An exercise room has six weight-lifting machines that have no motors and seven treadmills, each equipped with a 2.5-hp (shaft output) motor. The motors operate at an average load factor of 0.7, at which their efficiency is 0.77. During peak evening hours, all 13 pieces of exercising equipment are used continuously, and there are also two people doing light exercises while waiting in line for one piece of the equipment. Assuming the average rate of heat dissipation from people in an exercise room is 600 W, determine the rate of heat gain of the exercise room from people and the equipment at peak load conditions.

The rate of heat gain of the exercise room = 20.87kW

Explanation:

Given

Weight-lifting machines = 6

Number of treadmills = 7

Number of people doing exercise = 13 + 2 = 15

Heat Dissipation = 600W

Motor Load Factor = 0.7

Motor Efficiency = 0.77

Motor Power Output = 2.5hp

First, we calculate the motor total shaft power output

This is calculated as follows

W = Number of treadmills * Motor Power Output

W = 7 * 2.5hp Convert to Watt

W = 7 * 2.5 * 746W

W = 13,055W

Then we calculate the heat generated, Q from the motor.

This is calculated from efficiency formula:

Eff = W * Load Factor / Q --- Make Q the subject of formula

Q = W * Load Factor / Eff

Q = 13,055 *0.7 / 0.77

Q = 11868.18181818181

Q = 11868.18W

Then we Calculate the heat from people doing exercise

Q2 = number of people * heat dissipation

Q2 = 15 * 600

Q2 = 9000W

Total heat gain is then calculated as:

9000W + 11868.18W

= 20,868.18W or 20.87kW

The rate of heat gain of the exercise room = 20.87kW