The temperature of an incompressible substance of mass $m$ and specific heat $c$ is reduced from $T_0$ to $T\left(<T_0\right)$ by a refrigeration cycle. The cycle receives energy by heat transfer at $T$ from the substance and discharges energy by heat transfer at $T_0$ to the surroundings. There are no other heat transfers. Plot $\left(W_{\text {min }} / m c T_0\right)$ versus $T / T_0$ ranging from $0.8$ to $1.0$, where $W_{\min }$ is the minimum theoretical work input required.

In a heat-treating process, a 2-kg aluminum rod, initially at 1075 K, is quenched in a tank containing 150 kg of water, initially at 295 K. There is negligible heat transfer between the contents of the tank and their surroundings. Taking the specific heat of the metal rod and water as constant at $0.903 kJ/kg \cdot K$ and $4.2 kJ/kg \cdot K,$ respectively, determine (a) the final equilibrium temperature after quenching, in K, and (b) the amount of entropy produced within the tank, in kJ/K.

Care must be taken in preparing solutions of solutes that liberate heat on dissolving. The heat of solution of $\mathrm{NaOH}$ is $-44.5 \mathrm{~kJ} / \mathrm{mol} \mathrm{NaOH}$. To what maximum temperature may a sample of water, originally at $21^{\circ} \mathrm{C}$, be raised in the preparation of $500 \mathrm{~mL}$ of $7.0 \mathrm{M}$ $\mathrm{NaOH}$ ? Assume the solution has a density of $1.08 \mathrm{~g} / \mathrm{mL}$ and specific heat of $4.00 \mathrm{Jg}^{-1 \circ} \mathrm{C}^{-1}$.

A 0.8-m^3 rigid tank contains carbon dioxide gas at 250 K and 100 kPa. A 500-W electric resistance heater placed in the tank is now turned on and kept on for 40 min, after which the pressure of the carbon dioxide is measured to be 175 kPa. Assuming the surroundings to be at 300 K and using constant specific heats, determine (a) the final temperature of the carbon dioxide, (b) the net amount of heat transfer from the tank , and (c) the entropy generation during this process.

A constant pressure piston/cylinder contains $0.5 \mathrm{~kg}$ air at $300 \mathrm{~K}, 400\ \mathrm{kPa}$. Assume the piston/cylinder has a total mass of $1 \mathrm{~kg}$ steel and is at the same temperature as the air at any time. The system is now heated to $1600 \mathrm{~K}$ by heat transfer from a $1700 \mathrm{~K}$ source. Find the entropy generation using constant specific heats for air. Solve using appropriate table.

Ethanol

$$(C_2H_5OH)$$

melts at

$$- 114 ^ { \circ } \mathrm { C }$$

and boils at

$$78 ^ { \circ } \mathrm { C }$$

The enthalpy of fu sion of ethanol is 5.02 kJ/mol, and its enthalpy of vaporization is 38.56 kJ/mol. The specific heats of solid and liquid ethanol are 0.97 and 2.3 J/g-K, respectively. How much heat is required to convert the same amount of ethanol at

$$- 155 ^ { \circ } \mathrm { C }$$

to the vapor phase at

$$78 ^ { \circ } \mathrm { C }$$

A $10,000$ -microfarad $[\mu \mathrm{F}]$ capacitor is charged to $25$ volts [V]. If the capacitor is completely discharged through an iron rod $0.2$ meters long and $0.25$ centimeter in diameter, resulting in $90 \%$ of the stored energy being transferred to the rod as heat, how much does the temperature of the rod increase? Give your answer in kelvins. Data you may need: Specific gravity of iron: $S G=7.874$ Specific heat of iron: $C_{P}=0.450 \mathrm{~J} /(\mathrm{g} \mathrm{K})$

Someone has suggested that the air-standard Otto cycle is more accurate if the two isentropic processes are replaced with polytropic processes with a polytropic exponent $n=1.3$. Consider such a cycle when the compression ratio is $8, P_1=$ $95\ \mathrm{kPa}, T_1=15^{\circ} \mathrm{C}$, and the maximum cycle temperature is $1200^{\circ} \mathrm{C}$. Find the heat transferred to and rejected from this cycle and the cycle's thermal efficiency. Use constant specific heat at room temperature.

A certain heat lamp emits 200 W of mostly IR radiation averaging 1500 nm in wavelength. (a) What is the average photon energy in joules? (b) How many of these photons are required to increase the temperature of a person’s shoulder by $2.0^{\circ} \mathrm{C},$ assuming the affected mass is 4.0 kg with a specific heat of $0.83\ \mathrm{kcal} / \mathrm{kg} \cdot^{\circ} \mathrm{C}.$ Also assume no other significant heat transfer. (c) How long does this take?

A simple ideal Brayton cycle uses helium as the working fluid; operates with $12\ \mathrm{psia}$ and $60^{\circ} \mathrm{F}$ at the compressor inlet; has a pressure ratio of $14$, and a maximum cycle temperature of $1300^{\circ} \mathrm{F}$ when the isentropic efficiency of the compressor is $95$ percent. Find the power produced by this cycle when the rate at which the helium is circulated about the cycle is $100\ \mathrm{lbm} / \mathrm{min}$. Use constant specific heat at room temperature.

Consider the following data:

$$\begin{array}{lcc} \text { Metal } & \text { Al } & \text { Cu } \\ \hline \text { Mass }(\mathrm{g}) & 10 & 30 \\ \text { Specific heat }\left(\mathrm{J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C}\right) & 0.900 & 0.385 \\ \text { Temperature }\left({ }^{\circ} \mathrm{C}\right) & 40 & 60 \end{array}$$

When these two metals are placed in contact, which of the following will take place?

($a$) Heat will flow from $\mathrm{Al}$ to $\mathrm{Cu}$ because $\mathrm{Al}$ has $\mathrm{a}$ larger specific heat.
($b$) Heat will flow from $\mathrm{Cu}$ to $\mathrm{Al}$ because $\mathrm{Cu}$ has a larger mass.
($c$) Heat will flow from $\mathrm{Cu}$ to $\mathrm{Al}$ because $\mathrm{Cu}$ has a larger heat capacity.
($d$) Heat will flow from $\mathrm{Cu}$ to $\mathrm{Al}$ because $\mathrm{Cu}$ is at a higher temperature.
($e$) No heat will flow in either direction.

An air-standard dual cycle has a compression ratio of $18$ and a cutoff ratio of $1.1$. The pressure ratio during constantvolume heat addition process is $1.1$. At the beginning of the compression, $P_{1}=90 \mathrm{~kPa}, T_{1}=18^{\circ} \mathrm{C}$, and $V_{1}=0.003 \mathrm{~m}^{3}$. How much power will this cycle produce when it is executed $4000$ times per minute? Use constant specific heats at room temperature.

An ideal Stirling cycle uses energy reservoirs at $40^{\circ} \mathrm{F}$ and $640^{\circ} \mathrm{F}$ and uses hydrogen as the working gas. It is designed such that its minimum volume is $0.1\ \mathrm{ft}^{3},$ maximum volume is $1\ \mathrm{ft}^{3},$ and maximum pressure is 400 psia. Calculate the amount of external heat addition, external heat rejection, and heat transfer between the working fluid and regenerator for each complete cycle. Use constant specific heats at room temperature.

An electric heater is used to heat a room of volume $62 \mathrm{m}^{3}$. Air is brought into the room at $5^{\circ} \mathrm{C}$ and is completely replaced twice per hour. Heat loss through the walls amounts to approximately $850 \mathrm{kcal} / \mathrm{h}$. If the air is to be maintained at $20^{\circ} \mathrm{C}$, what minimum wattage must the heater have? (The specific heat of air is about $0.17 \mathrm{kcal} / \mathrm{kg} \cdot \mathrm{C}^{\circ}$)