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Unit Operations & Transport Phenomena
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The latter material is important beyond the several examples considered, because it forces the student to think about what can be said about the solution of linear problems without actually solving any equations. Also, we assume that the liquid is cooled so that re-radiation can be neglected. The lower plate is a stationary plate and a force is applied to the upper plate to move with a velocity V. Kennard, Kinetic Theory of Gases, McGraw-Hill, New York (1938), pp. 292-295,300-306. In highly turbu- lent flow this jet necks down to a minimum cross section at the vena con- tracts.

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The more complex problem of fluidized beds is not included in this chapter.' 6. 1 DEFINITION OF FRICTION FACTORS We consider the steadily driven flow of a fluid of constant density in one of two systems: (a) the fluid flows in a straight conduit of uniform cross section; (b ) the fluid flows around a submerged object that has an axis of symmetry (or two planes of symmetry) parallel to the direction of the approaching fluid.

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Momentum transfer In fluid mechanics, the most useful equations are based on the principle of mass balance or continuity equation. The equations are first written in differential form. Mass transfer with simultaneous chemical reaction and heat transfer. Now the instantaneous velocity U, is represented as a sum of the mean value uz and a superimposed fluctuation U: about the mean. The friction velocity for pipe flow is ** (6.55) where rW is from Eq. (4.80). Equation (10.8) simplifies to l/Cf)rn = 4.0[logI,,(rJe)] + B,/(2)“* - 2.53 (10.9) 414 APPLKATIONS OF TRANSPORT PHENOMENA 9 4 8 7 6L I I 0.2 0.4 0.6.

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Y.: in Miring of Liquids by Mechanical Agifation, J. As the velocity is further increased, a transition region occurs. The same kind of theory may be applied to concentrated solutions and molten polymers by examining the motion of a single bead spring model in the "mean force field" exerted by the surrounding molecules. C. 4: at z = L, ~ 1 1 = ~ " 1 (10.5-12) B. There, in 98, a discussion of nonorthogonal coordinates is given.

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Next, Eq. (15.24) is used to calculate the actual wall shear rate f*jd; these values are also given in Table 15.4, along with the correction factor (3n’+ 1)/(4n’). The balance equation given as Eq. (3.1) is repeated here: INPUT + GENERATION = OUTPUT + ACCUMULATION (3.1) which can be rearranged to: ACCUMULATION = -(OUTPUT - INPUT) + GENERATION (7.1) Equation (7.1) simply says that the property will accumulate if there is more flowing in than out when no property is generated within the volume.

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For fluids whose Prandtl numbers are unity, Eq. (12.27) reduces to Eq. (12.1). Equation (7.30) is the basis for making a force balance on a free body; a free body is the widely used term for a control volume in which all forces act on the body. In other cases it can be estimated ( 3 ) using thermodynamics. This interpre- tation is consistent with the molecular picture of momentum transport and the kinetic theories of gases and liquids. Estimate the normal force (in newtons) on a circular sign 8 ft in diameter during a hurricane whose winds are blowing at 120mph.

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For example, for a scalar function ~ ( x, y, z ) = +(r, 0, z): Note that we are careful to use different symbols ,y and I,//, since x is a different function of x, y, z than I,! As pointed out in the preceding section, the turbulent diffusivity is zero at the wall, and consequently does not appear in Eq. 21.4-2. There is, after all, a pronounced similarity between the heated wire problem and the viscous flow in a circular tube.

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He showed that the friction factor was a function of the Reynolds number and the relative roughness together, as can be clearly seen from a plot of his data. VT} and ' This appendix is very similar to Appendix A of R. This is done by multiplying both sides of the equation by sin mrr], where m is an integer, and then integrating over the physically pertinent range from r] = 0 to r] = 1, thus: m 1 lo1 (1 - r ] ) sin mrqdr) = C D. sin nrr] sin rnrr]dr] n = l 0 The left side gives l/m.rr; the integrals on the right side are zero when n f m and when n = m.

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What is the pipe length for (a) blackbody (b) gray (emissivity of 0.75 atm) at a total pressure of 1 atm and 1400 K.75 m. The left hand face is at TI and the right-hand face a T2. The capillary tubing is 0.0025 m inside diameter. Thus, the flow is fully turbulent; the parameter s is computed from Eq. (6.122): s = 0.585 + (3.172 x 10-3)(37 483)“.*33 = 21.066 The parameter m is obtained from Eq. (6.119): m = -0.617 + (8.211 x 10-3)(37 483)‘.‘= = 31.70 = 32 Thus, from Eqs. (6.115) and (6.116): a, = -0.3527 and a* = -0.6473 (xviii) Equation (6.118) is used to obtain perhaps the best possible estimate of 0,. max: c, max = (0.8467)/[1.0 + (-0.3527)/(2) + (-0.6473)/(32 + l)] = 1.053 m s-i = 3.455 ft s-’ (xix) (xvii) (xv) b4 As will be seen, this estimate of nz,, max is sometimes more reliable than that from the universal velocity equations [Eq. (ix)].

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There are, oddly enough, a number of such similarities between extraction and distillation. In essence, one of the components is then separated from the mix of the other component and extractive agent. At a given time the bottom plate moves with a velocity V.0 (1-7) where p is the fluid‘s viscosity (i. Complete tables of pipe and tubing data are in Table B.l and B.2, respectively. Calculate the steam condensed per hour per unit length if the tube walls are kept at 88°C. 7-21.