Jacksons - Supercritical Heat Transfer

8/12/2019 Jacksons - Supercritical Heat Transfer

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SUPERCRITICAL HEAT TRANSFER

Jackson, J. D.

DOI:10.1615/AtoZ.s.supercritical_heat_transfer

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Physical Properties Near the Critical Point

The principal difference in the behavior of a fluid as its temperature is raised above the

critical temperature is evident from the pressure-volume diagram shown in Figure 1. Below

the critical temperature Tc the variation of pressure p and volume v along an isotherm showsdiscontinuities where the isotherm intersects the saturation line. Phase change occurs at the

saturation line, and the horizontal constant pressure segment of the isotherm represents the

presence of vapor and liquid in varying proportions. The critical temperature isotherm has

zero slope at one point, the pressure there being the critical pressure, pc. Above the critical

temperature, isotherms show no discontinuity and from a macroscopic standpoint there is a

continuous variation from a liquid-like fluid to a gas-like fluid.

Figure 1.

Since the pressure variation across most thermal boundary layers is negligible, it is the

variation of properties along an isobar that is important. The variation of specific volume v,

specific heat c p, absolute viscosity η, thermal conductivity λ and specific enthalpy h for water

at pressure of 245 bar is shown in Figure 2 The temperature at which the specific heat reaches

a peak is known as the pseudocritical temperature, T pc. As the pressure is increased this

temperature increases, the maximum value of the specific heat falls, and the variation of the

other properties becomes less severe.

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may be made to traverse a wide range of values by adjusting the bulk mean temperature of the

fluid in the limiting case of a very small heat flux. There is no reason why constant property

equations should not work. Consider the well-known expression

(1)

where Nu is the Nusselt Number (= αD/λ), Re is the Reynolds Number (= D/η), and Pr is

the Prandtl Number (=ηc p/λ), in which α is Heat Transfer Coefficient and is mass flow rate

per unit area.

From this it follows that

(2)

Noting that the variations with temperature, of the thermal conductivity λ, and the viscosity η

are rather similar in form (Figure 2), one would not expect a major effect on the heat transfer

coefficient on this account. The specific heat, on the other hand, becomes very large in the

vicinity of the pseudocritical temperature, and one might expect the heat transfer coefficient

to behave in a similar manner. This, in fact, happens, as is well illustrated by the data of

Yamagata et al. for water (Figure 3). It will be noted that as the heat flux increases the

magnitude of the peak in the heat transfer coefficient decreases. This is because the region of

fluid at high Prandtl Number contracts as the heat flux, and hence the temperature gradient,

increases.

Figure 3.

I ntermediate and high heat fluxes

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Heat flux is not the sole determinant of the effects, which are about to be described. However,

for a given size and orientation of pipe and a given flow rate, the effects are very largely

determined by heat flux. Under some conditions the orientation of the pipe becomes

important, and very different results are obtained depending upon whether the pipe is

horizontal or vertical, and whether in the latter case the flow is upwards or downwards.

In cases where orientation is not important the deterioration in heat transfer coefficient with

heat flux that was evident in Figure 3 becomes more marked both with decreasing mass

velocity and increase of heat flux. Figure 4 shows the water data of Vikhrev et al. for a

constant mass flow and three levels of heat flux, the highest of which results in virtually no

enhancement of a as the bulk temperature traverses the pseudocritical temperature.

Figure 4.

Turning to data in which the orientation of the pipe is important, we find that under some

conditions of pipe diameter, mass flow and heat flux, heat transfer to a fluid is much better in

a vertical pipe when the flow is in the downward rather than upward. When it was first

observed, this behavior was difficult to reconcile with the currently held view that forced and

free convection reinforced each other when they acted in the same direction. An example of

this behavior is shown in Figure 5 where the data of Jackson and Evans-Lutterudt for carbon

dioxide at two different heat fluxes are shown for both upward and downward flow. It will beseen that there is a rapid deterioration in heat transfer following a relatively small increase in

heat flux with upward flow but not with downward flow.















Figure 5.

An interesting comparison between carbon dioxide data that is unaffected by the direction offlow and with data that is so affected is shown in Figure 6. All three sets of data have been

selected so that, in the absence of any gravitational effect, they would be expected to yield

similar distributions of wall temperature against bulk enthalpy. In the case of one set of data

(due to Shiralkar and Griffith), no difference between upward and downward flow was found,

whereas the other two sets (due to Jackson and Evans-Lutterudt and Bourke et al.) showed

such a difference. The sharp temperature peaks appear to be characteristic of upward heated

flows in which gravitation is important. Note that, following these peaks, heat transfer is

better than it would have been in the absence of gravitational effects, and also that in

downward flow there is substantially improved heat transfer with no wall temperature peaks.







Figure 6.

A Criterion for buoyancy affected flow

The examples of the effect of orientation of the pipe that have been quoted above confirm the

importance of interactions between forced and free convection. The sharp peaks in the wall

temperature that occur with upward heated flows were first attributed to "pseudofilm boiling" .

The true nature of the phenomenon was revealed when results for both upward and downward

flow became available, and it is now possible to say with some confidence whether it is likely

to occur in given circumstances. The exclusion of data affected by buoyancy is clearly

necessary preliminary before any attempt can be made to correlate data for pure forced

convection.

An explanation of the mechanism by which buoyancy affects heat transfer has been proposed

by Hall and Jackson (1969). The dominant factor is the modification of the shear stress

distribution across the pipe, with a consequential change in turbulence production.

It can be shown that a criterion for negligible buoyancy effects is

(3)



where is the Grashof Number (= g(ρ b − )D3/ ), in which is the integrated mean

density and the subscript b indicates physical properties evaluated at the local bulk

temperature.

The application of the above criterion is made difficult by the fact that evaluation of the

Grashof Number requires that the wall temperature Tw is known, so that in applications wherethe wall heat flux is specified a correlation describing forced convection heat transfer (with

negligible buoyancy effects) must be used in conjunction with the criterion.

Forced Convection in the Absence of Buoyancy

The enhancement of heat transfer coefficient at small temperature differences when the bulk

temperature is close to the pseudocritical temperature has already been noted and attributed to

the large value of c p in this region. As the temperature difference is increased, the proportion

of the flow experiencing this high specific heat shrinks, and the heat transfer coefficient is

reduced. If we take the temperature difference as the main factor affecting enhancement of theheat transfer coefficient (for a given fluid and bulk temperature) then it follows that the effect

should be characterized by a quantity such as . Attempts have been made to express the

ratio of the heat transfer coefficient to that for constant properties in terms of the quantity

; typical of these are the generalized curves of Lokshin for water at 250 bar (Figure 7). It

is seen that above a value of ≈ 0.7 no enhancement occurs and there is a monotonic

decrease in a as the fluid bulk temperature traverses the pseudocritical temperature.

Figure 7.

Many attempts have been made to correlate forced convection data in terms of conventional

dimensionless groups with modifications to allow for the effect of physical property variation.

Jackson et al. (1975) have made a study of virtually all known correlations and have

compared their performance when tested against forced convection data for water and for

carbon dioxide. The fact that they were able to exclude data that may have been affected by

buoyancy constituted an important advance over earlier attempts to correlate forced











convection data. They found that by far the most effective correlation was that published in

1965 by Krasnoschekov and Protopopov (1966). This utilizes the Petukhov-Kirrillov equation

for constant properties forced convection and accounts for property variation by additional

terms involving wall to bulk density ratio and integrated to bulk specific heat ratio, each

raised to suitable powers. It correlated 80% of the data points to within ± 15% and 91% to

within ± 20%. Because the Petukhov-Kirrillov equation was rather cumbersome to useJackson and Fewster also tried the following simpler version in which a Dittus-Boelter form

of constant properties forced convection equation is used instead:

(4)

where suffix w refers to conditions at the wail,

and

In this form the equation correlated 77% of the data points to ± 15% and 90% to within ±20%. Approximately 2000 data points were tested against the correlation.

Mixed Convection in Pipes

Vertical pipes

This topic has already received some discussion and a criterion for the onset of buoyancy

effects has been given. Also the tendency towards the sudden occurrence of localized peakingof the wall temperature in upward flow with increase of heat flux has been noted. In view of

this instability it is not surprising that the magnitude of the subsequent increase in wall

temperature (for conditions of specified heat flux) has proved difficult to predict. For a

detailed picture of the influences of buoyancy on heat transfer to fluids in vertical pipes,

including fluids at supercritical pressure, see Jackson and Hall (1980).

For a downward heated flow there is a continuous enhancement in heat transfer as buoyancy

becomes relatively stronger. This behavior has been found with many fluids at supercritical

pressure and also with other fluids. Not only is the heat transfer improved, but wall

temperatures are less sensitive to heat flux.

Horizontal pipes



As might be expected, buoyancy in uniformly heated horizontal pipes causes a progressive

increase in the temperature of the upper surface relative to the lower surface of the pipe. This

is undoubtedly due in part to a stratification of the flow, the hotter and less dense fluid

occupying the upper part of the pipe; there may also be an effect due to the damping effect of

the stabilizing density gradient on turbulence near the upper surface of the pipe. At the lower

surface heat transfer is frequently better than for forced convection alone, suggesting thatthere may be some amplification of turbulence by the destabilizing density gradient in this

region. For a detailed review of the effects of buoyancy on heat transfer to fluids flowing in

horizontal tubes, including fluids at supercritical pressure, see Jackson (1983).

A considerable number of measurements on horizontal pipes have been made using fluids at

supercritical pressure. A typical set of data presented by Belyakov et al. is reproduced in

Figure 8. The deterioration of the upper surface occurs progressively along the pipe and does

not show the sharp peaks that are obtained with upward flow.

Figure 8.

Free Convection

Free convection in supercritical pressure fluids has received a good deal less attention than

has forced convection, presumably because there have been fewer pressing industrial

applications. The situation was reviewed by Hall (1971), and little further data have been

reported since that time.






The general picture is one of enhancement of heat transfer at low heat fluxes with surface and

fluid temperature spanning the pseudocritical temperature. As with forced convection, the

enhancement is greatest when the bulk temperature is close to the critical temperature, but

there is no evidence of the impairment of heat transfer that is found at high heat fluxes with

forced convection.

Boiling

Nucleation at high subcritical pressures

Nucleation at a boiling site of a given size becomes easier as the pressure of a boiling liquid

increases. The decrease of surface tension means that a smaller difference between vapor and

liquid pressure is required in order to make a nucleus grow; moreover, a given pressure

difference can be achieved with a smaller superheat as the pressure level increases. For a

given superheat, therefore, it will be possible to activate smaller nuclei at higher pressures.

In addition to this effect, there is the possibility of spontaneous nucleation occurring as aresult of thermal fluctuations in a superheated liquid at a surface as p/pc approaches unity.

Indeed, it may be that the very "quiet" transition to film boiling that occurs at high pressure

may be a consequence of spontaneous nucleation.

Pool boiling

The effect of pressure on nucleate boiling is well established: as the pressure is raised towards

the critical pressure the heat transfer coefficient increases, and the limiting heat flux for

transition from nucleate to film boiling first increases, reaches a maximum at a value of p/pc

of about 0.35, and then decreases.

Pool boiling experiments at high pressures show very clearly how the range of conditions

under which nucleate boiling occurs is contracted by transition to film boiling.

Following transition to film boiling at pressures close to the critical value, a highly regular

flow of vapor from the heated surface is observed. At p/pc ≈ 0.9 uniformly spaced columns of

bubbles are observed to rise from a heated wire. As the critical pressure is approached these

columns change into "tubes" of vapor, again with very regular spacing, and finally change

further into what appears to be a sheet of vapor rising from the wire.

Flow boiling

As indicated earlier, the general picture of supercritical pressure forced convection has

become comparatively clear. In contrast, very few detailed investigations have been made at

high subcritical pressures.

Figure 9 shows data for water flowing upwards in a 20-mm diameter pipe for pressures

extending from subcritical to slightly supercritical, taken from the work of Herkenrath et al. It

is of interest to compare the subcritical curves, each of which involves a local heat transfer

crisis, with the curve for supercritical pressure, which does not. Note that the conditions are

such that buoyancy induced temperature peaks would not be expected in the case of the

supercritical data.






Figure 9.

It will be seen from Figure 9 that as the pressure is increased the position of the heat transfer

crisis shifts towards the region of lower quality; the magnitude of the wall temperature step at

first decreases, then increases and finally decreases as the critical pressure is approached.

Furthermore, the shape changes from a sharp to a more gradual increase in wall temperature.

Some data obtained by the present author and co-workers using carbon dioxide at a reduced

pressure, p/pc = 0.99, for both upward and downward flow showed an increase in walltemperature following transition which was rather gradual (the curve for p = 215 bar in

Figure 9) and that in upward flow peaks develop due to buoyancy influence as they do at

supercritical pressure. It was also apparent that at high subcritical pressures downward flow

was advantageous, just as at supercritical pressures: transition occurred later in downward

flow and the consequences were less severe. Experiments with carbon dioxide at lower

pressure show a sharp transition rather similar to those illustrated in Figure 9.

The beneficial consequences of buoyancy in downward flow do not appear to persist as the

pressure is dropped further, where the above mentioned trend appears to be reversed.

References:

Hall, W. B. (1971) Heat transfer near the critical points, Advances in Heat Transfer , Vol. 6,

Academic Press, New York.

Hall, W. B. and Jackson, J. D. (1969) Laminarisation of a turbulent pipe flow by buoyancy

forces, ASME Paper No. 69-HT-55.

Hall, W. B. and Jackson, J. D. (1978) Heat transfer near the critical point, Keynote Lecture,

Sixth International Heat Transfer Conference, Toronto, Canada.














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Jacksons - Supercritical Heat Transfer