Heat Transfer Handbook part 13 ppt

BOOKCOMP, Inc. — John Wiley & Sons / Page 110 / 2nd Proofs / Heat Transfer Handbook / Bejan 110 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 [110], (68) Lines: 2274 to 2297 ——— * 528.0pt PgVar ——— Normal Page * PgEnds: PageBreak [110], (68) TABLE 2.7 Thermophysical Properties of Fluids along the Saturation Line a (Continued) RC318 (Octafluorccyclobutane, FC-C318) (Continued) s (liq.) 0.8686 0.9488 1.0262 1.1014 1.1748 1.2110 1.2471 1.2833 1.3199 1.3579 1.3995 s (vap.) 1.3940 1.4078 1.4258 1.4461 1.4671 1.4773 1.4871 1.4959 1.5033 1.5080 1.5065 c v (liq.) 0.7084 0.7397 0.7739 0.8070 0.8385 0.8541 0.8699 0.8862 0.9040 0.9246 0.9524 c v (vap.) 0.6425 0.6803 0.7183 0.7571 0.7976 0.8189 0.8413 0.8652 0.8914 0.9215 0.9597 c p (liq.) 0.9819 1.0244 1.0699 1.1182 1.1752 1.2109 1.2557 1.3173 1.4133 1.6004 2.2182 c p (vap.) 0.6873 0.7291 0.7746 0.8270 0.8931 0.9360 0.9910 1.0680 1.1911 1.4390 2.2820 w (liq.) 654.66 579.81 504.72 430.19 355.56 317.77 279.32 239.80 198.61 154.70 105.83 w (vap.) 101.56 103.78 104.61 103.64 100.37 97.668 94.081 89.431 83.456 75.752 65.658 η (liq.) 506.03 375.90 287.28 222.61 172.66 151.41 131.94 113.78 96.438 79.162 60.232 η (vap.) 9.1935 9.9598 10.697 11.458 12.270 12.721 13.232 13.844 14.638 15.803 17.976 λ (liq.) 63.039 58.334 53.929 49.767 45.779 43.825 41.886 39.956 38.064 36.443 37.590 λ (vap.) 8.5065 9.7277 11.014 12.394 13.905 14.731 15.633 16.657 17.908 19.659 22.983 σ 0.0153 0.0128 0.0103 0.0080 0.0058 0.0047 0.0037 0.0028 0.0019 0.0011 0.0004 a T , temperature (K); p, pressure (MPa); ρ, density (kg/m 3 ); h, enthalpy (kJ/kg); s, entropy (kJ/kg ·K); c v , isochoric heat capacity (kJ/kg·K); c p , isobaric heat capacity (kJ/kg ·K); w, speed of sound (m/s); η, viscosity (µPa ·s); λ, thermal conductivity (mW/m ·K); σ, surface tension (N/m). BOOKCOMP, Inc. — John Wiley & Sons / Page 111 / 2nd Proofs / Heat Transfer Handbook / Bejan THERMOPHYSICAL PROPERTIES OF FLUIDS 111 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 [111], (69) Lines: 2297 to 2357 ——— 5.18057pt PgVar ——— Normal Page * PgEnds: Eject [111], (69) α(ρ,T ) RT = α(δ,τ) = α 0 (δ,τ) + α r (δ,τ) (2.3) where δ = ρ/ρ c , τ = T c /T , and the ideal gas constant R is 8.314472 J/mol ·K (Mohr and Taylor, 1999). The Helmholtz energy of the ideal gas (the fluid in the limit of noninteracting particles realized at zero density, where p = ρRT )isgivenby a 0 = h 0 − RT − Ts 0 (2.4) The ideal gas enthalpy is given by h 0 = h 0 0 +  T T 0 c 0 p dT (2.5) where h 0 0 is the enthalpy reference point at T 0 and c 0 p is the ideal gas heat capacity given by eq. (2.9). The ideal gas entropy is given by s 0 = s 0 0 +  T T 0 c 0 p T dT −R ln ρT ρ 0 T 0 (2.6) where s 0 0 is the entropy reference point at T 0 and p 0 , and ρ 0 is the ideal gas density at T 0 and p 0 . The values for h 0 0 and s 0 0 are chosen arbitrarily for each fluid. Combining these equations results in the following equation for the Helmholtz energy of the ideal gas: a 0 = h 0 0 +  T T 0 c 0 p dT −RT − T  s 0 0 +  T T 0 c 0 p T dT −R ln ρT ρ 0 T 0  (2.7) This ideal gas Helmholtz energy can be expressed in dimensionless form by α 0 = h 0 0 τ RT c − s 0 0 R − 1 + ln δτ 0 δ 0 τ − τ R  τ τ 0 c 0 p τ 2 dτ + 1 R  τ τ 0 c 0 p τ dτ (2.8) where δ 0 = ρ 0 /ρ c and τ 0 = T c /T 0 . In the calculation of the thermodynamic properties using an equation of state explicit in the Helmholtz energy, an equation for the ideal gas heat capacity, c 0 p ,is needed to calculate the Helmholtz energy for the ideal gas, α 0 . Equations for the ideal gas heat capacity are generally given in the form c 0 p R =  k N k T i k +  k N k u 2 k exp(u k )  exp(u k ) − 1  2 (2.9) where u k is M k /T . N k and M k are constants determined from spectroscopic or other experimental data. Table 2.4 gives values of the ideal gas isobaric heat capacity for BOOKCOMP, Inc. — John Wiley & Sons / Page 112 / 2nd Proofs / Heat Transfer Handbook / Bejan 112 THERMOPHYSICAL PROPERTIES OF FLUIDS AND MATERIALS 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 [112], (70) Lines: 2357 to 2401 ——— 2.6756pt PgVar ——— Normal Page PgEnds: T E X [112], (70) selected fluids. The values were calculated from the ideal gas heat capacity equations given in the references in Table 2.2. Unlike the equations for the ideal gas, dense fluid or residual behavior is often described using empirical models that are only loosely tied with theory. The dimen- sionless residual Helmholtz energy is generally given as α r (δ,τ) =  k N k δ i k τ j k +  k N k δ i k τ j k exp  −δ l k  +  k N k δ i k τ j k exp  −ϕ k (δ − 1) 2 − β k (τ − γ k ) 2  (2.10) Although the values of i k ,j k , and l k are empirical fitting constants, j k is generally expected to be greater than zero, and i k and l k are integers greater than zero. The terms in the third sum correspond to modified Gaussian bell-shaped terms, introduced by Setzmann and Wagner (1991) to improve the description of properties in the critical region. Calculation of Properties The functions used for calculating pressure, com- pressibility factor, internal energy, enthalpy, entropy, Gibbs energy, isochoric heat capacity, isobaric heat capacity, and the speed of sound from eq. (2.3) are given as eqs. (2.11 to 2.19). These equationswereusedin calculating values of thermodynamic properties given in the tables. p = ρRT  1 + δ  ∂α r ∂δ  τ  (2.11) Z = P ρRT = 1 + δ  ∂α r ∂δ  τ (2.12) u RT = τ  ∂α 0 ∂τ  δ +  ∂α r ∂τ  δ  (2.13) h RT = τ  ∂α 0 ∂τ  δ +  ∂α r ∂τ  δ  + δ  ∂α r ∂δ  τ + 1 (2.14) s R = τ  ∂α 0 ∂τ  δ +  ∂α r ∂τ  δ  − α 0 − α r (2.15) g RT = 1 + α 0 + α r + δ  ∂α r ∂δ  τ (2.16) c v R =−τ 2  ∂ 2 α 0 ∂τ 2  δ +  ∂ 2 α r ∂τ 2  δ  (2.17) c p R = c v R +  1 + δ(∂α r /∂δ) τ − δτ  ∂ 2 α r /∂δ ∂τ  2  1 + 2δ(∂α r /∂δ) τ + δ 2  ∂ 2 α r /∂δ 2  τ  (2.18) BOOKCOMP, Inc. — John Wiley & Sons / Page 113 / 2nd Proofs / Heat Transfer Handbook / Bejan THERMOPHYSICAL PROPERTIES OF FLUIDS 113 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 [113], (71) Lines: 2401 to 2440 ——— -1.27905pt PgVar ——— Normal Page PgEnds: T E X [113], (71) w 2 M RT = 1 + 2δ  ∂α r ∂δ  τ + δ 2  ∂ 2 α r ∂δ 2  τ −  1+δ ( ∂α r /∂δ ) τ − δτ  ∂ 2 α r /∂δ ∂τ  2 τ 2  ∂ 2 α 0 /∂τ 2  δ +  ∂ 2 α r /∂τ 2  δ  (2.19) Equations for additional thermodynamic properties are given in Lemmon et al. (2000). Table 2.3 displays the molecular weight, critical temperature, and critical density used in these equations for various fluids. The thermodynamic properties of various fluids along the saturated liquid and vapor lines are given in Table 2.7. Thermodynamic Properties of Mixtures The Helmholtz energy for mixtures of fluids can be calculated from a generalized mixture model using the equations of state for the pure fluids in the mixture and an excess function to account for the interactions between different species, as given by Lemmon and Jacobsen (1999). This model is used in the REFPROP program available from the Standard Reference Data Program of NIST for mixture calculations. (See the introduction for additional details on the NIST databases.) The Helmholtz energy of a mixture is a = a idmix + a E (2.20) where the Helmholtz energy for an ideal mixture defined at constant reduced temper- ature and density (similar to simple corresponding states) is a idmix = n  i=1 x i  a 0 i (ρ,T ) + a r i (δ,τ) + RT ln x i  (2.21) where n is the number of components in the mixture, a 0 i is the ideal gas Helmholtz energy for component i, and a r i is the pure fluid residual Helmholtz energy of com- ponent i evaluated at a reduced density and temperature defined below. The excess contribution to the Helmholtz energy from mixing is a E RT = α E (δ,τ, x) = n−1  i=1 n  j=i+1 x i x j F ij 10  k=1 N k δ d k τ t k (2.22) where the coefficients and exponents are obtained from nonlinear regression of ex- perimental mixture data. The same set of mixture coefficients (N k ,d k , and t k ) is used for all mixtures in the model, and the parameter F ij is a scaling factor that relates the excess properties of one binary mixture to those of another. Multicomponent mixtures can then be calculated without any additional ternary or higher interactionparameters. All single-phase thermodynamic properties [such as those given in eqs. (2.11 to 2.19)] can be calculated from the Helmholtz energy using the relations α 0 = n  i=1 x i  a 0 i (ρ,T) RT + ln x i  (2.23) BOOKCOMP, Inc. — John Wiley & Sons / Page 114 / 2nd Proofs / Heat Transfer Handbook / Bejan 114 THERMOPHYSICAL PROPERTIES OF FLUIDS AND MATERIALS 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 [114], (72) Lines: 2440 to 2473 ——— 5.97198pt PgVar ——— Normal Page PgEnds: T E X [114], (72) α r = n  i=1 x i α r i (δ,τ) + α E (δ,τ, x) (2.24) where the derivatives are taken at constant composition. The reducedvalues of density and temperature for the mixture are δ = ρ ρ red (2.25) τ = T red T (2.26) where ρ and T are the mixture density and temperature, respectively, and ρ red and T red are the reducing values, ρ red =   n  i=1 x i ρ c i + n−1  i=1 n  j=i+1 x i x j ξ ij   −1 (2.27) T red = n  i=1 x i T c i + n−1  i=1 n  j=i+1 x i x j ζ ij (2.28) The parameters ζ ij and ξ ij are used to define the shapes of the reducing temperature line and reducing density line. These reducing parameters are not the same as the critical parameters of the mixture and are determined simultaneously with the other parameters of the mixture model for each binary pair in the nonlinear fit of experi- mental data. 2.2.2 Transport Properties The equations used for calculating the transport properties of fluids are given in the references cited in Table 2.2. For many substances, equations for transport properties have not been developed or published, and alternative techniques must be used. A widely used method for calculating the transport properties of fluids is based on predictions using the extended corresponding states (ECS) concept. In Table 2.2, those fluids for which the ECS method is used to determine the best current values are identified. Values for some of the fluids have been improved by fitting experimental data, while others are used in a purely predictive mode. Extended Corresponding States The principle of corresponding states stems from the observation that the properties of many fluids are similar when scaled by their respective critical temperatures and densities. Extended corresponding states models modify this scaling by using additional shape factors to improve the repre- sentation of data. ECS methods may be used to predict both the thermodynamic and transport properties, especially for fluids with limited data. The method starts with BOOKCOMP, Inc. — John Wiley & Sons / Page 115 / 2nd Proofs / Heat Transfer Handbook / Bejan THERMOPHYSICAL PROPERTIES OF FLUIDS 115 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 [115], (73) Lines: 2473 to 2522 ——— -1.41756pt PgVar ——— Normal Page * PgEnds: Eject [115], (73) the ECS model of Huber et al. (1992) and Ely and Hanley (1983) and combines this predictive model with the best available thermodynamic equations of state (see Klein et al., 1997; McLinden et al., 2000). The viscosity of a fluid can be represented as the sum of the dilute gas and the real fluid contributions, η(T , ρ) = η ∗ (T ) + η r (T , ρ) (2.29) where η ∗ is the Chapman–Enskog dilute gas contribution described later. The ther- mal conductivity of a fluid can be represented as the sum of energy transfer due to translational and internal contributions. λ(T , ρ) = λ int (T ) + λ trans (T , ρ) (2.30) where the superscript “trans” designates the translational term (i.e.,contributions aris- ing from collisions between molecules) and the superscript “int” designates the con- tribution from internal motions of the molecule). The internal term is assumed to be independent of density. The translational term is divided into a dilute-gas contribution λ ∗ and a density-dependent term, which is further divided into a residual part (super- script r) and a critical enhancement (superscript “crit”). The thermal conductivity is thus the sum of four terms: λ(T , ρ) = λ int (T ) + λ ∗ (T ) + λ r (T , ρ) + λ crit (T , ρ) (2.31) Dilute-Gas Contributions The standard formulas for the dilute-gas contribu- tions that arise from kinetic theory and which have been used by Ely and Han- ley (1983), Huber et al. (1992), and others, but with an empirical modification, are used here. The transfer of energy associated with internal degrees of freedom of the molecule is assumed to be independent of density and can be calculated from the Eucken correlation for polyatomic gases given in Hirschfelder et al. (1967), λ int j (T ) = f int η ∗ j (T ) M j  c 0 p,j − 5 2 R  (2.32) where c 0 p is the ideal gas heat capacity, R the gas constant, and M the molar mass. The subscript j emphasizes that all quantities are to be evaluated for fluid j . The factor f int accounts for the energy conversion between internal and translational modes. The dilute-gas part of the translational term is given by λ ∗ j (T ) = 15Rη ∗ j (T ) 4M j (2.33) The Chapman–Enskog dilute-gas viscosity is given by standard kinetic gas theory from Hirschfelder et al. (1967): η ∗ j (T ) = (M j T) 1/2 σ 2 j Ω (2,2) (kT /ε j ) (2.34) BOOKCOMP, Inc. — John Wiley & Sons / Page 116 / 2nd Proofs / Heat Transfer Handbook / Bejan 116 THERMOPHYSICAL PROPERTIES OF FLUIDS AND MATERIALS 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 [116], (74) Lines: 2522 to 2562 ——— 2.8364pt PgVar ——— Normal Page PgEnds: T E X [116], (74) where σ j and ε/k are the molecular size and energy parameters associated with an intermolecular potential function such as the Lennard-Jones 12-6 potential, and Ω (2,2) is the collision integral (again for the Lennard-Jones fluid), which is a function of the reduced temperature, kT/ε j . The empirical function of Neufeld et al. (1972) is often used for Ω (2,2) . While the dilute-gas viscosity equation is derived from theory, the molecular size and energy parameters are most often evaluated from low-density viscosity data. This function can thus be treated as a theoretically based correlating function. Where experimentally based Lennard-Jones parameters are not available, they may be estimated by the relations suggested by Huber and Ely (1992): ε j /k = ε 0 k T crit j T crit 0 (2.35) σ j = σ 0  ρ crit 0 ρ crit j  1/3 (2.36) where the zero subscript refers to the reference fluid used in the ECS described below. Tables 2.5 and 2.6 give values for the dilute-gas thermal conductivity and viscosity. Density-Dependent Contributions The principle of corresponding states can be used to model the residual part of the thermal conductivity and viscosity. Such models have been applied to a wide variety of fluids by many authors, including Leland and Chappelear (1968), Hanley (1976), Ely and Hanley (1983), and Huber et al. (1992). This approach is especially useful for fluids for which limited experimental data exist. The simple corresponding states model is based on the assumption that different fluids are conformal; that is, they obey, in reduced coordinates, the same intermolec- ular force laws. A reduced property may be obtained by dividing the individual state values by the value of the property at the critical point. This assumption leads to the conclusion that with appropriate scaling of temperature and density, the reduced residual Helmholtz energies and compressibilities of the unknown fluid j and a ref- erence fluid 0 (for which an accurate, wide-ranging equation of state is available) are equal: α r j (T , ρ) = α r 0 (T 0 , ρ 0 ) (2.37) Z j (T , ρ) = Z 0 (T 0 , ρ 0 ) (2.38) The reference fluid is usually chosen as one that has a molecular structure similar to the fluid of interest. The conformal temperature T 0 and density ρ 0 are related to the actual T and ρ of the fluid of interest by T 0 = T f = T T crit 0 T crit j θ(T , ρ) (2.39) BOOKCOMP, Inc. — John Wiley & Sons / Page 117 / 2nd Proofs / Heat Transfer Handbook / Bejan THERMOPHYSICAL PROPERTIES OF FLUIDS 117 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 [117], (75) Lines: 2562 to 2610 ——— 8.44946pt PgVar ——— Normal Page * PgEnds: Eject [117], (75) ρ 0 = ρh = ρ ρ crit 0 ρ crit j φ(T , ρ) (2.40) where the multipliers 1/f and h are termed equivalent substance reducing ratios, or simply reducing ratios. Initially, the corresponding states approach was developed for spherically sym- metric molecules for which the reducing ratios are simple ratios of the critical pa- rameters (θ and φ both equal to 1). The ECS model extends the method to other types of molecules by the introduction of the shape factors θ and φ. These shape factors are functions of temperature and density, although the density dependence is often neglected. The ECS method has been applied to both the thermodynamic and transport prop- erties. In this model, the thermal conductivity is λ r j (T , ρ) = F λ λ r 0 (T 0 , ρ 0 ) (2.41) and the viscosity is η r j (T , ρ) = F η η r 0 (T 0 , ρ 0 ) (2.42) where F λ = f 1/2 h −2/3  M 0 M j  1/2 (2.43) F η = f 1/2 h −2/3  M j M 0  1/2 (2.44) Following the methods of Klein et al. (1997) and McLinden et al. (2000), the reducing ratios 1/f and h are further modified by empirical viscosity shape factors and ther- mal conductivity shape factors fitted to experimental data. The thermal conductivity approaches infinity at the critical point, and even well removed from the critical point the critical enhancement can be a significant portion of the total thermal conductivity. The transport properties of various fluids along the saturated liquid and vapor lines are given in Table 2.7. Transport Properties of Mixtures For mixtures, the calculation of thermal conductivity and viscosity from extended corresponding states principles uses the same terms as those given in equations for pure fluids. For the thermal conductivity, the internal and translational contributions are calculated using λ int mix (T , x) +λ ∗ mix (T , x) = n  j=1 x j  λ int j (T ) + λ ∗ j (T )   n i=1 x i ϕ ji (2.45) where BOOKCOMP, Inc. — John Wiley & Sons / Page 118 / 2nd Proofs / Heat Transfer Handbook / Bejan 118 THERMOPHYSICAL PROPERTIES OF FLUIDS AND MATERIALS 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 [118], (76) Lines: 2610 to 2681 ——— 0.8703pt PgVar ——— Long Page PgEnds: T E X [118], (76) ϕ ji =  1 +  η ∗ j η ∗ i  1/2  M j M i  1/4  2  8  1 + M j M i  −1/2 (2.46) All quantities in the dilute-gas terms are evaluated at the temperature of the mix- ture rather than at some conformal temperature. The residual viscosity includes an additional term, η r mix = ∆η 0  T f x , ρh x  F η + ∆η ∗ (ρ,x) (2.47) where ∆η 0 is the residual viscosity of the reference fluid and the additional term ∆η ∗ is an Enskog hard-sphere correction for effects of differences in size and mass in a mixture. The F λ term in the residual part of the transport equations is F λ = f 1/2 x h −2/3 x g 1/2 x (2.48) where the subscript x indicates a mixture quantity, and g 1/2 x = M 1/2 0 f 1/2 x h 4/3 x n  i=1 n  j=1 x i x j (f i f j ) 1/4  2 1/g i + 1/g j  −1/2  1 8  h 1/3 i + h 1/3 j  3  4/3 (2.49) The g i are g i = M 0  λ r 0 (T 0 , ρ 0 ) λ r j (T j , ρ j )  2 f j h −4/3 j (2.50) The reducing ratios are defined as f x = T/T 0 ,h x = ρ 0 /ρ,f j = T j f x /T , and h j = ρh x /ρ j . The conformal conditions for component j and the mixture are found by solving the equations α r j (T j , ρ j ) = α r mix (T , ρ,x) (2.51) Z j (T j , ρ j ) = Z mix (T , ρ,x) (2.52) α r j (T 0 , ρ 0 ) = α r mix (T , ρ,x) (2.53) Z j (T 0 , ρ 0 ) = Z mix (T , ρ,x) (2.54) where the mixture properties are calculated using the Helmholtz energy mixture model described earlier. 2.3 THERMOPHYSICAL PROPERTIES OF SOLIDS In this section we discuss the transport properties associated with heat transfer in solid materials. Starting with the equation for the conservation of energy, we demon- strate the need for constitutive models for the internal energy and heat flux. A brief BOOKCOMP, Inc. — John Wiley & Sons / Page 119 / 2nd Proofs / Heat Transfer Handbook / Bejan THERMOPHYSICAL PROPERTIES OF SOLIDS 119 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 [119], (77) Lines: 2681 to 2711 ——— -0.16792pt PgVar ——— Long Page * PgEnds: Eject [119], (77) discussion of the physical mechanisms that contribute to heat transfer in solids fol- lows. Next, several common measurement techniques are discussed to assist the reader in assessing their applicability should property measurements be needed. In practice, such measurements might be desired because the tabulated property values of a listed material may differ from those of the material obtained for use because of differences in alloying, level of impurities, or crystalline structure, for example. Introduction to the molecular modeling of thermal conductivity may be found in Klemens (1969) and Parrott and Stuckes (1975). Brief discussions of thermal conductivity, specific heat, and thermal expansion may be found in introductory texts in solid-state physics (e.g., Reif, 1965; Brown, 1967; Kittel, 1996). The general behavior of material properties with respect to temperature may also be found in Jakob (1955) and in introductory texts (e.g., Incropera and DeWitt, 1996). 2.3.1 Conservation of Energy The equation for the conservation of energy in the absence of stress power may be written as di dt =−∇· ˙ q + q  (2.55) where i the enthalpy per unit volume, t the time, ˙ q the heat flux vector, and q  the volumetric rate of heat generation due, for example, to chemical reaction or to absorption of electromagnetic radiation. The stress power is the rate of work done by the stress field during deformation. It is identically zero for rigid-body motion. In general, i may be a function of deformation and temperature and ˙ q may be a function of the temperature gradient as well. Two constitutive equations are needed for eq. (2.55), one for i and another for ˙ q. Changes in the specific internal energy may be written as di dt = ρ m c p dT dt (2.56) where c p is the specific heat per unit mass at constant pressure, ρ m the mass density, and T the temperature. For solids, it may easily be shown that the specific heat is nearly independent of pressure (i.e., c p ≈ c v , where c v is the specific heat at constant volume) (see, e.g., Reif, 1965). Similarly, it is usually assumed that c p is independent of internal stresses (Parrott and Stuckes, 1975). Fourier’s equation is most often assumed to relate ˙ q to temperature. It agrees well with most measurements and may be written as ˙ q =−λ ·∇T (2.57) where λ is the thermal conductivity tensor. In most common circumstances, the thermal conductivity is isotropic and eq. (2.57) may be rewritten as ˙ q =−λ ∇T . In the absence of internal heat generation and assuming isotropic thermal conduc- tivity, substituting eqs. (2.56) and (2.57) into eq. (2.55) yields . Inc. — John Wiley & Sons / Page 113 / 2nd Proofs / Heat Transfer Handbook / Bejan THERMOPHYSICAL PROPERTIES OF FLUIDS 113 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 [ 113] ,. (liq.) 506.03 375.90 287.28 222.61 172.66 151.41 131 .94 113. 78 96.438 79.162 60.232 η (vap.) 9.1935 9.9598 10.697 11.458 12.270 12.721 13. 232 13. 844 14.638 15.803 17.976 λ (liq.) 63.039 58.334. John Wiley & Sons / Page 111 / 2nd Proofs / Heat Transfer Handbook / Bejan THERMOPHYSICAL PROPERTIES OF FLUIDS 111 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 [111],