A 1/2-in-diameter water pipe is 60 ft long and delivers water at 5 gal/min at 20°C. What fraction of this pipe is taken up by the entrance region?

Question

A [latex]\frac{1}{2}[/latex]-in-diameter water pipe is 60 ft long and delivers water at 5 gal/min at 20°C. What fraction of this pipe is taken up by the entrance region?

Accepted Answer

Convert

Q = (5 gal/min)[latex]\frac{0.00223 ft^3/s}{1 gal/min} = 0.0111 ft^3/s[/latex]

The average velocity is

[latex]V = \frac{Q}{A} = \frac{0.0111 ft^3/s}{(\pi/4) [(\frac{1}{2}/12) ft]^2} = 8.17[/latex] ft/s

From Table 1.4 read for water [latex] u = 1.01 imes 10^{-6} m^2/s = 1.09 imes 10^{-5} ft^2/s[/latex]. Then the pipe Reynolds number is

[latex]Re_d = \frac{Vd}{ u} = \frac{(8.17 ft/s)[(\frac{1}{2}/12) ft]}{1.09 imes 10^{-5} ft^2/s} = 31,300[/latex]

Table 1.4 Viscosity and Kinematic Viscosity of Eight Fluids at 1 atm and 20°C
Fluid	µ, kg/(m·s)[latex]^†[/latex]	Ratio µ/µ([latex]H_2[/latex])	ρ, kg/[latex]m^3[/latex]	v, [latex]m^2/s^†[/latex]	Ratio v/v(Hg)
Hydrogen	9.0 E-6	1.0	0.084	1.05 E-4	910
Air	1.8 E-5	2.1	1.20	1.50 E-5	130
Gasoline	2.9 E-4	33	680	4.22 E-7	3.7
Water	1.0 E-3	114	998	1.01 E-6	8.7
Ethyl alcohol	1.2 E-3	135	789	1.52 E-6	13
Mercury	1.5 E-3	170	13,550	1.16 E-7	1.0
SAE 30 oil	0.29	33,000	891	3.25 E-4	2,850
Glycerin	1.5	170,000	1,260	1.18 E-3	10,300

[latex]^†[/latex]1 kg/(m·s) = 0.0209 slug/(ft·s); 1 [latex]m^2[/latex]/s = 10.76 [latex]ft^2[/latex]/s.

This is greater than 4000; hence the flow is fully turbulent, and Eq. (6.6) applies for entrance length:

[latex]\frac{L_e}{d} \approx 1.6 Re_d^{1/4} for Re_d \leq 10^7[/latex] (6.6)

[latex]\frac{L_e}{d} \approx 1.6 Re_d^{1/4} = (1.6)(31,300)^{1/4} = 21[/latex]

The actual pipe has L/d = (60 ft)/[([latex]\frac{1}{2}[/latex]/12)ft] = 1440. Hence the entrance region takes up the fraction

[latex]\frac{L_e}{L} = \frac{21}{1440} = 0.015 =[/latex] 1.5%

This is a very small percentage, so that we can reasonably treat this pipe flow as essentially fully developed.

Question 6.2: A 1/2-in-diameter water pipe is 60 ft long and delivers wate...

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