Question

An insulated piston-cylinder device contains 0.8 L of saturated liquid water at a constant pressure of 120 kPa. An electric resistance heater inside the cylinder is turned on, and electrical work is done on the water in the amount of 1400 kJ. Assuming the surroundings to be at \(25^{\circ} \mathrm{C}\) and \(100 \mathrm{kPa}\), determine \((a)\) the minimum work with which this process could be accomplished and ( \(b\) ) the exergy destroyed during this process.

Short answer

Answer: (a) To calculate the minimum work, we need the values of the electrical work done and the exergy destroyed. Once you have found these values, use the formula \(W_{min} = W_{electrical} - X_{destroyed}\) to calculate the minimum work. (b) To calculate the exergy destroyed, we need the value of entropy generation and the dead state temperature. Use the formula \(X_{destroyed} = T_0 \cdot S_{gen}\) to determine the exergy destroyed during this process.

Step-by-step solution

Find the initial and final states of the water

First, we need to find the initial and final states of the water. The initial state is given as saturated liquid at 120 kPa, and the volume is 0.8 L. Using the saturated liquid tables, we can find the specific volume and enthalpy at this initial state. The final state can be found using the energy balance and the electrical work done on the water.

Apply energy balance

We will apply the first law of thermodynamics to the system. The energy balance can be written as: \(\Delta E = Q - W_{electrical}\) Here, \(\Delta E\) is the change in energy, \(Q\) is the heat transfer, and \(W_{electrical}\) is the electrical work done. Since the piston-cylinder device is insulated, there is no heat transfer, and we can write: \(\Delta E = - W_{electrical}\) We are given that \(W_{electrical} = 1400 \,\text{kJ}\). So, \(\Delta E = -1400\, \text{kJ}\) The change in energy can be written in terms of specific internal energy as: \(\Delta E = m(u_2 - u_1)\) Where \(m\) is the mass of the water, \(u_1\) and \(u_2\) are the specific internal energies of the initial and final states, respectively. We can find the mass of the water using the specific volume at the initial state: \(m = \frac{V}{v_1}\) Now, we can find the final specific internal energy, \(u_2\).

Apply second law of thermodynamics

Now, we will apply the second law of thermodynamics to the system to determine the minimum work and exergy destroyed. The second law can be written as: \(\Delta S - S_{gen} = \frac{Q}{T_0}\) Since there is no heat transfer in this case, the equation becomes: \(\Delta S = S_{gen}\) Exergy destroyed is given by: \(X_{destroyed} = T_0 \cdot S_{gen}\) Minimum work, \(W_{min}\) can be determined using the exergy balance equation: \(W_{min} = W_{electrical} - X_{destroyed}\) We can now plug in the values to find the minimum work and exergy destroyed.

Calculate minimum work (\(a\))

Now we will calculate the minimum work required for this process. As mentioned earlier: \(W_{min} = W_{electrical} - X_{destroyed}\) With \(W_{electrical} = 1400\, \text{kJ}\) and the calculated values for \(X_{destroyed}\), we can determine \(W_{min}\).

Calculate exergy destroyed (\(b\))

To calculate the exergy destroyed, we will use the formula derived in step 3: \(X_{destroyed} = T_0 \cdot S_{gen}\) Using the calculated value for \(S_{gen}\) and \(T_0 = 25^{\circ} \mathrm{C}\), we can determine the exergy destroyed during this process.