The commercial steel pipe in the figure has a diameter of 3 in. and transfers glycerin from the large tank to the outlet at ”B”. If the tank is open at the top, determine the initial discharge at ”B” when the gate valve at ”C” is fully opened.

The initial discharge at B when the gate valve at C is fully opened can be calculated using Bernoulli's equation, which states that the total pressure of a fluid flowing through a pipe is constant. At point A, the fluid is at rest, so the velocity (v1) is zero. The pressure (p1) is atmospheric (since the tank is open to the air). At point B, the fluid is exiting the pipe, so the pressure (p2) is also atmospheric. The height difference between A and B is given as 10 ft. Thus, Bernoulli's equation can be written as: p1 + (1/2)?v1^2 + ?gh1 = p2 + (1/2)?v2^2 + ?gh2 where: p1 = atmospheric pressure v1 = 0 ? = density of glycerin g = acceleration due to gravity h1 = 0 p2 = atmospheric pressure v2 = velocity of fluid at point B h2 = 10 ft Solving for v2: v2 = ?(2gh2) Substituting in the given values: v2 = ?(2*32.2 ft/s^2 * 10 ft) ? 20 ft/s The initial discharge at B can then be calculated using the formula for volume flow rate: Q = Av where: Q = volume flow rate A = cross-sectional area of pipe v = velocity of fluid The cross-sectional area of a pipe with diameter D is given as: A = (?/4)D^2 Substituting in the given values: A = (?/4)(3 in)^2 = 7.07 in^2 Converting to feet: A = 7.07 in^2 * (1 ft/12 in)^2 ? 0.049 ft^2 Finally, the initial discharge at B is: Q = Av2 = (0.049 ft^2)(20 ft/s) ? 0.98 ft^3/s Therefore, the initial discharge at B when the gate valve at C is fully opened is \approx imately 0.98 ft^3/s.