A company markets ethylene glycol antifreeze in halfgal bottles. A machine fills and caps the bottles at a rate of 60 per minute. The \(68^{\circ} \mathrm{F}\) ethylene glycol \(\left(\rho=69.3 \mathrm{lb}_{\mathrm{m}} / \mathrm{ft}^{3}, v=1.93 \times\right.\) \(\left.10^{-4} \mathrm{ft}^{2} / \mathrm{s}\right)\) is pumped to the machine from a tank \(36 \mathrm{ft}\) away. The pressure at the discharge of the pump is 35 psig, and the pressure at the inlet to the filling machine must be at least 15 psig. Determine the diameter necessary for drawn copper pipe. The pipe has no elevation change.

Identify and list down the given variables in the problem. The rate of bottle filling is 60 bottles per minute, which is equivalent to 1 bottle per second since there are 60 seconds in a minute. Each bottle contains half a gallon of ethylene glycol. The distance from the pump to the machine is 36 feet. The discharge pressure at the pump end is 35 psig, while the pressure at the machine end must be at least 15 psig. The properties of ethylene glycol are given: \(\rho = 69.3 \, \text{lbm/ft}^3\) and \(v = 1.93 \times 10^{-4} \, \text{ft}^2/\text{s}\).

Calculate the mass flow rate (\(\dot{m}\)) using the given rate of bottle filling and the density (\(\rho\)) of ethylene glycol. For the volume flow rate (\(\dot{V}\)), since 1 bottle is being filled per second and each bottle is half a gallon, we get \(\dot{V} = 0.5 \, \text{gal/s} = 0.5/7.48 \, \text{ft}^3/\text{s}\) because there are 7.48 gallons in 1 cubic foot. The mass flow rate can be determined with the equation \(\dot{m} = \dot{V} \times \rho\).

Since we are using Bernoulli's equation later, we need the pressures in absolute terms, not gauge pressures. Atmospheric pressure at sea level is approximately 14.7 psi. So, the absolute pressures at the pump and machine are 35 psig + 14.7 psi = 49.7 psi and 15 psig + 14.7 psi = 29.7 psi, respectively. Convert these pressures to lb/ft^2 by multiplying by \(144 \, \text{ft}^2/\text{in}^2\) as we need to use consistent units in our calculations. Thus, we get pump pressure = 49.7 psi * 144 = 7156.8 lb/ft^2 and machine pressure = 29.7 psi * 144 = 4276.8 lb/ft^2.

Apply Bernoulli's equation, which equates total energy per unit mass along a streamline. This accounts for the pressure, kinetic, and potential energies. However, as per given data, the potential energy can be ignored as there is no elevation change. Thus, the equation simplifies to: \(\frac{P_1}{\rho g} + \frac{v_1^2}{2g} = \frac{P_2}{\rho g} + \frac{v_2^2}{2g} + h_f\), where \(P_1\) and \(P_2\) are the pressures at the pump and machine, respectively, \(v_1\) and \(v_2\) are the velocities, \(g\) is the acceleration due to gravity, and \(h_f\) is the frictional losses. Given that the pipe frictional loss is important, we can express it using the Darcy-Weisbach equation: \(h_f = \frac{f L v^2}{2 g D}\), where \(f\) is the friction factor, \(L\) is the length of the pipe, \(D\) is its diameter, and \(v\) is the velocity. This allows us to compute \(D\), but the friction factor \(f\) is also a function of \(D\). Therefore, this problem needs iterative solution.

Guess an initial value for \(D\). Substitute it in the equations from Step 4 to compute \(f\), and then re-calculate the updated \(D\). Repeat the process until \(D\) converges to a constant value. Use the Moodey chart or another tool to find \(f\). This step may require numerical or programming skills.