1. بنام خدا

2. Defining LMTD as
The total heat transfer rate for all single-pass flow arrangements (PF, CF, evaporator, condenser)
In case of CF HEX with Ch=Cc, DTlm is indeterminate
Therefore from L’Hospital: DTlm= DT1= DT2

3. In case of PF HEX, the same definition of LMTD (previous slide) is valid. But obviously
LMTD represents the maximum temperature potential for heat transfer that can only be obtained in a counter flow heat exchanger. Therefore, the surface area required to obtain a prescribed heat transfer rate Q is smaller for a counter flow arrangement than that for a parallel flow arrangement, assuming that U is the same for both cases.
Also note that Tc2 can exceed Th2 for counter flow but not for parallel flow.
Example 2.4 in book

4. Multipass and Crossflow HX
LMTD defined for single pass (PF or CF) HEX is not valid for multipass and crossflow HEX. This time by defining a mean temperature difference DTm
We can determine DTm in terms of LMTD of CF HEX, and two quantities P and R.

6. Meanings of Tlm,cf P and R
DTlm,cf is the log-mean temperature difference for a counter flow arrangement with the same fluid inlet and outlet temperatures.
P is a measure of the ratio of the heat actually transferred to the cold fluid to the heat which would be transferred if the same fluid were to be raised to the hot fluid inlet temperature; therefore is the temperature effectiveness of the HEX on the cold fluid side.
R is the ratio of the mcp value of the cold fluid to that of the hot fluid and it is called the heat capacity rate ratio.

7. Correction factor F
With the definition of the correction factor F heat transfer rate for a multipass or crossflow HEX can be given as
F is a nondimensional term which depends on the temperature effectiveness P, the heat capacity rate ratio R, and the flow arrangement.

8. Charts for correction factors
Depending on the flow arrangement there are series of charts to obtain F using calculated values of P and R.
Such figures are:
In our textbook: Figures 2.7 through 2.14.
In Incropera and DeWitt: Figures through
It is immaterial whether the cold fluid flows through the shell or inside the tubes
If temperature change of one fluid is negligible, F = 1
Some examples of charts are...

9. LMTD correction factor F for a shell-and-tube heat exchanger – one shell pass and two or multiple of two tube passes
Example 2.5 in book now, also study examples

12. Effectiveness-Number of Transfer Units (e-NTU) for HEX Analysis
When inlet and exit temperatures are unknown, a trial and error procedure may be needed. Instead, the method of number of transfer units (NTU) based on HEX effectiveness may be used.
The e - NTU method is based on the fact that the inlet or exit temperature differences of a heat exchanger are a function of UA/Cc and Cc/Ch.
The HEX heat transfer equations may be written in dimensionless form resulting in some dimensionless groups.

13. Dimensionless groups 1. Heat capacity rate ratio: , C*  1
2. HEX heat transfer effectiveness:
e is the ratio of the actual heat transfer rate in a HEX to the thermodynamically limited maximum possible heat transfer rate if an infinite heat transfer area were available in a counter flow HEX.

14. The actual heat transfer is obtained either by the energy given off by the hot fluid or the energy received by the cold fluid
If Ch > Cc, then (Th1-Th2) < (Tc2-Tc1)
If Ch < Cc, then (Th1-Th2) > (Tc2-Tc1)
The fluid that might undergo the maximum temperature difference is the fluid having the minimum heat capacity rate Cmin.

15. Maximum heat transfer:or
Therefore, HEX effectiveness can be written as
The above equation is valid for all heat exchanger flow arrangements. The value of e ranges between 0 and 1.
For a given e and Qmax, the actual HT rate is
Q = e (mcp)min(Th1-Tc1)

27. 3. Number of Transfer Units:
The third dimensionless number NTU shows the nondimensional heat transfer size of the HEX

28. Single pass heat exchanger Cc>Ch
Ch=Cmin and Cc=Cmax
We had obtained
By using definition of NTU
where the + is for counter flow and the – is for parallel flow

29. Using
and
Th2 and Tc2 can be eliminated to obtain for CF
If Cc < Ch, the result will be the same

30. For Cmin/Cmax=1, this result is indeterminate, but by applying L’Hospital’s rule, the following result is obtained for counter flow
and for parallel flow

31. For Cmin/Cmax=0, both CF and PF HXs
Similar expressions for all flow arrangements including multipass and crossflow can be obtained following the same approach. Some of those results are presented as tables like the following. Also there are charts like Fig 2.15 of the text book

32. e-NTU Expressions (Table 2.2 of the book, more detail in book)
Type of HEX
e(NTU,C*)
NTU(e,C*)
Counterflow
Parallel Flow
Cross flow, Cmin mixed and Cmax unmixed
Cross flow, Cmax mixed and Cmin unmixed
1 to 2 shell-and-tube HEX

35. LMTD e-NTU LMTD and e-NTU relations F = f (P, R, flow arrangement)
e = f (NTU, C*, flow arrangement)

36. Example 2.9
A two-pass tube, baffled single-pass shell, shell-and-tube HEX is used as an oil cooler. Cooling water flows through the tubes at 20oC at a flow rate of kg/s. Engine oil enters the shell side at a flow rate of 10 kg/s. The inlet and outlet temperatures of oil are 90oC and 60oC, respectively. Determine the surface are of the HEX using both the LMTD and e-NTU methods, if the overall heattransfer coefficient based on the outside tube area is 262 W/m2K. The specific heats of water and oil are 4179 J/kgK and 2118 J/kgK, respectively.

37. Heat Exchanger Sizing
If inlet temperatures, one of the outlet temperatures and mass flow rates are known, we can use LMTD method for sizing problem:
1. Calculate Q and the unknown temperature
2. Calculate LMTD and obtain F if necessary
3. Calculate U
4. Determine A from A=Q/(UFDTlm,cf)

38. Heat Exchanger Rating
For an available heat exchanger (size, mass flow rates, inlet temperatures and materials are known) using e-NTU method we can rate the heat exchanger:
Calculate C*=Cmin/Cmax and NTU=UA/Cmin
Determine e from appropriate charts or e-NTU equations
Calculate Q=e Cmin(Th1-Tc1)
Calculate outlet temperatures

39. Sizing Using e-NTU method
Calculate e using Cmin, Cmax and temperatures
Calculate C*=Cmin/Cmax
Calculate U
Determine NTU from charts or equations
When NTU is known calculate heat transfer area from A=(CminNTU)/U

40. Rating using LMTD
LMTD method can also be used for rating but for the unknown temperature trial and error approach is required.
In general LMTD is more appropriate for sizing and e-NTU is more appropriate for rating problems.
Variable U may complicate the calculations, next figure shows some cases where U is variable. Variation dependent on flow Reynolds number, HT surface geometry, fluid physical properties