This article is concerned with the transfer of thermal energy by the movement of fluid and, as a consequence, such transfer is dependent on the nature of the flow. Heat transfer by convection may occur in a moving fluid from one region to another or to a solid surface, which can be in the form of a duct, in which the fluid flows or over which the fluid flows. Convective heat transfer may take place in boundary layers, that is, to or from the flow over a surface in the form of a boundary layer, and within ducts where the flow may be boundary-layer-like or fully-developed. It may also occur in flows which are more complicated, such as those which are separated, for example, in the aft region of a cylinder in cross-flow or in the vicinity of a backward-facing step. The flow may give rise to convective heat transfer where it is driven by a pump and is referred to as forced convection, or arise as a consequence of temperature gradients and buoyancy, referred to as natural or free convection. Examples are given later in this Section and are shown in Figure 1 to facilitate introduction to terminology and concepts.
The boundary layer on the flat surface of Figure 1 has the usual variation of velocity from zero on the surface to a maximum in the free-stream. In this case, the surface is assumed to be at a higher temperature than the free-stream and the finite gradient at the wall confirms the heat transfer from the surface to the flow. It is also possible to have zero temperature gradient at the wall so there is no heat transfer to or from the surface but heat transfer within the flow. If the flow is laminar, heat transfer from the surface is given by the Fourier flux law, that is:
where q represents the rate of heat transfer per unit surface area, λ is the thermal conductivity, T is the temperature and y is distance measured from the surface. The same expression applies to any region of the flow and also in the case of the adiabatic wall where zero temperature gradient implies zero heat transfer. It should be noted that the surface can be horizontal as shown, with air flow driven by a fan or a liquid flow by a pump, and that it can equally be vertical, with buoyancy providing the driving force for the flow. In the latter case, the free-stream velocity would be zero so that the corresponding profile would have zero values at the wall and far from the wall.
The backward-facing step of Figure 1 results in a more complicated flow and several boundary layers can be identified within the flow as a consequence of separation and reattachment. The details of flows of this type are not well-understood so it is difficult to identify the characteristics of the boundary layers and it can be imagined that the shapes of the velocity and temperature profiles— and therefore of the local heat transfer within the fluid and to the wall—will vary considerably from one location to another. It is known, for example, that the rate of heat transfer can become high at the location of reattachment of the upstream flow on to the surface of the step, as is also the case at the leading edge of a cylinder in cross-flow, but the detailed mechanisms remain incompletely understood and research continues.
It is well known that even comparatively simple geometrical configurations, such as those of Figure 1, can give rise to heat transfer rates which vary considerably depending on the nature of the flow and of the surface. With laminar flows, heat transfer to or from the wall varies with distance from the leading edge of a boundary layer. Turbulent flows can give rise to heat transfer rates which are much larger than those of laminar flows, and are caused by the manner in which the turbulent fluctuations increase mixing; they also affect the heat transfer to and from the surface, especially where the free-stream fluid is able to penetrate to the wall even for short periods of time. The nature of the surface, for example the degree or type of roughness, usually affects heat transfer to or from it, and in some circumstances to a large extent. It is convenient, therefore, to represent the heat transfer at the wall by the expression
where again represents the rate of heat transfer from the wall, this time over unit area of surface; the temperature difference refers to that between the wall and the free stream; and α is the “heat transfer coefficient” which is a characteristic of the flow and of the surface. The two temperatures can vary with x-distance and it can be difficult to identify a free-stream temperature in some complex flows. Typical values of α are shown in Table 1, from which it can be seen that increases in velocity generally result in increases in heat transfer coefficient, so that α is smallest in natural convection and increases to 100 and more on flat surfaces with air velocities greater than around 50 m/s. The heat transfer coefficient is considerably greater with liquid flows and greater again with two-phase flows.
Table 1. Typical values of heat transfer coefficient
|Forced convection; low speed flow of air over a surface
|Forced convection; moderate speed flow of air over a surface
|Forced convection; moderate speed cross- flow of air over a cylinder
|Forced convection; moderate flow of water in a pipe
|Forced convection; boiling water in a pipe
|Free convection; vertical plate in air with 30°C temperature difference
It should be noted that the above equations are expressed in terms of dimensional parameters. And it is easy to see that a combination of the two will lead to a nondimensional parameter αx/λ, where α is a wall heat transfer coefficient, x is a characteristic distance and λ is the conductivity of the fluid; this is known as the Nusselt number and can readily be devised from dimensional analysis as well as from nondimensional forms of conservation equations as suggested in the following section. The heat transfer coefficients of Table 1 can be expressed in terms of this nondimensional number, the Nusselt number, and analytical and correlation equations are usually expressed in this way as will be shown below.
It is also useful to note that the heat transfer coefficient and the Nusselt number can be used to refer to local values at a location x on a surface, or to an integrated value up to the location x.
The concept of dimensional analysis gives rise to several nondimensional groups, to which reference will be made in this section, and it is convenient to introduce them here. In addition to the Nusselt number, reference will be made to the following:
|Pr = ηcp/λ
|Re = ρux/η
|Nu = αx/λ
|St = α/ρcp u = Nu/Pr Re
|Gr = gβ(Tw − T∞)y3/ν2
The Prandtl number is dependent only on fluid properties; the Reynolds number is a ratio of inertial to viscous forces and is relevant throughout the subject of fluid mechanics and convection; the Stanton number is a combination of Nu, Pr, and Re; and the Grashof number characterizes natural convection with the gravitational acceleration, g, and β, the coefficient of volumetric thermal expansion, and is a combination of inertial, u2/y, frictional, vu/y2, and buoyancy, gβΔT, scales. These nondimensional groups may be obtained from conservation equations and are convenient in the representation of results and correlations of experimental data.
It is useful to examine the equations which represent conservation of mass, momentum and energy and these are written below for rectangular Cartesian coordinates with simplification of uniform properties.
The three equations representing conservation of momentum and the equation representing conservation of energy have the same form, with the terms on the left-hand sides representing convection of momentum and energy. It should be noted that these convective terms are nonlinear, thereby presenting difficulties for any solution and that there are four individual parts of convection corresponding to variations in time and in the three directions. The terms on the right-hand side are slightly simplified forms of those representing transport by diffusion together with pressure forces and sources or sinks of thermal energy. Terms for buoyancy may be added as shown in a following section. It is easy to see that nondimensional velocities and distances in the momentum equations will lead to the inverse of Reynolds number and of the temperatures, velocities and distances in the energy equation to a nondimensional group which comprises (1/PrRe). In later sections, these equations will be simplified to deal with convective heat transfer in steady, laminar flows of forced and free convection.
It is evident from the above that there is some similarity between the equations for conservation of momentum and thermal energy so that the solutions of the two equations will have similar forms when the source terms are zero, the Prandtl number is unity and the solutions are presented in nondimensional form. The presence of buoyancy is often limited to the second momentum equation into which an additional term of the form ρβg(Tw − T∞) must be added. Where the surface which gives rise to the temperature difference—and therefore to the buoyant force—is not vertical, the angle of the surface to the direction of the gravitational force must be considered. This will lead to the resolution of forces so that part of the buoyancy term will appear in the first momentum equation with that in the second equation, multiplied by the sine of the angle to the vertical. This will give rise to an additional nondimensional group, the Grashof number.
In the absence of convection terms, the energy equation reduces to that for heat conduction and the momentum equations are no longer relevant where the conduction takes place in a stationary material. Many other simplifications of the above equations are possible, including those for two-dimensional flows and for boundary-layer flows, as will be seen below. Also, it is possible to integrate the equations and, in their simpler forms, this can have some merit; for example, in the integral momentum and energy equations where the dependent variable is devised so as to be represented in terms of one independent variable, and therefore solvable by simple numerical methods. More complicated forms may also exist as discussed in the following section.
Most flows in nature and in engineering equipment occur at moderately high Reynolds numbers, so they are described as turbulent. Thus, the properties of the flow at any point are time dependent with scales which vary from very small, the Kolmogorov scale, to that corresponding to the largest possible dimension of the flow. In a room, for example, the Kolmogorov scale may be of the order of a fraction of 1 mm or less than 1 ms time scale if the velocity is of the order of 1 m/s, and the largest, of the order of several meters or more than 103 larger. There are two important implications for this: the first, that the rate of heat transfer from a surface to a flow will be considerably greater than if the flow were laminar at the same Reynolds number; and secondly, that the conservation equations are even more difficult to solve than for the laminar flow since any numerical solution must now consider physical and time scales which encompass three orders of magnitude. The first means that turbulent convection is important, much more important than laminar convection; and the second, that the conservation equations cannot be solved in their general form except where the boundary conditions allow them to be reduced to simpler forms and even then, with additional problems. This conclusion has led to the widespread use of correlation formulas based on measurements and these, of necessity, encompass limited ranges of flow. Some examples are presented and discussed in the following section. It has also led to widespread attempts to solve complicated forms of the conservation equations with assumptions which represent the turbulent aspects of the flow. The following paragraphs provide an introduction to this approach.
The introduction of Reynolds averaging, that is, to rewrite the time-dependent variables as sums of mean and fluctuating components, to introduce the new dependent variable into the conservation equations and to average overall time results in equations of the form:
where the upper case symbols refer to time-averaged quantities; the lower case, to fluctuating quantities with q, the temperature fluctuations; κ is λ/ρcp; and the overbars, to averages of multiplications of two time-dependent quantities. The equations have been written in tensor notation to render them more compact, but the similarity between the conservation of the time-averaged momentum and energy equations is still evident. The terms representing convection are still on the left-hand side, with diffusion on the right-hand side. There are now two diffusion terms in each equation: one representing laminar diffusion; and the second, the correlations between fluctuating components. There are still five equations, but now there are more than five unknowns since the correlations imply six terms in the momentum equations and three in the energy equation. Thus, it is evident that these equations do not represent a soluble set without assumptions which reduce the number of unknowns to the number of equations. These require models for the Reynolds stresses, , and the turbulent heat fluxes, , and, as shown elsewhere, it is possible to derive equations for these correlation terms. Each gives rise to higher order correlations so that a decision must be made about closure as well as the introduction of model assumptions.
By analogy with laminar flow, it is possible to write the turbulent momentum flux and turbulent heat flux in the forms
and nondimensional forms of these expressions with turbulent viscosity and turbulent conductivity will lead to Reynolds and Prandtl numbers, where the latter is frequently referred to as a turbulent Prandtl number.
The turbulent Prandtl number has found considerable use in engineering calculations of convective heat transfer since it can be assigned a value of unity. With the laminar Prandtl number also near unity for air—and often of secondary importance since laminar diffusion is less important than turbulent diffusion—the momentum and energy equations can be solved once for flows where there is no pressure gradient and no sources or sinks of energy, with similar results if presented in nondimensional variables. This approach applies to complex flows with difficult numerical solutions and to simple boundary-layer flows as will be shown.
With assumptions of high Reynolds numbers and local equilibrium, so that the influence of one region of flow on another is small, it is possible to simplify the time-averaged conservation equations. Assuming two-dimensional boundary layers yields:
where Cμ and Ct are constants, lm is the mixing length for the transfer of momentum and lt is a corresponding mixing length for the transfer of thermal energy. These equations reduce to the effective viscosity and Prandtl number equations referred to above when the length scales and constants are equal and the Prandtl number is unity. Thus, the concept of a turbulent Prandtl number is limited in its applicability, as is that of a turbulent viscosity. But the range of acceptance for engineering calculations remains large.
As will be shown below, the exact solution of the equation appropriate to the laminar flow over a flat plate, where the free-stream and plate temperatures are constant and different, may be written as:
which recognizes the importance of the Reynolds and Prandtl numbers and expresses the heat transfer coefficient in terms of the Nusselt number. The corresponding result for laminar natural convection over a vertical plate with similar boundary conditions is:
In turbulent flows, approximations appropriate to a flat plate with forced convection have led to expressions of similar form; for example,
As a consequence, equations used to represent measurements of complex flows—where analytic and numerical solutions are either impossible or subject to large inaccuracy—tend to have this form. Several examples are provided in the following sections.
Forced convection is associated with flows which are driven by pumps and fans or by the movement of a body through stationary fluids, as in an aeroplane or ship where each has substantial means at its disposal to cause it to move. It is in contrast to natural convection where gravity provides the driving force, although it is possible to have mixed convection in a limited number of flows where the pressure and gravitational forces are of the same order of magnitude, that is Gr/Re2 is approximately unity. All exact analytical solutions are simplified forms of conservation equations and for laminar flows. Some other cases are discussed below.
Boundary Layer Heat Transfer is discussed in the relevant article.
The flow between flat plates is portrayed in Figure 2. It comprises boundary layers which begin at the leading edges, grow on each of the two surfaces until the potential core narrows to zero and then continues towards a fully-developed laminar flow, after which all gradients in the x-direction become zero.
The boundary layers are represented by the boundary layer equations
with boundary conditions
corresponds to a symmetry boundary condition.
In the initial region where the boundary layers are separated by a region of potential flow, the analysis is similar to that for a boundary layer, with the free-stream condition represented by the potential core velocity and temperature. Further downstream, the flow becomes fully-developed so that the velocity and temperature profiles will not change if expressed in terms of appropriate dimensionless quantities. This will be demonstrated below. It is useful to note, however, that there is an intermediate region where there is no potential core and where the flow is not fully-developed. In this region, it is necessary to solve the equations representing conservation of mass and momentum so that each is satisfied; this may require an interactive approach.
In the case of fully-developed laminar flow, the convective terms become zero since
and the momentum equation becomes
with the pressure gradient constant so that integration with the boundary conditions at the wall and on the symmetry line leads to:
and, if one plate moves parallel to the other with a constant velocity U, the solution becomes
In the former case, the temperature profile has the simple form
This, too, may be complicated by considering the effect of viscous heating, which requires the addition of a term of the form to the conservation equation for energy, and—for zero pressure gradient and constant values of U—leads to
This last result must be regarded as an approximation since no account has been taken of variations likely to occur in transport and thermodynamic properties.
The velocity profile for fully-developed laminar flow is a parabola when the walls are stationary, provided that the fluid properties are constant and the velocities are low; it is linear when the pressure gradient is absent and the wall moves with constant speed with respect to the other. The effect of a moving surface is to provide a force which can act against or with that exerted by pressure. This is reflected in the velocities which can be in positive and negative directions. The temperature profile is expressed in terms of surface temperatures and it is clear that the bulk temperature will increase if one or both of the walls is hotter than the initial temperature, T1 Thus, the temperature profile is often expressed in terms of the initial temperature and the bulk-mean temperature, defined as:
where U is the bulk velocity as discussed further below.
Flow and heat transfer in a pipe are of rather more importance than those between parallel plates since they are found more frequently in engineering practice. The flow may again begin at the leading edge so that laminar flow solutions can be obtained as for parallel plates, but this time to equations in cylindrical coordinates and without the prospect of one surface moving with respect to another. At small values of Reynolds number, ρud/η , the length required to achieve fully-developed laminar flow may be given by the expression
and originates from asymptotic solutions of the boundary layer equations. The flow in small-diameter pipes required to achieve these small Reynolds numbers comes from larger diameter pipes or from plenum chambers, so it is likely that boundary layers do not have their origins at the beginning of the small diameter pipe. Rather, there is a sudden contraction for which the flow is properly represented by more complete forms of the conservation equations than their boundary-layer forms. Indeed, the flow may separate inside the pipe with a more rapid movement towards fully-devel-oped conditions than would be the case with attached boundary layers.
The region of developing flow can be small in many cases and fully-developed flow is usually more important than the developing flow. The conservation equations in cylindrical coordinates may be reduced for fully-developed flow in the same way as between two plates, with the result
and this, with boundary conditions
may be integrated to yield
The Moody friction factor, defined as f = − (dp/dx)/0.5ρU2/D, commonly represents the relationship between pressure drop, geometry and fluid properties, and may be deduced for fully-developed laminar pipe flow as:
which is sometimes referred to as the Hagen-Poiseuille friction law.
The energy equation in cylindrical coordinates has the form
and this reduces for fully-developed flow to
where Tb is the bulk temperature, defined as:
Integration of the differential equation with boundary conditions corresponding to symmetry at the centre line, and for the particular condition that
which is independent of the Reynolds and Prandtl numbers, provided the flow remains laminar. An iterative solution is required to solve the equations for the boundary condition
and leads to the result
which shows that the solution depends on the thermal boundary condition.
Of course, the flow will remain laminar only if the Reynolds number is less than around 2 300 or to larger values if it is so free from disturbances that none are available to propagate and cause turbulent flow, as is usually the case. Where turbulent flow occurs because a disturbance has propagated and led to fluctuations in all regions of flow except in the viscous sublayer, the nature of the flow and of the problem has changed. It is possible to return to considering the consequences of the onset of transition and of the transitional region in the context of the boundary layer in the entrance region of the pipe. But the overall effect will be to induce turbulent flow rapidly so that the emphasis is again, and even more so, on the region of fully-developed flow—which now corresponds to turbulent and not laminar flow. It is possible to retain a boundary-layer flow, possibly with transitional regions for some distance, but the common shape of slightly-rounded entrance geometry usually leads to a fully-developed turbulent flow in distances not much more than 50 diameters, and in shorter distances for engineering calculations.
Correlation of measurements of pressure drop against bulk velocity and diameter has led Blasius to propose the expression
which, together with the laminar flow result of
allow Figure 3 to be drawn and in which the laminar-flow result can be extended to Reynolds numbers well in excess of 105, provided care is taken with the nature of the initial conditions, with the smooth surface of the pipe and with the absence of disturbances of any kind. More usually, laminar flow does not exist at Reynolds numbers larger than 2,300, above which transition to the turbulent-flow curve takes place with a transitional region which can be short or long depending on the nature of the disturbances. The friction factor thus varies with the Reynolds number, based on the diameter of the pipe, and with laminar, transitional and turbulent regions as shown in Figure 3.
The skin-friction coefficient (or Fanning friction factor) is related to the friction factor by
so that the coefficient for turbulent flow may be expressed as
with the constants stemming from consideration of experimental results and are, therefore, of limited applicability. Figure 3 shows the variation of friction factor with Reynolds number based on the pipe diameter, and the distinction between those for laminar and turbulent flow. At high Reynolds number, the results become less certain as indicated by the two lines, but the graph is adequate for many design purposes.
Consideration of the similar nature of the equations representing conservation of momentum and energy implies that the variation of Nusselt number will also be dependent upon Reynolds number, together with the Prandtl number where it is different from unity. An example of an expression describing the variation of Nusselt number with turbulent flow in a pipe is:
As with Figure 3 and the friction factor and skin-friction coefficient, uncertainty increases at high Reynolds numbers and also in the transitional region where the difference between the results for laminar and turbulent flows are widely divergent. This may occur over a range of Reynolds numbers depending on the initial and boundary conditions. It should be noted that rough surfaces increase the values of skin-friction coefficient and Nusselt number. Related calculations can be made for noncircular ducts with an hydraulic diameter replacing the geometric diameter.