Heat Transfer Handbook part 57 potx

Detailed studies of the different flow regimes and the corresponding heat transfer have been carried out, along with three-dimensional transport, for wide ranges of the governing paramete

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Figure 7.12 (a) Typical vertical rectangular enclosure; (b) calculated isotherms for Pr= 1.0

at Ra= 2 × 104 (From Elder, 1966.)

At small values of Ra, Ra ≤ 1000, there is little increase in the heat transfer

over that due to conduction alone, for which the Nusselt number Nu= hd/k = 1.

However, as Ra increases, several flow regimes have been found to occur, resulting

in a significant increase in the Nusselt number In laminar flow, these regimes include the conduction, transition, and boundary layer regimes The conduction regime is characterized by a linear temperature variation in the central region of the enclosure

In the boundary layer regime, thin boundary layers appear along the vertical walls, with horizontal temperature uniformity between the two layers In the transition regime, the two boundary regions are thicker and the isothermal interior region does not appear The characteristics are seen clearly in Figs 7.13 and 7.14 As Ra increases further, secondary flows appear, as characterized by additional cells in the flow, and transition to turbulence occurs at still higher Ra Detailed studies of the different flow regimes and the corresponding heat transfer have been carried out, along with three-dimensional transport, for wide ranges of the governing parameters, as reviewed by Gebhart et al (1988)

An interesting solution was obtained by Batchelor (1954) for large aspect ratios,

H/d → ∞ The flow can then be assumed to be fully developed, with the velocity

and temperature as functions only of the horizontal coordinatey This problem can

be solved analytically to yield the velocityU and temperature θ distributions as

U = Ra

12Y (1 − Y )(1 − 2Y ) and θ = 1 − Y (7.62)

0

0

20

10

10

20

20

20

30

x = 34.2 cm

8.9

y ⫻ 10 (m)3

layer regime (From Eckert and Carlson, 1961.)

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Figure 7.14 Measured velocity distributions at midheight in a rectangular enclosure at vari-ous values of Ra (From Elder, 1965.)

where the nondimensionalization given earlier for eqs (7.60) and (7.61) is used

Therefore, the temperature distribution is independent of the flow and the Nusselt number Nu is 1.0

Many other flow configurations in internal natural convection, besides the rectangu-lar enclosure, have been studied because of both fundamental and applied interest

Inclined and horizontal enclosures have been of interest in solar energy utilization and have received a lot of attention Horizontal layers, with heating from below, provide the classical Benard problem whose instability has been of interest to many researchers over many decades Convective flow does not arise up to a Rayleigh num-ber Ra, based on the layer thickness and temperature difference, of around 1700 The

INTERNAL NATURAL CONVECTION 555

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exact value was obtained as 1708 by several researchers, using stability analysis, as reviewed by Gebhart et al (1988) Therefore, the Nusselt number based on these characteristic quantities is 1.0 up to this value of Ra As the Rayleigh number in-creases, convective flow arises and different flow regimes, including turbulent flow, and different instabilities have been studied in detail, as reviewed by Gebhart et al

(1988) Hollands et al (1975) presented experimental results and correlated their own data and that from other studies for horizontal enclosures Similarly, Hollands et al

(1976) presented correlations for inclined enclosures heated at the bottom

Natural convection in cylindrical, spherical, and annular cavities has also been of interest Ostrach (1972) reviewed much of the work done on the flow inside horizontal cylindrical cavities Kuehn and Goldstein (1976) carried out a detailed experimental and numerical investigation of natural convection in concentric horizontal cylindrical annuli A thermosyphon, which is a fully or partially enclosed circulating fluid system driven by buoyancy, has been of interest in the cooling of gas turbines, electrical machinery, nuclear reactors, and geothermal energy extraction See, for instance, the review by Japikse (1973) and the detailed study by Mallinson et al (1981)

Partial enclosures are of importance in room ventilation, building fires, and cooling

of electronic equipment Several studies have been directed at the natural convection flow arising in enclosures with openings and the associated thermal or mass transport (Markatos et al., 1982; Abib and Jaluria, 1988) Figure 7.15 shows the schematic

of the buoyancy-driven flow in an enclosure with an opening Typical results for flow due to a thermal source on the left wall are shown in Fig 7.16 The numbers in the figure indicate dimensionless stream function and temperature values A strong stable stratification, with hot fluid overlying colder fluid, is observed Many other such natural convection flows and the resulting transport processes have been investigated

in the literature

Figure 7.15 Buoyancy-driven flow due to a thermal energy source in an enclosure with an opening

where Ra is based on the thermal source height and temperature difference from the ambient (From Abib and Jaluria, 1988.)

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7.6 TURBULENT FLOW 7.6.1 Transition from Laminar Flow to Turbulent Flow

One of the most important questions to be answered in convection is whether the flow is laminar or turbulent, since the transport processes depend strongly on the flow regime Near the leading edge of a surface, the flow is well ordered and well layered

The fluctuations and disturbances, if any, are small in magnitude compared to the mean flow The processes can be defined in terms of the laminar governing equations,

as discussed earlier However, as the flow proceeds downstream from the leading edge, it undergoes transition to turbulent flow, which is characterized by disturbances

of large magnitude The flow may then be considered as a combination of a mean velocity and a fluctuating component Statistical methods can generally be used to describe the flow field In several natural convection flows of interest, particularly in industrial applications, the flow lies in the unstable regime or in the transition regime

In a study of the transition of laminar flow to turbulence, it is necessary to deter-mine the conditions under which a disturbance in the flow amplifies as it proceeds downstream This involves a consideration of the stability of the flow, an unstable circumstance leading to disturbance growth These disturbances enter the flow from various sources, such as building vibrations, fluctuations in heat input, and vibra-tions in equipment Depending on the condivibra-tions in terms of frequency, location, and magnitude of the input disturbances, they may grow in amplitude due to a balance

of buoyancy, pressure, and viscous forces This form of instability, termed

hydrody-namic stability, leads to disturbance growth.

The disturbances gradually amplify to large enough magnitudes to cause signifi-cant nonlinear effects and secondary mean flows which distort the mean velocity and temperature profiles This leads to the formation of a shear layer, which fosters fur-ther amplification of the disturbances, and concentrated turbulent bursts arise These bursts then increase in magnitude and in the fraction of time they occur, eventually crowding out the remaining laminar flow and giving rise to a completely turbulent flow The general mechanisms underlying transition are shown in Fig 7.17 from the work of Jaluria and Gebhart (1974) Several books on flow stability, such as those by Chandrasekhar (1961) and Drazin and Reid (1981), are available and may be con-sulted for further information on the underlying mechanisms

7.6.2 Turbulence

Most natural convection flows of interest in nature and in technology are turbulent

The velocity, pressure, and temperature at a given point in these flows do not remain constant with time but vary irregularly at relatively high frequency There is consid-erable amount of bulk mixing, with fluid packets moving around chaotically, giving rise to the fluctuations observed in the velocity and temperature fields rather than the well-ordered and well-layered flow characteristic of the laminar regime Due to the importance of turbulent natural convection flows, a considerable amount of ef-fort, both experimental and analytical, has been directed at understanding the basic

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Figure 7.17 Growth of the boundary layer and the sequence of events during transition in natural convection flow over a vertical surface in water, Pr= 6.7 (From Jaluria and Gebhart,

1974.)

mechanisms and determining the transport rates The work done in forced flows has been more extensive, and in fact, much of our understanding of turbulent flows in natural convection is based on this work The transport mechanisms in turbulent flow are very different from those in laminar flow, and some of the basic considerations are given here For further information, books on turbulent transport, such as those

by Hinze (1975) and Tennekes and Lumley (1972), may be consulted

In describing a turbulent flow, the fluctuating or eddy motion is superimposed on

a mean motion The flow may, therefore, be described in terms of the time-averaged values of the velocity components (denoted as ¯u, ¯v, and ¯w) and the disturbance or

fluctuating quantities (u, v, andw) The instantaneous value of each of the velocity components is then given as

u = ¯u + u v = ¯v + v w = ¯w + w (7.63)

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Similarly, the pressure and temperature in the flow may be written as

The time averages are found by integrating the local instantaneous value of the particular quantity at a given point over a time interval that is long compared to the time period of the fluctuations For steady turbulence, the time-averaged quantities

do not vary with time Then, from the definition of the averaging process, the time averages of the fluctuating quantities are zero For unsteady turbulence, the time-averaged quantities are time dependent Here we consider only the case of steady turbulence, so that the average quantities are independent of time and allow the flow and the transport processes to be represented in terms of time-independent variables

If the instantaneous quantities defined by eqs (63) and (64) are inserted into the governing continuity, momentum, and energy equations and a time average taken, additional transport terms, due to the turbulent eddies, arise An important concept employed for treating these additional transport components is that of eddy viscosity

εMand diffusivityεH Momentum and heat transfer processes may then be expressed

as a combination of a molecular component and an eddy component This gives for

¯u(y) and T (y)

τ

ρ = (ν + ε M )

d ¯u

q

ρc p = (α + ε H )

d ¯T

whereτ is the total shear stress and qis the total heat flux For isotropic turbulence,

εM andεH are independent of direction and are of the form −uv/(∂ ¯u/∂y) and

vT/(∂ ¯T /∂y), respectively.

If the preceding relationships forτ and qare introduced into the governing equa-tions for the mean flow, obtained by time averaging the equaequa-tions written for the total instantaneous flow in boundary layer form, we obtain

∂ ¯u

∂x+

∂ ¯v

¯u ∂ ¯u

∂x + ¯v

∂ ¯u

∂y = gβ( ¯T − T∞) +

∂

∂y



(ν + ε M ) ∂ ¯u

∂y



(7.68)

¯u ∂ ¯T ∂x + ¯v ∂ ¯T ∂y =∂y ∂



(α + εH ) ∂ ¯T ∂y



(7.69)

where the viscous dissipation and energy source terms are neglected A replacement

ofν and α in the laminar flow equations by ν+εMandα+εH, respectively, yields the

governing equations for turbulent flow However,εMandεH are not properties of the fluid but functions of the flow As such, they are not known a priori and experimental

results or various turbulence models are used for approximating them

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Several turbulence models have been developed in recent years and employed for solving various turbulent natural convection flows of practical interest Among these, theκ–ε model, where κ is the turbulence kinetic energy and ε the rate of dissipation

of turbulence energy, has been employed extensively Both these quantities are calcu-lated from differential equations which describe their transport in the flow and which are similar to the vorticity and energy transport equations Launder and Spalding (1972) discuss this turbulence model, and similar models, in detail and also give the relevant constants and functions for these models

Various researchers have employed different turbulence models to simulate com-plex turbulent natural convection flows The algebraic eddy viscosity model has been

a popular choice because of its simplicity and because the empirical constants in higher-order models may not be available for a given flow circumstance A consid-erable amount of work has been done on recirculating turbulent flows in enclosures, such as those due to fire in a room (Abib and Jaluria, 1995) See, for instance, the review paper on such flows by Yang and Lloyd (1985)

Experimental work has also been done on turbulent natural convection flows and heat transfer However, the data available for different flow configurations and cir-cumstances are few Cheesewright (1968) measured velocity and temperature profiles and provided heat transfer data from a heated surface in air In turbulent flow, the gen-eralized temperature profiles were found to remain largely unchanged downstream

This aspect was employed by Cheesewright to determine the end of the transition regime The work of Vliet and Liu (1969) and of Vliet (1969) was directed at vertical and inclined uniform heat flux surfaces The turbulence level, which was defined as

u/ ¯u, was measured and found to be as high as 0.3 The velocity profile was found to

widen with a decrease in the nondimensional velocity, as also observed by Jaluria and Gebhart (1974) Both water and air were employed Vliet and Ross (1975) considered

an inclined surface with a constant heat flux In the Grashof number,g was replaced

byg cos γ, where γ is the angle at which the surface is inclined with the vertical This

is in line with the earlier discussion on inclined surfaces

In several problems of practical interest, the heat transfer and flow processes are so complicated that the analytical and numerical methods discussed earlier cannot be employed easily and one has to depend on experimental data Over the years, a con-siderable amount of information on heat transfer rates for various flow configurations and thermal conditions has been gathered Some of this information has already been presented earlier The present section gives some of the commonly used results for

a few important cases The results included here are only a small fraction of what is available in the literature, and the attempt is only to present useful results in a few common circumstances and to indicate the general features of the empirical relation-ships Unless mentioned otherwise, all fluid properties are to be evaluated at the film temperatureTf = (T w + T∞)/2.

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7.7.1 Vertical Flat Surfaces

To cover the range of Rayleigh number Ra from laminar to turbulent flow, the fol-lowing correlations have been employed over many years for isothermal surfaces (McAdams, 1954; Warner and Arpaci, 1968):

Nu= 0.59Ra

1/4 for104< Ra < 109 (laminar) (7.70) 0.10Ra1/3 for109< Ra < 1013 (turbulent) (7.71)

These equations are applicable for Pr values that are not very far from 1.0 Churchill and Chu (1975a) have recommended the following correlation, which may be applied over the entire range of Ra:

Nu= 0.825 + 0.387Ra1/6

1+ (0.492/Pr)9/168/27

2

(7.72)

For laminar flow, greater accuracy is obtained with the correlation [Churchill and Chu (1975a)]

Nu= 0.68 +  0.67Ra1/4

1+ (0.492/Pr)9/164/9 for 0< Ra
109 (7.73) Here Nu and Ra are both based on the plate heightL The preceding correlations are

generally preferred to the others, since they have the best agreement with experimen-tal data The local Nusselt numbers Nuxcan be obtained from the results given in the preceding references

For the uniform heat flux case, the heat transfer results given by Vliet and Liu (1969) in water yield the following relationships For laminar flow,

Nux,q = 0.60(Gr∗

x · Pr)1/5 for 105< Gr∗

x · Pr < 1013 (7.74)

Nuq = 1.25Nu L,q for 105< Gr∗

x · Pr < 1011 (7.75) For turbulent flow,

Nux,q = 0.568(Gr∗

x · Pr)0.22 for 1013< Gr∗

x · Pr < 1016 (7.76)

Nuq = 1.136Nu L,q for 2× 1013< Gr∗

x · Pr < 1016 (7.77) where