30 BASIC CONCEPTS
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in the cylindrical coordinate system as
" ! ! ! ∂Vr 1 ∂Vθ Vr ∂Vz
∂r r ∂θ r ∂z
! ! 1 ∂Vθ Vθ 1 ∂Vr 1 1 ∂Vz ∂Vθ
2 ∂r r r ∂θ 2 r ∂θ ∂z
!2 #
+ 2 ∂z + ∂rz − 3(∇ ·V)2 (1.96)
and in the spherical coordinate system as
∂Vr !2 1 ∂Vθ Vr !2 1 ∂Vθ Vr Vφ cotφ!2 ∂r r ∂φ r r sinφ ∂θ r r
" ! # " ! # 1 ∂ Vφ 1 ∂Vr 1 sinφ ∂ Vθ 1 ∂Vθ
2 ∂r r r ∂φ 2 r ∂φ r sinφ r sinφ ∂θ
" !#
+ 1 r sinφ ∂θ +r∂r Vθ − 3(∇ ·V)2 (1.97)
1.6 DIMENSIONAL ANALYSIS
Bejan(1995)providesadiscussionoftherulesandpromiseofscaleanalysis.Dimen-sional analysis provides an accounting of the dimensions of the variables involved in aphysicalprocess.Therelationshipbetweenthevariableshavingabearingonfriction loss may be obtained by resorting to such a dimensional analysis whose foundation lies in the fact that all equations that describe the behavior of a physical system must be dimensionally consistent. When a mathematical relationship cannot be found, or when such a relationship is too complex for ready solution, dimensional analysis may be used to indicate, in a semiempirical manner, the form of solution. Indeed, in considering the friction loss for a ﬂuid ﬂowing within a pipe or tube, dimensional analysis may be employed to reduce the number of variables that require investiga-tion, suggest logical groupings for the presentation of results, and pave the way for a proper experimental program.
OnemethodforconductingadimensionalanalysisisbywayoftheBuckingham-π theorem (Buckingham, 1914): If r physical quantities having s fundamental dimen-sions are considered, there exists a maximum number q of the r quantities which, in themselves, cannot form a dimensionless group. This maximum number of quanti-ties q may never exceed the number of s fundamental dimensions (i.e., q ≤ s). By combining each of the remaining quantities, one at a time, with the q quantities, n
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dimensionless groups can be formed, where n = r − q. The dimensionless groups are called π terms and are represented by π1,π2,π3, . . . .
The foregoing statement of the Buckingham-π theorem may be illustrated quite simply.Supposethereareeightvariablesthatareknownorassumedtohaveabearing on a cetain problem. Then r = 8 and if it is desired to express these variables in terms of 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 q quantities should be selected, if possible, so that each contains each of the physical quantities at least once. Thus, if q = 4, there will be n = r−q = 8−4 = 4 different πterms, and the functional relationship in the equation that relates the eight variables will be
f(π1,π2,π3,π4)
1.6.1 Friction Loss in Pipe Flow
It is expected that the pressure loss per unit length of pipe or tube will be a function of the mean ﬂuid velocity V, the pipe diameter d, the pipe roughness e, and the ﬂuid properties of density ρ and dynamic viscosity µ. These variables are assumed to be the only ones having a bearing on ∆P/L and may be related symbolically by
∆P = f(V,d,e, ρ,µ)
Noting that r = 6, the fundamental dimensions of mass M, length L, and time T are selected so that s = 3. This means that the maximum number of variables that cannot, by themselves, form a dimensionless group will be q = r −s = 6 −3 = 3.
The variables themselves, together with their dimensions, are displayed in Table 1.1. Observe that because mass in kilograms is a fundamental dimension, pressure must be represented by N/m2, not kg/m2. Pressure is therefore represented by F/A = mg/A and dimensionally by MLT−2/L2 = M/LT2.
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 be n = r −q = 6 −3 = 3 dimensionless π groups:
π1 = ∆P Vaρbdc
π2 = eVaρbdc π3 = µVaρbdc
In each of the π groups, the exponents are collected and equated to zero. The equations are then solved simultaneously for the exponents. For π1,
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32 BASIC CONCEPTS
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TABLE 1.1 Variables and Dimensions for the Example of Section 1.6.1, SI System
Variable Dimension
Pressure loss ∆P M/LT2 Length L L Velocity V L/T Diameter d L Roughness e L Density ρ M/L3 Viscosity µ M/LT Pressure loss per unit length ∆P/L M/L2T2
π1 = ∆P ˆaρbdc = L2T2 T a L3 b Lc
Then
M: 0 = 1 +b
L: 0 = −2 +a −3b +c T: 0 = −2 −a
A simultaneous solution quickly yields a = −2,b = −1, and c = +1, so that
∆P ˆ−2 −1 ∆P d ∆P
L ρLV2 (L/d)ρV2
For π2,
a π2 = eVaρbdc = L T L3 Lc
Then
M: 0 = b
L: 0 = 1 +a −3b +c T: 0 = −a
This time, the simultaneous solution provides a = b = 0 and c = −1, so that
π2 = ed−1 = d
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For π3,
π3 = µVaρbdc = LT T L3 Lc
Then
M: 0 = 1 +b
L: 0 = −1 +a −3b +c T: 0 = −1 −a
from which a = b = c = −1, so that
π3 = µV−1ρ−1d−1 = ρVd
the reciprocal of the Reynolds number. Let a friction factor f be deﬁned as
f = 2(∆P/L)d (1.98)
such that the pressure loss per unit length will be given by
L = ρ2d 2 (1.99)
Equation (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 containing pipe or tube. Hence,
!
f = 2g(∆P/L) = φ ρVd, L (1.100)
A representation of eq. (1.100) was determined by Moody (1944) (see Fig. 5.13).
1.6.2 Summary ofDimensionless Groups
A summary of the dimensionless groups used in heat transfer is provided in Table 1.2. A summary of the dimensionless groups used in mass transfer is provided in Table 1.3. Note that when there can be no confusion regarding the use of the Stanton and Stefan numbers, the Stefan number, listed in Table 1.2, is sometimes designated as St.
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34 BASIC CONCEPTS
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TABLE 1.2 Summary of Dimensionless Groups Used in Heat Transfer
Group Symbol Deﬁnition
Bejan number Be ∆PL2/µα Biot number Bi hL/k Colburn j-factor jh St · Pr2/3
Eckert number Ec V∞/cp(Tw −T∞) Elenbass number El ρ2βgcpz4∆T/µkL Euler number Eu ∆P/ρV2
Fourier number Fo αt/L2 Froude number Fr V2/gL Graetz number Gz ρcpVd2/kL Grashof number Gr gβ∆T L3/ν2
Jakob number Ja ρlcpl(Tw −Tsat)/ρggh g) Knudsen number Kn λ/L
Mach number Ma V/a Nusselt number Nu hL/k
Peclet number Pe Re · Pr = ρcpVL/k
Prandtl number Pr cpµ/k = ν/α Rayleigh number Ra Gr · Pr = ρgβ∆T L3/µα
Reynolds number Re ρVL/µ Stanton number St Nu/Re · Pr = h/ρcpV Stefan number Ste cp(Tw −Tm)/hsf Strouhal number Sr Lf/V
Weber number We ρV2L/σ
TABLE 1.3 Summary of Dimensionless Groups Used in Mass Transfer
Group Symbol Deﬁnition
Biot number Bi hDL/D Colburn j-factor jD STD · Sc2/3 Lewis number Le Sc/Pr = α/D Peclet number PeD Re · Sc = VL/D Schmidt number Sc v/D Sherwood number Sh hDL/D Stanton number StD Sh/Re · Sc = hD/V
1.7 UNITS
As shown in Table 1.4, there are seven primary dimensions in the SI system of units and eight in the English engineering system. Luminous intensity and electric current are not used in a study of heat transfer and are not considered further.
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