Question

A geothermal district heating system involves the transport of geothermal water at \(110^{\circ} \mathrm{C}\) from a geothermal well to a city at about the same elevation for a distance of \(12 \mathrm{~km}\) at a rate of $1.5 \mathrm{~m}^{3} / \mathrm{s}\( in \)60-\mathrm{cm}$-diameter stainless steel pipes. The fluid pressures at the wellhead and at the arrival point in the city are to be the same. The minor losses are negligible because of the large length-to-diameter ratio and the relatively small number of components that cause minor losses. (a) Assuming the pump-motor efficiency to be 65 percent, determine the electric power consumption of the system for pumping. \((b)\) Determine the daily cost of power consumption of the system if the unit cost of electricity is \(\$ 0.06 / \mathrm{kWh}\). (c) The temperature of geothermal water is estimated to drop \(0.5^{\circ} \mathrm{C}\) during this long flow. Determine if the frictional heating during flow can make up for this drop in temperature.

Short answer

Question: Calculate (a) the required power for the pumping system, (b) the daily cost of power consumption for the geothermal heating system, and (c) determine if frictional heating can compensate for the temperature drop during flow.

Step-by-step solution

Calculate the pressure loss due to flow in the pipes

We will use the Darcy-Weisbach equation to find the pressure loss due to the flow in the pipes. Here is the equation: \(\Delta P = f \cdot \frac{L}{D} \cdot \frac{1}{2} \rho v^2\) where \(\Delta P\) is the pressure loss, \(f\) is the friction factor, \(L\) is the length of the pipes, \(D\) is the pipe diameter, \(\rho\) is the fluid density, and \(v\) is the fluid velocity. Now we need to find the fluid velocity, given by: \(v = \frac{Q}{A}\), where \(Q\) is the flow rate and \(A\) is the cross-sectional area of the pipe. Then, we can use the pressure loss to calculate the required power to pump the water.

Calculate the required pump power

The required pump power can be calculated as follows: \(P_{required} = \frac{\Delta P \cdot Q}{\eta}\), where \(\eta\) is the pump-motor efficiency. With this equation, we can find the required power for the pumping system.

Calculate the daily cost of power consumption

We can calculate the daily cost of power consumption by multiplying the electric power consumption with the unit cost of electricity and the number of hours of operation in a day. \(Cost_{daily} = P_{required} \cdot cost \cdot hours\), where cost is given in $/kWh and hours = 24. Now we can find the daily cost of power consumption for the geothermal heating system.

Determine if frictional heating compensates for the temperature drop

Frictional heating can be calculated using the following formula: \(Q_f = c_p \cdot T_{drop} \cdot m\), where \(Q_f\) is the energy due to frictional heating, \(c_p\) is the specific heat capacity of the fluid, \(T_{drop}\) is the temperature drop, and \(m\) is the fluid mass flow rate. Now, we need to compare the energy loss due to the temperature drop (\(Q_{loss}\)) and the energy gain due to frictional heating. If \(Q_f >= Q_{loss}\), then frictional heating can compensate for the temperature drop during flow. (a) Following Step 2, we calculate the required power for the pumping system. (b) Following Step 3, we calculate the daily cost of power consumption for the geothermal heating system. (c) Following Step 4, we determine if frictional heating can compensate for the temperature drop during flow.