US 2802624 A
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C. F. KAYAN Aug. 13, 1957 APPARATUS FOR CALCULATING HEAT EXCHANGER PERFORMANCE Filed Aug. 17, 1951 2 Sheets-Sheet 1 FIG. 2
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2 Sheets-Sheet 2 APPARATUS FOR CALCULATING HEAT EXCHANGER PERFORMANCE Car/ 2 BY 4R5; If 47 4 7 v ATTOR YS Aug. 13, 1957 Filed Aug. 17 1951 FIG. 5
United States Patent O APPARATUS FOR CALCULATING HEAT EXCHANGER PERFORMANCE Carl F. Kayan, New York, N. Y. Application August 17, 1951, Serial No. 242,231 2 Claims. (Cl. 235--61) This invention relates to an improved method and apparatus for calculating the performance of industrial heat exchangers for transferring heat from'one fluid to another.
The calculation of the performance of heat exchangers by the methods now in use is laboriousand time-consuming, and such calculations present a serious problem when it is necessary to take into account varying conditions Within the heat exchanger, such as varying specific heat and changing over-all thermal conductance. Varying over-all conductance is a particular problem when the heat transfer process involves radiation. Such would be the case, for example, if one of the fluids were superheated steam in turbulent flow where the conductance on the steam side of the exchanger consists in part of convection and in part of gaseous radiation, and wherein the radiation conductance is distinctly responsive to temperature conditions.
In accordance with the invention an electrical circuit is arranged to simulate the conditions existing within the heat exchanger. In order to do this the electrical circuit comprises a network of electrical resistances which are made proportional to the several thermal resistances (the inverse of thermal conductance) involved in the heat exchange process. The over-all electrical potential difference (voltage difference) of the circuit corresponds to the over-all temperature difference of the heat exchanger, and differences in electrical potential between different parts of the electrical circuit represent ditferences in temperature between corresponding parts of the heat exchanger. The electrical current resulting from the over-all potential difference, or voltage, impressed upon the circuit represents the over-all heat flow of the heat exchanger.
The thermal resistances of the heat exchanger comprise the resistance to the warming up (heat absorption) of the cold fluid which is a function of the specific heat of that fluid, the resistance to the cooling off (heat releasing) of the hot fluid, a function of the specific heat of that fluid, and the thermal resistance of the heat exchanger structure. The last includes the total series resistance to heat flow between source and receiver fluids. According to the present invention a method of calculating heat exchanger performance has been provided which includes the step of representing the heat absorbing capacity of the cold fluid and the heat releasing capacity of the hot fluid each by means of a resistance to the flow of electrical current. The thermal resistance of the heat exchanger structure is also similarly represented.
The invention will be better understood from the consideration of the accompanying drawings and the following description in greater detail. In these drawings 'Fig. 1 is a diagrammatic view of a counterflow fluidto-fluid heat exchanger;
Fig. 2 is a diagram representing the variations in temperature of the hot and cold fluids as they travel through the heat exchanger;
Fig. 3 is a diagram of an electrical apparatus having a circuit in simulation of the heat exchanger of Fig. l, or a section of such exchanger;
Fig. 4 is an illustrative development of Fig. 3;
Fig. 5 is a further illustrative development;
Fig. 6 is a diagram of an apparatus similar to that of Fig. 3, butwhich simulates the heat exchanger of Fig. 5 divided into five sections;
Fig. 7 is a modification of Fig. 6; and i Fig. 8 is like Fig. 2, but produced by calculation employing the apparatus of Figs. 6 or 7.
The invention will be described and discussed in connection with the calculations for the counterflow type of heat exchanger as that type introduces more problems than the parallel flow type. Referring to Fig. l, the hot or heat source fluid enters the heat exchanger at 10 and travels toward the right through the heat exchanger shell 11, leaving by connection 12. The cold or heat absorbing .fluid travels from right to left through the tube bundle,
which is represented by the single tube .13.
The conventional relationships for a two-fluid heat exchanger are given herewith:
=AUAt 1 representing the transfer of heat from a hot source fluid to a cold receiver fluid;
representing the heat involved in the temperature change of the cold fluid;
representing the heat involved in the temperature change of the hot fluid;
q=w c At =E At where q=time-rate of gross heat exchange, B. t. u./hr.
A=total exchanger area, ft.
U=over-all heat transfer conductance between source and receiver fluids, B. t. u./ (hr. sq. ft. per F.)
At =l igarithmie mean temperature difference (ideally),
h in c out) 1.4 h ou c in with t and t representing hot and cold substance temperatures, distinguished as to in and out conditions The temperature relationships may follows:
also be expressed as h ou n in where e =the base of natural logarithms Analysis of steady-state heat flow through complex structures has been accomplished extensively by means of electrical analogy (1, 2, 3) The field of usefulness for the electrical analogy approach may be greatly extended through application to heat exchanger steady-state proc- Where The concept of equivalent thermal resistance may similarly be applied to the heat exchanger fluids themselves. V i
Thus, Equations 2 and 3 may be rewritten:
where v a Rc=1/Ec, 'F/(B. t. 11.
=thermal equivalent corresponding to the ,cold sub- L stance water equivalent, representing the resistance to the warming-up operation Rh=l/Eh, F./(B. t. u. hr.)
=thermal resistance corresponding to the hot substance water equivalent, representing the resistance to the cooling-off operation Thus, in accordance with this extension of the conventional heat transfer resistance concept, Equation 4 may now be shown to include heat exchange resistance:
q=Atm/(R/A.)=Atc/Rc=Ath/Rn (9) Therefore I v .Atm=q(R/A) (10) Atc=qRc 11 Ath=qRn An electrical network maybe set up to simulate heat transfer in a fluid-to-fluid heat exchanger, illustratively, the counterflow heat exchanger indicated in Fig. 1. Herein electrical resistances are chosen proportional to thermal resistances represented in Equations l0, l1, and 12, and electrical potentials are taken as representative of temperatures. To facilitate operations, the over-all temperature difference (th in 'tc in) which is represented by an over-"all voltage diflerence, is considered as 100 percent; thus intermediate potential diiferences are indicated as some fraction of the over-all 100 percent. The equivalent over-all heat flow thus may be determined by means of the net thermal resistance represented by the net electrical resistance of the simulatingcircuit. Total current flow on the. basis of the over-all impressed direct current voltage may, on the basis of, appropriate circuit scale factors, indicate the over-all heat flow rate. Thus the electrical analogy analysis will permit the prediction of thermal performance,' but, it should be noted,flonly in terms of the assumed basic data. It is recognized that calculations could likewise be directly carried out on the basis of the corresponding thermal resistance circuit, but, as will be evident later, electrical experimental operations often are more readily executed. I
. For the purposes of simplicity, the countercurrent heat exchanger of Fig. 1 has been'considered as divided into five equal-area sections. It will be apparent that any number of sections to suit the desired accuracy may be employed, i. e., the pattern of the circuit equations makes it possible to set them up for n sections. Fig. 2 indicates the five sections.
' For any one'section, linear change in temperature is assumed, and consequently,'linear. change in electrical potential in the equivalent electrical circuit. The temperatu'redifference 'at' the midpoint, of the section s representative of the heat flow through the section (arithmetic mean temperature difference), that is, as due to the thermal resistance for the section of area A/5 for a system of five sections and gross area A. Thus the contribution to total heat flow for this section is given by This heat flow is likewise represented in the change in temperatures for both the-hot andthe cold fluids. The change in temperature for the fluids will be given by Equations 11, Al'c=(Aq)Rc, and 1 2, Atn =(Aq)Rh. Circuitwise, these relationships may be incorporated into a network arrangement as given in Fig. 3. 7
Since the total change in the fluid temperatures for the section is desired, and since the heat transfer energy coincides with the heat exchange energy for the fluids, the circuit may be modified as given in Fig. 4, shown for thermal conditions, and Fig. 5, for electrical conditions. At represents the over-all difference in temperature for one section, and Ae the corresponding overall difference in electrical potential for the simulating electrical analogy circuit.
Fig. 4- thus represents thebasic element in the translation to an electrical analogy circuit; Here Ac, represents the over-all potential difference, RH: and Re the equivalent electrical resistance correspondingto the heat exchange resistances Ril/Eh 'and Ra l/En, and R0 represents the total overall resistance of the'circuit. If Ru represents the equivalent electrical resistance .corresponding to the heat transfer net thermal. resistance R/A=1/ (AU), then if n sections are used, the equivalent electrical resistance corresponding to the section heat transfer resistance will be nRU, i. e., fora five-section system, SRU.v Thus, Ro=5Ru+'(RH+Rc)/2. Potential differences Ae and Ae represent, to scale, the temperature changes for the hot and cold fluids, respectively.
In Figs. 3, 4, 5, 6 and 7 a source of adjustable electrical potential is indicated at 14 and an ammeter at 15. Also a slide-wire potentiometer is shown at 16 together with agalvanometer 17 and a-probing contact 18. By means of this device the relative potential of any desired point in the resistance network can be quickly determined.
If both the heat transfer and the heat exchange fluid thermal resistances are regarded as constant over the entirearea'of the exchanger :(constant U and constant fluid specific heats), the' five-section electrical network for theheat exchangersimulation may be shown as in Fl j lhe diflere'nt electrical resistances Ry and represent ballast resistances to compensate for the successive and additive fluid potential differences accomplished in each section, such'that the total fluid temperature changes will be simulated by Am and Aec for the hot and cold fluids, respectively. The total flow is representedby the total circuit current I, representing the sum of the separate current contributions from each of the five sections. The dotted lines in the figure, running from section to section, connect points of equal potential, in accordance with the additive Ae principle cited above. 7 V
In accordance with the stipulation of constant U and constant fluid specific heats, the following conditions are inforce:
Values for Ry and Rx for the different sections are given by the following relationships, in which.
Ballast resistances R Ballast resistances Rx:
Fig. 6 has been calculated. tion has been assumed: Ec
EH=1000 B. t. u./(hr. F.), AU:
i. e., combining A and are as follows: Rc=200 ohms, RH:
relationships of Equatwork corresponding to The following basic informa- 1000 B. t. u./(hr. F.), U. Proportional electrical values 100 ohms, Ru==100 ohms, Ro=5(100) 100-1-200) /2=650 ohms. The circuit values are given in Table 1:
Table 1 Section 1 2 3 4. 5
1, 636 1, 341 1, 098 see Relative potential values for each section are calculated and shown in Table 2, indicative of the fluid temperatures. d cold fluids are 1.00 and 0,
The end values of the hot an respectively.
Table 2 H- Ae iz/Ae over-all (1-AeH)/Ae over-2.111.000.
200 0.222 0 (end) 349 The combined resistance of the network is 258 ohms.
The circuit of Fig. 6 may circuit (Fig. 7) in which Rx are eliminated. The valu fled so that they will in turn as to potential drop as before.
joined to the adjoining sectio R0 and RH beyond RC1 and R115, respectively,
those of their own sections. A
current in addition to separate resistance element RF be replaced by an equivalent the ballast resistances Ry and es of R0 and RH are modiproduce the identical results us. The
Each section is electrically circuit elements of carry extra Ran-Rm) will be Iidentified (Fig. 7) as herewith indicated: R.F=Ro
It is to be note basically constant, and therefore RF1=RF2=RF3=RF4==RF5=RF Values for the modified ver be computed from 13 and 14.
RH5'=RH, i. e., original value d that Ru, R0 and RH are On the basis of the preceding equations, a new network corresponding to Fig. 7 was set up. The values for this network, based on the data as used for the ballast network case reported in Tables 1 and 2, are shown in Apparatus corresponding to Fig. 7 as well as apparatus corresponding to Fig. 6 have been constructed and used in heat exchanger calculations. The experimental results obtained from electrical measurements on the network of Fig. 7, adjusted as to values of RE and Re, follow in Table 4, and are to be compared with those cited in Table 2. It is to be noted that the agreement is acceptable.
Table 4 Section H 1 2 3 4 5 Asa/Ac over-all 0.770 0.673 0.556 0.405 0.218 0(end) 1-(Aeg5Ae over-all 1.00 0.952 0.891 0.817 0.724 .606
The combined resistance of the network measured 257 ohms.
Corresponding values for the temperatures of the hot and cold fluids, calculated by use of the apparatus of the invention and based on conventional algebraic heat exchanger relationships for the same conditions as represented by Equation 5, are shown in Table 5.
Table 5 Section 1 2 3 4 5 Aea/Ae over-all 0.775 0.676 0. 550 0.405 0.227 00 (end) 1EAeg)/Ae over-all 1.000 0.951 0.889 0.815 0.726 0.612
f It is to be noted that Tables 2, 4, and 5 show good agreement, it being recognized that Tables 2 and 4 are derived from a lumped circuit arrangement. Fig. 8 represents the results of Table 4 on a temperature basis.
Values for Figs. 6 and 7 (Tables 1 and 3) were set up in terms of constancy of values for R0 and RH, as well as for RU, for all of the difierent sections. It is, however, to be recognized that the corresponding thermal values for R0, RH and R (i. e., U), are themselves a function of temperature conditions, and hence, with changes indicated in temperatures, not constant. Since, for example, R=1/ U, and is, as indicated, composed of the hot-fluid component, the wall component, and the cold-fluid component, R will be subject to the variations in these component resistances as dependent on temperature. Thus hFh=hc+hr (20) where hrn=composite boundary conductance, B. t. u./(hr. sq.
hc=boundary conductance due to convection (forced),
B. t. u./(hr. sq. ft. F.)
h;;equivalent conductance of steam gaseous radiation corresponding to the existent temperature conditions, ivy in B. t. u./ (hr .;sq. ft. F.) "I Accordingly, hm and Rm (=1/hFh) will be particularly subject to the temperature conditions, hence variable throughout the traverse of the area of the heat exchanger. R: and Rh in turn depend on the realized values of specific heat, i. e., the existing temperatures. The electrical analogy approach lends itself especially to the solution of the heat exchanger problem with varying properties, such as for example, introduced through the variation of the gaseous radiation cited above, etc. Accordingly, the solution of the five-section counter-current heat exchanger problem has been developed to deal with the varying properties, using the electrical analogy approach. 7 Based on the ballast circuit of Fig. 6, values for Rx and R throughout the five-section circuit have been set up, noting that RmRc, and R are nolonger constant, but are considered to change from section to section. Let it be noted that Ro1=5RUi+ (Rc1+RH1)/2, Ro2=5RU2+ (RC2+RH2)/2, etc.
Values for P and Q change as the sections are traversed. The values for M and N change fromsection to section. Thus M1 RO1R'H1,' Mz'=Ro2-Rri2, etc. N1'= Rel-RC1; N2=Rv2-Rcz, etc.
The ballast resistance values for Rx and Ry at the difierent sections are shown In the case of the circuits designed for variable properties, the procedure is to start with estimated valuesfor the diiferent properties, and on determinationof resultant temperatures on this basis, to correct the different values accordingly. Repeated corrections can be intro.-
duced until the accuracy requirement is satisfied.
It is to be noted that if RH=0, the resultant solution represents the heat exchanger case of constant temperature source, such as is represented by the ideal condenser analogy.
78 or feed water heater case. If Rc= 0, the resultant solution represents the heat exchanger case of constant temperature receiver, such as is found in arefrigeration brine cooler. It is believed that theindicated method of electrical analogy can likewisebe successfully applied to nonorthodox cases, suchascross-flow exchangers.
The following specific example is illustrative of the application of the method and apparatusof the invention. Let us assume that 1100 lb./hr. of liquid, specific heat 0.91, are to be cooled from an initial temperature of F., by brine, 400 lb./hr., initial temperature=0 F., specific heat: 1.25, in a countercurrent heat exchanger of 10 sq. ft. heating surface area, for which U the coetficient of heat transfer, under the flow circumstances, is 100. (a) For constant specific heats and constant value of U, show the final outlet temperatures of the two fluids, the variation of the fluid temperature throughout the exchanger, and thetotal amount of heat transferred per hour. (b) What would be the outlet temperatures for 200 F. initial hot fluid temperature, and 60 F. initial cold fluid temperature. V
.- Here (as shown in column 5, paragraph 1).
A five-section network may be set up to represent the heat flow in the exchanger by the method of electrical These represent the relative basic resistances.
In accordance with the analogy relationships of Equations 15 and 16, the five section network corresponding to Fig. 6 has been calculated. For convenience in use of resistor values, the different resistances shown above are represented in ohms, as multiplied by a' factor of 100,000. Thus, with the relative values retained, proportional electrical values are as follows: Rn=100 ohms; Rc=200 ohms; Ru: 100 ohms, and Ro=5(100) +(100+200)/2= 650 ohms.
The five-section circuit values are given in Table 1. The
measured overall resistance of the network constructed in accordance With-the values of Table 1 measures 258 ohms. By means of the potential dividing slide-wire device 16 and galvanorneter 17, the relativepotential at various points in the network with respect to the over-all (100%) may be determined. These values are shown in Table 2.
The thermal resistance corresponding to the measured over-all resistance of 258 ohms is 258/ 100,000=.00258.
For100 volts over-all potential corresponding to a temperature difierence of 1000=100 F., the current flow shows 1388amps. corresponding to 38,800 B. t. u./hr. For volts, corresponding to an over-all temperature difference of 20060=140 F., the current flow shows .542 amps. corresponding to 54,200 B. t. u./ hr.
The temperature of the fluids at any point may be established from the measured relative potential values obtained from the circuit. Thus, the temperature may be calculated tFlu1d=fcold, inltial-i-CAtover-all where C =relative potcntial value, with C=0 taken at the low or cold side of the circuit, and Atover-al1=difi6leI1CQ between incoming hot fluid and initial co ld fluid, tcold, initial. 7
Thus, based on Table 2 and t, 1main=0 F., the outgoing cold fluid temperature=tc, orif=0+.775 100=77.5 F. The outgoing hot fluid temperature'=0+0.611 100= 61.1 F. p
In a similar fashion, the temperatures of the fluids at the diflerent sections may readily be calculated.
Fig. 7 shows the plot of fluid temperatures as calculated above.
The equivalent values for the second case with 140 F. over-all temperature difierence are readily calculated. Thus, the outgoing cold fluid temperature will be: tnmm=60+.775 140=60+108.2=168.2 F. The outgoing hot fluid temperature=60+.61l 140=145.5 F.
in place of the five-section network as used above and corresponding to Fig. 6, the equivalent electrical circuit corresponding to Fig. 7 may be used with similar results to the above. In addition, although these two previous networks of Figs. 6 and 7 are predicated on constant values of Rn, Re and Ro, since average specific heats are dependent on the temperature ranges encountered, and the over-all conductance U may change with temperature conditions, the network circuit values corresponding to Fig. 6 may be modified on the basis of Equations 23 and 24, and corresponding to Fig. 7, by Equations 25 and 26.
These modifications permit the analogy network calculater of the invention to provide analytical results which would be extremely diflicult to establish by orthodox calculation methods. For example, the calculations just described require only a few minutes, and if it is desired to ascertain the effect of changes in specific heat or of the heat exchanger conductance, re-calculation using the new values of these factors can be done in a similarly short space of time.
1. In an apparatus for calculating the performance of a heat exchanger transferring heat between two fluids which are continuously changing in temperature as they pass through the heat exchanger but wherein, at any given point in the apparatus, the temperatures of the fluids and the intervening separating wall are substantially constant, a plurality of electrical circuits each representing a section of said heat exchanger, each of said circuits comprising a plurality of electrical resistances, said resistances corresponding respectively to the warming-up of the cold fluid passing through said exchanger, the structural thermal resistance of the heat exchanger and the thermal resistance to the cooling-E of the hot fluid passing through 10 said exchanger, the resistances in each of said circuits being arranged in series connection, each of said resistances being adjustable, said plurality of circuits being connected in parallel so as to cause the electrical currents flowing in them to be added together and thus represent the heat exchanger as a whole, means for measuring the total current flowing in said parallel circuits so as to determine the total heat. flowing between said fluids, and an adjustable potential source for impressing on said parallel circuits an electrical potential corresponding to the over-all temperature difierence in the particular heat exchanger to be calculated.
2. Heat exchanger calculating apparatus as set forth in claim 1 wherein means are provided for determining the electrical potential of any point in any of said circuits as an indication of the temperature of the corresponding part of the heat exchanger.
Reterences Cited in the file of this patent UNITED STATES PATENTS 1,206,968 Wilsey Dec. 5, 1916 2,040,086 Goodwillie May 12, 1936 2,519,615 Wannamaker Aug. 22, 1950 2,598,267 Kayan May 27, 1952 2,630,968 Muskat Mar. 10, 1953 FOREIGN PATENTS 605,822 Great Britain July 30, 1948 252,807 Switzerland Oct. 16, 1948 OTHER REFERENCES Heat-Transfer Problems Solved With Roomful of R-C Networks, Electronics, April 1943; pp. 181-183.
The Theory of Mathematical Machines, F. J. Murray, revised edition, Kings Grown Press, New York (pp. III- 17 relied upon).
The Accuracy of Measurements in Lumped R-C Cable Circuits as Used in the Study of Transient Heat Flow (Paschkis and Heisler), AIEE Transactions, volume 63, April 1944, page 165.