The term advection is used when referring to a situation where heat is transferred only by the macroscopic motion of the fluid. In reality, in addition to this macroscopic motion, there are still also random molecular level motions in the substance which affect the heat transfer.
The term convection is used when referring to the combined effect of these two mechanisms.
Regarding the relative strengths of advection and conduction mechanisms in convection heat transfer, conduction has a strong effect near the surface where velocity is low as will be illustrated later, and the effect of advection increases farther from the surface. Convection can be classified as either natural or forced convection. In natural convection, the movement of the fluid is caused by buoyancy forces which are generated by density differences in the fluid. Differences in density, in turn, are due to differences in temperature. In forced con-vection, the movement of the fluid is caused by some external force. For example, the move-ment of the fluid can be caused by a pump. (Incropera et al. 2007, 6-8)
Convection is usually associated with situations where heat is transferred between a solid surface and a moving fluid. Essential terms related to such situations are velocity and thermal boundary layers. Consider a situation in which a fluid flow having constant velocity and temperature profiles in the y-direction begin to flow over an isothermal flat plate in the x-direction. The situation is illustrated in figure 7. (Incropera et al. 2007, 348-350)
Figure 7. Development of velocity and thermal boundary layers on a flat plate. (Incropera et al. 2007, 348-350)
The velocity of the fluid molecules in contact with the surface can be assumed to be zero and due to shear stresses, the velocity of the adjacent fluid layer in the y-direction also de-creases. When the flow proceeds in the x-direction, more and more fluid layers are affected by earlier fluid layers. A region where there are velocity gradients in the y-direction is called
a velocity or hydrodynamic boundary layer and its thickness increases until it is so thick that the effect of the previous fluid layer on the next one is insignificant. In addition to the ve-locity boundary layer, a thermal boundary layer with temperature gradients also begins to develop if surface and fluid temperatures differ from each other. Fluid molecules in contact with the surface acquire the surface temperature, and then the heat is transferred to the next fluid layers. Like the thickness of the velocity boundary layer, also the thickness of the ther-mal boundary layer increases as the flow proceeds in the x-direction as heat is further trans-ferred to the fluid in the y-direction. (Incropera et al. 2007, 348-350)
The velocity boundary layer can be either laminar or turbulent in nature. In a laminar flow, viscous forces are strong enough to keep the flow regular so fluid molecules move along certain streamlines. In a turbulent flow, viscous forces cannot keep the flow regular so oc-casional vortices are generated. Those vortices mix the flow so that higher speed molecules from the free stream are carried towards the surface and slower speed molecules from the surface are carried towards the free stream. For this reason, the shape of the velocity profile is flatter in turbulent than in the laminar boundary layer which can be seen from figure 8.
The turbulent boundary layer has three different layers, which are a turbulent region, a buffer layer, and a viscous or laminar sublayer. Usually, when the velocity boundary layer is de-veloping on some surface, the boundary layer is first laminar, then comes transition zone where the flow has both laminar and turbulent properties which share varies as a function of time and finally flow turn to be fully turbulent. In figure 8, boundary layer development from laminar to turbulent on a flat plate is illustrated. (Incropera et al. 2007, 359-361)
Figure 8. Turbulent and laminar velocity boundary layers on a flat plate. (Incropera et al. 2007, 360)
A dimensionless parameter called a Reynolds number describes the nature of the flow, that is, whether the boundary layer has time to change from the laminar to turbulent or whether it is constantly laminar. The equation for the Reynolds number is as follows (Incropera et al.
2007, 360):
π π =ππ€πΏπ
π (3.6)
where π π is the Reynolds number (-), π is the density (kg/m3), π€ is the velocity (m/s), πΏπ is the characteristic length (m) and π is the dynamic viscosity (Paβ s).
Dimension used as a characteristic length in the above equation depends on the case under consideration. For example, in the foregoing case where the fluid flows over the flat plate, a length of the plate is used as a characteristic length. On the other hand, in cases where the fluid flows over or inside a tube, the diameter of the tube is usually used (Incropera et al.
2007, 424, 487). Determination of the flow velocity also differs depending on the flow situ-ation.
Basically, the Reynolds number tells the relative strengths of the inertial and viscous forces acting in the fluid. The boundary layer is called to be laminar when the Reynolds number is small and when the Reynolds number is large enough, the boundary layer is turned to be turbulent. A critical Reynolds number is a term used to describe the border between these two flow types. Because in reality change from laminar to turbulent flow is not sharp but the amount of turbulence grows slowly, any fixed value does not tell the absolute truth. The value range of the critical Reynolds number also depends on the case. For example, if the flow proceeds over the flat plate, the range is from 105 to 3β 106 and the typical fixed value used in calculations is 5β 105. (Incropera et al. 2007, 359-361)
The so-called Newtonβs law of cooling describes the amount of heat transferred by convec-tion as follows (Incropera et al. 2007, 8):
πβ²β² = βπ(ππ β ππ) (3.7)
where βπ is the convective heat transfer coefficient (W/m2K), ππ is the surface temperature (K) and ππ is the fluid temperature (K).
The heat transfer rate is obtained when the heat flux is multiplied by the heat transfer area as follows (Incropera et al. 2007, 99):
π = βππ΄π (ππ β ππ) = ππ βπ1 π βππ΄π
(3.8)
where π΄π is the surface area (m2).
As in the case of conduction, also in the case of convection, the thermal resistance can be found from the electricity analogy. Thus, resistance is as follows:
π ππππ£ = 1
βππ΄π (3.9)
where π ππππ£ is the thermal resistance for convection (K/W).
The value of the convection heat transfer coefficient is dependent on the conditions in the boundary layers. Conditions are affected by surface geometry, nature of the flow, and fluid properties like density, viscosity, thermal conductivity, and specific heat capacity. In gen-eral, convection heat transfer coefficients are higher for forced than for free convection and higher for liquids than for gases. (Incropera et al. 2007, 8, 355)
In a region where thicknesses of the boundary layers are increasing, the heat flux and so also the convection heat transfer coefficient are not constant, but they decrease due to decreasing velocity and temperature gradients near the surface. The heat transfer coefficient is bigger for turbulent than for laminar flow because due to turbulent mixing velocity gradients and then also temperature gradients are higher near the surface. The value of the convection heat transfer coefficient can be calculated as follows (Incropera et al. 2007, 350, 361, 371):
βππππ£= ππ’β π
πΏπ (3.10)
where ππ’ is the Nusselt number (-).
A dimensionless parameter called a Nusselt number describes the dimensionless temperature gradient at the surface and it is a function of the Reynolds number and the Prandtl number (Incropera et al. 2007, 371). Nusselt number correlations for different flow situations have been defined by performing experimental tests (Incropera et al. 2007, 403). The Prandtl
number which is a dimensionless parameter is defined as follows (Incropera et al. 2007, 376):
ππ =πππ
π (3.11)
where ππ is the Prandtl number (-) and ππ is the speciο¬c heat capacity at constant pressure (J/kgK).
In the following subchapters 3.2.1 and 3.2.2, relevant flow situations relating to the calcula-tion model development are covered. Those situacalcula-tions are flow over the tubes and tube banks, and flow inside the tubes.
3.2.1 Flow over tubes and tube banks
Consider a situation where the tube is in a cross-flow as in figure 9.
Figure 9. Tube in a cross-flow. (Incropera et al. 2007, 423)
At the front of the tube, there is a so-called forward stagnation point where the fluid velocity is zero and the pressure reaches its maximum. When the flow proceeds from that point to-wards the zenith, pressure decreases, velocity increases, and boundary layers develop as in the case of a flat plate. At the zenith, the pressure reaches its minimum and velocity its max-imum. After that, pressure starts to increase, velocity starts to decrease, and boundary layers continue their development. Because velocity decreases after the zenith, at some point ve-locity gradient becomes zero at the surface and the fluid near the surface starts to flow re-versal direction forming vortices. At this so-called separation point the velocity boundary layer comes off from the surface and a wake is generated behind the tube where the flow is irregular. This is illustrated in figure 10. (Incropera et al. 2007, 423)
Figure 10. Development of velocity boundary layer on a tube surface. (Incropera et al. 2007, 423)
The location of the separation point depends on the Reynolds number. If the Reynolds num-ber is big, the laminar boundary layer becomes turbulent before the separation point which leads to that separation occurs later because in the turbulent boundary layer fluid has more momentum to resist velocity decreasing. If the Reynolds number is small, the boundary layer is laminar all the time before separation. This is illustrated in figure 11. The critical Reynolds number for this kind of case is 2β 105. (Incropera et al. 2007, 423-424)
Figure 11. Turbulent and laminar velocity boundary layers on a tube surface. (Incropera et al. 2007, 424)
In industrial heat exchangers, several tubes form a tube bank. The tube arrangement of the tube bank can be either staggered or aligned which is illustrated in figure 12.
Figure 12. Aligned and staggered tube bank arrangements. (Modified from source: VDI 2010, 726).
Parameters π π and π πΏ shown in the earlier figure are used to describe perpendicular distances between centerlines of the adjacent tube rows. They are called transverse and longitudinal pitch, respectively (Incropera et al. 2007, 437). When pitches are divided by a tube outer diameter ππ, a transverse pitch ratio π = π π/ππ and a longitudinal pitch ratio π = π πΏ/ππ are obtained (VDI 2010, 726).
In a tube bank, the efficiency of the heat transfer varies. When the flow proceeds inside the tube bank, the efficiency of the heat transfer increases due to increasing turbulence of the flow. The amount of turbulence increases approximately between the first and fifth tube rows and after that stays constant. Heat transfer enhancement inside the tube bank due to turbu-lence also depends on the tube configuration. Heat transfer is more efficient in staggered arrangements because tubes of the next row are not hidden behind the tubes of the previous row, as is the case of aligned arrangements. Therefore, a larger amount of tube surface is directly exposed to the flow. Aligned tube arrangements, where π πΏ/π π < 0,7, that is, adja-cent tube rows in a longitudinal direction are very close to each other, are especially bad from the perspective of heat transfer. (Incropera et al. 2007, 441)
3.2.2 Flow inside tubes
Then, consider a situation where the flow is inside the tubes. When the flow enters the tube, hydrodynamic and thermal boundary layers start to develop as in the case of a flat plate. The hydrodynamic boundary layer is growing until layers from opposite sides of the tube join at the centerline of the tube which is illustrated in figure 13. (Incropera et al. 2007, 486-487)
Figure 13. Development of velocity boundary layer inside a tube. (Incropera et al. 2007, 487)
The beginning part of the tube where the hydrodynamic boundary layer is developing is called a hydrodynamic entrance region. After that region, the flow is called to be hydrody-namically developed laminar or turbulent flow depending on that whether the boundary layer became turbulent before the opposing layers merged. For the flow inside the tubes, the crit-ical Reynolds number range is from 2300 to 10000 and the commonly used fixed value is 2300. The velocity profile of the developed flow is parabolic in the case of laminar flow as in figure 13 but in the case of turbulent flow, it is flatter as was discussed already in chapter 3.2. In a turbulent case, the profile looks similar to what the profile is in the entrance region in figure 13. The entrance region is much shorter in the case of turbulent flow and its end-point cannot be determined so accurate than in the case of laminar flow due to velocity pro-files are so similar in the entrance and developed regions. (Incropera et al. 2007, 486-487) In the case of the thermal boundary layer, similar boundary layer development occurs at the beginning part of the tube than in the case of the hydrodynamic boundary layer. In this case, the beginning part of the tube is called a thermal entrance region. If the surface condition is fixed to constant temperature or heat flux, the flow will reach thermally developed condi-tions at some point. The temperature profile of the developed flow depends on the surface condition. In figure 14 profiles are shown for both constant surface conditions. (Incropera et al. 2007, 492)
Figure 14. Development of thermal boundary layer inside a tube. (Incropera et al. 2007, 492)