What is the physical significance of the Reynolds number? How is it defined for (a) flow in a circular pipe of inner diameter D and (b) flow in a rectangular duct of cross section $a imes b?$

Show that the Reynolds number for flow in a circular pipe of diameter D can be expressed as $\mathrm{Re}=4 \dot{m} /(\pi D \mu).$

Consider the flow of air and water in pipes of the same diameter, at the same temperature, and at the same mean velocity. Which flow is more likely to be turbulent? Why?

Air enters a 10-m-long section of a rectangular duct of cross section $15 cm imes 20 cm$ made of commercial steel at 1 atm and $35^{\circ} \mathrm{C}$ at an average velocity of 7 m/s. Disregarding the entrance effects, determine the fan power needed to over­come the pressure losses in this section of the duct.

The velocity profile in fully developed laminar flow in a circular pipe of inner radius R = 10 cm, in m/s, is given by $u(r) = 4(1 − r^2/R^2).$ Determine the mean and maximum velocities in the pipe, and the volume flow rate.

Water at $10^{\circ} \mathrm{C}\left(\rho=999.7 \mathrm{kg} / \mathrm{m}^{3} \text { and } \mu=1.307 \times\right. \left.10^{-3} \quad \mathrm{kg} / \mathrm{m} \cdot \mathrm{s}\right)$ is flowing steadily in a 0.12-cm-diameter, 15-m-long pipe at an average velocity of 0.9 m/s. Determine (a) the pressure drop, (b) the head loss, and (c) the pumping power requirement to overcome this pressure drop.

Water at $15^{\circ} \mathrm{C}\left(\rho=999.1 \mathrm{kg} / \mathrm{m}^{3} \text { and } \mu=1.138 \times\right. \left.10^{-3} \quad \mathrm{kg} / \mathrm{m} \cdot \mathrm{s}\right)$ is flowing steadily in a 30-m-long and 5-cm-diameter horizontal pipe made of stainless steel at a rate of 9 L/s. Determine (a) the pressure drop, (b) the head loss, and (c) the pumping power requirement to overcome this pressure drop.

The water needs of a small farm are to be met by pumping water from a well that can supply water continuously at a rate of 4 L/s. The water level in the well is 20 m below the ground level, and water is to be pumped to a large tank on a hill, which is 58 m above the ground level of the well, using 5-cm internal diameter plastic pipes. The required length of piping is measured to be 420 m, and the total minor loss coefficient due to the use of elbows, vanes, and so on, is estimated to be 12. Taking the efficiency of the pump to be 75 percent, determine the rated power of the pump that needs to be purchased, in kW. The density and viscosity of water at anticipated operation conditions are taken to be $1000 \mathrm{kg} / \mathrm{m}^{3}$ and $0.00131 \mathrm{kg} / \mathrm{m} \cdot \mathrm{s},$ respectively. Is it wise to purchase a suitable pump that meets the total power requirements, or is it necessary to also pay particular attention to the large elevation head in this case? Explain.

How does surface roughness affect the pressure drop in a pipe if the flow is turbulent? What would your response be if the flow were laminar?

A 0.3 m by 0.5 m rectangular air duct carries a flow of 0.45 m$^{3}$/s at a density of 2 kg/m$^{3}$. Calculate the mean velocity in the duct. If the duct tapers to 0.15 m by 0.5 m size, what is the mean velocity in this section if the density is 1.5 kg/m$^{3}$ there?

Air enters a 10-m-long section of a rectangular duct of cross-section $15 \mathrm{~cm} \times 20 \mathrm{~cm}$ made of commercial steel at $1 \mathrm{~atm}$ and $35^{\circ} \mathrm{C}$ at an average velocity of $5 \mathrm{~m} / \mathrm{s}$. Disregarding the entrance effects, determine the fan power needed to overcome the pressure losses in this section of the duct.

Air enters a $7 -\mathrm{m}$-long section of a rectangular duct of cross section $15 \mathrm{~cm} \times 20 \mathrm{~cm}$ at $50^{\circ} \mathrm{C}$ at an average velocity of $7 \mathrm{~m} / \mathrm{s}$. If the walls of the duct are maintained at $10^{\circ} \mathrm{C}$, Find $(a)$ the outlet temperature of the air, $(b)$ the rate of heat transfer from the air, and $(c)$ the fan power needed to overcome the pressure losses in this section of the duct.

A channel of triangular cross section, which is 25 m long and 1 m deep, is used for the storage of water. The water and the surrounding air are each at a temperature of $25^{\circ} \mathrm{C}$, and the relative humidity of the air is 50%. (a) If the air moves at a velocity of 5 m/s along the length of the channel, what is the rate of water loss due to evaporation from the surface? (b) Obtain an expression for the rate at which the water depth would decrease with time due to evaporation. For the above conditions, how long would it take for all the water to evaporate?

In an air heating system, heated air at $40^{\circ} \mathrm{C}$ and 105 kPa absolute is distributed through a $0.2 \mathrm{m} imes 0.3 \mathrm{m}$ rec­tangular duct made of commercial steel at a rate of $0.5 m^3/s$. Determine the pressure drop and head loss through a 40-m- long section of the duct.