Problem 4.8 in the text describes a hydraulic power storage system in which water is pumped from a river up to an elevated storage reservoir through two 30-inch ID pipes, each carrying 20,000 gal/min. Power is consumed to pump the water uphill. Later, the water flows by gravity at the same flow rate from the reservoir back down to the river through two 30-inch ID pipes. As the water enters the river, it passes through two power-generation turbines (one for each pipe). The objectives of the problem are (a) to calculate the electrical power consumed to pump the water up to the reservoir, (b) to calculate the electrical power generated by the turbines as the water returns, and (c) to calculate the overall power-storage efficiency of the system by dividing the electrical power generated by the turbines by electrical power consumed by the pump.

Your assignment is to solve a modified version of Problem 4.8 with the following changes in the problem statement. First, only a single 30-inch ID pipe carries the water up to the storagereservoir at a rate of 25,000 gal/min, and a single 30-inch ID pipe carries the water back to the river at the same flow rate. Second, the frictional energy loss is estimated to be 35 ft of water instead of 15 ft.

Hint 1: Inefficiency in a pump or turbine results in energy being lost as energy is interconverted between electricity and mechanical energy. A pump converts electricity into a fluids mechanical energy, and a turbine does the opposite. As described on page 92 of the text, the amount of electrical energy consumed by the pump per unit mass of fluid (Wp) is greater than the increase in the fluids mechanical energy per unit mass. To account for pump inefficiency, the Wp term in the modified Bernoulli equation is multiplied by the pump efficiency (), which ranges between 0 and 1. On the other hand, when a turbine is used, the energy flows in the opposite direction (from mechanical energy in the fluid to electrical energy). Due to inefficiency in the turbine, the amount of mechanical energy per unit mass of fluid lost from the fluid as it drives the turbine is greater than amount of electrical energy per unit mass of fluid produced by the turbine (Wp). Thus, the Wp term in the modified Bernoulli equation should be divided by the turbine efficiency. In both cases (pump and turbine, after the Wp term is calculated, the electrical power (either consumed by the pump or generated by the turbine) is then calculated by multiplying by the fluids mass flow rate (Power = mW p).

Hint 2: The problem statement says the friction loss is equivalent to 35 ft of water. The modified Bernoulli equation (Eq. 4.74) can be written in terms of head by dividing it by g/gc to obtain

0=( gc/g P2/ + z2 + u2²/2 g ) ( gc/g P1/ + z1+ u1²/2 g ) gc/g W p+ gc/g hf

Each term in this equation has dimensions of length and can be thought of as the height of a fluid column that would have equivalent amount of mechanical energy. For example, the term last term in the equation is called the friction head term. Thus, the problem the statement that the friction loss is equivalent to 35 ft of water could be written in the form of an equation as gc/g hf =3 5 ft

Plugging in g = 32.2 ft/s² and gc = 32.2 ft lbm/(s² lbf) and rearranging gives hf = 35 ft lbf/lbm.

Hint 3: Often homework problems in the text book deliberately dont provide all the details students would like to have. The intent is to help students develop skill in making the sort of reasonable engineering assumptions that are commonly needed to solve real-world problems. For example, in Problem 4.8, insufficient information is given about the waters entry and exit points for the pressure and velocity terms to be specified in the modified Bernoulli equation. The following logic is an example of making reasonable engineering assumptions to solve the problem. For the first flow path (from the river, through the pump, through the pipe, and into the reservoir), you might assume that the pipe inlet is submerged in the river. Thus, point 1 in the modified Bernoulli equation could be taken to be the top of the river, where the velocity is approximately zero and the pressure is 1 atm. Also, you might assume the pipe outlet is just above the water level of the reservoir. Thus, at point 2 there would be a pressure of 1 atm and a equal to the fluids velocity in the pipe. For return flow (from the reservoir, through the pipe and turbine, and into the river), you might assume that point 1 is submerged in the reservoir, and point 2 is submerged in the river. Note that other assumptions could be made. For example, you might assume that point 2 is just above the river water level. It is good practice when you make an engineering assumption to check the assumption and/or determine whether the assumption significantly affects the calculated results. Here, you could repeat the calculation for point 2 being above and below water level to see how sensitive the calculation is to that assumption