Heat Transfer – 06-92-328
The first law of thermodynamics or the energy balance for any system undergoing any process can be expressed
as:
Under steady conditions and in the absence of any work, the amount of net heat transfer to or from the system is
given by:
Fourier’s law of heat conduction:
Newton’s law of cooling:
)
The net rate of radiation heat transfer between two surfaces is given by:
where is the surface emissivity, = 5.67 x10-8 W/m2 · K4, is the Stefan–Boltzmann constant,
The one-dimensional heat conduction equation in rectangular, cylindrical, and spherical coordinate systems for
the case of constant thermal conductivities are expressed as:
where the property
is the thermal diffusivity of the material.
Under steady conditions, the surface temperature of a plane wall of thickness 2L, a cylinder of outer radius ,
and a sphere of radius
in which heat is generated at a constant rate of per unit volume in a surrounding
medium at T can be expressed as:
The maximum temperature rise between the surface and the midsection of a medium is given by:
One-dimensional heat transfer through a simple or composite body exposed to convection from both sides to
mediums at temperatures
and
can be expressed as:
This relation can be extended to plane walls that consist of two or more layers by adding an additional resistance
for each additional layer. The elementary thermal resistance relations can be expressed as follows:
Conduction resistance (plane wall):
Conduction resistance (cylinder):
Conduction resistance (sphere):
:
Convection resistance:
Interface resistance:
Radiation resistance:
Once the rate of heat transfer is available, the temperature drop across any layer can be determined from:
Heat Transfer – 06-92-328
Adding insulation to a cylindrical pipe or a spherical shell will increase the rate of heat transfer if the outer
radius of the insulation is less than the critical radius of insulation, defined as:
Fins:
Very long fin:
Adiabatic fin tip:
Very long fin:
Adiabatic fin tip:
x=0
x=0
Fins exposed to convection at their tips can be treated as fins with insulated tips by using the corrected length
instead of the actual fin length.
The temperature of a fin drops along the fin, and thus the heat transfer from the fin will be less because of the
decreasing temperature difference toward the fin tip. To account for the effect of this decrease in temperature on
heat transfer, we define fin efficiency as:
When the fin efficiency is available, the rate of heat transfer from a fin can be determined from
The performance of the fins is judged on the basis of the enhancement in heat transfer relative to the no-fin case
and is expressed in terms of the fin effectiveness
, defined as:
Here,
is the cross-sectional area of the fin at the base and
represents the rate of heat transfer from this
area if no fins are attached to the surface. (
for uniform section fin
Fin efficiency and fin effectiveness are related to each other by:
The horizontal axis is :
Heat Transfer – 06-92-328
Transient Heat Transfer (Chapter-4)
The temperature of a lumped body of arbitrary shape of mass m, volume V, surface area , density , and specific heat
initially at a uniform temperature that is exposed to convection at time t = 0 in a medium at temperature
with a heat
transfer coefficient h is expressed as:
(characteristic length)
The error involved in lumped system analysis is negligible when:
The Fourier number is given by
Using a one-term approximation, the solutions of one-dimensional transient heat conduction problems are expressed
analytically as: (These equations are valid for Fourier number > 0.2)
Plane Wall
Cylinder
Sphere
where the constants
and are functions of the Bi number only, and their values are listed in Table 4–1 against the Bi number
for all three geometries. The error involved in one term solutions is less than 2% when
.
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Fundamentals of Convection (Chapter-6): (all analysis in this chapter is for laminar flow)
If Ts is the surface temperature and T∞ is the free-stream temperature. The convection coefficient is expressed as:
If k is the thermal conductivity of the fluid and
is the characteristic length. The Nusselt number, which is the
dimensionless heat transfer coefficient, is defined as:
The friction force per unit area is called shear stress, and the shear stress at the wall surface is expressed as:
(or)
where is the dynamic viscosity, is the upstream velocity, and
force over the entire surface is determined from:
is the dimensionless friction coefficient. The friction
The relative thickness of the velocity and the thermal boundary layers is best described by the dimensionless Prandtl
number, defined as:
;
The continuity, momentum, and energy equations for steady two-dimensional incompressible flow with constant properties
are determined from mass, momentum, and energy balances to be:
Continuity:
:
where the viscous dissipation function
is:
X-momentum:
Energy:
:
Using the boundary layer approximations and a similarity variable, these equations can be solved for parallel steady
incompressible flow over a flat plate, with the following results (laminar flow):
Velocity boundary layer thickness:
Local friction coefficient:
Local Nusselt number:
Thermal boundary layer thickness:
The Nusselt number can be expressed by a simple power-law relation of the form:
where m and n are constant exponents, and the value of the constant C depends on geometry. The Reynolds analogy relates
the convection coefficient to the friction coefficient for fluids with Pr ≈ 1, and is expressed as:
The analogy can be extended to other Pr by the modified Reynolds analogy or Chilton– Colburn analogy, expressed as:
Heat Transfer – 06-92-328
External Forced Convection (Chapter-7):
The drag coefficient CD is a dimensionless number that represents the drag characteristics of a body, and is defined as:
where is the frontal area for blunt (bluff) bodies, and surface area for parallel flow over flat plates or thin airfoils. For
flow over a flat plate, the Reynolds number is:
Transition from laminar to turbulent occurs at the critical Reynolds number of:
Laminar
Smooth & rough
Turbulent
Smooth
Smooth
Turbulent
Rough
Note: The Nusselt number in this table is for uniform
temperature (isothermal surface)
The average friction coefficient and Nusselt number
relations for parallel flow over a flat plate are
Combined
Laminar
Turbulent
Smooth
Smooth & rough
The local friction coefficient and Nusslet number relations
for parallel flow over a flat plate are
For isothermal surfaces with an unheated starting section of length ξ, the local Nusselt number and the average convection
coefficient relations are:
Flat plate with an unheated starting length
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When a flat plate is subjected to uniform heat flux, the local Nusselt number is given by:
For uniform heat flux surfaces with an unheated starting section of length ξ, the local Nusselt number and the average
convection coefficient relations are:
Flat plate with an unheated starting length
The average Nusselt numbers for cross flow over a cylinder and sphere are
The fluid properties are evaluated at the film temperature
in the case of a cylinder, and at the free-stream
temperature
(except for
, which is evaluated at the surface temperature ) in the case of a sphere. Although these
correlations for uniform wall-temperature heating, these correlations can be used for uniform heat-flux surface to obtain an
approximate solution.
In tube banks, the Reynolds number is based on the maximum velocity
In-line & Staggered with
that is related to the approach velocity V as
,
Staggered with
The average Nusselt number for cross flow over tube banks is expressed as:
where the values of the constants C, m, and n depend on value of maximum
Reynolds number. Such correlations are given in the table below. All properties
except
are to be evaluated at the arithmetic mean of the inlet and outlet
temperatures of the fluid defined as:
Heat Transfer – 06-92-328
NL : Number of pipes in longitudinal direction
NT : Number of pipes in transverse direction
N: Total number of tubes
Use
in these formulae
The average Nusselt number for tube banks with less than 16 rows is expressed as
correction factor whose values are given in the table below.
. Where F is the
The heat transfer rate to or from a tube bank is determined from:
where
is the logarithmic mean temperature difference defined as:
The exit temperature of the fluid
is:
Where
is the heat transfer surface area and
is the mass flow rate of the fluid. The pressure
drop
and the pumping power required for a tube bank is expressed as:
where f is the friction factor and
is the correction factor, both given in the figures below.
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Heat Transfer – 06-92-328
Internal Forced Convection (Chapter-8):
The mean velocity and mean temperature for a circular tube of radius R are expressed as:
The Reynolds number for internal flow and the hydraulic diameter are defined as:
The flow in a tube is laminar for Re < 2300, turbulent for Re
10,000, and transitional in between.
The length of the region from the tube inlet to the point at which the boundary layer merges at the centerline is the
hydrodynamic entry length . The region beyond the entrance region in which the velocity profile is fully developed is the
hydrodynamically fully developed region. The length of the region of flow over which the thermal boundary layer develops
and reaches the tube center is the thermal entry length . The region in which the flow is both hydrodynamically and
thermally developed is the fully developed flow region. The entry lengths are given by:
For
For
where
is the logarithmic mean temperature difference defined as:
The pressure drop
Flows)
and required pumping power
for a volume flow rate of
are: (Laminar & Turbulent
The head loss can be calculated from
For fully developed laminar flow in a circular pipe, the velocity profile, friction factor and volume flow rate is given as:
Heat Transfer – 06-92-328
For fully developed laminar flow, the Nusselt number is calculated from:
Circular tube, laminar
Circular tube, laminar (
Non-circular tube, laminar, the Nusselt number for isothermal and constant heat flux surfaces, and the friction factor are
calculated from the table.
For developing laminar flow in the entrance region with constant surface temperature, we have
When the difference between the surface and the fluid temperatures is large
All properties are evaluated at the bulk mean fluid temperature
surface temperature.
, except for
, which is evaluated at the
For fully developed turbulent flow with smooth surfaces, we have
0.023
The most common and convenient correlation that is used to evaluate the Nu in turbulent flow is:
When the variation in properties is large due to a large temperature difference
All properties are evaluated at the bulk mean fluid temperature
surface temperature.
, except for
, which is evaluated at the
For liquid metal flow in the range of 104 < Re < 106 we have:
For fully developed turbulent flow with rough surfaces, the friction factor f is determined from the Moody chart or
Heat Transfer – 06-92-328
For a concentric annulus, the hydraulic diameter is
, and the Nusselt numbers are expressed as:
where the values for the Nusselt numbers are given in the table below for fully-developed laminar flow with one surface
isothermal and the other adiabatic :
For fully developed turbulent flow, the inner and outer convection coefficients are approximately equal to each other, and
the tube annulus can be treated as a noncircular duct with a hydraulic diameter of
,. The Nusselt number in
this case can be determined from a suitable turbulent flow relation such as the Gnielinski equation.
Heat Transfer – 06-92-328
Moody Chart
Heat Transfer – 06-92-328
Heat Exchanger (Chapter-11):
Heat transfer in a heat exchanger usually involves convection in each fluid and conduction through the wall separating the
two fluids. In the analysis of heat exchangers, it is convenient to work with an overall heat transfer coefficient U or a total
thermal resistance R, expressed as:
where the subscripts i and o stand for the inner and outer surfaces of the wall that separates the two fluids, respectively.
When the wall thickness of the tube is small and the thermal conductivity of the tube material is high, the last relation
simplifies to:
where.
accounted for by:
where
those surfaces.
. The effects of fouling on both the inner and the outer surfaces of the tubes of a heat exchanger can be
are the areas of the inner and outer surfaces and
and
are the fouling factors at
In a well-insulated heat exchanger, the rate of heat transfer from the hot fluid is equal to the rate of heat transfer to the cold
one. That is,
&
where the subscripts c and h stand for the cold and hot fluids, respectively, and the product of the mass flow rate and the
specific heat of a fluid
is called the heat capacity rate.
Of the two methods used in the analysis of heat exchangers, the log mean temperature difference (or LMTD) method is best
suited for determining the size of a heat exchanger when all the inlet and the outlet temperatures are known. The
effectiveness–NTU method is best suited to predict the outlet temperatures of the hot and cold fluid streams in a specified
heat exchanger. In the LMTD method, the rate of heat transfer is determined from:
is the log mean temperature difference, which is the suitable form of the average temperature difference for use in the
analysis of heat exchangers. Here
and
represent the temperature differences between the two fluids at the two ends
(inlet and outlet) of the heat exchanger.
For cross-flow and multipass shell-and-tube heat exchangers, the logarithmic mean temperature difference is related to the
counter-flow one
as:
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where F is the correction factor, which depends on the geometry of the heat exchanger and the inlet and outlet temperatures
of the hot and cold fluid streams.
(F) is calculated from the figure
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The effectiveness of a heat exchanger is defined as
Where
and
is the smaller of
effectiveness relations or charts.
and
. The effectiveness of heat exchangers can be determined from
Condenser or boiler (one of the streams has a constant temperature, i.e.,
heat exchangers reduces (regardless the type of H.E) to :
As
, i.e.,
): The effectiveness of the
, the effectiveness of the H.E approaches to a minimum value
Heat Transfer – 06-92-328
Heat Transfer – 06-92-328
Radiation Heat Transfer:
The view factor Fi → i represents the fraction of the radiation leaving surface i that strikes itself directly; Fi → i = 0 for plane
or convex surfaces and Fi → i = 0 for concave surfaces. For view factors, the reciprocity rule is expressed as:
The sum of the view factors from surface i of an enclosure to all surfaces of the enclosure, including to itself, must equal
unity. This is known as the summation rule for an enclosure. The superposition rule is expressed as the view factor from a
surface i to a surface j is equal to the sum of the view factors from surface i to the parts of surface j. The symmetry rule is
expressed as if the surfaces j and k are symmetric about the surface i then Fi → j = Fi → k.
The rate of net radiation heat transfer between two black surfaces is determined from:
The net radiation heat transfer from any surface i of a black enclosure is determined by adding up the net radiation heat
transfers from surface i to each of the surfaces of the enclosure:
The total radiation energy leaving a surface per unit time and per unit area is called the radiosity and is denoted by J. The
net rate of radiation heat transfer from a surface i of surface area Ai is expressed as
is the surface resistance to radiation. The net rate of radiation heat transfer from surface i to surface j can be expressed as
is the space resistance to radiation. The network method is applied to radiation enclosure problems by drawing a
surface resistance associated with each surface of an enclosure and connecting them with space resistances. Then the
problem is solved by treating it as an electrical network problem where the radiation heat transfer replaces the current and
the radiosity replaces the potential. The direct method is based on the following two equations:
The first and the second groups of equations give N linear algebraic equations for the determination of the N unknown
radiosities for an N-surface enclosure. Once the radiosities J1, J2, …, JN are available, the unknown surface temperatures and
heat transfer rates can be determined from the equations just shown.
The net rate of radiation transfer between any two gray, diffuse, opaque surfaces that form an enclosure is given by
Heat Transfer – 06-92-328