Question

A utility manager is trying to determine which hp motor to purchase for a pump station. A 400 hp motor with a wire-to-water efficiency of \(65 \%\) can pump 3,000 gpm. Similarly, a 250 hp motor with a wire-to-water efficiency of \(75 \%\) can pump 2,050 gpm. With an electrical rate of \(\$ 0.155\) per \(\mathrm{kW}-\mathrm{Hr}\), how much would it cost to run each motor to achieve a daily flow of 2 MG? Which one is less expensive to run?

Short answer

The 250 hp motor is less expensive to run.

Step-by-step solution

Determine the Daily Flow Requirement

First, convert the daily flow requirement from million gallons (MG) to gallons. Since 1 MG = 1,000,000 gallons, a daily flow of 2 MG equals 2,000,000 gallons.

Calculate Hours Required for Each Motor

Calculate how many hours each motor needs to run to achieve the daily flow of 2,000,000 gallons.For the 400 hp motor: \[\text{Hours} = \frac{2,000,000 \, \text{gallons}}{3,000 \, \text{gpm} \times 60 \, \text{minutes/hour}}\]For the 250 hp motor: \[\text{Hours} = \frac{2,000,000 \, \text{gallons}}{2,050 \, \text{gpm} \times 60 \, \text{minutes/hour}}\]

Convert Motor Power to kW

Convert horsepower to kilowatts since the electrical rate is given in kW-Hr. Use the conversion factor: 1 hp = 0.746 kW.For the 400 hp motor: \[\text{kW} = 400 \, \text{hp} \times 0.746 \, \text{kW/hp} \]For the 250 hp motor: \[\text{kW} = 250 \, \text{hp} \times 0.746 \, \text{kW/hp} \]

Adjust for Efficiency

Adjust the power consumption for the efficiency of each motor.For the 400 hp motor with 65% efficiency:\[\text{Efficient kW} = \frac{\text{kW}}{0.65}\]For the 250 hp motor with 75% efficiency:\[\text{Efficient kW} = \frac{\text{kW}}{0.75}\]

Calculate Total Energy Used

Calculate the total energy consumption by multiplying the efficient kW by the hours required for each motor.For the 400 hp motor:\[\text{Energy (kWh)} = \text{Efficient kW} \times \text{Hours}\]For the 250 hp motor:\[\text{Energy (kWh)} = \text{Efficient kW} \times \text{Hours}\]

Calculate Total Cost

Use the given electrical rate to calculate the total cost to run each motor.For the 400 hp motor:\[\text{Cost} = \text{Energy (kWh)} \times 0.155 \, \text{\textbackslash\(} \text{/kW-Hr}\]For the 250 hp motor:\[\text{Cost} = \text{Energy (kWh)} \times 0.155 \, \text{\textbackslash\)} \text{/kW-Hr}\]

Compare Costs

Compare the costs calculated for both motors to determine which one is less expensive to run.

Key concepts

Daily Flow Requirement

To determine the cost of running a pump motor, you first need to understand the daily flow requirement. In this exercise, the goal is to pump 2 million gallons (MG) of water each day. Since 1 MG equals 1,000,000 gallons, the daily flow requirement is 2,000,000 gallons. This is essential as it determines how long the motor needs to run to pump the required volume of water.

Motor Efficiency

Motor efficiency measures how effectively a motor converts electrical power into mechanical power to pump water. It's expressed as a percentage. In our example, the 400 hp motor has an efficiency of 65%, and the 250 hp motor has an efficiency of 75%. This means that the 400 hp motor converts 65% of the electrical power into water-pumping power, while the 250 hp motor converts 75%. High efficiency means less energy wasted as heat and more energy used for pumping.

Energy Consumption Calculation

Energy consumption is calculated by determining the kilowatt-hours (kWh) used by each motor to pump the required volume of water. First, convert motor power from horsepower (hp) to kilowatts (kW) using the conversion factor 1 hp = 0.746 kW.
Then, adjust the power consumption for efficiency. For the 400 hp motor, you divide the calculated kW by 0.65, and for the 250 hp motor, you divide by 0.75.
Finally, multiply the efficient kW by the hours each motor needs to run to find the total energy consumption in kWh.

Conversion of Units

Unit conversion is crucial in this problem. We convert the motor power from hp to kW because the electrical rate is given in kW-Hr. Since 1 hp equals 0.746 kW, we multiply the motor's hp by 0.746 to get kW.
Additionally, converting the daily flow requirement to gallons and the motor's pumping rate to minutes helps determine how long the motors need to run.

Cost Comparison

To find out which motor is cheaper to operate, we calculate the total running cost using the energy consumption and the electrical rate. Multiply the total energy (kWh) by the electrical rate of 0.155 dollars per kW-Hr to get the cost.
Compare the costs of running both motors to identify the less expensive option. Even if one motor has a higher power, its higher efficiency might lead to overall lower costs due to reduced energy consumption.