Question

Reynolds number E. What is the mean velocity u. (ft/s) and the Reynolds number Re = pu., D/ for 35 gpm (gallons per minute) of water flowing in a 1.05- in. ID. pipe if its density is p = 62.3 lb/ft and its viscosity is = 1.2 cP? What are the units of the Reynolds number?

Answer 1

The Reynolds number (Re) and mean velocity (u) for water flowing through a pipe can be calculated using the flow rate, pipe dimensions, fluid density, and viscosity. The Reynolds number is a dimensionless quantity that helps predict the flow pattern in the pipe.

Step-by-step explanation:

The question relates to the calculation of the Reynolds number and the mean velocity (u) for a given flow rate of water through a pipe. First, to find the mean velocity u, the flow rate needs to be converted to cubic feet per second (ft³/s) and then divided by the cross-sectional area of the pipe. The Reynolds number Re is a dimensionless number used to predict flow patterns in different fluid flow situations. It is calculated using the formula Re = ρuD/μ, where ρ is the fluid density, u is the mean velocity, D is the pipe diameter, and μ is the dynamic viscosity of the fluid.

To proceed with the calculation, the given flow rate of 35 gallons per minute (gpm) is converted to cubic feet per second, the pipe's internal diameter is converted to feet, the density of water (62.3 lb/ft³) is used, and the viscosity (1.2 cP) is converted to lb/(ft·s). The mean velocity u is then calculated, and subsequently, the Reynolds number Re is determined.

The units of the Reynolds number are indeed unitless, as demonstrated by the cancellation of units in its definition, ensuring it is a dimensionless quantity.

Answer 2

The mean velocity is 13 ft/s.

The Reynolds number is 88,583 and it is dimensionless.

Step-by-step explanation:

We have water flowing in a pipe of 1.05 in diameter.

The density is ρ=62.3 lb/ft and the viscosity is 1.2 cP.

The mean velocity can be calculated as

`u=(Q)/(A)=(Q)/(\pi*D^2/4)=(35gpm )/(3.14*(1.05in)^2/4)\\\\  u=(35)/(0.865)*(gal)/(min)(1)/(in^2)*(231in^3)/(1gal)*(1)/(60s) \\\\    u=156\,in/s=13\,ft/s`

The Reynolds number now can be calculated for this flow as

`Re=(\rho*u*D)/(\mu)`

being ρ: density, u: mean velocity of the fluid, D: internal diameter of the pipe and μ the dynamic viscosity.

To simplify the calculation, we can first make all the variables have coherent units.

Viscosity

`\mu=1.2cP=(1.2)/(100)(g)/(cm*s)*(1lb)/(453.6g)*(30.48cm)/(1ft)= 0.0008(lb)/(ft*s)`

Diameter

`D=1.05in*((1ft)/(12in) )=0.0875ft`

Then the Reynolds number is

`Re=(\rho*u*D)/(\mu)\\\\Re=62.3(lb)/(ft^3)*13(ft)/(s) *0.0875ft*(1)/(0.0008)*(ft*s)/(lb)\\\\Re=88,583`