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BOOKCOMP, Inc. — John Wiley & Sons / Page 271 / 2nd Proofs / Heat Transfer Handbook / Bejan DEFINITIONS OF SPREADING AND CONSTRICTION RESISTANCES 271 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 [271], (11) Lines: 548 to 574 ——— 0.60939pt PgVar ——— Long Page * PgEnds: Eject [271], (11) Figure 4.2 Heat flow lines and isotherms for steady conduction from a finite heat source into a half-space. (From Yovanovich and Antonetti, 1988.) doubly connected areas (e.g., circular or annular area). The free surface of the half- space is adiabatic except for the source area. If heat enters the half-space, the flux lines spread apart as the heat is conducted away from the small source area (Fig. 4.2); then the thermal resistance is called spreading resistance. If the heat leaves the half-space through a small area, the flux lines are constricted and the thermal resistance is called constriction resistance. The heat transfer may be steady or transient. The temperature field T in the half-space is, in general, three- dimensional, and steady or transient. The temperature in the source area may be two- dimensional, and steady or transient. If heat transfer is into the half-space, the spreading resistance is defined as (Car- slaw and Jaeger, 1959; Yovanovich, 1976c; Madhusudana, 1996; Yovanovich and Antonetti, 1988) R s = T source − T sink Q (K/W) (4.16) where T source is the source temperature and T sink is a convenient thermal sink tem- perature; and where Q is the steady or transient heat transfer rate: Q = A q n dA = A −k ∂T ∂n dA (W) (4.17) where q n is the heat flux component normal to the area and ∂T /∂n is the temperature gradient normal to the area. If the heat flux distribution is uniform over the area, Q = qA. For singly and doubly connected source areas, three source temperatures have been used in the definition: maximum temperature, centroid temperature, and area-averaged temperature, which is defined according to Yovanovich (1976c) as T source = 1 A A T source dA (K) (4.18) BOOKCOMP, Inc. — John Wiley & Sons / Page 272 / 2nd Proofs / Heat Transfer Handbook / Bejan 272 THERMAL SPREADING AND CONTACT RESISTANCES 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 [272], (12) Lines: 574 to 603 ——— 1.02235pt PgVar ——— Normal Page PgEnds: T E X [272], (12) where A is the source area. Because the sink area is much larger than the source area, it is, by convention, assumed to be isothermal (i.e., T sink = T ∞ ). The maximum and centroid temperatures are identical for singly connected axisymmetric source areas; otherwise, they are different (Yovanovich, 1976c; Yovanovich and Burde, 1977); Yovanovich et al., 1977). For doubly connected source areas (e.g., circular annulus), the area-averaged source temperature is used (Yovanovich and Schneider, 1977). If the source area is assumed to be isothermal, T source = T 0 . The general definition of spreading (or constriction) resistance leads to the follow- ing relationship for the dimensionless spreading resistance: k LR s = L A A θ dA A − (∂θ/∂z) z=0 dA (4.19) where θ = T (x,y) − T ∞ , the rise of the source temperature above the sink tem- perature. The arbitrary characteristic length scale of the source area is denoted as L. For convenience the dimensionless spreading resistance, denoted as ψ = kLR s (Yovanovich, 1976c; Yovanovich and Antonetti, 1988), is called the spreading resis- tance parameter. This parameter depends on the heat flux distribution over the source area and the shape and aspect ratio of the singly or doubly connected source area. The spreading resistance definition holds for transient conduction into or out of the half- space. If the heat flux is uniform over the source area, the temperature is nonuniform; and if the temperature of the source area is uniform, the heat flux is nonuniform (Carslaw and Jaeger, 1959; Yovanovich, 1976c). The relation for the dimensionless spreading resistance is mathematically identical to the dimensionless constriction re- sistance for identical boundary conditions on the source area. For a nonisothermal singly connected area the spreading resistance can also be defined with respect to its maximum temperature or the temperature at its centroid (Carslaw and Jaeger, 1959; Yovanovich, 1976c; Yovanovich and Burde, 1977). These temperatures, in general, are not identical and they are greater than the area-averaged temperature (Yovanovich and Burde, 1977). The definition of spreading resistance for the isotropic half-space is applicable for single and multiply isotropic layers which are placed in perfect thermal contact with the half-space, and the heat that leaves the source area is conducted through the layer or layers before entering into the half-space. The conductance h cannot be defined for the half-space problem because the corresponding area is not defined. 4.2.2 Spreading and Constriction Resistances in Flux Tubes and Channels If a circular heat source of area A s is in contact with a very long circular flux tube of cross-sectional area A t (Fig. 4.3), the flux lines are constrained by the adiabatic sides to “bend” and then become parallel to the axis of the flux tube at some distance z = from the contact plane at z = 0. The isotherms, shown as dashed lines, are everywhere orthogonal to the flux lines. The temperature in planes z = √ A t BOOKCOMP, Inc. — John Wiley & Sons / Page 273 / 2nd Proofs / Heat Transfer Handbook / Bejan DEFINITIONS OF SPREADING AND CONSTRICTION RESISTANCES 273 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 [273], (13) Lines: 603 to 635 ——— 0.57106pt PgVar ——— Normal Page * PgEnds: Eject [273], (13) Flux Tube or Channel Heat Source (Contact Area, )A s z z = z =0 A t Figure 4.3 Heat flow lines and isotherms for steady conduction from a finite heat source into a flux tube or channel. (From Yovanovich and Antonetti, 1988.) “far” from the contact plane z = 0 becomes isothermal, while the temperature in planes near z = 0 are two- or three-dimensional. The thermal conductivity of the flux tube is assumed to be constant. The total thermal resistance R total for steady conduction from the heat source area in z = 0 to the arbitrary plane z = is given by the relationship QR total = T s − T z= (K) (4.20) where T s is the mean source temperature and T z= is the mean temperature of the arbitrary plane. The one-dimensional resistance of the region bounded by z = 0 and z = is given by the relation QR 1D = T z=0 − T z= (K) (4.21) The total resistance is equal to the sum of the one-dimensional resistance and the spreading resistance: R total = R 1D + R s or R total − R 1D = R s (K) (4.22) By substraction, the relationship for the spreading resistance, proposed by Mikic and Rohsenow (1966), is R s = T s − T z=0 Q (K/W) (4.23) BOOKCOMP, Inc. — John Wiley & Sons / Page 274 / 2nd Proofs / Heat Transfer Handbook / Bejan 274 THERMAL SPREADING AND CONTACT RESISTANCES 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 [274], (14) Lines: 635 to 664 ——— 1.00616pt PgVar ——— Normal Page * PgEnds: Eject [274], (14) where Q is the total heat transfer rate from the source area into the flux tube. It is given by Q = A s −k ∂T ∂z z=0 dA s (W) (4.24) The dimensionless spreading resistance parameter ψ = k LR s is introduced for convenience. The arbitrary length scale L is related to some dimension of the source area. In general, ψ depends on the shape and aspect ratio of the source area, the shape and aspect ratio of the flux tube cross section, the relative size of the source area, the orientation of the source area relative to the cross section of the flux tube, the boundary condition on the source area, and the temperature basis for definition of the spreading resistance. The definitions given above are applicable to singly and doubly connected source areas; however, A s /A t < 1 in all cases. The source area and flux tube cross-sectional area may be circular, square, elliptical, rectangular, or any other shape. The heat flux and temperature on the source area may be uniform and constant. In general, both heat flux and temperature on the source area are nonuniform. Numerous examples are presented in subsequent sections. 4.3 SPREADING AND CONSTRICTION RESISTANCES IN AN ISOTROPIC HALF-SPACE 4.3.1 Introduction Steady or transient heat transfer occurs in a half-space z>0 which may be isotropic or may consist of one or more thin isotropic layers bonded to the isotropic half-space. The heat source is some planar singly or doubly connected area such as a circular annulus located in the “free” surface z = 0 of the half-space. The dimensions of the half-space are much larger than the largest dimension of the source area. The “free” surface z = 0 of the half-space outside the source area is adiabatic. If the source area is isothermal, the heat flux over the source area is nonuniform. If the source is subjected to a uniform heat flux, the source area is nonisothermal. 4.3.2 Circular Area on a Half-Space There are two classical steady-state solutions available for the circular source area of radius a on the surface of a half-space of thermal conductivity k. The solutions are for the isothermal and isoflux source areas. In both problems the temperature field is two-dimensional in circular-cylinder coordinates [i.e., θ(r, z)]. The important results are presented here. Isothermal Circular Source In this problem the mixed-boundary conditions (Sneddon, 1966) in the free surface are BOOKCOMP, Inc. — John Wiley & Sons / Page 275 / 2nd Proofs / Heat Transfer Handbook / Bejan SPREADING AND CONSTRICTION RESISTANCES IN AN ISOTROPIC HALF-SPACE 275 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 [275], (15) Lines: 664 to 720 ——— 1.70782pt PgVar ——— Normal Page * PgEnds: Eject [275], (15) z = 00≤ r<a θ = θ 0 0 r>a ∂θ ∂z = 0 (4.25) and the condition at remote points is: As √ r 2 + z 2 →∞, then θ → 0. The temperature distribution throughout the half-space z ≥ 0 is given by the infinite integral (Carslaw and Jaeger, 1959) θ = 2 π θ 0 ∞ 0 e −λz J 0 (λr) sin λa dλ λ (K) (4.26) where J 0 (x) is the Bessel function of the first kind of order zero (Abramowitz and Stegun, 1965) and λ is a dummy variable. The solution can be written in the following alternative form according to Carslaw and Jaeger (1959): θ = 2 π θ 0 sin −1 2a (r − a) 2 + z 2 + (r + a) 2 + z 2 (K) (4.27) The heat flow rate from the isothermal circular source into the half-space is found from Q = a 0 −k ∂θ ∂z z=0 2πrdr = 4kaθ 0 ∞ 0 J 1 (λa) sin λa dλ λ = 4kaθ 0 (W) (4.28) From the definition of spreading resistance one finds the relationship for the spreading resistance (Carslaw and Jaeger, 1959): R s = θ 0 Q = 1 4ka (K/W) (4.29) The heat flux distribution over the isothermal heat source area is axisymmetric (Car- slaw and Jaeger, 1959): q(r) = Q 2πa 2 1 1 − (r/a) 2 0 ≤ r<a (W/m 2 ) (4.30) This flux distribution is minimum at the centroid r = 0 and becomes unbounded at the edge r = a. Isoflux Circular Source In this problem the boundary conditions in the free surface are BOOKCOMP, Inc. — John Wiley & Sons / Page 276 / 2nd Proofs / Heat Transfer Handbook / Bejan 276 THERMAL SPREADING AND CONTACT RESISTANCES 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 [276], (16) Lines: 720 to 777 ——— 0.16327pt PgVar ——— Normal Page PgEnds: T E X [276], (16) z = 00≤ r<a ∂θ ∂z =− q 0 k 0 r>a ∂θ ∂z = 0 (4.31) where q 0 = Q/πa 2 is the uniform heat flux. The condition at remote points is identical. The temperature distribution throughout the half-space z ≥ 0 is given by the infinite integral (Carslaw and Jaeger, 1959) θ = q 0 a k ∞ 0 e −λz J 0 (λr)J 1 (λa) dλ λ (K) (4.32) where J 1 (x) is the Bessel function of the first kind of order 1 (Abramowitz and Stegun, 1965), and λ is a dummy variable. The temperature rise in the source area 0 ≤ r ≤ a is axisymmetric and is given by (Carslaw and Jaeger, 1959): θ(r) = q 0 a k ∞ 0 J 0 (λr)J 1 (λa) dλ λ (K) (4.33) The alternative form of the solution according to Yovanovich (1976c) is θ(r) = 2 π q 0 a k E r a 0 ≤ r ≤ a (K) (4.34) where E(r/a) is the complete elliptic integral of the second kind of modulus r/a (Byrd and Friedman, 1971) which is tabulated, and it can be calculated by means of computer algebra systems. The temperatures at the centroid r = 0 and the edge r = a of the source area are, respectively, θ(0) = q 0 a k and θ(a) = 2 π q 0 a k (K) (4.35) The centroid temperature rise relative to the temperature rise at the edge is greater by approximately 57%. The values of the dimensionless temperature rise defined as kθ(r/a)/(q 0 a) are presented in Table 4.1. TABLE 4.1 Dimensionless Source Temperature r/a kθ(r/a)/q 0 a r/a kθ(r/a)/q 0 a 0.0 1.000 0.6 0.9028 0.1 0.9975 0.7 0.8630 0.2 0.9899 0.8 0.8126 0.3 0.9771 0.9 0.7459 0.4 0.9587 1.0 0.6366 0.5 0.9342 BOOKCOMP, Inc. — John Wiley & Sons / Page 277 / 2nd Proofs / Heat Transfer Handbook / Bejan SPREADING AND CONSTRICTION RESISTANCES IN AN ISOTROPIC HALF-SPACE 277 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 [277], (17) Lines: 777 to 834 ——— -0.27078pt PgVar ——— Normal Page PgEnds: T E X [277], (17) The area-averaged source temperature is θ = 1 πa 2 q 0 a k a 0 ∞ 0 J 0 (λr)J 1 (λa) dλ λ 2πrdr (K) (4.36) The integrals can be interchanged, giving the result (Carslaw and Jaeger, 1959): θ = 2q 0 k ∞ 0 J 2 1 (λa) dλ λ 2 = 8 3π q 0 a k (K) (4.37) According to the definition of spreading resistance, one obtains for the isoflux circular source the relation (Carslaw and Jaeger, 1959) R s = θ Q = 8 3π 2 1 ka (K/W) (4.38) The spreading resistance for the isoflux source area based on the area-averaged tem- perature rise is greater than the value for the isothermal source by the factor (R s ) isoflux (R s ) isothermal = 32 3π 2 = 1.08076 4.3.3 Spreading Resistance of an Isothermal Elliptical Source Area on a Half-Space The spreading resistance for an isothermal elliptical source area with semiaxes a ≥ b is available in closed form. The results are obtained from a solution that follows the classical solution presented for finding the capacitance of a charged elliptical disk placed in free space as given by Jeans (1963), Smythe (1968), and Stratton (1941). Holm (1967) gave the solution for the electrical resistance for current flow from an isopotential elliptical disk. The thermal solution presented next will follow the analysis of Yovanovich (1971). The elliptical contact area x 2 /a 2 + y 2 /b 2 = 1 produces a three-dimensional temperature field where the isotherms are ellipsoids described by the relationship x 2 a 2 + ζ + y 2 b 2 + ζ + z 2 ζ = 1 (4.39) The three-dimensional Laplace equation in Cartesian coordinates can be transformed into the one-dimensional Laplace equation in ellipsoidal coordinates: ∇ 2 θ = ∂ ∂ζ f(ζ) ∂θ ∂ζ = 0 (4.40) where ζ is the ellipsoidal coordinate for the ellipsoidal temperature rise θ(ζ) and where f(ζ) = (a 2 + ζ)(b 2 + ζ)ζ (4.41) BOOKCOMP, Inc. — John Wiley & Sons / Page 278 / 2nd Proofs / Heat Transfer Handbook / Bejan 278 THERMAL SPREADING AND CONTACT RESISTANCES 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 [278], (18) Lines: 834 to 903 ——— 2.35738pt PgVar ——— Long Page PgEnds: T E X [278], (18) The solution of the differential equation according to Yovanovich (1971) is θ = C 2 − C 1 ∞ ζ dζ √ f(ζ) (K) (4.42) The boundary conditions are specified in the contact plane (z = 0) where θ = θ 0 within x 2 a 2 + y 2 b 2 = 1 ∂θ ∂z = 0 outside x 2 a 2 + y 2 b 2 = 1 (4.43) The regular condition at points remote to the elliptical area is θ → 0asζ →∞. This condition is satisfied by C 2 = 0, and the condition in the contact plane is satisfied by C 1 =−Q/4πk, where Q is the total heat flow rate from the isothermal elliptical area. The solution is, therefore, according to Yovanovich (1971), θ = Q 4πk ∞ ζ dζ (a 2 + ζ)(b 2 + ζ)ζ (K) (4.44) When ζ = 0, θ = θ 0 , constant for all points within the elliptical area, and when ζ →∞, θ → 0 for all points far from the elliptical area. According to the definition of spreading resistance for an isothermal contact area, we find that R s = θ 0 Q = 1 4πk ∞ 0 dζ (a 2 + ζ)(b 2 + ζ)ζ (K/W) (4.45) The last equation can be transformed into a standard form by setting sin t = a/ a 2 + ζ. The alternative form for the spreading resistance is R s = 1 2πka π/2 0 dt {1 − [(a 2 − b 2 )/a 2 ] sin 2 t} 1/2 (K/W) (4.46) The spreading resistance depends on the thermal conductivity of the half-space, the semimajor axis a, and the aspect ratio of the elliptical area b/a ≤ 1. It is clear that when the axes are equal (i.e., b = a), the elliptical area becomes a circular area and the spreading resistance is R s = 1/(4ka). The integral is the complete elliptic integral of the first kind K(κ) of modulus κ = (a 2 − b 2 )/a 2 (Byrd and Friedman, 1971; Gradshteyn and Ryzhik, 1965). The spreading resistance for the isothermal elliptical source area can be written as R s = 1 2πka K(κ) (K/W) (4.47) The complete elliptic integral is tabulated (Abramowitz and Stegun, 1965; Magnus et al. (1966); Byrd and Friedman, 1971). It can also be computed efficiently and very accurately by computer algebra systems. BOOKCOMP, Inc. — John Wiley & Sons / Page 279 / 2nd Proofs / Heat Transfer Handbook / Bejan SPREADING AND CONSTRICTION RESISTANCES IN AN ISOTROPIC HALF-SPACE 279 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 [279], (19) Lines: 903 to 931 ——— 0.90663pt PgVar ——— Long Page * PgEnds: Eject [279], (19) TABLE 4.2 Dimensionless Spreading Resistance of an Isothermal Ellipse a/b k √ AR s a/b k √ AR s 1 0.4431 6 0.3678 2 0.4302 7 0.3566 3 0.4118 8 0.3466 4 0.3951 9 0.3377 5 0.3805 10 0.3297 4.3.4 Dimensionless Spreading Resistance of an Isothermal Elliptical Area To compare the spreading resistances of the elliptical area and the circular area, it is necessary to nondimensionalize the two results. For the circle, the radius appears as the length scale, and for the ellipse, the semimajor axis appears as the length scale. For proper comparison of the two geometries it is important to select a length scale that best characterizes the two geometries. The proposed length scale is based on the square root of the active area of each geometry (i.e., L = √ A) (Yovanovich, 1976c; Yovanovich and Burde, 1977; Yovanovich et al., 1977). Therefore, the dimensionless spreading resistances for the circle and ellipse are (k √ AR s ) circle = √ π 4 (k √ AR s ) ellipse = 1 2 √ π a b K(κ) where κ = 1 − (b/a) 2 . The dimensionless spreading resistance values for an iso- thermal elliptical area are presented in Table 4.2 for a range of the semiaxes ratio a/b. The tabulated values of the dimensionless spreading resistance reveal an inter- esting trend beginning with the first entry, which corresponds to the circle. The di- mensionless resistance values decrease with increasing values of a/b. Ellipses with larger values of a/b have smaller spreading resistances than the circle; however, the decrease has a relatively weak dependence on a/b. For the same area the spread- ing resistance of the ellipse with a/b = 10 is approximately 74% of the spreading resistance for the circle. 4.3.5 Approximations for Dimensionless Spreading Resistance Two approximations are presented for quick calculator estimations of the dimension- less spreading resistance for isothermal elliptical areas: k √ AR s = √ πα ( √ α + 1) 2 for 1 ≤ α ≤ 5 1 2 √ πα ln 4α for 5 ≤ α < ∞ (4.48) BOOKCOMP, Inc. — John Wiley & Sons / Page 280 / 2nd Proofs / Heat Transfer Handbook / Bejan 280 THERMAL SPREADING AND CONTACT RESISTANCES 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 [280], (20) Lines: 931 to 1006 ——— 0.74892pt PgVar ——— Normal Page PgEnds: T E X [280], (20) where α = a/b ≥ 1. Although both approximations can be used at α = 5, the second approximation is slightly more accurate, and therefore it is recommended. 4.3.6 Flux Distribution over an Isothermal Elliptical Area The heat flux distribution over the elliptical area is given by (Yovanovich, 1971) q(x,y) = Q 2πab 1 − x a 2 − y b 2 −1/2 (W/m 2 ) (4.49) The heat flux is minimum at the centroid, where its magnitude is q 0 = Q/2πab, and it is “unbounded” on the perimeter of the ellipse. 4.4 SPREADING RESISTANCE OF RECTANGULAR SOURCE AREAS 4.4.1 Isoflux Rectangular Area The dimensionless spreading resistances of the rectangular source area −a ≤ x ≤ a, −b ≤ y ≤ b with aspect ratio a/b ≥ 1 are found by means of the integral method (Yovanovich, 1971). Employing the definition of the spreading resistance based on the area-averaged temperature rise with Q = 4qab gives the following dimensionless relationship (Yovanovich, 1976c; Carslaw and Jaeger, 1959): k √ AR s = √ π sinh −1 1 + 1 sinh −1 + 3 1 + 1 3 − 1 + 1 2 3/2 (4.50) where = a/b ≥ 1. Employing the definition based on the centroid temperature rise, the dimensionless spreading resistance is obtained from the relationship (Carslaw and Jaeger, 1959) k √ AR s = √ π 1 sinh −1 + sinh −1 1 (4.51) Typical values of the dimensionless spreading resistance for the isoflux rectangle based on the area-average temperature rise for 1 ≤ a/b ≤ 10 are given in Table 4.3. Table 4.3 Dimensionless Spreading Resistance of an Isoflux Rectangular Area a/b k √ AR s a/b k √ AR s 1 0.4732 6 0.3950 2 0.4598 7 0.3833 3 0.4407 8 0.3729 4 0.4234 9 0.3636 5 0.4082 10 0.3552 . 2nd Proofs / Heat Transfer Handbook / Bejan DEFINITIONS OF SPREADING AND CONSTRICTION RESISTANCES 271 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 [271],. transient heat transfer rate: Q = A q n dA = A −k ∂T ∂n dA (W) (4.17) where q n is the heat flux component normal to the area and ∂T /∂n is the temperature gradient normal to the area. If the heat. 272 / 2nd Proofs / Heat Transfer Handbook / Bejan 272 THERMAL SPREADING AND CONTACT RESISTANCES 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 [272],
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