F Convection- Internal Flow

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Islamic Azad UniversityKaraj Branch

Chapter 8

Convection: Internal Flow

Entrance Conditions

Must distinguish between entranceand fully developed regions. Hydrodynamic Effects: Assume laminar flow with uniform velocity profile at inlet of a

circular tube.

Velocity boundary layerdevelops on surface of tube and thickens with increasingx. Inviscid region of uniform velocity shrinks as boundary layer grows.

Does the centerline velocity change with increasingx? If so, how does it change? Subsequent to boundary layer merger at the centerline, the velocity profile becomes

parabolicand invariant withx. The flow is then said to be hydrodynamicallyfully

developed. How would the fully developed velocity profile differ for turbulent flow?2


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Mean Velocity

Velocity inside a tube varies over the cross section. For every differentialarea dAc:

Overall rate of mass transfer through a tube with cross section Ac:

where umis the mean (average)velocity

! Can determine average velocity at any axial location (along the x-direction), from knowledge of the velocity profile

(8.1)

(8.2)

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Velocity Profile in a pipe

Recall from fluid methanics that for laminar flow of an incompressible,constant property fluid in the fully developed region of a circular tube (pipe):

(8.3a)

(8.3b)

(8.3c)


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Thermal Considerations: Mean

Temperature

We can write Newtons law ofcooling inside a tube, byconsidering a meantemperature, instead of T

!

where Tmis the mean (average) temperature

(8.4)

For constant rand cp, Tmis defined:

(8.5)

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Fully Developed Conditions

For internal flows, the temperature, T(r), as well as the meantemperature, Tmgenerally vary in the x-direction, i.e.


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A fully developed thermally region is possible, if one of two possiblesurface conditions exist :

Uniform wall temperature (Ts=constant) Uniform heat flux (qx=const)

Thermal Entry Length :

Although T(r) changes with x, the relative shape of the temperatureprofile remains the same: Flow is thermally fully developed.

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It can be proven that for fully developed conditions, the localconvection coefficient is a constant, independent of x:



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Mean temperature variation

along a tube

We are still left with the problem of knowing how the mean temperature

Tm(x), varies as a function of distance, so that we can use it in Newtons

law of cooling to estimate convection heat transfer.

Recall from Chapter 1, page 10 that by simplifying the energy balance for

flow inside a control volume

where Tm,iand Tm,oare the mean temperatures of the inlet and outlet

respectively

For flow inside a pipe: (8.6)

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Mean temperature variation

along a tubeP=surface perimeter

For a differential control volume:

where P=surface perimeter

=pD for circular tube,

=width for flat plate

! Integration of this equation will result in an expression for the variationof Tmas a function of x.

(8.7)


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Case 1: Constant Heat Flux

Integrating equation (8.7):(8.8)

where P=surface perimeter

=pD for circular tube,

=width for flat plate

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Case 2: Constant Surface

Temperature,Ts=constant

From eq.(8.7)

Integrating for the entire length of the tube:

where(8.10) (8.11)

(8.9)

Asis the tube surface area, As=P.L=pDL,

DTlmis the log-mean temperature difference

with Ts-Tm=DT:


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Case 3: Uniform External

Temperature

" Replace Tsby and by (the overall heat transfer coefficient,which includes contributions due to convection at the tube inner andouter surfaces, and due to conduction across the tube wall). Equations

(8.9) and (8.10) become:

(8.11) (8.12)

Reminder from Chapter 3, p. 19

lecture notes

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Laminar Flow in Circular Tubes

For cases involving uniform heat flux:

For cases involving constant surface temperature:

(8.13)

(8.14)

1. Fully Developed Region

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2. Entry Region: Velocity and Temperature arefunctions of x

Thermal entry lengthproblem: Assumes thepresence of fully developedvelocity profile

Combined (thermal andvelocity) entry length

problem: Temperature andvelocity profiles developsimultaneously

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For constant surface temperature condition:

Thermal Entry Length case or combined entry with Pr"5

Combined Entry Length case

All properties, except msevaluated at average value of mean

temperature

(8.15)

(8.16)

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Turbulent Flow in Circular Tubes

For a smooth surface and fully turbulent conditions theDittus Boelter equation may be used for small to moderate

temperature differences Ts-Tm:

n=0.4 for heating (Ts>Tm)

and 0.3 for cooling (Ts


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The Gnielinski correlation takes into account the friction factor:

Friction factors may be obtained from the Moody diagram.

(8.19)

For small Pr numbers 3x10-3#Pr #5x10-2(i.e. liquid metals)(8.20)

(8.21)

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Turbulent Flow in Circular Tubes

Example (Problem 8.55)

Repeat Problem 8.55. This time the values of the heat transfer

coefficients are not provided, therefore we need to estimate them.

Water at a flow rate of 0.215 kg/s is cooled from 70C to 30C by

passing it through a thin-walled tube of diameter D=50 mm and

maintaining a coolant at 15C in cross flow over the tube.

(a) What is the required tube length if the coolant is air and its velocity isV=20 m/s?

(b) What is the required tube length if the coolant is water is V=2 m/s?

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Non-Circular tubes

Use the concept of the hydraulic diameter:

where Acis the flow cross-sectional area and P the wetted perimeter

! See Table 8.1 textbook for typical values of Nusselt numbers forvarious cross sections

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Example (Problem 8.80)

You have been asked to perform a feasibility study on the design of a

blood warmer to be used during the transfusion of blood to a patient. It is

desirable to heat blood taken from the bank at 15C to a physiological

temperature of 37C, at a flow rate of 200 ml/min. The blood passes

through a rectangular cross-section tube, 6.4 mm by 1.6 mm, which is

sandwiched between two plates held at a constant temperature of 40C.Compute the length of the tubing required to achieve the desired outlet

conditions at the specified flow rate. Assume the flow is fully developed

and the blood has the same properties as water.

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Summary

Numerous correlations exist for the estimation ofthe heat transfer coefficient, for various flow

situations involving laminar and turbulent flow. Always make sure that conditions for which

correlations are valid are applicable to your

problem.

!Summary of correlations in Table 8.4 of textbook

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