Chapter 3.1: Heat Exchanger Analysis Using LMTD method
Slide 2
Steady Flow Steady Flow Energy Equation (SSSF-EE) Neglecting
potential and kinetic energy changes and in the absence of the
external work, the SSSF-EE is reduced to Multiplying the two sides
by For the hot fluid side, is negative (heat rejected) For the cold
side, is positive (heat added) Steady Flow Steady Flow Energy
Equation (SSSF-EE) Neglecting potential and kinetic energy changes
and in the absence of the external work, the SSSF-EE is reduced to
Multiplying the two sides by For the hot fluid side, is negative
(heat rejected) For the cold side, is positive (heat added) h h h
h
Slide 3
Note that eqns.(1) and (2) are independent of the flow
arrangement and heat exchanger type. Another useful expression can
be established using the overall heat transfer coefficient U that
is composed of the thermal resistances of inside flow, separating
wall material and outside flow. This rate equation is of the form:
Where is an appropriate mean temperature difference. Note that
eqns.(1) and (2) are independent of the flow arrangement and heat
exchanger type. Another useful expression can be established using
the overall heat transfer coefficient U that is composed of the
thermal resistances of inside flow, separating wall material and
outside flow. This rate equation is of the form: Where is an
appropriate mean temperature difference.
Slide 4
Logarithmic Mean Temperature Difference Parallel-Flow Heat
Exchanger: The hot and cold fluid temperature distributions
associated with a parallel-flow heat exchanger are shown in this
figure. The temperature difference T is initially large but decays
rapidly with increasing x. The outlet temperature of the cold fluid
never exceeds that of the hot fluid. The form of T m may be
determined by applying an energy balance to differential elements
in the hot and cold fluids. Each element is of length dx and heat
transfer area dA. Logarithmic Mean Temperature Difference
Parallel-Flow Heat Exchanger: The hot and cold fluid temperature
distributions associated with a parallel-flow heat exchanger are
shown in this figure. The temperature difference T is initially
large but decays rapidly with increasing x. The outlet temperature
of the cold fluid never exceeds that of the hot fluid. The form of
T m may be determined by applying an energy balance to differential
elements in the hot and cold fluids. Each element is of length dx
and heat transfer area dA.
Slide 5
Assumptions: 1. The heat exchanger is insulated from its
surroundings, in which case the only heat exchange is between the
hot and cold fluids. 2. Axial conduction along the separating wall
is negligible. 3. Potential and kinetic energy changes are
negligible. 4. The fluid properties are constant. 5. The overall
heat transfer coefficient is constant. Let, Applying an energy
balance to each of the differential elements, it follows that and
Assumptions: 1. The heat exchanger is insulated from its
surroundings, in which case the only heat exchange is between the
hot and cold fluids. 2. Axial conduction along the separating wall
is negligible. 3. Potential and kinetic energy changes are
negligible. 4. The fluid properties are constant. 5. The overall
heat transfer coefficient is constant. Let, Applying an energy
balance to each of the differential elements, it follows that
and
Slide 6
The heat transfer across the surface area dA may be given by
where, T = T h T c is the local temperature difference between the
hot and the cold fluids., by differentiation Substituting eqns. (4)
and (5) into eqn.(7) results in Substituting for from eqn.(6) into
eqn.(8) and integrating across the heat exchanger, we obtain The
heat transfer across the surface area dA may be given by where, T =
T h T c is the local temperature difference between the hot and the
cold fluids., by differentiation Substituting eqns. (4) and (5)
into eqn.(7) results in Substituting for from eqn.(6) into eqn.(8)
and integrating across the heat exchanger, we obtain
Slide 7
Substituting for C h and C c from eqns.(1) and (2) into
eqn.(9), it follows that For the parallel-flow heat exchanger:
Comparing the above expression with eqn.(3), we conclude that the
appropriate average temperature difference is a Substituting for C
h and C c from eqns.(1) and (2) into eqn.(9), it follows that For
the parallel-flow heat exchanger: Comparing the above expression
with eqn.(3), we conclude that the appropriate average temperature
difference is a
Slide 8
logarithmic mean temperature difference ( T lm ) or (LMTD)
where For the parallel-flow heat exchanger, the endpoint
temperature differences are defined as logarithmic mean temperature
difference ( T lm ) or (LMTD) where For the parallel-flow heat
exchanger, the endpoint temperature differences are defined as
Slide 9
Counter-flow Heat Exchanger: The hot and cold fluid temperature
distributions associated with a counter-flow heat exchanger are
shown in the figure. Note that the temperature of the cold fluid T
c,o may now exceed the outlet temperature of the hot fluid T h,o.
The following equations are used for the counter-flow arrangement
Counter-flow Heat Exchanger: The hot and cold fluid temperature
distributions associated with a counter-flow heat exchanger are
shown in the figure. Note that the temperature of the cold fluid T
c,o may now exceed the outlet temperature of the hot fluid T h,o.
The following equations are used for the counter-flow
arrangement
Slide 10
where, For the counter-flow heat exchanger the endpoint
temperature differences must be now defined as For the special case
of T 1 = T 2, But by the application of LHospital rule where, For
the counter-flow heat exchanger the endpoint temperature
differences must be now defined as For the special case of T 1 = T
2, But by the application of LHospital rule
Slide 11
Example 1: In a concentric double pipe heat exchanger, the
inlet and outlet temperatures of the hot fluid are, respectively, T
h,i = 260 o C and T h,o = 140 o C, while for the cold fluid they
are T c,i = 70 o C and T c,o = 125 o C. Calculate the logarithmic
mean temperature difference for (a) parallel-flow arrangement, and
(b) counter-flow arrangement. Data: Find: (a), (b) Solution: The
temperature profiles for the counter-flow and parallel-flow
arrangements are illustrated in the sketch. (a) For the
parallel-flow arrangement heat exchanger: Example 1: In a
concentric double pipe heat exchanger, the inlet and outlet
temperatures of the hot fluid are, respectively, T h,i = 260 o C
and T h,o = 140 o C, while for the cold fluid they are T c,i = 70 o
C and T c,o = 125 o C. Calculate the logarithmic mean temperature
difference for (a) parallel-flow arrangement, and (b) counter-flow
arrangement. Data: Find: (a), (b) Solution: The temperature
profiles for the counter-flow and parallel-flow arrangements are
illustrated in the sketch. (a) For the parallel-flow arrangement
heat exchanger:
Slide 12
(b) For the counter-flow arrangement heat exchanger: (b) For
the counter-flow arrangement heat exchanger:
Slide 13
Comment: For the same inlet and outlet temperatures, the log
mean temperature for counter-flow is larger than that for
parallel-flow: Hence the surface area required to achieve a
prescribed heat transfer rate is smaller for the counter-flow than
for the parallel-flow arrangement, assuming the same value of U.
Arithmetic Mean Temperature Difference It is of interest to compare
the LMTD of T 1 and T 2 with their arithmetic mean: Comment: For
the same inlet and outlet temperatures, the log mean temperature
for counter-flow is larger than that for parallel-flow: Hence the
surface area required to achieve a prescribed heat transfer rate is
smaller for the counter-flow than for the parallel-flow
arrangement, assuming the same value of U. Arithmetic Mean
Temperature Difference It is of interest to compare the LMTD of T 1
and T 2 with their arithmetic mean:
Slide 14
A comparison of the logarithmic and arithmetic mean temperature
differences as a function of the ratio ( T 1 / T 2 ) is presented
in the following table: Special Operating Conditions: 1.Equal Heat
Capacity Rates: The figure shows a counter-flow heat exchanger for
which the heat capacity rates are equal (C c =C h ), or. The
temperature difference T must then be a constant throughout the A
comparison of the logarithmic and arithmetic mean temperature
differences as a function of the ratio ( T 1 / T 2 ) is presented
in the following table: Special Operating Conditions: 1.Equal Heat
Capacity Rates: The figure shows a counter-flow heat exchanger for
which the heat capacity rates are equal (C c =C h ), or. The
temperature difference T must then be a constant throughout the
321.71.51.21 T1/T2T1/T2 1.101.041.0231.01371.00281 T am / T lm
Slide 15
heat exchanger, thereby. 2.Infinite Heat Capacity Rates: a)
Condensation: (in a condenser) In case of condensing a vapor, the
hot fluid undergoes a change of phase from gas to liquid phase. Its
temperature remains constant throughout the heat exchanger
(isothermal process), while the temperature of the cold fluid
increases. During condensation, the hot fluid loses Latent
heat,while the cold fluid gains sensible heat. heat exchanger,
thereby. 2.Infinite Heat Capacity Rates: a) Condensation: (in a
condenser) In case of condensing a vapor, the hot fluid undergoes a
change of phase from gas to liquid phase. Its temperature remains
constant throughout the heat exchanger (isothermal process), while
the temperature of the cold fluid increases. During condensation,
the hot fluid loses Latent heat,while the cold fluid gains sensible
heat.
Slide 16
a) Evaporation: (in an evaporator or boiler) In an evaporator
or a boiler, the cold fluid undergoes a change in phase from liquid
to gas phase. The cold fluid gains A latent heat for evaporation,
while the hot fluid loses sensible heat. a) Evaporation: (in an
evaporator or boiler) In an evaporator or a boiler, the cold fluid
undergoes a change in phase from liquid to gas phase. The cold
fluid gains A latent heat for evaporation, while the hot fluid
loses sensible heat.
Slide 17
Correction Factor (F): The logarithmic mean temperature
difference developed above is not applicable for the heat transfer
analysis of cross-flow and multi-pass heat exchangers. Therefore, a
correction factor (F) must be introduced so that the simple LMTD
for the counter- flow regime can be adjusted to represent the
corrected temperature difference T corr for the cross-flow and
multi-pass arrangements as The rate of heat transfer is given then
by The correction factor is less than unity for cross-flow and
multi- pass arrangements. It is unity for true counter-flow heat
exchanger. It represents the degree of departure of true mean
temperature difference from the LMTD for the counter-flow.
Correction Factor (F): The logarithmic mean temperature difference
developed above is not applicable for the heat transfer analysis of
cross-flow and multi-pass heat exchangers. Therefore, a correction
factor (F) must be introduced so that the simple LMTD for the
counter- flow regime can be adjusted to represent the corrected
temperature difference T corr for the cross-flow and multi-pass
arrangements as The rate of heat transfer is given then by The
correction factor is less than unity for cross-flow and multi- pass
arrangements. It is unity for true counter-flow heat exchanger. It
represents the degree of departure of true mean temperature
difference from the LMTD for the counter-flow.
Slide 18
Algebraic expressions for the correction factor F have been
developed for various shell-and-tube and cross-flow heat exchanger
configurations, and the results are represented graphically.
Selected results are shown in the following figures for common heat
exchanger configurations. T = the fluid temperature in the shell
side, t = the fluid temperature in the tube-side. With this
convention it does not matter whether the hot fluid or the cold
fluid flows through the shell or the tubes. If the temperature
change of one fluid is negligible,. Such would be the case of phase
change (evaporation or condensation) Algebraic expressions for the
correction factor F have been developed for various shell-and-tube
and cross-flow heat exchanger configurations, and the results are
represented graphically. Selected results are shown in the
following figures for common heat exchanger configurations. T = the
fluid temperature in the shell side, t = the fluid temperature in
the tube-side. With this convention it does not matter whether the
hot fluid or the cold fluid flows through the shell or the tubes.
If the temperature change of one fluid is negligible,. Such would
be the case of phase change (evaporation or condensation)
Slide 19
shell-and-tube heat exchanger with one shell and any multiple
of two tube passes shell-and-tube heat exchanger with one shell and
any multiple of two tube passes
Slide 20
shell-and-tube heat exchanger with two shell passes and any
multiple of four tube passes shell-and-tube heat exchanger with two
shell passes and any multiple of four tube passes
Slide 21
Single pass, cross-flow heat exchanger with both fluids unmixed
Single pass, cross-flow heat exchanger with both fluids
unmixed
Slide 22
Single pass, cross-flow heat exchanger with one fluid mixed and
the other unmixed Single pass, cross-flow heat exchanger with one
fluid mixed and the other unmixed