Hot air at 60^∘ C leaving the furnace of a house enters a 12-m-long section of a sheet metal duc

Hot air at 60^∘ C leaving the furnace of a house enters a...

Hot air at $60^{\circ} \mathrm{C}$ leaving the furnace of a house enters a $12-\mathrm{m}$-long section of a sheet metal duct of rectangular cross section $20 \mathrm{~cm} \times 20 \mathrm{~cm}$ at an average velocity of $4 \mathrm{~m} / \mathrm{s}$. The thermal resistance of the duct is negligible, and the outer surface of the duct, whose emissivity is $0.3$, is exposed to the cold air at $10^{\circ} \mathrm{C}$ in the basement, with a convection heat transfer coefficient of $10 \mathrm{~W} / \mathrm{m}^{2}$. . Taking the walls of the basement to be at $10^{\circ} \mathrm{C}$ also, determine $(a)$ the temperature at which the hot air will leave the basement and $(b)$ the rate of heat loss from the hot air in the duct to the basement. Evaluate air properties at a bulk mean temperature of $50^{\circ} \mathrm{C}$. Is this a good assumption?

Key Concepts

Energy Balance in Flow Systems

This concept involves applying the conservation of energy to a fluid that flows through a duct or channel. The inlet energy minus the energy lost to the surroundings over the length of the duct equals the outlet energy. In problems like these, integrating the heat transfer along the duct allows determination of the fluid's temperature change and the overall heat loss from the fluid to the environment.

Forced Convection Heat Transfer

Forced convection is the process by which heat is transferred between a surface and a fluid that is in motion, typically characterized by a convective heat transfer coefficient. In many engineering applications, including duct flows, the forced convection mechanism is employed to estimate the rate at which heat is removed or added to the fluid, and it is fundamental for calculating temperature profiles along the flow.

Radiation Heat Transfer

Radiation heat transfer is the process by which thermal energy is emitted by a surface and transmitted in the form of electromagnetic waves. It is dependent on the emissivity of the surface, the surface temperature, and the temperature of the surroundings. In problems involving ducts or exchanges with the ambient environment, radiation can be an important additional mechanism for heat loss when the temperature differences are large.

Thermal Resistance and Overall Heat Transfer Coefficient

Thermal resistance is a method of quantifying the opposition to heat flow within materials or across interfaces. When multiple mechanisms such as conduction, convection, and radiation coexist, the overall heat transfer coefficient (often obtained by combining individual resistances in series or parallel) provides a single parameter to describe the net rate of heat loss. This concept simplifies the modeling of complex heat transfer systems.

Bulk Mean Temperature Concept

The bulk mean temperature is an average temperature of the fluid flow which is used to evaluate the fluid properties like density, viscosity, and specific heat. In heat transfer calculations, these properties can vary with temperature but are often taken at a representative bulk temperature to make the problem tractable, which is acceptable if the temperature variation is not extremely high.

Evaluation of Fluid Properties

In thermal systems, it is common to assume that fluid properties are constant and equal to those at a bulk or mean temperature. This simplification helps in the analytical or numerical solution of the problem. Such an assumption is typically valid when the temperature range is moderate and the resultant variations in the fluid properties do not significantly affect the heat transfer analysis.

Transcript

00:01 Hi everyone, so what we have is a house of volume 400 cubic meters that has an internal pressure of 95 kpa.

00:08 It is desired that the temperature go from 14 degree c to 24 degrees c through a heating system.

00:15 The heating system consists of ducts and a 250 kilowatt fan and a 30 kilowatt heater.

00:23 And we know that every time this air passes through this duct, the temperature raises by 5 degrees c.

00:29 We know the house is losing heat at a transfer rate of 450 kilojoules per minute, and what we're asked to solve for is the time it takes to heat this house up to 24 degrees celsius and the mass flow rate of air through this duct system.

00:45 So what we're going to assume first is that we're operating at steady state conditions, so properties will not be changing the time.

00:52 We're also going to assume that air is an ideal gas.

00:57 Next, we're going to assume that any changes in kinetic energy.

01:00 Energy and potential energies are negligible.

01:04 Now what we're going to do is go ahead and write down the properties of our air.

01:09 So we can say that since we are dealing with air and it's an ideal gas, we can say that r is equal to 0 .287 kilojoules per kilogram kelvin.

01:19 We can say that the specific heats of air, cp, is equal to 1 .005 kilojoules per kilogram kelvin, and our cv is equal to 0 .0.

01:30 1 .718 kilojoules per 2 gram kelvin.

01:34 The first thing we're going to do is calculate the total mass of air that's in our building.

01:38 So we can say that m is equal to p1 v1 over r t1 and this gives us an initial mass in the building equal to 461 .3 kilograms.

01:52 Now we're going to take the entire system or the entire building as our system for our energy balance...

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