Question

A piston-cylinder device initially contains \(0.35-\mathrm{kg}\) steam at \(3.5 \mathrm{MPa}\), superheated by \(7.4^{\circ} \mathrm{C}\). Now the steam loses heat to the surroundings and the piston moves down, hitting a set of stops at which point the cylinder contains saturated liquid water. The cooling continues until the cylinder contains water at \(200^{\circ} \mathrm{C}\). Determine \((a)\) the final pressure and the quality (if mixture), \((b)\) the boundary work, \((c)\) the amount of heat transfer when the piston first hits the stops, \((d)\) and the total heat transfer.

Short answer

Based on the step by step solution given above, formulating a short answer question can be as follows: Question: Given a piston-cylinder device containing superheated steam initially, where the steam loses heat to the surroundings and the piston moves down, determine the following for when the temperature of water reaches 200°C: (a) the final pressure and quality (if a mixture), (b) the boundary work, (c) the amount of heat transfer when the piston first hits the stops, (d) and the total heat transfer. Consider the following information: - Initial mass of steam (m): 0.35 kg - Initial pressure (P1): 3.5 MPa - Superheat temperature: 7.4°C

Step-by-step solution

Determine the initial state of the steam

Given initial mass (m) of 0.35 kg, initial pressure (P1) of 3.5 MPa, and superheat temperature of 7.4°C, we first need to calculate the actual initial temperature (T1), of the steam. We will use steam tables to obtain the saturation temperature (Ts) at 3.5 MPa and then add the superheat temperature to find T1: \(T_1 = T_s + 7.4\) Using the steam tables, find Ts for the initial pressure of 3.5 MPa and then calculate T1.

Determine the final state of the water

We are asked to determine the final pressure and quality when the water is at 200°C. At this temperature, the water is a compressed liquid mass of 0.35 kg. Again, by using the steam tables, we can find the specific volume (v2), internal energy (u2), and enthalpy (h2) for the compressed liquid at this temperature.

Calculate the final pressure and quality

Now that we have the property values for the final state of the water (at 200°C), we can determine the final pressure and quality. Since we know the specific volume (v2) and the mass (m), we can calculate the final volume (V2): \(V_2 = m * v_2\) Now, to determine the final pressure (P2), we can use the steam tables and look for the pressure corresponding to the given specific volume (v2) and temperature (200°C). If the steam is a mixture (saturated liquid-vapor mixture) at the final state, we also need to calculate the quality (x2) using the specific volumes of saturated liquid (vf) and saturated vapor (vg) at the final pressure: \(x_2 = \frac{v_2 - v_f}{v_g - v_f}\)

Calculate the boundary work

Boundary work (Wb) can be calculated as follows: \(W_b = P1 * (V_1 - V_2)\) Calculate V1 using the initial specific volume (v1) and mass (m): \(V_1 = m * v_1\) Now, find v1 from the steam table using the initial temperature (T1) and pressure (P1). Finally, calculate the boundary work (Wb).

Calculate the heat transfer when the piston first hits the stops

At this point, the steam loses heat, and the system contains saturated liquid water. Calculate the change in internal energy (ΔU) for this process: \(\Delta U = m * (u_2 - u_1)\) Now, use the first law of thermodynamics to find the heat transfer (Q) when the piston first hits the stops: \(Q = \Delta U + W_b\)

Calculate the total heat transfer

Total heat transfer (Qtotal) can be defined as the sum of the heat transfer when the piston first hits the stops (Q) and the heat transfer during the cooling process at constant volume (Qcv): \(Q_\text{total} = Q + Q_\text{cv}\) To find Qcv, use the constant volume specific heat (Cv) of liquid water and the change in temperature (ΔT): \(Q_\text{cv} = m * C_v * \Delta T\) Now, calculate the total heat transfer (Qtotal).