Computational Fluid Mechanics and Heat Transfer Third Edition_10 docx

Now we wish to undertake a sequence of progressivelyharder problems of forced convection heat transfer in more complicatedflow configurations.Incompressible forced convection heat transfer

The bed was soft enough to suit me .But I soon found that there came such a draught of cold air over me from the sill of the window that this plan would never do at all, especially as another current from the rickety door met the one from the window and both together formed a series of small whirlwinds in the immediate vicinity of the spot where I had thought

to spend the night Moby Dick, H Melville, 1851

7.1 Introduction

Consider for a moment the fluid flow pattern within a shell-and-tube heat

exchanger, such as that shown in Fig.3.5 The shell-pass flow moves up

and down across the tube bundle from one baffle to the next The flow

around each pipe is determined by the complexities of the one before it,

and the direction of the mean flow relative to each pipe can vary Yet

the problem of determining the heat transfer in this situation, however

difficult it appears to be, is a task that must be undertaken

The flow within the tubes of the exchanger is somewhat more tractable,

but it, too, brings with it several problems that do not arise in the flow of

fluids over a flat surface Heat exchangers thus present a kind of

micro-cosm of internal and external forced convection problems Other such

problems arise everywhere that energy is delivered, controlled, utilized,

or produced They arise in the complex flow of water through nuclear

heating elements or in the liquid heating tubes of a solar collector—in

the flow of a cryogenic liquid coolant in certain digital computers or in

the circulation of refrigerant in the spacesuit of a lunar astronaut

We dealt with the simple configuration of flow over a flat surface in

341

Chapter6 This situation has considerable importance in its own right,and it also reveals a number of analytical methods that apply to otherconfigurations Now we wish to undertake a sequence of progressivelyharder problems of forced convection heat transfer in more complicatedflow configurations.

Incompressible forced convection heat transfer problems normallyadmit an extremely important simplification: the fluid flow problem can

be solved without reference to the temperature distribution in the fluid.Thus, we can first find the velocity distribution and then put it in theenergy equation as known information and solve for the temperaturedistribution Two things can impede this procedure, however:

• If the fluid properties (especially µ and ρ) vary significantly with

temperature, we cannot predict the velocity without knowing thetemperature, and vice versa The problems of predicting velocityand temperature become intertwined and harder to solve We en-counter such a situation later in the study of natural convection,where the fluid is driven by thermally induced density changes

• Either the fluid flow solution or the temperature solution can, itself,

become prohibitively hard to find When that happens, we resort tothe correlation of experimental data with the help of dimensionalanalysis

Our aim in this chapter is to present the analysis of a few simpleproblems and to show the progression toward increasingly empirical so-lutions as the problems become progressively more unwieldy We beginthis undertaking with one of the simplest problems: that of predictinglaminar convection in a pipe

7.2 Heat transfer to and from laminar flows in pipes

Not many industrial pipe flows are laminar, but laminar heating and ing does occur in an increasing variety of modern instruments and equip-ment: micro-electro-mechanical systems (MEMS), laser coolant lines, andmany compact heat exchangers, for example As in any forced convectionproblem, we first describe the flow field This description will include anumber of ideas that apply to turbulent as well as laminar flow

cool-Figure 7.1 The development of a laminar velocity profile in a pipe.

Development of a laminar flow

Figure7.1shows the evolution of a laminar velocity profile from the

en-trance of a pipe Throughout the length of the pipe, the mass flow rate,

where A c is the cross-sectional area of the pipe The velocity profile, on

the other hand, changes greatly near the inlet to the pipe A b.l builds

up from the front, generally accelerating the otherwise undisturbed core

The b.l eventually occupies the entire flow area and defines a velocity

pro-file that changes very little thereafter We call such a flow fully developed.

A flow is fully developed from the hydrodynamic standpoint when

∂u

at each radial location in the cross section An attribute of a dynamically

fully developed flow is that the streamlines are all parallel to one another

The concept of a fully developed flow, from the thermal standpoint,

is a little more complicated We must first understand the notion of the

mixing-cup, or bulk, enthalpy and temperature, ˆ h b and T b The enthalpy

is of interest because we use it in writing the First Law of

Thermodynam-ics when calculating the inflow of thermal energy and flow work to open

control volumes The bulk enthalpy is an average enthalpy for the fluid

flowing through a cross section of the pipe:

T b ≡ rate of flow of enthalpy through a cross section

rate of flow of heat capacity through a cross section

Thus, if the pipe were broken at any x-station and allowed to discharge

into a mixing cup, the enthalpy of the mixed fluid in the cup would equalthe average enthalpy of the fluid flowing through the cross section, and

the temperature of the fluid in the cup would be T b This definition of T b

is perfectly general and applies to either laminar or turbulent flow For

a circular pipe, with dA c = 2πr dr , eqn (7.5) becomes

A fully developed flow, from the thermal standpoint, is one for which

the relative shape of the temperature profile does not change with x We

state this mathematically as

where T generally depends on x and r This means that the profile can

be scaled up or down with T w − T b Of course, a flow must be namically developed if it is to be thermally developed

hydrody-Figure 7.2 The thermal development of flows in tubes with

a uniform wall heat flux and with a uniform wall temperature

(the entrance region).

Figures7.2and7.3show the development of two flows and their

sub-sequent behavior The two flows are subjected to either a uniform wall

heat flux or a uniform wall temperature In Fig.7.2we see each flow

de-velop until its temperature profile achieves a shape which, except for a

linear stretching, it will retain thereafter If we consider a small length of

pipe, dx long with perimeter P , then its surface area is P dx (e.g., 2π R dx

for a circular pipe) and an energy balance on it is1

Figure 7.3 The thermal behavior of flows in tubes with a

uni-form wall heat flux and with a uniuni-form temperature (the

ther-mally developed region).

This result is also valid for the bulk temperature in a turbulent flow

In Fig 7.3 we see the fully developed variation of the temperatureprofile If the flow is fully developed, the boundary layers are no longer

growing thicker, and we expect that h will become constant When q w is

constant, then T w − T b will be constant in fully developed flow, so thatthe temperature profile will retain the same shape while the temperature

rises at a constant rate at all values of r Thus, at any radial position,

In the uniform wall temperature case, the temperature profile keeps

the same shape, but its amplitude decreases with x, as does q w Thelower right-hand corner of Fig.7.3has been drawn to conform with thisrequirement, as expressed in eqn (7.7)

The velocity profile in laminar tube flows

The Buckingham pi-theorem tells us that if the hydrodynamic entry length,

x e, required to establish a fully developed velocity profile depends on

uav, µ, ρ, and D in three dimensions (kg, m, and s), then we expect to

find two pi-groups:

x e

D = fn (Re D )

where ReD ≡ uavD/ν The matter of entry length is discussed by White

[7.1, Chap 4], who quotes

x e

The constant, 0.03, guarantees that the laminar shear stress on the pipe

wall will be within 5% of the value for fully developed flow when x >

x e The number 0.05 can be used, instead, if a deviation of just 1.4% is

desired The thermal entry length, x e t , turns out to be different from x e

We deal with it shortly

The hydrodynamic entry length for a pipe carrying fluid at speeds

near the transitional Reynolds number (2100) will extend beyond 100

di-ameters Since heat transfer in pipes shorter than this is very often

im-portant, we will eventually have to deal with the entry region

The velocity profile for a fully developed laminar incompressible pipe

flow can be derived from the momentum equation for an axisymmetric

flow It turns out that the b.l assumptions all happen to be valid for a

fully developed pipe flow:

• The pressure is constant across any section.

• ∂2u ∂x2is exactly zero

• The radial velocity is not just small, but it is zero.

• The term ∂u ∂x is not just small, but it is zero.

The boundary layer equation for cylindrically symmetrical flows is quite

similar to that for a flat surface, eqn (6.13):

For fully developed flows, we go beyond the b.l assumptions and set

v and ∂u/∂x equal to zero as well, so eqn (7.13) becomes

1

r

d dr



r du dr



= 1µ

dp dx

We integrate this twice and get

u =

1

4µ

dp dx

u = R24µ



− dp dx

This is the familiar Hagen-Poiseuille2 parabolic velocity profile We can

identify the lead constant ( −dp/dx)R2 4µ as the maximum centerline velocity, umax In accordance with the conservation of mass (see Prob-lem7.1), 2uav= umax, so

Thermal behavior of a flow with a uniform heat flux at the wall

The b.l energy equation for a fully developed laminar incompressibleflow, eqn (6.40), takes the following simple form in a pipe flow wherethe radial velocity is equal to zero:

2The German scientist G Hagen showed experimentally how u varied with r , dp/dx,

µ, and R, in 1839 J Poiseuille (pronounced Pwa-zói or, more precisely, Pwä-z´¯e e) did the same thing, almost simultaneously (1840), in France Poiseuille was a physician interested in blood flow, and we find today that if medical students know nothing else about fluid flow, they know “Poiseuille’s law.”

For a fully developed flow with q w = constant, T w and T bincrease linearly

with x In particular, by integrating eqn (7.10), we find

d dr



r dT dr

The first b.c on this equation is the symmetry condition, ∂T /∂r = 0

at r = 0, and it gives C1 = 0 The second b.c is the definition of the

mixing-cup temperature, eqn (7.6) Substituting eqn (7.19) with C1 = 0

into eqn (7.6) and carrying out the indicated integrations, we get

C2= T b − 7

24

q w R k

Equation (7.22) is surprisingly simple Indeed, the fact that there isonly one dimensionless group in it is predictable by dimensional analysis.

In this case the dimensional functional equation is merely

h = fn (D, k)

We exclude∆T , because h should be independent of ∆T in forced tion; µ, because the flow is parallel regardless of the viscosity; and ρu2

convec-av,because there is no influence of momentum in a laminar incompressibleflow that never changes direction This gives three variables, effectively

in only two dimensions, W/K and m, resulting in just one dimensionlessgroup, NuD, which must therefore be a constant

Example 7.1

Water at 20◦C flows through a small-bore tube 1 mm in diameter at

a uniform speed of 0.2 m/s The flow is fully developed at a point

beyond which a constant heat flux of 6000 W/m2 is imposed Howmuch farther down the tube will the water reach 74◦C at its hottestpoint?

Solution. As a fairly rough approximation, we evaluate properties

 u

avk 4q w α

0.2(0.6367) 4(6000)1.541(10) −7 = 1785

so the wall temperature reaches the limiting temperature of 74◦C at

L = 1785(0.001 m) = 1.785 m

While we did not evaluate the thermal entry length here, it may be

shown to be much, much less than 1785 diameters

In the preceding example, the heat transfer coefficient is actually

The high h is a direct result of the small tube diameter, which limits the

thermal boundary layer to a small thickness and keeps the thermal

resis-tance low This trend leads directly to the notion of a microchannel heat

exchanger Using small scale fabrication technologies, such as have been

developed in the semiconductor industry, it is possible to create

chan-nels whose characteristic diameter is in the range of 100 µm, resulting in

heat transfer coefficients in the range of 104 W/m2K for water [7.2] If,

instead, liquid sodium (k ≈ 80 W/m·K) is used as the working fluid, the

laminar flow heat transfer coefficient is on the order of 106 W/m2K — a

range that is usually associated with boiling processes!

Thermal behavior of the flow in an isothermal pipe

The dimensional analysis that showed NuD = constant for flow with a

uniform heat flux at the wall is unchanged when the pipe wall is

isother-mal Thus, NuD should still be constant But this time (see, e.g., [7.3,

Chap 8]) the constant changes to

NuD = 3.657, T w = constant (7.23)for fully developed flow The behavior of the bulk temperature is dis-

cussed in Sect.7.4

The thermal entrance region

The thermal entrance region is of great importance in laminar flow

be-cause the thermally undeveloped region becomes extremely long for

higher-Pr fluids The entry-length equation (7.12) takes the following form for

the thermal entry region3, where the velocity profile is assumed to be

fully developed before heat transfer starts at x = 0:

x e t

Thus, the thermal entry length for the flow of cold water (Pr  10) can be

over 600 diameters in length near the transitional Reynolds number, andoil flows (Pr on the order of 104) practically never achieve fully developedtemperature profiles

A complete analysis of the heat transfer rate in the thermal entry gion becomes quite complicated The reader interested in details shouldlook at [7.3, Chap 8] Dimensional analysis of the entry problem shows

re-that the local value of h depends on uav, µ, ρ, D, c p , k, and x—eight

variables in m, s, kg, and J K This means that we should anticipate fourpi-groups:

NuD = fn (Re D , Pr, x/D) (7.25)

In other words, to the already familiar NuD, ReD, and Pr, we add a new

length parameter, x/D The solution of the constant wall temperature

problem, originally formulated by Graetz in 1885 [7.6] and solved in venient form by Sellars, Tribus, and Klein in 1956 [7.7], includes an ar-rangement of these dimensionless groups, called the Graetz number:

con-Graetz number, Gz≡ ReD Pr D

Figure 7.4 shows values of NuD ≡ hD/k for both the uniform wall

temperature and uniform wall heat flux cases The independent variable

in the figure is a dimensionless length equal to 2/Gz The figure also

presents an average Nusselt number, NuDfor the isothermal wall case:

NuD ≡ hD

k = D k

1

L

0 NuD dx (7.27)

3The Nusselt number will be within 5% of the fully developed value if xe t 

0.034 ReDPrD for Tw= constant The error decreases to 1.4% if the coefficient is raised from 0.034 to 0.05 [Compare this with eqn ( 7.12 ) and its context.] For other situations,

the coefficient changes With qw = constant, it is 0.043 at a 5% error level; when the

ve-locity and temperature profiles develop simultaneously, the coefficient ranges between about 0.028 and 0.053 depending upon the Prandtl number and the wall boundary con- dition [ 7.4 , 7.5 ].

Figure 7.4 Local and average Nusselt numbers for the

ther-mal entry region in a hydrodynamically developed laminar pipe

flow

where, since h = q(x) [T w −T b (x)], it is not possible to average just q or

∆T We show how to find the change in T b using h for an isothermal wall

in Sect.7.4 For a fixed heat flux, the change in T bis given by eqn (7.17),

and a value ofh is not needed.

For an isothermal wall, the following curve fits are available for the

Nusselt number in thermally developing flow [7.4]:

For fixed q w, a more complicated formula reproduces the exact result

for local Nusselt number to within 1%:

Example 7.2

A fully developed flow of air at 27◦ C moves at 2 m/s in a 1 cm I.D pipe.

An electric resistance heater surrounds the last 20 cm of the pipe and

supplies a constant heat flux to bring the air out at T b = 40 ◦C What

power input is needed to do this? What will be the wall temperature

7.3 Turbulent pipe flow

Turbulent entry length

The entry lengths x e and x e t are generally shorter in turbulent flow than

in laminar flow Table 7.1 gives the thermal entry length for various

values of Pr and ReD, based on NuDlying within 5% of its fully developed

value These results are based upon a uniform wall heat flux is imposed

on a hydrodynamically fully developed flow

For Prandtl numbers typical of gases and nonmetallic liquids, the

en-try length is not strongly sensitive to the Reynolds number For Pr > 1 in

particular, the entry length is just a few diameters This is because the

heat transfer rate is controlled by the thin thermal sublayer on the wall,

which develop very quickly Similar results are obtained when the wall

temperature, rather than heat flux, is changed

Only liquid metals give fairly long thermal entrance lengths, and, for

these fluids, x e t depends on both Re and Pr in a complicated way Since

liquid metals have very high thermal conductivities, the heat transfer

rate is also more strongly affected by the temperature distribution in the

center of the pipe We discusss liquid metals in more detail at the end of

this section

When heat transfer begins at the inlet to a pipe, the velocity and

tem-perature profiles develop simultaneously The entry length is then very

strongly affected by the shape of the inlet For example, an inlet that

in-duces vortices in the pipe, such as a sharp bend or contraction, can create

Table 7.1 Thermal entry lengths, x et /D, for which Nu Dwill be

no more than 5% above its fully developed value in turbulent

Table 7.2 Constants for the gas-flow simultaneous entry

length correlation, eqn (7.31), for various inlet configurations

NuD

Nu∞ = 1 + (L/D) C n for Pr= 0.7 (7.31)where Nu∞ is the fully developed value of the Nusselt number, and C and

n depend on the inlet configuration as shown in Table7.2.Whereas the entry effect on the local Nusselt number is confined to

a few ten’s of diameters, the effect on the average Nusselt number maypersist for a hundred diameters This is because much additional length

is needed to average out the higher heat transfer rates near the entry.The discussion that follows deals almost entirely with fully developedturbulent pipe flows

Illustrative experiment

Figure7.5shows average heat transfer data given by Kreith [7.9, Chap 8]for air flowing in a 1 in I.D isothermal pipe 60 in in length Let us seehow these data compare with what we know about pipe flows thus far.The data are plotted for a single Prandtl number on NuD vs ReD

coordinates This format is consistent with eqn (7.25) in the fully oped range, but the actual pipe incorporates a significant entry region.Therefore, the data will reflect entry behavior

devel-For laminar flow, NuD  3.66 at Re D = 750 This is the correct value

for an isothermal pipe However, the pipe is too short for flow to be fullydeveloped over much, if any, of its length Therefore NuDis not constant

Figure 7.5 Heat transfer to air flowing in

a 1 in I.D., 60 in long pipe (afterKreith [7.9])

in the laminar range The rate of rise of NuDwith ReDbecomes very great

in the transitional range, which lies between ReD = 2100 and about 5000

in this case Above ReD  5000, the flow is turbulent and it turns out

that NuD  Re 0.8

D

The Reynolds analogy and heat transfer

A form of the Reynolds analogy appropriate to fully developed turbulent

pipe flow can be derived from eqn (6.111)

Stx = h

ρc p u ∞ = C f (x) 2

1+ 12.8Pr0.68 − 1 4C f (x) 2

(6.111)

where h, in a pipe flow, is defined as q w /(T w − T b ) We merely replace

u ∞ with uav and C f (x) with the friction coefficient for fully developed

pipe flow, C f (which is constant), to get

St= h

ρc p uav = C f 2

1+ 12.8Pr0.68 − 1 4C f 2 (7.32)This should not be used at very low Pr’s, but it can be used in either

uniform q w or uniform T w situations It applies only to smooth walls

The frictional resistance to flow in a pipe is normally expressed in

terms of the Darcy-Weisbach friction factor, f [recall eqn (3.24)]:

f ≡  head losspipe length

D

u2 av

2

 =  ∆p

L D

ρu2av

2

where∆p is the pressure drop in a pipe of length L However,

τ w = frictional force on liquid

surface area of pipe = ∆p

The friction factor is given graphically in Fig.7.6as a function of ReDand

the relative roughness, ε/D, where ε is the root-mean-square roughness

of the pipe wall Equation (7.35) can be used directly along with Fig 7.6

to calculate the Nusselt number for smooth-walled pipes

Historical formulations A number of the earliest equations for the

Nusselt number in turbulent pipe flow were based on Reynolds analogy

in the form of eqn (6.76), which for a pipe flow becomes

u ∞ with uav and C f (x) with the friction coefficient for fully developed

pipe flow, C...

The friction factor is given graphically in Fig.7.6as a function of ReDand

the relative roughness, ε/D, where ε is the root-mean-square roughness

of