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}=4180 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\( is then discharged at \)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 with 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

Question: Calculate the heat transfer rating of a water-to-water heat exchanger and the amount of money saved per year due to natural gas savings, given the hot and cold water temperatures, water flow rate, furnace efficiency, and natural gas cost as described above. Answer: The heat transfer rating of the heat exchanger is \(Q_\text{exchanger}\) and the amount of money saved per year due to natural gas savings is \(\text{Cost}_\text{saving}\).

Step-by-step solution

Find the initial heat transfer rate (Q_initial)

First, we will calculate the initial heat transfer rate without the heat exchanger. This can be found using the formula: \(Q = mc_p\Delta T\) where \(m = 8 \frac{\text{kg}}{\text{min}}\) is the flow rate of the water, \(c_p = 4180 \frac{\text{J}}{\text{kg} \cdot\text{K}}\) is the specific heat capacity of water, \(\Delta T = (60 - 14) \ \text{K}\) is the temperature difference between the hot water and the cold water. \(Q_\text{initial} = 8\cdot4180\cdot(60-14)\)

Calculate the heat transfer rate after installing the heat exchanger (Q_final)

Since the heat exchanger can recover 72 percent of the available heat in the hot water, the heat transfer rate after installing the heat exchanger will be: \(Q_\text{final} = Q_\text{initial}\times(1 - 0.72)\)

Calculate the heat transfer rating of the heat exchanger (Q_exchanger)

We can now calculate the heat transfer rating of the heat exchanger, which is the difference between the initial heat transfer rate and the final heat transfer rate: \(Q_\text{exchanger} = Q_\text{initial} - Q_\text{final}\)

Calculate the cost saved per year due to natural gas savings (Cost_saving)

First, we need to convert the heat transfer rates to their energy equivalents per year in kJ, taking into account the working hours and days, and then we can find the difference in energy usage: \(E_\text{initial} = Q_\text{initial} \times \frac{60 \ \text{min}}{\text{hour}}\times 8 \ \text{hours} \ \text{day}\times 5 \ \text{days} \text{week}\times 52 \ \text{weeks}\times \frac{1 \ \text{kJ}}{1000 \ \text{J}}\) \(E_\text{final} = Q_\text{final} \times \frac{60 \ \text{min}}{\text{hour}}\times 8 \ \text{hours} \ \text{day}\times 5 \ \text{days} \text{week}\times 52 \ \text{weeks}\times \frac{1 \ \text{kJ}}{1000 \ \text{J}}\) Then, we need to factor in the furnace efficiency of 78 percent, which will impact the amount of natural gas needed to provide the necessary heat: \(E_\text{initial, ng} = \frac{E_\text{initial}}{0.78\times105500 \frac{\text{kJ}}{\text{therm}}}\times\$1.00 \ \text{therm}\) \(E_\text{final, ng} = \frac{E_\text{final}}{0.78\times105500 \frac{\text{kJ}}{\text{therm}}}\times\$1.00 \ \text{therm}\) Finally, we can find the cost saved per year: \(\text{Cost}_\text{saving} = E_\text{initial, ng} - E_\text{final, ng}\) Now we can plug in values into the formula to get the final results.