Turbulent Heat Transfer Based on Fluid-Friction Analogy
When heat is transferred between a solid and a turbulent fluid heat transfer
rate can be increased by supplying more energy to the pump handling the
fluid
P U1.8 (approx)
h U0.8
The increase in wall shear stress (skin friction) is accompanied by an
increase in rate of heat transfer.
Reynolds Analogy
Reynolds postulated that the laws governing momentum and heat transfer
were the same.
According to his theory the movement of heat between a surface and a fluid
follows the same law as the movement of momentum between the surface
and a fluid
Based on the Reynolds analogy the temperature distribution of the fluid
1
inside a pipe can be derived as
Tw − Tb1 2 fL Nu f
ln = St = =
Tw − Tb2 dw Re Pr 2
Reynolds analogy agrees well with turbulent heat transfer data on fluids
which have a Prandtl number close to 1.
2
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Prandtl Analogy
Reynolds analogy does not take into account the velocity distribution
across the tube.
Prandtl extended Reynolds work by considering the velocity distribution in
the laminar sublayer.
The thickness of the laminar sublayer is δ1
Velocity at the edge of the laminar layer is u δ1
The temperature of the wall is Tw
Temperature of the fluid at the edge of the laminar layer is T δ
1
In the laminar sublayer H and M are both = 0
In the turbulent core H = M
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Prandtl Analogy
f /2
St =
1 + 5 f / 2 (Pr − 1)
When Prandtl number is 1 this will be equal to Reynolds analogy.
Temperature distribution is given by:
Tw − Tb1 2 fL 1
ln =
Tw − Tb2 dw 1 + 5 f / 2 (Pr − 1)
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The Von Karman Analogy
Von Karman obtained a further improvement of Prandtl’s analogy by
making use of the universal velocity distribution.
f /2
St =
1 + 5 f / 2{Pr − 1 + ln[1 + 5 / 6(Pr − 1)]}
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Colburn’s Analogy
The Colburn’s analogy between fluid friction and heat transfer is easier
to apply and yields results which are in agreement with experiment and
of simpler form
f
St Pr 2/3
=
2
Colburn investigated a large number of convection-heat transfer and
pressure-drop data and arrived at the above correlation
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Comparison of various
analogies between heat
transfer and momentum
transfer
When Pr = 1 all equations become
identical.
All equations agree well at high
Prandtl numbers
At low Prandtl numbers there is wide
divergence. Prandtl and Von Karman
gives much lower Nusselt numbers.
Hence new equations have been
developed for predicting heat transfer
coefficient for such fluids
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The Martinelli Analogy
Martinelli emphasized the calculation of Nusselt numbers for liquid
metals but presented relationships which applies to all fluids at all
Reynolds numbers.
The conditions of flow and heat transfer for which Martinelli developed
relationships are:
Velocity and temperature profiles are fully developed
The fluid properties are independent of temperature
There is uniform heat flux along the tube wall
Martinelli assumed that H was proportional to M and determined M
using the universal velocity distribution
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The Martinelli Analogy
Temperature distribution: Three different temperature distribution
corresponding to laminar sublayer, buffer layer and turbulent core.
εH f Tw − Tc
St = εM 2 Tw − Tb
εH εH Re f
5[ Pr + ln(1 + 5 Pr) + 0.5F1 ln ]
εM εM εM 2
Where F1 is a constant and depends on Re & Pr.
The above equation depends on the value of H/ M, which, lacking
actual experimental data is usually taken as unity.
Martinelli assumed H/ M to be independent of the distance from the
wall. This assumption is now known to be incorrect.
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Comparison of Martinelli, Colburn and von Karman
Analogies
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Calculation of Eddy Diffusivities
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Experimental values of Eddy Diffusivities
To simplify calculations H/ M is taken to be unity in most work.
However experimental investigation has shown that H/ M to vary between 1.0
and 1.6.
The ratio H/ M is a function not only of the Reynolds number but of the
position in the cross section of the pipe.
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Effect of Prandtl number on temp distribution in a tube
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