Mechanical Engineers Handbook 2011 Part 10 ppt

The maximum value of the velocity results byimposing that 3.152 veri®es 3.151 integrated averaged over the ductcross section, namely, 3.2.4 HEAT TRANSFER IN THE FULLY DEVELOPED REGION Th

where u0is the maximum velocity (in the axis) The problem then reduces to

®nding u0

It is obvious that u…y; z† thus obtained does not identically match thetrue solution to the problem The maximum value of the velocity results byimposing that (3.152) veri®es (3.151) integrated (averaged) over the ductcross section, namely,

3.2.4 HEAT TRANSFER IN THE FULLY DEVELOPED REGION

The key problem of heat transfer in duct ¯ows is to determine the ship between the wall-to-stream temperature drop and its associated heattransfer rate For ¯ow in a circular duct of radius r0 and with averagelongitudinal velocity U , the mass ¯ow rate is _m ˆ rpr2U (Fig 3.20, [10]).The heat transfer rate from the wall to the stream should equal the change inthe enthalpy of the stream To verify this we write the ®rst principle for acontrol volume of length dx,

relation-q002pr0dx ˆ _m…hx‡dxÿ hx†: …3:158†

In the perfect gas limit of the ¯uid …dh ˆ cPdTm†, or in the limit of anincompressible ¯ow and under a negligible pressure variation …dh ˆ cPdTm†,this balance equation implies

ture ®eld in the stream is not uniform Its de®nition results from the ®rstprinciple applied to a stream tube, that is, q002pr0dx ˆ d„ArucPTdA, as

Tmdefˆrc1

PU

1A

For ducts with uniform-temperature walls, this means

Tmˆ 1U

where q00 is positive if the heat transfer is wall $ stream

3.2.5 THE FULLY DEVELOPED TEMPERATURE PROFILE

The heat transfer coef®cient may be evaluated provided the temperature

®eld T …x; y†, and hence the energy boundary value problem for the speci®edboundary conditions, is solved ®rst For example, for the stationary laminar

¯ow in a straight circular duct, the energy equation is

1a

This equation indicates the balance

Axial convection  Radial conduction; Axial conductionwith the following scales:

Ua

q00DrcPU

…3:165†Note that en route to this scaling relation we used the relation

hDk

 2

aDU

 2:Longitudinalz‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚}|‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚{Conduction

…3:167†The bottom line to this analysis is that for large PeÂclet numbers, PeD, theaxial conduction heat transfer may be negligibly small, and the energyequation becomes

u…r †a

Furthermore, from the balance

Axial convection  Radial conduction;

it follows that the Nusselt number is a constant of order 1,

At this point it is important to note that, in the literature, the fullydeveloped temperature pro®le is de®ned through

r ˆr 0

Twÿ Tm 1: …3:171†

Consequently, the variation of @T =@r jr ˆr0with respect to x is identical to that

of Tw…x† ÿ Tm…x†, and since @T =@r is a function of x and r , it follows that

Here f1, f2, and f3 are functions of r =r0 and x

3.2.6 DUCTS WITH UNIFORM HEAT FLUX WALLS

When q00 is not a function of x, the ordinary differential equation (3.168)admits an analytical solution,

Consequently, the temperature at a particular location is a linear function of

x, and its slope is proportional to q00(Fig 3.21), after [10] On the other hand,the r -variation of T , respectively f…r =r0†, may be found by solving theenergy equation for the thermally fully developed ¯ow Using (3.168), thetemperature pro®le (3.174) and the Hagen±Poiseuille velocity pro®le (3.144)yield

with NuDˆ hD=k now emerging explicitly Integrating this equation twiceand using the boundary condition f0jr*ˆ0ˆ finite results in

f…r *; NuD† ˆ C2ÿ 2NuD r *

2

 2

ÿ r24

1 ˆ 4NuD

…10

…1 ÿ r2†r *dr * ˆ1148NuD; …3:180†which means that the Nusselt number for the thermally fully developedHagen±Poiseuille region is

NuDˆ48

This result is in good agreement with the scale analysis (3.169)

Table 3.3 gives the Nusselt number, NuDˆ hDh=k, for different types ofducts, obtained by integrating the energy equation

ua

3.2.7 DUCTS WITH ISOTHERMAL WALLS

Figure 3.22, after [10], shows a qualitative sketch of the temperature pro®lefor a duct with an isothermal, Tw, wall If the average temperature of the ¯ow

at coordinate x1in the fully developed region is T1, then the heat transfer is

driven by the temperature difference Twÿ T1where the ¯ow temperature isincreasing downstream, monotonously Consequently, the heat ¯ux

q00…x† ˆ h‰Twÿ Tm…x†Š …3:184†

exhibits the same trend On the other hand, by scale analysis (3.169), h wasshown to be constant Now, eliminating q00between (3.184) and (3.159) andthen integrating it yields

which, if we observe that the sign of f…r *† is reversed, is similar to (3.176)

The corresponding boundary conditions may be

If we apply the heat transfer coef®cient de®nition (3.162), the Nusseltnumber then results as

3.2.8 HEAT TRANSFER IN THE ENTRANCE REGION

The previous results are valid for laminar internal forced ¯ow, when bothvelocity and temperature are fully developed, that is, for x > maxfX ; XTg.The length XT is that particular value of the x-coordinate where dT reachesthe value of the hydraulic diameter

The scale analysis may be used to produce order of magnitude estimates,and Fig 3.23a shows, qualitatively, the in¯uence of Pr on the scaling

dT…XT†  Dh As seen previously, the ratio d=dT increases monotonouslywith Pr; hence, X =XT has to vary conversely

Figure 3.23

The internal

forcedconvection heat

For ¯uids with Pr  1, in virtue of (3.71), dT grows faster than d,

Dh(Fig 3.23) Hence, in the temperature boundary layer the scale of u is U ,and it may be shown that dT…x†  xReÿ1=2x Prÿ1=2, that is, a result that isidentical to the one obtained for ¯uids with Pr  1 The scaling relations(3.193) and (3.191) lead to

XT

a valuable, general conclusion that is valid for any Pr

The local Nusselt number in the thermally developing region …x  XT†scales as

NuD ˆhDh

q00DT

¯uid layer is entrained into a slow, upward motion that, provided the ¯uid

reservoir is large enough, does not perturb the ¯uid away from the wall.Because the hydrostatic pressure in the stagnant ¯uid reservoir decreaseswith altitude, a control volume of ¯uid conveyed in this ascending motion Ð

in fact, a wall jet Ð expands while traveling upward

By virtue of the mass conservation principle (the reservoir contains a

®nite amount of ¯uid), the ¯uid control volume will eventually return to thebottom of the warm wall, entrained by a descending stream In this closingsequence of its travel the ¯uid control volume is cooled and compressed (itsdensity is increased) by the increasing hydrostatic pressure

Summarizing, the ¯uid control volume may be seen as a system thatundergoes a cyclic sequence of heating, expansion, cooling, and compres-sion processes, which is in fact the classical thermodynamic work-producingcycle (Fig 3.24) [2] Here, unlike the classical thermodynamic cycles, theheating and expansion, on one hand, and the cooling and compression, onthe other hand, are (respectively) simultaneous processes Ð neither atconstant volume (the control volume expands=compresses while heatingup=cooling down) nor isobaric (the hydrostatic pressure varies continuouslywith the altitude) The work potential produced by this cycle is ``consumed''through internal friction between the ¯uid layers, which are in relativemotion

In natural convection heat is transferred from the heat source (e.g., thewarm, vertical wall) to the adjacent ¯uid layer by thermal diffusion, then byconvection and diffusion within the ¯uid reservoir When the ¯uid reservoirthat freely convects the heat is external to the heat source, the convectionheat transfer is called external

3.3.1 THE THERMAL BOUNDARY LAYER

The ¯uid region where the temperature ®eld varies from the wall ture to the reservoir temperature and where, in fact, motion exists is calledthe thermal (temperature) boundary layer (Fig 3.25)

tempera-The central object of the thermal analysis is, again, to evaluate the heattransferred from the wall to the reservoir, and the bottom line to it is ®ndingthe heat transfer coef®cient

hydefˆ qw;y00

Twÿ T0ˆ ÿ

k@T@x

xˆ0

and the energy equation is

The temperature gradient at the wall scales as

@T

@x

xˆ0DTd

T ; where DT ˆ Twÿ T0; …3:200†and, since the thermal boundary layer is a slender region, it makes sense toassume that

This means that, in the governing equations, the second-order derivativeswith respect to y may be neglected with respect to the second-orderderivatives with respect to x:

Another important feature revealed by the scaling is that pressure doesnot vary signi®cantly across the dT region, that is,

P…x; y† ' P…y† ˆ P0…y†: …3:202†Furthermore, if we observe that the pressure distribution in the reservoir

is essentially hydrostatic, the pressure gradient in the y-direction may bereplaced by

These form the reduced set of boundary layer equations

As shown by the thermodynamic equation of state r ˆ r…T ; P†, thedensity of the ¯uid is a function of temperature and pressure Consequently,

by Taylor expansion, it follows that

r ' r0‡ …T ÿ T0†@r

@T

... betterrepresented by the condition Gry  10< small>9 for all ¯uids within the range

10< small>ÿ3< Pr < ;10< small>3

1 The velocity and... (Fig 3.26) Traditionally [4], the threshold limit is

Ray 10< small>9, regardless of the particular Pr number of the working ¯uid Thiscriterion, which does not depend... hexagonal shape [2] Forenclosures that are not shallow, depending on the particular length toheight aspect ratio, this threshold may depart from the critical value (3.263)and the ¯ow structure may be