Fourier’s law of conduction states that the heat flux , or the transfer rate per unit surface area k q′′ q k A through which heat is transferred by conduction, is proportional to the te
Trang 1Chapter 1:
Concept Review Solutions
1.1 Briefly respond to the following to review the primary concepts learned in this
chapter:
(a) Explain how heat transfer by conduction is different from that by radiation
Conduction is a diffusion process and requires a medium (matter which could be solid, liquid, or gas) for the transport of thermal energy or heat from a high
temperature region to that at a lower temperature Radiation, on the other hand, is an electromagnetic wave transmission process and hence, does not require an
intervening material medium (between the high and low temperature regions) for the transport of heat
(b) State Fourier’s law of conduction for a one-dimensional system in words and write its heat transfer rate equation
Fourier’s law of conduction states that the heat flux , or the transfer rate per unit surface area
k
q′′
) (q k A through which heat is transferred by conduction, is proportional
to the temperature gradient normal (or perpendicular) to the surface Mathematically, this can be expressed as
dn
dT
q k′′∞ or
dn
dT A
q k∞
where(dT dn) is the temperature gradient in a one dimensional system, and A is the
surface area Note that in a two- or three-dimensional system, the gradient would be expressed by a partial differential, or as (∂T ∂n) The proportionality constant gives
the definition of the thermal conductivity k of a homogeneous medium and
dn
dT kA
q k =−
which is the general form of Eq (1.1)
(c) How is the convective heat transfer coefficient defined?
The convective heat transfer coefficient is defined as the ratio heat flux (or the transfer rate per unit area
k
q′′
) (q k A of the surface with which a fluid (gas or liquid) exchanges heat) and the temperature difference between the surface and some reference location in the fluid The average convection heat transfer coefficient can be calculated from Eq (1.10) as
Trang 2T A
q T
q
h c
∆
=
∆
′′
=
(d) Write the equation for heat loss from a spherical satellite in space when the satellite surface has an emissivity of 50%
The spherical satellite can be modeled as a gray body of diameter d s, which has a
surface temperature T s and an emissivity ε, and the outer space (or surrounding) as a
black enclosure at a temperature of T ∞ Thus, the radiation heat loss can be calculated from Eq (1.17) as
) )(
5 0 )(
( )
∞
−
where σ is the Stefan-Boltzmann constant
(e) Identify the thermo-physical property that characterizes a metal and a thermal insulating material, and comment upon the magnitude of the two
The thermal conductivity k is the thermo-physical property that characterizes the
difference between a metal (a good conductor of heat) and an insulating material (a poor conductor of heat) The relative magnitude or value of their respective thermal conductivity is such that
insulation metal k
k >>
Often for an insulating material, because of its make up (packed fibers, cellular foams, etc which may have entrapped gas), an effective thermal conductivity of
insulation k eff is specified which encapsulates a combination of heat flow mechanisms through the material
1.2 In both thermodynamics and heat transfer, energy conservation with transfer of heat
is considered
(a) Explain the differences in modeling heat transfer in the two cases
The thermodynamic model looks at a device as an open (or closed) system (or control volume) and simply considers the overall energy balance; how or by what mechanism the energy or heat is transferred is not modeled The heat transfer model,
on the other hand, considers the mechanism(s) or mode(s) of heat transfer in the energy balance and the relative contribution of heat transfer by conduction, convection, and radiation is calculated
(b) Pick a simple engineering system or device of your choice and describe both the thermodynamics and heat transfer models for the transfer of heat and energy conservation in this system/device
As an example, consider a closed feedwater heater, which is typically a shell-and-tube heat exchanger in a steam power plant wherein steam extracted from a turbine
Trang 3flows on the shell-side of the exchanger and boiler feedwater flows inside the tubes
As the steam, which is at a higher temperature than that of the feedwater, condenses
on the outside of the tubes it heats up the feedwater
In the Thermodynamic model, all we can calculate is the rate heat transfer from the condensing steam received by the boiler feedwater This can be expressed by applying energy conservation (First Law of Thermodynamics) to the control volume
of the heat exchanger (an open system) for steady-state conditions Assuming no heat loss to the surroundings, the following is then obtained:
feedwater steam q
or
feedwater in
out p feedwater
in out steam
h
In the heat transfer model, we can calculate how or by what mechanism heat is transferred from the condensing steam to the feedwater Energy conservation in this case tells us that heat transfer rate by convection due to steam condensation on the outside of the tubes in the heat exchanger is equal to the heat transfer rate by conduction through the material of the tubes This in turn is equal to the heat transfer rate by convection from the tube wall to the flowing feedwater, or
feedwater c tubewall k steam
q
q= , = , = ,
where
)
, ,steam c s o s w o
q = − convection from steam to outer surface of tubes,
) (
) ln(
2
, ,
r
w w
tubewall
r r
L k dA
dr
dT k q
o
i
−
=
−
) ( ,
, ,feed water c fw i w i fw
q = − convection from inner tube surface to feedwater
T s is the average steam temperature, A o is the outside surface area of the tubes, T w,o is
the outside surface temperature of the tube walls, T w,i is the inner surface temperature
of the tube walls, r i and r o are the inner and outer radii of the tubes, A i is the inside
surface area of the tubes, and T fw is the average bulk temperature of feedwater Note that the term after the second equality sign in the conduction equation will be derived
in Chapter 2 and the student need not be concerned with it at this stage
Based on this example, students should identify a few other heat exchangers in different applications and develop both the thermodynamic and heat transfer models for each