Heat Transfer Handbook part 54 pps

Conjugate Heat Transfer from an Isolated Heat Source in a Plane Wall, in Fundamentals of Forced Convection Heat Transfer, ASME-HTD-210, M.. The Effect of Variations in Stream-wise Spacin

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turbulent

x x-coordinate direction

y y-coordinate direction

z z-coordinate direction

Greek Letter Subscripts

∆p pressure loss

δ thickness of boundary layer, dimensionless

Superscripts

+ normalized variable

∗ normalized variable

 first derivative

 second derivative

 third derivative

Other

∂ partial derivative

∇ vectoroperator

REFERENCES

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Churchill, S W., and Bernstein, M (1977) A Correlating Equation for Forced Convection from

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Surface-Mounted Block to Forced Convective Air Flow in a Channel, J Heat Transfer, 118, 301–

309

Nakayama, W., Matsushima, H., and Goel, P (1988) Forced Convective Heat Transfer from

Arrays of Finned Packages, in Cooling Technology for Electronic Equipment, W Aung,

ed., Hemisphere Publishing, New York, pp 195–210

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in Air Cooling Technology for Electronic Equipment, S J Kim and S W Lee, eds., CRC

Press, Boca Raton, FL, pp 103–171

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Heat Source on a Plane Conducting Surface: A Benchmark Experiment, in Heat Transfer

in Electronic Systems, ASME-HTD-292, ASME, New York, pp 25–36.

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987–990

Roeller, P T., Stevens, J., and Webb, B (1991) Heat Transfer and Turbulent Flow

Characteris-tics of Isolated Three-Dimensional Protrusions in Channels, J Heat Transfer, 113, 597–603.

Smith, A G., and Spalding, D B (1958) Heat Transferin a LaminarBoundary Layerwith

Constant Fluid Properties and Constant Wall Temperature, J R Aeronaut Soc., 62, 60–64.

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AIChE J., 18, 361–371.

Wirtz, R A., and Dykshoorn, P (1984) Heat Transfer from Arrays of Flat Packs in Channel

Flow, Proc 4th International Electronics Packaging Society Conferences, New York, pp.

318–326

Wirtz, R A., Sohal, R., and Wang, H (1997) Thermal Performance of Pin-Fin Fan-Sink

Assemblies, J Electron Packag., 119, 26–31.

Womac, D J., Ramadhyani, S., and Incropera, F P (1993) Correlating Equations for

Impinge-ment Cooling of Small Heat Sources with Single Circular Liquid Jets, J Heat Transfer, 115,

106–115

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J P Hartnett and T F Irvine, eds., Vol 8, Academic Press, New York

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S Kakac¸, R K Shah, and W Aung, eds., Wiley, New York

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CHAPTER 7 Natural Convection

YOGESH JALURIA Mechanical and Aerospace Engineering Department Rutgers University

New Brunswick, New Jersey

7.1 Introduction 7.2 Basic mechanisms and governing equations 7.2.1 Governing equations

7.2.2 Common approximations 7.2.3 Dimensionless parameters 7.3 Laminar natural convection flow over flat surfaces 7.3.1 Vertical surfaces

7.3.2 Inclined and horizontal surfaces 7.4 External laminar natural convection flow in other circumstances 7.4.1 Horizontal cylinder and sphere

7.4.2 Vertical cylinder 7.4.3 Transients 7.4.4 Plumes, wakes, and other free boundary flows 7.5 Internal natural convection

7.5.1 Rectangular enclosures 7.5.2 Other configurations 7.6 Turbulent flow

7.6.1 Transition from laminar flow to turbulent flow 7.6.2 Turbulence

7.7 Empirical correlations 7.7.1 Vertical flat surfaces 7.7.2 Inclined and horizontal flat surfaces 7.7.3 Cylinders and spheres

7.7.4 Enclosures 7.8 Summary Nomenclature References

The convective mode of heat transfer involves fluid flow along with conduction, or diffusion, and is generally divided into two basic processes If the motion of the

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fluid arises from an external agent, for instance, a fan, a blower, the wind, or the motion of the heated object itself, which imparts the pressure to drive the flow, the

process is termed forced convection If, on the other hand, no such externally induced

flow exists and the flow arises “naturally” from the effect of a density difference, resulting from a temperature or concentration difference in a body force field such

as gravity, the process is termed natural convection The density difference gives rise

to buoyancy forces due to which the flow is generated A heated body cooling in ambient air generates such a flow in the region surrounding it The buoyant flow arising from heat or material rejection to the atmosphere, heating and cooling of rooms and buildings, recirculating flow driven by temperature and salinity differences

in oceans, and flows generated by fires are other examples of natural convection

There has been growing interest in buoyancy-induced flows and the associated heat and mass transfer over the past three decades, because of the importance of these flows in many different areas, such as cooling of electronic equipment, pollution, materials processing, energy systems, and safety in thermal processes Several books, reviews, and conference proceedings may be consulted for detailed presentations on this subject See, for instance, the books by Turner (1973), Jaluria (1980), Kakac¸ et

al (1985), and Gebhart et al (1988)

The main difference between natural and forced convection lies in the mechanism

by which flow is generated In forced convection, externally imposed flow is generally known, whereas in natural convection it results from an interaction of the density difference with the gravitational (or some other body force) field and is therefore inevitably linked with and dependent on the temperature and/or concentration fields

Thus, the motion that arises is not known at the onset and has to be determined from

a consideration of the heat and mass transfer process which are coupled with fluid flow mechanisms Also, velocities and the pressure differences in natural convection are usually much smaller than those in forced convection

The preceding differences between natural and forced convection make the ana-lytical and experimental study of processes involving natural convection much more complicated than those involving forced convection Special techniques and methods have therefore been devised to study the former, with a view to providing information

on the flow and on the heat and mass transfer rates

To understand the physical nature of natural convection transport, let us consider the heat transfer from a heated vertical surface placed in an extensive quiescent medium at a uniform temperature, as shown in Fig 7.1 If the plate surface tem-perature T w is greater than the ambient temperature T∞, the fluid adjacent to the vertical surface gets heated, becomes lighter (assuming that it expands on heating), and rises Fluid from the neighboring areas moves in, due to the generated pressure differences, to take the place of this rising fluid Most fluids expand on heating, result-ing in a decrease in density as the temperature increases, a notable exception beresult-ing water between 0 and 4°C If the vertical surface is initially at temperatureT∞, and then, at a given instant, heat is turned on, say through an electric current, the flow undergoes a transient before the flow shown is achieved It is the analysis and study

of this time-dependent as well as steady flow that yields the desired information on the heat transfer rates, flow and temperature fields, and other relevant process variables

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w

x

x

Flow

Flow

Velocity boundary layer

Entrainment

Entrainment

T⬁, ␳⬁

T⬁, ␳⬁

T w>T⬁

T w<T⬁ y

y

Body force (gravity field)

Figure 7.1 Natural convection flow over a vertical surface, together with the coordinate system

The flow adjacent to a cooled surface is downward, as shown in Fig 7.1b, provided

that the fluid density decreases with an increase in temperature

Heat transfer from the vertical surface may be expressed in terms of the commonly used Newton’s law of cooling, which gives the relationship between the heat transfer rateq and the temperature difference between the surface and the ambient as

where ¯h is the average convective heat transfer coefficient and A is the total area of the

vertical surface The coefficient ¯h depends on the flow configuration, fluid properties,

dimensions of the heated surface, and generally also on the temperature difference, because of which the dependence ofq on T w −T∞is not linear Since the fluid motion becomes zero at the surface due to the no-slip condition, which is generally assumed

to apply, the heat transfer from the heated surface to the fluid in its immediate vicinity

is by conduction It is therefore given by Fourier’s law as

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q = −kA

∂T

∂y

 0

(7.2)

Here the temperature gradient is evaluated at the surface,y = 0, in the fluid and k is

the thermal conductivity of the fluid From this equation it is obvious that the natural convection flow largely affects the temperature gradient at the surface, since the remaining parameters remain essentially unaltered The analysis is therefore directed

at determining this gradient, which in turn depends on the nature and characteristics

of the flow, temperature field, and fluid properties

The heat transfer coefficient ¯h represents an integrated value for the heat transfer

rate over the entire surface, since, in general, the local valueh xwould vary with the vertical distance from the leading edge,x = 0, of the vertical surface The local heat

transfer coefficienth x is defined by the equation

q= hx (T w − T∞) (7.3) whereqis the rate of heat transfer per unit area per unit time at a locationx, where the

surface temperature difference isT w − T∞, which may itself be a function ofx The

average heat transfer coefficient ¯h is obtained from eq (7.3) through integration over

the entire surface area Both ¯h and h xare generally given in terms of a nondimensional

parameter called the Nusselt number Nu Again, an overall (or average) value Nu, and

a local value Nux, may be defined as

Nu= ¯hL

h x x

where L is the height of the vertical surface and thus represents a characteristic

dimension

The fluid far from the vertical surface is stationary, since an extensive medium is considered The fluid next to the surface is also stationary, due to the no-slip condition

Therefore, flow exists in a layer adjacent to the surface, with zero vertical velocity on either side, as shown in Fig 7.2 A small normal velocity component does exist at the edge of this layer, due to entrainment into the flow The temperature varies fromT w

toT∞ Therefore, the maximum vertical velocity occurs at some distance away from the surface Its exact location and magnitude have to be determined through analysis

or experimentation

The flow near the bottom or leading edge of the surface is laminar, as indicated

by a well-ordered and well-layered flow, with no significant disturbance However,

as the flow proceeds vertically upward or downstream, the flow gets more and more disorderly and disturbed, because of flow instability, eventually becoming chaotic

and random, a condition termed turbulent flow The region between the laminar and turbulent flow regimes is termed the transition region Its location and extent

depend on several variables, such as the temperature of the surface, the fluid, and the nature and magnitude of external disturbances in the vicinity of the flow Most of the processes encountered in nature are generally turbulent However, flows in many

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Figure 7.2 Velocity and temperature distributions in natural convection flow over a vertical surface

industrial applications, such as those in electronic systems, are often in the laminar

or transition regime A determination of the regime of the flow and its effect on the flow parameters and heat transfer rates is therefore important

Natural convection flow may also arise in enclosed regions This flow, which is

generally termed internal natural convection, is different in many ways from the

ex-ternal natural convection considered in the preceding discussion on a vertical heated

surface immersed in an extensive, quiescent, isothermal medium Buoyancy-induced flows in rooms, transport in complete or partial enclosures containing electronic equipment, flows in enclosed water bodies, and flows in the liquid melts of solidify-ing materials are examples of internal natural convection In this chapter we discuss both external and internal natural convection for a variety of flow configurations and circumstances

7.2.1 Governing Equations

The governing equations for a convective heat transfer process are obtained by con-siderations of mass and energy conservation and of the balance between the rate of momentum change and applied forces These equations may be written, for constant viscosityµ and zero bulk viscosity, as (Gebhart et al., 1988)

Dρ

Dt =

∂ρ

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ρDV

Dt = ρ

∂V

∂t + V · ∇V



= F − ∇p + µ∇2V+µ

ρc p DT

Dt = ρc p

∂T

∂t + V · ∇T



= ∇ · (k∇T ) + q+ βT Dp

Dt + µΦv (7.7)

where V is the velocity vector,T the local temperature, t the time, F the body force

per unit volume, c p the specific heat at constant pressure,p the static pressure, ρ

the fluid density,β the coefficient of thermal expansion of the fluid, Φv the viscous dissipation (which is the irreversible part of the energy transfer due to viscous forces), andqthe energy generation per unit volume The coefficient of thermal expansion

β = −(1/ρ)(∂ρ/∂T ) p, where the subscriptp denotes constant pressure For a perfect

gas,β = 1/T , where T is the absolute temperature The total, or particle, derivative

D/Dt may be expressed in terms of local derivative as ∂/∂t + V · ∇.

As mentioned earlier, in natural convection flows, the basic driving force arises from the temperature (or concentration) field The temperature variation causes a difference in density, which then results in a buoyancy force due to the presence

of the body force field For a gravitational field, the body force F = ρg, where g

is the gravitational acceleration Therefore, it is the variation ofρ with temperature

that gives rise to the flow The temperature field is linked with the flow, and all the preceding conservation equations are coupled through variation in the densityρ

Therefore, these equations have to be solved simultaneously to determine the velocity, pressure, and temperature distributions in space and in time Due to this complexity

in the analysis of the flow, several simplifying assumptions and approximations are generally made to solve natural convection flows

In the momentum equation, the local static pressurep may be broken down into

two terms: one,p a, due to the hydrostatic pressure, and other other,p d, the dynamic pressure due to the motion of the fluid (i.e., p = p a + p d) The former pressure component, coupled with the body force acting on the fluid, constitutes the buoyancy force that is driving mechanism for the flow Ifρ∞is the density of the fluid in the ambient medium, we may write the buoyancy term as

F− ∇p = (ρg − ∇p a ) − ∇p d = (ρg − ρ∞g) − ∇pd = (ρ − ρ∞)g − ∇p d (7.8)

If g is downward and thex direction is upward (i.e., g = −ig, where i is the unit

vector in thex direction and g is the magnitude of the gravitational acceleration, as

is generally the case for vertical buoyant flows), then

and the buoyancy term appears only in thex-direction momentum equation

There-fore, the resulting governing equations for natural convection are the continuity equa-tion, eq (7.5), the energy equaequa-tion, eq (7.7), and the momentum equaequa-tion, which becomes

ρDV Dt = (ρ − ρ∞)g − ∇p d+ µ∇2V+µ

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7.2.2 Common Approximations

The governing equations for natural convection flow are coupled, elliptic, partial differential equations and are therefore of considerable complexity Another problem

in obtaining a solution to these equations lies in the inevitable variation of the density

ρ with temperature or concentration Several approximations are generally made to

simplify these equations Two of the most important among these are the Boussinesq and the boundary layer approximations

The Boussinesq approximations involve two aspects First, the density variation

in the continuity equation is neglected Thus, the continuity equation, eq (7.5), be-comes ∇ · V = 0 Second, the density difference, which causes the flow, is

ap-proximated as a pure temperature or concentration effect (i.e., the effect of pressure

on the density is neglected) In fact, the density difference is estimated for thermal buoyancy as

These approximations are employed very extensively for natural convection An im-portant condition for the validity of these approximations is thatβ(T − T∞) 1

(Jaluria, 1980) Therefore, the approximations are valid for small temperature dif-ferences ifβ is essentially unchanged However, they are not valid near the density

maximum of water at 4°C, whereβ is zero and changes sign as the temperature varies

across this value (Gebhart, 1979) Similarly, for large temperature differences en-countered in fire and combustion systems, these approximations are generally not applicable

Another approximation made in the governing equations is the extensively em-ployed boundary layer assumption The basic concepts involved in using the bound-ary layer approximation in natural convection flows are very similar to those in forced flow The main difference lies in the fact that the pressure in the region outside the boundary layer is hydrostatic instead of being the externally imposed pressure, as is the case in forced convection The velocity outside the layer is only the entrainment velocity due to the motion pressure and is not an imposed free stream velocity How-ever, the basic treatment and analysis are quite similar It is assumed that the flow and the energy, or mass, transfer, from which it arises, are restricted predominantly

to a thin region close to the surface Several experimental studies have corroborated this assumption As a consequence, the gradients along the surface are assumed to be much smaller than those normal to it

The main consequences of the boundary layer approximations are that the down-stream diffusion terms in the momentum and energy equations are neglected in com-parison with the normal diffusion terms The normal momentum balance is neglected since it is found to be of negligible importance compared to the downstream balance

Also, the velocity and thermal boundary layer thicknesses,δ and δT, respectively, are given by the order-of-magnitude expressions

δ

L = O

1

Gr1/4



(7.12)