Free-Convective Heat Transfer - Startseite€¦ · To describe free-convective motion and heat...
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Free-Convective Heat Transfer
Transcript of Free-Convective Heat Transfer - Startseite€¦ · To describe free-convective motion and heat...
Free-Convective Heat Transfer
Oleg G. Martynenko Pavel P. Khramtsov
Free-ConvectiveHeat TransferWith Many Photographs ofFlows and Heat Exchange
ABC
Professor Oleg G. MartynenkoBelarus Academy of SciencesHeat and Mass Transfer InstituteP. Brovka str. 15220072 MinskBelarus
Dr. Pavel P. KhramtsovInternational Center of Excellencefor Research Eng. and Technology (ICERET)Auf dem Gossberg55471 WüschheimGermanyEmail: [email protected] Academy of SciencesHeat and Mass Transfer InstitutePhysical and ChemicalHydrodynamics LaboratoryP. Brovka str. 15220072 MinskBelarus
Library of Congress Control Number: 2005921210
ISBN-10 3-540-25001-8 Springer Berlin Heidelberg New YorkISBN-13 978-3-540-25001-2 Springer Berlin Heidelberg New York
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The authors express gratitude to Irina A. Shikh and Tatyana A. Baranova fortheir help in preparing and editing the text of the book and also for adviceconcerning its style and presentation. Thanks also to Natalya K. Shveeva andGreta R. Maljavskaya for their help in editing the English version of the book.The authors acknowledge the help from Sergey V. Volkov and Victor S. Burakin carrying out the research work.
Oleg G. MartynenkoPavel P. Khramtsov
Contents
1 Basic Statements and Equations of Free Convection . . . . . . . 11.1 Equations and Uniqueness Conditions . . . . . . . . . . . . . . . . . . . . . . 21.2 Boussinesq Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Method of Generalized Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Free-Convective Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . 121.6 Integral Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.7 Loss of Stability and Transition to Turbulence . . . . . . . . . . . . . . 231.8 Outer and Inner Flow Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371.9 Experimental Methods in Free Convection . . . . . . . . . . . . . . . . . . 381.10 Processing of Experimental
and Calculated Data on Heat Transfer . . . . . . . . . . . . . . . . . . . . . 671.11 Basic Similarity Criteria and Parameters
of Free-Convective Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . 69References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2 Free Convection on a Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812.1 Vertical Flat Plate
with a Constant Wall Temperature . . . . . . . . . . . . . . . . . . . . . . . . 882.2 Constant Heat Flow on a Vertical Plane Surface . . . . . . . . . . . . . 942.3 Plane Vertical Plate with a Variable Surface Temperature . . . . 972.4 Plane Vertical Plate with a Variable Heat Flux on a Surface . . 1042.5 Free Convection on a Vertical Surface
in Stratified Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1142.6 Conjugated Problems on Vertical Surface . . . . . . . . . . . . . . . . . . . 1282.7 Discontinuity of Boundary Conditions
on the Vertical Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1392.8 Free Convection Near a Vertical Surface
in a Variable Field of Mass Forces . . . . . . . . . . . . . . . . . . . . . . . . . 1632.9 Free-Convective Heat Transfer
on a Plane Inclined Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
VIII Contents
2.10 Horizontal and Almost Horizontal Surfaces . . . . . . . . . . . . . . . . . 1762.11 Spatial Flow on a Plane Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 1952.12 Compressibility and Variability
of Thermophysical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1982.13 Energy Dissipation and the Work of Compression . . . . . . . . . . . . 2042.14 Effect of Volumetric Heat Generation
on Free Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2082.15 Injection and Suction on a Plane Surface . . . . . . . . . . . . . . . . . . . 208References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
3 Free Convection on Curved Surfaces . . . . . . . . . . . . . . . . . . . . . . . 2193.1 Vertical Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2203.2 Horizontal Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2293.3 Inclined Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2433.4 Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2473.5 Vertical Needle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2523.6 Cylinder of Arbitrary Cross Section and Prism . . . . . . . . . . . . . . 2523.7 Sphere and Spheroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2573.8 Curved Surface of Complex Geometry . . . . . . . . . . . . . . . . . . . . . . 263References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
4 Natural Convection in Enclosures . . . . . . . . . . . . . . . . . . . . . . . . . . 2794.1 Spherical and Cylindrical Cavities . . . . . . . . . . . . . . . . . . . . . . . . . 2794.2 Rectangular Cavities and Interlayers . . . . . . . . . . . . . . . . . . . . . . . 2914.3 Cylindrical Interlayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3104.4 Spherical Interlayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3224.5 Cavities of Complex Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
5 Free Convection in Tubes and Channels,on Ribbed Surfaces and in Tube Bundles . . . . . . . . . . . . . . . . . . 3455.1 Rectangular Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3455.2 Cylindrical Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3595.3 Finned Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3645.4 Tube Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3765.5 Panels with Cellular Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
6 Nonstationary Processes of Free Convection . . . . . . . . . . . . . . . 3936.1 Main Dependences for Calculation
of Unsteady Free Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3936.2 Free Convection in Oscillating Flows . . . . . . . . . . . . . . . . . . . . . . . 411References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
Contents IX
7 Heat Transfer by Mixed Convection . . . . . . . . . . . . . . . . . . . . . . . 4297.1 Effect of Radiation on Free-Convective Heat Transfer . . . . . . . . 4297.2 Combined Free and Forced Convection . . . . . . . . . . . . . . . . . . . . . 434References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
8 Heat Transfer in Mediawith Special Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4758.1 Water at Extreme Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4758.2 Critical and Supercritical State of a Substance . . . . . . . . . . . . . . 4868.3 Rarefied Gases and Evacuated Liquids . . . . . . . . . . . . . . . . . . . . . 4958.4 Convection Induced by Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . 4998.5 Biosystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5018.6 Solidifying Melt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512
1
Basic Statements and Equationsof Free Convection
A body which is brought into a fluid having another temperature is a sourceof disturbance of the equiprobable state of the medium. The elements of thefluid bordering on the body surface assume its temperature, and the process ofheat distribution in the fluid by molecular thermal conductivity – the processof conduction – begins. For a small difference of temperatures, this is the basicmechanism of heat transfer. The arising temperature nonuniformity which isconnected with the nonuniformity of density ∆ρ leads to the occurrence ofupward (downward) flows or convection which transfers heat from the object.
In general, natural-convection heat transfer occurs in a nonuniform fieldof mass forces:
∆→F = ∆(
→ρgt) = ∆ρ
→gt +ρ∆
→gt .
If the density nonuniformity ∆ρ is due to the temperature nonuniformity,then the occuring motion is referred to as thermal gravitational convection.A change in the density can also be due to nonuniform distribution of theconcentration of any mixture component or to chemical reactions (in this casewe speak of concentration diffusion, or convection), to the presence of phaseswith different densities or to surface tension forces at the phase interface, etc.
Natural-convection flows can be induced by both gravitational and othermass forces (centrifugal, Coriolis, electromagnetic, etc.). For example, in ro-tating gas-filled channels the nonuniform mass force field is caused not onlyby density difference, but also by a nonuniform acceleration field.
Motion and heat exchange occurring in an infinite space are called freeconvection. The pressure in the field of thermal nonuniformity and in the zoneof convective flow can be considered constant. Motion and heat exchange ina bounded volume is called natural convection.
Natural-convection flows can be laminar and turbulent. Experimental datashow that in free convection the basic area of thermal and hydrodynamic dis-turbances is concentrated in a rather thin boundary layer of fluid near theheat transfer surface. For example, at the bottom of a heated vertical platea laminar boundary layer is formed. With increase in the height of the plate
2 1 Basic Statements and Equations of Free Convection
the boundary layer thickness increases and thus heat transfer decreases. Ata certain height, the laminar motion is disturbed and becomes turbulent. Inthis region, the flow represents random motion of the masses of the fluidwhose characteristics are described by stochastic functions of space and timevariables. For a part of the heated surface, where the characteristics of ther-mal turbulence become statistically identical, the heat transfer coefficient isindependent of the body dimensions.
For free convection one cannot consider separately thermal and hydrody-namic boundary layers, since the motion of fluid is fully determined by theprocess of heat transfer.
1.1 Equations and Uniqueness Conditions
To describe free-convective motion and heat transfer, the laws of momentum,mass, and energy conservation in the fluid moving under the action of mass,surface, and inertial forces are used. In a rectangular coordinate system theconservation equations have the form [1.1]
ρ
(∂ui
∂τ+ uj
∂ui
∂xj
)= ρFi +
∂
∂xj
[µ
(∂ui
∂xj+
∂uj
∂xi
)]
− 23
∂
∂xj
(µ
∂uj
∂xj
)− ∂p
∂xi, (1.1.1)
∂ρ
∂τ+
∂
∂xj(ρuj) = 0 , (1.1.2)
ρcp
(∂T
∂τ+ uj
∂T
∂xj
)= Qν + βT
(∂p
∂τ+ uj
∂p
∂xj
)+
∂
∂xj
(λ
∂T
∂xj
)
+ µ
[12
(∂ui
∂xj+
∂uj
∂xi
)2
− 23
(∂uj
∂xj
)2]
. (1.1.3)
Closure of system (1.1.1)–(1.1.3) is achieved through the thermodynamicequation of state
ρ = ρ (p, T ) (1.1.4)
and the equations relating the coefficients of viscosity, heat capacity, heatconduction, and volumetric expansion to pressure and temperature.
The system of differential equations should be augmented with uniquenessconditions to single out the considered process from the whole class of thephenomena described by the system of differential equations (1.1.1)–(1.1.3).
The geometric conditions specify the form and the linear dimensions ofthe body in which the process proceeds.
1.1 Equations and Uniqueness Conditions 3
Initial conditions are necessary in the problems of nonstationary free con-vection. They represent the distribution of velocities and temperatures at acertain initial moment of the time τ = τ0:
ui = ui0(xi) ,(1.1.5)
T = T0(xi) .
Boundary conditions specify the values of the required functions at theboundaries of the region considered and can be described in a number ofways.
For a solid body in a viscous fluid flow, when the free path of molecules inthe fluid is mach smaller than the characteristic size of the body, the velocityof the particles of the fluid on a fixed surface is equal to zero, whereas on amoving one it coincides with the velocity of the points of the surface (no-slipcondition):
uiw (τ, xiw) = 0 . (1.1.6)
In weakly rarefied gases the sleep velocity on a solid surface is proportionalto the derivative of the tangent velocity component with respect to the normalto the surface. If there is mass transfer on the surface, the normal velocitycomponent is determined by the rate of absorption (release) of substance bythe wall. The boundary conditions also include setting specification of thevelocity far from the body immersed in a flow. A large variety of boundaryconditions exist for temperature.
The boundary condition of the 1st kind consists in specification of tem-perature distribution over the heating surface at any instant of time:
T = Tw (τ, xiw) . (1.1.7)
The boundary condition of the 2nd kind specifies the heat flux density foreach point of the body surface as a function of time:
−λjw
(∂T
∂nj
)= qjw (τ, xiw) , (1.1.8)
where j is the number of continuous boundary surfaces of the body.The simplest boundary conditions are the constancy of temperature or of
the heat flux density on the body surface:
T = Tw = const ,
−λjw
(∂T
∂nj
)w
= qw = const . (1.1.9)
The boundary condition of the 3rd kind characterizes the law of convectiveheat exchange between the body surface and environment. In this case, theheat flux density is directly proportional to the difference of temperaturesbetween the body surface and the environment:
4 1 Basic Statements and Equations of Free Convection
−λjw
(∂T
∂nj
)w
= αj (Tjw − T∞) . (1.1.10)
Relation (1.1.10) holds only for constant Tjw. In the majority of casesTjw changes along the surface depending on the body properties. Therefore,for free-convective heat transfer relation (1.1.10) can be accepted as a first-approximation boundary condition.
The boundary condition of the 4th kind corresponds to heat exchange ofthe body surface with the surrounding medium or to heat exchange betweenadjoining bodies when the temperature on the boundary of these bodies isthe same. When a solid body is immersed in a flow of fluid, in addition to theequality of the temperatures:
Tjw(xiw, τ) = [T∞(xi, τ)]jw (1.1.11)
the equality of heat fluxes also holds:
−λjw
(∂T
∂nj
)w
= −λj∞
(∂T∞∂nj
)w
. (1.1.12)
The boundary condition of the 4th kind leads to conjugate problems offree-convective heat transfer.
1.2 Boussinesq Approximation
When pressure and temperature differences in a flow are small, then it followsfrom the thermodynamic equation of state (1.1.4) that a change in the densityis also small:
ρ(p, T )ρav
= 1 +(
1ρ
∂ρ
∂T
)pav
(T − Tav) +(
1ρ
∂ρ
∂p
)Tav
(p − pav)
+12
(1ρ
∂2ρ
∂T 2
)av
(T − Tav)2 +12
(1ρ
∂2ρ
∂p2
)av
(p − pav)2
+12
(1ρ
∂2ρ
∂T∂p
)av
(T − Tav)(p − pav) + . . . (1.2.1)
Here, the coefficients of volumetric expansion and isothermal compressionprecede the temperature and pressure differences. For the majority of workingbodies the numerical values of ε1 = βθ0 and ε2 = βp∆p are rather small (forwater ε1 = 1.5 · 10−4θ0 and ε2 = 4.8 · 10−8l; for air ε1 = 3.5 · 10−3θ0
and ε2 = 1.2 · 10−6l). This allows one to neglect the quadratic terms in theexpansion
ρ = ρav (1 − βθ + βp∆p) . (1.2.2)
The smallness of the ratio ε2/ε1 = (10−3−10−4)l/θ0 permits the assertionthat for the overwhelming majority of real geometrical sizes and temperature
1.2 Boussinesq Approximation 5
differences a change in the pressure in free motion of the fluid does not exerta substantial influence on the change in the density. In view of the above, weget
ρ = ρav (1 − βθ) . (1.2.3)
The motion driven by the temperature difference, subject to (1.2.3), is theelementary model of thermal convection.
The system of free-convective motion and heat transfer equations is [1.2,1.3]
ρav
(∂ui
∂τ+ uj
∂ui
∂xj
)= −βρθavFi +
∂
∂xj
[µ
(∂ui
∂xj+
∂uj
∂xi
)]− ∂p
∂xi, (1.2.4)
∂uj
∂xj= 0 , (1.2.5)
ρavcp
[∂θ
∂τ+ uj
∂θ
∂xj+ uj
(∂Tav
∂xj− βT
Fj
cp
)]= ∆Qv + βT
∂p
∂τ
+ βTuj∂p
∂xj+
∂
∂xj
(λ
∂θ
∂xj
)+
µ
2
(∂ui
∂xj+
∂uj
∂xi
)2
. (1.2.6)
In (1.2.4) the component of the buoyancy O(ε1) is preserved. It is shownin [1.4] that this kind of approximation for the momentum equation is valid ifthe acceleration ρav∂ui/∂τ) is small in comparison with the lifting force |Fi|(equal to the Earth gravity force).
The system of equations of free-convective heat transfer in the form(1.2.4)–(1.2.6) represents the Boussinesq approximation. The comparison ofthe solutions in this approximation with experimental data shows that thissystem correctly reflects the basic special features of thermal convection.We note that the Boussinesq approximation does not impose restrictions onchanges in thermophysical characteristics and on the effect exerted by thework of compression and dissipation of energy on the flow and heat transfer.
The state equation in the form of (1.2.3) was first applied for research offree convection in the atmosphere by Oberbeck in 1879 [1.5].
The Boussinesq approximation can also be applied to complex thermody-namic systems with the state equation
ρ = ρ (p, T, ai) (1.2.7)
for small changes in the of parameters ai, pressure, and temperature:
ρ = ρav
(1 − βθ + βp∆p +
∑βi∆ai + . . .
). (1.2.8)
As an example, we can mention homogeneous multicomponent systems orsolutions with a small concentration of the corresponding component [1.6].
For dropping liquids it is necessary to take into account the quadraticdependence of density on temperature difference in (1.2.3) [1.2, 1.7–1.9].
6 1 Basic Statements and Equations of Free Convection
1.3 Method of Generalized Variables
A complex system of nonlinear partial differential equations is used in in-vestigation of the processes of motion and free-convective heat transfer. Forexample, in Boussinesq’s approximation it includes five equations for deter-mining the unknown variables ui, p, T and supplementary equations for thedependences of the thermophysical parameters ρ, β, cp, and λ on temperatureand pressure.
To impart a generalized form to the results of a numerical or experimentalsolution and also to reduce the number of the parameters of a problem, amethod of generalized variables is used [1.10, 1.11]. It rests on the replace-ment of individual parameters of the problem by their certain combinationsrepresenting generalized variables. The structure of the parameters-complexesdepends on the form of the differential operators used in equations.
The efficiency of the method of generalized variables can be demonstratedon the equations in the Boussinesq approximation (1.2.4)–(1.2.6). Considerfree-convective motion in the field of the gravity forces Fi = −gi near a surfacewith a given temperature Tw. The thermophysical characteristics are assumedto be constant and independent of temperature. The free convection equationshere have the form
∂ui
∂τ+ uj
∂ui
∂xj= −giβθ + ν
∂2ui
∂x2j
− 1ρ
∂p
∂xi,
∂uj
∂xj= 0 , (1.3.1)
∂θ
∂τ+ uj
∂θ
∂xj+ uj
(∂T∞∂xj
− gjβT
cp
)=
∆Qν
ρcp+
βT
ρcp
∂p
∂τ
+βTuj
ρcp
∂p
∂xj+ a
∂2θ
∂x2j
+ν
2cp
(∂ui
∂xj+
∂uj
∂xi
)2
.
We reduce (1.3.1) to a dimensionless form. For this purpose, we intro-duce characteristic scales whose role can be conveniently played by the valuesentering into the uniqueness conditions. For a linear scale, we select any char-acteristic dimension of the body l, for the velocity u0, and for temperature,pressure, time, gravity force, and volumetric heat generation θ0, p0, τ0, g, Qν0,respectively. After making the system of (1.3.1) dimensionless, we get
l
τ0u0
∂ui
∂τ+ uj
∂ui
∂xj=
gβθ0l
u20
giθ +ν
u0l
∂2ui
∂x2j
− p0
ρu20
∂p
∂xi,
∂uj
∂xj= 0 , (1.3.2)
1.3 Method of Generalized Variables 7
l
τ0u0
∂θ
∂τ+ uj
(∂θ
∂xj+
∂θ∞∂xj
)+
βgl
cpuigi (θ + θ∞) =
Qv0 l
ρcpu0ϑ0∆Qv
+p0β
ρcp(θ + θ∞)
(l
τ0u0
∂p
∂τ+ uj
∂p
∂xj
)+
a
u0l
∂2θ
∂x2j
+12
νu0
cpθ0l
(∂ui
∂xj+
∂uj
∂xi
)2
.
Equation (1.3.2) involve the parameters and scales, which cannot be foundfrom the uniqueness conditions and can be selected proceeding from the phys-ical nature of the problem. First of all, this concerns the characteristic velocityu0. We refer to (1.3.2) and assume the coefficient at the volumetric force inthe momentum equation to be equal to unity:
u0 =√
gβθ0l . (1.3.3)
The characteristic velocity can also be obtained by comparing the volumet-ric and viscosity forces (u0 = gβv0l
2/ν) or the inertia forces in the momentumequation (u0 = l/τ0). Moreover, it is possible to introduce the characteristicvelocity from the equation of energy.
The pressure scale p0 is usually selected to be equal to the doubled valueof the dynamic pressure ρu2
0.To define the characteristic velocity we make use of formula (1.3.3) and
rewrite the system (1.3.2) in the form
l
Zh Gr1/2
∂ui
∂τ+ uj
∂ui
∂xj= giθ + Gr−1/2 ∂2ui
∂x2j
− ∂p
∂xi,
∂uj
∂xj= 0 ,
1
ZhGr1/2
∂θ
∂τ+ uj
(∂θ
∂xj+
∂θ∞∂xj
)+ Ec uigi(θ + θ∞) (1.3.4)
=Os
Pr Gr1/2∆Qν + Ec ε1(θ + θ∞)
(Gr1/2
Zh∂p
∂τ+ uj
∂p
∂xj
)
+1
Pr Gr1/2
∂2θ
∂x2j
+12
EcGr1/2
(∂ui
∂xj+
∂uj
∂xi
)2
.
The solution of system of (1.3.4) is determined by six dimensionless com-plexes: Zh, Gr, Os, Ec, Pr and ε1. We note that the choice of characteristicscales determines the system of similarity numbers which governs the solutionof the problem [1.34]. Using the above numbers, it is possible to constructanother system of dimensionless complexes, but their number will not exceedsix.
8 1 Basic Statements and Equations of Free Convection
Generally, the expression for velocity, temperature, and pressure are
ui = fi(xi, τ, Zh, Gr ,Pr, Os, Ec, ε1; Pk) ,
θ = f4(xi, τ, Zh, Gr, Pr, Os, Ec, ε1; Pk) , (1.3.5)∆p = f5(xi, τ, Zh, Gr, Pr, Os, Ec, ε1; Pk) .
The group of dimensionless numbers Pk includes the ratios that determinethe geometric properties of the system, the scale values of the excess temper-ature and of the same kind of constants that characterize the properties ofthe dissimilar parts of the system.
To calculate heat transfer, we avail ourselves of the boundary conditionson a separation surface:
αθw = −λ
(∂θ
∂xi
)w
.
After making the above equation dimensionless,
Nu θw = −(
∂θ
∂xi
)w
(1.3.6)
we obtain the new complex Nu = αl/λ.The form of the generalized equation for the dimensionless heat-transfer
coefficient can be determined from (1.3.5):
Nu = f6(xi, τ ,Zh, Gr, Pr, Os, Ec, ε1; Pk) . (1.3.7)
In determining the mean coefficient of heat transfer, the coordinates andtime drop out of the arguments:
Nu = f7(Zh, Gr, Pr, Os, Ec, ε1; Pk) . (1.3.8)
The quantity of the governing dimensionless numbers can be reduced if wedivide the process of heat transfer into the groups differing in the uniquenessconditions. For stationary processes, the numbers containing the time scale τ0
(i.e., the Zhukowski homochromism criterion Zh) drop out of the governingones. The Gr number contains the parametric criterion ε1. In free convection,heat generation due to Joule dissipation and work of compression exert a littleinfluence on heat transfer. Moreover, for heat transfer in a medium withoutinternal heat sources (1.3.8) it is possible to write
Nu = f8(Gr,Pr; Pk) . (1.3.9)
The dependence (1.3.9) can be simplified further by using the similaritytheory methods [1.12]. In the case of slow motion of relatively viscous fluidsthe inertia forces in the motion equations are small in comparison with otherforces acting in the fluid, and they can be neglected. If we select the value
1.4 Dimensional Analysis 9
u0 = gβv0l2/ν as the characteristic velocity, then (1.3.4) in the simplest case
of stationary motions transforms to
giθ +∂2ui
∂x2j
− ∂p
∂xi= 0 ,
∂uj
∂xj= 0 , (1.3.10)
uj
(∂θ
∂xj+
∂θ∞∂xj
)=
1Ra
∂2θ
∂x2j
.
As (1.3.10) contain only the number Ra, dependence (1.3.9) can be pre-sented in the form
Nu = f9(Ra; Pk) . (1.3.11)
This motion occurs, when Pr � 1.Another limiting case is a model of an ideal fluid when the internal friction
forces in the equations of motion are negligibly small in comparison with thevolumetric and inertial forces. The motion and heat transfer (1.3.4) for anideal fluids are
∂ui
∂τ+ uj
∂ui
∂xj= giθ − ∂p
∂xi,
(1.3.12)∂uj
∂xj= 0 ,
and heat transfer can be defined as
Nu = f10(Gr Pr2; Pk) . (1.3.13)
The dependence (1.3.13) describes convection processes in liquid metals(Pr � 1).
Nusselt was the first to formulate the physical similarity laws for theprocesses of free-convective heat transfer [1.13]. Thereafter, the similaritytheory in free-convective heat transfer were extended in [1.14, 1.15]. At thesuggestion of Greber, in 1931 three basic similarity criteria in free convec-tion Gr, Pr and Nu were given the names of the famous scientists [1.16]. Thedesignations of other criteria were introduced as they appeared. The basic nat-ural convection similarity criteria and their standard designations are given inAppendix 1.11.
1.4 Dimensional Analysis
In studying the transfer processes the mathematical description of which doesnot exist, an effective tool for investigation is the experiment. It is expedientto represent experimental results in a generalized form. Similarity numbers
10 1 Basic Statements and Equations of Free Convection
can be found by the dimensional analysis method, but for this it is necessaryto define a list of the physical quantities which are essential for the processconsidered.
All the physical quantities can conventionally be divided into primary andsecondary ones.
For heat transfer processes it is convenient to select the length L, massM, time T, the amount of heat Q, and the temperature Θ as the primaryquantities.
The secondary quantities are expressed in terms of the primary ones ac-cording to definitions or physical laws. The dimensional formulas for the sec-ondary quantities ϕ have the form of exponential monomials:
[ϕ] = Ln1 Mn2 Tn3 Qn4 Θn5 . (1.4.1)
The dimensionality of the secondary quantity relative to the given primaryone is determined by the value of the exponent ni at this primary quantity.
Let us compose a product of dimensional formulas, raised to certain pow-ers, for physical quantities, which are essential of the process. We assumethat the dimensionality of the exponential monomial is equal to zero. Thenthe exponential monomial can be represented as a product of dimensionlesscomplexes of dimensional quantities.
Let us consider the application of the dimensional analysis to the problemsof free convection. For free-convective heat transfer the following quantities(the dimensionality is specified in parentheses) are essential:
l(L), g(LT−2), β(Θ−1), µ(MT−1L−1), ρ(ML−3), λ(QL−1T−1Θ−1), θ(Θ),cp(QM−1Θ−1), α(QL−2T−1Θ−1), τ(T ), Qv(QL−3), Jg(L2MT−2Q−1) .
Dimensionless variables must have the form
tagbβcµdρeλfθhckpαlτmQn
v Jgp , (1.4.2)
where a, b, c, . . . , p denote exponents. If we substitute the dimensionality ofeach of the quantities into (1.4.2), then for the dimensionality of the variablewe get
La(LT−2)bΘ−c(MT−1L−1)d(ML−3)e(QL−1T−1Θ−1)fΘh(QM−1Θ−1)k
(QL−2T−1Θ−1)Tm(QL−3)n(L2MT−2Q−1)p .
The equality to zero of the sum of the exponents at each of the symbolsof the primary quantities is the condition for the entire expression to be di-mensionless. According to the number of the primary quantities of the systemof units selected we obtain five equations for determining 12 exponents. Thenumber of the exponents, for which the values can be selected arbitrarily isequal to 12−5 = 7. Then the remaining exponents are defined in terms of thefirst ones. After transformations we obtain a system of seven dimensionless
1.4 Dimensional Analysis 11
complexes: Nu, Zh, Gr, Pr, Os, Ec, and ε1. The required functional depen-dence connects these seven complexes:
F (Nu, Zh, Gr, Pr, Os, Ec, ε1) = 0 ,
orNu = f7(Zh, Gr, Pr, Os, Ec, ε1)
and it has already the known form of (1.3.8).In going over to the dimensionless quantities the number of the variables
decreased from 12 to 7. This corresponds to the π-theorem: the number ofdimensionless complexes is equal to the number of physical quantities essentialfor the given process minus the number of primary quantities.
We may analyze special cases from the previous paragraph by the dimen-sional method. Using the assumptions made in deriving formula (1.3.9), theessential physical quantities must include l, gβ, ρ, µ, λ, θ, cp and α. Therequired dependence must connect here three (8 − 5 = 3) dimensionless com-plexes Nu, Gr, and Pr.
When the viscosity is small, the coefficient µ must be excluded from theessential physical quantities. The number of dimensional physical quantitiesdecreases to 7, and with the same five primary quantities we obtain only twodimensionless exponential products. This result coincides with dependence(1.3.13).
The inertial forces acting in a medium are determined in the dimensionalanalysis by the density ρ. However, even of the case of small inertial forcesthe density cannot be excluded completely from consideration because it iscontained in the mass force and in the enthalpy of the system. Then theessential physical quantities must include l, µ, ρcp, λ, θ, ρgβ, and α. Weobtain two dimensionless complexes Nu and Ra and dimensionless dependence(1.3.11).
The latter two cases correspond to intermediate asymptotics in the limitingcases of the number Pr [1.17]. In a Cartesian coordinate system we can takeinto account the direction (Lx, Ly, Lz) and the direction sign of the velocityvector (Lv and L−v) [1.18]. In multiphase media the vector character of thelinear dimensions must be taken into account separately for each phase. Thismakes it possible to investigate the conjugate problems of free-convective heattransfer using the dimensional analysis [1.19].
As an example we consider stationary free convection on a vertical planesurface in laminar fluid flow. We assume that there is no volumetric heatgeneration and the work of the forces of compression and energy dissipationare negligibly small. We select a rectangular coordinate system so that theaxis x is directed along the plate in the direction of the volumetric force andthe axis y along the normal to the surface. Seven quantities are essential forthe given process:
l(Lx), gβ(LxT−2Θ), µ(L−1x LyMT−1), ρ(L−1
x L−1y M),
λ(L−1x Ly QT−1Θ−1), θ(Θ), cp(QM−1Θ−1), α(L−1
x QT−1Θ−1) .
12 1 Basic Statements and Equations of Free Convection
There are two (8 − 6 = 2) independent dimensionless complexes in the givenproblem: Nu/Gr1/4 and Pr, between which we can establish the dependence
Nu/Gr1/4 = f11(Pr) (1.4.3)
instead of the generalized formula (1.3.9).For Pr � 1
Nu = C1(Gr Pr2)1/4, (1.4.4)
for Pr � 1Nu = C2Ra1/4 . (1.4.5)
Dependences (1.4.3)–(1.4.5) are more convenient for processing experimen-tal results than similar formulas (1.3.9), (1.3.11) and (1.3.13).
Replacing the heat transfer coefficient α by the heat flux q, we transformthe simple dependence (1.3.9) to [1.20]
Gr∗ = f12(Gr, Pr) ,(1.4.6)
Gr∗ = Nu Gr .
If we exclude from consideration the heat transfer coefficient α, then (1.4.6)will relate to the boundary conditions of the 1st kind. For the boundary con-ditions of the 2nd kind, on the surface [1.21]
Gr = f13(Gr∗,Pr) . (1.4.7)
1.5 Free-Convective Boundary Layer
The main ideas of the boundary layer approximation for natural and forcedconvections are similar [1.22]. The main difference is that the pressure outsidea boundary layer is not determined by the main stream conditions and is hy-drostatic, and the velocity beyond the boundary layer is equal to zero [1.23].And it is assumed that free-convective flow and mass and energy transfersby this flow are concentrated in the main in a thin layer near the surface.Outside this layer, the fluid is assumed to be immobile, which is confirmedby numerous experimental studies. This entails the assumption that the gra-dients along the surface are much smaller than those along the normal to it.The scales of the boundary layer are determined by some characteristic quan-tities δ and δT , where δ is the thickness of a hydrodynamic boundary layerequal to the distance over which the main change in the velocity occurs andδT is the thickness of a thermal boundary layer equal to the distance overwhich the temperature of the flow changes from the wall temperature T0 tothe ambient temperature T∞. In general, δ �= δT . If δT < δ, then the motionoutside the thermal layer, where the buoyancy is absent, is determined by vis-cous interaction between the moving fluid layers. When the thickness of the
1.5 Free-Convective Boundary Layer 13
viscous layer δν is smaller than that of the thermal one (δν < δT ), the motionoutside the viscous boundary layer is potential and differs from zero withinthe limits of the thermal layer δT [1.24]. For a free-convective flow, the mo-tion in the boundary layer is primary, whereas the main stream is secondary.For free convection we cannot consider thermal and hydrodynamic boundarylayers separately, since fluid motion is fully determined by the process of heattransfer.
The concept of the boundary layer in free convection in comparison withthe forced one is extended due to the substantial influence of buoyancy andforces of compression in addition to viscosity and inertia.
In deriving approximate boundary layer equations use is mainly madeof the concept of smallness of the thermal and dynamic boundary layerthicknesses in comparison with the characteristic size of the system at largeGrashoff numbers [1.2, 1.23, 1.25–1.27]. Let us consider the system of Boussi-nesq equations which describes the simplest vertical free-convective flows inwhich the thermophysical parameters of transfer µ(T ) and λ(T ) do not changeappreciably in the region of flow. In practice, this occurs when µ and λ areweak functions of temperature or at small temperature differences:
∂u
∂x+
∂v
∂y= 0 , (1.5.1)
u∂u
∂x+ ν
∂u
∂y= ν
(∂2u
∂x2+
∂2u
∂y2
)− 1
ρ
∂p
∂x+ gβ (T − T∞) , (1.5.2)
u∂v
∂x+ ν
∂v
∂y= ν
(∂2v
∂x2+
∂2v
∂y2
)− 1
ρ
∂p
∂y, (1.5.3)
u∂T
∂x+ ν
∂T
∂y= a
(∂2T
∂x2+
∂2T
∂y2
)+
βT
ρcp
(u
∂p
∂x+ ν
∂p
∂y
)
+µ
ρcpΦ +
q
ρcp. (1.5.4)
A detailed analysis of the relative influence of different physical processesin this system [1.28] allows one to obtain the equations that describe a two-dimension stationary free-convective vertical flow in a boundary layer:
u∂u
∂x+ ν
∂u
∂y= ν
∂2u
∂y2+ gβθ ,
∂u
∂x+
∂v
∂y= 0 , (1.5.5)
u∂θ
∂x+ ν
∂θ
∂y= a
∂2θ
∂y2+
qv
ρcp+
ν
cp
(∂u
∂y
)2
+βT
ρcpu
∂p
∂x.
The boundary and initial conditions depending on the character of heattransfer on the surface can be written as
14 1 Basic Statements and Equations of Free Convection
θ = θw or∂θ
∂y= −qw
λ; or
∂θ
∂y=
λb
λ
∂θb
∂yfor y = 0 ;
(1.5.6)u = 0, θ = 0 or y → ∞;u = u0(y), θ = θ0(y) for x = x0 .
The dimensionless system of the equations of free-convective heat transfernear a vertical plate with a constant temperature Tw for a stationary case inthe absence of volumetric heat sources is [1.2]
u∂u
∂x+ v
∂u
∂y= θ + Gr−1/2
(∂2u
∂x2+
∂2u
∂y2
)− ∂∆p
∂x,
u∂v
∂x+ v
∂v
∂y= Gr−1/2
(∂2v
∂x2+
∂2v
∂y2
)− ∂∆p
∂y,
∂u
∂x+
∂v
∂y= 0 , (1.5.7)
u∂θ
∂x+ ν
∂θ
∂y+ u
∂θ∞∂x
+ Ec u (θ + θ∞)
= Ec ε1 (θ + θ∞)(
u∂∆p
∂x+ ν
∂∆p
∂y
)+ Pr−1Gr−1/2
(∂2θ
∂x2+
∂2θ
∂y2
)
+ Ec Gr−1/2 ×[(
∂u
∂x
)2
+(
∂ν
∂y
)2
+(
∂u
∂y+
∂ν
∂x
)2]
.
Using as a basis the assumption about the finite nature of increments fortemperature and longitudinal velocity in a boundary layer of thickness δ andmeasuring the longitudinal coordinate in the scale y = δY , from the continuityequation we get
ν = δV . (1.5.8)
Substituting (1.5.8) into the system of (1.5.7) and preserving the values ofthe same order of magnitude lead to the equation [1.2]
u∂u
∂x+ V
∂u
∂Y= θ + δ−2
ν Gr−1/2 ∂2u
∂Y 2− ∂∆p
∂x,
∂∆p
∂Y= 0 ,
∂u
∂x+
∂V
∂Y= 0 , (1.5.9)
u∂θ
∂x+ δνδ−1
T V∂θ
∂Y+ u
∂θ∞∂x
+ Ec u (θ + θ∞) = Ec ε1 (θ + θ∞) u∂∆p
∂x
+Pr−1Gr−1/2δ−2T
∂2θ
∂Y 2+ Ec Gr−1/2δ−2
ν
(∂u
∂Y
)2
.
The second equation of motion points to the constancy of the pressure inthe section of the boundary layer normal to the body surface. The motion is
1.5 Free-Convective Boundary Layer 15
determined by three dimensionless criteria: Gr, Pr, and Ec. Depending on theorder of magnitude of the similarity numbers, various relationships are possiblebetween the thicknesses of the viscous δν and thermal δT boundary layers. Inthe majority of cases, the work of compression and viscous dissipation ofenergy exert a weak influence on the process of free-convective heat transfer,and the corresponding components are usually neglected. If Pr = O(1), thenunder the boundary conditions of the 1st kind the thicknesses of the thermaland dynamic boundary layers coincide and are the values of the order ofGr−1/4. Then (1.5.9) take the form:
u∂u
∂x+ V
∂u
∂Y= θ +
∂2u
∂Y 2− ∂∆p
∂x,
∂u
∂x+
∂V
∂Y= 0 , (1.5.10)
u∂θ
∂x+ V
∂θ
∂Y+ u
∂θ∞∂x
= Pr−1 ∂2θ
∂Y 2.
The heat transfer is determined by the dimensionless relation
Nu/Gr1/4 = f (Pr) . (1.5.11)
For small Prandtl numbers the thickness of the viscous boundary layeris much smaller than that of the thermal one (δν/δT = Pr1/2). The thermalboundary layer in this case consists of a thin viscous near-wall layer and azone of inviscid flow.
The motion is induced by the temperature difference, which is observedover the entire thermal layer. The thickness of the hydrodynamic layer δ co-incides then with the thickness of thermal layer δT and is determined fromthe heat balance equation
δT = δ = Pr−1 Gr−1/2 . (1.5.12)
The condition of the applicability of the boundary-layer approximation inthis case is Gr Pr2 � 1. The system of boundary layer (1.5.1)–(1.5.4) in thelimit Pr → 0 is
u∂u
∂x+ ν
∂u
∂y= gβθ ,
∂u
∂x+
∂ν
∂y= 0 , (1.5.13)
u∂θ
∂x+ ν
∂θ
∂y= a
∂2θ
∂y2
and the system of dimensionless (1.5.10) is transformed to
u∂u
∂x+ V
∂u
∂Y= θ − ∂∆p
∂x,
∂u
∂x+
∂V
∂Y= 0 , (1.5.14)
u∂θ
∂x+ V
∂θ
∂Y+ u
∂θ∞∂x
=∂2θ
∂Y 2.
16 1 Basic Statements and Equations of Free Convection
The dimensionless relation for heat transfer is
Nu(Gr Pr2
)1/4= const . (1.5.15)
This model corresponds to the thermal boundary layer of an ideal fluidnear the surface.
At large Prandtl numbers, the thermal boundary layer is thin comparedwith the viscous one (δν/δT = Pr1/2). The buoyancy in this case acts onlywithin the limits of the thermal layer. In the viscous layer θ ≡ 0, and the flowin it occurs due to the entrainment of the fluid by friction. The equations ofboundary layer (1.5.5) for this case are
ν∂2u
∂y2+ gβθ = 0 ,
∂u
∂x+
∂v
∂y= 0 , (1.5.16)
u∂θ
∂x+ ν
∂θ
∂y= a
∂2θ
∂y2+
ν
cp
(∂u
∂y
)2
.
Neglect of the convective terms in the equation of motion leads to singulardisturbances of the system (1.5.16), since the condition u = 0 for y → ∞is not satisfied. It is replaced by the condition of velocity limitation, whichgives ∂u/∂y = 0 for y → ∞. The viscous flow outside the boundary layer isdetermined by the equations [1.25]
u∂u
∂x+ ν
∂u
∂y= ν
∂2u
∂y2,
∂u
∂x+
∂ν
∂y= 0 , (1.5.17)
which satisfy the boundary condition u = 0 for y → ∞. The boundary con-dition on the wall for a longitudinal velocity is its equality to longitudinalvelocity for the internal thermal layer.
Selecting the expression
u0 =gβθ0L
2
ν(1.5.18)
as the characteristic scale of velocity and forming, on its basis, the “Reynoldsnumber” for free convection, the square of which is equal to the Grashofnumber
GrL =u2
0L2
ν2=
gβL3(TW − T∞)ν2
,
from (1.5.10) we obtain the dimensionless equations for a thermal layer [1.2]:
1.5 Free-Convective Boundary Layer 17
θ +∂2u
∂Y 2− ∂∆p
∂x= 0 ,
u∂θ
∂x+ V
∂θ
∂Y+ u
∂θ∞∂x
=∂2θ
∂Y 2, (1.5.19)
∂u
∂x+
∂V
∂Y= 0 .
The condition for the applicability of the approximation is Gr Pr � 1.This condition can be satisfied due to large values of Pr at small Gr numbers.For example, for an aqueous solution of glycerine the Prandtl number canreach the values ∼106. The considered boundary layer has essential featuresconnected with the thermophysical characteristics of the object investigated.The dimensionless relation for heat transfer is [1.29]
Nu
Ra1/4= const , (1.5.20)
and the viscous flow outside the thermal boundary layer is described then bythe equations
u∂u
∂x+ V
∂u
∂Y=
∂2u
∂Y 2,
(1.5.21)∂u
∂x+
∂V
∂Y= 0 .
Equations (1.5.21) satisfy the boundary condition u = 0 for Y → ∞.The system of boundary-layer equations for large Eckert numbers Ec
is constructed similarly. The quantity Ec characterizes the amount of heatevolved due to friction and compression work. The large values of the Ecnumber occur in atmospheric phenomena and astrophysics at small temper-ature differences and large linear dimensions of the system. In the limitingcase Ec → ∞, the equations of motion and heat transfer (1.5.1)–(1.5.4) areconverted to the form:
ν∂2u
∂y2+ gβθ = 0 ,
∂u
∂x+
∂v
∂y= 0 , (1.5.22)
a∂2θ
∂y2+
ν
cp
(∂u
∂y
)2
= 0 .
Outside the thermal boundary layer there is a viscous boundary layer ofthickness δν = δT (Pr Ec)1/2, where the flow is independent of buoyancy andin the limit Ec → ∞ is determined by (1.5.17).
The boundary-layer thickness and the characteristic scale of velocity atlarge Ec numbers are evaluated from the conditions
18 1 Basic Statements and Equations of Free Convection
δ = (Gr Pr Ec)−1/4, u0 = Pr−1/2Ec−1/2 (1.5.23)
and the dimensionless system of the boundary-layer equations in this case is
θ +∂2u
∂Y 2− ∂∆p
∂x= 0 ,
u (θ + θ∞) = ε1 (θ + θ∞) u∂∆p
∂x+
∂2θ
∂Y 2+(
∂u
∂Y
)2
, (1.5.24)
∂u
∂x+
∂V
∂Y= 0 .
The dimensionless relation for calculating the heat transfer coefficient is
Nu
(Pr Gr Ec)1/4= const . (1.5.25)
The conditions for the applicability of the approximation of such aboundary-layer model are Gr Pr Ec � 1 and Ec � 1 without any additionalrestrictions on the values of Gr and Pr.
The boundary-layer equations in application to natural convection can alsobe obtained as the zero approximation of the method of small disturbances.The dimensionless stream function f = Ψ/(νG) and the temperature θ canbe represented as an expansion in terms of the small parameter ε:
f = f0(η) + εf1(η) + ε2f2(η) + . . . ,
θ = θ0(η) + εθ1(η) + ε2θ2(η) + . . . ,
where ε(x) = 1/G, η =y
x4√
Grx, and G = 4[gβx3(T − T∞)/4ν2
]1/4.
Substituting these expansions into full partial-differential equations andisolating the terms containing ε0, we obtain the system of boundary-layerequations for a free-convective flow [1.23].
Similarly we can write the corresponding equations for axisymmetric flows,for example, for a flow near a vertical cylinder or a flow in a wake above a pointheat source and also for a number of other cases of flow. If the boundary-layerthickness δ is not a small value in comparison with x, the boundary-layer ap-proximation becomes inappropriate. These conditions exist in the regions nearthe leading edge of a vertical surface and also at small surface temperaturesand at some particular values of the physical parameters of a fluid [1.23].
1.6 Integral Methods
One of the commonest approximate methods used for calculating free-convective heat transfer is the integral method allowing one to determine thebehavior of the velocity and temperature fields near the body surface; this
1.6 Integral Methods 19
behavior characterizes heat transfer between the body and the medium. Theapplication of the method requires satisfaction of the summarized relationsobtained from the differential equations of boundary layer, without determin-ing velocity and temperature at each point. Moreover, the satisfactions theboundary conditions are also required.
In the majority of cases, the main integrated equations are the Karmanmomentum and the heat balance equations. There are two ways of obtainingthese equations: one is based on the momentum principle in the Euler formand on the heat balance on the surface; the second is analytical consisting ofthe transformation of the boundary-layer equations. In the first case, a certaincontrol surface is isolated near the plate, and the balance of various charac-teristics on it is considered. The momentum flux through such an immobilesurface is equal to the sum of the integral of volumetric forces and the fric-tion resistance of a part of the plate from its leading edge to the section withthe coordinate x. The change in the total amount of heat in the boundarylayer is determined by heat transfer due to thermal conductivity through theimmobile fluid layer near the body surface.
We refer to the boundary-layer equations for the case of viscous developedmotion in the absence of volumetric heat generation, negligibly small energydissipation, and constant ambient temperature [1.30]:
∂u
∂x+
∂v
∂y= 0 ,
u∂u
∂x+ ν
∂u
∂y= gβθ + ν
∂2u
∂y2, (1.6.1)
u∂θ
∂x+ ν
∂θ
∂y= a
∂2θ
∂y2,
with the boundary conditions
u = 0, v = 0, θ = θw or qw = −λ∂θ
∂yfor y = 0 ,
(1.6.2)u = 0, θ = 0 for y → ∞ .
Using the continuity equation, we rewrite the second and the third equa-tions of the system (1.6.1) as
∂u2
∂x+
∂uν
∂y= gβθ + ν
∂2u
∂y2,
(1.6.3)∂uθ
∂x+
∂νθ
∂y= a
∂2θ
∂y2.
We integrate the obtained equations over y from 0 to ∞ or to a certainfinite boundary-layer thickness δmin. In the latter case, the asymptotic bound-ary conditions are replaced by approximated ones:
20 1 Basic Statements and Equations of Free Convection
u = 0,∂u
∂y= 0, θ = 0,
∂θ
∂y= 0 at y = δm . (1.6.4)
The integration yields
δmin,∞∫0
∂u2
∂xdy + (uν)|δmin,∞
0 = gβ
δmin,∞∫0
θdy + ν
(∂u
∂y
)∣∣∣∣δmin,∞
0
,
(1.6.5)δmin,∞∫
0
∂uθ
∂xdy + (νθ)|δmin,∞
0 = a
(∂θ
∂y
)∣∣∣∣δmin,∞
0
.
Using boundary conditions (1.6.4) and assuming it possible to change theorder of differentiation and integration, we get
d
dx
δmin,∞∫0
u2dy = gβ
δmin,∞∫0
θdy + ν
(∂u
∂y
)0
,
d
dx
δmin,∞∫0
uθdy = −a
(∂θ
∂y
)0
. (1.6.6)
The quantity δmin represents the least of the boundary-layer thicknesses δv
and δT . Equations (1.6.6) are the main integral equations of the free-convectiveboundary layer [1.31].
The essence of the method used to investigate boundary-layer equations,based on application of integral equations, rests on approximation of anunknown velocity or temperature profile by some function satisfying theboundary conditions and containing a free parameter (for example, theboundary-layer thickness) which thereafter is determined from the momen-tum equation (heat balance equation).
In free convection, the main results were obtained by the Karman-Pohlhausen integral method at equal thermal and hydrodynamic boundary-layer thicknesses and when low degree polynomials are used as approximationsof the velocity and temperature profiles [1.31].
It is necessary to note the special features of the application of the integralmethods for two different cases: Pr > 1 and Pr < 1, since on transition throughthe values Pr ∼ 1 the structure of the boundary layer changes substantially.For the case Pr > 1 the following functions can be used as temperature andvelocity approximations [1.32]:
T − T∞ = ∆T exp(−x/δT ) , (1.6.7)
ν = V exp(−x/δ) [1 − exp(−x/δT )] , (1.6.8)
where V, δ, and δT are the unknown functions of the coordinate y and ∆T =T0 − T∞ = const. Substituting the selected profiles as integrated functionsinto the momentum and energy integrals:
1.6 Integral Methods 21
d
dy
X∫0
ν2dx = −ν
(∂ν
∂x
)x=0
+ gβ
X∫0
(T − T∞) dx , (1.6.9)
d
dy
X∫0
ν (T∞ − T ) dx = a
(∂T
∂x
)x=0
, (1.6.10)
and assuming X → ∞, we obtain
d
dy
[V 2δq2
2 (2 + q) (1 + q)
]= −νV q
δ+ gβ∆T
δ
q, (1.6.11)
d
dy
[V δ
(1 + q) (1 + 2q)
]=
a
δ, (1.6.12)
where q is a function of Pr:
q(Pr) =δ
δT. (1.6.13)
In (1.6.11) and (1.6.12) three functions are unknown: V (y), δ(y), andq(Pr). The third equation can be selected in various ways. Sometimes, thecondition δ = δT (q = 1) is used as a closing equation [1.33]. However, inthis case a large number of problems dealing with a situation with variableδ/δT drops out of consideration; therefore, it is more expedient to assumethat δ �= δT and to use as a closing equation other conditions following fromthe physical statement of the problem. It is necessary to bear in mind that(1.6.11)–(1.6.12) are approximate, and the initial equations can be satisfiedat various approximations of integrated functions. This affords some freedomin applying the integral method. Directly near the surface, under the non-slipcondition the inertia terms in the Navier-Stokes equation can be neglectedin comparison with buoyancy and viscous forces irrespective of the Prandtlnumber, and then the equation
0 = ν∂2ν
∂x2+ gβ (T0 − T∞) (1.6.14)
can be used for closing the system (1.6.11)–(1.6.12). The solution of the threeequations obtained leads to the following dependence q(Pr) [1.32]:
Pr =56q2 q + 1/2
q + 2. (1.6.15)
Here, we have the following formula for Nu:
Nu =[38
q3
(q + 1) (q + 1/2) (q + 2)
]1/4
Ra1/4y . (1.6.16)
In the limit Pr → ∞
22 1 Basic Statements and Equations of Free Convection
δ
δT=(
65
Pr)1/2
and Nu = 0.783 Ra1/4y . (1.6.17)
In case where Pr < 1, with an exponentially decreasing temperature profile(1.6.7), the other velocity profile can be selected as an integrated function:
ν = V1 exp (−x/δT ) [1 − exp (−x/δν)] , (1.6.18)
where V1, δT , and δν are unknown functions of y. In this case, the solution ofthe system of (1.6.9)–(1.6.10) and (1.6.14) leads to other dependences [1.32]:
Pr =53
(q1
1 + q1
)2
, q1 =δT
δν, (1.6.19)
Nu =(
38
)1/4 (q1
2q1 + 1
)1/2
Ra1/4y . (1.6.20)
In the limit Pr → ∞δν
δT=(
35
Pr)1/2
,
Nu = 0.689 (Pr Ray)1/4. (1.6.21)
The calculation of the Nusselt number depends to some extent on a choiceof analytical functions for velocity and temperature profiles, which must cor-respond to the physically acceptable form of the profile and at the same timelead to the least complex analytical transformations. In the foregoing analysisthe choice of the exponential of form approximation for the temperature andvelocity profiles implies rather simple transformations. The integral analysismade in [1.33] and based on polynomial distribution for the correspondingprofiles at δ = δT also predicts correct Nusselt numbers for a wide range ofPr, although the condition δ = δT restricts this range by the values near 1(Pr ∼ 1).
The integral form of the momentum equation can be obtained by multi-plying it by uk with subsequent integration (k ≥ 1) [1.25]:
1k + 1
d
dx
δ,∞∫0
uk+2dy = gβ3
δmin,∞∫0
θukdy − νk
δ,∞∫0
uk−1
(∂u
∂y
)2
dy . (1.6.22)
When k = 1, (1.6.22) represents an integral Leibenson’s relation of me-chanical energy balance for free convection [1.34].
Similarly, from the heat balance equation we have
1k + 1
d
dx
δ,∞∫0
uθk+1dy = a
δT ,∞∫0
θk ∂2θ
∂y2dy . (1.6.23)
1.7 Loss of Stability and Transition to Turbulence 23
Integral relations are widely applied in numerical methods to check theaccuracy of calculation and the convergence of the results obtained. Usingthe integral equations, one can find averaged characteristics which determinethe scales of initial quantities. In a number of cases, in assigning the distrib-ution of the unknown parameters from experiments or theoretical calculationsthe integral equations allow one to determine the averaged characteristics ofthe process irrespective of the internal features of the structure of the flowand of heat transfer in the volume investigated. Most frequently, the inte-gral relations are used in approximate practical calculations. These methodsmade it possible to solve many problems in free-convective heat transfer, how-ever, there are no proofs of convergence or error estimations for the generalcase. Recently various velocity and temperature distributions are applied toincrease the accuracy of the integral methods. Their accuracy is evaluated bycomparing them with analytical solutions or with asymptotic dependences.However, the very choice of suitable velocity and temperature profiles remainsarbitrary, and one cannot claim with certainty which one leads to the mostaccurate results.
1.7 Loss of Stability and Transition to Turbulence
For the majority of free-convective flows, the intensity of transfer processesdepends strongly on the mode of flow, therefore investigation of the boundary-layer stability and conditions of transition from a laminar to a turbulentflow determines the accuracy of calculation of transfer characteristics. Thetransition to a turbulent regime occurs as a result of the effect of externaldisturbances leading to the development of flow instability. The sources ofdisturbances can be, for example, vibration of buildings and equipment, fluc-tuations in heat supply to a heated surface, instability in the stratificationof the environment, etc. Depending on the conditions of their occurrence,buoyancy magnitude, physical and geometrical characteristics of the processconsidered, the amplitude of these disturbances can increase on mutual effectof pressure, buoyancy, and viscosity.
The first stage in the occurrence and development of instability, for manyflows is the initial growth of small disturbances [1.23, 1.28, 1.35]. It has beenestablished that the disturbance developing in a natural-convection flow oftenhas the form of a periodic wave moving downstream. If during mutual theeffect of hydrodynamic and gravitational forces these waves receive additionalenergy, their amplitude grows, and the motion becomes unstable. However, ifthe disturbance has a fixed frequency, this occurs at different distances fromthe leading edge depending on the Grashof number Grx. The disturbance ofany given frequency is unstable here only in a certain range of the Grashofnumbers and, consequently, of the corresponding values of the longitudinaldistance [1.28].
