Most often the term “tube” in the theory of heat transfer implies a duct with the round (circular) cross section.
In flow at low Reynolds Number (Re) in a long straight tube with a constant cross section and smooth walls, each particle moves with a constant velocity along a rectilinear trajectory. Due to viscosity, the fluid particles, flow near the walls at a lower rate than those far from the walls. The flow is an orderly motion of layers relative to each other. This flow is known as laminar. A shear stress which is determined by the Newton law of viscosity τ_{1} = η(du/dy) originates between these layers. However, at higher Re numbers the flow is no longer orderly. An intense cross mixing of fluid changes the flow into a turbulent one.
O. Reynolds was the first to study consistently the different patterns in laminar and turbulent flows. According to his investigations, the transition of the laminar flow pattern to the turbulent one occurs at a critical Reynolds number Re_{c} = (ū d/ν)_{c} = 2300. Re_{c} significantly depends on the conditions at the tube inlet and fluid flow to the inlet. V. V. Ekman, by carefully reducing disturbances at the tube inlet, obtained Re_{c} = 40000. There also exists a lower limit of Re_{c} corresponding to about 2000 below which even the strongest disturbances decay.
The experimental studies have shown that in a definite Re range the flow in the vicinity of Re_{c} is of an intermittent nature, i.e., first laminar and then turbulent. This flow can be described by the intermittence coefficient γ indicating over which fraction of the time the flow is turbulent. Hence, the flow is turbulent at γ = 1 and laminar at γ = 0.1. Rotta established the dependence of γ on the dimensionless distance from the tube inlet x/D (here, x is the distance and D the tube diameter) for different Re (see Figure 1). At constant Re, γ monotonically grows with increasing distance x/D. The tube length which is necessary for laminar-turbulent transition diminishes with increasing Re.
Other flow characteristics, including velocity distribution over the cross section, also change over some length of the tube beginning with the inlet. If the fluid flows into the tube out of a big tank, the velocity distribution at the inlet is uniform, but with the distance from the inlet it gradually becomes elongated, under the action of friction forces, taking, at some distance from the inlet, a final shape remaining henceforward unchanged. The tube length, over which the velocity profile changes, is referred to as the entry or initial hydrodynamic length. For laminar fluid flow this length strongly depends on Re and can be determined by the Boussinesq relation l_{in}/D = 0.065 Re.
For turbulent flow, the entry hydrodynamic length is about 50 tube diameters. In what follows we will analyze the developed flow in the tube.
Transition of the laminar flow to the turbulent flow drastically changes the velocity profile and the friction drag law. In laminar flow, the velocity distribution over the cross section is parabolic (Figure 2, curve 1), while in the turbulent flow, as a result of pulsation exchange in the transverse direction, it is substantially more uniform (curve 2). The turbulent velocity profile can be divided into four characteristic regions (Figure 3): a viscous region (or a linear sublayer) immediately near the tube wall (0 ≤ y^{+} ≤ 5, y^{+} = yu_{f}/μ, ) in which the velocity variation is linear and determined by the coefficient of molecular viscosity; an intermediate, or buffer, layer (5 ≤ y^{+} ≤ 30) in which viscous and turbulent stresses are commensurate in quantity; a fully turbulent (logarithmic) layer (y^{+} > 30, y/r_{0} < 2) in which the molecular viscosity can be neglected and, finally, a turbulent core or the region of validity of the law of the wake in the tube center. The former three regions form the wall layer with a universal (Re-independent) velocity distribution in u^{+} – y^{+} coordinates. The point at which the velocity profile in the flow core deviates from the logarithmic one in the coordinates u^{+} = u/u_{r} = f(y^{+}) depends on Re. In the u/ū = f(y/r_{0}) coordinates it approximately equals y/r_{0} = 0.15 and remains nearly unchanged over a considerable Re range.
For turbulent flow, the shear stress is expressed as the sum of molecular and turbulent stresses
Variation of turbulent shear stresses τ_{t} and turbulent viscosity ν_{t} with y^{+} is shown in Figure 4. The turbulent shear stress depends linearly on y almost everywhere except for the narrow region near the wall (Figure 4, curve 1) since τ_{t}/τ_{w} = (1 – y/r_{0}) – τ_{l}/τ_{w}. The molecular viscosity is commensurate with the turbulent viscosity only near the wall (Figure 4, curve 2).
Using the Prandtl hypothesis for the mixing length l, we can represent turbulent viscosity by the relation
The quantity l characterizes the local properties of the turbulent flow and can be considered as local turbulence scale. The curves of dimensionless mixing path length l/r0 are graphed in Figure 5. Prandtl assumed that the mixing path length increases linearly with increasing distance from the wall: 1 = ky (Figure 5, curve 1). Nikuradze determined experimentally the mixing length for flowing in a tube. The results of his measurements (Figure 5, curve 2) are described by the equation
In the case of short distances from the wall, all the terms in Eq. (3) of a higher order than y/r_{0} can be omitted. This results in 1 = 0.4y. Thus, by Nikuradze's measurement results, k known as the von Karman or turbulence constant turns out to be 0.4. Manifold subsequent measurements of this constant also yield values close to 0.4. Thus, in the wall region the coefficient of turbulent kinematic viscosity is
Now we turn back to the velocity profile (Figure 3). In the viscous sublayer the dimensionless velocity is equal to a dimensionless coordinate u^{+} = y^{+}. Prandtl derived for the wall turbulent region
Equation (5) follows from Eqs. (1) and (4) under the assumption that τ = τ_{1} + τ_{t} = τ_{t} = const. The reliability of this approximation can be estimated using Figure 4. The constants entering Eq. (5) for the first time were determined based on Nikuradze's experiments. As was noted, k = 0.4 and the constant A was found to be equal to 5.5. Hence, the velocity distribution in the turbulent zone is described by the logarithmic dependence
T. von Karman introduced the concept of an intermediate, or buffer, layer. Retaining for the viscous sublayer and for the turbulent zone the above expressions for the velocity profile, he derived the equation
for the range of y^{+} values from 5 to 30.
Reichardt's formula is extensively used for Re > 2 × 10^{4} range to describe the velocity distribution over the entire cross section of the round tube by a unified relation
the coefficient of turbulent viscosity being described by
for y^{+} ≤ 50 and by
for y^{+} > 50.
Van Driest extended Prandtl’s Equation (4) for turbulent viscosity to a viscous layer introducing into the expression for the mixing path length the damping factor
Figure 6 shows the variation of root-mean-square pulsations along the tube radius. The regions of the flow near the axis and near the wall are depicted in two different figures. In one region the velocity scale is u_{0}, the maximum velocity which can be achieved on the tube axis, in the other, the dynamic velocity u_{f}. In the entire region axial pulsations are most intensive. Their maximum value = 0.08 is achieved inside the buffer layer at y^{+} = 15. Radial velocity pulsations are the minimum ones. As is seen from Figures 4 and 6, inside the wall layer turbulence is neither homogeneous, for which the condition must hold, nor isotropic, with condition = 0 would be valid.
For flow in a tube it is customary to define a dimensionless friction factor in terms of the pressure gradient associated with friction (see also Friction Factors, Pressure Drop, Single Phase)
which defines the Moody (or Weissbach) friction factor. The alternative definition, f = c_{f} = τ_{w}/(1/2 ρū^{2} ), called the Fanning friction factor, is also used. Note that = 4f.
For the laminar flow the friction factor may be calculated using the Hagen-Poiseuille relation
where K = 64 for round tubes but depends on the cross section shape of the channel for other channel shapes. The K values for some shapes of cross section other than round are presented in Channel Flow.
If the flow is turbulent, Blasius' equation (for Re < 10^{5})
or Filonenko’s relation (for 10^{4} < Re < 5 × 10^{6})
can be used for determination of .
The above laws hold only for isothermal motion when the fluid temperature and, hence, the physical properties (ρ, η, λ, c_{p}) in all the points of the flow retain each the same value. With heat transfer the fluid temperature varies over both the tube section and length. The temperature variation over cross section brings about the change of physical properties and, as a consequence, the change of the velocity profile and pressure loss. The variation of physical properties of liquids and gases with temperature is different. In liquids, the dynamic velocity coefficient η is the most temperature-dependent value, while other properties depend relatively slightly on temperature. Therefore, in the case of liquids we can confine ourselves to considering only viscosity variation assuming other properties to be constant and equal to the values at some mean temperature. In gases temperature substantially changes density ρ, the dynamic viscosity η, and the thermal conductivity λ. The specific heat cp varies relatively slightly. Therefore, in the case of gas flow at fairly high temperature drops it is necessary to take into account the dependence of on temperature, (ρ, η, and λ, and, sometimes, also c_{p}). The most widespread method of taking into account nonisothermal nature of flow in calculating the pressure loss consists in introducing correction factors into friction factor determined for the case of isothermal flow. The dependence / = (η_{w}/η_{b})^{n} is used for liquids. Here and are the friction factor for nonisothermal and isothermal flows, respectively, η_{w} and η_{b} are the dynamic viscosity at the wall and bulk temperature, respectively. The exponent n depends on the flow regime, the direction of the flux, and other factors. In order to calculate the friction factor for nonisothermal turbulent gas flow we can make use of Kutateladze’s approximate relation / = (T_{b}/T_{w})^{0.5} where T_{b} is the bulk and T_{w} the wall temperature and is determined as the bulk temperature.
In practice, especially at high Re the tubes cannot be always considered hydraulically smooth since elements of roughness can occur on their surface. Roughness can be both natural, due to production process and service conditions, or artificial, i.e., purposefully made on the tube surface (thread, transverse projections, grooves, etc.) in order to augment heat transfer. Two main types of roughness are commonly distinguished: three-dimensional and two-dimensional. The former is produced by roughness elements on the surface such as projections of various shapes about which the flow is three-dimensional. The latter type are projections of various shapes which are continuous around the tube perimeter. The flow around these projections is two-dimensional. In general, the surface roughness is characterized by the height and shape of the elements of roughness closely adjoining one another, e.g., sand grains; one parameter, the height of the element of sand-like roughness ks is sufficient to characterize this roughness. Therefore, in this case the character of flow and resistance depends only on one additional parameter, viz. the roughness ratio k_{s}/r_{0}. In turbulent flow in rough tubes Schlichting distinguished three flow regimes, depending on the dimensionless roughness height or the so-called Reynolds number of the roughness :
A surface with no manifestation of roughness (0 ≤ ≤ 5) in which roughness elements do not project beyond the viscous sublayer and the surface behaves as a hydraulically smooth one. In this case the friction factor is determined from the relations for smooth tubes as given above.
The transition regime (5 ≤ ≤ 70) when surface roughness projects beyond the viscous sublayer and the friction factor depends not only on Re, but on roughness ratio as well.
The fully rough regime ( > 70) under which resistance predominantly consists of resistances of single roughness elements dominate the resistance. In this region, fluid viscosity plays no role and the friction factor becomes a function of roughness ratio alone.
According to Nikuradze, the friction factor of tubes with sand-like in the fully rough regime is expressed as
Near the wall the velocity profile in the rough tube builds up less steeply than in a smooth tube. Distribution of the mixing length along the radius of a rough tube absolutely coincides with the same distribution in smooth tubes. Here too, the distribution of the. mixing length is determined by Eq. (3). In particular, near the wall 1 = ky = 0.4y. This means that the logarithmic law of velocity distribution (Eq. (5)) is also applicable to rough tubes (Figure 7). One should only shift the y coordinate depending on the value of the element of sand-like roughness k_{s}. Consequently, the velocity profile takes the form
Here R( ) is the so-called roughness hydrodynamic function.
The shaded region in Figure 7 depicts the transition region from a hydraulically smooth surface to the one with well-pronounced roughness.
The comparance of velocity profiles for a smooth (6) and a rough (16) walls was used to derive
for sand-like roughness.
Nikuradze's measurement results, for the fully rough regime with sand-like roughness, yield R( ) = 8.5. Figure 8 shows roughness function R( ) versus the dimensionless value of sand-like rouhgness. Zone 1 corresponds to hydraulically smooth surface, zone 2, to the region, and zone 3, to the fully rough regime. Curve 1 is plotted from Eq. (17) and curve 2, corresponds to R ( ) = 8.5.
As was noted above, in contrast to sand-like roughness, real roughness cannot be determined by assigning one parameter of the k_{s} type. However, a technical roughness can be determined as Schlichting suggested, in terms of equivalent sand-like roughness. Equivalent sand-like roughness means the height of the sand-like roughness element k_{s} such that the tube with this roughness which has the same friction factor in the fully rough regime as the tube with a given technical roughness. The equivalent sand-like roughness is found experimentally or from published data. The equivalent sand-like roughness is calculated by measuring the friction factor for a tube with a given roughness type (Eq. (16)).
Figure 9 presents the Moody diagram for commercial tubes in which roughness is expressed in terms of the equivalent sand-like roughness. The dashed curve represents the boundary between the regimes of partial and fully rough regimes, curves 1 and 2 correspond to laminar (Equation (12)) and turbulent (Equations (13) and (14)) flows in smooth tubes. In this case, the friction factor for all the three flow regimes in tubes with natural roughness is well described by Colebrook and White's equation
For k_{e}/r_{0} → 0 Eq. (18) takes the form of Prandtl's formula for hydraulically smooth tubes
which virtually coincides with Filonenko’s Equation (14) though the latter is more convenient for calculations. For Re → ∞ Eq. 18 reduces to Eq. 15 for the fully rough regime.
REFERENCES
Schlichting, H. (1968) Boundary Layer Theory, 6th edn., McGraw-Hill, New York.
Arpaci, V. S. and Larsen, P. S. (1984) Convection Heat Transfer, Prentice-Hall, Inc., Englewood Cliffs.
Les références
- Schlichting, H. (1968) Boundary Layer Theory, 6th edn., McGraw-Hill, New York.
- Arpaci, V. S. and Larsen, P. S. (1984) Convection Heat Transfer, Prentice-Hall, Inc., Englewood Cliffs.