Forced flow in tubes is used widely in practical applications (see Tubes, Single-Phase Flow In). If the velocity and temperature distribution of fluid are uniform at the inlet to the tube, dynamic and thermal boundary layers begin to develop symmetrically along the wall (Figure 1). The boundary layer thickness increases along the tube length and gradually fills the entire flow section. After the dynamic boundary layers are joined, a constant velocity distribution sets in which is parabola-shaped for the laminar flow (at Re ≤ 2300) or a distribution characteristic of the turbulent flow (see Tubes, Single-Phase Flow In). The distance from inlet to this section is known as entrance length or hydrodynamic stabilization section.

In heat transfer, as the fluid flows along the tube, the wall layers are heated or cooled. In this case, at the entrance region of the tube, the fluid core retains a temperature equal to that at the inlet T_{0} and does not participate in heat transfer. Temperature variation occurs in the wall layers. Thus, near the tube surface in the entrance region a thermal boundary layer is initiated whose thickness, as it becomes farther away from the inlet, grows. The thermal boundary layers are joined at a certain distance from the inlet known as the thermal entrance region L_{T} and afterwards all the fluid participates in heat transfer. After that, the dimensionless temperature profiles θ = (T − T_{w})/(T_{0} − T_{w}) become similar, i.e., the temperature in different sections differs only in magnitude, while the law according to which it varies across the radius remains the same.

For the turbulent flow P. L. Kirillov and co-authors obtained, by analogy with the *universal velocity profile* (see Tubes, Single-Phase Flow In), a *universal temperature distribution of* the form (Figure 2)

Here A = Pr_{t}/k = 2.3, k is the turbulence constant (k = 0.4), Pr_{t} the turbulent Prandtl number taken here equal to 0.9. B(Pr) and T^{+} is given by the dimensionless temperature (T^{+} = (T_{w} – T)ρc_{p}u_{r}/q_{w}).

The coefficient B(Pr) increases greatly with increasing Pr due to diminishing fluid heat conduction. As Pr increases, an ever greater fraction of the thermal resistance occurs in the viscous sublayer.

Kader’s formula is convenient to use which describes the temperature profile by a unified relation

where Г =
and β(Pr) = (3.8Pr^{1/3} − 1.3)^{2} + 2.12ln Pr.

In a manner similar to the coefficient of turbulent viscosity

we can introduce a parameter for turbulent heat transport, i.e., the coefficient of turbulent thermal conductivity. In a thin shear layer, the most important of the three fluctuating components of turbulent heat transport, is ρc_{p}v'T'
, which is the rate of turbulent enthalpy transfer along y. The turbulent thermal diffusivity for this value κ_{t} =
. There is virtually no absolute analogy in turbulent flow between heat and momentum transfer, since the term containing the fluctuating pressure component enters the equations defining the velocity field, but does not enter the temperature-field-defining equations. However, the ratio ν_{t}/κ_{t} which is said to be the turbulent Prandtl number (in analogy with the molecular number Pr = ν/κ = ηc_{p}/λ) is commonly assumed as the value of the order of one

The heat transfer coefficient α is determined by the temperature difference between the bulk fluid temperature T_{b} and the wall temperature T_{w}

where
is the heat flux at the wall. The bulk temperature is given by

Owing to the fact that in the thermal entrance region the temperature gradient decreases faster than the temperature drop, the heat transfer coefficient α = –λ(dT/dy)_{w}/(T_{b} − T_{w}) diminishes tending to a constant value that is characteristic of the fully developed flow (see Figure 3a). If we have a laminar flow consecutively developing into a turbulent one, variation of the heat transfer coefficient over the tube length is different as in Figure 3b, where the heat transfer coefficient decreases in the laminar flow region and the grows again before reducing to its fully developed (constant) value for the turbulent region.

It has been shown analytically that for laminar flow of a fluid with constant physical properties and uniform temperature at the inlet, for T_{w} = const in the heated region and where the flow is hydrodynamically fully developed before heating commences, L_{T} is given by

i.e., at a given Re, the length of *thermal entrance region* is determined by the value of Pr. In calculating heat transfer coefficient under these conditions for gases and liquids the relationships may be used

where Pe = RePr.

When Pe^{−1} x/D → ∞,
corresponding to the region of fully developed heat transfer.

In laminar flow, heat transfer depends on the thermal boundary condition. Thus, when the heat heat flux on the wall is constant (
= const), which is often the case in practice, the limiting Nusselt number Nu = 4.36, the length of the thermal entrance region L_{T} = 0.07Pe, and heat transfer in this case is generally by approximately 20% higher than under the boundary condition T_{w} = const.

The correlations for laminar flow presented above describe heat transfer when there is fully developed flow at the inlet. If the heated region starts at the entrance of the tube and the hydrodynamic and heat transfer boundary layers develop stimultaneously (as in the case for the results shown in Figure 1) then heat transfer in the developing region will be slightly higher than during fully developed flow.

A number of generalizing dependences are suggested for calculating fully developed heat transfer in the turbulent flow. For practical calculations it is advisable to use an equation suggested by B. S. Petukhov and V. V. Kirillov:

where
= (1.82ln Re – 1.64)^{−2}, k = 1.07.

This relationship was refined by Petukhov and co-authors, k = 1.07 was replaced by k = 1 + 900/Re. With allowance for this correction the equation may be used for 10^{4} ≤ Re ≤ 5·10* and 0.2 ≤ Pr ≤ 200.

Empirical dependences of the form

where c, m, and n are constants, are also extensively used. For instance, c = 0.023, m = 0.8, n = 0.4 according to *Dittus and Boelter's* data, while c = 0.021, m = 0.8, n = 0.43 according to Mikheev's data.

The length of the thermal entrance region during turbulent flow is by far shorter than during laminar flow. With increasing Pr the Nusselt number in the thermal entrance region diminishes and differs slightly from Nu_{∞} at Pr >> 1 (Figure 4).

In laminar flow in tubes of noncircular cross section heat transfer varies drastically with changing shape. For turbulent flow Nu can be calculated by the equations for circular tubes if an equivalent diameter D_{e} = 4S/P is used as a characteristic parameter, where S is the cross sectional area and P the wetted perimeter. However, this is justified only in the case where heat transfer occurs across the entire wetted perimeter P.

The direction of heat flux (fluid heating or cooling) effects heat transfer if the physical properties of fluid within the temperature range considered vary substantially. For heated gases this influence can be taken into account using an additional factor (T_{w}/T_{b})^{n}, where (T_{w}/T_{b}) is the temperature factor. Within 0 < T_{w}/T_{b} < 2 for monatomic and diatomic gases the exponent n = −0.5 in turbulent flow. The effect of temperature factor on heat transfer to polyatomic gases is slightly weaker. For gas cooling (T_{w}/T_{b} < 1) it can be assumed that
. For liquids, the effect of temperature-dependent properties on heat transfer can be taken into account using the factor (η_{w}/η_{b})^{m}, where η_{b} is the dynamic viscosity of the bulk fluid temperature, η_{w} the value at the wall temperature. The constant m depends on the direction of heat flux. For fluid heating, i.e., at η_{w}/η_{b} < 1, m =−0.11, for fluid cooling, i.e., η_{w}/η_{b} > 1, m = −0.25.

For the effect of thermogravitational convection on the flow and heat transfer in tubes see Mixed convection.

Artificial two- and three-dimensional roughness is often used for augmentation of heat transfer. As a result of flow separation, a vortex zone appears behind the projection whose extension to the point of reattachment to the surface is 6–8 k, where k is the height of roughness element. A strong turbulent flow is initiated behind this zone. A small vortex region with 1–2 k extension also is initiated before the next projection. If the distance between projections is small, smaller than 6 k, then the vortex zones fill all the width of recession and heat transfer augmentation is ineffective. Investigations evidence that in order to afford the most effective heat transfer the distance between projections must be within 10–12 k. The higher the Prandtl number, the smaller the height of roughness elements for effective augmentation. Investigations show that in the fully rough regime (see Tubes, Single-Phase Flow In) heat transfer rate is virtually independent of the height of roughness elements, though the pressure loss monotonically falls with diminishing k. This means that in designing heat exchangers it is advisable to use projections of the minimum height which still affords the effect of roughness to manifest itself to a full degree. Achieving rational artificial roughness makes it possible to increase the heat transfer coefficient two- to threefold. It is desirable, however, that heat transfer enhancement not involve a too great increase of pressure loss. Because of this, the elements of two-dimensional roughness with sharp edges (of rectangular or triangular shape) are not advantageous. Powetful vortex zones and high resistance of the shape arising behind them lead to high energy losses. It is more advantageous to use correctly streamlined roughness elements whose resistance is much lower and for which heat transfer in the fully rough regime remains at the same level.

Secondary flows initiated in *bent tubes* substantially augment *heat transfer* due to the action of centrifugal forces. An especially high (severalfold) enhancement of heat transfer is observed at Re corresponding to the region of laminar-vortex flow which in straight channels is appropriate to the laminar or the transition flow regime. In curvilinear channels centrifugal forces are oriented in a different way with respect to the areas of heat-transfer surface, therefore, the heat transfer coefficient varies greatly around the perimeter of the channel. In laminar-vortex flow heat transfer on the outward generatrix of the coil tube may exceed severalfold heat transfer on the inward generatrix. This ratio for the turbulent flow is lower.

#### REFERENCES

Arpaci, V. S. and Larsen, P. S. (1984) *Convection Heat Transfer*, Prentice-Hall, Inc., Englewood Cliffs.

Kakac, S., Shah, R, K., and Aung, W. (1987) *Handbook of Single-Phase Convective Heat Transfer*. DOI: 10.1016/0255-2701(88)87007-7