Question

An ideal Stirling cycle filled with air uses a \(75^{\circ} \mathrm{F}\) energy reservoir as a sink. The engine is designed so that the maximum air volume is \(0.5 \mathrm{ft}^{3},\) the minimum air volume is \(0.06 \mathrm{ft}^{3},\) and the minimum pressure is 15 psia. It is to be operated such that the engine produces 2 Btu of net work when 5 Btu of heat are transferred externally to the engine. Determine the temperature of the energy source, the amount of air contained in the engine, and the maximum air pressure during the cycle.

Short answer

In this exercise, we analyzed an ideal Stirling cycle filled with air and determined the energy source temperature, the amount of air contained in the engine, and the maximum air pressure during the cycle. We found the following: 1. Energy source temperature (T_H) is approximately 1059.1 R (849.3 K). 2. The amount of air contained in the engine is approximately 0.045 lbm. 3. The maximum air pressure during the cycle is approximately 206.34 psia.

Step-by-step solution

Compute the isothermal heat addition Q_A, and the isothermal heat rejection Q_R

We know that 5 Btu of heat are transferred externally to the engine and that the net work produced by the engine is 2 Btu. Using energy conservation, we can compute the isothermal heat addition (Q_A) and isothermal heat rejection (Q_R). _net work = Q_A - Q_R_ Plugging in known values, we can solve for Q_A: _2 Btu = Q_A - 5 Btu_ _Q_A = 7 Btu_ Now that we have the heat addition, we can find the heat rejection: _Q_R = Q_A - 2 Btu = 7 Btu - 2 Btu = 5 Btu_

Calculate the energy source temperature (T_H)

We have Q_A = 7 Btu and know the energy reservoir sink temperature, T_C = 75°F. We can use the isothermal heat addition equation to find the energy source temperature (T_H). _Isobaric heat addition:Q_A∝T_H-T_C_ _ 7 Btu ∝ T_H - 75°F_ To solve for T_H, we need to find the proportionality constant: _proportionality constant = Q_A / (T_H - T_C)_ Since we don't have the value of T_H, we'll proceed to step 3 and return to step 2 with the missing information.

Calculate the volume and pressure ratios

Given the maximum and minimum air volumes V_max = 0.5 ft³ and V_min = 0.06 ft³, we can find the volume ratio (r_v): _r_v = V_max / V_min = (0.5 ft³) / (0.06 ft³)_ _r_v ≈ 8.333_ Given the minimum air pressure P_min = 15 psia, we can find the pressure ratio (r_p) using the volume ratio and the Stirling Cycle equation: _r_p = r_v^γ_ γ = 1.4 (for air) _r_p = (8.333)^{1.4} ≈ 13.756_

Calculate the maximum air pressure (P_max)

Now that we have the pressure ratio (r_p), we can find the maximum air pressure (P_max) using the minimum pressure (P_min): _P_max = P_min * r_p = 15 psia * 13.756 ≈ 206.34 psia_

Determine the amount of air contained in the engine (n)

Now that we have found P_max, we can go back to step 2 and determine the proportionality constant. _proportionality constant = Q_A / (T_H - T_C)_ First, we will determine the temperatures in Kelvin (T_C and T_H): T_C = 75°F = (75 + 459.67) ∗ (5/9) K ≈ 535.67_R ≈ 297.6K Now, we can use the ideal gas law to relate the temperatures and pressure with the amount of air: _nRT_C = P_min * V_min_ _nRT_H = P_max * V_max_ Dividing the two equations, we get: _nR(T_H/T_C) = (P_max * V_max) / (P_min * V_min)_ We can cancel the R (gas constant) and n from both sides: _((P_max * V_max) / (P_min * V_min)) = (T_H / T_C)_ Solving for T_H: _T_H = T_C * (P_max * V_max) / (P_min * V_min)_ _T_H = (297.6 K) * (206.34 psia * 0.5 ft³) / (15 psia * 0.06 ft³) ≈ 849.3 K ≈ 1059.1 R_ We can go back to step 2 and find the proportionality constant: _proportionality constant = Q_A / (T_H - T_C) = 7 Btu / (1059.1 R - 75°F)_ _proportionality constant ≈ 0.00954_ Now that we have the proportionality constant, we can finally find the amount of air (n) using the ideal gas law equation: _n = (P_max * V_max) / (RT_H)_ And using the Ideal Gas constant R=53.35 (ft∗lbf)/(lbm∗R) _n = (206.34 psia * 0.5 ft³) / (53.35 (ft∗lbf)/(lbm∗R) * 1059.1 R) ≈ 0.045 lbm_ #Summary# To summarize, we found that: 1. Energy source temperature, T_H = 1059.1 R (849.3 K) 2. Amount of air contained in the engine = 0.045 lbm 3. Maximum air pressure during the cycle = 206.34 psia

Key concepts

Isothermal Heat Addition in Thermodynamics

In thermodynamics, isothermal heat addition is a process where heat is transferred to or from a system at a constant temperature. Understanding this concept is crucial when analyzing thermodynamic cycles, such as the Stirling cycle.

During the isothermal heat addition phase of the Stirling cycle, the gas inside the engine absorbs heat while maintaining a constant temperature. This process is critical for the engine's efficiency, as it determines the amount of work output in relation to the heat input.

In the given exercise, the heat added to the Stirling cycle engine is calculated as 7 Btu, derived from the energy conservation principle where the net work output (2 Btu) is subtracted from the total heat input (5 Btu). Isothermal processes are one of the key aspects of the Stirling cycle, which ensures efficient energy conversion by maintaining constant temperatures during heat exchange.

Pressure Volume Ratio in the Stirling Cycle

Another significant aspect of the Stirling cycle is the pressure volume ratio. The pressure volume ratio is defined as the ratio of the maximum to minimum volume (or vice versa for pressure) of the working fluid during the cycle. It is a critical factor in determining the efficiency and the work produced by the Stirling engine.

In the exercise, we're provided with the maximum volume of 0.5 cubic feet and a minimum of 0.06 cubic feet. The volume ratio is then simply the larger volume divided by the smaller, yielding a value of about 8.333.

This ratio is taken into account when determining the maximum pressure in the engine using the ideal gas law and properties specific to the working medium, which in this case is air with a specific heat ratio (gamma, \(\gamma\)) of 1.4. This relationship is a cornerstone in understanding how the engine scales pressure in response to volume changes, an important consideration for engineers designing such systems.

The Ideal Gas Law in Engine Calculations

The ideal gas law is an equation of state of a hypothetical ideal gas and is a good approximation to the behavior of many gases under varying conditions. The law combines Charles's Law, Boyle's Law, and Gay-Lussac’s Law and is generally stated as \(PV = nRT\), where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.

For the given exercise, the ideal gas law helps in determining the amount of air within the engine and, subsequently, the maximum pressure. Once we've found the maximum and minimum pressures and volumes, we can deduce the temperature of the energy source and the mass of air (\(m\)) contained in the engine.

This law is an invaluable tool in solving thermodynamic problems, as it allows engineers to relate physical properties of the gas within the cycle, such as temperature and pressure, to the amount of substance present, thereby enabling them to design engines for optimal performance.