1. Chapter 3.1: Heat Exchanger Analysis Using LMTD method

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) h h h h

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.

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 Tm 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.

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

6. The heat transfer across the surface area dA may be given by
where, T = Th – Tc 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

7. Substituting for Ch and Cc from eqns. (1) and (2) into eqn
Substituting for Ch and Cc 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

8. logarithmic mean temperature difference (Tlm) or (LMTD)
where
For the parallel-flow heat exchanger, the endpoint temperature differences are defined as

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 Tc,o may
now exceed the outlet
temperature of the hot
fluid Th,o. The following
equations are used for the
counter-flow arrangement

10. where,
For the counter-flow heat exchanger the endpoint temperature differences must be now defined as
For the special case of T1 = T2 ,
But by the application of L’Hospital rule

11. Example 1:
In a concentric double pipe heat exchanger, the inlet and outlet temperatures of the hot fluid are, respectively, Th,i = 260oC and Th,o = 140oC, while for the cold fluid they are Tc,i = 70oC and Tc,o = 125oC. 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:

12. (b) For the counter-flow arrangement heat exchanger:

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 T1 and T2 with their arithmetic mean:

14. Special Operating Conditions: Equal Heat Capacity Rates:
A comparison of the logarithmic and arithmetic mean temperature differences as a function of the ratio (T1/T2) is presented in the following table:
Special Operating Conditions:
Equal Heat Capacity Rates:
The figure shows a counter-flow heat
exchanger for which the heat capacity
rates are equal (Cc =Ch), or .
The temperature difference T must
then be a constant throughout the 3 2
1.7
1.5
1.2 1
T1/T2
1.10
1.04
1.023
1.0137
1.0028
Tam/Tlm

15. heat exchanger, thereby.
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.

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.

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 Tcorr 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.

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)

19. shell-and-tube heat exchanger with one shell and any multiple of two tube passes

20. shell-and-tube heat exchanger with two shell passes and any multiple of four tube passes

21. Single pass, cross-flow heat exchanger with both fluids unmixed

22. Single pass, cross-flow heat exchanger with one fluid mixed and the other unmixed