 Incompressible water at 20°C flows through a 1½ inch diameter pipe at a rate of 5 gallons per minute. What is the length of the entrance region in feet? Well, the first question we have to answer is, is this flow laminar or turbulent? We have equations for both, but we have to know which equation to use. For that, we'll have to calculate the Reynolds number and compare it to the critical Reynolds number for internal flow through a circular pipe, which is 2,300. I'm going to use my Reynolds number calculation with respect to kinematic viscosity to avoid having to look up two properties instead of one. I know the velocity to use in this equation is going to be the average velocity corresponding to 5 gallons of flow per minute. That means I'm going to use average velocity times cross sectional area, therefore average velocity is equal to volumetric flow rate divided by area to write volumetric flow rate times diameter divided by cross sectional area times kinematic viscosity. I recognize that for a circular pipe, the area can be represented in terms of diameter. So I'll write volumetric flow rate times diameter divided by pi over 4 times diameter squared. And then I have to multiply by kinematic viscosity again. The diameter in the numerator cancels one of the diameters in the denominator, leaving me with volumetric flow rate times 4 divided by pi times diameter times kinematic viscosity. That gives me 4 times 5 gallons per minute divided by pi times the diameter, which is one half inch, times the kinematic viscosity for water. I know it's incompressible water at 20 degrees Celsius, which means I'm going to go into my property tables, specifically table A1. On table A1, I have metric data and imperial data, and the row stretches the entire way. So at 20 degrees Celsius, which is an imperial temperature of 68 degrees Fahrenheit, my density of water is 998 kilograms per cubic meter, and is also 1.937 slugs per cubic foot. Note that by using kinematic viscosity in place of density and dynamic viscosity, we have completely circumvented having to deal with slugs. Yet another reason that using kinematic viscosity is so convenient. But the kinematic viscosity for water at 20 degrees Celsius can be expressed as 1.082 times 10 to the negative fifth square feet per second. And that, again, came from table A1. So I'll write 1.082 times 10 to the negative fifth square feet per second. Now I can begin to cancel units because I want a dimensionless number at the end. So first I can write one minute is 60 seconds. I can write there are 12 inches in one foot, and then I have to convert cubic feet to gallons and vice versa. For that, I'm going to need a little bit more space. The conversion from gallons to cubic feet is going to be one gallon is equal to 0.13368 cubic feet. So 0.13368 cubic feet, gallon cancels gallon, cubic feet, cancels feet, and square feet. That means I'm left with a unitless proportion at the end, which is what I want. So calculator, if you'd be so kind, 4 times 5 times 12 times 0.13368 divided by high times 0.5 times 1.082 E negative sine 5 times 60 yields a Reynolds number of 31461.5. We're comparing that number to our critical Reynolds number, which is 2,300. I see that my Reynolds number is greater than that. Therefore, I have turbulent flow, or more accurately, we're characterizing this as turbulent flow. For turbulent flow, we're going to use the equation. Entrance length divided by diameter is equal to 1.6 times the Reynolds number raised to the 1 quarter power. Therefore, entrance length can be expressed as 1.6 times diameter times the Reynolds number raised to the 1 quarter power. That means I have 1.6 times 31461.5 to the 1 quarter power, multiplied by our diameter. And the convenient thing about this is because 1.6 and the Reynolds number are dimensionless, whatever unit I plug in for diameter is going to be the unit that I get out at the end. So if I plug in 1.5 inch, I will get out an entrance length in inches. So 1.6 times 31641.5, excuse me, 31461.5, raised to the 0.25 power times 0.5 yields 10.6545. So my entrance length is about 10 and 5 eighths inches. Then a follow-up question. If the length of the pipe is 60 feet, the entrance region represents one fraction of the pipe. So 10.6545 divided by 12 is 0.887879 feet. And in fact, I think I asked for the entrance length in feet so I can correct that. And then I'm taking that number divided by 60 to yield a percentage. And I see that only one and a half percent of my pipe is entrance region. That's such a small number that we could probably get away with neglecting the entrance effect, depending on the circumstances for the environment that we're calculating this number for. If we need a real accurate number for some purpose or another, we may include it, but we can probably get away with neglecting it.