Question

Air is compressed steadily by a compressor from \(100 \mathrm{kPa}\) and \(17^{\circ} \mathrm{C}\) to \(700 \mathrm{kPa}\) at a rate of \(5 \mathrm{kg} / \mathrm{min}\). Determine the minimum power input required if the process is (a) adiabatic and ( \(b\) ) isothermal. Assume air to be an ideal gas with variable specific heats, and neglect the changes in kinetic and potential energies.

Short answer

Question: Calculate the minimum power input required for adiabatic and isothermal compression of air from an initial state of 100 kPa and 290 K to a final state of 700 kPa. The mass flow rate is 5 kg/min. Answer: The minimum power input required for (a) Adiabatic Compression is approximately 186.9 kW, and (b) Isothermal Compression is approximately 145.3 kW.

Step-by-step solution

Adiabatic Compression

For an adiabatic process, the heat exchange is zero. In this case, the first law of thermodynamics simplifies to: ΔU = -W The relationship between the initial and final pressures and temperatures for an adiabatic process can be expressed as: \((\dfrac{P_1}{P_2})^{\frac{k-1}{k}}=\dfrac{T_1}{T_2}\) where \(k\) is the ratio of specific heat capacities for air (1.4). Using this equation, we can find the final temperature \(T_2\): \((\dfrac{100\,\mathrm{kPa}}{700\,\mathrm{kPa}})^{\frac{1.4-1}{1.4}}=\dfrac{290\,\mathrm{K}}{T_2}\) \(T_2 = 580.84\,\mathrm{K}\) Now we can use the definition of specific heat capacity at constant volume (\(C_v\)) to find the change in internal energy: ΔU = \(mC_v(T_2 - T_1)\) To find the minimum work done, we can use the equation ΔU = -W: \(W_{min}=-ΔU\) Now that we have the expression for the work done, we can find the required minimum power input for the adiabatic compression by dividing the work done by the time taken.

Isothermal Compression

For an isothermal process, the temperature remains constant, so \(T_2=T_1=290\,\mathrm{K}\). The work done can be expressed as: \(W_{min}=mRT\mathrm{ln}\frac{P_2}{P_1}\) Again, we can use this equation to find the minimum power input required for isothermal compression by dividing the work done by the time taken. Putting the numbers in the equation, we can get the result for both cases. (a) Adiabatic Compression: \(W_{min}\approx-5\,\mathrm{kg/min}\times1000\,\mathrm{J/(kg\cdot K)}\times(580.84\,\mathrm{K}-290\,\mathrm{K})\times\mathrm{min}/60\,\mathrm{s}\) \(W_{min}\approx186.9\,\mathrm{kW}\) (b) Isothermal Compression: \(W_{min}\approx5\,\mathrm{kg/min}\times287\,\mathrm{J/(kg\cdot K)}\times290\,\mathrm{K}\times\mathrm{ln}\frac{700\,\mathrm{kPa}}{100\,\mathrm{kPa}}\times\mathrm{min}/60\,\mathrm{s}\) \(W_{min}\approx145.3\,\mathrm{kW}\)

Key concepts

Adiabatic Process

An adiabatic process is a fundamental concept in thermodynamics in which a system does not exchange heat with its surroundings. This implies that any work done on or by the system goes into changing its internal energy without affecting the temperature of the surroundings.

In the context of compression, an adiabatic process would mean that the temperature of the air will rise as it is compressed. This is due to the work being done on the air which increases its internal energy, resulting in a higher temperature. The ideal gas law, which relates pressure, volume, and temperature, is modified by incorporating the specific heat values to account for the adiabatic condition.

Diving deeper, the ratio of specific heat capacities (referred to as gamma or k) comes into play. For air, this value is typically around 1.4. The relationship between pressure and temperature in an adiabatic process is described by the formula \( (\frac{P_1}{P_2})^{\frac{k-1}{k}} = \frac{T_1}{T_2} \), establishing how the temperature will change relative to the change in pressure during the compression.

Isothermal Process

In contrast to adiabatic processes, an isothermal process keeps the temperature of the system constant, even as compression or expansion happens. This process requires perfect thermal contact between the system and the environment so that heat can enter or leave the system as needed to maintain a steady temperature.

For an isothermal compression of a gas, the work done on the gas is precisely balanced by the heat exchanged with the surroundings. The ideal gas law becomes particularly useful here as the product of pressure and volume (PV), remains constant throughout the process. Consequently, the work done during isothermal compression can be calculated using the equation \( W_{min} = mRT\mathrm{ln}\frac{P_2}{P_1} \), where R is the gas constant and m is the mass of the gas. This equation highlights the logarithmic relationship between pressure changes and the work required to maintain isothermal conditions.

Ideal Gas Law

The ideal gas law is a cornerstone of thermal physics, encapsulating the relationship between pressure (P), volume (V), temperature (T), and the amount of gas (n) through the equation \( PV = nRT \). Here, R stands for the ideal gas constant. This law assumes that the gas particles behave ideally, meaning they have no volume of their own and no interactions with each other besides collisions.

When solving thermodynamic problems, the ideal gas law provides a means to understand how adjusting one of the variables (keeping the others constant) will affect the state of the gas. In the case of compression—either adiabatic or isothermal—the volume decreases, which typically leads to an increase in pressure. The gas constant R is key to determining the work done during isothermal processes and also aids in understanding the behavior of the gas under adiabatic conditions when combined with the concepts of internal energy and specific heat capacities.

Specific Heat Capacities

Understanding specific heat capacities is vital for analyzing thermal processes. Specific heat capacity is a material's intrinsic property that measures the amount of heat required to change the temperature of a unit of mass by one degree. There are two types of specific heat capacities—\(C_p\), the specific heat capacity at constant pressure, and \(C_v\), the specific heat capacity at constant volume.

For ideal gases, the specific heats are related to the gas constant and each other by the equation \(C_p - C_v = R\). In adiabatic processes where there is no heat exchange, the specific heat capacity at constant volume, \(C_v\), is used in calculations to find changes in internal energy, as in \(\Delta U = mC_v(T_2 - T_1)\). These properties greatly influence the behavior of gases during thermodynamic processes, such as compression, and are crucial for the accurate calculation of work and heat in various cycle processes.