Air enters a 0.5 -ft inside diameter duct with p1=20 psia, T1=80^∘ F, and V1=200 ft / s . What f

step 1 I can write the value of MA1 is equal to v1 by c1 which is equal to v1 by under root RT1K which

Air enters a 0.5 -ft inside diameter duct with p1=20 psia,...

Air enters a 0.5 -ft inside diameter duct with $p_{1}=20$ psia, $T_{1}=80^{\circ} \mathrm{F},$ and $V_{1}=200 \mathrm{ft} / \mathrm{s} .$ What frictionless heat addition rate in $\mathrm{Btu} / \mathrm{s}$ is necessary for an exit gas temperature $T_{2}=$ $1500=^{\circ} \mathrm{F} ?$ Determine $p_{2}, V_{2},$ and $\mathrm{Ma}_{2}$ also.

Key Concepts

Ideal Gas Law and Specific Heats

The behavior of air in duct flow is typically modeled as an ideal gas, where the relationship between pressure, temperature, and density is given by the ideal gas law. Additionally, the specific heat at constant pressure (cp) is used to relate changes in temperature to changes in enthalpy. Understanding these properties is essential for converting between energy additions in Btu/s and temperature changes in °F or Rankine.

Rayleigh Flow

Rayleigh flow refers to a type of compressible flow in which heat is added (or removed) in a constant area duct while friction is negligible. This process results in changes to the flow properties (such as pressure, temperature, density, and velocity) in a way that is governed by specific Rayleigh flow relations. It is particularly useful for analyzing the impact of heat addition on compressible gas dynamics, allowing one to predict exit conditions from known inlet conditions.

Conservation of Energy in Compressible Flow

In the analysis of compressible flows with heat addition, the conservation of energy principle plays a central role. The energy balance, often expressed in terms of enthalpy, relates the increase in thermal energy (due to heat addition) and the changes in kinetic energy of the flow. For a frictionless process, the heat addition directly raises the specific enthalpy, which in turn affects the static temperature and velocity of the gas as it flows through the duct.

Mach Number

The Mach number is a key dimensionless quantity in compressible flow, defined as the ratio of the flow velocity to the local speed of sound. It is critical for determining the regime of the flow (subsonic, sonic, or supersonic) and influences the behavior of the gas under thermal and dynamic processes. In the context of heat addition, variations in the Mach number can affect and be affected by changes in temperature, pressure, and density.

Continuity and Mass Conservation

The continuity equation, which expresses the conservation of mass in a flow, is fundamental for relating the velocity, density, and cross-sectional area of the duct. In a constant area duct, the mass flow rate must remain constant, meaning that any change in density due to heating must be compensated by a corresponding change in velocity. This principle is used to determine how the exit velocity adapts to meet the requirements imposed by the added heat.

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