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EF-Conference on Compact Heat Exchangers and Enhancement Technology for the Process Industries, Banff, Canada, July 18-23, 1999
ECONOMIC OPTIMIZATION OF COMPACT HEAT EXCHANGERS
Holger Martin
Universität Karlsruhe (TH), Thermische Verfahrenstechnik, D-76128 Karlsruhe
Fax: +49 721 608 3490; e-mail: holger.martin@ciw.uni-karlsruhe.de
ABSTRACT

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EF-Conference on Compact Heat Exchangers and Enhancement Technology for the Process Industries, Banff, Canada, July 18-23, 1999
ECONOMIC OPTIMIZATION OF COMPACT HEAT EXCHANGERS
Holger Martin
Universität Karlsruhe (TH), Thermische Verfahrenstechnik, D-76128 Karlsruhe Fax: +49 721 608 3490; e-mail: holger.martin@ciw.uni-karlsruhe.de
ABSTRACT
Rough estimation of an economically optimal flow velocity still seems to be general engineering practice in heat exchanger design. The more rational alternative, a tediously detailed economic optimization, may not often be chosen because of excessive engineering costs. An intermediate path is shown in the present paper. Using some simplifying assumptions, a dimensionless function FC(Re) may be derived, which is proportional to the sum of the annual costs of investment C
I
(assumed to be proportional to the surface area) and the costs of operation C
O
(proportional to the pumping power required). The minimum of this function at the optimal Reynolds number (Re=Re
opt
) depends on the type of heat exchanger chosen, and on a new dimensionless quantity, called Re
eco
, that contains all necessary economic input parameters. The minimum can be easily found by standard methods. The question of an economically optimal efficiency
ε
opt
can also be answered in a simple way. So the influence of changing economic situations may be easily taken into account even at an early stage of heat exchanger design.
ECONOMIC FLUID VELOCITY
The economic design of heat transfer equipment usually starts with assuming a value of the flow velocity, which is thought to be close to an economic optimum value. In some textbooks (e.g. in (Martin,1992)) one may find ranges of recommended flow velocities of say 0.2 m/s <
w
liquid
< 2.0 m/s, and 5 m/s <
w
gas, atmospheric pressure
< 50 m/s. Usually these flow velocities are roughly choosen depending on the individual experience of the designer. A more rational alternative to this rough engineering practice would be the detailed economic optimization of each heat exchanger during design. An example of such a detailed step-by-step procedure is given in (Martin,1992) for a double-pipe heat exchanger. Many authors, as
e
.
g
., Gregsrc (1959), fourty years ago, or, more recently, Hewitt and Pugh (1998) have tried to improve and to simplify the economic design of heat exchangers. In these sources, and especially in the later, more extended version of Gregsrc’s book (Gregsrc, 1973), one may find a number of additional references on the topic. The present paper suggests a solution, which may be useful in the early stages of heat exchanger design. More or less experienced guessing of a value for the flow velocity should be replaced by calculation, based on a rational approach (see also: Martin, 1998). A simple explicit formula for the optimal flow velocity will be derived, which is very easy to apply. Taking the relevant economic parameters into account, the most economic cross-sectional area of an apparatus may thus be found.
ASSUMPTIONS
The
annual costs of investment
C
I
or capital costs are taken to be proportional to the surface area Aand the amortization
a
*
(of say 10%/year).
∗
⋅=
aACC
AI
(1) The price per unit area C
A
will of course depend on the type of apparatus, on the material needed, and on the size (
i.e.,
, on the surface area A) itself. It is generally well known, that the price of equipment does not linearly increase with its size. So Eq. (1) should be regarded as a linearization of a more appropriate empirical power law (C=constA
n
), with an exponent n
less than one. If the prices of heat exchangers of a given type for different sizes are known, one may find C
A
= C
A0
(A/A
0
)
(n-1)
, to account for the degressive increase of price with equipment size.
EF-Conference on Compact Heat Exchangers and Enhancement Technology for the Process Industries, Banff, Canada, July 18-23, 1999
The
costs of operation
C
O
will be taken to be proportional to the pumping power required to overcome the flow resistances in the exchanger
( )
PtelO
pVx1k C
η∆+τ=
(2) The following variables do have an influence on the costs of operation: the price of electrical energy k
el
, the hours of operation per year
τ,
the factor x accounting for the pumping power required on the other side of the heat exchanger (for symmetrical operation, x would be equal one in a plate heat exchanger for example), the pressure drop
∆
p, the volumetric flowrate V
t
,
and last but not least, the efficiency of the pump (or fan)
η
P
. In many cases, C
O
from Eq. (2) will be the main part of the cost of operation, at least for a heat exchanger with process fluids on both sides,
i
.e., not a heater or a cooler in the sense of the pinch-point-method of energy integration. For heaters and coolers the (thermal) energy cost of the utilities will usually be much more important.
TOTAL COST FUNCTION
Starting from these assumptions one can easily show, that the total cost, i.e., the sum C =
C
I
+
C
O
measured in an appropriate currency unit per year [ACU/year], when nondimensionalized (FC = C/C
N
) by
NTUdVcaCC
t pA N
λρ=
∗
(3) leads to a relatively simple cost function
ov3eco
Nu1ReRe2f )x1(1(Re)FC
++=
(4) where the volume flowrate V
t
, density, specific heat capacity, and conductivity (
ρ
,
c
p
,
λ
) of the process fluid on one side (chosen for the design), the diameter
d
, and the Number of Transfer Units NTU
have only to be known if absolute values in [ACU/year] are required. In Eq. (4) Re = wd/
ν
, f(Re) is the Fanning friction factor, the “velocity” in the so called “economic Reynolds number” Re
eco
is to be calculated from the specific economic parameters and the fluid density
ρ
:
νρ⋅τη=
∗
dk aCRe
3/1elPAeco
(5) The term Nu
ov
stands for a dimensionless overall heat transfer coefficient Nu
ov
= kd/
λ
.
The latter can be expressed as
∗
++=
f ,wov
R Nuy1 Nu1
(6) where the factor y stands for a dimensionless transfer resistance of the other side (just as x had been introduced in Eq. (2) to account for the pumping power on the other side) and R*
w,f
includes the resistances of the solid wall, and fouling resistances if necessary. As a first example, Fig. 1 shows the results of an optimization of this kind applied to the chevron-type plate heat exchangers, using the equations given in Martin (1996) for pressure drop and heat transfer. These equations are given in a comprehensive form in the
Appendix
. Varying the angle
ϕ
of the corrugation pattern from 30° to 80°, measured against the main flow direction, the pressure drop at a fixed flowrate would increase by a factor of about 20. So with increasing angle, the economically optimal Reynolds number varies from about Re
opt
(30°) = 4300 to Re
opt
(80°) = 1500. The total cost function FC from Eq. (4) has an absolute minimum at an angle of about 60°.
0.000.050.10100 1000 10000
FC
3045607580
Re
Fig. 1
Total Cost Function FC vs Re for Chevron-Type Plate Heat Exchangers, Parameter: Chevron angle,
ϕ
=30°, 45°, 60°, 75°, 80° (see Table 1) This can be seen better in Fig. 2, where the values of FC
min
from Fig. 1 are plotted versus
ϕ.
The
numerical values used in this optimization are listed in Table 1. The curves in Fig.1, showing the total cost function FC vs. Re,
i
.e., the sum of C
I
(falling with Re) and C
O
(rising with Re) clearly show the minima of the total costs. The Reynolds numbers at these minima are the economically optimal ones. They can be found graphically or by standard procedures. From Eq. (1) we can see, that increasing price per m
2
of heat transfer surface C
A
, and increasing interest rates (
i.e
., a* increasing) would lift the falling branch of the curve,
i.e.
, move the optimum velocity to higher values.
EF-Conference on Compact Heat Exchangers and Enhancement Technology for the Process Industries, Banff, Canada, July 18-23, 1999
0.030.040.0510 30 50 70 90
FC
min
ϕ
Fig. 2
Minima of the Total Cost Function FC
min
vs Chevron-Angle
ϕ
Equation (2) tells us that higher price of electrical energy k
el
, higher number of hours of operation
τ
(max: 8760 h/year), and lower pump efficiency
η
P
would lift the rising branch of the curve, and thus move the optimum velocity to lower values.
Table 1
Economic Optimization of Plate Heat Exchangers - numerical input, and results. Equations used: see Appendix.
Pr 3 x 1 R*
wf
0.003 Re
eco
3000 y 1
ϕ
30° 45° 60° 75° 80° Re
opt
4287 3334 2518 1750 1517
FC
min
0.0423 0.0372 0.0356 0.0378 0.0408 So far the method (Eqs. (1) through (6)) had already been presented by Martin (1998). In the following, a shortcut solution will be derived, that allows for an explicit closed-form calculation of the optimal fluid velocity.
THE SHORTCUT SOLUTION
The Fanning friction factor f (=
ξ/4)
and the overall Nusselt number Nu
ov
may be approximated by simple power laws in many practical cases.
nF
Recf
−
=
(7)
mhov
Re(Pr)c Nu
=
(8) Here the factors c
F
and c
h
, as well as the exponents n and m are constants. The exponent m in Eq. (8) may be chosen a little bit smaller than the corresponding exponent in an equation for the Nusselt number of one side, as the overall heat transfer coefficient contains a wall (and fouling) resistance, which do not depend on the flow rate. So if the Reynolds exponent in the Dittus-Boelter equation for turbulent tube flow is 0.8, one may take m in Eq. (8) to be about 0.7 or 0.6, depending on the relative importance of the wall resistance. The total cost function FC from Eq. (4), with Eqs. (7) and (8) leads to a relatively simple function of the Reynolds number F*(Re)=FC
.
c
h
.
mn33ecoFm
ReRe2c)x1(
ReF
−−−∗
++=
(9) The derivative of F* with respect to Re, when put equal to zero, yields an explicit formula to calculate the optimal Reynolds number, w
opt
d/
ν
, or the optimal flow velocity:
)n3/(1
F3ecoopt
c)x1)(mn3(
Re2mdw
−
+−−⋅⋅ ν=
(10) It is clear that the factor c
h
(Pr) has no influence on the value of the optimal flow velocity. From Eq. (10) with Eq. (5) for Re
eco
, one can find that the optimal flow velocity, under the above mentioned assumptions, depends on the exponents (n, m) of the friction and (overall) heat transfer laws, and on two physical properties of the fluid (the density
ρ
, and the dynamic viscosity
µ
=
νρ)
. The diameter d of the channel has an effect on w
opt
, too.
)n3/(1
)n1(nn
opt
dmn3mw
−−−−
ρµ⋅−−∝
(11) Using the well-known Blasius equation for turbulent tube flow, n = 1/4, and the optimal flow velocity depends on the tube diameter and on the viscosity with the weak exponents 1/11, or –1/11 respectively, while the fluid density enters with an exponent of –3/11. For fully developed laminar tube flow, n = 1,
i.e
., the density has no effect on w
opt
, while d and
µ
enter with exponents of ½ and - ½. Similar approximate explicit solutions of the economic optimization problem had been found much earlier (see
e
.g., Gregsrc, 1959), but they seem to have been greatly ignored in industrial practice.
EXAMPLES
Using typical sets of input data for Eqs. (5) and (10) the results may be compared to the values known from experience. C
A
400 Euro/m² price/m
2
a* 10%/year amortization
EF-Conference on Compact Heat Exchangers and Enhancement Technology for the Process Industries, Banff, Canada, July 18-23, 1999
η
P
0.5 pump efficiency
τ
6500 h/year hours of operation k
el
30 Euro/MWh price of electrical energy x
1 ratio of pumping powers
Water in the tubes of a shell-and-tube hx:
ρ
997 kg/m³ density
ν
8.93.10
-7
m²/s kinematic viscosity d 12 mm tube diameter c
F
0.3164/4 turbulent tube flow n 0.25 Blasius law m 0.7 Re exponent of ov htc Re
eco
= 6296 Re
opt
=23734
w
opt
=1.77 m/s Air
in the tubes of a shell-and-tube hx
:
ρ
1.168 kg/m³ density
ν
1.58.10
-5
m²/s kinematic viscosity d 12 mm tube diameter c
F
0.3164/4 turbulent tube flow n 0.25 Blasius law m 0.7 Re exponent of ov htc Re
eco
= 3372 Re
opt
=12009
w
opt
=15.8 m/s Water
in a
chevron-type plate hx:
ρ
997 kg/m³ density
ν
8.93.10
-7
m²/s kinematic viscosity d 6 mm hydraulic diameter c
F
18.2/4
ϕ
=71° (hard plate)* n 0.25 as in Blasius law m 0.6 Re exponent of ov htc Re
eco
= 3148 Re
opt
=2372
w
opt
=0.35 m/s Air
in a
chevron-type plate hx:
ρ
1.168 kg/m³ density
ν
1.58.10
-5
m²/s kinematic viscosity d 6 mm hydraulic diameter c
F
18.2/4
ϕ
=71° (hard plate)* n 0.25 as in Blasius law* m 0.6 Re exponent of ov htc Re
eco
= 1687 Re
opt
=1201
w
opt
=3.17 m/s
* The values for c
F
=18.2/4 and n = 0.25 have been taken from (Martin,1992, p.72, Fig. 2.29). The results for water (w
opt
=1.77 m/s) and air (15.8 m/s) respectively in a conventional shell-and-tube heat exchanger do in fact agree very well with the well-known recommended values for liquids (0.2 to 2.0 m/s) and for gases at normal pressure (5 to 50 m/s) as given at the beginning of the paper. The considerably lower optimal velocities of only w
opt
=0.35 m/s for water (or 3.17 m/s for air) for a compact chevron-type plate heat exchanger are a result of the much higher flow resistance of this type of exchanger.
ECONOMICALLY OPTIMAL EFFICIENCIES
Once the cross-sectional area of the heat exchanger has been found from the given volume flowrate, and the optimal flow velocity, the remaining question is to fix the length, or the number of transfer units NTU. Usually, the efficiency
ε
of a heat exchanger is assumed to be given, when starting the design procedure. In that case NTU is also fixed, if the flow configuration has been chosen. Heat recovery, however, has an economic value opposite to the total costs C, which may be written as the savings, S: S=S
max
ε
(12) with S
max
=
ρ
c
p
V
t
(T
h,in
-T
c,in
)
τ
k
therm
(13) where the maximal possible savings are proportional to the maximal possible change of enthalpy flow, and k
therm
is the price of thermal energy, often roughly estimated to be one third of the price of electrical energy (k
therm
=k
el
/3). The difference (S-C), with C as the total costs C=C
A
a*
ρ
c
p
V
t
(d/
λ)
NTU
FC(Re, Re
eco,
...) (14) should be maximized for optimal economic design. It is clear that under the assumptions made in this paper, the savings are proportional to the efficiency, while the cost are proportional to the NTU (see bold terms in Eqs. (12) and (14)). This idea has been brought forward only recently by Chawla (1999). The break even point,
i.e
. the situation, where the savings just equal the costs (S-C=0), naturally leads to a second dimensionless criterion (besides Re
eco
) connected with the economic design of heat exchangers:
*aCd
k )T-(T
GT
Atherminc,inh,
⋅⋅τλ=
(15) which may be called a „thermal gain number“. For the break-even situation, the Eqs. (12) to (15) require
GT,...)Re(Re,FC
NTU
eco00
=Θ= ε
(16) or FC < GT. Otherwise the installation of the heat exchanger would only lead to economic losses. The ratio of efficiency
ε
to NTU is the normalized mean temperature difference (NMTD),
Θ
(see Martin, 1992).

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