Question

An insulated piston-cylinder device initially contains \(0.02 \mathrm{m}^{3}\) of saturated liquid-vapor mixture of water with a quality of 0.1 at \(100^{\circ} \mathrm{C} .\) Now some ice at \(-18^{\circ} \mathrm{C}\) is dropped into the cylinder. If the cylinder contains saturated liquid at \(100^{\circ} \mathrm{C}\) when thermal equilibrium is established, determine (a) the amount of ice added and ( \(b\) ) the entropy generation during this process. The melting temperature and the heat of fusion of ice at atmospheric pressure are \(0^{\circ} \mathrm{C}\) and \(333.7 \mathrm{kJ} / \mathrm{kg}\).

Short answer

Based on the given step-by-step solution, answer the following question: Question: Calculate the mass of ice added to the cylinder and the total entropy generation during the process. Answer: To calculate the mass of ice added and the total entropy generation during the process, follow these steps: 1. Determine the initial mass of the water contained in the cylinder using the initial specific volume (vi), initial volume (Vi), and steam tables. 2. Determine the final mass of saturated liquid in the cylinder using the final specific volume (vf) and volume (Vf). 3. Calculate the mass of ice added (m_ice) by subtracting the initial mass from the final mass. 4. Calculate the entropy change in ice, liquid, and vapor during the process using the given formulas and corresponding specific heats and temperatures. 5. Calculate the total entropy generation (S_gen) by adding all of the entropy changes during the process.

Step-by-step solution

Determine the initial mass of the water contained in the cylinder

To find the initial mass of the water, we need to use the initial volume, quality, and temperature provided. Using the steam tables, find the initial specific volume \(v_{i}\) from the given parameters: \(v_i = x v_g + (1-x) v_f\) Where \(x\) is the quality, \(v_g\) is the specific volume of water vapor, and \(v_f\) is the specific volume of saturated liquid at the given temperature. Using the steam tables, we find \(v_g = 0.1944 \,\mathrm{m}^3/\mathrm{kg}\) and \(v_f = 0.001043 \,\mathrm{m}^3/\mathrm{kg}\), and we already have \(x=0.1\). Now we can plug in these values to compute \(v_i\). Now, calculate the initial mass based on the initial specific volume \(v_i\): \(m_i = \frac{V_i}{v_i}\) where \(m_i\) is the initial mass of saturated liquid-vapor mixture, and \(V_i\) is the initial volume of the mixture in the piston-cylinder device.

Determine the final mass of saturated liquid in the cylinder

To determine the final mass of the saturated liquid, we can calculate it by considering that the final specific volume of saturated liquid is \(v_f\) at \(100^{\circ} \mathrm{C}\): \(m_f = \frac{V_f}{v_f}\) where \(m_f\) is the final mass of saturated liquid in the cylinder, and \(V_f = V_i\) (the volume remains constant during the process).

Calculate the mass of ice added to the cylinder

To calculate the mass of ice added, subtract the initial mass of mixture from the final mass: \(m_{ice} = m_f - m_i\)

Calculate the entropy change in ice, liquid, and vapor during the process

Entropy change in ice is given by: \(\Delta S_{ice} = m_{ice}c_{p_{ice}}\ln\frac{T_f}{T_{ice}} + m_{ice}\frac{h_{fus}}{T_f}\) Where \(c_{p_{ice}}\) is the specific heat of ice, \(h_{fus}\) is the heat of fusion, \(T_{ice}=-18^{\circ} \mathrm{C}\) converted to Kelvin, and \(T_f=100^{\circ} \mathrm{C}\) converted to Kelvin. Entropy change in the liquid phase is given by: \(\Delta S_{liq} = m_{liq}c_{p_{liq}}\ln\frac{T_f}{T_i}\) Entropy change in the vapor phase is given by: \(\Delta S_{vap} = m_{vap}c_{p_{vap}}\ln\frac{T_f}{T_i}\) Where \(m_{liq}\) and \(m_{vap}\) are the masses of liquid and vapor phases respectively, and \(c_{p_{liq}}\) and \(c_{p_{vap}}\) are the specific heats of liquid and vapor phases respectively.

Calculate the total entropy generation

To calculate the total entropy generation, we need to add all of the entropy changes during the process: \(S_{gen} = \Delta S_{ice} + \Delta S_{liq} + \Delta S_{vap}\)