Air flows steadily between two sections in a duct. At...

Air flows steadily between two sections in a duct. At section (1) the temperature and pressure are $T_{1}=80^{\circ} \mathrm{C}, p_{1}=301 \mathrm{kPa}(\mathrm{abs})$ and at section $(2),$ the temperature and pressure are $T_{2}=180^{\circ} \mathrm{C}$ $p_{2}=181 \mathrm{kPa}(\mathrm{abs}) .$ Calculate the (a) change in internal energy between sections (1) and $(2),(\mathbf{b})$ change in enthalpy between sections (1) and $(2),(\mathrm{c})$ change in density between sections (1) and (2) (d) change in entropy between sections (1) and $(2) .$ How would you estimate the loss of available energy between the two sections of this flow?

Air flows steadily between two sections in a duct. At section (1) the temperature and pressure are $T_{1}=80^{\circ} \mathrm{C}, p_{1}=301 \mathrm{kPa}(\mathrm{abs})$ and at section $(2),$ the temperature and pressure are $T_{2}=180^{\circ} \mathrm{C}$ $p_{2}=181 \mathrm{kPa}(\mathrm{abs}) .$ Calculate the (a) change in internal energy between sections (1) and $(2),(\mathbf{b})$ change in enthalpy between sections (1) and $(2),(\mathrm{c})$ change in density between sections (1) and (2)
(d) change in entropy between sections (1) and $(2) .$ How would you estimate the loss of available energy between the two sections of this flow?

Key Concepts

Exergy (Available Energy) Loss

Exergy analysis quantifies the loss of available energy (or usefulness) in a system due to irreversibilities such as friction, heat losses, and other non-ideal effects. It involves comparing the actual state of the system with a reference environmental state to evaluate the maximum useful work that can be extracted. Analyzing exergy losses provides insight into the efficiency of energy conversion processes and helps in optimizing system design for improved performance.

Enthalpy

Enthalpy represents the total heat content of a system and is particularly useful for flow processes occurring at constant pressure. For an ideal gas, enthalpy is calculated using the specific heat at constant pressure (cp) with the relation h = cpT. The change in enthalpy, ?h = cp?T, is directly linked to heat transfer and is crucial in energy balance calculations in ducts and similar devices.

Entropy Change

Entropy is a measure of disorder or randomness in a system, and its change is an important indicator of the irreversibilities in a process. For ideal gases, the change in entropy can be calculated using the relation ?s = cp ln(T2/T1) - R ln(p2/p1), which accounts for variations in temperature and pressure. This concept is vital for assessing the quality of energy transformations and for determining the performance of thermodynamic cycles.

Density and the Ideal Gas Law

The density of an ideal gas is related to its pressure and temperature through the ideal gas law (? = p/(RT)). In flow problems, changes in pressure and temperature lead to corresponding changes in density, which can affect the dynamics of the flow. Understanding this relation is essential for analyzing compressible fluids and for designing systems where pressure and temperature variations are significant.

Internal Energy

Internal energy for an ideal gas is a function of temperature only and is typically calculated using the specific heat at constant volume (cv). The change in internal energy when a gas experiences a change in temperature during flow is determined by ?u = cv?T. This property is fundamental in understanding energy conservation in processes and serves as a primary measure of the energy stored internally within the gas.

Ideal Gas Behavior

This concept assumes that the gas undergoing flow behaves ideally, meaning that its molecules do not interact and its properties depend solely on temperature and pressure. Under this assumption, the thermodynamic properties like internal energy, enthalpy, and density can be expressed with simple relations involving specific heat capacities and the ideal gas law (p = ?RT), which significantly simplifies the calculations for many engineering problems.

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