Heat transfer to or from a fluid flowing through a tube
R. Shankar Subramanian
A common situation encountered by the chemical engineer is heat transfer to fluid flowing
through a tube. This can occur in heat exchangers, boilers, condensers, evaporators, and a host
of other process equipment. Therefore, it is useful to know how to estimate heat transfer
coefficients in this situation.
We can classify the flow of a fluid in a straight circular tube into either laminar or turbulent flow.
It is assumed from hereon that we assume fully developed incompressible, Newtonian, steady
flow conditions. Fully developed flow implies that the tube is long compared with the entrance
length in which the velocity distribution at the inlet adjusts itself to the geometry and no longer
changes with distance along the tube.
Reynolds number
The value of the Reynolds number permits us to determine whether the flow is laminar or
turbulent. We define the Reynolds number as follows.
Reynolds number Re =
DV ρ
μ
=
DV
ν
Here, D is the inside diameter of the tube (or pipe), V is the average velocity of the fluid, ρ is
the density of the fluid and μ is its dynamic viscosity. It is common to use the kinematic
viscosity ν = μ / ρ in defining the Reynolds number. Another common form involves using the
mass flow rate m instead of the average velocity. The mass flow rate is related to the volumetric
flow rate Q via m = ρ Q , and we can write Q =
π
4
D 2V . Therefore, the Reynolds number also
can be defined as
Re =
4m
πμ D
The flow in a commercial circular tube or pipe is usually laminar when the Reynolds number is
below 2,300. In the range 2,300 < Re < 4, 000 , the status of the flow is in transition and for
Re > 4, 000 , flow can be regarded as turbulent. Results for heat transfer in the transition regime
are difficult to predict, and it is best to avoid this regime in designing heat exchange equipment.
By the way, turbulent flow is inherently unsteady, being characterized by time-dependent
fluctuations in the velocity and pressure, but we usually average over these fluctuations and
define time-smoothed or time-average velocity and pressure; these time-smoothed entities can be
steady or time-dependent (on a time scale much larger than that of the fluctuations), and here we
only focus on steady conditions when we discuss either laminar or turbulent flow.
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Principal differences between heat transfer in laminar flow and that in turbulent flow
In discussing heat transfer to or from a fluid flowing through a straight circular tube, it is useful
to distinguish between the axial or main flow direction, and the directions that lie in a plane
perpendicular to the tube axis. In that plane, transverse heat flow can be broken into radial and
azimuthal components.
The principal difference between laminar and turbulent flow, as far as heat transfer is concerned,
is that an additional mechanism of heat transfer in the radial and azimuthal directions becomes
available in turbulent flow. This is commonly termed “eddy transport” and is intense, providing
much better transfer of energy across the flow at a given axial position than in laminar flow,
wherein conduction is typically the only mechanism that operates in the transverse directions (an
exception occurs when there are secondary flows in the transverse direction, such as in coiled
tubes). Another difference worthwhile noting is the extent of the “thermal entrance region” in
which the transverse temperature distribution becomes “fully developed.” This region is
relatively short in turbulent flow (precisely because of the intense turbulent transverse transport
of energy), whereas it tends to be long in laminar flow. Heat transfer correlations, based on
experimental results, are typically divided into those applicable in the thermal entrance region,
and those that apply in the “fully developed” region. In the case of laminar flow, it is important
to be aware of this distinction, and normally a laminar flow heat exchanger is designed to be
short, to take advantage of relatively high heat transfer rates that are achievable in the thermal
entrance region. In the case of turbulent flow, the thermal entrance region is short, as noted
earlier, and typically heat transfer occurs mostly in the “fully developed” region. Therefore,
turbulent heat transfer correlations are commonly provided for the latter region.
Laminar heat transfer correlations
A variety of correlations are in use for predicting heat transfer rates in laminar flow. From
dimensional analysis, the correlations are usually written in the form
Nu = f ( Re, Pr,")
μC p ν
hD
is the Nusselt number, f is some function, and Pr =
is the Prandtl
=
k
α
k
number. Here, h is the heat transfer coefficient, k is the thermal conductivity of the fluid, and
C p is the specific heat of the fluid at constant pressure. As you can see, the Prandtl number can
where Nu =
be written as the ratio of the kinematic viscosity ν to the thermal diffusivity of the fluid α . The
ellipses in the right side of the above result stand for additional dimensionless groups such as
L / D , which is the ratio of the tube length to its diameter, and other groups that we’ll discuss as
they occur.
As we noted before, efficient heat transfer in laminar flow occurs in the thermal entrance region.
A reasonable correlation for the Nusselt number was provided by Sieder and Tate.
2
⎛ D ⎞ ⎛ μb ⎞
⎟
⎜ ⎟ ⎜
⎝ L ⎠ ⎝ μw ⎠
1/ 3
Nu = 1.86 Re
1/ 3
Pr
1/ 3
0.14
You can see that as the length of the tube increases, the Nusselt number decreases as L−1/ 3 . This
does not, however, imply that the Nusselt number approaches zero as the length becomes large.
This is because the Sieder-Tate correlation only applies in the thermal entrance region. In long
tubes, wherein most of the heat transfer occurs in the thermally fully-developed region, the
Nusselt number is nearly a constant independent of any of the above parameters. When the
boundary condition at the wall is that of uniform wall temperature, Nu → 3.66 . If instead the
flux of heat at the wall is uniform, Nu → 4.36 , but in this case we already know the heat flux
and a heat transfer coefficient is not needed. Remember that the purpose of using a heat transfer
coefficient is to calculate the heat flux between the wall and the fluid. In the case of uniform
wall flux, we can use an energy balance directly to infer the way in which the bulk average
temperature of the fluid changes with distance along the axial direction.
μb
appears in the above laminar flow heat transfer correlation. We have
μw
defined μ as the viscosity of the fluid. The subscripts “b” and “w” stand for “bulk” and “wall,”
respectively. We know that the bulk temperature of the fluid will change along the tube. The
wall temperature may be constant, or it may vary along the length of the tube. In all cases, we
can use an arithmetic value of the average between the extreme values that occur in the system.
Because the exponent (0.14) is small, the effect of this term on the Nusselt number is not large –
it is only a small correction, and this averaging is quite justified. In fact, for all the other
physical properties such as density, thermal conductivity, and specific heat, we should estimate
values at the average temperature of the fluid between the inlet and outlet.
Notice that a ratio
The Reynolds and Prandtl numbers are raised to the same power in the laminar flow correlation.
Therefore, we can write the correlation as
Nu = 1.86 ( Re Pr )
1/ 3
⎛ D ⎞ ⎛ μb ⎞
⎟
⎜ ⎟ ⎜
⎝ L ⎠ ⎝ μw ⎠
1/ 3
0.14
= 1.86 ( Pe
)
1/ 3
⎛ D ⎞ ⎛ μb ⎞
⎟
⎜ ⎟ ⎜
⎝ L ⎠ ⎝ μw ⎠
1/ 3
0.14
where we have introduced a new group Pe called the Peclet number.
Pe = Re Pr =
DV ν
ν α
=
DV
α
The Peclet number Pe plays a role in heat transfer that is similar to that of the Reynolds number
in fluid mechanics. Recall that the physical significance of the Reynolds number is that it
represents the ratio of inertial forces to viscous forces in the flow, or equivalently, the relative
importance of convective transport of momentum compared with molecular transport of
momentum. Thus, the Peclet number tells us the relative importance of convective transport of
thermal energy when compared with molecular transport of thermal energy (conduction).
3
The author of the textbook recommends the following laminar flow heat transfer correlation
from a book by D.K. Edwards, V.E. Denny, and A.F. Mills for the average Nusselt number for a
tube of length L .
Nuaverage =
haverage D
k
D
L
= 3.66 +
2/3
D⎞
⎛
1 + 0.04 ⎜ Re Pr ⎟
L⎠
⎝
0.065 Re Pr
Unlike the correlation of Sieder and Tate, this result can be used for short or long tubes. Note
that as the length becomes very large, Nuaverage → 3.66 , which is the result for a uniform wall
temperature when the temperature field is fully developed.
The textbook also provides useful information about entrance lengths. For example, the
hydrodynamic entrance length Lef for the friction factor to decrease to within 5% of its value for
fully developed laminar flow conditions is given as
Lef
D
≈ 0.05 Re
Likewise, if the velocity profile in laminar flow is fully developed and we then apply a uniform
wall temperature boundary condition, the thermal entrance length can be estimated from
Leh
≈ 0.033 Re Pr
D
When both the velocity and temperature fields develop with distance simultaneously, the
problem is more involved.
Turbulent flow
The entrance lengths are much shorter for turbulent flows, because of the additional transport
mechanism across the cross section. Thus, typical hydrodynamic entrance lengths in turbulent
flow are 10-15 tube diameters, and the thermal entrance lengths are even smaller. Therefore, for
L
most engineering situations wherein
≥ 50 , we use correlations for fully developed conditions.
D
Correlations for turbulent flow are classified based on whether the interior wall of the tube is
smooth or whether it is rough.
Smooth tubes
The earliest correlations for turbulent heat transfer in a smooth tube are due to Dittus and
Boelter, McAdams, and Colburn. A common form to be used for fluids with Pr > 0.5 is
4
Nu = 0.023 Re0.8 Pr 0.4
The usual recommendation is to use this correlation for Re > 10, 000 , but in practice it is used
even when the flow is in transition between laminar and turbulent flow for lack of better
correlations. A modern correlation that is slightly more accurate is recommended in the textbook
for your use.
Nu =
( f / 8 )( Re− 1, 000 ) Pr
1/ 2
1 + 12.7 ( f / 8 ) ( Pr 2 / 3 − 1)
Mills suggests using this correlation for Reynolds numbers between 3,000 and 106 . Of course, it
should not be used if Pr = 1 , but there are no fluids with that precise value of the Prandtl number.
For low Prandtl number liquid metals, the textbook provides special correlations to be used for
uniform wall temperature and uniform wall flux boundary conditions.
Physical properties to be used in these correlations are evaluated at the average of the inlet and
exit temperatures of the fluid. The friction factor f is the Darcy friction factor, and you can use
Petukhov’s formula for evaluating it.
f =
1
⎡⎣ 0.790 ln ( Re ) − 1.64 ⎤⎦
2
This result is good for turbulent flow in smooth pipes for Re ≤ 5 × 106 .
Rough tubes and pipes
In the case of commercial pipes, roughness of the interior surface is inevitable, whereas drawn
tubes tend to be less rough. The extent of roughness depends on the nature of the surface. Mills
provides a discussion of heat transfer in turbulent flow in rough pipes in Section 4.7.1. The heat
transfer rate is predicted in this case by using a group called the Stanton number
Nu
Nu
.
St =
=
Re Pr
Pe
St =
f
8
1/ 2
⎛ f ⎞
0.9 + ⎜ ⎟
⎝8⎠
⎡ g ( h + , Pr ) − 7.65⎤
⎣
⎦
where the friction factor f is calculated using
5
⎧
⎡k / R
5.02
13 ⎞ ⎤ ⎫
⎛k /R
−
+
f = ⎨−2.0 log10 ⎢ s
log10 ⎜ s
⎟ ⎬
Re
Re ⎠ ⎥⎦ ⎭
⎝ 7.4
⎣ 7.4
⎩
−2
In the above correlations, ks is known as the “equivalent sand grain roughness”
Values of
ks for a variety of pipes, tubes, and other types of surfaces can be found in Table 4.8 in the
textbook. The symbol R represents the inside radius of the pipe.
The function g ( h + , Pr ) is
tabulated in Table 4.9, but you will first need to convert h to h + which is dimensionless. The
symbol h stands for the average height of protrusions from the surface. For equivalent sand
grain roughness, we can use h = ks . For a pipe, the relationship between the dimensionless
quantities (+ variables) and the physical variables is given in Equation (4.140a). For example,
1/ 2
k
+
s
Vk s ⎛ f ⎞
=
ν ⎜⎝ 8 ⎟⎠
so that
1/ 2
h+ =
Vh ⎛ f ⎞
ν ⎜⎝ 8 ⎟⎠
Recall that V is the average velocity of flow, and Mills uses the symbol ub to denote this
quantity in Equation (4.140a).
Non-circular cross-sections
To handle non-circular cross-sections such as annular, triangular, rectangular, and the like, we
use the concept of hydraulic diameter Dh . This is defined as
Dh =
4A
P
where A is the cross-sectional area and P is the wetted perimeter. For a circular tube of
π D2
, and P = π D , so that the above definition yields Dh = D . For turbulent
diameter D , A =
4
flow in non-circular cross-sections, we can use the correlations for circular tubes. The hydraulic
diameter is used in place of the diameter in these correlations. Results for fully developed heat
transfer in laminar flow are given in Table 4.5 for a variety of cross-sections, and a correlation
for flow between parallel plates is given in Equation (4.51) of the text.
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Physical property variations
The Sieder-Tate correlation for laminar flow contains a correction for the variation of viscosity
with temperature. The other correlations given here do not contain an explicit correction. Mills
discusses how to accommodate property variations at the end of Section 4.2, on page 300. The
approach is different for liquids and gases.
For liquids, a correction is made to the value of the friction factor and that of the Nusselt number
by multiplying the values calculated from the correlation using the bulk fluid properties
(evaluated at an arithmetic average temperature between the inlet and outlet temperatures for the
stream) by a suitable factor. This factor, for the Nusselt number correction, is a viscosity ratio
n
⎛ μs ⎞
⎜ ⎟ , wherein the subscripts s and b refer to the surface and bulk, respectively. In the case of
⎝ μb ⎠
the bulk, an arithmetic average between the inlet and the outlet is to be used, and if the surface
temperature varies from the inlet to the outlet, an arithmetic average is to be used as well.
Likewise, the friction factor correction is made by multiplying the value obtained using bulk
m
⎛μ ⎞
average temperatures by ⎜ s ⎟ . Suitable values for use in these corrections are given in Table
⎝ μb ⎠
4.6.
n
⎛T ⎞
In the case of gases, the correction factor for the Nusselt number is ⎜ s ⎟ and that for the
⎝ Tb ⎠
m
⎛T ⎞
friction factor is ⎜ s ⎟ .
⎝ Tb ⎠
7