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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
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For a fully developed temperature profile,
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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
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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
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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
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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
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,
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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.
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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
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