BOOKCOMP, Inc. — John Wiley & Sons / Page 30 / 2nd Proofs / Heat Transfer Handbook / Bejan 30 BASIC CONCEPTS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [30], (30) Lines: 1462 to 1497 ——— -3.0659pt PgVar ——— Normal Page PgEnds: T E X [30], (30) in the cylindrical coordinate system as Φ = 2 ∂ ˆ V r ∂r 2 + 1 r ∂ ˆ V θ ∂θ + ˆ V r r 2 + ∂ ˆ V z ∂z 2 + 1 2 ∂ ˆ V θ ∂r − ˆ V θ r + 1 r ∂ ˆ V r ∂θ 2 + 1 2 1 r ∂ ˆ V z ∂θ + ∂ ˆ V θ ∂z 2 + 1 2 ∂ ˆ V r ∂z + ∂ ˆ V z ∂r 2 − 1 3 (∇· ˆ V) 2 (1.96) and in the spherical coordinate system as Φ = 2 ∂ ˆ V r ∂r 2 + 1 r ∂ ˆ V θ ∂φ + ˆ V r r 2 + 1 r sin φ ∂ ˆ V θ ∂θ + ˆ V r r + ˆ V φ cot φ r 2 + 1 2 r ∂ ∂r ˆ V φ r + 1 r ∂ ˆ V r ∂φ 2 + 1 2 sin φ r ∂ ∂φ ˆ V θ r sin φ + 1 r sin φ ∂ ˆ V θ ∂θ 2 + 1 2 1 r sin φ ∂ ˆ V r ∂θ + r ∂ ∂r ˆ V θ r 2 − 2 3 (∇· ˆ V) 2 (1.97) 1.6 DIMENSIONAL ANALYSIS Bejan (1995) provides a discussion of the rules and promise of scale analysis. Dimen- sional analysis provides an accounting of the dimensions of the variables involved in a physical process. The relationship between the variables having a bearing on friction 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. One method for conducting a dimensional analysis is by way of the Buckingham-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 BOOKCOMP, Inc. — John Wiley & Sons / Page 31 / 2nd Proofs / Heat Transfer Handbook / Bejan DIMENSIONAL ANALYSIS 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [31], (31) Lines: 1497 to 1552 ——— 9.91005pt PgVar ——— Normal Page PgEnds: T E X [31], (31) dimensionless groups can be formed, where n = r − q. The dimensionless groups are called π terms and are represented by π 1 , π 2 , π 3 , The foregoing statement of the Buckingham-π theorem may be illustrated quite simply. Suppose there are eight variables that are known or assumed to have a bearing 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 π terms, and the functional relationship in the equation that relates the eight variables will be f(π 1 , π 2 , π 3 , π 4 ) 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 ˆ 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/Land may be related symbolically by ∆P L = f( ˆ 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/m 2 , not kg/m 2 . Pressure is therefore represented by F/A = mg/A and dimensionally by MLT −2 /L 2 = 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 be n = r − q = 6 −3 = 3 dimensionless π groups: π 1 = ∆P L ˆ V a ρ b d c π 2 = e ˆ V a ρ b d c π 3 = µ ˆ V a ρ b d c In each of the π groups, the exponents are collected and equated to zero. The equations are then solved simultaneously for the exponents. For π 1 , BOOKCOMP, Inc. — John Wiley & Sons / Page 32 / 2nd Proofs / Heat Transfer Handbook / Bejan 32 BASIC CONCEPTS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [32], (32) Lines: 1552 to 1597 ——— 2.88019pt PgVar ——— Short Page * PgEnds: Eject [32], (32) TABLE 1.1 Variables and Dimensions for the Example of Section 1.6.1, SI System Variable Dimension Pressure loss ∆P M/LT 2 Length LL Velocity VL/T Diameter dL Roughness eL Density ρ M/L 3 Viscosity µ M/LT Pressure loss per unit length ∆P/L M/L 2 T 2 π 1 = ∆P L ˆ V a ρ b d c = M L 2 T 2 L T a M L 3 b L c 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 π 1 = ∆P L ˆ V −2 ρ −1 d = ∆Pd ρL ˆ V 2 = ∆P (L/d)ρ ˆ V 2 For π 2 , π 2 = e ˆ V a ρ b d c = L L T a M L 3 b L c 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 π 2 = ed −1 = e d BOOKCOMP, Inc. — John Wiley & Sons / Page 33 / 2nd Proofs / Heat Transfer Handbook / Bejan DIMENSIONAL ANALYSIS 33 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [33], (33) Lines: 1597 to 1737 ——— 3.78221pt PgVar ——— Short Page PgEnds: T E X [33], (33) For π 3 , π 3 = µ ˆ V a ρ b d c = M LT L T a M L 3 b L c Then M:0= 1 +b L:0=−1 +a − 3b +c T:0=−1 −a from which a = b = c =−1, so that π 3 = µ ˆ V −1 ρ −1 d −1 = µ ρ ˆ Vd the reciprocal of the Reynolds number. Let a friction factor f be defined as f = 2(∆P /L)d ρ ˆ V 2 (1.98) such that the pressure loss per unit length will be given by ∆P L = ρf ˆ V 2 2d (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) ρ ˆ V 2 d = φ ρ ˆ Vd µ , e L (1.100) A representation of eq. (1.100) was determined by Moody (1944) (see Fig. 5.13). 1.6.2 Summary of Dimensionless 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. BOOKCOMP, Inc. — John Wiley & Sons / Page 34 / 2nd Proofs / Heat Transfer Handbook / Bejan 34 BASIC CONCEPTS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [34], (34) Lines: 1737 to 1743 ——— -0.7849pt PgVar ——— Normal Page PgEnds: T E X [34], (34) TABLE 1.2 Summary of Dimensionless Groups Used in Heat Transfer Group Symbol Definition Bejan number Be ∆PL 2 /µα Biot number Bi hL/k Colburn j -factor j h St · Pr 2/3 Eckert number Ec ˆ V 2 ∞ /c p (T w − T ∞ ) Elenbass number El ρ 2 βgc p z 4 ∆T/µkL Euler number Eu ∆P/ρ ˆ V 2 Fourier number Fo αt/L 2 Froude number Fr ˆ V 2 /gL Graetz number Gz ρc p ˆ Vd 2 /kL Grashof number Gr gβ∆TL 3 /ν 2 Jakob number Ja ρ l c pl (T w − T sat )/ρg g h fg ) Knudsen number Kn λ/L Mach number Ma ˆ V/a Nusselt number Nu hL/k P ´ eclet number Pe Re · Pr = ρc p ˆ VL/k Prandtl number Pr c p µ/k = ν/α Rayleigh number Ra Gr · Pr = ρgβ∆TL 3 /µα Reynolds number Re ρ ˆ VL/µ Stanton number St Nu/Re · Pr = h/ρc p ˆ V Stefan number Ste c p (T w − T m )/h sf Strouhal number Sr Lf / ˆ V Weber number We ρ ˆ V 2 L/σ TABLE 1.3 Summary of Dimensionless Groups Used in Mass Transfer Group Symbol Definition Biot number Bi h D L/D Colburn j -factor j D ST D · Sc 2 /3 Lewis number Le Sc/Pr = α/D P ´ eclet number Pe D Re · Sc = ˆ V L/D Schmidt number Sc v/D Sherwood number Sh h D L/D Stanton number St D Sh/Re · Sc = h D / ˆ 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. BOOKCOMP, Inc. — John Wiley & Sons / Page 35 / 2nd Proofs / Heat Transfer Handbook / Bejan UNITS 35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [35], (35) Lines: 1743 to 1777 ——— 1.96013pt PgVar ——— Normal Page * PgEnds: Eject [35], (35) TABLE 1.4 Primary Dimensions and Units for the SI and English Engineering Systems Unit and Symbol Dimension (Quantity) SI System English System Mass kilogram (kg) pound-mass (lb m ) Length meter (m) foot (ft) Time second (s) second (s) Temperature kelvin (K) rankine (°R) Amount of substance mole (mol) mole (mol) Luminous intensity candela (Cd) candle Electric current ampere (A) ampere (A) Force newton (N) pound-force (lb f ) 1.7.1 SI System (Syst`eme International d’Unit´es) The SI system of units is an extension of the metric system and has been adopted in many countries as the only system accepted legally. The primary dimensions used in a study of the thermal sciences embrace the first five entries in the center column of Table 1.1. There are standards for all these units. For example, the standard for the second is the duration of 9,192,631,770 periods, corresponding to the transition states between two levels of the ground state of the cesium-133 atom. The mole is defined as the molecular weight of a substance expressed in the appropriate mass unit. For example, a gram-mole (g-mol) of nitrogen contains 28.01 grams (g) and 1 kg-mol of nitrogen contains 28.01 kilograms (kg). The number of moles of a substance, N , is related to its mass m and molecular weight by the simple expression N = m M (1.101) In the SI system, force is a secondary dimension. The unit of force is therefore a secondary or derived unit and is the newton (N), which can be obtained from Newton’s second law, F = ma,as 1N= 1kg(m/s 2 ) = 1kg·m/s 2 and the constant of proportionality is unity, or 1N·s 2 kg ·m = 1 (1.102) Because pressure is force per unit area, P = F/A, the unit of pressure, the pascal (Pa), can be expressed as 1Pa= 1N/m 2 = (1kg·m/s 2 )(1/m 2 ) = 1kg m · s 2 (1.103) BOOKCOMP, Inc. — John Wiley & Sons / Page 36 / 2nd Proofs / Heat Transfer Handbook / Bejan 36 BASIC CONCEPTS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [36], (36) Lines: 1777 to 1859 ——— -1.23988pt PgVar ——— Long Page * PgEnds: Eject [36], (36) A work interaction, or work, is represented by δW = Fx The unit of work or energy is the joule (J), defined as W = 1J= 1N· m (1.104) and because power P is the rate of doing work, the unit of power is the watt: P = dW dt = ˙ W = 1J/s = 1N· m/s (1.105) Note that quantities that pertain to a rate can be designated by a dotted quantity. Observe that the weight of a body is equal to the force of gravity on the body. Hence, weight always refers to a force, and in the SI system this force is always in newtons. The mass of the body can always be related to its weight via W = mg (1.106) where g, the local gravitational acceleration, has a mean value at sea level of g = 9.807 m/s 2 and is a function of location. This shows that the weight of a body may vary, whereas the mass of the body is always the same. 1.7.2 English Engineering System (U.S. Customary System) The English engineering system (sometimes referred to as the U.S. customary system of units) is often used in the United States. This system takes the first five entries and the last entry in the right-hand column of Table 1.4. Here both mass and force are taken as primary dimensions and the pound is used as the unit of mass (the lb m ) and the unit of force (the lb f ). This leads to more than a little confusion when this system is used. Because there are now six primary dimensions to be used with Newton’s second law, it must be written as F ∝ ma When the proportionality constant g c is inserted, the result is F = ma g c (1.107) with g c = 32.174 ft/s 2 taken as the standard acceleration of gravity. This means that a force of 1 lb f will accelerate a mass of 1 lb m at a rate of 32.174 ft/s 2 . Thus, 1lb f = (32.174 ft/s 2 )(1lb m ) g c or BOOKCOMP, Inc. — John Wiley & Sons / Page 37 / 2nd Proofs / Heat Transfer Handbook / Bejan UNITS 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [37], (37) Lines: 1859 to 1924 ——— 0.25612pt PgVar ——— Long Page * PgEnds: PageBreak [37], (37) g c = 32.174 lb m -ft lb f -s 2 (1.108) Thus Newton’s second law must be written as F = ma 32.174 (1.109) It is important to remember that the SI system of units does not require this conversion factor. 1.7.3 Conversion Factors Conversion factors from English engineering units to SI units are given in Table 1.5. Conversion factors for commonly used heat transfer parameters are given in Table 1.6. TABLE 1.5 Conversion Factors from English Engineering Units to SI Units To Convert from: To: Multiply by: Acceleration ft/sec 2 m/s 2 3.048 × 10 −1 Area ft 2 m 2 9.2903 × 10 −2 in 2 m 2 6.4516 × 10 −4 Density lb m /in 3 kg/m 3 2.7680 × 10 4 lb m /ft 3 kg/m 3 16.018 Energy, heat, and work Btu J 1.0544 × 10 3 ft-lb f J 1.3558 kW-hr J 3.60 × 10 6 Force lb f N 4.4482 Length ft m 3.048 × 10 −1 in. m 2.54 × 10 −2 mi km 1.6093 Mass lb m kg 4.5359 × 10 −1 ton kg 9.0718 × 10 2 Power ft-lb f /min W 2.2597 × 10 −2 horsepower (hp) W 7.457 ×10 2 Pressure atm Pa 1.0133 × 10 5 lb f /ft 2 Pa 47.880 lb f /in 2 Pa 6.8948 × 10 3 Velocity ft/sec m/s 3.048 × 10 −1 mi/hr m/s 4.4704 × 10 −1 BOOKCOMP, Inc. — John Wiley & Sons / Page 38 / 2nd Proofs / Heat Transfer Handbook / Bejan 38 BASIC CONCEPTS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [38], (38) Lines: 1924 to 2038 ——— 2.0997pt PgVar ——— Normal Page PgEnds: T E X [38], (38) TABLE 1.6 Conversion Factors for Heat Transfer Parameters from English Engineering Units to SI Units To Convert from: To: Multiply by: Heat flux btu/hr-ft 2 W/m 2 3.1525 kcal/h · m 2 W/m 2 1.163 Heat transfer coefficient Btu/hr-ft 2 -°F W/m 2 · K 5.678 kcal/h · m 2 · °C W/m 2 · K 1.163 Heat transfer rate Btu/hr W 0.2931 Mass flow rate lb m /hr kg/s 1.26 × 10 −4 lb m /sec kg/s 4.536 × 10 −1 Specific heat Btu/lb m -°F J/kg · K4.187 × 10 3 Surface tension lb m /ft N/m 1.4594 × 10 1 Temperature 1°R K 0.5555 Thermal conductivity Btu/hr-ft-°F W/m · K 1.731 kcal/h · m · °F W/m · K 1.163 Thermal diffusivity ft 2 /sec m 2 /s 9.29 × 10 −2 ft 2 /h m 2 /s 2.581 × 10 −5 Thermal resistance °F-hr/Btu K/W 1.8958 Viscosity (dynamic) lb m /ft-sec N · s/m 2 1.4881 centipoise N · s/m 2 1 × 10 3 Viscosity (kinematic) ft 2 /hr m 2 /s 9.29 × 10 −2 ft 2 /hr stoke 929 NOMENCLATURE Roman Letter Symbols A cross-sectional area, m 2 a square root of heat source area, m 2 speed of sound, m/s C constant, dimensionless c specific heat, W/m · K D substantial differential, dimensionless mass diffusivity, m 2 /s BOOKCOMP, Inc. — John Wiley & Sons / Page 39 / 2nd Proofs / Heat Transfer Handbook / Bejan NOMENCLATURE 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [39], (39) Lines: 2038 to 2038 ——— 0.00604pt PgVar ——— Normal Page PgEnds: T E X [39], (39) d differential, dimensionless diameter, m e unit vector, dimensionless e roughness, m specific energy, J/kg · K F force vector, N F force, N F radiation factor, dimensionless f frequency, m −1 G mass velocity, kg/m 2 · s g acceleration of gravity, m/s 2 H microhardness, N/m 2 h heat transfer coefficient, W/m 2 · K specific enthalpy, J/kg k thermal conductivity, W/m · K L path length, m physical dimension, dimensionless M gas parameter, m physical dimension, kg m surface slope, dimensionless mass, kg ˙m mass flow rate, kg/s n number of moles, dimensionless number of dimensionless groups, dimensionless n normal direction, dimensions vary P pressure, N/m 2 p wetted perimeter, m q heat flow, W maximum number of quantities, dimensionless q g heat generation, W/m 3 q heat flux, W/m 2 q heat generation, W/m 3 q heat flux vector, W/m 2 R thermal resistance, K/W region, dimensionless r radial direction, m radius, m number of physical quantities, dimensionless S surface area, m 2 s specific entropy, J/kg · K scale factor, dimensions vary number of fundamental dimensions, dimensionless T temperature, K t time, s physical dimension, s . of heat transfer and are not considered further. BOOKCOMP, Inc. — John Wiley & Sons / Page 35 / 2nd Proofs / Heat Transfer Handbook / Bejan UNITS 35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [ 35] ,. / Heat Transfer Handbook / Bejan UNITS 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [37], (37) Lines: 1 859 . Proofs / Heat Transfer Handbook / Bejan 32 BASIC CONCEPTS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [32], (32) Lines: