nonstationary heat transfer in upflowing water in vertical, electri-


800

600

400

200

2. 'z

a b

0 10 200 0 20 30

FIG. 22. Temperature of outside surface of the pipe and normalized coefficient of nonstationary heat transfer under (a) decreased or (b) increased heat load.

city heated 109-cm-long pipes with internal and external diameters of 0.408 and 0.81 5 cm, respectively. Nonstationary conditions were created by step changes of heat power in the wall by means of variation of the electrical load. The experiments have been carried out over the ranges P = 25-30 MPa, m = (0.4-1.2) x lo3 kg/m2 sec at heat flux on the inside surface of channel reaching 1 MW/m2. At mass velocities (m = 400-500 kg/m2 sec), regimes with the local degraded heat transfer have been observed. In all the experiments the fluid temperature remained lower than the pseudocritical one.

Figure 22 illustrates the character of changes of wall temperature and the normalized Nusselt number Nu/Nu, for three cross-sections of a pipe for the regime with locally degraded heat transfer under stationary conditions, which is determined by the value Nu,. In the initial stage of the experiment (z = 0), some value qws = const. was given and the stationary distribution t,, was set. In Fig. 22a, three values t,, for the coordinates x /d = 16.5, 215, and 240 are shown. The coordinate x /d = 215 corresponds to a local maximum of wall temperature. In the moment t = 0, heat power in the wall of the pipe drops stepwise to some constant level of heat load. At this level of load a new distribution t, corresponding to Nu/Nu, = 1 establishes during 20 sec. This distribution t , is monotonic. Then the heat load increases stepwise to its initial value qws, and during 30 sec the transition process develops, as is shown in Fig. 22b. After 30 sec, at the end of second transition process, the initial distribution t,, with a local maximum, again establishes. When the heat load decreases, the zone with local degraded heat transfer moves to the outlet; this is reflected in the sharp fall of value Nu/Nu, at the coordinate x /d = 240. At that

46 A. F. POLYAKOV

time, the rate of change of wall temperature sharply decelerates. Such behavior of the relative coefficient of heat transfer is also observed under an increasing heat load.

As is shown by the example in Fig. 22, variable physical properties of SCP media lead to substantially greater peculiarities of heat transfer under nonstationary conditions as compared to constant physical properties.

Under the real conditions, the development of transition processes in fluid is connected with changing temperature fields in the wall and with the necessity to solve the conjugated problem. Even under stationary conditions, the influence of the wall on heat transfer to SCP fluids can have a significant value in the realization of wall temperature distributions with local maxima. Watson [73] has showed that the interaction of wall thermal conductivity along the flow and local heat transfer dependence on wall temperature can lead to a disturbance of the stationary state and to a realization of distributions t , with several peaks.

2. Thermoacoustic Perturbations

As the experimental investigation of Daney et al. [74] showed, under thermal conditions (ti, .c t,, < t,,,) in flows of SCP fluid (helium) in heated channels, when a transition from pseudoliquid state to pseudogas one takes place, the development of pulsation regimes and the propagation of density waves along the pipe are possible. The amplitude of thermal perturbation moving from inlet to outlet of the heated channel was established to increase. Labountsov and Mirzoyan [75] have theoretically investigated the problem of the thermal stability of helium flow in heated channels of small diameter (i.e., without manifestation of buoyancy forces). The analysis carried out on a basis of the basic theory of hydrodynamic stability resulted in the determination of a diagram of flow regimes. Experimental data [74] are plotted in Fig. 23 over the coordinates APi,,/APou, - u,,,/uin (APi, and AP,,, are the pressure losses at the inlet and outlet; uin and u,,, are specific volumes of fluid in input and output cross-sections). In general, theoretical and experimental data are in good agreement qualitatively and quantitatively.

Thermoacoustic oscillations generated in flows of fluid under nearcritical parameters and in the presence of heat transfer are of special scientific and practical interest. In turn, thermoacoustic oscillations have an important influence on the character of heat transfer.

As a result of theoretical investigations [76-791, it was established that under the temperature condition t , > t,, > t b , the anomalous improvement of heat transfer, which occurs at a certain combination of parameters and is accompanied with high-frequency fluctuations of pressure, has an acoustic character. This has been determined by the presence of the wall layer of fluids


‘JoutlVi n

FIG. 23. Diagram of stability of pipe flow of SCP helium on the basis of experimental data (points) [74] and theoretical results (line) [75].

with a low sound velocity and distortion of the acoustic wave profile in the region of high gradients of sound velocity, thus resulting in the sharp growth of velocity in standing sound waves and the intensification of heat transfer in turbulent flows. Such regimes are realized, provided that some conditions are satisfied, in particular, certain acoustic properties of the channel. Some results on amplitude-frequency features of oscillations have been obtained. Estima- tions of the limits of the existence of regimes with acoustic oscillations and of the degree of heat transfer increases have been made. Theoretical data have agreed with experimental data.

When conducting the analysis and generalization of data for regimes with acoustic oscillations, i t is necessary to allow for the anomalous minimum value of sound velocity, as was discovered by Erohin an Kalyanov [80], under the temperature that slightly exceeds the pseudocritical temperature.

Experimental data on the development of thermoacoustic oscillations in turbulent flow of SCP water in a heated pipe (d = 0.44 cm) and in an annular channel ( d , : d , = 1:0.6 cm) have been presented by Maevskiy et al. [Sl]. In this work the lower boundaries of existence of the thermoacoustic oscillations have been stated; the oscillations had frequencies of 100 and 200 Hz and amplitudes up to 0.6 MPa in the annular channel.

B. HEAT TRANSFER AUGMENTATION UNDER CONDITIONS CORRESPONDING TO DEGRADED HEAT TRANSFER

The complex investigation of heat transfer and hydraulic resistance is an advantage of the work [82] in comparison with other works on heat transfer intensification for SCP fluid.

48 A. F. POLYAKOV

In previous studies, investigations have focused on the influence on heat transfer of swirled flow using helix ribbons and internal helix finning of the pipe. The result of applications of such techniques was the elimination of wall temperature peaks at the same qw and rn, as for a smooth pipe. The possibility of a twofold increase of admissible heat flux (under the same maximum value of wall temperature) is indicated. The installation of a ribbon in horizontal pipes makes it possible to equalize heat transfer over the circumference of the pipe. However, note that when heat transfer degrades due to thermal acceleration, in swirled flow the trend to heat transfer deterioration is conserved under sufficiently high heat loads.

A method for heat transfer augmentation based on turbulization of the wall region of flow has been studied experimentally [82]. Augmentation has been accomplished using spiral wire inserts; the wires had diameters from 0.02 to 0.06 cm and were inserted on the inner surface of a heated pipe with a diameter of 0.8 cm. Experiments have been carried out with carbon dioxide under a pressure of 7.7 MPa, Rein = (2.3-3.4) x lo5, and heat flux on the wall up to 1.5 MW/m2. It was stated that all the above-considered positive results of augmentation can be achieved by using spiral inserts; under conditions corresponding to degraded local heat transfer in smooth pipes, spiral inserts with a relatively big step(s/d = 3-5) are the most effective ones. The limitation of data obtained in this way did not allow suggesting general recommenda- tions on the optimization of augmentation devices.

Another type of augmentation device has been subjected to a test [83]: transverse annular bosses were directed into the pipe. Such devices have been widely tested for gases and liquids with weakly varying properties and have demonstrated their efficiency. Experiments [83] have been carried out with hydrocarbonaceous fluids under SCP in heated pipes with a diameter of 0.4 cm. Local heat transfer and hydraulic resistance were measured. It was found that at low values [Re = (1.1-5.2) x lo3], the increase of the normalized heat transfer coefficient Nu/Nu,, attains large values (> 5) , and the efficiency of augmentation determined by the ratio (Nu/Nu,,)/( jd/fd,,,) exceeds one, attaining the value 1.5. For ordinary fluids under the same conditions, the maximum value of efficiency was 1.01. Here Nu,, and id,,, are the values of the Nusselt number and of drag coefficient for smooth pipes.

VI. Concluding Remarks

Based on the material discussed in this review, some conclusions and pro- posals about directions of further investigations can be made.

During the past 10-12 years, the studies of heat transfer grew beyond the fully empirical state. Physically based analyses of the nature of the develop-


ment of some peculiarities of turbulent flow and heat transfer (in particular, for regimes with degraded local heat transfer and for regimes with pulsating flow and with thermoacoustic perturbations) have been carried out. Methods of mathematical modeling of turbulent heat transfer in vertical pipes have been developed, and numerical calculations for water, carbon dioxide, and helium have been carried out under different boundary conditions over a wide range of change of the definitive parameters. Systematic experimental investigations of hydraulic resistance, velocity, and temperature fields have been undertaken. The importance of a number of new problems has been demonstrated, and some preliminary data have been obtained on the temporal and spatial nonuniformity of boundary conditions, and on problem solution in conjugated formulation of “wall fluid.” A number of other scientific and practical problems are being solved: refinement of fluid thermophysical properties at supercritical state parameters, specifically, on thermal conductivity, and their application for refinement of analysis and generalization of data on heat transfer; search for optimum methods to augment heat transfer, viz. in order to eliminate regimes with local wall temperature peaks; accumulation of experimental data on heat transfer to helium under different conditions to satisfy the needs of cryogenic engineering; and obtaining more perfect computational dependences.

The achievements indicated above do not completely resolve the enumer- ated problems, nor do they encompass the diversity of problems of heat exchange and hydrodynamics under SCP. Current investigations may develop in the following directions: experimental study of turbulent flow structure and heat transport; development of methods of three-dimensional mathematical modeling of heat transport, including a conjugate formulation of problem; experimental and theoretical computational investigations of nonstationary heat transfer, including generation of internal (in particular) thermoacoustic oscillations; accounting for the refined peculiarities of the thermophysical properties of media and their contamination with solute components; development of generalized methods of calculations based on a combined analysis of hydrodynamic and thermal features of flow; and development of effective procedures of heat transfer augmentation and a search for its optimization, with elimination of regimes with unfavorable thermomechanical actions on the surface.

NOMENCLATURE

EL,, parameter of threshold value, f friction factor [u,/(m2/8p)]

C, specific heat at constant pressure f. inertial resistance factor, Eq. (38) d diameter of the tube (= Zr,) or Y acceleration due to gravity

Eq. (11) f* drag coefficient, Eq. (39)

diameter of the wire Ga Gallileo number (gd3/v2)

A. F. POLYAKOV

Gr Grashof number (gpATd3/v2) Gr, modified Grashof number

h heat transfer coefficient (qw/Tw - Tb) i specific enthalpy J dimensionless parameter of acceler-

(eP4wd4/kv2)

k K m

Nu,% P Pe Pr

P

E

0

b

in, out PC T th

C

W

ation defined in Eq. (12) thermal conductivity turbulence energy mass velocity averaged through

tube cross-section CfS:(Pu)R dR1

local and average Nusselt number pressure Peclet number (= Pr Re) Prandtl number (pC,/k)

Pr, q r r0 R

Ra Re St t T U

V

V X

Y

Greek Symbols

volume expansion coefficient P C-(1/P)(aP/at),I V

dissipation of turbulent energy P dimensionless temperature defined u

in Eq. (46) T

Subscripts

bulk 0 critical value inlet or outlet (section), respectively *

value for turbulent flow threshold value value at the wall

pseudocritical temperature a,

turbulent Prandtl number heat flux radial coordinate tube radius dimensionless radial coordinate

W O )

Rayleigh number (= Pr Gr) Reynolds number (md/p) Stanton number (= Nu/Re Pr) temperature ("C) absolute temperature (K) axial velocity radial velocity specific volume (= l/p) axial coordinate radial distance (= ro - r)

viscosity kinematic viscosity (p/p) density shear stress time

scale, reference, characteristic, ini-

variable physical properties value taken far from the surface in

the flow (outside the boundary layer) or far from the start of heating when in a pipe

tial, or axial value

Superscripts

+ dimensionless value

Abbreviation

SCP supercritical pressure

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