Free convection, or natural convection, is a spontaneous flow arising from nonhomogeneous fields of volumetric (mass) forces (gravitational, centrifugal, Coriolis, electromagnetic, etc.):
If density variation Δρ is caused by spatial nonuniformity of a temperature field, then a flow arising in the Earth gravitational field is called thermal gravitation convection. The density variability may also result from nonuniform distribution of concentration of any component in a mixture or from chemical reactions, difference in phase densities or from surface tension forces at the phase interface (in this case concentration diffusion or convection is implied).
Free-convective flows may be laminar and turbulent. A flow past a solid surface, the temperature of which is higher (lower) than that of the surrounding flowing medium, is the most widespread type of free convection. Figures 1 and 2 schematically illustrate characteristic examples of free convection. At the beginning of heating of a vertical surface (x = 0) (Figure 1a) a laminar boundary layer is formed. The layer thickness grows along the flow direction and at a certain distance, corresponding to x_{c1}, the fluid flow becomes unstable changing within the range from x_{c1} to x_{c2} from laminar to turbulent. To this character of flow structure variation there corresponds the change in the coefficient of heat transfer α_{x} which in the case of the developed turbulent FC remains constant along the plate length where the characteristics of thermal turbulence become statistically equal. The pictures of FC development in the flow past a hot sphere or horizontal cylinder are qualitatively similar (Figures 1b, c). On bodies of large diameters (Figure 1c) a turbulent boundary layer develops thus forming an ascending turbulent thermal plume in a trailing edge. From hot bodies of small diameters a laminar thermal plume ascends (Figure 1b) which at some distance from a body becomes turbulent. In narrow and closed cavities a FC flow is much more complex (Figure 2), due to the interaction between near-wall fluid flows formed on the heat exchanging surfaces. On heating one vertical wall (temperature T_{h}) and cooling the other (temperature T_{c1}) the modes with a common fluid flow are possible through the entire cavity that involve local secondary flows near vertical walls as is exemplified in Figure 2a by FC in a square cavity. A flow in narrow slots between parallel vertical plates is formed in the form of periodic circulations (Figure 2b). In a horizontal fluid layer between cold upper (T_{c1}) and hot lower (T_{h}) walls, the fluid flow has a cellular form with hexagonal cells (Benard cells) in the center of which fluid ascends from the hot surface to the cold one whereas in the periphery, it descends (Figures 2c, d). Such a form of the fluid flow was first observed by Benard in 1901. With increasing heat flux the cells are destroyed and the flow converts to a turbulent one.
In the theoretical analysis of FC flows and heat transfer the laws of momentum, mass and energy conservation at certain boundary conditions are used. The Boussinesq approximation of "weak" thermal convection is widely applied, i.e., density deviations from a mean value are considered to be negligible in all the equations, except for the equation of motion where they are taken into account in the buoyancy force term. For small temperature drops in a flow the relation ρ(T) may be considered linear
where ρ_{0} is the fluid density at temperature T_{0}, β = − [∂ρ/∂T]_{p}/p is the volumetric coefficient of thermal expansion.
Numerical values of β are usually small (water: β = 1.5 × 10^{−4}, air: β = 3.5 × 10^{−3} at T = 273 K), therefore the density variation is taken into account only in those cases where it affects the gravitational forces. The Boussinesq approximation correlates the coefficient of volumetric expansion of a medium β with the gravity acceleration g; they enter into the governing equations only as a product. Physical substantiation of the Boussinesq approximation is based on the smallness of accelerations in FC flows as compared with the acceleration due to gravity.
Comparison with vast experimental material indicates the fact that the Boussinesq approximation well reflects the main specific features of thermal gravitation convection in a wide class of real convective flows.
As is shown by experimental data, in many cases of FC the main variations of the characteristics of thermal and hydrodynamic fields are concentrated in relatively narrow boundary layers near the heat transfer surface where viscous forces are commensurable with inertial and volumetric forces. The smallness of a boundary layer thickness as compared with characteristic dimensions of bodies allows one to introduce additional simplifications into the equations of motion and heat transfer.
The concept of a boundary layer is far more complex for FC than for forced convection, because thermal and hydrodynamic problems cannot be treated separately due to the fact that the fluid flow is completely determined by heat transfer. The main motive force (the difference between wall and surrounding temperatures) noticeably manifests itself only in a thin near-wall zone. This region of a temperature field with the thickness δ_{T} is called a thermal boundary layer.
The difference of temperature in a boundary layer creates a volumetric buoyancy force which causes motion. At the surface, the fluid is stationary (the "no-slip" condition). With distance from a wall the velocity u gradually grows to a maximum and then, under the effect of viscous friction, it vanishes (Figure 1a). Beyond the limits of this dynamic boundary layer there is a region of inviscid (potential) flow. The distance along the normal from the wall to the place, where the velocity differs from zero by 1 per cent of the value of u_{max}, is taken as the dynamic boundary layer thickness δ.
When δ_{T} < δ, the motion outside the thermal layer, where the buoyancy force is absent, is determined by the forces of dynamic and turbulent interaction between separate fluid layers.
When δ < δ_{T}, outside the dynamic boundary layer and within the thermal layer δ_{T} the flow may be considered as potential.
A flow in a boundary layer makes a main contribution into the transfer processes, whereas the induced outer flow is secondary and provides only higher order correction. This is the manifestation of the secondary effect of a boundary layer on the flow in the surrounding medium.
It follows from the dimensional analysis that a relative boundary layer thickness δ/x has the order of Gr^{−0.25}, where Gr = gβ(T_{w} – T_{∞})x^{3}/ν^{2}. At very large Grashof numbers characteristic of practical applications of the FC boundary layer theory, the boundary layer thickness is usually very small compared to the body size. Comparatively thick boundary layers take place for media with small Prandtl numbers (Pr) and with small differences between the body and surrounding temperatures.
In the Boussinesq approximation for an incompressible fluid and a steady-state regime, the equations of momentum, mass and energy conservation for laminar FC in a plane boundary layer are as follows
The system of equations (1) allows the determination of the both velocity components (u, v) and of the temperature field (T) for various boundary conditions.
To generalize the solution results or experimental data as well as to reduce the quantity of problem parameters, similarity theory is used.
Some problem parameters are substituted by their combinations, the so-called generalized variables. Their structure depends on the form of differential operators entering into Eqs. (1). We shall reduce the equations to the dimensionless form. It is convenient to use the quantities entering into the unambiguity conditions (boundary conditions) as the reduction scales. As a linear scale we shall take some characteristic dimension of a body L, for a temperature it is convenient to use, for instance, the relation θ = (T – T_{∞})/(T_{w} – T_{∞}), where T_{w} is the body surface temperature, T_{∞}, is the surrounding temperature, T is the local temperature. The characteristic velocity may be obtained from the comparison of volumetric and viscosity forces u_{0} = βgΔTL^{2}/ν or from the estimates of the type u_{0} = L/τ_{0}, where τ_{0} is the time scale.
The dimensionalization yields
The Grashof number Gr = βgΔTL^{3}/ν^{2} is the main governing criterion and the most important characteristic of FC heat transfer. It is the measure of the relation between the buoyancy forces in a nonisothermal flow and the forces of molecular viscosity. It also determines the mode of medium motion along the heat transfer surface. In its physical meaning, it is similar to the Reynolds number for a forced flow.
At small Gr numbers a FC flow is absent and heat transfer is carried out by molecular thermal conduction. In particular, in a horizontal layer (Figure 2c) this takes place at Ra_{δ} = Gr_{δ}Pr = βg(T_{h} – T_{c})δ^{3}/νa < 1708. When Ra_{δ} = 1708, the stability of a horizontal layer is disturbed and a FC fluid flow develops in the form of Benard cells (Figures 2c, d). At Ra_{x} = Gr_{x}Pr ≈ 10^{9} on a vertical plate there takes place the transition from a laminar to turbulent flow (Figure 1a).
Free convection heat transfer, similar to that under forced convection, is characterized by the Nusselt number Nu = αL/λ. This is usually an unknown quantity since it involves the heat transfer coefficient α which should be found. Thus, the dimensionless form of the heat transfer coefficient, Nu, depends on the dimensionless numbers Pr, Gr and the coordinate X = x/L
In the theory of a FC boundary layer a wide use is made of integral relations obtained by averaging the motion and energy equations over the boundary layer thickness. For stationary conditions, with dissipation and compression work being neglected, these equations have the form
The system of equations (3) is not suitable for use in approximate calculations. Multiplying the boundary layer equations by the velocity and integrating we obtain the balance of mechanical energy
subsequent integration. For example, the equations of the first moment are
The most widely used method for processing the results of calculations and experiments is the application of the exponential function between the similarity criteria
where C, m, and n are the constant dimensionless numbers. If, in logarithmic coordinates, all the points fall on a straight line, this forms the basis of the practical method for constructing the exponential function. If the test points fall on a curve, then the single line is substituted by a segmented line. For separate segments of such a curve, the values of C, m, n are different.
To expand the applicability region of the relation of type (4), it may be presented in the form of the sum
If the physical properties of a medium depend on temperature, then equations determining the form of these relations should be among the main equations. In this case, the similarity criterion should be treated as the arguments of correlations. Here the application of the generalized analysis is impossible and one has to restrict oneself to approximate solutions. In particular, if thermophysical characteristics can be represented by exponential functions of temperature, an additional parametric criterion, introduced into relation (4), is presented in the form of the surrounding medium-to-wall temperature ratio, viz.: (T_{∞}/T_{w})^{1}^{l}. Physical parameters should be referred to one of the two characteristic temperatures. This method is applicable to gases.
The dependence of liquid heat transfer on the heat flux direction and temperature difference are approximately allowed for by the introduction of an additional multiplier (Pr_{∞}/Pr_{w})_{1}_{2} into the similarity equation. For fluid heating Pr_{∞}/Pr_{w} > 1; for cooling Pr_{∞}/Pr_{w} < 1. The ratio Pr_{∞}/Pr_{w} the more is different from zero the higher is the temperature head. The variability of physical parameters may be taken into account by parametric simplexes of the type λ_{∞}/λ_{w}, η_{∞}/η_{w}, c_{p∞}/c_{pw}, c_{p}/c_{pw}, etc. as well as by the introduction of the temperature, which is determining for the given process.
Nusselt suggested the averaging of the physical parameters by the equation
and to calculate the determining temperature as log-mean one
When , relation (5) is presented in the form of the power series of ratio T_{∞}/T_{w}. If we restrict ourselves to the first term of the series, then ; in the case of restriction to two terms T = (T_{∞} + T_{w})/2. In the FC problems the determining temperature is often chosen in the form of a linear combination of wall and surrounding medium temperatures:
where a and b are the coefficients varying from 0 to 1: (a + b) = 1.
The determining linear size is usually taken to be that which to a greater extent corresponds to a physical essence of the process (e.g. the plate height, cylinder or sphere diameter, gap or boundary layer thickness, etc.). The other dimensions enter into the similarity equation in the form of simplexes P_{Lk} = L_{k}/L (the width and thickness of a plate, the height of a vertical or length of a horizontal cylinder, gap height). In a number of cases, a combination of heterogeneous physical quantities entering into the unambiguity conditions (the length scale in the asymptotic theory L/Gr_{1/4}, the linear dimension in the case of jet convection) is taken as the determining linear size. To universalize computational relations and eliminate parametric criteria, a common characteristic dimension is introduced. As an example, we shall give: 1/L = 1/a + 1/b for a horizontal plate; πD for a horizontal cylinder; πD/2 for a sphere; D_{hyd} = 4S/P_{l} is the hydraulic diameter for a horizontal channel of an arbitrary cross-section (S is the cross-section area, P_{l} is the wetted perimeter).
For applied problems in the calculation of heat transfer from surfaces of an arbitrary shape in an infinite fluid the equation
is suggested, or in the dimensional form
where the quantities C, A and n depend on Ra_{L} ( = L^{3}gp(T_{∞} – T_{w})β/η^{2}) and the body shape. A mean boundary layer temperature is taken to be the determining temperature. Correction factors are introduced for inclined and horizontal surfaces.
A specific feature of laminar FC on a vertical plate at a constant wall temperature (Figure 1a) is the fact that it allows a self-similar solution if a new variable is introduced into Eqs. (1) in the form
The boundary conditions are
Having represented the stream function and the dimensionless temperature as
we obtain Eqs. (1) in the form
where f'(η_{s}) = df/dη_{s}.
The boundary conditions are
Local heat transfer rate at a distance x from the plate edge may be determined by the formula Nu_{x} = αx/λ = (Gr_{x}/4)^{1/4}H(Pr) (where H(Pr) = 0.75Pr^{1/2}/(0.609 + 1.22Pr^{1/2} + 1.28Pr)^{1/4}) valid for 0 ≤ Pr < ∞. The presented correlation reflects two noteworthy physical facts:
In the case of laminar FC, the coefficient of heat transfer along a vertical surface varies according to the law α(x) = Ax^{−1/4}.
In the limiting cases of Pr → 0 and Pr → ∞, the dependence of the dimensionless coefficient of heat transfer on the Prandtl number has different characters, viz., when Pr → 0 and when Pr → ∞.
The case of large Prandtl numbers corresponds to a very high viscosity and consequently to a slow flow usually called a creeping flow. For such flows the inertia terms in the equation of motion may be neglected and the relation for the Nusselt number Nu has the form F(GrPr). The case of Pr → 0 corresponds to small viscosity thus allowing the neglection of viscous effects in the equation of motion and the relation for the Nusselt number acquires the form F(Gr Pr^{2}).
The mean Nusselt number on a plate with a length x = L is
or .
The constancy of heat flux on a wall ( ) that corresponds to constant heat supply to the heat transfer surface (e.g., in electric heating devices, in the elements of radioelectronic equipment) is a boundary condition important in practice. This case can be easily realized in practice by heating a thin metal foil of a constant thickness with an electric current, therefore this is often used in experiments. In these problems, the wall temperature T_{w} is an unknown quantity. By virtue of the above, a modified Grashof number calculated from the heat flux on a wall, viz.: is taken to be the determining dimensionless parameter instead of the traditional . Here the temperature difference entering into the ordinary Grashof number is replaced by the multiplier . The wall temperature grows along the flow as (T_{w} – T_{∞}) ~ x^{1/5} and, consequently, the dimensionless coefficient of heat transfer, the Nusselt number, at depends on as which compares to the relation for T_{w} = const (as will also be seen by substitution of ). For the given case of in Figures 3 and 4 examples are presented of velocity and temperature distributions in a laminar boundary layer at various Prandtl numbers constructed based on the results by Sparrow and Gregg (J. Heat Transfer. 1956, v. 78, p. 435). These theoretical results are in a good agreement with experimental data.
Figure 3. Dimensionless velocity profiles in free convection boundary layers on flat plates with constant heat flux.
Figure 4. Dimensionless temperature profiles in free convection boundary layers on a flat plate with constant heat flux.
Many FC flows occurring in nature and technology are turbulent, i.e. they have irregular pulsatory character. As compared with a large amount of theoretical and experimental studies of turbulence in forced turbulent flows, turbulence with FC is studied considerably less. However, the basic mechanisms of a turbulent flow are quite similar. Their main difference is in the fact that in FC flows the values of averaged velocities are smaller, while the levels of disturbances are higher than in forced flows. The flow field is related to the temperature field and to study turbulence with FC one requires simultaneous diagnostics of the both fields. This relation greatly complicates both the theoretical analysis and measurements. Experimental data on local heat transfer with developed turbulent FC on vertical and inclined surfaces at within the range of are described by Wlitt and Ross (J. Heat Transfer. 1975, v. 97, p. 549) as
where γ is the angle of surface inclination to the vertical plane. It follows from the equation given that the coefficient of heat transfer with turbulent FC is independent of x. Some data indicate the possibility of the presence of a weak dependence α(x).
For engineering calculations of heat transfer from bodies of different geometry and orientation in space, a variety of dimensionless empirical relations have been suggested.
For a vertical plate surface (Figure 1a) at Rayleigh numbers Ra_{L} = Gr_{L} Pr, varying within the range from 10^{4} to 10^{13} and covering both the laminar and turbulent flow zones, a length-mean Nusselt number is equal to
This equation may be used for liquid metals (Pr < 0.1) when Ra_{L} is substituted by Gr_{L} Pr^{2} according to the above considered case of limiting values of Pr. This expression for may be applied to the determination of mean heat transfer from a vertical cylinder with a height H, if the boundary layer thickness is much smaller than the cylinder diameter D, i.e., D/H ≥ 35/Gr^{1/4}. The effects caused by the body curvature are especially substantial at small and moderate Grashof numbers.
For a horizontal cylinder (Figures 1b, c) mean heat transfer may be determined by the formula
which is valid for Ra_{D} < 10^{12}.
When a cylinder is inclined at some angle j to the horizontal, there appears an axial velocity component and the flow becomes three-dimensional. The growth of the boundary layer with cylinder inclination weakens convective heat transfer. Heat transfer near the lower end of an inclined cylinder of a finite length is determined by relations characteristic of the flow along the cylinder. In the upper portion of the cylinder the flow approaches the case of that around a horizontal cylinder. For small inclination angles this effect is insignificant. For example, decreases by 8 per cent with φj growing from 0° to 45°. When the cylinder axis approaches the normal heat transfer from a cylinder decreases sharply.
The correlation for calculating averaged heat transfer from a sphere to a surrounding medium is presented in the following form
for Pr = 0.7 – 6, Ra_{D} = 10^{−6} – 10^{4}, .
Heat transfer from a horizontal flat surface greatly depends on its position (upwards or downwards) relative to the direction of buoyancy force and also on body dimensions. The greatest coefficient of heat transfer with a free flow near a horizontal plate should be expected at the places of maximum flow rate, i.e., at the plate ends. At rather large plate dimensions the average coefficient of heat transfer ceases to depend on the end effect. The mean heat transfer coefficient may be determined by the formulas
If the hot heat transfer surface is facing down, the resultant flow occurs from the center to the edges and the averaged values of the coefficient of heat transfer are calculated by the expression
King (Mech. Eng. 1932, v. 54, p. 347) obtained a practically important result and suggested a general equation for approximate determination of heat transfer from a body of an arbitrary shape that was found during generalized studies of heat transfer from plates, cylinders, bars, spheres and bodies of other geometric forms. This equation is similar in form to the formula for laminar flow near vertical surfaces. This formula may be used in the absence of more definite data for a body of the given form. The King formula is
The characteristic body dimension L is found from the relation
where L_{h} and L_{v} are the body dimensions along the horizontal and vertical lines. Thus, for a vertical plate it is equal to the height, and for a sphere it is equal to the radius.
In a fluid confined between two vertical surfaces (Figures 2a, b) heat transfer at small Grashof numbers (approximately up to 2000) is mainly performed by heat conduction and the Nusselt number is equal to 1. At large Grashof numbers 10_{6} < Pr < 10^{9}, 1 < Pr < 20 for the cavity aspect ratio H/δ = l – 40 the averaged heat transfer between the vertical surfaces (the horizontal sides of the cavity are thermally insulated) is described by the relation
In a fluid between horizontal surfaces at small Grashof numbers (Gr < 1700) the heat conduction mode is set and the Nusselt number is equal to unity both in the case when the heated surface is below and in the case when it is above. At large Grashof numbers corresponding to the turbulent flow mode which begins when Gr ≈ 5 × 10^{4} there is no dependence on δ and mean heat transfer between the upper heated and lower cooled plates is described by the relation
which is valid for 0.02 < Pr < 8750, 3 10^{5} < Ra < 7 10^{9}.
Study of the dynamics of a thermal FC plume over the heated elements is a special branch of the FC theory. The two idealized models (Figure 5) are considered: in the form of a two-dimensional plume (2) from a linear heat source (1) or in the form of an axisymmetric plume (2) from a point source (1). The plumes arise as a result of continuous heat supply. If heat is released only during a short period of time, then the thus originating flow is called a thermic.
In practice the problem of the interaction between ascending flows having free boundaries and other flows and surfaces is often encountered. In particular, this refers to the cooling of elements of electronic equipment when the flows produced by heat sources placed at different places interact. Therefore, it is important to position these sources in such an order that maximum heat removal can be obtained. Elements with heat sources may be located on a surface which is usually thermally insulated and the resultant flow is caused by the interaction of flows formed by sources arranged in different places (Figure 6). In many of the production processes related to heating one has also to deal with the interaction of flows created by a system of heated elements, in particular, by water-cooling towers, pipe-lines for transportation of hot liquid. Figures 7a and b illustrate the interaction of plumes with equal and different heat supplies and Figures 7c and d show the effect of vertical and curvilinear surfaces on a flow in a plane plume adjacement to them.
REFERENCES
Gebhart, B. (1973) Natural Convection Flows and Stability: Advances in Heat Transfer, v. 9, Academic Press.
Jaluria, Y. (1980) Natural Convection. Heat and Mass Transfer. Pergamon Press.
Referências
- Gebhart, B. (1973) Natural Convection Flows and Stability: Advances in Heat Transfer, v. 9, Academic Press.
- Jaluria, Y. (1980) Natural Convection. Heat and Mass Transfer. Pergamon Press.