Water flows down a long, straight, inclined pipe of diameter D and length L (Fig. P9–125). There is no forced pressure gradient between points 1 and 2; in other words, the water flows through the pipe by gravity alone, and P₁ = P₂ = P_atm. The flow is steady, fully developed, and laminar. We adopt a coordinate system in which x follows the axis of the pipe. 

(a) Use the control volume technique of Chap. 8 to generate an expression for average velocity V as a function of the given parameters ρ, g, D, Δz, μ, and L.

(b) Use differential analysis to generate an expression for V as a function of the given parameters. Compare with your result of part (a) and discuss. 

(c) Use dimensional analysis to generate a dimensionless expression for V as a function of the given parameters. Construct a relationship between your ∏’s that matches the exact analytical expression.

FIGURE P9–125

Transcribed Image Text:

Az P₁ D P, H α V L P2 too