In distribution manifolds it is often desired to have the same flow rate Q_B in each branch. If the branches are identical, the inlet pressure at each would need to be the same. There are at least two strategies for making those pressures uniform. If viscous losses along the main pipe are negligible, a series of diameter reductions could keep the velocity from falling and the pressure from rising, as shown in Fig. P12.9(a). If the losses are not negligible, a constant pipe diameter D₀ and a certain length L between branches could be chosen so that the pressure recovery at a branch is exactly balanced by the pressure drop in the next pipe segment, as shown in Fig. P12.9(b). Suppose that the flow is turbulent everywhere and let the initial diameter, flow rate, and velocity for both cases be D₀, Q₀, and v₀, respectively.

(a) For the case with negligible losses and variable pipe diameter, let D_n be the diameter after branch n. How should D₁ differ from D₀? What should D_n be in general (n≥1)?

(b) For the case where the viscous pressure drops balance the pressure recoveries, derive a relationship to guide the choices of D₀ and L. Explain how it could be used.

(c) For air flow with v₀ = 12.3 m/s, D₀ = 100 mm, D_B = 25 mm, L = 1.8 m, P1 = 1.025 atm, and P₀ = 1.000 atm, how would the pressure drop in the first segment compare with the pressure recovery at the first branch? Which of the two approaches would be most applicable in optimizing such a manifold?

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Qo QB Do D₁ 요 (a) D₂ سيدا Qo QB Do L L L (b) Figure P12.9 Distribution manifolds in which the pressure recoveries at the side openings are offset by: (a) diameter reductions in the main pipe (if viscous losses are negligible); or (b) viscous losses along a constant-diameter main pipe. The approximate pressure variations along the main pipe are shown below each design. An experimental pressure profile like that in (b) is shown in Denn (1980, p. 129).