Chapter 3

Two kilograms of Refrigerant $134a$, initially at 2 bar and occupying a volume of $0.12{\mathrm{m}}^3$, undergoes a process at constant pressure until the volume has doubled. Kinetic and potential energy effects are negligible. Determine the work and heat transfer for the process, each in $\mathrm{kJ}$.

A system consisting of $2\mathrm{kg}$ of ammonia undergoes a cycle composed of the following processes: Process 1-2: constant volume from $p_1=10$ bar, $x_1=0.6$ to saturated vapor Process 2-3: constant temperature to $p_3=p_1,Q_{23}=+228\mathrm{kJ}$ Process 3-1: constant pressure Sketch the cycle on $p-v$ and $T-v$ diagrams. Neglecting kinetic and potential energy effects, determine the net work for the cycle and the heat transfer for each process, all in $\mathrm{kJ}$.

A system consisting of $1\mathrm{kg}$ of ${\mathrm{H}}_2\mathrm{O}$ undergoes a power cycle composed of the following processes: Process 1-2: Constant-pressure heating at 10 bar from saturated vapor. Process 2-3: Constant-volume cooling to $p_3=5$ bar, $T_3={160}^{\circ }\mathrm{C}$. Process 3-4: Isothermal compression with $Q_{34}=-815.8\mathrm{kJ}$ Process $4-1:$ Constant-volume heating. Sketch the cycle on $T-v$ and $p-v$ diagrams. Neglecting kinetic and potential energy effects, determine the thermal efficiency.

A well-insulated copper tank of mass $13\mathrm{kg}$ contains $4\mathrm{kg}$ of liquid water. Initially, the temperature of the copper is ${27}^{\circ }\mathrm{C}$ and the temperature of the water is ${50}^{\circ }\mathrm{C}$. An electrical resistor of neglible mass transfers $100\mathrm{kJ}$ of energy to the contents of the tank. The tank and its contents come to equilibrium. What is the final temperature, in ${}^{\circ }\mathrm{C}?$

A system consists of a liquid, considered incompressible with constant specific heat $c$, filling a rigid tank whose surface area is A. Energy transfer by work from a paddle wheel to the liquid occurs at a constant rate. Energy transfer by heat occurs at a rate given by $\dot{Q}=-\mathrm{ha}(T-T_0)$, where $T$ is the instantaneous temperature of the liquid, $T_0$ is the temperature of the surroundings, and $\mathrm{h}$ is an overall heattransfer coefficient. At the initial time, $t=0$, the tank and its contents are at the temperature of the surroundings. Obtain a differential equation for temperature $T$ in terms of time $t$ and relevant parameters. Solve the differential equation to obtain $T(t)$.

Determine the temperature, in $\mathrm{K}$, of $5\mathrm{kg}$ of air at a pressure of $0.3\mathrm{MPa}$ and a volume of $2.2{\mathrm{m}}^3.$ Verify that ideal gas behavior can be assumed for air under these conditions.

Compare the densities, in $\mathrm{kg}/{\mathrm{m}}^3$, of helium and air, each at $300\mathrm{K},100\mathrm{kPa}$. Assume ideal gas behavior.

Consider a gas mixture whose apparent molecular weight is 33 , initially at 3 bar and $300\mathrm{K}$, and occupying a volume of $0.1{\mathrm{m}}^3$. The gas undergoes an expansion during which the pressure-volume relation is $p{V}^{1.3}=$ constant and the energy transfer by heat to the gas is $3.84\mathrm{kJ}$. Assume the ideal gas model with $c_v=0.6+(2.5\times {10}^{-4})T$, where $T$ is in $\mathrm{K}$ and $c_{\mathfrak{v}}$ has units of $\mathrm{kJ}/\mathrm{kg}\cdot \mathrm{K}$. Neglecting kinetic and potential energy effects, determine (a) the final temperature, in $\mathrm{K}$. (b) the final pressure, in bar. (c) the final volume, in ${\mathrm{m}}^3$. (d) the work, in $\mathrm{kJ}$.

Two kilograms of a gas with molecular weight 28 are contained in a closed, rigid tank fitted with an electric resistor. The resistor draws a constant current of $10\mathrm{amp}$ at a voltage of $12\mathrm{V}$ for $10\min$. Measurements indicate that when equilibrium is reached, the temperature of the gas has increased by ${40.3}^{\circ }\mathrm{C}$. Heat transfer to the surroundings is estimated to occur at a constant rate of $20\mathrm{W}$. Assuming ideal gas behavior, determine an average value of the specific heat $c_p$, in $\mathrm{kJ}/\mathrm{kg}\cdot \mathrm{K}$, of the gas in this temperature interval based on the measured data.

A gas is confined to one side of a rigid, insulated container divided by a partition. The other side is initially evacuated. The following data are known for the initial state of the gas: $p_1=5$ bar, $T_1=500\mathrm{K}$, and $V_1=0.2{\mathrm{m}}^3$. When the partition is removed, the gas expands to fill the entire container, which has a total volume of $0.5{\mathrm{m}}^3$. Assuming ideal gas behavior, determine the final pressure, in bar.