Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 2

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Hence,DhDu 1 DPP Dρ=+− 2DtDtρ Dtρ Dt(1.77)Moreover, the heat fluxes qx and qy can be expressed in terms of local temperaturegradients through use of Fourier’s law:q = −k ∇TThus, the combination of eqs. (1.74), (1.77), and (1.78) results inDhP DρDPρ= ∇ · (k ∇T ) + q ++ µΦ −+ ρ ∇ · V̂DtDtρ Dt(1.78)(1.79)Here, too, eq. (1.53) points out that the terms in parentheses in eq. (1.79) are equal tozero, so that eq. (1.75) reduces toρDhDP= ∇ · (k ∇T ) + q ++ µΦDtDt(1.80)Bejan (1995) points out that the change in specific enthalpy for a single-phase fluidis given byBOOKCOMP, Inc.

— John Wiley & Sons / Page 27 / 2nd Proofs / Heat Transfer Handbook / Bejan28123456789101112131415161718192021222324252627282930313233343536373839404142434445BASIC CONCEPTSdh = T ds + v dP = T ds +dPρ(1.81)where T is the absolute temperature and ds is the specific entrophy change: dsdsdT +ds =dP(1.82)dT PdP TThe last of the Maxwell relations given by Bejan (1997) ∂(1/ρ)1 ∂ρβ∂s=−= 2=−∂P T∂Tρ∂TρPPwhere β is the volumetric coefficient of thermal expansion,1 ∂ρβ=−ρ ∂T P(1.83)[28], (28)(1.84)Lines: 1328 to 1411With cp taken as the specific heat at constant pressures, it can be shown thatcpds=(1.85)dT PTand eqs. (1.81) through (1.85) can be combined,1dh = cp dT + (1 − βT ) dPρ(1.86)so that the left-hand side of eq. (1.80) can be written asρDhDTDP= ρcp+ (1 − βT )DTDtDt(1.87)Thus, the temperature formulation of the first law of thermodynamics isρcpDTDT= ∇ · (k ∇T ) + q + βT+ µΦDtDt(1.88)with the special forms for the ideal gas where β = 1/T ,ρcpDTDP= ∇ · (k ∇T ) + q ++ µΦDtDt(1.89a)and for an incompressible fluid where β = 0,ρcpDT= ∇ · (k ∇T ) + q + µΦDt(1.89b)Most convection problems concern an even simpler model where the fluid hasconstant thermal conductivity k, neglible viscous dissipation Φ, zero internal heatBOOKCOMP, Inc.

— John Wiley & Sons / Page 28 / 2nd Proofs / Heat Transfer Handbook / Bejan———7.50035pt PgVar———Long PagePgEnds: TEX[28], (28)29CONSERVATION OF ENERGY123456789101112131415161718192021222324252627282930313233343536373839404142434445generation (q = 0), and a negligible compressibility effect, βT (DP /Dt) ≈ 0.

Theenergy equation for this model is simplyρcpDT= k ∇ 2TDt(1.90)or, in the rectangular coordinate system,ρcp∂T∂T∂T∂T+ V̂x+ V̂y+ V̂z∂t∂x∂y∂z=k∂ 2T∂ 2T∂ 2T++ 222∂x∂y∂zin the cylindrical coordinate system,∂T∂T∂TV̂θ ∂Tρcp+ V̂r++ V̂z∂t∂rr ∂θ∂z1 ∂∂ 2T∂T1 ∂ 2T=kr+ 2 2 + 2r ∂r∂rr ∂θ∂zand in the spherical coordinate system,∂TV̂φ ∂TV̂θ ∂T∂Tρcp+ V̂r++∂t∂rr ∂φr sin φ ∂θ1 ∂1∂T1 ∂ 2T∂2 ∂T=k 2r+ 2sin φ+r ∂r∂rr sin φ ∂φ∂φsin2 φ ∂θ2(1.91)[29], (29)(1.92)———9.43025pt PgVar———Long Page* PgEnds: Eject(1.93)If the fluid can be modeled as incompressible then, as in eq. (1.89b), the specificheat at constant pressure cp is replaced by c. And when dealing with extremelyviscous flows, the model is improved by taking into account the internal heating dueto viscous dissipation,ρcpDT= k ∇ 2 T + µΦDt(1.94)In the rectangular coordinate system, the viscous dissipation can be expressed as2 2 2 ∂ V̂y∂ V̂x∂ V̂z Φ = 2++∂x∂y∂z+∂ V̂y∂ V̂x+∂y∂x2+2 2∂ V̂y∂ V̂z∂ V̂x∂ V̂z+++∂z∂y∂x∂z2∂ V̂y∂ V̂z2 ∂ V̂x+−+∂y3 ∂x∂zBOOKCOMP, Inc.

— John Wiley & Sons / Page 29 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 1411 to 1462(1.95)[29], (29)30123456789101112131415161718192021222324252627282930313233343536373839404142434445BASIC CONCEPTSin the cylindrical coordinate system asΦ=2∂ V̂r∂r22 2V̂r1 ∂ V̂θ∂ V̂z+++r ∂θr∂z22V̂θ1 ∂ V̂r∂ V̂θ1 ∂ V̂θ1 1 ∂ V̂z−++++2 ∂rrr ∂θ2 r ∂θ∂z21 ∂ V̂r1∂ V̂z+− (∇ · V̂)2+2 ∂z∂r3(1.96)and in the spherical coordinate system as2 2 2 ∂V̂cotφ11V̂V̂V̂∂V̂∂V̂rθrθrφΦ=2 +++++∂rr ∂φrr sin φ ∂θrr+1∂r2∂rV̂φr+12sin φ ∂r ∂φV̂θ1 ∂ V̂θ+r sin φr sin φ ∂θ 2  21 ∂ V̂r1∂ V̂θ+− (∇ · V̂)2+r 32 r sin φ ∂θ∂r r+1 ∂ V̂rr ∂φ22[30], (30)Lines: 1462 to 1497———-3.0659pt PgVar(1.97)———Normal PagePgEnds: TEX[30], (30)1.6DIMENSIONAL ANALYSISBejan (1995) provides a discussion of the rules and promise of scale analysis.

Dimensional analysis provides an accounting of the dimensions of the variables involved ina physical process. The relationship between the variables having a bearing on frictionloss may be obtained by resorting to such a dimensional analysis whose foundationlies in the fact that all equations that describe the behavior of a physical system mustbe dimensionally consistent. When a mathematical relationship cannot be found, orwhen such a relationship is too complex for ready solution, dimensional analysismay be used to indicate, in a semiempirical manner, the form of solution. Indeed, inconsidering the friction loss for a fluid flowing within a pipe or tube, dimensionalanalysis may be employed to reduce the number of variables that require investigation, suggest logical groupings for the presentation of results, and pave the way for aproper experimental program.One method for conducting a dimensional analysis is by way of the Buckingham-πtheorem (Buckingham, 1914): If r physical quantities having s fundamental dimensions are considered, there exists a maximum number q of the r quantities which, inthemselves, cannot form a dimensionless group.

This maximum number of quantities q may never exceed the number of s fundamental dimensions (i.e., q ≤ s). Bycombining each of the remaining quantities, one at a time, with the q quantities, nBOOKCOMP, Inc. — John Wiley & Sons / Page 30 / 2nd Proofs / Heat Transfer Handbook / BejanDIMENSIONAL ANALYSIS12345678910111213141516171819202122232425262728293031323334353637383940414243444531dimensionless groups can be formed, where n = r − q.

The dimensionless groupsare called π terms and are represented by π1 , π2 , π3 , . . . .The foregoing statement of the Buckingham-π theorem may be illustrated quitesimply. Suppose there are eight variables that are known or assumed to have a bearingon a cetain problem. Then r = 8 and if it is desired to express these variables in termsof four physical dimensions, such as length L, mass M, temperature θ, and time T ,then s = 4. It is then possible to have q = r − s = 8 − 4 = 4 physical quantities,which, by themselves, cannot form a dimensionless group.The usual practice is to make q = s in order to minimize labor.

Moreover, the qquantities should be selected, if possible, so that each contains each of the physicalquantities at least once. Thus, if q = 4, there will be n = r − q = 8 − 4 = 4 differentπ terms, and the functional relationship in the equation that relates the eight variableswill be[31], (31)f (π1 , π2 , π3 , π4 )Lines: 1497 to 15521.6.1Friction Loss in Pipe Flow———It is expected that the pressure loss per unit length of pipe or tube will be a functionof the mean fluid velocity V̂ , the pipe diameter d, the pipe roughness e, and the fluidproperties of density ρ and dynamic viscosity µ. These variables are assumed to bethe only ones having a bearing on ∆P /L and may be related symbolically by∆P= f (V̂, d, e, ρ, µ)LNoting that r = 6, the fundamental dimensions of mass M, length L, and time Tare selected so that s = 3.

This means that the maximum number of variables thatcannot, by themselves, form a dimensionless group will be q = r − s = 6 − 3 = 3.The variables themselves, together with their dimensions, are displayed in Table1.1. Observe that because mass in kilograms is a fundamental dimension, pressuremust be represented by N/m2, not kg/m2. Pressure is therefore represented by F /A =mg/A and dimensionally by MLT −2 /L2 = M/LT 2 .Suppose that ν, ρ, and d are selected as the three primary quantities (q = 3).These clearly contain all three of the fundamental dimensions and there will ben = r − q = 6 − 3 = 3 dimensionless π groups:π1 =∆P a b cV̂ ρ dLπ2 = eV̂ a ρb d cπ3 = µV̂ a ρb d cIn each of the π groups, the exponents are collected and equated to zero. Theequations are then solved simultaneously for the exponents.

For π1 ,BOOKCOMP, Inc. — John Wiley & Sons / Page 31 / 2nd Proofs / Heat Transfer Handbook / Bejan9.91005pt PgVar———Normal PagePgEnds: TEX[31], (31)32123456789101112131415161718192021222324252627282930313233343536373839404142434445BASIC CONCEPTSTABLE 1.1 Variables and Dimensions for the Exampleof Section 1.6.1, SI SystemVariableDimensionPressure loss ∆PLength LVelocity VDiameter dRoughness eDensity ρViscosity µPressure loss per unit length ∆P /LM/LT 2LL/TLLM/L3M/LTM/L2 T 2[32], (32) a bLM∆P a b cMV̂ ρ d = 2 2π1 =LcLL TTL3Lines: 1552 to 1597———Then2.88019pt PgVarM:0=1+bL:0 = −2 + a − 3b + cT:0 = −2 − aA simultaneous solution quickly yields a = −2, b = −1, and c = +1, so thatπ1 =∆P −2 −1∆P d∆P=V̂ ρ d =LρLV̂ 2(L/d)ρV̂ 2For π2 ,π2 = eV̂ a ρb d c = L a bMLLcTL3ThenM:0=bL:0 = 1 + a − 3b + cT:0 = −aThis time, the simultaneous solution provides a = b = 0 and c = −1, so thatπ2 = ed −1 =BOOKCOMP, Inc.

— John Wiley & Sons / Page 32 / 2nd Proofs / Heat Transfer Handbook / Bejaned———Short Page* PgEnds: Eject[32], (32)33DIMENSIONAL ANALYSIS123456789101112131415161718192021222324252627282930313233343536373839404142434445For π3 ,π3 = µV̂ a ρb d c =MLT a bLMLcTL3ThenM:0=1+bL:0 = −1 + a − 3b + cT:0 = −1 − afrom which a = b = c = −1, so thatπ3 = µV̂ −1 ρ−1 d −1 =µ[33], (33)ρV̂dLines: 1597 to 1737the reciprocal of the Reynolds number.Let a friction factor f be defined asf =———2(∆P /L)dρV̂ 23.78221pt PgVar(1.98)———Short PagePgEnds: TEX(1.99)[33], (33)such that the pressure loss per unit length will be given by∆Pρf V̂ 2=L2dEquation (1.99) is a modification of the Darcy–Fanning head-loss relationship,and the friction factor defined by eq. (1.95), as directed by the dimensional analysis,is a function of the Reynolds number and the relative roughness of the containingpipe or tube.

Hence,2g(∆P /L)ρV̂ d ef ==φ,(1.100)µ LρV̂ 2 dA representation of eq. (1.100) was determined by Moody (1944) (see Fig. 5.13).1.6.2Summary of Dimensionless GroupsA summary of the dimensionless groups used in heat transfer is provided in Table1.2. A summary of the dimensionless groups used in mass transfer is provided inTable 1.3. Note that when there can be no confusion regarding the use of the Stantonand Stefan numbers, the Stefan number, listed in Table 1.2, is sometimes designatedas St.BOOKCOMP, Inc. — John Wiley & Sons / Page 33 / 2nd Proofs / Heat Transfer Handbook / Bejan34123456789101112131415161718192021222324252627282930313233343536373839404142434445BASIC CONCEPTSTABLE 1.2Summary of Dimensionless Groups Used in Heat TransferGroupBejan numberBiot numberColburn j -factorEckert numberElenbass numberEuler numberFourier numberFroude numberGraetz numberGrashof numberJakob numberKnudsen numberMach numberNusselt numberPéclet numberPrandtl numberRayleigh numberReynolds numberStanton numberStefan numberStrouhal numberWeber numberTABLE 1.3DefinitionBeBijhEcElEuFoFrGzGrJaKnMaNuPePrRaReStSteSrWe∆P L2 /µαhL/kSt · Pr2/32V̂∞/cp (Tw − T∞ )2ρ βgcp z4 ∆T /µkL∆P /ρV̂ 2αt/L2V̂ 2 /gLρcp V̂ d 2 /kLgβ ∆T L3 /ν2ρl cpl (Tw − Tsat )/ρgg hf g )λ/LV̂ /ahL/kRe · Pr = ρcp V̂ L/kcp µ/k = ν/αGr · Pr = ρgβ ∆T L3 /µαρV̂ L/µNu/Re · Pr = h/ρcp V̂cp (Tw − Tm )/ hsfLf/V̂ρV̂ 2 L/σSummary of Dimensionless Groups Used in Mass TransferGroupBiot numberColburn j -factorLewis numberPéclet numberSchmidt numberSherwood numberStanton number1.7SymbolSymbolDefinitionBijDLePeDScShStDhD L/DSTD · Sc2/3Sc/Pr = α/DRe · Sc = V̂ L/Dv/DhD L/DSh/Re · Sc = hD /V̂UNITSAs shown in Table 1.4, there are seven primary dimensions in the SI system of unitsand eight in the English engineering system.

Luminous intensity and electric currentare not used in a study of heat transfer and are not considered further.BOOKCOMP, Inc. — John Wiley & Sons / Page 34 / 2nd Proofs / Heat Transfer Handbook / Bejan[34], (34)Lines: 1737 to 1743———-0.7849pt PgVar———Normal PagePgEnds: TEX[34], (34)UNITS12345678910111213141516171819202122232425262728293031323334353637383940414243444535TABLE 1.4 Primary Dimensions and Units for the SI and EnglishEngineering SystemsUnit and SymbolDimension(Quantity)SI SystemEnglish SystemMassLengthTimeTemperatureAmount of substanceLuminous intensityElectric currentForcekilogram (kg)meter (m)second (s)kelvin (K)mole (mol)candela (Cd)ampere (A)newton (N)pound-mass (lbm)foot (ft)second (s)rankine (°R)mole (mol)candleampere (A)pound-force (lbf)[35], (35)1.7.1SI System (Système International d’Unités)Lines: 1743 to 1777The SI system of units is an extension of the metric system and has been adopted in———many countries as the only system accepted legally.

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