A hot-air home heating system takes 500 ft^3 / min (cfm) air at 14.7 psia, 65 F into a furnace a

A hot-air home heating system takes 500 ft^3 / min (cfm) air...

A hot-air home heating system takes $500 \mathrm{ft}^{3} / \mathrm{min}$ (cfm) air at 14.7 psia, 65 F into a furnace and heats it to $130 \mathrm{F}$ and delivers the flow to a square duct 0.5 ft by 0.5 ft at 15 psia. What is the velocity in the duct?

Key Concepts

Mass Flow Rate

Mass flow rate is the product of fluid density, flow velocity, and cross-sectional area. It is a conserved quantity in steady flow systems, which means that even when temperature and pressure changes affect the density of the gas, the mass flow rate through one section of the system must equal the mass flow rate through any other section. This conservation principle is fundamental when calculating parameters like the velocity in varying duct geometries.

Continuity Equation

The continuity equation is a fundamental principle in fluid dynamics that states that for steady flow, the mass flow rate remains constant along a streamline. This principle requires that any change in the cross-sectional area of the flow must be compensated by a corresponding change in the velocity and density of the fluid. It is crucial for analyzing flows in ducts and pipes where the geometry changes.

Ideal Gas Law

The ideal gas law links the pressure, temperature, and density of a gas through the equation p = ?RT. This relationship is essential for problems involving air or other gases, as heating or compressing the gas changes its density, which in turn affects flow characteristics such as velocity. It provides a first-order approximation for understanding the behavior of gases under different thermodynamic conditions.

Transcript

00:01 A hot air home heating system takes in air at a specific state, puts it into a furnace, and heats it to a higher temperature.

00:10 It then delivers the flow to a square duct of a given area and at a given pressure.

00:14 You want to calculate the velocity of the air in the duct.

00:18 So we'll first write our continuity equation for the heating system.

00:25 The inlet air has a mass flow rate m .i, which can be written as the volume flow rate, v .i, over the specific volume little v i and this is the same as the exit mass flow rate m .e and we can write this as the area of the duct ae times the exit velocity capital v e over the specific volume little vee so if we use an ideal gas behavior we can find the inlet and exit specific volumes so for an ideal gas little v i is equal to r t i over pi where r is the gas constant t i is the inlet temperature and p i is the inlet pressure these values are all known so we can calculate this specific volume that's 53 .34 times 500 and 25 over the pressure 14 .7 and this is in pounds per square inch so we have to times it by 12 squared 144 and so we get the inlet specific volume to be 13 .23 and american units which is cubic feet per pound.

02:09 Now we can do the same and find ve.

02:11 So the specific exit volume is rte, the exit temperature over the exit pressure pe.

02:22 And again these values are known.

02:25 So that's r for air is 53 .34 multiplied by the exit temperature of 130 plus four hundred and six hundred and six.

02:40 To get our answer in rankin over the exit pressure 15 again multiplied by 144.

02:49 And so we get our exit specific volume to be 14 .57 cubic feet per pound mass.

03:01 Now the last thing we need is the mass flow rate, mass flow rate m.

03:07 So m .i is equal to m .e, so we'll simply call it m...

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