Question

In a chemical plant, a certain chemical is heated by hot water supplied by a natural gas furnace. The hot water \(\left(c_{p}=\right.\) \(4180 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K})\) is then discharged at \(60^{\circ} \mathrm{C}\) at a rate of \(8 \mathrm{~kg} / \mathrm{min}\). The plant operates \(8 \mathrm{~h}\) a day, 5 days a week, 52 weeks a year. The furnace has an efficiency of 78 percent, and the cost of the natural gas is \(\$ 1.00\) per therm ( 1 therm \(=105,500 \mathrm{~kJ})\). The average temperature of the cold water entering the furnace throughout the year is \(14^{\circ} \mathrm{C}\). In order to save energy, it is proposed to install a water-to-water heat exchanger to preheat the incoming cold water by the drained hot water. Assuming that the heat exchanger will recover 72 percent of the available heat in the hot water, determine the heat transfer rating of the heat exchanger that needs to be purchased and suggest a suitable type. Also, determine the amount of money this heat exchanger will save the company per year from natural gas savings.

Short answer

Answer: The heat transfer rating of the proposed heat exchanger is approximately 17.74 kW, and it will save the company approximately $678.62 per year in natural gas savings.

Step-by-step solution

Calculate the energy required to heat the water

Given, the mass flow rate of hot water, \(m = 8 \, kg/min\). We also know the specific heat capacity of water, \(c_p = 4180 \, J/(kg \cdot K)\), the initial temperature of the cold water, \(T_{cold} = 14^{\circ} C\), and the final temperature of the hot water, \(T_{hot} = 60^{\circ} C\). We can calculate the energy required to heat the water, \(Q\), with the formula: \(Q = m \cdot c_p \cdot (T_{hot} - T_{cold})\)

Convert the mass flow rate from kg/min to kg/s

Since the energy consumption is usually calculated in \(J/s\), we need to convert the mass flow rate to \(kg/s\): \(m'= \frac{8 \, kg}{1 \, min} \cdot \frac{1 \, min}{60 \, s} = \frac{2}{15} \, kg/s\)

Calculate the energy required to heat the water in J/s (or Watts)

Now that we have the mass flow rate in \(kg/s\), we can compute the energy required to heat the water in \(J/s\) (or \(Watts\)): \(Q' = m' \cdot c_p \cdot (T_{hot} - T_{cold}) = \frac{2}{15} \cdot 4180 \cdot (60 - 14) \approx 24640 \, J/s = 24.64 \, kW\)

Calculate the energy recovered by the heat exchanger

It is given that the heat exchanger will recover 72 percent of the available heat in the hot water. So, the energy recovered by the heat exchanger, \(Q_{recovered}\), is: \(Q_{recovered} = 0.72 \cdot Q' = 0.72 \cdot 24.64 \approx 17.74 \, kW\)

Calculate the yearly energy consumption of the plant without the heat exchanger

The plant operates for 8 hours a day, 5 days a week, and 52 weeks a year. Also, the furnace has an efficiency of 78 percent. So, the yearly energy consumption, \(E_{year}\), without the heat exchanger is: \(E_{year} = 8h/day \cdot 5days/week \cdot 52weeks/year \cdot \frac{1}{0.78} \cdot Q' = 8 \cdot 5 \cdot 52 \cdot \frac{24.64}{0.78} \approx 104145.9 \, kWh/year\)

Calculate the yearly cost of natural gas without the heat exchanger

The cost of natural gas is \(1.00\) per therm, and 1 therm is equal to 105,500 kJ. So, the yearly cost of natural gas without the heat exchanger, \(C_{year}\), is: \(C_{year} = \frac{E_{year} \cdot 1 \, therm \cdot 1000 \, J}{105500 \, kJ/therm} = \frac{104145.9 \cdot 1}{105.5} \approx \$ 987.3/year\)

Calculate the yearly energy consumption and cost with the heat exchanger

Now that we know the heat exchanger recovers 17.74 kW of energy, we can calculate the remaining energy consumption and cost with the heat exchanger: \(E_{year}^{'} = 8h/day \cdot 5days/week \cdot 52weeks/year \cdot \frac{1}{0.78} \cdot (Q' - Q_{recovered}) = 8 \cdot 5 \cdot 52 \cdot \frac{24.64 - 17.74}{0.78} \approx 32568.9 \, kWh/year\) \(C_{year}^{'} = \frac{E_{year}^{'} \cdot 1 \, therm \cdot 1000 \, J}{105500 \, kJ/therm} = \frac{32568.9 \cdot 1}{105.5} \approx \$ 308.68/year\)

Calculate the yearly cost savings due to the heat exchanger

Lastly, we can calculate the yearly cost savings provided by the heat exchanger, \(S_{year}\): \(S_{year} = C_{year} - C_{year}^{'} = \$ 987.3 - \$ 308.68 = \$ 678.62/year\) The heat transfer rating of the heat exchanger that needs to be purchased is approximately \(17.74 \, kW\), and it will save the company approximately \(678.62\) per year in natural gas savings. Suitable types of heat exchangers for this application might include plate heat exchangers or shell and tube heat exchangers.

Key concepts

Heat Transfer Rating

The heat transfer rating of a heat exchanger is a measure of its capacity to transfer heat between two fluids at different temperatures. It is commonly expressed in kilowatts (kW) or British Thermal Units per hour (BTU/hr). To calculate the heat transfer rating, one must consider the flow rate of the fluids, their specific heat capacities, and the temperature difference between the incoming and outgoing fluids.

From the exercise, we can extract an example of how the heat transfer rating is calculated for a real-world scenario. Here, the hot water used to heat the chemical had a specific heat capacity of 4180 J/kg⋅K. The heat transfer rating was then calculated by determining the energy required to elevate the water temperature from the initial to the final state, adjusting for the efficiency of the heat exchanger. This value becomes crucial in selecting the right heat exchanger to meet the required level of heat recovery.

A higher heat transfer rating indicates a more efficient heat exchanger, capable of transferring more heat energy over a given period. It's one of the essential criteria for selecting a heat exchanger, along with factors such as compatibility with the fluids, operational pressure range, and maintenance requirements.

Natural Gas Cost Savings

Natural gas cost savings are achieved when a process like a heat exchanger improves the energy efficiency of a system that uses natural gas as its energy source. In our exercise, installing a water-to-water heat exchanger allows the plant to recycle heat that would otherwise be wasted, thus reducing the amount of natural gas needed to heat the incoming water and saving money on energy bills.

Savings are calculated by comparing the cost of natural gas required to run the plant without the heat exchanger to the cost after the heat exchanger installation. By recovering a percentage of the heat lost in the discharged water, less energy from natural gas is needed to maintain the desired temperature levels. The effectiveness of the heat exchanger directly correlates to the amount of savings: a more efficient heat exchanger can significantly reduce operating costs and create a shorter payback period on the investment. By calculating the yearly cost savings, as shown in the exercise, businesses can make informed decisions about the potential financial benefits of upgrading their heating systems.

Energy Consumption Calculation

Energy consumption calculation is a fundamental aspect of determining the efficiency of thermal systems like heaters and heat exchangers. This calculation typically involves measuring or estimating the amount of energy a system uses over a specific period, which can then be compared against the energy savings achieved through upgrades or improvements.

In our exercise, the energy consumption of the furnace and the subsequent reduction in energy usage due to the proposed heat exchanger are calculated using the mass flow rate of the water, its specific heat capacity, the temperature rise, and the existing efficiency of the furnace. These values are critical for understanding how much energy (and therefore money) can be saved by installing a heat exchanger.

An accurate energy consumption calculation helps businesses forecast the economic impact of implementing energy-saving measures and supports strategic decisions aimed at reducing operational costs and carbon footprints. It also serves as a baseline to monitor the continuous performance of the installed systems and identify any areas where further efficiency gains can be made.