Chapter 8INTERNAL FORCED

CONVECTION

Heat Transfer

Universitry of Technology Materials Engineering DepartmentMaE216: Heat Transfer and Fluid

ObjectivesObtain average velocity from a knowledge of velocity profile, and average temperature from a knowledge of temperature profile in internal flow.

Have a visual understanding of different flow regions ininternal flow, and calculate hydrodynamic and thermal entrylengths.

Analyze heating and cooling of a fluid flowing in a tube under constant surface temperature and constant surface heat flux conditions, and work with the logarithmic mean temperature difference.

Obtain analytic relations for the velocity profile, pressure drop, friction factor, and Nusselt number in fully developed laminar flow.

Determine the friction factor and Nusselt number in fully d l d t b l t fl i i i l l ti d

TRODUCTIONuid or gas flow through pipes or ducts is commonly used in heating and

oling applications and fluid distribution networks. e fluid in such applications is usually forced to flow by a fan or pump through ow section.hough the theory of fluid flow is reasonably well understood, theoreticalutions are obtained only for a few simple cases such as fully developed

minar flow in a circular pipe. erefore, we must rely on experimental results and empirical relations for most d flow problems rather than closed-form analytical solutions.

For a fixed surface area, the circular tube gives the most heat transfer for the leastpressure drop.

luid velocity in a pipe changes zero at the wall because of the p condition to a maximum at the center.

d flow, it is convenient to work an average velocity Vavg, which ins constant in incompressiblewhen the cross-sectional area of ipe is constant.

average velocity in heating and ng applications may change ewhat because of changes in ty with temperature.

n practice, we evaluate the fluid erties at some average erature and treat them as tants.

value of the average (mean) velocity at some streamwise cross-section is:

average velocity for incompressiblen a circular pipe of radius R is:

ERAGE VELOCITY AND TEMPERATURE

d flow, it is convenient to work with an age or mean temperature Tm, whichns constant at a cross section. The mean

erature Tm changes in the flow direction ever the fluid is heated or cooled.

minar and Turbulent Flow in Tubesin a tube can be laminar or turbulent, depending on the flow itions.flow is streamlined and thus laminar at low velocities, but turns lent as the velocity is increased beyond a critical value. sition from laminar to turbulent flow does not occur suddenly;

er, it occurs over some range of velocity where the flow fluctuates een laminar and turbulent flows before it becomes fully turbulent. pipe flows encountered in practice are turbulent. nar flow is encountered when highly viscous fluids such as oils n small diameter tubes or narrow passages.sition from laminar to turbulent flow depends on the Reynoldsber as well as the degree of disturbance of the flow by surface hness, pipe vibrations, and the fluctuations in the flow. flow in a pipe is laminar for Re < 2300, fully turbulent for Re > 00 and transitional in between

ds number for flow in a circular tube

w through noncircular tubes, the ds number as well as the Nusselt r, and the friction factor are on the hydraulic diameter Dh

Under most practical conditions, the flow in a pipe is laminar for Re < 2300, fully turbulent for Re > 10,000, and transitional in between.

E ENTRANCE REGIONty boundary layer (boundary layer): The region of the flow in which the effects of the s shearing forces caused by fluid viscosity are felt.pothetical boundary surface divides the flow in a pipe into two regions: ary layer region: The viscous effects and the velocity changes are significant.onal (core) flow region: The frictional effects are negligible and the velocity remains ally constant in the radial direction.dynamic entrance region: The region from the pipe inlet to the point at which the y profile is fully developed.dynamic entry length Lh: The length of the hydrodynamic entrance region.dynamically fully developed region: The region beyond the entrance region in which ocity profile is fully developed and remains unchanged.

the entrance region is hydrodynamically ping flow since this ision where the velocity develops.

uid properties in internal flow are usually evaluated at the bulk mean fluid erature, which is the arithmetic average of the mean temperatures at the and the exit: Tb = (Tm, i + Tm, e)/2.

The pment of

mal entrance region: The region of flow over which the thermal boundary layer ps and reaches the tube center.

mal entry length: The length of the thermal entrance region.mally developing flow: Flow in the thermal entrance region. This is the region

the temperature profile develops. mally fully developed region: The region beyond the thermal entrance region in

the dimensionless temperature profile remains unchanged.developed flow: The region in which the flow is both hydrodynamically andally developed.

thermally fully developed region of a the local convection coefficient isant (does not vary with x).fore, both the friction (which is relatedl shear stress) and convection cients remain constant in the fullyoped region of a tube.ressure drop and heat flux are higher in trance regions of a tube, and the effect

Variation of the frictionfactor and the convection

heat transfer coefficient in the flow direction for flow in a tube (Pr > 1).

odynamically fully developed:

mally fully developed:

ce heat flux:

y gths

riation of local Nusselt umber along a tube in turbulent flow for both

if f

e Nusselt numbers and thus h values are much higher in the entrance region.e Nusselt number reaches a constant value at a distance of less than 10 meters, and thus the flow can be assumed to be fully developed for x > 10D.e Nusselt numbers for uniform surface perature and uniformface heat flux ditions are identical

he fully developed ons, and nearly

ntical in the entrance ons.

NERAL THERMAL ANALYSISThe thermal conditions at the surface can be approximated to be • constant surface temperature

(Ts= const) or • constant surface heat flux (qs = const).The constant surface temperature condition is realized when a phase change process such as boiling or condensation occurs at the outer surface of a tube. The constant surface heat flux condition is realized when the tube is subjected to radiation or electric resistance heating uniformly from all directions.We may have either Ts = constant or qs = constant at the surface of a tube,

he local heat transfer coefficient.

ce heat flux

of heat transfer

stant Surface Heat Flux (qs = constant)

Variation of the tube surface and the mean fluid

temperatures along the

fluid temperature tube exit:

of heat transfer:

ce temperature:

h f th t t fil i

Energy interactions for a differential control volume in a tube.

ar tube:

nstant Surface Temperature (Ts = constant)of heat transfer to or from a fluid flowing in a tube

suitable ways of expressing Tavg: Arithmetic mean temperature difference

ogarithmic mean temperature difference

hmetic mean temperature difference:

k mean fluid temperature: Tb = (Ti + Te)/2

using arithmetic mean temperature difference, we assume that the mean temperature varies linearly along the tube, which is hardly ever the case n Ts = constant.

s simple approximation often gives acceptable results, but not always.

rating from x = 0 (tube inlet, Ti) to x = L (tube exit, Tm = Te)

The variation of the mean fluidtemperature along the tube for thecase of constant temperature.

An NTU greater than 5 indicates thatfl id fl i i t b ill h th

Log mean temperature difference

Number of transfer units. A ure of the effectiveness of the ransfer systems. TU = 5, Te = Ts, and the limit for ransfer is reached.ll value of NTU indicates more tunities for heat transfer.an exact representation of the

ge temperature difference en the fluid and the surface.Te differs from Ti by no more 0 percent, the error in using the etic mean temperature nce is less than 1 percent.

MINAR FLOW IN TUBES

The maximum velocity occurs at the centerline, r = 0:

it fil

Pressure Drop

ntity of interest in the analysis of pipe flow is the pressure drop P sincerectly related to the power requirements of the fan or pump to maintain flow.

In laminar flow, the friction factor is a function of the Reynolds number only and is independent of the roughness of the pipe surface.

Head loss

e losses are ly expressed

ad loss hL represents the additional height that the fluid to be raised by a pump in order to overcome the frictional in the pipe. The head loss is caused by viscosity, and it is related to the wall shear stress.

Poiseuille’s law

specified flow rate, the pressure drop and he required pumping power is proportionallength of the pipe and the viscosity of the

but it is inversely proportional to the fourth r of the radius (or diameter) of the pipe.

quired pumping power to me the pressure loss:

The average velocity for laminar flow

The differential volume elementused in the derivation of energy

rate of net energy transfer to theol volume by mass flow is equal

perature Profile and the Nusselt Number

nstant Surface Heat Flux

ying the boundary conditions x = 0 at r = 0 (because of metry) and T = Ts at r = R:

Therefore, for fully developed laminar flow in a circular tube subjected to constant surface heat flux, the Nusselt number is a constant.

There is no dependence on the Reynolds or the Prandtl numbers.

nstant Surface Temperature

aminar flow in a tube with constant ace temperature, both the friction

thermal conductivity k for use in the Nu relations should be evaluatedhe bulk mean fluid temperature.laminar flow, the effect of surface roughness on the friction factor and heat transfer coefficient is negligible.

Laminar Flow in Noncircular TubesNusselt number relations are given in Table 8-1 for fully developed laminarflow in tubes of various cross sections.The Reynolds and Nusselt numbers for flow in these tubes are based on the hydraulic diameter Dh = 4Ac/p.Once the Nusselt number is available, the convection heat transfer coefficient

eloping Laminar Flow in the Entrance Region

n the difference between the surface and the fluid temperatures is large, y be necessary to account for the variation of viscosity with temperature:

All properties are evaluated at the bulk mean fluid temperature, except for s, which is evaluated at the surface temperature.

average Nusselt number for the thermal entrance region of between isothermal parallel plates of length L is

a circular tube of length L subjected to constant surface temperature, theage Nusselt number for the thermal entrance region can be determined from:

average Nusselt number is larger at the entrance region, and it oaches asymptotically to the fully developed value of 3.66 as L → .

RBULENT FLOW IN TUBES

First Petukhov equationChilton–Colburn analogy

Colburn equation

Dittus–Boelter equation

n the variation in properties is large due to a large temperature difference:

Second Petukhov equation

Gnielinskirelation

e relations above are not very sensitive to the thermal conditions at thee surfaces and can be used for both Ts = constant and qs = constant.

gh Surfaces

rbulent flow, wall roughness increases the heat transfer coefficient hfactor of 2 or more. The convection heat transfer coefficient for rough s can be calculated approximately from the Gnielinski relation or

riction factor in fully developed turbulent pipe flow depends on theolds number and the relative roughness /D, which is the ratio of the height of roughness of the pipe to the pipe diameter.

Colebrook equation

dy chart is given in the appendix as Fig. A–20. sents the Darcy friction factor for pipe flow as a function of the Reynolds er and /D over a wide range.

An approximate explicit relation for f wasgiven by S. E. Haaland.

eloping Turbulent Flow in the Entrance Regionentry lengths for turbulent flow are typically short, often just 10 tubeeters long, and thus the Nusselt number determined for fully developedent flow can be used approximately for the entire tube.

simple approach gives reasonable results for pressure drop and heatfer for long tubes and conservative results for short ones.lations for the friction and heat transfer coefficients for the entrance regionsvailable in the literature for better accuracy.

bulent Flow in Noncircular Tubes

In turbulent flow, the velocityprofile is nearly a straight line in the core region and any

sure drop and heat transfercteristics of turbulent flow in tubes are

nated by the very thin viscous sublayer o the wall surface, and the shape of the region is not of much significance.urbulent flow relations given above for ar tubes can also be used forrcular tubes with reasonable accuracy l i th di t D i th l ti

w through Tube Annulus

A double-tube heat exchanger that consists of two concentric tubes.

Hydraulic diameter of annulus

aminar flow, the convection coefficients for theand the outer surfaces are determined from

ully developed turbulent flow, hi and hopproximately equal to each other, and theannulus can be treated as a noncircular with a hydraulic diameter of Dh = Do − Di.Nusselt number can be determined from a

le turbulent flow relation such as thenski equation. To improve the accuracy, elt numbers can be multiplied by the ing correction factors when one of the

walls is adiabatic and heat transfer is gh the other wall:

t Transfer Enhancements with rough surfaces have much

er heat transfer coefficients thans with smooth surfaces. transfer in turbulent flow in a tube

be increased by as much as 400 ent by roughening the surface. ghening the surface, of course, increases the friction factor and the power requirement for the p or the fan.

convection heat transfer icient can also be increased by cing pulsating flow by pulse rators, by inducing swirl by ting a twisted tape into the tube, inducing secondary flows by

ummaryIntroductionAverage Velocity and Temperature Laminar and Turbulent Flow in Tubes

The Entrance Region Entry Lengths

General Thermal Analysis Constant Surface Heat Flux Constant Surface Temperature

Laminar Flow in Tubes Pressure Drop Temperature Profile and the Nusselt Number Constant Surface Heat Flux Constant Surface Temperature Developing Laminar Flow in the Entrance Region

Turbulent Flow in Tubes Rough Surfaces Developing Turbulent Flow in the Entrance Region Turbulent Flow in Noncircular Tubes Flow Through Tube Annulus