(21 22) internal convection part2

8/13/2019 (21 22) Internal Convection Part2

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INTERNAL FORCED CONVECTION

Associate Professor

IIT Delhi

E-mail: [email protected]


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Constant surface heat flux

con t on

In fully developed region:

as h is constant

P.Talukdar/Mech-IITD 2


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For a fully developed temperature profile,


The shape of the temperature profile remains unchanged in the fully

developed region of a tube subjected to constant surface heat flux


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Constant temperature

boundary condition ,

flowing in a tube can be expressed as

In the constant surface temperature (Ts = constant) case, Tave can be

expressed approximately by the arithmetic mean temperature

difference Tam as


o a goo way s empera ure oes no ncrease near y

Remedy?


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Constant temperature

boundary conditionEner balance ives:

constant

Integrating from x = 0 to x = L

=s

Possible to find out temperature at any x

by replacing As = pxP.Talukdar/Mech-IITD 6


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Note that the temperature difference

exponentially in the flow direction, and the

rate of decay depends on the

magnitude of the exponent hAx /m.Cp

This dimensionless parameter is called the

number of transfer units, denoted by

NTU, and is a measure of the effectiveness

of the heat transfer systems



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Logarithmic mean

temperature difference

is the logarithmic mean temperature difference. Note


i s - i e s - e

differences between the surface and the fluid at the inlet

and the exit of the tube, respectively.


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Laminar Flow in TubesEnergy balance gives,



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Velocity distribution

Hence,

f(r) and g(x)

B.C.

Solution:



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Velocity profile for fully

developed flow

Mean Velocity

Maximum at r = 0

Pressure drop


22

328

D

LV

R

LVp mm

=

=


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Friction Factor

= P

In practice, it is found convenient to

express the pressure drop for all types of

internal flows laminar or turbulent flows,

2

2mV

D

L

circular or noncircular tubes, smooth or

rough surfaces) as328 LVLV mm =

=

DR



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Poiseuilles Law

Pumping Power

Volume flow rate

o seu e s aw

For a specified flow rate, the pressure drop andthus the required pumping power is proportional

to the length of the tube and the viscosity of the


,

power of the radius (or diameter) of the tube


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In 1939 to cover the transitionally rough range, Colebrook

com ne e smoo wa an u y roug re a ons n o a

clever interpolation formula:

.

was plotted in 1944 by Moody into what is now called the

Moody chart for pipe friction . The Moody chart is probably

.accurate to 15 percent for design calculations over the full

range



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Temperature profile for a fully

developed flow



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Integrating, Applying the B.Cs.



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Circular tube, laminar (constant heat flux)


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friction factor

non-circular tubes



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For a circular tube of length L subjected to constant surface

tem erature the avera e Nusselt number for the thermal entrance

region can be determined from (Edwards et al., 1979)

This relation assumes that the flow is hydrodynamically developed ,

approximately for flow developing hydrodynamically

When the difference between the surface and the fluid temperatures

is large [Sieder and Tate (1936)]



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For smooth tubes the friction factor in turbulent flow can be

determined from the explicit first Petukhov equation[Petukhov (1970)]

The Nusselt number in turbulent flow is related to the friction factor

through the ChiltonColburn analogy expressed as

Once the friction factor is available, this equation can be used conveniently


.


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For fully developed turbulent flow in smooth tubes, a simple relation for the

Nusselt number can be obtained by substituting the simple power law relation

f = 0.184 Re-0.2 for the friction factor into

The accuracy of this equation can be improved by modifying it as

where n = 0.4 for heating and 0.3 for cooling of the fluid flowing through

the tube. This equation is known as the DittusBoelter equation [Dittus

and Boelter 1930


(21 22) internal convection part2

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