At numerous places in chemical plants there is a need for a pipe of length L to carry a given fluid at a specified flow rate Q. An appropriate pipe diameter D must be chosen by the designer. Larger-diameter pipes are more expensive to buy and install but cheaper to operate. More precisely, the total annual cost C of a pipe can be expressed as

Embedded in the dimensional constant K₁ are the purchase, installation, and maintenance costs of the pipe and associated fittings, each per unit length and each put on an annual basis by assuming a certain lifetime for the system (Peters et al., 2003, pp. 401–406). For steel pipe, n ≅ 1.5. The power that a pump must deliver for flow through a straight pipe is Q|Δ℘|. The coefficient K₂ is the product of the electric rate, the operating time per year, and factors that account for pump efficiency and pressure drops in fittings. The fixed costs increase with D but the pumping costs decrease, such that there is an optimal value of D.

(a) In economic calculations the friction factor can be approximated as

where a and b are constants. Peters et al. (2003) suggest a= 0.04 and b = 0.16 for turbulent flow in new steel pipes; Eqs. (2.2-6) or (2.2-9) also could be used, depending on the range of Re. Show that the diameter D* that minimizes C is

and derive an expression for the dimensionless constant c.

(b) By examining the exponents in Eq. (P2.8-3), show that there is only a weak dependence of D* on ρ and μ, for either laminar or turbulent flow. Thus, for a given Q there is one fairly narrow range of D* for low-viscosity liquids and another for gases.

(c) Let U* be the mean velocity when D = D*. Show that

for turbulent flow. Thus, unlike D*, U* is nearly independent of Q. For Schedule 40 steel pipe, U* for liquids is typically between 1.8 and 2.4 m/s (Tilton, 2007).

Transcribed Image Text:

C = K₁D"L+K₂Q|AP|. (P2.8-1)