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'-NATIONAL ADVISORY COMMITTEEFOR AERONAUTICS

TECHNICAL NOTE 3016

ANALYSIS OF TURBULENT HEAT TRANSFER AND FLOW

IN THE ENTRANCE REGIONS OF SMOOTH PASSAGES

By Robert G. Deissler

Lewis Flight Propulsion LaboratoryCleveland, Ohio

Washington

October 1953

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

TECHNICAL NOTE 3016

ANALYSIS OF TURBULENT HEAT TRANSFER AND FLOW IN THE

ENTRANCE REGIONS OF SMOOTH PASSAGES

4By Robert G. Deissler

SUMMARY

A previous analysis for fully developed turbulent heat transfer andflow with variable fluid properties is extended and applied to theentrance regions of smooth tubes and parallel flat plates. Integralheat-transfer and momentum equations are used for calculating the thick-nesses of the thermal and flow boundary layers. The effect of variableproperties is determined for the case of uniform heat flux, uniforminitial temperature distribution, and fully developed velocity distri-bution. A number of other cases in which the fluid properties are con-

-sidered constant are analyzed. The predicted Nusselt numbers for airwith a uniform wall temperature and uniform initial temperature andvelocity distributions agree closely with experimentally determinedvalues.

INTRODUCTION

Flow and heat transfer in the entrance regions of channels havebeen subjects of considerable interest. The laminar flow case hasbeen analyzed in references 1 (pp. 299-310), 2, and 3 (pp. 451-464),for instance, and the turbulent case, in references 4 and 5. In theseanalyses constant fluid properties were assumed, and uniform walltemperature was postulated in the cases in which heat transfer was con-sidered. In the turbulent flow cases the flow was assumed to be tur-bulent at all points along the passage.

The analysis in reference 4 is for a Prandtl number of 1 and Isbased on an assumed 1/7-power velocity profile and the Blaoius rei:-tance formula. The relative changes in heat-transfer coefficient alongthe tube predicted in that analysis are approximately correct, but theabsolute values of the heat-transfer coefficient at the end of theentrance region are about 20 percent too low. The analysis given inreference 5 is applicable to cases in which the heat transferred byturbulence can be neglected, as might occur at very low Prandtl numbers.

2 NACA TN 3016

The analysis given herein, which was made at the NACA Lewis labora-tory, is an extension of the analysis given in reference 6 for fullydeveloped heat transfer and flow with variable fluid properties tothe entrance regions of smooth passages. In calculating the growth ofthe thermal and flow boundary layers in the entrance region, integralequations for heat transfer and momentum are used. For obtaining thevelocity and temperature distributions in the bounidary layers, the sameassumptions are made for solving the turbulent transfer equationias were made in reference 6.

U)CD

The flow is assumed to be turbulent at all points along the

passage, as in previous analyses, so that the analysis should be

applicablu when disturbances occurring at the entranco are sufficientto produce a turbulent boundary layer. According to Prandtl's assump-tion, which is in reasonable agreement with experiment, in the presenceof a laminar boundary layer near the entrance, the turbulent portion ofthe boundary layer behaves as th u.Th >e bundary.° layer wero turbulentall the way from the entrance (ref. 7, p. 74). The present calculations

may therefore be applicable to the turbulent portion of the boundarylayer even when a laminar boundary layer exists near the entrance.

The cases considered in the present analysis are given in thefollowing table:

Phenomenon Wall Initial Initial Passage Prandtl Proper-boundary temperature velocity number tiencondition distribution distribution

Heat Uniform Uniform Fully Tube 0.73 Variabletransfer heat flux developed

Heat Uniform Uniform Fully Parallel .73 Variabletransfer heat fluxl developed flat

platesHeat Uniform !Uniform Fully Tube .73 Constanttransfer wall tem- developed

peratureHeat Uniform Uniform Uniform Tube .73 Constanttransfer heat flux

Heat Uniform Uniform Uniform Parallel .73 Constanttransfer heat flux flat

pLates

Heat Uniform Uniform Uniform Tube .73 Constanttransfer wall tem-

peratureHeat Uniform Uniform Fully Tube .01 Constanttransfer heat flux developed

Friction --------- ------------ Uniform Tube ... ConstantFriction --------------------- Uniform Parallel Constant

flat___ ___ ___ ___ ___ _ _ ___ ___ __ ___ ___ ___ Plates ____

NACA TN 3016 3

Although most of-the cases in the preceding table are for constant fluidproperties, the greater portion of the report is concerned with thefirst two cases, in which the properties are variable.

ANALYS IS

For calculating hunt transfer and friction in the entrance regionof ducts, the usual boundary layer assumptions are made; that is, it isCOassumed that the effects of heat transfer and friction are confined tofluid layers close to the surface (thermal and flow boundary layers,respectively). The temperature and velocity distributions outside theboundary layers are asounied uniform, and the total temperature and totalpressure are ccnstant along the length of the duct for the region outsidethe boundary layers. More exact analyses (ref. 2) indicate that theseassumptions are valid, even for laminar flow, except in the region at adistance from the entrance where the bzundary layer fills a large portionof the tube. For that region, however, the Nusselt numbers and friction

- factors-have valuus very close to the values for fully developed flow.0

In the following analysis, flow in round tubes and between parallelflat plates will be considered. The effuct of frictional heating will

0 be neglected and then later investigated to some extent in the sectionEffect of frictiLnal heating on entrance length.

Velocity and Temperature Distributions in Boundary Layers for Gases

For obtaining the velocities and temperatures in the flow andthermal boundary layeri, the differential equations for shear stressand heat tranofer can be written as follows:

du du-- + dy

q kdt PCpg dtdy - ceh dy

(The symbols used in this report are defined in appendix A.) In thepreceding equations e and Eh are the eddy diffusivities for momentumand heat transfer, respectively, the values for which are dependent on

the amount and kind-of turbulent mixing at a point. When written indimensionless form, these equations become

_t _ du+ P e du+0 O dy + O 70P d,+U -

4 NACA TN 3016

and

C 1 P 0 dt5

where the subscript 0 refers to values at the wall.

Assumptions. - The following assumptions are made in the use of theprevious equations for obtaining velocity and temperature distributionsCin the flow and thermal boundary layers:

1. The eddy diffusivities for momentum & and heat transfer chare equal. Previous analyses for fully developed flow in tubes based onthis assumption yielded heat-transfer coefficients and friction factorsthat agree with experiment for Reynolds numbers above 15,000 (ref. 6).At low Reynolds or Peclet numbers (Pe = RexPr), the ratio of eddy

_ diffusivities appears to be a function of P-eclet number_(ref_ 8); butfor Reynolds numbers above 15,000, it is nearly constant for gases, atleast for fully developed boundary layers. The same assumption is madehere for developing boundary layers.

2. The eddy diffusivity & is given by

t = n2uy

in the region close to the wall (y+< 26), and by the Karman relation

X2 (dUjdy)i 22

(d 2u/dy 2 )

in the region at a distance from the wall (y+ >26). These expressionshave been experimentally-verified for fully developed boundary layerswith variable properties in reference 6. It is assumed here that theyapply also to developing boundary layers. The quantities n and x areexperimental constants-having the values 0.109 and 0.36, respectively.

3. The variations across the flow and thermal boundary layers ofthe shear stress r and the heat transfer per unit area q have anegligible effect on the velocity and temperature distributions. It isshown in figure 12 of reference 6 that the assumption of a linear varia-tion of shear stress and heat transfer across the boundary layers(- or q = 0 at the edge of the thermal or flow boundary layer) givesvery nearly the same velocity and temperature profiles as those obtainedby assuming uniform shear stress and heat transfer across the boundary

1

NACA TN 3016

layers for values of b+ or 5h+ between 500 and 5000.1 For small

values of bh+, such as occur very near the entrance, the effect of

variable heat transfer will be checked in appendix B and in the sectionEffect of variation of heat transfer across boundary layer on Nusseltnumber for small 5h+ and constant fluid properties.

4. The molecular shear stress and heat-transfer terms in theequations can be neglected in the region at a distance from the wall

C(ref. 6, fig. 14).

5. The static pressure can be considered constant across thepassage.

6. The Prandtl number and spucific heat can be considered constantwith temperature variation. The variations with temperature of thespecific heat and Prandtl number of gases are of a lower order of magni-tude than the variations of viscosity, thermal conductivity, anddensity. The variations of viscosity and thermal conductivity withtemperature are assumed to be given by the equations

p/p0-= k/k0 ' (t/to)0 "68

From the definitions of the quantities involved it can be shown that

t/t0 = 1 - Pt +

Solution of equations. - The details of thi solution of the equa-tions for shear stress and heat transfer under the foregoing assumptionsare given in reference 6. The resulting relations between t+ and y+and u+ and y+ are given in figures 1 and 2. Positive values ofcorrespond to heat addition to the gas; negative values, to heatextraction. For the cases in which the fluid properties are consideredconstant, the curves for 0 = 0 are used. 2 The plotted values of t+

and u+ are to be used in the thermal and flow boundary layers, thatis, for values of y+< 5h+ or y+< 6+. For values of y+ greater

1Figure 12 of reference 6 applies to developing boundary layers ifthe symbols r 0 + in the figure are replaced by 6

+ or 5h

2The case in which 3 = 0 is a limiting one which can be approachedas closely as desired by making the temperature difference small. Inthe limit, where the temperature difference is zero or infinitesimal, theproperties must be constant.

6 NACA TN 3016

than these values, t+ and u+ are constant and have the values at b

+

and b + . Thu relations between 6h + and X/D and 6 + and X!D will

be obtainud in the following sections.

Devulopment of Thermal Boundary Layers

Heat-flow equations for thermal boundary layers. - The heat-flowuquation for the thermal boundary layer in a round tube can be obtainedfrom the following diagram:

FlowX

" "" 7 dV qC

Enurgy flows into the differential annulus by convection through plane 1and leaves through plane 2. In addition to this energy there is a radialflow of energy (hert) from the tube surface by conduction and a radial

flow by convection at bh. No radial heat flow due to temperature

gradient takes place at bh because the temperature gradient is, by

definition of the thermal boundary layer, zero at the edge of theboundary layer.,

Equating the heat energy entering the annulus to that leavinggives, for constant-ch

q0 29 r 0 dX + tPu cp g 2Ar dy +

f hP Ad2- Pu 2or dy-r Cp g tb

tPu cp g 2ir d)I

NACA TN 3016 7

where the subscriLpts on the parentheses refer to planes I and 2 in the,UuLrumor

qL) r _ d\X- d c tPu-(ro - Yd

Cri~ii C li dpu(ru, - d],

,herL h ift'.- i .thh. inl-ra1 inlrat chanc, in the X-direction.Equtitiin (i) 1-. thv 1,-- irt.- v-quaticri fir flow in tube : relatinL-z the thermal-

c -n~,'-1&crthik~..to thc Ai.Juno1 alonv, the- tubQ.

An equation cropni tu equuitlon W 0-) be easily dterived forflo'w b,.twut-n fiat plutoo. Tris tqu'mtic~n -is

qV- - d 1 cp " ztPu d: c C th g d 6hPjud~ (2)

Equation (I) n tho lncc rnpreso Itl1 form has been used extensivelyfor calculat in- thu ;ir,-wth of thu thormal bounldary layt'r on a flatplate (r, f. 7, p.

Uniform heat r'luAt, variably fluid propertie:3 (gases). - The totaltumpurature Dutsidu the thermal boundary ]iiyur doe not vary with XInaior'uch ao n3 hoat penotratesa the region outside, the boundary layer.(Thu differunces butwtee-n tela.,-l and static tempurature are small exceptat hlv-h Mach numbers.) For uniform heut flux at the wall, equation (1)can then be intO 7Ut~d tij give

J,)r) ~ (t - ttb) Pu(ro - y)dy(3Equation (3) can be written in dimensionless form as

(tX 1- t+) P, u(r 0 + y +)dy+ (4)

2r0 +

NACA TN 3016

where, from the definitions of the various quantities involved and theperfect gas law (constant pressure),

-Po Z I/( -t + ) (5)

When equation (4) I tL' bU used, values of ro + and must first be

fixed. Values of X/D can then bu calculated for various values ofbh+. The relations between t" and y+ and u + and y+ are obtained

from figurus 1 and 3.

Tie equation for fiat plates corresponding to equation (4) is:)tOiinud from equation (2) and is found to btu

X 1 (t + - t+) L u + dy+ (6)D Pr,+

where D Is the ,quivalent diamuter f-r flow between flat plates, ortwice the platu .1pacin .

Uniform wall tumrpuriture, constant fluid proprtiec. - For flow intubes, equation (1) can be written as

__] -d (t tb) Pu(rD - y)d (7)

where the quantity in bratckets is also thu upper limit on the integral.For constant fluid proptrrtios and uniform wall temperature, equation (7)can be. written in dinri ile.-i f~rm as

d K fbh (th - t)P u(r 0 o y+) y (8)ro ro r,+

where the shear stress TA is allowed to vary with X. But

tb/t 0 ti/t u - tb+

a since tb does not vary with X. Therefore,

= (I/tb+)(l-ti/tO)

NACA TN 3016 9

Substituting this expression for P into equation (8) results in

1+ t t +X 1 + _d u+(ro+-y (9)D + to+ t + t°(

Equation (1,4) givue tho rulatin ,utwo,:i , /T and bh+ for round tubeswith uniform wAll temperature and constant fluid properties. Its useiLl similar to that of ,z'qution (4).

Nusoult numborc and Reo:.ioldo numbur-i. - It can easily be shown(seou ref. )) frorm the definitions of the quantities involved that theNussult number ard Ru-ynulds number (propurtiei variable or constant)f -.r circulAr tubes with thv fluid proportius evaluated at the wall tem-poraturo uru givon Ly

+ Pr

tt

mdut +% r + (I

whou + + +

+ r - t +tb

dyo + tl13t + dy ++u( 0 y)d± 12

* o 1-Pt ~ 1-pt5

++-rt + u + (r o + - + ) t r o

+ dy + + + U+r 0 -y )dv+

t+ r- t0 +(12)

u+(ro - ) i + +,, +

+ y + _t+ Jh + u or y y

and r r o+ y Y

ro++

NACA TN 3016

whic, by combination with uquation (12), becomes

ub + "I (I-!ptb b +( ) +_ dy + + -1+ + +(rO+Y +)dy+

r +

Thi f1ro-oinz_ uquutions uru for a circular tube. The correspondingoquationo fir flow bUttwoun flat phltos are

Nu- = Iro+ + Pr (14)NU- + (4tb

and

Re,) u 4ut+ ro+ (15)

++wh rt° + +d:.+ + ut d

+ f h + (16)+ + 1 +_+ d-;+ + u ,

and

= I +) U-__y+ + U+ d (17)-+

l-t +f dY + +r0-t h /

When values of Nu0 and Re are calculated, values of r0+ and

must be fixed as in uquation (4). Values of Nu0 and Re0 are thus

obtained for given values of X/D and 0.3

3Unfortunately, 0 varies along the passage. It will later be seenthat q1 ' is more convrnient for use as a heat-flux parameter.

NACA TN 3016 Ii

The temperature and velocity distributions from figures 1 and 2 canbe used for those quantities except for the region outside the thermalboundary layer. Thu temperature distribution is, of course, uniformoutside the thurmal boundary layer (t+ = tb+). When the velocity distri-

bution in that region is calculated for the case in which the velocitydistribution is fully developed, it should be remembered that theproperties are constant and have the values at bh+. For values ofh+>-, the velocity distribution curves outside the thermal boundary

Cr on the sexmilogaritlimic plot are straight lines having the slope of the

curves at bh+.

Nunselt numbiro and Reynolds numbers with properties based on tem-peratures othur than the will tumperaturo can be obtained by use of therolut ions

tb/t 0) 1-otb +

t 03 +

tpb/POi /(l-tb

/ kJlki o (1-Ott,+) d

and oo forth, whort, the purfect gas law with constant static pressureacross the tube, constant Prandtl number, and constant specific heathave boen assumed us in reforencu 6. The uxponent d is obtained fromviscosity daita and is found to be about 0.6z8 for air and most gases.The following. relatons- for Reynolds numbers then result:

1Reb = Re 0 d+l (18)i (tb/tO)d~

(tb/to

Rei Rub (ti/to)d (19)

where the continuity relation Pb Ub = Pi ui has been used. Similarly,

Nu0NNub (20)

(tb/to)d

12 NACA TN 3016

andNub

Nui - bd (21)(ti/tbd

Another useful parameter is qj', which can be calculated from

- (tb/tO) (22)

ub (ti/tO)(

Equation (22) can be obtained from the definition of qi' and the

perfect gas law for constant pressure, to/tb = pb/PO, for Pb ub is

constant along the tube. The important property of qi' is that

it does not vary along the length of the tube.

It is desired to calculate the variation of bulk temperature and walltemperature along the tube for uniform heat flux. It can be shown, by useof the expressions qo h(to-tb) and iDXqo pg(t/4)D2 ub cp(tb-t) ,

that dimensionless bulk temperature and wall temperature parameters aregiven by

(tb - t i 1 (23)

and

an to ti) I. Re r 4x(24)tl l Nu I D

Equations (23) and (24) can be used for flow between flat plates as

well as for tubes.

The average Nusselt number for a given X/D is calculated from

X/Dfv X d (X/D) (25)

Nui~avNui

This equation is consistent with the definition

|I

NACA TN 3016 13

X(t0-tb)dX

(to - tb)av- X b (26)

which is the usual way of defining the average difference between walland bulk temperatures for uniform heat flux. For uniform wall temperatureequation (26) is replaced by

0oX []b)/(qO/qa(to - tb)av = X

which gives a Nusselt numbt.r consistent with equation (25). For pointsbeyond the entrance region the Nusselt numbers and temperature distri-butions aro calculated as outlined in appendix C.

Development of Flow Boundary Layer

Mumentum equations for flow boundary layers. - The momentum equa-tion for the fluid in a given space can be written for steady flow invector notation as

F J (Pv . dA) (27)or the force acting on the fluid in a space is equal to the rate offlow of momentum out of the space (ref. 10, p. 233). When the X com-ponent of equation (27) is written for the previously illustrateddifferential annulus (5h replaced by b), there results

dFX d pu2 2iTr d - u d Pu 2or d (

where the differential of the second integral is the radial mass flowinto the annulus at 5. The force dFX is composed of pressure forces

on planes 1 and 2 and a shear force at the wall. There is no shearforce at 5 because the velocity gradient is, by definition of theboundary layer, zero at the edge of the boundary layer. The force dFX

is then equal to

-[tr 0 2 - (r 0 -5)2]dp - 2iT r 0 T0 dX

Substituting for dFX gives, for equation (28),

14 NACA TN 3016

&(2r o-b)dp + 2ro 0 dX -2ub d f Pu(ro-Y)dYJ - 2d Pu2(rO-Y)dY]

(29)

In the regiin outsidu thu boundary layer, the flow is frictionless, sot hat LO

CO

dp - Pub dub (30) O

Substitutlng -lquation (BY) in equation (29), rearranging, and integratingbetween the tubu ertr:nc,- and X give

X (2r-6) d d[ b Pu2(ro)r)dy]ro f

f0To roUj au b d [J ' Pu(roj-Y)dy

'" 0 r(

whore the bcundary-layer thicme:s has been taken as zero at X = 0.Equation (31) is the desrud equation for flow in tubes relating theboundary-luyer thickneso to distance along the tube.

Equations corresponding to equations (29) and (31) can be derivedfor flow betwuun-flat plates. Those uquations are

5 dp + ro dX = u5 d Pu d -df Pu2 d (3)

and /f

S;u 5p+ d Pu dyl 1 1 u dy]

0 'T o'T O 0 0 'T(

(33)

..........

!

NACA TN 3016 15

Equation (32) was first obtained by von Karman and has been used exten-sively in the incompressible form for calculating the growth of theboundary layer for fluids flowing over surfaces (ref. 7, p. 66).

The value of 5 must be known in terms of u to solve equa-

tions (31) and (33). This relation will be obtained in the sectionConservation of mass, constant properties.

Dimensionless form of momentum uquations for constant properties. -(C Equation (31) for tubes c'n be written in dimensionless form for constant

fluid propertiei as

x 1 Ub+ ro+ (o+/rCo+)0 + -A(-i+ o")+ +-rc) 0

++El+ "i (r +_:+ )dy:+ ] dU (rO+- d

u f + d u + dy

+ rL

Ya

t (3,4)

Equation (34) can be verifled by substitutint the definitions of thedimensionloos quantities into thu equation. Equation (34) can be putinto a form more convenient for computiti,n by using the definition forthe differential of a product. Thus

+ + (o-y+) ++ + + (ro+-y+) +r 0 ub dLJf u + (r0 -y+) dY] dr ub u + dy -

0ro + f r,0

b+ (ro+_-y+)

d(ro ub+) u + + dy (35)

Substitution of equation (35) into the last term of equation (34)(the substitution can be more easily made if eq. (34) is first differen-tiated with respect to X) and simplification give

16 NACA TN 3016

x 'I f++ , o ++ +(ro+.Y+)dy d(rO+ u+)Lj \r+ r0 + 2ro

+ 3J

f o (U + - u+)u+(ro+ - y+)dy (36)2r,+

Equation (356) relates X/D to 5* for various values of Re

for tubes. The relation butween u+ and y+ is obtained from the

curve for p 0 in figure 2. The valuu of ro+ for a given Re and

6+ will be obtained in the next section. The corresponding equation

for flow between flat plates is fuund from equation (33) to be

x U + r,) t+ +66 u + d d(ro+ u t+) +

+ +

Ru + dto+ u )-

I Ei[] l + d r + ( + - ')u+ dy (37)( Ub -u 1p u

[11 + fo ~u ~(7

Conservation of mass, constant properties. - In order to solve

equations (36) and (37), u I must be known in terms of Re. This

relation can be obtained from the law of conservation of mass. For a

constant-area duct and constant density, the law of conservation of mass

gives

ui =b (38)

or the bulk wlocty is constant and equal to the uniform initialvelocity. For a tube, equation (38) can be written

Ui = u(r 0 - y)dyro2

S2 6' u E

2 P-5o u(r -y)dy + (r 0 8 2 (39)r 0

2 r02 0

NACA TN 3016 17

Equation (3d) can be writtun in dimensionless form as

4 +u+ (r 0 ~ - 2Re = J - u+(r + - y()dy + 6+) (40)t+ loo+dy

The equation c¢rrosponding to equation (40) for flow between flat platesis

4Re. J 4 u+ dy+ + 4ub+(ro + - b+) (41)Equation (40) Dr (41) :,iv, the rt.,lation between Re and 6+ or u6+

for a given valu,_ ._)f r,

flussvlt numbors, Re nrolds numbers, and friction factors; constantfluid properties. - Nusselt numbers for a developing velocity distribu-ticn as well as a developing temperature distribution can be calculatedfrom _-quations (10) and (12) or (14) and (16) as before. If the assump-tion is made that the thermal- and flow-boundary-layer thicknesses areequal (bh+= 5+ for Pr- l)4, equation (12) for tubes can be simplified,

and f..,r constant fluid properties (p O) becomesSb

4 ++ +C-) b + + + + t6+ ub+t+ u( - Y)dY + r (ro+ - 5h+)

t t 5h+ +(r + + u4+ + h+)2 (42)

u r - Y4 )dy 4 + -n-- (r0 - h

A similar equatic,n cmna be obtained for flow between flat plates.

Reynolds numbers fur flow in tubes and between plates can be calcu-lated from uquations (40) and (41). For constant fluid properties theReynolds number li constant along the length of the tube.

The pressure drop between thu entrance and a given oection, which isrequirud to calculate the friction factor, io obtained by integratinguquation ("O).

Pub" Pui'-P- P (43)

Or, in dimensionless form,

P- 2(ub ro+)2 - I R(2 (44)

4This assumption is used only for determining tb+ for the case where

the initial velocity and temperature distributions are uniform.

18 NACA IN 3016

for tubes, or

p, B(ub+ ro+) T e2 (45)

for flat plates. The pressure-drop parameter p' in these equations isdefined as

(p1 - PD2p ' (46)Z2O

The average friction factor based on thu pressure drop for a given X/D .I

is defined by

- D(pj -p)fayV - "' (47)2XPu

or

fav 2 (X/D)(48)

This equation includuo_ momentum changes caused by velocity profiledevelopment.

The local friction factor basd on the local pressure gradient is

defined by

f= - D(dp/dX) (49)

"Pub

or

f dp'/d(X/D) (50)2 Re

When f is to be found from equation (50), the curve for p' against X/Dcan be differentiated; or the curves;, calculated from equation (36) or (37),

for u + ro+ against X/D can be differentiated and dp'/d(X/D) cal-

culated from equation (44). The latter procedure proved to be the moreaccurate.

The local friction factor based on the local shear stress at thewall and excluding momentum changes due to a developing velocity profileis defined by

2r0 (51)

SUb

NACA TN 3016 19

or

_ 2 (52)+2ub

and, since Re is constant along the tube and equal to 2 ub+ r 0+ ,

8r 0 +of- =8o (53)

Re ('cn

whure the relation betweon re+ and Re for a given 6 + is obtained

from equatin (4,_:) or (41).

RESU-LTS AID DISCUSSION

Heat Transfer with Unif,_,rr Heat Flux, Uniform Initial

o T~emp.ratur, Distribution, Fully Developed Velocity

Distributi',n, and Variabl- Fluid Properties

f-r Ga,--; Prandtl 11umter of 0.73

Dovelopmunt of thermal boundary layer. - Figure 3 illustrates thedevelopment of the temperature profile in a tube for uniform heat fluxand a fully developed velocity profile. The distance along the tubefor various thermal-boundary-layer thic1aiessvo is calculated from equa-tion (4) and the temperature distribution in figure 1. The temperaturedistribution in the thermal boundary layer is obtained from figure 1;the temperature outside the boundary layer is uniform, inasmuch as noheat penetrates that region. The Reynolds numbers are calculated fromequations (11) anid (13). Figure 3 indicates that for a fixed tubediameter at a given dintance from the entrance, the boundary-layerthic1laeso is less for high Reynolds numbers than for low ones, or thedistance required fur a given degree of temperature profile development

is greater for high Reynolds numbers than for low ones.

Although the temperature distributions in figure 3 are for = 0,or constant fluid properties, similar distributions are obtained forvariable fluid properties with large amounts of heat transfer, as maybe inferred from figure 4. The variation of the thermal entrancelength, defined as the length required for the thermal boundary layersfrom the opposite walls to meet at the center of the passage, withReynolds number and heat flux ql' is shown in figure 4. The quantity

qi' was calculated from equation (22). The quantity qi' rather than

is used as a heat-flux parameter inasmuch as it is constant alongthe length of the tube. For the range of qi' shown, to/tb varies

from about 0.4 to 3. Figure 4 indicates that heat flux has but aslight effect on the thermal entrance length.

20 NACA TN 3016

Local Nusselt numbers for tubes. - Local Nusselt numbers hD/k iwith properties ,valuated at the inlet temperature are plotted againstX/D for various values of Rei and qi' (heat flux) in figure S. The

heat-transfer coefficlnt in the Nussult number is based on the differ-once between wall and bulk temperature (,;uL appendix A). Furobtaining the curves Nusselt numbers wore calculated from equations (10),(2:), and (i1); Remolds numbers, from ,uquations (11), (18), and (19);qi', from equation (22); and X/D, from equation (4) for various values

,f thu parameters bh' ro+, and P. Interpolation was necessary forobtaining the vwrition of Nui with X/D for constant values of Re1 r

and qi'.

The curvu In figure ., indicate, as might be expected, that theNuss,lt nu'bero (hut-tran.ifor coefficients) close to the entrance arevery high in oI.ptrison with the fully developed values farther down thetube at tht. ends of tht_ curveu.. Thu -,, v.Ilu2.e.: _re hlLt _ e,-uethe thermal t- undary' layer is thin and the temperature gradients conse-quent ly ire sv_,ru near t.h,.e entrance. At X/D = 0 the boundary-layerthicrmocss bh ID zero, so that the hoat-transfer coefficient is infinite

at the ,ntrance. Fo:r uniform heat flux this means that the temperaturedifference t0 - tb must be zero at X/D = 0; for a finite temperature

difference at X/D = 0 the heit transfer per unit area could not beuniform, since it would be infinite at the entrance.

Ir, -sXti a ,f the curve.s in fiture ' shows that the Nusselt numbersv'ry rxarl ,' :i.h Ih, ir Lull , dev lope1d values long before the thermal

1;unr:iac,"y.':rs iha,:, ,t at th. ,-nds c-,f th,_. curves. For instance, forC 0 (.-:,on.tant fliid r,,_,rtts) nd 100,000, the Nusselt number

Xt ;/D -f 9 is within 2 pricnt ot its fully developed value at X/D.:2 v".-.. Lhis indi,-:t,'s that, f,,r practial purposes, the thermal entrance

ei -%hs ,rin b- c. :nsidered to- be lss than half the values given in2 i ;-r,'

C,Jmpario,:,ri f the curves for qi' 0 0, 0.0u4, and -0.0025 indicates

a decrease in Nul at a ,.,iven X/D for heat addition to the gas and

an increast for heat cxtracticn from the gas. For both heat additionand uxtraction, Nu1 increases with X/D for large values of X/D.

These trends are clearly shown in figure 6, where Nui is plotted forqi' ( i,. _,,4 and -C.10:25 for large values of X/D. For qi' = 0 (constant

properties) the Nusselt numbers are constant beyond the entrance length.The Nunselt number; for values of X/D beyond the entrance region werecaiculated by the method described in appendix C. For heat extractionfrom the gas the curves arc cut at the points shown because the walltemperature reaches absolute zero at those points. The portions of thecurves for which the temperature is below the liquefaction temperatureof the gas should, of course, be disregarded because the properties nolonger have the assumed variations with temperature. The increases in

INACA TN 3016 21

Nussult number with X/D are caused by the variation of properties alongthe tube and across the tube. The increases are due principally to thevariation of to/tb, which decreases along the tube for both heat addition

to and extraction from the gas.

It is clear from figure 6 that fully developed heat transfer, inthe sense that the heat-transfer coefficient becomes independent of X/D,cannot be obtained when the properties are variable. Fully developed heat

N) transfer can, however, be obtained in the sense that the relation betweenCn the Nussult number and Reynolds number becomes independent of X/D for

sufficiently large values of X/D when the properties are evaluated atthe proper reference tempurature, as shown in figure 7. The curves infigure 7 were obtained by cross-plotting the curves for qij = 0 from

figmre. 4. They can be used for qi' / 0 by evaluating the properties,

including density, in the Nusselt number and Reynolds number at thereference temperatures t. = x(t0 - tb) + tb given in figure 8. The

values of x were computed by selecting values of tx such that the

values of Nux and Reo. calculatud from the curves in figure 5, from

equations (18) to (23), and from the definition of tx , fall on the lines

for q = 3 in figure 7. The curves in figure 8 indicate that the

referenci2 temprature for points close to the entrance is slightlyclostr t:. the bulk temperature than the reference temperature for fullydeveloped heat transfor, although its variation with X/D is not large.

Effect of variation :f heat transfer across boundary layer onNuoselt nuwnber fcr small b,+ and constant fluid properties. - For large

values of bh+ (ro+ for fully developed flow), the effect of the radial

variation of heat transfur on temperature distributions was checked in

reference 2 and found to be small. However, for low values of 8h+,

such as occur very close to the entrance, the flow is nearly laminar,so that the temperature profile in the boundary layer is not flat as it

is for higher values of bh+. The temperature distribution in the

thermal boundary layer for laminar heat transfer from a flat plate canbe closely represented by a cubic parabola (ref. 7, p. 89), which meansth,t the heat transfer per unit area varies as 1 - y/5hC, as shown in

appendix B. This variation of heat transfer is used herein for turbulent

heat transfer for s=all 6 h+ in order to obtain the effect of variation

of q/q0 on the Nusselt numbers in the entrance region. The expression

for the temperature distribution with this variation of q/q0 is obtained

in appendix B.

I.

22 NACA TN 3016

The variation of Nu with X/D for variable heat transfer perunit area across the boundary layer (q/qo = I - y2/6h 2 ) is shown in fig-ure 9. For comparison, the curves from figure 5(a) for constant heat

transfer (q/qu = 1) are shown by dotted lines. The curves in figure 9

indicate that the variation of heat transfer per unit area across theboundary layer has a negligible effect on the relation between Nusseltnumber and X/D, so that the use of the temperature distributions infigure 1 appears justified, uvn for low values of bh+

uO

Wall and bulk temperature distributions for tubos. - Wall andbulk temperatures alcng a tube with uniform heat transfer as calculatedfrom equations (23) and (24) are plotted in dimensionless form infigure lo. As mentioned previously, the wall and bulk temperatures areequal at the untrance for uniform heat transfur. The bulk temperaturevaries linearly with X/D inasmuch as the specific heat is assumedconstant. For qij = C the wall and bulk temperature curves are

parallel beyond the entrance region. For both heat addition(qi' = C). uc'4) and heat uxtrrction (q 1 ' - .0025) from ti w gas, thewall and bulk temperatture curves convergu buy-.nd the entrance regionbecause of the increase in het-transfor coefficient with X/D forthose portions of th, curves.

Average Nusselt numbers fcr tubes. - The variation of averageNusselt number Nu,av with X/D a,, calculated from equation (25) is

shown in figre 11. The trends are similar to those for local Nusseltnumbers but the changes are mzro gradual.

The relation between Nux,av and Rex,av for various values of

X/D is given in figure 12. The curves were obtained by cross-plottingthe curves fr- qi' I 0 In figure 11(a) and can be used for qi' / 0

by evaluating the properties in the Nusselt number and Reynolds numberat tx,av - x(t0,av - tb,av) + tbav, where x is given in figure 13.

The values of x were computed by selecting values of txav such that

the values of Nux,av and Rex,av calculated as outlined in appendix C

fall on the lines for qi' = 0 in figure 12. Figure 13 indicates that

the value of x increases continuously with X/D for both heat additionto and extraction from the gas. These trends are indicated experimentallyfor heat addition In reference 11 , whore a constant reference temperatureused for a range of values of X/D was found to give average Nusseltnumbers for high heat fluxes at low values of X/D which were slightlyhigher than those for low heat fluxes. The values of x given infigure 13 indicate that for large values of X/D, x> 1, so that for heataddition tx,av is higher than the average wall temperature although

not necessarily higher than the exit wull teraperature. Tfie variationof x with X/D is caused by the variation of properties along thetube. A more satisfactory correlation would perhaps be obtained by

NACA TN 3016 23

using x only to account for the variation of properties across thetube and using another factor to account for the variation along thetube. The curves in figures 12'and 13 might, however, provide a con-venient means of utilizing the results of the analysis for some purposes.

Nusselt numbers and teuiperature distributions for flow between flatplates. - Nusselt numbers, wall temperature distributions, and bulk tem-purature distributions for flow between flat plates with uniform heattransfer and fully developed velocity distribution are plotted in

0figures 14 to 16. The curves agree closely with the corresponding curvesfor flow in a tube. For instance, the curves for local Nusselt numbersplotted in figuru 14 are between and 5 percent higher in the fully

developed portions of the curves than the corresponding curves for tubesin figure 5. The portions of the curves close to the entrance agreevon m:ru cluseiy. This agreement indicates that results for turbulent

heat transfer in tubes can be used, with small error, for flat plateswhen an uquivalent diameter equal to twice the plate spacing is used.

It should be L.mphasized that in applying the results of the fore-going analysis the sae assumptions regarding the variation of propertieswith temperature should be made as were made in the analysis, that is,constant specific hat, and viscosity and thermal conductivity bothprop:,rtic:nal to to. 6 8

It should ai.ou be mentioned that the results in this analysis donot include the low Peclet number effect described in reference 8 inas-much as the temperature distributions ir figure 1 do not include thateffect. Thu Nusselt numbers at Rei = 10,UL0 may therefore be slightly

high, but the ratios of local to fully developed Nusselt numbers shouldbe accurate.

Effect of frictional heating on entrance length. - The effect offrictional heating on the X/D for a given 5 h+ can be approximately

accounted for by replacing tb+ and t+ by T8+ and T+, respectively,

in equation (4) and setting P/Po = 1/(l - PT+ _au+2) (ref. 9). Inusing that procedure it is assumed that the total-temperature gradient iszero at the edge of the thermal boundary layer, a good assumption forPrandtl numbers close to unity.

The variation of thermal entrance length (X/D)he with Reynolds

number and heat flux with frictional heating (c = 0.00025) is shownin figure 17. Mach numbers for air at the tube center at (X/D)he are

indicated in the figure. The Mach numbers were calculated from

24 NACA TN 3016

which can be obt,,ined from the definitions of M, To, uc , and m. Com-

parison of figure 17 with figure 4 indicates that frictional heatingincrvases thu entrance length, although the increase is not large forsubsonic flow. Innsmuch as Nussolt numbers for fully developed heattransfer are not affected by frictional heating (ref. 9), the effectof frictional heating on the relations between Nusselt number and X/Dprobably is slight when the total rather than the static bulk tempera-ture is used in the dufiniticn of the heat-transfer coefficient.

H(at Transfer with Uniform Wall Temperature, Uniform Initial

Temperature Diotribution, Fully Developed Velocity

Distributicn, and Constant Fluid Properties for

Case; Prandtl Number of C.73

Relationj am,:ng Nusselt number, Reynolds number, and X/D for thecase of tubes, as calculated from equations (9) to (11), are plotted infigure 18. Cmparls-n of figure 18 with figure 5 indicates that theNussult numbers for unif rm watll temperature are very slightly lowerthan those for uniform heat flux. The heat transfer per unit area q0must be infinite at X/D - C) for uniform wall temperature. This occursonly for an infinitusimal distance dX, however, so that a finite amountDf heat is transferred.

Heat Transfer with Uniform Heat Flux, Uniform Initial Velocity and

Temperature Distributions, and Constant Fluid Properties

for Oases; Prandtl Number of 0.73

For calculating the variation of Nusselt number with X/D forthis case the flow-boundary-layer thickness as well as the thermal-boundary-layer thickness should be known for various values of X/D.However, in the calculation of Nusselt numbers, it is consideredsufficiently accurate to assume the thicknesses of the thermal andflow boundary layers to be equal, inasmuch as the Prandtl number isclose to unity.

Relations among Nusselt number, Reynolds number, and X/D fortubes, as calculated from equations (10), (42), (40), and (4), areplotted in figure 19. Comparison of figure 19 with figure 5 indicates

NACA TN 3016 25

that the Nusselt numburo for a uniform initial velocity distribution are

higher than those for a fully developed velocity distribution. This

result might be expected because of the higher friction in the case of

the uniform initial velocity profile.

Average Nusselt numbers as calculated from equation (25) are plottedin figure 20. Figure 20(b), which is a logarithmic plot, compares thepredicted relations between Nuav and X/D with the empirical relation

Nu a(X/D)- 'l. Curve having a slope of -0.1 on the log-log plot agreeclosely with the predicted lines for X/D between 6 and 60. The agree-ment of the data given in reference 11 with the empirical relation forvalues of X/D as hig~h as 120 might be due to the use of a constantrather than a variable ref,,rnc temperature as given in figure 13.

Nus ult numbers for flow between flat plates are given in figure 21.Ccmpairisun of figure 21 with figure 19 indicates very close agreementbetween the Nusselt nu=bri for tubes and flat plates.

Huat Tranofur with Uniform Wall Temperature, Uniform Initial

o Velocity and Temperature Distributions, and Constant Fluid

Properties for Gajus; Prandtl Number of 0.7".;

and Ctmpariscn with Experimental Data

The variatin of Nusselt number with X/D for this case is givenin figuro 22. The curveii agree closely with those for the corres-ponding case for uniform htat flux given in figure 19. The Nusseltnumbers for uniform wall temperature are slightly lower than those foruniform heat flux.

A comparisr:-n between analytical and experimental results for uni-form wall temperature and uniform initial velocity and temperature dis-tributions is given in figure 23. The experimental curves representdata from reference. 12 for air flowing in a tube at uniform temperaturewith a bellmouth entrance and a screen at the entrance and should repre-sent the case analyzed in this section. The screen at the entranceshould insure a turbulent boundary layer throughout the tube. Thefigure shows that the agreement between analytical and experimentalresults for this case is very good.

The lack of a comparison of the present analytical results for auniform wall temperature and a fully developed velocity distributionwith the data from reference 12, which were taken with a long approachsection at the entrance, should perhaps be explained. It appears thatthe sharp edge at the beginning of the long approach section produced

26 NACA TN 3016

large disturbances which were not damped out in the length of approachsection used. The data in reference 12 for heat-transfer coefficientswith a sharp entrance without an approach section indicate such results.For that case the heat-transfer coefficients are higher throughout thetube than those obtained with a bellmouth entrance.

Average Nusselt numbers calculated from equation (25) are plottedin figure 24. As in the case of local Nusselt numbers, the averageNusselt numbers for a unifrm wall temperature in figure 24 are slightlylower than those for a uniform heat flux in figure 20.

Go

Huat Transfer with Uniform Heat Flux, Uniform Initial

Temperature Distribution, Fully Developed Velocity

Distribution, and Constant Fluid Properties for

Liquid Metali; Prandtl Number of O.01

Temperature distributions for fully developed flow calculated asdescribed in reference 8 and plotted in figure 25 were used in thethermal boundary layer for calculating the variation of Nusselt numberwith X/D for liquid metals. The velocity distributions were obtainedfr,-m the curve for 0 K i-i in figure 2. The curves for Nu against:/D in figure 26 wore calculated from equations (4) and (10) to (12)

= = o). An unuCxpected feature of these curves is the slight increaseof Nusselt number with X/D at large values of X/D and Reynolds num-ber. This increase is duo to the change in shape of the temperatureprcfile with bh+ for low Prandtl numbers. Thus, as 5h+ increases,

the tempurature gradient at the wall incre%ses slightly in someinstances because of the added turbulence in the boundary layer athighor values of bh+. This effect is indicated experimentally by

the data in reference 1.

A plot of Nusselt number against Pe/(X/D) for various Pecletnumbers is given in figure 27. If the heat transfer contributed byeddy diffusion were neglected, the curves for various Reynolds numberswould fall essentially on a single line as in reference 5. Examinationof the curves in figure 27 indicates that for Reynolds numbers below1O0 the curves could be represented approximately by a single line.This line is slightly higher than the one obtained in reference 5 inas-much as a uniform wall temperature rather than a uniform heat flux waepostulated in that reference.

NACA TN 3016 27

Friction Factors with Uniform Initial Velocity Distribution

and Constant Fluid Properties

Throe friction factors aru discussed in this section: fr, basedon the shear stress at the wall; f, based on the static-pressure gradient;and f..' based on the pressure drop along the tube. The friction factor

f, is calculatud from uquation (5,); f, f-ot-un equation (50); and fav

from equation (48). The Reynolds numbur:. at1i va±ues of X/D corres-ponding to thesu friction factors are ca.o.u.r,', a frcim equations (41) and(36). The velocity distribution3 in the fjoA IrTundiry layer areobtained from the curve for 0 z 0 in figure 2.

The fricticn fact,_rs for a tube based on the shear ztress at thewall and on the static preoouru gradient are plotted in figures 28 and

- respectively. Th,, friction factcr ' based on the pressure gradientare hi;,Ihur than thus,_- based on the shear stress in the entrance regionbecausZ! the former includo the pressure decrease caused by the increasein momentum of the fluid as thu velocity profile develops as well as

0 the pressure docrvasu caused by the shear stress at the wall. Forfully devloped flew the two friction factors are identical. Averagefriction factc ro based c-n thc pressuro drop are plotted in figure 30and are, of course, hig.her than the local values.

Thu c,_mputed vlues of f and fay are subject to some inaccuracy

becau.i, -f errurs associatd with measuring the slope of a curve (eq.(5)) and becAu~o the two terms in equati,:n (44), which are subtracted,are nearly equal close t-) the entrance. Thu results presented should,however, be weil within the limits of accuracy of experimentally deter-mined friction factors.

Friction factors for flow between flat plates are plotted infigures 31 to and are found to be very similar to those for a tube.

Comparison of Lcal Nusselt Numbers and Friction Factors

A comparison of local Nusselt numbers and friction factors forvarious cases is presented in figure 74, where the ratio of localNusselt number or friction factor to the fully developed value isplotted against X/D for a Reynolds number of 100,000. The quantityNu/Nud is smaller than f/fd for a given X/D in all cases except

that of heat transfer to a liquid metal (Pr = 0.01). The larger valuesof Nu/Nud for a liquid metal are apparently caused by the fact that

the temperature profile at low Prandtl numbers is similar to that for

'liz

I

NACA TN 3016

laminar flow. At vry high Reynolds numbers, where the dip occurs inthe curves for Nusselt number plotted against X/D, and at very lowReynolds numbers, the values of Nu/Nud may be lower than f/fd for

liquid metals (fig. £9). For gases the values of Nu/Nud are lower

than f/fd for a given value of X/D because of the momentum pressureloss for velocit:, profile development included in the friction factorf; there is no countenrpart for this momentum pressure loss in heattransfer.

It should be mentioned that thu preceding conclusions may applyonly to the case analyzed, that is, to thu case where the boundary layer CQ

ip turbulent throughout the tube. The turbulent boundary layer might beproduced by a bollmouth with a strip of sand paper at the entrance orby screens. For flow in which the boundary layer is partially laminar,or for which there are large disturbancen at the entrance, such as mightbe causvd by a right-angle-edge entrance, the conclusions might bealtered.

SU IARY OF RESULTS

The following results were obtained from the analytical investiga-tion of the entrance region for turbulunt heat transfer and flow insmooth tubes and botweun. parallel flat plates:

1. The thin thurmal boundary layers and consequently severe tem-perature grudionts at the wall near the entrance produced high heattransfer near the entrance. Approximately fully developed heat transferwas, in general, attained in an entranco length less than 10 diameters.

2. The effect of variable properties on the local Nusselt numbercorrelation for gases with fully developed velocity profiles was todecrease the Nusselt .number with conductivity based on the inlet tem-perature for he't addition to the gas and to increase it for heatextraction from thu gas at a given ratio of distance from the entranceto tube diameter X/D and inlet Reynolds number. For large values ofX/D, the Nusselt number with conductivity based on the inlet temperatureincreased continuously with X/D for both heat addition to and extrac-tion from the gas. A heat-transfer coefficient which is independent ofX/D is therefore not obtained for heat transfer with variable properties,even in the fully developed region.

3. The reference temperature used for evaluating the propertiesin order to eliminate the effect of heat flux on the curves for localNusselt number against Reynolds number at various values of X/D variedonly slightly with X/D. The reference temperature for average Nusseltnumbers, however, increased continuously with X/D for heat additionto the gas and was higher than the average wall temperature but lower

than the exit wall temperature at very large values of X/D.

I'NACA UNT 3016 29

.1. The variatiun of Nuoult number with X/D for a gas with ui-f orm inIti.,il tumporaturo ,tnd voliocity diotributions and uniform heat fluxa Tocd with the oxpurirnjuntally deturmined rulation, Nuo (/D)-Cl, forye [Ue1: of X/D butwuti f, and 6u.

F. The predicted v-Ariaticri of NusSL it number with X/D for airwith a uniform will ieumperatur,_ and uniform initial velocity arnd tern-pieraturo distributluo '±udvtery wcll with the oxperimental values

N) "r-m rcferencu 12.cr,

t.. Thu varDiutic of I~ocioit numtc, r with Qrautz numk, r for liquid"'Lt as (Prenmdti nuirb,-:r _f oi±) wasi repreisentud approximitely by a singleine f' :r ?c-oiet u2tr'bol,_w 10,J

7. Tht. ratio f' fricti n I'acttr buoud on the static-pressure]'L~t~~tin th itaxc r_~ t_> thu fully developed value was greater

than the c~rre.-j1_ndn, r~jtic -, Nu.sult, numbtmrn for gases in the entrance- c nbteciuok )f the n mentiLa, pru,.-u irso aioociated with the velocity

o. E ntialy thio"Fint, Verriati,:fis of Nusselt number and frictionfactor witli X/D wer -Ituiid for ficw btu.twuun parallel flat plates'a' w,-r,- _btailrud f--r a tube wheon th,- equivrilent diameter equal to twicetho plat,. lp-tcirI.-, w-, U:.o 'LI fr the lt.

L-.wi.- Fli-,ht Pr 'pulciz'n LbrtI~atnaiAdvIirr >mfit cior Air onaut ics

'>Peln~d CI,,, July; 21, _X7~

30 NACA TN 3016

APPENDIX A?

S'.5BOLS

The t'oliowing symbols are uoud ixi this report:

A area, s.q f t

C ~spucific huat of f~uid at cunsrtarit pressure, t/i,(F

c P) specific huit ,f fluid fit ccristint preosurti at wall,Btu/(lb)( F)

D ins ido d lumu-tor --f tubus, ft

- d cuxpn,-nt; v.luu dupinds -n± variuticri of viocosity of fluidwith tt rnporturo

ICUbat lor due ' ~~l;,/ t/sc

h bcalii..rA-trurf-r ecruffici-nt q~/t, - ti,, Btu/(sec) (sq ft) (F)

tLv.~r-ut: ht-±t -tran~ifur c. eifficient,~ av/(tC) - tl~av

.1mucr--tzi t-ulu itnt :cf hoit , 77H ft-ib/Btu

tht-riia- c -nductivit.; of fluid, Btu/(rec)(sq ft)(0'F/ft)

r"b tiiormrul conductivity of fluid evaluutd at tb,Btu/(nu c) (sq ft) (DF/ft)

k t hurma l cc~rnductivitj cf fluid evaluated attiBtu/(sec) (sq ft) (cF/ft)

thurmal conductivity of fluid evaluated atto

Btu/ (otec) (sq ft.) (OF/f t)

k ~thermal conductivity of fluid evaluated attBtu/(soc) (sq ft)(OF/ft)

kP11 thermal conductivity of fluid evaluated at xa

Btu/(soc) (sq ft) (OF/ft)

NACA TN 3016 31

M4 Mnch iiumbor at tube center at X/D

nl conotant

p stzitic jre.-oure

pi static preiljuru at inlut, lb/sq ft abo

q rateuf !ie.at transfer tuward tube center per unit area,

q, ritt .)f heat transft-r at insido wall toward tube center perunit atrea, Btu/(.-uc)(oq ft)

q,, %v,)r% t rate f h-%t. trarnsfer att inside wall toward tube% v cuintur per unit rrea fur t -Ivon X/D, Btu/(sec) (sq ft)

r radiuo, disitaiic - fro tubu cl-rtur, ft

r ~ in. ild,- tutuv r-idin -cr on'-hulf distance between plates, ft

T ab~~ t t - t t umprature,

TC fttaI-lut0 t. tLa ttcznpuraturo at tutu coriter, 0 R

I.tcta I tompturaituru -,utoi d'- thermal tUcunidary layer, OR

t abs clutte otatic it-rmperaiturO, '-R

tt, tulk, static temipuraiture ~f fluid at cross section cf tube, OR

tb,av avrae uik te-mperature, -R

ti ~Iniet staitic tfumperaturc if fluid, OR

tx reference temperature for local Nuoselt and Reynolds numbers,- b) + tb, OR

t refurencu temperature for average Nusselt and Reynolds numbers,x~avX(t~a - Lbav tbav

tb temporature if fluid outside thermal boundary layer

9 NACA TN 3016

to absoluto wall temiperature, OR

t C 'AV vora, uC w.il tunyuraturo, LipR

(t.,-tb),,1 I avora,-e diffurt-ncu butwo~ur wall anid bulk temperature, 0 R

u tiwce-avL'ratu v(--c~city parzallui t(-_ axis of channel, ft/sec

UL, bulk vuL->cit -y at cross vcocti x1 -f tubu, ft/sOCI

111 ~voiccity of flulda nuf/-

Ut~volocit~y t*,utsid-i fl,,-w bundary luyur, ftlouc

dlo'truncu- fr,-rn untrinco, ft

rlumtL'r u--- d tn uattr arbitrary ttinperature in tube,

IV diditanc fr,-m wail, ft

6 flcw-boundary .!%yer thicnvst- , ft

6hra 1 u~ay au thickness , ft

c--ufficiunt -f -dd.,, diffusivity, s'q ft/sOc

% b', lutlL vlsc,->lt y c-f, f'luid, (lb)(sec)/sq ft

ablutLu vioccsity of fluid evaluated at t b, (lb)(sqec)/sq ft

Pi abs4lute visccoilty of fluid evaluatted at t, (lb)(sec)/sq ft

p y 11bs1 ut v I qcsi tyv of fluid evaluated at tx, (ib)(sec)/n-q ft

absolutQ visco.7lty of fluid evaluatud at xa-

(lb)(oec)/sq ft

absolute viscosity of fluid evaluatud at to, (lb)(sec)/sq ft

p mass dcnsity of fluid, (lb)(sec2,)/ft4

NACA TN 3016 33

Pb mass density of fluid evaluated at tb, (lb)(sec2 )/ft4

Pb~av mass density of fluid uvaluated at tbav' (lb)(sec2)/ft

4

Pi mass density of fluid evaluated at ti, (lb)(sec2 )/ft4

p. mass density of fluid evaluated at tx, (lb)(sec2 )/ft4

Cn , maso density of fluid evaluated at tx,av, (lb)(sec2)/ft4

POI mass donsity of fluid evaluated at to, (lb)(sec2 )/ft4

shear otrLs In fluid, lb/sq ft

T.C shear struus in fluid at wall, lb/sq ft

Dimunsioni trcuss:

f frictivn fuctor besed on static-proosure gradient, D dL/dx-P U~2p ubD(pi p)

fay friction fact:,r based cn static-pressure drop, D2 x Pub

fd full, d,velop,d friction factor

fT fricti-n factor based on shear stress at wall, 2TO/(p utj)

Gz Gra.,tz nur.fter, Re Pr/(X/D)

Nu Nusselt numbur, h D/k

Nuav averutg Nusselt number for cnstant properties defined by'2q. (2&)

Nab 11usselt number with thermal conductivity evaluated at tb,

h D/kb

Nud fully developed Nusselt number

Nui Nussult number with conductivity evaluated at t,

34 NACA TN 3016

NUi )av average Nusselt number defined by eq. (25) with conduc-tivity evaluated at ti

Nux Nusselt number with conductivity evaluated at tx

NUx,av average Nussult number with conductivity evaluated at txav

NuO Nusselt number with thermal conductivity evaluated at t0LO

p preasure-drp parameter, (p - p)pD2/ 2

Pe Peclet number, Re Pr

Pr Prandtl numbur, c p C' ilk

Pr o Prandtl number with proportieos evaluated at t)

qi' hjat-tranafer parrnuter, q,/(c p g PiUlti)

Re Reynolds number, P ub D/

Reb Reynolds number with density and viscosity evaluated at tb,%% b '/bD!

Rei Reynolds number with density and viscosity evaluated at ti,

PIUbD

Rex Reynolds number with density and viscosity evaluated at tx,Pxub D/px

Rex,av Reynolds number with density and viscosity evaluated at tRe i ,avPb,av x,av

Re 0 Reynolds number with density and viscosity evaluated at to,Poub D/.0

r 0 + tube radius parameter, 40p0 r 00

OFP0

#

NACA TN 3016 35

(t o - T)c g o 1.- T/t 0T+ total-temperature parameter, -

t paameer,(t o - t)cpg~ l - tito+temperature prqo- 7

C,,t + bulk-temperature parameter, - b)

u+ velocity parameter, t

ub bulk-velocity parameter, b

N

uc+ value of u+ at tube center

U'.'u6+C

(X) thermal entrance length divided by diameterW hey+ wall distance parameter, y

a compressibility or frictional hcatintv parameter, O/(2cqJcpt 0 0 )

heat-transfer parameter, qoVT-/(ep I t )

5+ dimensionless flow-boundary-layer thickne)s, 6 olt

6h+ dimensionless thermal-boundary-layer thickness, -%O/PO 5h

An integral sign having open brackets for the upper limit S

indicates that the variable of integration should be used as the''1 -upper limit.

36 NACA TN 3016

APPENDIX B

ANALYSIS INCLUDING EFFECT OF VARIATION OF HEAT TRANSFER

ACROSS BOUNDARY LAYER FOR SMALL bh+ WITH

CONSTANT FLUID PROPERTIES

LO)

As was mentioned in the section Effect of variation of heat transfer u0across boundary layer on Nusselt number for small bh+ and constant

fluid properties, the effect of variation of q/q0 across the boundary

layer for small 5h+ should be checked. For laminar flow over a flat

plate the temperature distribution in the thermal boundary layer can beclosely approximated by a cubic parabola. For a cubic parabola,q/q0 = 1 - y 2/h

2 as shown in the following:

For laminar flow,

q = - k dt/dy = q0 (l - y2/bh2) (Cl)

Integration of equation (Cl) gives

k (tio t) (C2)55h 2 =

Evaluation of equation (C2) at the edge of the boundary layer gives

(tu - t6) I (C3)

Combination of equations (C2) and (03) results in

tO -t 3 v 3 (04)tO -b t 6- h (6]5

which is the same as the equation on page 89 in reference 7. The

assumption that q/q0 = a - y2/5h2 therefore leads to a cubic parabola

for the temperature distribution for laminar flow. The same assumption

is used herein for turbulent flow for small h The equation for

turbulent heat transfer can be written

NACA TN 3016 37

q - (k + Cp g Pe) dt/dy (Cs)

With the assumption for q/qo and e = n2uy for flow close to the wall

C (ref. 6), equation (C5) becomes, in dimensionless form,

1 -+2 + n2u+y dt 06)h+2 67+r dy+

where t+ ' represents the temperature distribution with

q/qo = - Y+2/bh+2 as distinguished from t+ , which is the dimensionless

temperature parameter for q/qO = 1. Separation of variables and

integration of equation (C6) result in

t+V y dy + i .. y+2 dy +(C-~--- - ~(07)

1 + nu+y+ 5h+2 1 + n2u+y +

-0 Pr h 0 Pr

The first term on the right side of equation (07) is t+ for constantproperties (ref. 6), so that

t+ = t+ Y+dySh+2 1 + n2u+ ()h f

The values of t+ are obtained from figure 1. For obtaining theNusselt numbers the values of t+ ' from equation (CB) are used in

- equations (12) and (4) in pluc'e of t+ .

I

38 NACA TN 3016

APPENDIX C

CALCULATION OF NUSSELT NUMBERS, REYNOLDS NUMBERS, AND TEMPERATURE

DISTRIBUTIONS FOR FLOW BEYOND ENTRANCE REGION

For points beyond the entrance region the relation between Nusseltnumber and Reynolds number with the properties, including density,evaluated at the reference temperature tx = x(t 0 - tb) + tb, where -x = 0.4 for heat addition and 0.6 for heat extraction from the gas,can be obtained from the curve for fully developed flow in figure 7.From equations (18) to (21) and the definition of tx it can be shown

that

Rex =Re ixt ) d+l d (Bl)

and

Nu Nu . tb t0 ] d+

NUi-Nx )+1(B2)

where d = 0.68 for air and most other gases. The value of Nui for

a given value of Rai can be obtained from equations (Bl) and (B2)

provided the values of tb/ti and to/tb are known. The quantities

tb/ti and to/tb can be written in terms of the quantities plotted in

figure 10 as

tb t - 1 Xt= qi ; i + 1 =4 + 1(B3)

(see eq. (23)) and

to q1 t- T -7 + 1

tb tbti) l7_

In the procedure for obtaining the value of t0/tb, the quantity in

brackets in the numerator is first determined by extrapolating a known

NACA TN 3016 39

curve for that quantity plotted against X/D. The value of Nui canthen be obtained from equation (B2). With that value of Nui a new and

more accurate value of [(to-ti)/ti 11 can be calculated from equa-tion (24). The calculation of Nui can then be repeated using the new

value of

~[(to-ti)/ti] 1ilcoi

Cn

Average Nusselt numbers with the conductivity evaluated at theinlet temperature Nuiav can be calculated from equation (25), where

the values of Nui are calculated as outlined previously. It is

desirable to calculate average Nusselt numbers and Reynolds numbers withproperties :valuated at the average bulk or the average wall temperaturefor given values of X/D. The average bulk temperature rise for a givenX/D is one-half the temperature rise at X/D, since the specific heatis assumed constant. Therefore,

tbav - t i 1 (tb t l-,

oror tb~av qi' ~b - ti-) l .

t i 2 t} j+i (B5)

From the definitions of hay, Nui,av, Rei, and Pr, it can be shown that

to, tbav q1 ' Rei P (B6)

t t + Nuiav

Average Nusselt numbers and Reynolds numbers with properties evaluatedat tbav) toav, or any arbitrary temperature

txav x(t - tbav) + tb~av

can be calculated by use of equations (B5) and (B6).

40 NACA TN 3016

REFERENCES

1. Goldstein, Sidney: Modern Developments in Fluid Dynamics. ClarendonPress (Oxford), 193n.

2. Langhaar, H. L.: Steady Flow in the Transition Length of a Straight

Tube. Jour. Appl. Mech., vol. 9, no. 2, June 1942, pp. A55-A58.

3. Jacob, Max: Heat Transfer. Vol. 1. John Wiley & Sons, Inc., 1929.

4. Latzko, H.: Heat Transfer in a Turbulent Liquid or Gas Stream.NACA TM 1068, 1944.

5. Poppendiek, H. F., and Palmer, L. D.: Forced Convection Heat Trans-fer in Thermal Entrance Regions, Part II. ORNL 914 Metallurgy andCeramics, Reactor Exp. Eng. Div., Oak Ridge National Lab., OakRidge (Tenn.). May 26, 1952. (Contract No. W-7405, eng. 26.)

6. Deissler, R. G., and Eian, C. S.: Analytical and Experimental Inves-tigation of Fully Developed Turbulent Flow of Air in a Smooth Tubewith Heat Transfer with Variable Fluid Properties. NACA TN 2629,1952.

7. Eckert, E. R. G.: Introduction to the Transfer of Heat and Mass.First ed., McGraw-Hill Book Co., Inc., 1950.

8. Deissler, Robert G.: Analysis of Fully Developed Turbulent HeatTransfer at Low Peclet Numbers in Smooth Tubes with Application toLiquid Metals. NACA RM E52FO5, 1952.

9. Deissler, Robert G.: Analytical Investigation of Turbulent Flow inSmooth Tubes with Heat Transfer with Variable Fluid Properties forPrandtl Number of 1. NACA TN 2242, 1950.

10. Prandtl, L., and Tietjens, 0. G.: Fundamentals of Hydro and Aero-Mechanics. First ed., McGraw-Hill Book Co., Inc., 1931.

11. Weiland, Walter F., and Lowdermilk, Warren H.: Measurements of Heat-Transfer and Friction Coefficients for Air Flowing in a Tube ofLength-Diameter Ratio of 15 at High Surface Temperatures. NACARM E53E04, 1953.

12. Boelter, L. M. K., Young, G., and. Iversen, H. W.: An Investigationof Aircraft Heaters. XXVII - Distribution of Heat-Transfer Ratein the Entrance Section of a Circular Tube. NACA TN 1451, 1948.

13. Stromquist, W. K., and Boarts, R. M.: Effect of Wetting on LiquidMetal Heat Transfer. Prog. Rep., Dept. Chem. Eng., The Universityof Tennessee, Sept. 30, 1952. (AEC Contract No. AT-(40-1)-1310.)

NACA TN 3016 41

I I

cr, C,\ _______

..

D

cn.

ib,

"Nt "

~1

42 NACA ~TI 3016U,

C, I________________________________________________ - ,,.

I _____C' -, . C' 6 b

~ 0**'*.____ ____ ____ ' LI)

0

4'

C.

C

4-,"2

C,

4-

C

4:

4.

4-,

_____________ 4-,

4,

4-,

;-. '

__________ 4-

4-,

__________ __________ __________ __________ __________ __________ 4.

_________ _________ _________ _________ 4.4;

"2

4-.

________ ________ ________ ________ ________ ________ C' V

4-,

"2________________ ________________ I,

N

4.____________ ____________ ____________ ___________ 4.

"244-.

___________________ ___________________ ___________________ ___________________ ____________________ ___________________ 4.'

4,

,".441in Cl .,, C IC C' C' UIi) .4

1.61

0d/

0L~' / r..

S

NACA 2~ 3016 43 j

~4 - t

1I -~ L - -1-~-~

- t1,1- -

7. t I'(L~~ I -c~) * _

.1 I

* I I -I *

,-* .1~2I I

* .. I 1**I j II I

>:

1- A.

I I I

f I I -___ - -

*

I II _ _ I __ -~I I H -I_ I ~

1. I I _j HC-,

... 1 ~H a-h----- H TT 4- t,~ .4~ -j-o I H H

---9-- &, -9-'

I I a - Vo ~ ~ - - _________0 ~ 4* 0 I.) 4 H 4-.... o:11 ~O.1/~A = O.I/~ 0 .~/A 0.z,~

44 NACA TN 3016

v0

P4

4) id

43

__

4, NACA ~ 3016 45

NI4 [ 4..,

-4-

.t ~, 4-4 1

I HiC.', V -~1 4 4 ±7YTTi 172 '-4

-4-

4 1

- . 4 ~- . t . 4-

* - -t- -~

-4-- -----4 - 4 -4--- -1 ---- -- i- LI~414

(

I -a -, -~4, '-, ,~4 ...- 4 -4

444 44 .'~ -4

0'

I.- ,--.~ ~.

46 NACA TN 3016

_

--- pI~?1

* (11T jI1 i...* . 4 4 C

4-.

C, 4.44~

t - -4---.--

'-4

I-

44F* ii

~, ;1..~

4 ~-

- - 4.- - - -. -4 4.'- 2

C- )-

~ '-4

'-4-

ii

4.- 4*'-441.41..,

1.,-.*4* 4 4.- 1 .- - - -.

.2

(4

(4-.

1.4

('.4

4'

C)

-- tillU"

NACA TN 3016 47

00

0 4J.,0

II

iif

*0

("I' 0 I 43

to'

48 NACA 1W 3016

4;- , W, s w

V0

oP

t 1.

. ..

In-

nit-

~ -7- ~ .

NACA TN 3016 49

0CD 0

0 0 H

0 (U zto

0 m0 C)U-)- r 0

>

,0H

to to.*

4J0

0 :

>)44-)

-... 4

-0r-I 0~0

= +)4 +W

,on

o- -0 0+)4-

InK P4

50 NACA TN 3016

1000 ____ _Dzzc z :

t -n

1O,~X~O100,(X 1,000,000

Fipijre 7. -Viri'itl r, 4r I'-ic-Fi1 Nussel+ n~umber with TR'yn ids numbefr for varicus vnlues

f X/D fir gns iin In ni tube with pr-o'.rt jes evnlimted at tx - xt)tbwhe'xIidt~rnd~rifgr Pr. 0.73. Unifrn, heat flux, uni'orrhe, x nInk-iedf fgr Y +t+1 1 t 7-i-rnturp Jistributi-n, nnd fully deve] ped velocity distributi~n.

NACA TN 3016 51

U-))

Nl 0 '-4 '-4- r.

4-Y ~

C.)

U) 41

C.C,

-O 4-)

M a3

52 NACA TNI 3016

t)

z V4

CQ

NACA TN~ 3016 53

co

4- IW

V. 4

T~I -- ;

54 NACA TK 3016

Lt- .. .t - , V -t - "-- - - - - -t--- - 4 - -jU

I D

.4n

4- +.

E; L

I T45,

NACA TN 3016 55

c-n Q

4-' -

to 4-'-18- - - 0

C, (z 4E W-

4) )to- 0

) 400

o ko

4 4) -4 fr0

o N -4 0 .)"ot - -- 0

o to

004, 0

q) '4-

56 NACA TN 3016

8~~T _ ~_ 8 0 0c8 w

I ]-4 *0

=8h

+4

-4-- r4-

, >

,I i 4J". ,

g,

4S-41

T- a-

Aw InaV

NACA TN 3016 57

{r -°---

1/- '.-4 g:

i ' i'

--- ... .-1--

' 0'o 4,

I ~4$,

co ** 1- --

t ~ C vC - ~ .

S-, -~- --4-.1

- . - . ---- - . .~~-,I

Q , * >4

PO - t 4 *,--

All9.

58 NAA TN 3016

4-'

A

-4-

0 1

AIR' 4.

NACA TN 3016 59

----4- -

co

------- A

AD' 4-----

*~Itt r.-- f-.- - -A-

460 NACA TN 3016

r xii'-.

-.. . .

F -, t 4-

- -i- , , j I

* I ,

C'' F g4' F F

--i. . -,--- - -- - , - '3+ 4 I0

61NACA TN 3016

-Poo4o~ 1~

4-4

1. __ ___+

()q, 0 (cornotamnt mrrtiC!).

Fi~'jr& 14. - ~1.1 Nuaspit tnumb~r with X/ and ppyn-,Idr, rumber rXor gs flowing between parallel

tintVint!. r, I~5.t11'r~ eultt'lx. ~w lttI" teper~ture distributImi, and fully devel--red vel-city 41cstribullv

62 NACA TN 3016

f1')

*~~1 . I ..,

14 144

l'ijture 14. -C,ntinupl. VFkri tlion of Russelt rui.+er with X/D find iF-yn

I -- NACA TN 3016 63

700

00-

ReIZOO X 10 3

4 C,)-aoo \

200

05

100 - --

-- 10

0 2 4 6 8 10 12X/D

(c) q' = -0.0025 (heat extraction from gas).

Figure 14. - Concluded. Variation of Nusselt number with X/D and Reynolds number

for gas flowing between parallel flat plates. Pr, 0.73. Unieorm heat flux, uni-

form initial temperature distribution, and fully developed velocity distribution.

III-. 1

64 NACA TN 3016

,0

4'NACA 1W 3016 65

a.)

66 NACA 7N 3016

~0

H

Lo LO.-

.4-) Q U

(aJ 0

0- 1-1

H- H

Q 0 1.4-4

.440 0 P

co Qo

v, 0'T0 +

-S0 0

4-4i

lb, xo04))

NACA TN 3016 67

CD

0n

Fig r -

f _____________________IT__ ri rr.h~n f *.Y lf .r Ini n tempera

*ur,- dsr i,,t!-r n'd . .J... *1,;'i -.. ,~I tv 1 t~~

68 NACA TN 3016

, I f )

SX P' ° 1

4

Y,2' 1-.

I It

4 4-" t

-"'--4-4 ;)

/ . I

to o

r,

NACA TN~ 3016 69

G)

01

X0 X/10

(c) q' -0 -. 00:' (he.nt extractl r, from gns).

Figure 16). -Concluded. Varint1-on of' nverage Nusselt number defined by equation(2)with X/D and Reynolds number f'or gns flowing between parallel flat plates.

Pr, 0.73. Uniform heat flux, unifrrm Initial temperature distribution, andfully developed velocity distribution.

70 NACA 2TN 3016

.-.

-~c 0 t-I*

"",( v..

NACA TN 3016 71

CD

Lr~ -- ~~0

cn I V1~~~~ *8CQ4

-4

CD 4

C; 4

LO in"IA

- - - -- - 'nu-~~~~f - N -- -- - =4

72 NACA TN 3016

CD,

n0

~Q

0 CI

*~~4

4

.. . .

nmC

NACA TN 3016 73

K Ci

k i

0+

or

ot

1)

* 4

S L

-4-- 14 0~

2 e

NACA TN 301674

nNN

NACA TN 3016 75

CD,

o C-

o~4 C:4 .-04

o5 4'z ,0

-1 41 ILI- 4 -

0441

76 -- NACA Thf 3016

CC

+ 41

nK 1~

NACA TN 3016 77

r - '~-- w'.

CD ---- ~-- -Eni

-4i coLi

_ _ -1

S.. 1 -1-

* 4'4

'4: ~ 4:0

Il ,,-4 1

78 NACA TN 3016

'4 -4-T U. -

- 4-4

* * - 4- - ' C

;

I i I ." -. .

ol t NACA T1W 3016 7

CQC

4-'

cl, '-4

NMI.

4) (

4-)

$.4 -D

80 NACA TN 3016

oro

- 42.

1

rr41

.li2

NACA Th 3016 8

4ow

co4

(.n

'-1 41

jiQj

14 IN

nu0

82 NACA T!J 3016

4q- I

-4-1

.44

-4)

-~ .'~ - .*~ .- .- .-

NACA TN 3016 83

0

41

+4

-43

45

v too 0 0

84 NACA Tff 3016

S4#

F4J0u

C'Cb

----o$7 ~TV --t-- H- j -

I7, 7;ffj T

NACA TN 3016 85

V 0

H

-coJ

0

0 3

,4to (

00HH f4I

4-- X-

u co

> 0

- -L u4) -

4 4+

86 NACA TN 3016

.06 -(nCD~~~~

.05-

.04-

f .03\

.0?

.01Re

60

X/D

Figure 32. -Variation of friction factor based on static-pressure gradient withX/D and Reynolds number for flow between parallel flat plates. Constantproperties.

NACA TN 3016 87

.071 -

t4

04. 1-4

06__ __ _ ___

100

4 C, 10 12

X /D

Figure 33. -V'nrintir~n o"' nvernge frictin factor based on static-pressure dro-p

with X/D nd eynolds number for flow between parallel lat plates. Constant

properties. I

88 NACA TN 3016

CQ

0r

LL

LL

4:

P. 1; ao P/ -. jj

NACA.LMWW~s . 10441 - 100

t Tf'

!2 IS 9

3 j1~~JRl15- rn coC

',- -o C C !WIC6* V' o. -;

15: -0- Q

Cc,~ ~ ~ c p...- 2,-.bt i1.Z

pz '4 E 60

0 ev

te

Aoa.Z

IL -4

24Q. -Z

L" ~ ~ 5 En 2Q6&. w.2' i , li6-c=-F.2Ec -4 a ! a o

c.a. , >Eb,22 S

> 0I = .1..

v C 41 ;

- t(* 'C 4 ;

-< -L L

41 -Z. ~ 4

oz~

~T t

t4 2 2%4

S '.2 S

L1 CL o

u) o 1" 71=

Ic i