Rev Gin Therm (1996) 35, 517-525 @ Elsevier, Paris

A minimum entropy generation procedure for the discrete pseudo-optimization

of finned-tube heat exchangers

Enrico Sciubba

University of Roma 1, La Sapienza, dept of mechanical engineering, Roma, Italy

(Received 22 May 1995; accepted 30 August 1996)

Summary - This paper presents a novel method which can be helpful in assessing the optimal configuration of finned-tube heat exchangers. The method is an extension of the local irreversibilities method [17], and it is based on the determination on a local basis of the two components of the entropy generation rate: the one caused by viscous dissipations and the one due to thermal irreversibilities. Depending on the engineering purpose for which a technical device was designed, it can be argued that the optimal configuration will be that in which either one (or both) of these two entropy generation rates is minimized. For a heat exchanging device, it is important to minimize thermal irreversibilities, but more important is to minimize the mechanical power lost in achieving a prescribed heat-exchange performance: to this purpose, one can form a relative irreversibility index (named Bejan number here and in [17] because the original seed of this procedure can be found in [l]), and use it to assess the merit of a given configuration. In the procedure presented here, a circular, single-tube, finned heat exchanger configuration is considered: the velocity and temperature fields are computed (via a standard finite-element package, FIDAP) for a realistic value of the Reynolds number and for a variety of geometric configurations (various fin external diameters and fin spacing); then, the entropy generation rate is calculated from the flowfield, and is examined both at a local level, to detect possible bad design spots (ie, locations which correspond to abnormally high entropy generation rates, which could be cured by design improvements), and at an overall (integral) level, to assess the entropic performance of the heat exchanger. Optimal curves are given, and the optimal spacing of fins is determined using alternatively the entropy generation rate and the total heat transfer rate as objective functions: different optima arise, and the differences as well as the similarities are discussed in detail.

Nomenclature Red = UL/v

St Be = - St + SW

Bejan number (local)

integral)

Br= j@$$ Brinkman number C unit (per weight) cost of

finned tube.. . . c d D

total cost of finned tube. tube outer diameter thermal diffusivity coeffi- cient

k thermal conductivity. . . . . e fin length . . . . L reference length ti mass flow rate. P pressure................. Pr = v/a Prandtl number 0 heat flux . .

$2 m

W.m-‘.K-l m

kg.? N.m-’

W.mp2

B t T u, v, w

u W 5, Y1 z

Reynolds number based on external tube diameter fin spacing . . . . . entropy generation rate. . fin thickness. . . . . . temperature . . . . . . . . x - y and z components of the fluid velocity. . . reference velocity. . . finned-tube weight. Cartesian coordinates

W.mp3.K’ m K

m.s-l m.s-l

N m

Greek symbols

o! thermal diffusivity . m2X1 CL dynamic viscosity. kg.m-‘.s-’ V kinematic viscosity . . . . m2.s -1

P density.................. kg.mp3

Indices

a air f fin T thermal V viscous W wall

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1 I INTRODUCTION

At present, second-law based methods are widely employed to evaluate the performance characteris- tics of cycles and components, and researchers as well as design engineers have finally agreed that en- tropic (and exergetic) analyses allow a better under- standing of the dissipative phenomena which affect real processes. Most of the published papers and re- ports in second-law applications though, deal with cycle- and plant analysis, and very few studies can be found in the current literature which tackle the problem from the component point of view, ie, from a global perspective. It is universally accepted that the difference between ill- and well designed compo- nents can be quantified by calculating a parameter, generally called (second-law) eficiency, which is in principle defined as the ratio between useful output and used input: if several different devices perform the same technical transformation, the one which is affected by the least overall dissipations will have the highest efficiency. In particular, for a heat ex- change the correct thermal performance parameter is the so-called heat-exchange effectiveness, defined as:

actual E _ Q

Qideal

The difference between the heat rate ideally transferrable, Qideal, and that actually transferred, Q actual 7 is due to the fact that there is by necessity a finite AT across which heat is transferred.

When designing a heat exchange, another loss parameter is of paramount importance: the relative pressure drop across the device, which is propor- tional to the pumping power required by the heat exchanger:

&he = f (h, AT, 6?id, geometry) (24

ThAp phe = p

It is well-known that correct sizing of any heat exchanger should take into account both the thermal effectiveness and the pumping power: it can be shown [l to 31 that this leads to a formulation of an entropy generation rate minimization problem.

The effects of dissipative phenomena in fluid- handling devices can be assessed by calculating the entropy generation rate in the device itself: for all practical purposes (excluding processes which are inherently transient or which are constituted by continuously, rapidly variable bursting subpro- cesses which never reach steady-state), the entropy generation in the hardware can be neglected, and that in the working fluids can be taken to character- ize the goodness of the process under consideration.

Entropy can be generated in a fluid by three mechanisms: viscous dissipation, molecular ther- mal dissipation and chemical (reaction) dissipation:

in our study, we will consider only the first two, specifically excluding reacting fluids from our anal- ysis. Though the suggestion to analyze the entropy generation rate in a local sense can be found in Gaggioli I131 (see also Evans & von Spakovsky [121), Bejan [1,21 was the first to give a compre- hensive, unified treatment of entropy generation in fluid flow: he derived the basic expressions, and applied them to some simple analytical flowflelds tie, phenomena for which a closed-form solution ex- ists for both velocity and temperature fields). Later, Paoletti et al [16] applied the entropy analysis to compact heat exchanger geometries, in which the velocity and temperature fields had to be obtained by a numerical simulation; Drost & Zaworski [S], Drost & White [71, and Cafaro & Saluzzi [51 also applied the local entropy generation rate calcu- lation method to numerical simulations of realis- tic, complex flowfields. With specific regard to heat exchanger applications, second-law analyses have been performed by de Oliveira et al [6] and Sekulic 1171, but these authors used a lumped rather than a local approach.

The study at the origin the present paper was intended to calculate the dependence of the entropy generation rates (produced by thermal and viscous dissipations) on some typical design parameters (aspect ratio for a heat exchanger, geometric arrangement of the cooling passages in a gas turbine blade, etc), so that, using a second-law based criterion, an optimal design might be chosen.

This optimal design point is not (nor, by its heuristic nature, could be) obtained by the usual mathematical optimization procedures: it results rather from the direct scanning of a finite (and in fact quite small) solution set. In fact, it is more similar to a (single or multiple parameter) sensitivity study: this is not a disadvantage, as it may seem at first glance, because a small solution set is advantageous from at least two points of view:

a) an expert designer uses in practice exactly the same approach: he scans mentally only a very small number of feasible alternatives, and concentrates his attention on some detail which he knows to be important for designing a device with good performance characteristics;

b) expert systems which try to mimic the design process by reproducing the mental operations which an expert designer performs while designing, are at their best when they can search for a solution in a small (and even fuzzily defined) solution set.

This paper describes the procedure employed in optimizing a simple heat exchanger basic configura- tion (an externally finned-circular tube); extensions of the method to more complicated geometries are straightforward.

Paragraphs 2 and 3 contain the statement of the problem and the description of the solution pro- cedure. Paragraph 4 presents the results obtained for the finned-tube configuration, and draws some comparisons between the heat flux/fin spacing rela- tionship obtained here and the values reported in the literature; also in this paragraph, the calculated

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Optimization of finned-tube heat exchangers

values for the viscous and thermal entropy gener- ation rates are examined, and their dependence on design parameters is discussed; paragraph 5 contains the conclusions and some suggestions for practical application of the method.

2 I PROBLEM STATEMENT

Consider a finned tube (fig 1) of cylindrical cross- section, inside which a hot fluid flows: to augment the heat transfer characteristics of the outside surface (the cooling fluid being air), circular fins have been added. The heat transfer problem can be solved in closed form [4,103 and a mathematical optimization can be performed to calculate the optimal fin spacing, ie, the spacing which maximizes the heat flux per meter of tube for a given fin shape and thickness. In the configuration chosen here, let d be the tube outside diameter, 1 the fin radial projection from the tube surface (so that the fin outside diameter is df = d + 2 Z), t the fin thickness and s the interfln spacing (axial distance between the radial centerlines of two consecutive fins).

Fig. 1. Geometry definition for the finned-tube problem (the shaded volume represents the computational domain).

The dimensions shown in figure 1 and table I are those pertaining to the reference case; in the numerical simulation, 1 and s have been varied (but the tube diameter has been kept constant, so that Red = 966 throughout the calculations), and the velocity and temperature fields around the tube and the fins have been computed so that the local as well as the global heat exchange parameters could be calculated (local heat transfer coefficient, total heat transfer rate from the finned tube, for example).

The fluid inside the tube has been assumed to have an internal heat transfer coefficient substantially higher than the external one, so that the tube surface could be considered isothermal. The cooling fluid is air at far field conditions T, = 293 K and U, = 0.6 m.s-‘, so that the external flow problem remains laminar. The air properties are listed in table II, together with some material properties relevant to the calculations.

The problem can be stated as follows: a) is it possible to consistently compute the local

entropy production rate for different fin thicknesses, radii and spacing?

b) is it possible to deduce the existence of an optimal configuration (a combination of a value of 1, one of s and one of t) for which the entropy generation rate attains a minimum value?

c) can one draw analogies and/or make compari- sons between the classical optimization with respect to the released heat flux and the one proposed here, based on the minimum dissipation?

It turns out that all three questions can be an- swered in the affirmative, as it will be demonstrated in the following paragraphs and discussed in the conclusions.

Given the axisymmetrical nature of the problem, only a portion of the domain can be modeled: in this case, it was chosen to extend the computational domain to a slice of axial length equal to (s + t)/2 and radial width equal to twice the fin external radius (fig 1 and 2).

The boundary conditions are the following:

u = u, - channel inlet: v = w = 0

1 on surface A

T=T,

- tube and u = v = w = 0 on surfaces E, G, H fin surface: T = Tw on E

- symmetry planes:

?T = 0

0 on surfaces B & I sz= su 6v - outflow: Gz=iG

SW ST =-x-x 6x 6x

0 on surface C

- perfect thermal contact between tube and fins.

3 I SOLUTION PROCEDURE

The problem is described by the (laminar steady) Navier-Stokes equations for the fluid(‘) (2):

,.6”, _ SP ; 1 s2ui 3 6Xj SXi Re 6xj6xj

i = 1,2,3 j = 1,2,3 (3)

(1) Throughout the paper, summation cwer repeated indices has been adopted. (2) Re is constant throughout and equal to Rr,l.

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TABLE I Synoptic table of the analyzed configurations

Configuration Z

Case 1 2 3 4 5 6 7 8 9 10

d (mm) 3 3 3 3 3 3 3 3 3 3 e (mm) 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 s (mm) 0.30 0.10 0.15 0.20 0.22 0.25 0.40 0.60 0.80 0.95 t (mm) 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 Red 966 966 966 966 966 966 966 966 966 966

Configuration ZZ

Case 1 2 3 4 5 6 7 8

d (mm) 3 3 3 3 3 3 3 3 e (mm) 1 1 1 1 1 1 1 1 s (mm) 0.10 0.20 0.22 0.25 0.30 0.40 0.60 0.80 t (mm) 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 Red 966 966 966 966 966 966 966 966

Configuration ZZZ

Case 1 2 3 4 5 6 7 8 9 10

d (mm) 3 3 3 3 3 3 3 3 3 3 e (mm) 2 2 2 2 2 2 2 2 2 2 s (mm) 0.05 0.10 0.20 0.22 0.25 0.27 0.30 0.40 0.60 0.80 t (mm) 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 Red 966 966 966 966 966 966 966 966 966 966

Configuration ZV

Case 1 2 3 4 5 6

d (mm) 3 3 3 3 3 3 e (mm) 2.5 2.5 2.5 2.5 2.5 2.5 s (mm) 0.05 0.10 0.20 0.22 0.25 0.27 t (mm) 0.05 0.05 0.05 0.05 0.05 0.05 Red 966 966 966 966 966 966

L

TABLE II Finned tube, reference case: air and material properties

Far field temperature T, = 293 K Far field velocity U, = 0.6 m.ss’ Conductivity k, = 0.026 W.m-l.K-’ Kinematic viscosity v = 18.6.10M6 m2.sP1 Specific heat cP = 1000 J.kggl.K-’ Tube outside wall T, = 333 K

temperature Reference temperature AT = T,,, - T, = 40 K

difference Conductivity of the fin kF = 55 W.m-‘.K-1

material (steel) Density of the fin material pF = 7800 kg.m-” 7 ST kAT -=- -7 s, a2

G5 155 000

augmented by the energy equation:

.:T =-D s2T z sxj sxj sxj j = 1,2,3 (4)

7 8

3 3

t

2.5 2.5 0.30 0.40 0.05 0.05 966 966

9 1 10

3 3 2.5 2.5 I 0.60 0.80

0.05 0.05 966 966

Fig. 2. The computational domain and a sample mesh.

and by the entropy generation rate equation:

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Optimization of finned-tube heat exchangers

where the dimensionless coefficients Re, D and Br are computed at standard conditions (see table II).

All five equations (each component of equation (3) has been considered as one separate equation) have been made dimensionless, with the scalings (overbars indicate dimensional quantities):

For the derivation of equation (5), see [21 and [17]. Notice that equation (5) can be rewritten by

(6)

defining the local Bejan number, Be 1181 as:

Be = local entropy generation rate due to thermal effects total local entropy generation rate

St z-

it +.i,

1

‘I++* (2+&y (g+$) 2 ($GJ (7)

Notice also that, while the local Bejan number defined by equation (7) contains information on the local mechanism of entropy generation (Be tends to zero with vanishing thermal gradients, and tends to l/T2 with vanishing frictional effects), the average or global Bejan number [3,17]:

1 Be = ~

l+Br (8)

similarly contains information on the global, aver- age mechanisms of entropy generation: it goes to 0 in the limit of vanishing average thermal gradient AT, and to 1 in the limit of vanishing mean velocity gradients U/L.

Equations (3) and (4), with the proper boundary conditions, can be solved as a coupled system: for their solution, a finite-element commercial package, FIDAP (registered trademark of FDI Inc, Evanston, Illinois) was employed here.

For the details of the finite-element method in general and of the specific solution strategy, see [3,14,17]. Since FIDAP does not have a built-in provision to compute the two entropy generation rates St and SW, they were computed separately, according to the following procedure:

a) first, the solution space is scanned and the SU.

nodal values of the various derivatives 2 and &i

ST ~ are extracted; 6Xj

b) then, the nodal values of $t and Sv are computed using equation (3);

c) the average value of the entropy generation rates in each element is then obtained by proper weighing of the nodal values pertaining to the nodes belonging to that element (specifically, the entropy generation rate has been considered as a scalar value assigned to the centroid of each element: thus, the weighing functions are the shape functions of the FE discretization);

d) this average value is multiplied by the element volume and re-scaled to calculate the absolute entropy generation rates (in W.me3.Ke1);

e) when the global average is needed, it is obtained by summing all elemental contributions and dividing by the volume of the entire domain.

4. FINNED TUBE: RESULTS AND DISCUSSION

The flowfield computed for the reference con- figuration (tables I and II) is shown in figures 3 and 4. These results are in excellent agreement (for details, see [3]) with an unrelated, previously published numerical simulation of flow over finned tubes [14], particularly for what concerns the wake structure downstream of the tube and the ther- mal wake of the fin. The vortex street (Von Kar- man wake) for flow over a cylinder is suppressed here, in spite of the relatively high Reynolds num-

ber (Red = z = 966 in this case) by the vis-

cous squeezini of the fluid caused by the fins. A horseshoe-like vortex forms downstream of the tube, and interacts with the secondary (wake) flow to enhance thermal convection.

Fig. 3. Finned tube: velocity field in the midplane between fins (reference case).

293 “K 333 “K

Fig. 4. Finned tube: temperature field in the midplane between fins (reference case).

The heat flux from a single fin, &, is shown in figure 5: for a fixed fin thickness t, it increases

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as expected with the fin diameter df , and also in- creases with the fin spacing, tending asymptotically to the isolated fin limit. As a result, the heat flux per meter of tube, given by:

exhibits a maximum which corresponds appro- ximately (for all fin diameters) to a spacing of 0.025 + 0.030 m, which is identical with the optimal value calculated analytically in [4]. The heat exchange coefficient on the fin surface is shown in figure 6: it varies both radially and in the flow direction, as expected, and its values are in excellent agreement with the experimental findings reported in [16].

The viscous dissipation is - at these relatively low Re - much lower than its thermal counterpart; the thermal dissipation is higher in areas of high thermal gradient, while the viscous dissipation contours closely follow the vorticity contours; it turns out that the thermal dissipation is of four orders of magnitude larger than the viscous dissipation: this result can be justified on a global basis (ie, considering the integrals of the viscous and of the thermal entropy generation

rates, s

$dx dy dz and .I

>Tdz dy dz), from

dimensiozal analysis consider%ions. In fact, these global thermal and viscous entropy generation rates can be scaled as [2,17]:

so that the order of magnitude of their ratio is (see table II):

3.50,

0 1 2 3 4 5 6 7 6 Inidln npaclng (mm)

Fig. 5. Heat flux from a single fin

(11)

flow direction present calculation

experimental [ 161

26 16

Fig. 6. Heat exchange coefficient over the fin surface (W.m-2 .K-I).

The corresponding ratio between the computed values (fig 7) is about the same. The local viscous entropy generation rate on a plane just above the upper surface of the fin, &,, are shown in figure 7~: it closely follows the vorticity field (fig 8), as expected; 9, attains its maximum value in the proximity to the solid boundaries (tube and fin) and in the pseudo-mixing layer originated by the intersection of the horseshoe vortex shed by the tube and the (recirculating) secondary flow. The local thermal entropy production, correspondingly, attains its maximum in the thermal boundary layer and at the borders of the thermal wake where thermal gradients are higher (fig 7b).

With increasing fin spacing, the velocity gradi- ents decrease, and so does the entropy generated by viscous dissipation: figure 9 shows that the inte- gral of the viscous dissipation over the whole field, calculated per meter of tube, decreases markedly with increasing fin spacing and - for fixed spacings - increases with the diameter df, as expected.

The integral of the thermal dissipation over the entire field, calculated per meter of tube, displays a maximum which, for each fin diameter, is attained for a much lower fin spacing than the corresponding maximum in the heat flux (f;g 10): this fact can be explained as follows. As the fin spacing decreases below a certain limit, which depends on Re and can be readily identified by inspection of the thermal field (not shown here, but see fig 11 for an example), the fluid temperature becomes almost constant in the gap between two fins, except for a small zone on the upstream side where the entire temperature gradient is felt.

Thus, while for larger fin spacings a thermal boundary layer develops inside the gap, for lower spacings the flow is, so to speak, thermally devel- oped, with correspondingly lower overall thermal gradients in most of the domain, except at the up- stream border of the fin, where the entire AT is felt by the fluid. Consequently, the global thermal entropy is lower for extremely small fins spacings.

With increasing spacing, the thermal boundary layer does not develop fast enough to fill the spac- ing, and a portion of the fluid (towards the center of the gap) flows quasi-isothermally, so that the overall thermal gradients (and the global entropy

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Optimization of finned-tube heat exchangers

flow direction

viscous entropy generation thermal entropy generation

Fig. 7. Finned tube, reference configuration: viscous and thermal entropy generation rates.

flow direction

Fig. 8. Finned tube, reference configuration: vorticity field.

l I=ts(cmj t.3 * I=l(om)

Fig. 9. Global viscous entropy generation rate per meter df -d

of tube (l = z).

production) decrease again with fins spacing. The temperature profiles across the channel shown in figure 11 substantiate the preceding explanation.

6 I CONCLUSIONS

A method to compute the local entropy genera- tion in complex fluid problems was developed and implemented into an existing CFD finite-element

THERMAL ENTROPY PER METER

HEAT FLOW PER METER

* l=2.5(cm)

0 I=akm)

H

l I=l.S(cm)

* I=l(cln)

Fig. 10. Global thermal entropy generation rate and total

heat flux per meter of tube (e = q.

code. From a numerical analysis viewpoint, the re- sults of this research confirmed the findings of three previous studies [3,7,171.

a) The numerical calculation (with a finite- element procedure in this case) of the viscous and thermal local entropy generation rates (Equ 5) is straightforward and does not significantly affect the run time (because the calculation itself is not coupled with the solution of the velocity and

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299 ‘O’ i 320.

':= 08024 ) ~sL="~o'g)

311.

2% 302.

A -I-

O 0.2 0.4 0.6 0.8 I

A 0.2 0.4 0.6 - O%--- 7

X/L X/L

a) s = 0.008 m; b) s = 0.001 m

Fig. 11. Temperature profiles across the fin-to-fin gap (note: the X coordinate spans line A- A in figure I).

temperature fields, but can be run in sequence to that solution).

b) For these low Reynolds numbers, ST is substantially larger than &, everywhere in the domain (Be >> 0).

From a theoretical point of view, two remarks are of some importance:

c) local irreversibilities can be quantified in a consistent manner, so that different but similar flow processes (eg, with and without heat transfer) can be compared to assess their relative merit;

d) the local values of the entropy generation rates can be easily integrated over the entire domain of interest, thus recovering the results of a lumped approach calculation.

From the design engineer’s point of view, it is important to note that:

e) the optimal fin spacing, which would be usually chosen - for the configuration analyzed here - around 25 mm (to maximize the heat flux per meter of tube, see fig 10) brings about a very high thermal dissipation (viscous entropy dissipation is very low in this case, see figs 9 and 10). From a second-law point of view, it would be better to use a larger fin spacing, thus reducing the thermal entropy generation rate, and to accept the corresponding reduction in the heat flux. One possible criterion (but not the only one, see point f below) could be that of choosing for each fin diameter df the fin spacing for which & is half of its peak value: this would still allow reasonable heat fluxes per meter of tube, as shown in table III;

0 in the design of real components, economic considerations have to be introduced. Here, increas- ing the fin spacing as suggested by table III would reduce the cost of the finned tube (less fins per meter), but increase the total length required to obtain a specified heat transfer. Though economic optimization is outside of the purpose of this paper, table IV shows how the comparison between the installation costs of the two different configurations can be done. For these calculations, the following assumptions have been made:

- unit cost of finned tube ($/N), installed c = 0.1

Ptherm - total length of tube (m) L = ___ &

- design requirement for total heat transfer P therm = 1000 kW

- weight per meter of tube:

d”-(d-pj2+(d$-d2)& PF I

(where p is the tube thickness and the fol- lowing values have been assumed: p = 1 mm, pF = 76 500 N.mm3)

- total finned tube cost: C = c. W.

The results presented in table IV (though of limited validity because they were derived by a sensitivity study based on an educated guess, heuristic approach), show that a larger fin spacing (37 vs 28 mm) would reduce installation costs by 6% and allow for the very same total heat transfer;

g) any process for which a numerical calculation can be reasonably performed to compute fluid velocities and temperatures, can be evaluated using the local- or global entropy generation rate as an objective function. Thus, different configurations of a fluid-handling device can be compared, and the one with the least global irreversibility chosen.

Based upon past experience [3,7] and on the results of this study, the suggestion can be made that comprehensive numerical tests on different geometries and various processes be run, to build a documented data-base on irreversibilities in mechanical thermal components and their effective causes.

TABLE III Optima/ (SO) and suboptimal (ss) fin spacings

df Fin spacing Qpeak &d Fin spacing at Qs;+~~ rib /2 S, AQ A.!?,

at peak Q, SQ St = 3tpcJ2, ss (mm) (mm) (W.m-l ) (W.m-‘.K-‘1 (mm) (W.m-’ 1 (W.m-l.K-’ ) (%I (%)

50 25 290 60 46 250 37.5 14 37.5 60 26 465 78 42 390 46 16 41 70 27 585 80 41 500 49 14.5 39 80 28 740 80 37 670 61 9.5 24

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Optimization of finned-tube heat exchangers

TABLE IV installation costs of the optimal (CQ) and suboptimal (C,) configuration

df Fin spacing Fin spacing at Tube length Total weight Tube length Total weight Total cost

at peak Q f&r = ST~,,~/~ at peak Q at peak Q at 3~~~~~12 at $pea,./2 Peak Q &peak /2 SQ

&zId (“m”, WQ

21 WS CQ CS

(mm) (mm) (N) (N) ($1 ($9

50 25 46 3.448 74.890 4.000 73.260 7.489 7.326 60 26 42 2.150 56.810 2.564 55.930 5.681 5.593 70 27 41 1.709 54.100 2.000 51.520 5.410 5.152 80 28 37 1.351 50.490 1.493 47.480 5.049 4.748

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