Theoretical Solution for the Crossflow

8/19/2019 Theoretical Solution for the Crossflow

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Theoretical solution for the cross-flow heatexchanger

• A. Hofmann

Abstract

The theory of the cross-flow heat exchanger first was treated by Nußelt [1, ! on the base ofheat balances for the two interacting fluids o"er an exchanging area element. This leads to a

#artial differential e$uation. %ut the solution was an infinite row which could not be

ex#ressed in a com#act formula. &n this #a#er it will be shown that a com#act formula for theinfinite row is #ossible. All tem#eratures of the interacting fluids, for exam#le the local fluidtem#eratures and the mean outlet tem#eratures as well as the local tem#erature difference andthe mean tem#erature difference o"er the com#lete exchanger are now being a"ailable insim#le formulae which ha"e the form of an infinite sum. The summation has to be sto##ed ata finite "alue with a negligible de"iation.

As all "ariables in the formulae are dimensionless, normali'ed diagrams are de"elo#ed whichare generally "alid and gi"e a good o"er"iew o"er a wide range of exchanger conditions.


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Crossflow Air HX

Introduction

Crossflow heat exchangers are commonly used in gas heating or cooling. A tube bundle carriesa heating or cooling fluid (either gas or liquid), normally perpendicular to a gas flow whichpasses over the tubes and allows heat to be transferred between the fluids. Crossflow heatexchangers can be of the mixed or unmixed type (see figure 1). he mixed type is the simplerof these designs in which the gas is mixed and not separated into channels.

Figure 1: Crossflow heat exchanger types ( Cengel, 2012 )

Heat exchanger design

he exchanger design consists of several stainless steel tubes embedded in a boxed housing(see figure !). his unit is fitted into the stove pipe above the stove such that flue gas passesvertically through the housing (driven by natural convection) and passes over the tubes. Anelectrically powered fan sited at the rear of the tubes creates a high velocity air flow toimprove heat transfer characteristics (as opposed to those obtained by natural convection).

he performance of a heat exchanger of this type could be modified through varying tubenumber, dimensions and spacing and fan volume capacity (i.e. how much air the fan canmove).

Parameter Size [mm]

ube length "##

ubediameter

"#

$%

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$p

Table 1: Heat exchanger tube geo etry

Figure 2: !asic crossflow heat exchanger "esign ( #agic Heat $eclai er, 201% )

Modelling crossflow heat exchanger

he method adopted for modelling the heat transfer performance of a crossflow heatexchanger integrated into a domestic biomass stove flue pipe follows that of the ' %method.

- Logarithmic mean temperature difference (LM !"

A theoretical explanation of this method can be found elsewhere and will not be explainedhere ( olman, !##! ). *n general terms, the method quantifies the temperature differencebetween two exchange fluids by the log mean temperature difference, + m(C ). his can beexpressed as a function of the inlet and outlet temperatures of the fluids as-

where + a and + b are defined as the difference between the hot fluid inlet temperature andcold fluid inlet temperature, and hot fluid outlet temperature and cold fluid inlet temperaturerespectively. he brac eted term (C ) denotes that this ' % is for the case of a counterflowheat exchanger. or other types of heat exchanger (i.e. crossflow) a correction factor mustbe applied.

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- #orrection factor$ %

he correction factor is obtained from the calculation of two ratios / and 0 associated withthe inlet and outlet temperatures of the two fluids as shown in figure ". he value of is thenread from the graph. he value of + m is then ta en as-

Figure %: Correction factor for a single&pass cross&flow heat exchanger with one flui" ixe" ( Cengel, 2012 )

- #on&ection correlations

*n order to determine the heat transfer coefficients between the fluids and the heat exchangersurface, convection correlations must be used. hese correlations allow the calculation of thedimensionless usselt number u, which can consequently be used to calculate the heattransfer coefficient for the fluid. or the case of turbulent flow through a pipe as is the casefor the forced air flow-

he 0eynolds number 0e is defined-

where the terms are defined as fluid density 2, fluid velocity u, fluid dynamic viscosity μµ, andcharacteristic dimension l (pipe diameter). he /randtl number /r is a function of the fluidproperties and can be ascertained according to temperature. he convection heat transfercoefficient is then calculated-

he temperature at which fluid properties are evaluated is ta en to be the average of the inletand outlet temperatures.

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or the case of flow over a ban of tubes (i.e. flue gas flow), the 3rimson correlations are used( olman, !##! ). or a flow normal to an in4line tube ban the maximum flow velocity occursthrough the minimum fontal area ($ n4d). hus the maximum flow velocity is expressed

where u is the free stream velocity, d is the pipe diameter as previously and $ n is as shown infigure 5.

Figure ': Heat exchanger tube geo etry

he 0eynold6s number is then calculated using the equation previously described, with ureplaced by u max and the characteristic dimension l being the tube diameter d. or this analysisit is normally recommended that properties should be evaluated at the film temperature f which is the average of the bul fluid temperature (prior to the tube ban ) and the tubesurface temperature. owever, in order to simplify calculations, properties have beenevaluated at the average between the bul fluid temperature the average air temperatureinside the tubes. his will give a lower heat transfer coefficient than would otherwise beobtained, and thus the model can be considered a worst4case scenario in this respect. he

usselt number can then be calculated by the 3rimson correlation-

he constants C and n are obtained according to 3rimson as C7#.!85 and n7#.&"! ( 3rimson,19": ).

*n the case of a tube ban with less than 1# rows a correction factor must be appliedaccording to table !.

' ) * + , . / 0 1

2atio #.&; #.:8 #.;" #.;9 #.9! #.98 #.9: #.9; #.99 1.##

Table 2: $atio of con ection coefficient h for rows "eep to that for 10 rows "eep ( Hol an, 2002 )

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he convection heat transfer coefficient can again be calculated as before. Assuming that thewalls of the tubes are thin4walled and therefore conduction can be neglected, the total heattransfer coefficient < is expressed-

3iven specified inlet and outlet temperatures, the heat transfer can be calculated-

he mass flow rate of flue gas is dependent upon the stove, and the air flow rate upon the fanpower used. he flue gas specific heat capacity is assumed to be the same as air at the sametemperature. hen the total heat exchanger area A is related to the heat transfer by therelation-

he total heat exchanger area A is also expressed as-

ere n p is the number of pipes and d and l are as previously defined.

- Pressure drop

According to =asu (1999) the pressure drop across tube bundles can be expressed-

where > is nown as the loss coefficient. or the case of a series of inline pipes this is

calculated by the relation-

where n is the number of rows and > # is the loss coefficient for a single row of tubes. or thespacing of tubes in this design, the loss coefficient is given by /er ov (19&8 in =asu 1999) as-

ere s 1 7 $ p?d and f 7 ($ p4d)?($ n4d), see figure 5.



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here are additional relations for the pressure drop at the expansion on entrance to the heatexchanger and exit. rom =asu (1999) the loss coefficients for expansion and contraction arefound to be > expansion?contraction 7 #.5". his is calculated according to the ratio of cross4sectional areasof the flue pipe to the @ box.

he pressure drop for the expansion and contraction can then be expressed-

23S4L S

he performance of the heat exchanger was investigated for the case of a range of differentstoves. A basic study of the effects of heat exchanger design was also performed by modifyingfan power, and hence air flow rate through the tubes. *n all cases the paybac period iscalculated as shown in the economic analysis section.

A basic investigation was made into the optimum number of tubes and fan power based onthese tube dimensions and a & low efficiency stove.

- u5e num5er

irstly, the optimum number of tubes was investigated. ubes numbers were chosen as shownin table " and the heat recovery and paybac calculated. ote that each row of the heatexchanger consists of three tubes, therefore an additional " tubes is equivalent to oneadditional row.

'um5er of tu5es Heat transfer [6] Pa75ac8 period[7ears]

9 85! 1.1"

1! &"" 1.#&

18 :#" 1.#5

1; :&1 1.#5

!1 ;1# 1.1#

!5 ;8! 1.11

Table %: *ffect of tube nu ber on heat reco ery an" paybac+ perio"

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his shows that a heat exchanger with a tube number of either 18 or 1; gives the bestpaybac , though there are only small variations between all the systems tested. Althoughincreasing the number of tubes increases the overall heat exchange surface area and hencethe total heat transfer, it also reduces the velocity through the tubes, and increases materialcosts. hus increasing tube numbers beyond 1; does not offer any advantages in terms ofpaybac despite higher heat transfer.

- Motor power

he fan motor power was varied according to models available from ebm4papst ( ebmpapst,!#1" ). 0esults are shown in table 5.

Motor consumption[6]

Motor 9uoted flowrate [m *:s]

Heat transfer [6] Pa75ac8 period[7ears]

!& #.#8; 5;5 1.11

"# #.1!; :#" 1.#5

8; #.!&5 9#! #.95

;# #.";" 99" 1.##

1## #.5:! 1#"9 1.##

Table ': *ffect of fan otor power on heat reco ery an" paybac+ perio"

As can be seen, a higher air flow rate results in greater heat transfer due to the higher heattransfer coefficient associated with a higher velocity. or example the heat transfercoefficient between the air and tube wall using a fan motor with consumption !& is &.&8

?m ! .>, and for a 8; fan motor it is !!.!& ?m ! .>. owever, due to the higher powerconsumption and higher initial cost of a larger fan motor, increasing the motor above 8; doesnot reduce the paybac period and therefore it is concluded that 8; provides the optimumair flow rate.

- ;ptimised s7stem

a ing the results of these simple investigations into account, it was can be concluded that acrossflow heat exchanger with 18 tubes and a 8; fan motor would provide the shortestpaybac period. A system with these parameters was tested across a range of stove types andthe results obtained are shown in table 8.



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Sto&epower

[86]

Sto&eefficienc7

Sto&e fuel 3nerg7reco&ered

[6]

3fficienc7impro&emen

t [


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*t is expected that the heat exchanger of the unit will require periodic cleaning in order toremove creosote from the heat exchange tubes. his could be done by hand or by an additionalpiece of sheet metal that can be moved over the tube surface using a small handle on thefront of the unit.

2eferences

=asu, /., >efa, C., Kestin, '. (1999) =oilers and =urners, %esign and heory, ew Lor - $pringer

=$*. (1999). M $ound insulation and noise reduction for buildings. Code of practice.M =$ ;!"",'ondon,

=uilding 0esearch Hstablishment (=0H) N Construction *ndustry 0esearch and *nformationAssociation (C*0*A) $ound control for homes (199").

Cengel, L. A., urner, 0. ., Cimbala, K. . (!#1!) undamentals of hermal4fluid $ciences, 5thedition, 'ondon- c3raw ill igher Hducation

ebmpapst (!#1") E4motor multifunction motor technical specification sheet. FonlineG Availableat- http-??docs4europe.electrocomponents.com?webdocs?#8!f?#9##:&&b;#8!f"b9.pdf FAccessed- !! Apr !#1"G

3rimson, H. %. (19":) Correlation and arc ub, %. (!##") undamentals of oise and Qibration Analysis for Hngineers, !ndedition, Cambridge-


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• The Crossflow Heat Exchanger

A car radiator and an air conditioner evaporator coil are exam#les of crossflow heat exchangers. In both cases heattransfer is taking place between a liquid flowing inside a tube or tubes and air flowing past the tubes. With a carradiator, the hot water in the tubes is being cooled by air flowing through the radiator between the tubes. With anair conditioner evaporator coil, air flowing past the evaporator coils is cooled by the cold refrigerant flowing insidethe tube(s of the coil. !rossflow heat exchangers are typically used for heat transfer between a gas and a liquidas in these two exa"ples.

Understanding Heat Exchangers- Cross-fow, Counter-fow (Rotary/Wheel) andCross-counter-fow Heat Exchangers

The core of heat reco"ery is the heat exchanger. There are "arious ty#es of heat exchangersa"ailabe including cross-flow, counter-flow 9includes rotary wheel: and cross-counter-flow.

http://info.zehnderamerica.com/blog/understanding-heat-exchangers-cross-flow-counter-flow-rotarywheel-and-cross-counter-flow-heat-exchangershttp://info.zehnderamerica.com/blog/understanding-heat-exchangers-cross-flow-counter-flow-rotarywheel-and-cross-counter-flow-heat-exchangershttp://info.zehnderamerica.com/blog/understanding-heat-exchangers-cross-flow-counter-flow-rotarywheel-and-cross-counter-flow-heat-exchangershttp://info.zehnderamerica.com/blog/understanding-heat-exchangers-cross-flow-counter-flow-rotarywheel-and-cross-counter-flow-heat-exchangers


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Through a heat exchanger, fresh-filtered air flows into the house continuously and an e$ual"olume of stale air simultaneously flows out of the home. These airflows are allowed to #ass

by each other - se#arated only by a thin membrane. The longer the two streams flow #asteach other, the higher the efficiency.

How a Cross-fow Heat Exchanger WorksThe su##ly air does not reach $uite as high a tem#erature as with the counter-flow heatexchanger, as at two corners tem#eratures with high differences between them encounter eachother. The efficiency is therefore less, e"en when a "ery large exchange surface area is

#ro"ided. &n an o#timal scenario it reaches about ;*< efficiency.

The large tem#erature difference at one #art of the surface means this form of heat exchangereaches its maximum e"en with small surface areas. The de"ices can be constructed to be"ery com#act. This is in contrast to the counter-flow heat exchanger, which is more effecti"ethe longer it is.

How a Counter-fow Heat Exchanger WorksThese can be distinguished in that the counter-flow 'one com#rises the largest #art of thede"ice. At the beginning and end of the de"ice there are "ery small 'ones with crossed airstreams, and here too the #roblem of 6oining streams must be resol"ed. The efficiency in thecase of "ery long dimensions is entirely de#endent on the a"ailable surface area and in

#ractice reaches =>


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How a Rotary/Wheel Heat Exchanger WorksA rotary wheel heat exchanger is considered a counter-flow heat exchanger. The ad"antage of the rotary heat exchanger is that the air distribution is more straight-forward than in someother counter-flow heat exchangers. &n the latter de"ice there is the issue of the com#lexguiding of air from one side into the counter-flow 'one, and then on the other leading outfrom the counter-flow 'one. The counter-flow heat exchanger also in"ol"es relati"elycom#lex membranes and com#lex #roblems of maintaining air tightness between themembranes, as the two air flows are not to mix. Howe"er it?s on exactly this #oint thatadrawback to the rotary/wheel exchanger arises. 8hile the other de"ices #resented abo"etoo/ #ains to ensure the air flows do not come in contact, the rotary exchanger design acce#tsthis will ha##en. The cell through which the return air flows will ha"e outside air flowingthrough it 6ust a short time later. Therefore the airtightness is not good due to the mo"ing

wheel and it?s more susceptible to leakage between the fresh air and stale airstream .

How a Cross-Counter-fow Heat Exchanger WorksThe thermally wasteful corners are omitted than/s to the counter-flow 'one. The remainingcross-flow 'ones do not #lay as crucial a role if the counter-flow 'one has sufficient surfacearea. 8ith this geometry too, an efficiency of u# to =>< can be achie"ed.

@igurati"ely s#ea/ing, the cross-flow heat exchanger is di"ided in the middle and #ulleda#art.

The #rinci#le of airstreams #assing each other 9counter-flow: is a##lied to the s#ace created.The result is the cross-counter-flow heat exchanger.

• This design has se eral !ene"ts #

• a relati ely s$all si%e can !e achie ed&


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• the crossing o' the air strea$s at the 'ront and !ehind resol es thero!le$ o' oining the $ulti le strea$s&

• the counter-fow area in the $iddle gi es high e*ciency

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Theoretical Solution for the Crossflow

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