Turb Pipe Flow

7/27/2019 Turb Pipe Flow

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Heat Transfer, Ted D. Bennett

Copyright 2013 13-1 (last proof read: 4/13)

13 Turbulent convection

13.1 Reynolds averaged equations of transport

The so called Reynolds averaged equations of turbulent transport describe time averagedconditions of a flow. To obtain time averaged equations, all of the variables describing the flow

are decomposed into mean and fluctuation components. For example:

average fluctuating

x x x = + ,

average fluctuating

T T T= + , and so forth. (13-1)

After decomposition, transport equations are time averaged such that the resulting equations

describe only the time averaged conditions of transport. These turbulent equations look very

similar to the laminar equations of transport, except for two important differences. First, time

averaged variables are used to describe the behavior of the flow. Time averaging captures the net

effect of transport in turbulence, without detailing all of the turbulent behavior. For example, the

time averaged velocity x would not reflect any rapid fluctuations in the flow speed caused by

turbulence, but would reflect average rate at which the flow is advancing in the x direction.

The second difference arises from the influence of turbulent fluctuations on the time averaged

transport of quantities like heat and momentum. To illustrate, consider the Reynolds averaged

transport equations momentum and heat transfer in an incompressible boundary layer flow:

Mom:

( )

small scalediffusionturbulent avection

x x xx y y x

dP

x y y y dx

+ = +

(13-2)

Heat:

( )

small scalediffusionturbulent avection

x y y

T T TC k CT

x y y y

+ = +

(13-3)

Both equations are in a form suitable for boundary layers because streamwise diffusion terms (in

the x direction) have been neglected (see Section 11.2). In both equations, large scale

advection in the flow is described with respect to the time averaged velocities (x

,y ).

However, velocity fluctuations can also advect heat and momentum on a much smaller scale.For example, ( )y x , appearing in Eq. (13-2), describes the average rate at which the x -

component of momentum (per unit volume) is advected in the y -direction by turbulent

fluctuations in the fluid velocity. The over bar is meant to indicate that there should be some

correlation in the fluctuations fory and x to have a net effect on transport (in this context,

primes do not reflect derivatives). Likewise, ( )y CT , appearing in Eq. (13-3), describes the


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average rate at which heat (per unit volume) is advected in the y -direction by turbulent

fluctuations in the fluid motion. Notice that if the small scale turbulent advection terms were

dropped from Equations (13-2) and (13-3), and the notation for time averaged variables replaced

with standard notation, these equations would appear equivalent to the laminar boundary layer

equations.

To put the significance of turbulence into perspective, consider the fact that for a laminar

boundary layer, the ability to bring heat away from the surface relies entirely upon molecular

diffusion. However, in a turbulent boundary layer, the small scale rolling motion of turbulence

can rapidly bring heat away from the surface. In the time averaged perspective of the flow, this

process is viewed as diffusion since the small-scale turbulent motion permits differences in

concentrations of heat and momentum to interact in an analogous fashion to molecular diffusion.

To formalize this analogy, the small scale turbulent advection fluxes appearing in Eqs. (13-2)

and (13-3) can be equated to the prototypical form of diffusion fluxes:

( )( )

( )( )

x xy x M M

y H H

d d

dy dy

d u dT CT C

dy dy

= = = =

(turbulent transport) (13-4)

In these expressionsM and H are the turbulent diffusivities for momentum and heat

transfer. Notice the similarity of the turbulent diffusion laws to their molecular counterparts:

( )( )

( )( ) x x x

d u dT dT heat C k

dy dy dyd d d

momentumdy dy dy

= = =

= = =

(molecular transport) (13-5)

13.2 The mixing length model for turbulent diffusivity

The mixing length model is among the simplest for estimating

the turbulent diffusivities. In this model, the turbulent

diffusion flux is related to a mixing length scale . The

fluctuations in turbulence can be pictured as a rolling motion

on a scale . Suppose that this turbulent motion exists in the

flow illustrated in Figure 13-1. When a fluid element

undergoes this rolling motion, it interacts with changes in the

average flow speed over a length scale . One can expect

that the fluid element will experience a fluctuation in velocity

that will scale with the difference in average flow speeds

sampled with the scale of the rolling motion:

y

x

x

( )x y

Figure 13-1: Illustration of

Prandtls mixing length scale.


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( / )y x x y (13-6)

From this scaling arguments fory and x , the small scale advection of momentum can be

estimated as:

2

( ) x x xy x

y y y

=

(13-7)

The turbulent momentum diffusivity M can be defined by equating the momentum flux given in

Eq. (13-7) with the expected form of a diffusion flux given in Eq. (13-4). The result is:

2

x xM

y y

=

, (13-8)

or2 x

My

=

. (13-9)

Equation (13-9) is the expression for turbulent diffusivity (of momentum) used in the mixing-

length model. One shortcoming of the mixing length model is that it suggests that turbulent

diffusivity goes to zero at points of symmetry in the flow (where / 0x

y = ). (This effectively

suggests that the center line of duct flow is laminar.) Although this is not true, this fallacy in the

model can often be overlooked since turbulent transport fluxes will go to zero along lines of

symmetry (even though diffusivities do not).

To complete the mixing length model requires quantification of the mixing length. One

approach is to use the van Driest function:

[1 exp( / )]y y A + +

= ; ( R ) (13-10)

where/

wyy

+= . (13-11)

The van Driest function is well supported by experimental data on turbulent flows. In the van

Driest function, the wall coordinate y+

is a dimensionless presentation of distances from the wall

that uses the wall shear stress w as a scaling factor. Since the wall shear stress is unknown, it

must be determined as part of the flow solution.


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There are three empirical constants

that are needed to use the van

Driest function (13-10). The first

concerns the geometric restriction

that R . In a pipe, the largest

scales of the mixing length should

not exceed the pipe radius R . (In a

boundary layer, R would be

replaced by scale of the boundary

layer thickness .) The best

empirical choice of the constant

0.09 suggests that that the

length scale of mixing should

remain an order of magnitude

smaller than the pipe radius. The second constant is von Krmn constant 0.4. As shown inFigure 13-2, over an intermediate region of distances from the wall, the dimension of the mixing

length scales with distance from the wall. The von Krmn constant is the proportionality

constant for this scaling. The final constant used in the van Driest function (13-10) is the

damping coefficient A+ . Approaching the wall the mixing length must vanish. As a

consequence there is a thin region of laminar flow near the wall, known as the viscous sublayer.

The extent of the viscous sublayer is specified by the parameter 26A+

.

13.2.1 Turbulent heat diffusivity

It is reasonable to suspect that turbulent mixing is relatively indiscriminate as far as the transportof fluid properties (heat and momentum) is concerned. Therefore, to first order one might

suspect turbulent heat diffusivity is roughly the same as turbulent momentum diffusivity

H M . Although away from the viscous sublayer, this argument is reasonably good,

approaching the surface this assumption fairs less well. As an attempt to correct for this, a

turbulent Prandtl number can be defined, which is the ratio of turbulent momentum diffusivity to

turbulent heat diffusivity. An empirical correlation for the turbulent Prandtl number is given by:

1 exp( / )Pr 0.85

1 exp( / )

Mt

H

y A

y A

+ +

+ +

=

. (13-12)

0.0001

0.001

0.01

0.1

1

0.001 0.01 0.1 1

y

R=

Damping (A+ parameter )

/y R

R

Figure 13-2: Characteristics of the mixing length scale

(confined to a pipe).

Distance from pipe wall:


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This correlation uses a ratio of the van Driest

damping functions quantifying the relative

extent of the viscous sublayer (using the A+

parameter) in comparison to an analogous heat

diffusion sublayer (using the A+

parameter).Although the extent of the viscous sublayer ( A

+

) is pretty much independent of fluid properties,

the heat diffusion sublayer parameter A+ does

depend on the fluids (molecular) Prandtl

number, as indicated in Figure 13-3. For air (

Pr 0.7= ) the best choice of the damping

constant is about 32A+

= . For smaller Prandtl

numbers A+ increases steeply. For Pr 30> , A

+

decreases steeply. However, for Prandtl

numbers 0.9 Pr 30 , the damping constant falls in the range 30 31A+

.

13.3 Momentum equation for turbulent pipe flow

The steady-state turbulent equations for an incompressible pipe flow are:

Continuity: ( )1

0z rrz r r

+ =

(13-13)

Mom:1 1

( ) ( )z z z zr z M M dP

rr z r r r z z dz

+ = + + +

(13-14)

Because turbulent diffusivity changes spatially, the terms expressing the sum of the molecular

and thermal diffusivities are left inside the spatial derivatives.

For fully developed turbulent flow, the velocity profile is characterized by an axial velocity that

does not change with downstream distance ( 0z z = ). Additionally, there is no flow in the

radial direction ( 0r = ). Therefore, the momentum equation for fully developed turbulent pipe

flow simplifies to:

1 10 ( ) zM

Pr

r r r z

= +

(13-15)

The first integration step can be accomplished analytically:

2

1

1( )

2

zM

Pr r C

r z

+ =

(13-16)

0.1 1 10 100

26

28

30

32

34

Figure 13-3: Heat transfer damping function

constant dependency on Prandtl number.

A+

Pr


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Since, at r R= the turbulent diffusivity must go to zero ( 0M = ), the first integration constant

can be evaluated as:

2

1

1

2

z

r R

PC R R

r z

=

=

(13-17)

Therefore, the momentum equation becomes:

( )2 21

( ) 02

z zM

r R

Pr R R r

r r z

=

+ + =

(13-18)

A force balance between the wall shear stress and the pressure gradient requires:

2 2 z

r R

PR R

z r

=

=

or

1 1

2

z

r R

P

z R r

=

=

(13-19)

Using Eq. (13-19) the momentum equation (13-18) can be express in terms of the shear at the

wall, rather than in terms of the pressure gradient. In this manner, the momentum equation

becomes:

( )2 21

( ) 0z z zM

r R r R

r R R r r r R r

= =

+ + =

, (13-20)

which simplifies to

(1 ) 0M z zr R

r

r R r

=

+ = . (13-21)

Defining the dimensionless variables:

z

m

u

= and 1

r

R = , (13-22)

the momentum equation can be expressed as:

0

(1 ) (1 ) 0M u u

=

+ =

(13-23)

or

(1 ) (0)

1 M

uu

=

+

, (0) 0u = . (13-24)


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Equation (13-24) is a first order ordinary differential equation on which the no slip condition at

the wall (0) 0u = is imposed. However, a second condition is required to determine the

unknown value of (0)u appearing in the momentum equation. The second condition is related

to the fact that the velocity variable is normalized by the mean velocity ( /z mu = ). This

requires that:

2

02

R

z mrdr R = , or2

02

R

u rdr R= , or1

2

02 (1 )u R Rd R = (13-25)

Or,

1

02 (1 ) 1u d = (13-26)

Therefore, the momentum equation (13-24) is integrated with values of (0)u that are guessed

using the shooting method until the integration constraint (13-26) is satisfied.

13.3.1 Solution with the mixing-length model

To solve the momentum equation (13-24) still requires evaluation of the turbulent diffusivity

using the mixing length model expression (13-9). The ratio of turbulent to molecular diffusivity

is evaluated with the mixing-length model:

22 22(2 ) Re

2 2

m mM z DRu u

ur R R

= = = =

(13-27)

Substitution of Eq. (13-27) for the turbulent diffusivity into the momentum equation for pipe

flow (13-24) yields an equation that can be algebraically solved for u , yielding:

2

2

1 2 Re (0)(1 ) 1

Re

D

D

udu

d

+ =

(13-28)

This is the final form of the momentum equation in the mixing length model. During integration

of this equation, the dimensionless mixing length is evaluated from the van Driest function

(13-10). To determine the wall coordinate y+, the wall shear stress is determined from the

variables of the solution using

0

(0)w m mz

r R

d duu

dr R d R

= =

= = = . (13-29)

Consequently,


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/ (2 ) (0)(0) (0) Re / 2

2

w m mD

y Ry y uy u u

R R

+

= = = = , (13-30)

and the van Driest function becomes, in dimensionless form:

Re1 exp (0)

2

D

v

uR A

+

= =

; ( ) (13-31)

Equation (13-31) for the van Driest function is evaluated with Script 13-1.

Script 13-1: Dimensionless mixing length scale (LamEq.m)

function Lam = LamEq(eta)

global du0 ReD Ap Kappa Gamma

yp = eta*sqrt(ReD*du0/2.); % Eq. (13-30)

Lam = Kappa*eta*(1-exp(-yp/Ap)); % Eq. (13-31)

if (Lam>Gamma)

Lam = Gamma;

end

Turbulent pipe flow is solved by integrating momentum equation (13-28) coupled to the

integration constraint (13-26) that ensures the velocity variable is properly normalized. The

integrand of the integral (13-26) defines the differential equation:

1 2 (1 )dg

ud

= , 1(0) 0g = (13-32)

Therefore, normalization of the velocity variable u requires that1(1) 1g = . The turbulent pipe

flow is solved by simultaneously integrating Eq. (13-28) for u and Eq. (13-28) for 1g . These

coupled equations are defined in Script 13-2.

Script 13-2: Fully-developed turbulent pipe flow equation (TurbPipeEq.m)

function dU = TurbPipeEq(eta,U) % where U(1)=u U(2)=g1

global du0 ReD

dU = zeros(2,1);

Lam=LamEq(eta); % Script 13-1

dU(1) = (sqrt(1+2*Lam^2*ReD*du0*(1-eta))-1)/Lam^2/ReD; % Eq. (13-28)

dU(2) = 2*U(1)*(1-eta); % Eq. (13-32)

The correct value of (0)u appearing in the momentum equation (13-28) is guessed by the

shooting method, where the integration condition 1(1) 1g = is used to evaluate each guess. This

shooting routine is performed by Script 13-3.

Script 13-3: Shooting script for solving turbulent pipe flow (SolveTurbPipe.m)

function [eta, U]= SolveTurbPipe(eta)

global du0;


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du0_low=0.; % low bound on du(0)

du0_high=1e6; % high bound on du(0)

initU = [0, 0];

iter=0;

while ( iter < 200 && (du0_high-du0_low)>1.0e-6 )

du0=(du0_low+du0_high)/2.;

[eta,U] = rk4(@TurbPipeEq,eta,initU); % (rk4 can be replaced by ode45)

% Bi-section method (slow but stable)

if U(end,2) > 1 du0_high=du0;

else du0_low= du0;

end

iter=iter+1;

end

After solving for the turbulent pipe flow velocity profile u , the friction factor can be evaluated

from the solution. The definition for the friction factor is given by:

2( / )/ 2mdP dz Df

= (13-33)

Using Eq. (13-19), the pressure gradient needed to evaluate the friction factor can be determined

from the shear stress at the wall:

0

( / ) 1 1

2

mz

r R

dP dz u

R r R R

= =

= =

(13-34)

Substituting Eq. (13-34) into Eq. (13-33), the friction factor can be determined from the

dimensionless velocity gradient at the wall by:

2

0

4 ( / ) 1616 (0)

2 (2 ) Rem m D

D dP dz uf u

R

=

= = =

. (13-35)

The friction factor, calculated from the mixing length model, may be determined as a function of

the Reynolds number using the commands:

octave:1> global Kappa=0.4 Gamma=0.09 Ap=26

octave:2> global ReD du0

octave:3> N=200; % numerical mesh

octave:4> LogEta0=-6; % log of eta distance to first node

octave:5> LogEtaN=0; % log of eta distance to centerline

octave:6> etaspan=logspace(LogEta0,LogEtaN,N);

octave:7> ReD_span=logspace(2,7,20);

octave:8> n=1;

octave:9> for ReD=ReD_span,

> [eta,U]=SolveTurbPipe(etaspan); % Script 13-3

> fMix(n)=16*du0/ReD; % Eq. (13-35)


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> n=n+1;

> endfor

octave:10> fCor=(.8686*log(ReD_span./(1.964*log(ReD_span)-3.8215))).^-2; % Eq. (13-36)

octave:11> loglog(ReD_span,fMix,ReD_span,fCor,ReD_span,64./ReD_span);

octave:12> axis([100 1e7 .001 1])

Notice that integration is done with a mesh evenly divided in log-space. This is required to get a

sufficient number of nodes close enough to the wall to resolve the steep changes that occur there.

In Figure 13-4, the results of the mixing length model are contrasted with the limiting behavior

of laminar flow, given by 64/Relam

Df = , and the turbulent flow correlation:

2

Re0.8686ln

1.964 1nRe 3.8215

turb D

D

f

=

(13-36)

The mixing length model demonstratesagreement with the limiting behavior of

both the laminar flow result and the

turbulent correlation. However, the mixing

length model does not bridge the laminar

to turbulent flow transition with a

physically correct picture. Generally, one

does not expect a gradual transition

between laminar and turbulent behavior to

occur over a large spread of Reynolds

numbers. Instead, as the average velocity

of a laminar flow is slowly increased, one

would expect the friction factor to follow

the laminar line until an abrupt jump to

turbulence occurs.

13.4 Heat Transfer in turbulent pipe flow of constant wall temperature

The heat equation written for a steady turbulent flow through a pipe is:

1

( ) ( )r z H H

T T T T

rr z r r r z z

+ = + + +

(13-37)

Further simplifications to the heat equation can be made by observing that for fully developed

conditions, 0r = , and by making the boundary layer scaling arguments, the axial diffusion term

may be neglected relative to the radial diffusion term. Therefore, the equation for fully-

developed heat transfer in turbulent pipe flow simplifies to:

0.001

0.01

0.1

1

102 103 104 105 106 107

Correlation Eq. (13-36)

laminar flow

Mixing length model

ReD

Figure 13-4: Friction factor calculated from mixing

length model contrasted with laminar behavior and

turbulent correlation (13-36).


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1( )z H

T Tr

z r r r

= +

(13-38)

The same concept of a fully developed temperature profile developed in Section 12.1, for

laminar flows applies to turbulent flows. Furthermore, the fully-developed temperature profile

leads to the same relationship between the axial change in temperature /dT dz and axial change

in mean temperature /mdT dz as found for laminar conditions (see Eq. 12-19) for constant wall

temperature conditions. This relation expressed for turbulent flows is:

0s s m

s m s m

T T T T dT dT

z T T dz T T dz

= =

, when .sT const = (13-40)

Likewise, streamwise changes in mean temperature in turbulent flows are governed by the same

energy balance use for laminar flows in Section 12.3 (see Eq. 12-24). That energy balance leads

to the following relation for turbulent flows that expresses the change in mean temperature inrelation to the convection of heat from the walls of the pipe:

2 ( )m s m s m

p net m p

dT T T h T T

dz mC r R C

= =

. (13-42)

Combining Eq. (13-40) with Eq. (13-42) gives

2 ( )s

p m

h T TdT

dz R C

= , (13-43)

which when applied to the heat equation (13-38) yields:

2(2 )

( ) 1z Hsm

h R R T T T r

k r r r

= +

(13-44)

The heat equation is made dimensionless by defining the variables:

1 r

R = , z

m

u

= , m

s m

T T

T T

=

,

(2 )D

h RNu

k= . (13-45)

In terms of these new variables, the heat equation becomes:

(1 )(1 ) (1 )(1 )HDu Nu

= +

. (13-46)

The heat equation may be integrated once analytically for:


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0 0

(1 )(1 ) (1 ) 1 (1 ) 1 (0)H HDNu u d

= + = +

(13-47)

However, the Nusselt number is related to the wall temperature gradient:

(2 ) 2 22 (0)

( )

sD

s m s m r R

qh R R R T Nu

k k T T T T r

=

= = = =

(13-48)

Therefore, with the relationship between (0) and DNu , as expressed by Eq. (13-48), (0) can

be eliminated from Eq. (13-47), which puts the heat equation into the form:

( )2 ( ) 1

2(1 )(1 / )

D

H

Nu g

=

+, where 2

0

( ) 2 (1 )(1 )g u d

= . (13-49)

The normalized velocity variable u and temperature variable used in the heat equation willimpose a constraint on the problem with respect to the newly defined function 2( )g . By

definition, the mean dimensional temperature associated with the flow is expressed by:

1 1

0 0 0

1 1

0 0 0

2 (1 ) (1 )

2 (1 ) (1 )

R

z z m

m R

z z m

T rdr T R Rd Tu d

T

rdr R Rd u d

= = =

. (13-50)

However, changing the temperature variable in Eq. (13-50) to yields:

1 1

0 0

1 1

0 0

( ) (1 ) (1 )

( )

(1 ) (1 )

m s m

m m s m

T T T u d u d

T T T T

u d u d

+ = = +

. (13-51)

It is clear the Eq. (13-51) can only be satisfied if:

1

0

(1 ) 0u d = . (13-52)

Therefore, looking at the constraint function 2( )g , as defined in the heat equation (13-49), it

becomes clear that integration of this function must yield:


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01

1 1 1 1

22

0 0 0 0

(1) 2 (1 )(1 ) 2 (1 ) 2 (1 ) 1g

g d u d u d u d

==

= = = =

(13-53)

In Eq. (13-53), the term evaluating to 1, does so by Eq. (13-26), and the term evaluating to 0,

does so by Eq. (13-52). Therefore, to solve the heat equation (13-49) will require determining

the correct value of DNu such that the constraint 2 (1) 1g = is satisfied.

The last impediment to integrating the heat equation (13-49) is the need to evaluate the turbulent

heat diffusivity. This can be done by relating the turbulent heat diffusivity to the momentum

diffusivity with the definition of turbulent Prandtl number Pr /t M H = , such that

Pr Pr

Pr Pr

t tH H M M M

M

= = = . (13-54)

The turbulent Prandtl number is determined from correlation (13-12), and the turbulent

momentum diffusivity is determined from its definition using Eq. (13-27) for pipe flow. The M-

File defining the turbulent Prandtl number correlation is given in Script 13-4

Script 13-4: Turbulent Prandtl number correlation (PrtEq.m)

function Prt = PrtEq(eta,Ap,ApH)

global du0 ReD

yp = eta*sqrt(ReD*du0/2.); % Eq. (13-30)

Prt=0.85*(1-exp(-yp/Ap))/(1-exp(-yp/ApH)); % Eq. (13-12)

The system of equations that solves for turbulent flow heat transfer in a constant wall

temperature pipe is summarized by:

2

2

1 2Re (0)(1 ) 1

Re

D

D

uu

+ =

, (0) 0u = (13-55)

1 2 (1 )dg

ud

= , 1(0) 0g = , with (0)u yielding 1(1) 1g = (13-56)

( )2 ( ) 1

2(1 )(1 / )

D

H

Nu g

=

+, (0) 1 = (13-57)

2 2 (1 )(1 )g

u

=

, 2 (0) 0g = , with DNu yielding 2 (1) 1g = (13-58)

This system of equations is comprised of the turbulent momentum equation (13-55) and heat

equation (13-57) for flow through a constant wall temperature pipe. Coupled to these equations


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are an equation (13-56) for the constraint function 1g that is required by the normalization of the

velocity variable u , and an equation (13-58) for the constraint function 2g that is required by the

additional normalization of the temperature variable . If the turbulent momentum equation has

been solved prior to seeking a solution to the heat equation, then (0)u is presumed to be known

and Eq. (13-56) for 1g may be dropped from the reaming system of equations to be solved. This

is assumed to be the case in defining equations in Script 13-5 needed to solve for heat transfer in

turbulent pipe flow.

Script 13-5: Fully-developed turbulent pipe flow heat equation (TurbPipeTEq.m)

function dUT = TurbPipeTEq(eta,UT) % where UT(1)=u, UT(2)=T, UT(3)=g2

global du0 ReD NuD Pr Ap ApH Kappa Gamma

dUT = zeros(3,1);

Lam=LamEq(eta); % Script 13-1

dUT(1) = (sqrt(1+2*Lam^2*ReD*du0*(1-eta))-1)/Lam^2/ReD; % Eq. (13-55)

if (eta1.0e-6 )

NuD=(NuD_low+NuD_high)/2;

[eta,UT] = rk4(@TurbPipeTEq,eta,initUT); % (rk4 can be replaced by ode45)

% Bi-section method (slow but stable)

if max(UT(:,3)) > 1 NuD_high=NuD;

else NuD_low= NuD;

end

iter=iter+1;

end


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Solutions for the Nusselt obtained from the mixing length model can be checked again the

turbulent correlation:

( )( )

( ) ( )1/2 2/ 3

/ 8 Re 1000 Pr

1 12.7 / 8 Pr 1

D

D

fNu

f

=

+ (13-59)

This comparison can be made with the following script commands:

octave:1> global Kappa=0.4 Gamma=0.09 Ap=26

octave:2> global Pr ReD du0 NuD ApH

octave:3> N=200; % steps in numerical domain

octave:4> LogEta0=-6; % log of eta distance to first node

octave:5> LogEtaN=0; % log of eta distance to centerline

octave:6> etaspan=logspace(LogEta0,LogEtaN,N);

octave:7> Pr=7;

octave:8> ApH=31;

octave:9> ReD_span=logspace(4,6,3);

octave:10> n=1;

octave:11> for ReD=ReD_span,

> [eta,U]=SolveTurbPipe(etaspan); % Script 13-3

> fMix(n)=16*du0/ReD; % Eq. (13-35)

> [eta,UT]=SolveTurbPipeT(etaspan); % Script 13-5

> NuDMix(n)=NuD;

> n=n+1;

> endfor

octave:12> loglog(ReD_span,(NuDMix./Pr).*(1+12.7*(fMix/8).^0.5*(Pr^(2/3)-1)),'o');

octave:13> hold onoctave:14> loglog(ReD_span,fMix.*(ReD_span-1000)/8);

octave:15> hold off

A favorable comparison between the

mixing length model calculation of

DNu , and its relationship to the friction

factor asserted by correlation (13-59),

is made for a range of Prandtl numbers

and Reynolds numbers in Figure 13-5.

101

102

103

104

104 105 106(NuD/Pr)[1+1

2.7

(f/8)1(Pr

-1)]

3( / 8)(Re 10 )Df

Pr = 0.7 ( ) 32

7.0 ( ) 31

70. ( ) 28

A+

Figure 13-5: Correlation for turbulent pipe flow

Nusselt number.

ReD


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13.5 Problems

Problem 1

Turb Pipe Flow

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Transcript of Turb Pipe Flow