APPLIED INDUSTRIAL ENERGY AND ENVIRONMENTAL MANAGEMENT
Z. K. Morvay, D. D. Gvozdenac
Part III:
FUNDAMENTALS FOR ANALYSIS AND CALCULATION OF ENERGY AND
ENVIRONMENTAL PERFORMANCE
Applied Industrial Energy and Environmental Management Zoran K. Morvay and Dusan D. Gvozdenac © John
Wiley & Sons, Ltd
Toolbox 11
HEAT EXCHANGER OPERATING POINT
DETERMINATION
1. Heat Exchanger Definition: In a two-fluid heat exchanger two fluids are separated by a thin wall
(parting sheets or tube walls) through which heat flows. This exchanger is also known as a
recuperator.
The first law of thermodynamics has to be satisfied in any exchanger design procedure both at the
macro and micro levels. The overall energy balance for any two-fluid heat exchanger is, in an explicit
form, as follows:
m h c ph Th ,in
Th ,out
m c c pc Tc,out
Tc,in
(11.1)
This equation certainly satisfies the ‘macro’ energy balance under the assumptions that are usual
for the basic design theory of heat exchangers. The main assumptions assume that the Overall heat
transfer coefficient (U) and the Isobaric specific heat of fluids (cp) are constant. The next important
assumptions are that Heat losses are negligible and that Flow rates of fluids are constant. However, it
is often not very obvious that the ε – NTU – ω relation in its general form:
( NTU, , flow arrangemen t )
(11.2)
is the statement that expresses the ‘micro’ energy balance for a particular two-fluid heat exchanger
under the same assumptions. This particularity is due to the uniqueness of the solution of the
governing differential equations and the boundary conditions for a particular flow arrangement. These
differential equations, describing the fluid-temperature fields in the heat exchanger core, are the
statements of the ‘micro’ energy balances for an arbitrary differential control volume of that particular
core. The boundary conditions specify where the fluids at temperatures T h,in and Tc,in enter the core in
a particular flow arrangement. The solution of such a mathematical model, which introduces the
overall heat transfer coefficient U and the total heat transfer surface A, gathered in the overall
conductance UA, enables the evaluation of both fluid outlet temperatures (T h,out and Tc,out) for the
particular flow arrangement. Owing to the simplifying classical assumptions underlying the theory,
the mathematical model is linear and tractable by the available methods of calculus. This means that
the effectiveness relationship of Equation (2) can be derived for any heat exchanger, no matter how
complicated the flow arrangement is. This fact makes the ε – NTU – ω analysis universal.
1
Part III – Toolbox 11:
2
HEAT EXCHANGER OPERATING POINT DETERMINATION
In the case of a condenser and evaporator, one of the fluids changes the phase and the isobaric
specific heat becomes infinite. This indicates that product (m · cp) is then also infinite and the fluid
temperature passing the heat exchanger is constant. In these cases, effectiveness depends only on the
number of transfer units (NTU). The micro and macro energy balances are equal.
2. Heat Exchanger’s Dimensionless Groups: The second important feature of the ε – NTU – ω
method is the thermodynamic significance of the dimensionless groups appearing in the analysis.
The fluid heat capacity rate ratio is defined as follows:
Wm in
Wm ax
(11.3)
In this equation, Wmin means the lower heat capacity of two streams (Wmin = min (Wh, Wc)). If the
stream with lower heat capacity is recognized, then the other will be Wmax. One of the special cases
appears when Wh = Wc (the temperature changes of both streams are equal).
Simply, the ratio of the smaller to the larger heat capacity rate for the two fluid streams is as
follows:
0
1
(11.4)
and represents the dimensionless group suitable for understanding the overall fluid temperature
changes. The condition ω = 0 indicates the tendency of the strong stream towards the isothermal
change, while ω = 1 the trend of each stream to undergo the same temperature change from the inlet
to the outlet of the exchanger (balanced streams).
The temperature profiles of counter-flow heat exchanger for 0
1 are presented pictorially
in Fig. 11.1.
Th,out
Wh > Wc
Flow length
Tc,in
Th,in
Tc,out
Th,in
Th,out
Wh = Wc
Flow length
Temperature
Tc,out
Temperature
Temperature
Th,in
Wh < Wc
Tc,out
Th,out
Tc,in
Tc,in
Flow length
Figure 11.1: Temperature Distribution in Counter-Flow Heat Exchanger
Similarly, thermodynamic reasoning can be associated with the second dimensionless group, the
number of heat transfer units:
NTU
UA
Wm in
(11.5)
Simply, it is the ratio of the overall conductance UA and the smaller heat capacity rate (Wmin).
The range 0 NTU < in practice has a finite upper limit, but thermodynamically speaking, the
higher the NTU (higher overall conductance and smaller weak stream capacity), the smaller are the
local temperature differences across the heat transfer surface area and consequently the irreversibility
is lower. This means that better heat exchanger flow arrangements must have a monotonically
increasing effectiveness with NTU.
Part III – Toolbox 11:
3
HEAT EXCHANGER OPERATING POINT DETERMINATION
3. Heat Exchanger Effectiveness: The effectiveness (ε) of any two-fluid heat exchanger is
essentially a dimensionless measure of the heat quantity which is actually transferred between two
streams normalized with the maximum possible fluid enthalpy change in the system. This
hypothetical quantity of heat can be seen as the enthalpy change of the weak stream (stream with
lower heat capacity) undergoing the maximum possible temperature change (T h,in - Tc,in) without any
losses. The heat exchanger effectiveness is then simply defined as:
Q act
Q m ax
Wh (Th ,in
Wm in (Th ,in
Th ,out )
Wc (Tc,out
Tc,in )
Tc,in )
Wm in (Th ,in
Tc,in )
(11.6)
and it is a unique measure of its thermal performance. Uniqueness in this context means that the same
effectiveness, ε, is obtained by writing Qact either in terms of hot fluid parameters or in terms of cold
fluid parameters. Effectiveness is to be obtained from the solution of the mathematical model
mentioned above and will, thus, depend on two dimensionless groups which are the heat exchanger
parameters NTU and ω.
A summary of the necessary steps to derive an effectiveness relationship is as follows:
Write the differential equations describing the local heat transfer in the heat exchanger core
and specify the inlet (boundary) conditions.
Identify (between the hot and cold fluid stream) the weak stream, and assign a subscript min
to all its entities, assign max to the corresponding entities of the other fluid.
Solve the mathematical model by some appropriate method of calculus to obtain the
temperature distributions within the heat exchanger core.
Heat exchanger effectiveness calculation.
4. Heat Exchanger Operating Point: The operating point of an exchanger is the set of ε, NTU and ω
values that satisfy identically both its ‘macro’ and ‘micro’ energy balance. The flow arrangement as
an argument of the ε –NTU – ω relation makes the heat exchanger operating point unique for the
particular flow arrangement. Here, uniqueness implies that the three values (ε, NTU, and ω) are the
ordered ones for the specified flow arrangement. Different flow arrangements have different operating
points even for the same values of two chosen arbitrarily out of three corresponding parameters (ε,
NTU, and ω). If this is not the case, the flow arrangements are said to be equivalent.
In practice, a designer is faced with the problem of seven physical entities (for a specific flow
arrangement and for 0
1 ) that have to satisfy just two equations, namely Equations (11.1) and
(11.2). These equations state an unambiguous relationship of the type:
.
.
f (Th ,in , Th ,out , Tc,in , Tc,out , UA, (m c p ) h , (m c p ) c , flow arrangemen t )
0
(11.7)
For an arbitrary, but specified flow arrangement, any five of the seven variables must be known
for heat exchanger operating point determination. Depending on the combination of the two
unknowns that have to be determined in order to satisfy Equations (11.1) and (11.2), there are 21
possible problems for the determination of the heat exchanger operating point. They are shown in
Table 11.1 classified in six groups.
It can be stated that data on mass flow rates and fluid types are included in Wi m c p i (i = h, c)
or so cold ‘strongness’ of fluid streams. The units of these heat capacities are the same as for UA
[W/oC]. The dimensionless heat exchanger groups: NTU and ω, are combinations of these dimension
values. As overall heat transfer coefficient (U) can be defined independently of the size of heat
transfer surface area (A), complex UA has not to be divided into constituents. However, complexes
Wi have different nature. Known Wi assumes that both mass flow rate (mi) and isobaric specific heat
of fluid (cp,i) are known. If one of these two values is not known, this means that the heat capacity is
not known.
Part III – Toolbox 11:
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HEAT EXCHANGER OPERATING POINT DETERMINATION
By defining all seven basic heat exchanger parameters according to the energy balance it is
possible to define the Heat Exchanger Operating Point (HEOP).
The ‘sizing’ problems in groups I and II, and the ‘rating’ problems in groups III and IV can
readily be recognized. However, the problems in groups V and VI may be described as ‘regime’
problems. They are the most difficult to solve because there is no possibility for the identification of
fluid streams according to their relative strongness and Equations (11.1) and (11.2) must be treated
and resolved simultaneously based upon a guess made for the Wmin stream. Also, problems 14 to 21
always have one trivial (ω = 0) solution.
Table 11.1: Twenty-One Problems to Determine the Heat Exchanger Operating Point
Group
No.
Problem
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
I
II
III
IV
V
VI
UA
[W/K]
???
???
???
???
???
???
mhcph
[W/K]
mccpc
[W/K]
Th,in
[K]
???
Th,out
[K]
Tc,in
[K]
Tc,out
[K]
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
In the case when = 0, there are two special types of heat exchangers named Condensers and
Evaporators.
A condenser is a device in which the hot stream is converted from vapor to liquid by using the
cold stream. The hot stream heat capacity rate is then Wh
and the temperature of the vapor-liquid
mixture is constant and equal to Tcon = Th,in = Th,out (isothermal change). As the cold stream heat
capacity rate is greater than zero and less than infinity (0 < Wc < ) and as the temperature of the inlet
cold fluid is less than outlet temperature (Tc,in < Tc,out), the heat capacity rate ratio is:
Wc
Wh
0
(11.8)
The effectiveness of any flow arrangement of evaporators and condensers is as follows:
1 exp( NTU)
The list of possible problems that can appear is given in Table 11.2.
(11.9)
Part III – Toolbox 11:
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HEAT EXCHANGER OPERATING POINT DETERMINATION
Table 11.2: Five Problems to Determine the Condenser’s Operating Point
Group
No.
I & II
III & IV
Problem
No.
1
2
3
4
5
UA
[W/K]
???
mccpc
[W/K]
Tcon
[K]
Tc,in
[K]
Tc,out
[K]
???
???
???
???
An evaporator is a device in which the cold stream is converted from liquid to vapor by using the
hot stream. The cold stream heat capacity rate is then Wc
and the temperature of vapor-liquid
mixture is constant an equal to Teva = Tc,in = Tc,out (isothermal change). The hot stream heat capacity
rate is greater than zero and less than infinity (0 < Wh < ) and thus the heat capacity rate ratio is =
0. The list of possible problems that can appear is given in Table 11.3.
Table 11.3: Five Problems to Determine the Evaporator’s Operating Point
Group
No.
I &I I
III & IV
Problem
No.
1
2
3
4
5
UA
[W/K]
???
mhcph
[W/K]
Th,in
[K]
Th,out
[K]
Teva
[K]
???
???
???
???
5. In design work, the mean overall coefficient of heat transfer U is generally specified by the use of
design correlations for the mean coefficients of heat transfer for both sides of the heat exchanger (hh
[W/(m2 oC)] and hc [W/(m2 oC)]) in terms of Reynolds, Prandtl, etc. numbers. The nature of these
correlations is dependent on geometry, flow arrangement, and laminar or turbulent flow.
In the thermal analysis of existing heat exchangers, U can be readily determined on the basis of
recorded measurements for terminal temperatures and flow rates by the use of solutions for heat
exchanger effectiveness (ε). For example, ε can be calculated from Equation (11.7) and ω can be
obtained from the defining relation (Eq. (8)). Known ε and ω and ε – NTU – ω curve, which
corresponds to the geometry and flow arrangement of selected heat exchanger, can be used to
calculate the number of transfer units NTU. Now U can be calculated by using Equation (11.5) in the
form:
U
Wmin
NTU
A
(11.10)
Calculations for U obtained in either of these ways account for all of the various complicating
factors that may exist, such as fouling, fins and baffles.
6. When energy was cheap, little attention was paid to designing heat exchangers which made the
optimum use of available energy resources. Then, the primary constraints concerned the heat
exchanger duty (the amount of required heat exchange and the required heat transfer surface area to
satisfy this duty). The duty is a thermal constraint, while the size is an economic constraint since the
costs of materials and manufacturing are directly related to the heat transfer area.
Potential energy savings, along with economic incentives, have led to increased efforts to produce
heat exchanger configurations that can be used to:
reduce the size of a heat exchanger for a specified heat duty;
upgrade the capacity of an existing heat exchanger;
reduce the approach temperature difference for process streams;
reduce pressure drop and pumping power.
Part III – Toolbox 11:
6
HEAT EXCHANGER OPERATING POINT DETERMINATION
7. Software for Heat Exchanger’s Operating Point Determination can be used for solving any of
the 21 tasks ( 0
1 ) given in Table 11.1 and any of the five tasks for solving condensers (Table
11.2) and evaporators (Table 11.3). It is possible to choose one of the following flow arrangements:
Parallel Flow
Counter Flow
Shell & Tube: One shell pass, 2, 4, 6, … tube passes
Shell & Tube: Two shell passes, 4, 8, 12, … tube passes
Cross Flow (both fluid unmixed in pass)
Cross Flow (both fluid mixed in pass)
Cross Flow (hot fluid (wh) mixed and cold fluid (wc) unmixed in pass)
Cross Flow (Hot fluid (Wh) unmixed and Cold fluid (Wc) mixed in pass
The schemes and formulae for effectiveness calculation for all of these flow arrangements are
given in Table 11.4.
The heat flow rate is defined as follows:
Q
Wh (Th ,in
Th ,out )
Wc (Tc,out
Tc,in )
UA
T
(11.11)
The biggest temperature difference between streams (∆T) of all flow arrangements appears in the
counter flow heat exchanger (for the same inlet and outlet conditions) and it is called the logarithmic
mean temperature difference. For this flow arrangement it is:
Tln
(Th ,in
Tc,out ) (Th ,out
Th ,in Tc,out
ln
Th ,out Tc,in
For Wh = Wc or
Tln
(Th ,in
Tc,in )
(11.12)
= 1 it is:
Tc,out ) (Th ,out
For condensers (Tc = Th,in = Th,out and Wh
Tln
) it is:
(Tcon Tc,out ) (Tcon Tc,in )
Tcon Tc,out
ln
Tcon Tc,in
For evaporators (Teva = Tc,in = Tc,out and Wc
Tln
(11.13)
Tc,in )
(Th ,in
Teva ) (Th ,out Teva )
Th ,in Teva
ln
Th ,out Teva
(11.14)
) it is:
(11.15)
Software calculates this temperature difference which can be compared with the real temperature
difference for the desired flow arrangement which is also calculated by software. The ratio between
real and counter-flow temperatures is always less than one and gives the designer an opportunity to
estimate the ‘quality’ of the heat transfer of the desired heat exchanger flow arrangement as compared
to the best one (counter-flow).
The mean temperatures of both fluids are also calculated by software. These temperatures can be
used for the calculation of mean isobaric specific fluid heat and mass flow rates. The mean pressures
of fluids are assumed to be known.
Part III – Toolbox 11:
7
HEAT EXCHANGER OPERATING POINT DETERMINATION
For condensers and evaporators, the mean temperature differences are defined by Equations
(11.14) and (11.15), respectively, and are equal for any flow arrangement.
The first FORM that appears in the software is used for the selection of the heat exchanger. The
option buttons offer three possibilities: 1. General (both fluids change temperatures passing the heat
exchanger), 2. Condenser (only cold fluid changes temperature passing the heat exchanger and hot
fluid is at a constant temperature changing the phase, and 3. Evaporator (only hot fluid changes
temperature passing the heat exchanger and cold fluid is at constant temperature changing the phase).
After selecting the General type of heat exchanger and pressing the NEXT button the window
SELECTION OF FLOW ARRANGEMENT will be opened. Eight types of heat exchanger
arrangements (listed above) are offered. Using the Horizontal Scroll Bar the flow arrangement can
be selected. By pressing the button NEXT, the window PROBLEM DEFINITION is opened. By
entering five known values, any of 21 problems can be solved. By using the Horizontal Scroll Bar,
the problem can be selected. Text boxes with unknown values will contain the text ‘UNKNOWN’.
After data input by clicking on the button Calculation, the program will calculate the unknown
data and display all the data including the performance of heat exchanger and heat flow rates.
This program assumes that fluid stream heat capacity rates are known (in some of the problems).
This means that specific heats and mass flow rates of fluids have to be known. It is not always
possible to know these values. This means that they have to be assumed, and after performing the
calculation in the next window, the correction has to be made and the calculation has to be repeated.
The final window is used for the calculation of mass and volume flow rates for given fluids. The
program offers nine fluids to be selected and, if none corresponds to the actual fluid, the user can
input its own density and specific heat. The fluids offered by the program are:
1. Hydrogen
2. Nitrogen-clear
3. Oxygen
4. Carbon Monoxide
5. Carbon Dioxide
6. Sulfur Dioxide
7. Dry Air
8. R717 (Ammonia) – Liquid
9. R22 – Liquid
After selecting the fluids by using the Combo Boxes and inputting the mean pressure of fluids by
clicking on the button Calculation, the results appear in the proper Text Boxes.
For any correction of the calculation performed, the button Previous has to be used.
For printing the results, after finishing the calculation, the button Printing has to be used.
A very similar procedure is used for the calculation of either the Condenser or Evaporator. Of
course, other fluids are offered by the program and the enthalpy of saturated vapor and boiling fluid
are calculated.
Table 11.4: Effectiveness Relations for Various Heat Exchangers
Name and Scheme
( NTU, , flow arrangemen t )
=0
0<
<1
=1
Single-pass parallel flow
Wc, tc,in
1 exp
Wh, th,in
Wc, tc,out
Wh, th,out
Single-pass counter-flow
2
Wc, tc,out
Wh, th,in
Wh, th,out
Wc, tc,in
Shell & Tube: one shell pass,
2, 4, 6,… tube passes
3
1 exp( NTU)
1
1
1 exp
1
1
2 NTU
2
NTU
1 NTU
2
Wc, tc,out
Wc, tc,in
)
1 exp NTU(1
)
exp NTU(1
)
Wh, th,out
Wh, th,in
NTU(1
1
2
1 exp
NTU
1
2
1 exp
NTU
1
2
2
2
2
1 exp
NTU
2
1 exp
NTU
2
Part III – Toolbox 11:
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HEAT EXCHANGER OPERATING POINT DETERMINATION
Shell & Tube: two shell passes,
4, 8, 12,… tube passes
1
1
Wc, tc,in
4
2
1
1
Wh, th,in
1
1
1
2
2
1
where * is computed for the scheme 3 for
NTU/2
Wh, th,out
Wc, tc,out
and
Cross-flow: both fluids unmixed
Wc, tc,in
5
Wh, th,in
1 exp
Wh, th,out
(1
) NTU
I 0 2 NTU
1
I1 2 NTU
n/2
I n 2 NTU
1 exp
I0 2 NTU
2 NTU
I1 2 NTU
n 2
Wc, tc,out
Cross-flow: both fluids mixed
Wc, tc,in
NTU
6
Wh, th,in
Wh, th,out
NTU
1 exp( NTU)
1 exp(
NTU
NTU)
1
1
NTU
NTU
exp( NTU)
1
Wc, tc,out
Cross-flow: Wc mixed, Wh
unmixed
Wc, tc,in
7
Wh, th,in
Wh, th,out
1 exp
1 exp
1 exp( NTU)
1 exp(
NTU)
1 exp(
NTU)
(Wc
Wmax)
( Wh
Wm ax)
( Wh
Wm ax)
(Wc
Wmax)
1 exp
1 exp( NTU)
1 exp
1 exp( NTU)
Wc, tc,out
Cross-flow: Wc unmixed, Wh
mixed
1 exp
Wc, tc,in
8
Wh, th,in
1 exp
Wh, th,out
1 exp( NTU)
Wc, tc,out
8. Example: A cross flow heat exchanger (both fluids unmixed) is designed to heat 2.5 kg/s of dry air
at 1 bar from 15 oC to 30 oC. Hot water at 52.5 oC is used for this purpose. The outlet water
temperature is 24 oC (Fig. 11.2). The mean overall coefficient of the heat transfer surface is 300
W/(m2 oC)]. The other performance indicators of heat exchanger have to be determined.
Air,
Mc = 2.5 kg/s
pc = 1 bar
Tc,in = 15.0 oC
Water,
Th,in = 52.5 oC
Water,
Th,out = 24.0 oC
Air,
Th,out = 30.0 oC
Figure 11.2: Cross Flow Heat Exchanger (both fluids unmixed)
The software accompanying this Toolbox can be used for problem solving. Firstly, we select the
GENERAL option (Option Button) and by clicking on the button NEXT the second window is
opened offering eight types of heat exchanger arrangement. We select flow arrangement No. 5 (Cross
Part III – Toolbox 11:
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HEAT EXCHANGER OPERATING POINT DETERMINATION
Flow Heat Exchanger: both fluids unmixed) and after that, the software switches automatically to the
Table containing possible problems. We select problem No. 5. However, in our case only the mass
flow rate of air is given and we have to find the heat capacity of this stream. In the case of dry air, the
specific heat is approximately 1 kJ/(kg oC) which means that the heat capacity is around that value. It
will be the first iteration. In the next step, the program calculates the mean fluid temperatures and
specific heat of fluids. Multiplication of this specific heat and given air mass flow rate gives the new
and different heat capacity of the cold stream. By selecting the previous window and typing in the
new air heat capacity the calculation has to be repeated. In Fig. 11.3, the input data and calculated
results are presented. It must be noticed that at this stage of the calculation the type of fluids does not
have to be known. This simplifies the problem solving as the definition of the weak and strong stream
is not necessary.
Figure 11.3: PROBLEM No. 5 – Step 3
After the input of all the data and by clicking on the button CALCULATION, the results will
appear in the frame Output Data. Now, we know the fluid heat capacities, four temperatures and the
size of the heat exchanger represented by UA. Effectiveness, NTU and ω are 0.7600, 2.32 and 0.5263,
respectively.
The heat transfer flow rate is 37.7 kW and as we know the mean overall coefficient of the heat
transfer, we can calculate the required surface in the following way (Equation (11.9)):
A
Q
U
Tf
37.7 1000
300 12.3
10.2 [m 2 ]
(11.16)
It is always necessary to know how the value U = 300 [W/(m2 oC)] is specified (on finned or unfinned surface, hot or cold fluid side, etc.).
The mean temperature difference between the fluids is in this case 12.3 oC and it is less than the
logarithmic mean temperature difference (LMTD) which is 14.7 oC. The reason for that is the use of
the cross flow heat exchanger although the best solution is a counter flow heat exchanger.
The next step (step 4) in the calculation is the definition of the fluids and the determination of
fluid properties. Actually, we have already assumed the specific heat of the cold fluid and performed
the calculation in step 3. The mass flow rate of the cold stream is given and with a couple of iterations
we have to adjust the cold stream heat capacity to fit the given flow rate. For this, we have to know
the pressure of the air (cold fluid). By selecting hot fluid (Water) and its pressure (1 bar), the program
calculates the hot fluid flow rate. As cold fluid is air at the pressure of 1 bar, the program calculates
the appropriate flow rate. The density and specific heat are calculated for the mean cold fluid
Part III – Toolbox 11:
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HEAT EXCHANGER OPERATING POINT DETERMINATION
temperature. As the calculated flow rate is practically the same as the given one (2.5 ≈ 2.4996), the
calculation is finished.
Figure 11.4: PROBLEM No. 5 – Step 4
The final results can also be presented in the EXCEL worksheet. From Excel it can be printed out
(Fig. 11.5).
Figure 11.5: EXCEL Report
Part III – Toolbox 11:
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HEAT EXCHANGER OPERATING POINT DETERMINATION
Notation:
A
total heat transfer surface area, [m2]
cp
specific heat of fluids at constant pressure, [J/(kg oC]
m
mass flow rate, [kg/s]
NTU number of heat transfer units, [–]
T
temperature, [oC]
T temperature difference between streams, [oC]
U
overall heat transfer coefficient, [W/(m2 oC)]
UA overall heat transfer conductance, [W/oC]
W
fluid stream heat capacity rate, [W/oC]
ε
heat exchanger effectiveness, [–]
heat capacity rate ratio, [–]
Subscripts:
c
cold
h
hot
min refers to fluid with (mcp)min
max refers to fluid with (mcp)max
con refers to condenser
eva refers to evaporator
References
Baclic, B.S, Gvozdenac, D.D, Sekulic, D.P. (1982) Study of the Thermal Performances of
Compact Heat Exchangers, Vol. 2 of the Research Project No. 01.804/3-81, Institute of
Energy and Process Engineering, University of Novi Sad, Yugoslavia.
Bosnjakovic, F. (1976) Nauka o toplini, II dio, IV izdanje, Tehnička knjiga, Zagreb, Croatia.
Kays, W.M., London, A.L. (1984) Compact Heat Exchangers, 13th ed., McGraw-Hill Book
Company, New York.
Thomas, L.C (1982) Heat Transfer, Prentice Hall, New Jersey.