Question

Consider a thermal energy reservoir at \(1500 \mathrm{K}\) that can supply heat at a rate of \(150,000 \mathrm{kJ} / \mathrm{h}\). Determine the exergy of this supplied energy, assuming an environmental temperature of \(25^{\circ} \mathrm{C}\).

Short answer

Question: Determine the exergy of the thermal energy supplied by a reservoir at 1500 K, with a heat supply rate of 150,000 kJ/h when the environmental temperature is 25°C. Answer: The exergy of the supplied energy by the thermal energy reservoir at 1500 K is 120,240 kJ/h.

Step-by-step solution

Convert given temperatures to Kelvin

First, we need to convert the temperatures from Celsius to Kelvin, since we will be working in Kelvin for our calculations. Environmental temperature: \(T_0 = 25\,^{\circ}\mathrm{C} + 273.15 = 298.15\mathrm{K}\) Reservoir temperature: \(T_1 = 1500\mathrm{K}\)

Calculate the supplied heat rate in kW

We are given the heat supply rate in kJ/h. We will convert it to kW for easier calculations later on. Heat supply rate: \(Q_{supply} = \frac{150000\,\mathrm{kJ/h}}{3600\,\mathrm{s/h}} = 41.67\,\mathrm{kW}\)

Determine the exergy formula for a thermal energy transfer process

For a thermal energy transfer process, the formula for exergy (E) can be given as: \(E = Q \times (1 - \frac{T_0}{T_1})\) Where: E - Exergy Q - Heat transfer rate \(T_0\) - Environmental temperature \(T_1\) - Reservoir temperature

Calculate the exergy of the supplied energy

Using the formula determined in Step 3 and the values calculated in Steps 1 and 2, we can now find the exergy of the supplied energy. \(E = 41.67 \times (1 - \frac{298.15}{1500}) = 41.67 \times (1 - 0.19877) = 41.67 \times 0.80123 = 33.40\,\mathrm{kW}\)

Convert the exergy back to the desired units (kJ/h)

Since the original heat supply rate was given in kJ/h, we should convert the exergy back to kJ/h. Exergy: \(E' = 33.40\,\mathrm{kW} \times 3600\,\mathrm{s/h} = 120240\,\mathrm{kJ/h}\) So, the exergy of the supplied energy by the thermal energy reservoir at 1500 K is \(120240\,\mathrm{kJ/h}\).