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I‘d. on]: ‘1).0J’r'3: r5 #838 SITY LIBRARIES ll \l llllllLollllll lllllllllll \lll 1293 ll l l .'| This is to certify that the thesis entitled TSCAD: A HEAT EXCHANGER ANALYSIS AND DESIGN SOFTWARE PACKAGE presented by JON THELEN has been accepted towards fulfillment of the requirements for MASTERS MECHANICAL ENGINEERING degree in @W/szz [Major professor Date Fe-AYUaf/LQ'Z /7C// 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution PLACE IN RETURN BOX to remove We checkout from your record. TO AVOID FINES return on or before due due. DATE DUE DATE DUE DATE DUE L_ll I l—_ll l MSU Ie An Affirmative Action/Equal Opportunity Institution cumulus-9.1 L TSCAD: A HEAT EXCHANGER ANALYSIS AND DESIGN SOFTWARE PACKAGE By Jon J. Thelen A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Mechanical Engineering ABSTRACT TSCAD: A HEAT EXCHANGER ANALYSIS AND DESIGN SOFTWARE PACKAGE By Jon J. Thelen A software package has been developed for the analysis and design of heat exchangers. A variety of heat exchanger types can be evaluated including double pipe heat exchangers, crossflow.heat exchangers, and fifteen different; compact heat exchangers. The program can do either a traditional sizing calculation or a rating calculation. It can also handle cases where both mass flow rates are unknown, the mass flow rate of one fluid stream and the exit: temperature of the other fluid stream are unknown, and the mass flow rate and exit temperature of the same fluid streanl are unknown. Overall heat transfer coefficients are calculated within the program using state of the art Nusselt number correlations. Property values for air, water, steam, and oil are calculated within the program from curve fit equations. Several examples are performed to demonstrate the use of the program. A user's manual for the software is included. Table of Contents List of Tables..........................................vi List of Figures........................................vii Symbols and Abbreviations...............................ix Introduction.............................................1 Literature Review........................................5 Overview of the Program................. .............. ..lo Background..............................................13 Effectiveness - NTU................................l3 Overall Heat Transfer Coefficient..................17 Fouling.......................................18 Pinned Surfaces.............. ...... ...........19 Unfinned Surfaces.............................20 Method of Solution......................... ....... ......23 Effectiveness - NTU................................23 Estimating the Overall Heat Transfer Coefficient...29 Internal Flow in a Circular Pipe.. ......... ...30 A Cylinder in Cross-Flow......................32 Flow Inside an Annulus................. ....... 32 Fin-Side Heat Transfer in Compact Heat EXChangerSeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee37 PropertieSOOOOOOOOO00.......OOOOOOOOOOOOOO....0000040 Experimental Verification of TSCAD Performance..........49 Closure... ..... .0........................ OOOOOOOOOOOOOOO 56 Table of Contents - Continued Appendix A Property Functions.........................59 Appendix B Graphs of Property Functions...............63 Appendix C Program TSCADOOOOOeeeeeeeeeeeeeeee00000000085 Appendix D - user's Gaideeeeeeeeeeeeeeeeeeeeeeeeeeeeeeell? List Of References.....0...O....................0.0.0.0126 LIST 0' TABLES Table 1: Representative fouling factors [9]..............19 Table 2: Heat transfer data for heating from the core tube of an annulus (Di/Do-O.50) Data from Rays and Leung [4]....................33 Table 3: Variables for Nu-CRen...........................35 Table 4: Properties of air...............................43 Table 5: Operating Conditions for Verification...........54 Table 6: TSCAD Program Verification Data......... ..... ...54 Table 81: Properties of Water.................... ..... ...66 Table 82: Properties of Dry Steam........................73 Table B3: Properties of Air..............................77 Table B4: Properties of Engine Oil.......................83 vi Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure LIST OF TIGURES TSCAD fIOWChart. . . O . O 0 . . O O ......... O ....... O O 12 Equivalent thermal circuit....... ...... ......18 EffiCiency Of straight fins . . . O . . . . . O . . . . O . O O 2 o Efficiency of annular fins of rectangular protile.O......O...........O.............O...20 Nu VSe Re for Pr=0070000000 ..... O... ..... 0.0.34 Dependence of C and n on Pr..... ..... ........36 Comparison of Equation 5.26 with Rays and Leung Data [4]............ .......... 38 Plot of Colburn Factor (j ) vs. Re From Rays & mndon, 1984 3]....O.......O....39 Specific heat of air.............. .......... .42 First third of temperature domain... ......... 42 Middle third of temperature domain.. ...... ...44 Final third of temperature domain. ........ ...44 Comparison of calculated vs. tabulatedCPdata....O......OOOOOOIOOOOOOOOOO48 Comparison of calculated U vs. measured U....55 Density of water........ .................. ...63 Specific heat of water. .......... . ....... ....64 Viscosity of water............ ............. ..64 Thermal Conductivity of Water ............. ...65 Prandtl Number of Water.. ...... . ..... ........65 vii Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 86: 87: 88: 89: 810: 811: 812: 813: 814: 815: 816: 817: 818: 819: 820: LIST 0' TIGURES mneity or stem............OOOOOOOOOO0.0.0.070 Specific heat of steam.. .............. .......71 Viscosity of steam..........................71 Thermal conductivity of steam...............72 Prandtl number of steam....... ........ . ..... 72 Density of air..... ......................... 74 Specific heat of air ........................ 75 Viscosity of air........... ............... ..75 Thermal conductivity of air ................. 76 Prandtl number of air.......................76 Density of oil..............................80 Specific heat of oil.. ...... .... ..... .......81 Viscosity of oil............................81 Thermal conductivity of oil.................82 Prandtl number Of 011........OOOOOOOOOOOOOOOBZ viii cmin SYNBOLS AND ABBREVIATIONS Meaning area of heat exchanger fin surface area unknown parameter in parameter estimation heat capacity constant in compact exchanger data maximum heat capacity minimum heat capacity cmin/Cmax specific heat change in enthalpy inside diameter outside diameter change in temperature effectiveness of a heat exchanger convective heat transfer coefficient enthalpy Colburn factor thermal conductivity degrees Kelvin kilograms kiloJoules length viscosity ix m1 '0 .0 qmax R”f RB SYNBOLS AND ABBREVIATIONS Meaning mass flow rate (2h/kt)1/2: variable in fin analysis Newtons: unit of work dummy variable: exponent in correlations fin efficiency fin effectiveness Number of Transfer Units Nusselt number perimeter of fin density Prandtl number heat transfer rate maximum heat transfer resistance due to fouling Reynold's number critical Reynold's number ~2300 for pipe flow' wall conduction resistance percent standard deviation in error analysis Stanton number temperature fin thickness overall heat transfer coefficient weighting factor in transition flow analysis X m1 Subscript AAAAAA )c )h )in )out )lam )turb SYIBOLS AND ABBREVIATIONS Kenning Watts sensitivity coefficient in parameter estimation Meaning cold fluid stream hot fluid stream inlet flow condition outlet flow condition laminar flow turbulent flow xi I. INTRODUCTION The areas of mechanical design and systems and controls have made considerable use of the capabilities of computers in recent years. The areas of finite element analysis and optimization of mechanical systems are two examples. The area of heat transfer, however, has not made such efficient. use of the vast potential ability of computers. There has been more of an effort to close the gap in computer-aided design between the heat transfer field and other areas of engineering lately. This package is a part of the push to increase the computer-aided design capability in heat transfer. TSCAD is a software package that was written with the intent of providing an expert system which could carry out the calculations involved in heat exchanger analysis and design. A heat exchanger is a device which exchanges energy, in the form of heat, between two fluids that are separated by a. solid wall. Heat exchangers can be classified according to the geometry of the flow. According to this method of classification, three types of heat exchangers are 1 2 considered in TSCAD: concentric tube, cross-flow, and compact heat exchangers. Concentric tube heat exchangers consist of two fluids flowing in a pair of concentric tubes. Cross-flow consists of transverse flow of one fluid over a tube containing the other fluid. Compact heat exchangers are characterized by a very large surface area to volume ratio, usually achieved by using a bank of finned tubes within a small volume. Typically, one of the fluids in a compact heat exchanger is a gas. A heat exchanger has six main operating conditions: the inlet temperature, exit temperature, and mass flow rate for each fluid stream. These operating conditions, along with the heat exchange area and the overall heat transfer coefficient, describe the performance of a heat exchanger. TSCAD has the ability to solve for up to three unknown parameters for a given heat exchanger problem. Many of the problems encountered can be solved by the use of effectiveness-NTU (E-NTU) relations. 8y constructing some simple algorithms and making use of the standard E-NTU relations, all problems involving just the operating conditions can be solved, as well as those that involve just: one of the variables U or A. In the case of both the heat exchange area (A) and the overall heat transfer coefficient (U) unknown, the overall heat transfer coefficient must be calculated using the kinematic and thermal properties of the fluid, the flow velocity, and the geometry of the two flows. Established 3 Nusselt number correlations were used to calculate the convective coefficient for internal flow in a circular pipe (Dittus-Boelter [1]) and cross-flow over a cylinder (Churchill 5 Bernstein [2]). Heat transfer data from Keys and London [3] were used for the Colburn factor correlations for compact heat exchangers. No widely accepted correlations are available for flow through an annulus with a convective condition at the inner surface. However, Keys and Leung [4] have assembled experimental heat transfer data for the condition of turbulent flow in an annulus with a constant heat flux at the inner wall. Since these data were: similar to that collected in the lab for an inside convective condition, it was used to construct a correlation: for heat transfer from an annulus in a concentric tube heat exchanger. The resulting equation, accurate to about 15% it! the worst case, is a function of both Reynold's number and Prandtl number and is similar in form to the Dittus-Boelter> correlation for internal flow. The third major portion of this work involves the fluiri properties. In order to simplify the software and to save the time spent in opening and closing files, the properties are calculated within the program rather than being stored in a data file. The process of changing from data files to property equations involved plotting the set of available property data and deriving an equation or set of equations that describe the property as a function of temperature. This was done for the density, specific heat, thermal 4 conductivity and kinematic viscosity of each of the four types of fluid, water, steam, air, and oil. To demonstrate the capability of the system, a number of problems were solved using TSCAD. These are problems taken from selected heat transfer textbooks, as well as problems that make direct use of data obtained in the laboratory. II. LITERATURE RBVIS' Recently, much work has been done in the area of computer-aided design of thermal systems. A good deal of that work is designed for the purpose of instruction. One such example is the set of software used by the mechanical engineering department at Mississippi State University [11]. The curriculum at MSU makes use of more than twenty programs for thermal science instruction. Six of these deal specifically with heat exchangers. Of the heat exchanger software, the program Ex is the simplest. HX contains explicit functions of effectiveness as a function of NTU, and NTU as a function of E for some common heat exchanger arrangements. The problems that can be solved are very simple: given E calculate NTU, and given NTU, find E. Ex is nearly identical to the E-NTU portion of’ program TSCAD. TSCAD contains the effectiveness equations in function EFF, and the NTU equations in function TRNS. A second heat exchanger program used at M50 is CFHX. CFHX deals with cross flow of a fluid over finned tubes containing a second fluid. The main application for this 6 program is in the analysis of compact heat exchangers. In TSCAD, COMPACT solves problems involving compact heat exchangers, while CROSSFLOW solves problems with transverse flow over a single bare tube. Problems in shell and tube heat exchanger design cannot, be solved by TSCAD yet. These types of problems, however, have been the subject of other heat exchanger software work. The curriculum at MSU uses one such program, STAN. STAN can. calculate the outlet temperature of both fluid streams of a shell and tube heat exchanger, as well as estimate the pressure drop. TSCAD can calculate both outlet temperatures for concentric tube, cross-flow, and compact heat exchangers, but does not have the ability to calculate pressure drops. Another example for shell and tube heat exchangers is the expert system created by wang and Soler [12]. This program uses human expertise to carry out computations involved in heat exchanger analysis, and all decisions depend both on empirical and scientific knowledge. Like TSCAD, it is divided into modules. Among these are modules dealing with heat exchanger type, materials, dimensions, component types, parameter determination and calculation. wang and Soler's program has the capability to perform mechanical design of a heat exchanger by making use of a data base containing ASME codes. The program uses four data bases, including one containing physical data. .7 Program MATCODE [13] is another example of the use of data bases. MATCODE stores information dealing only with metals. The properties are stored as functions of temperature, and the program can interpolate between values. The information from the data base is stored in a DOS file which can then be accessed by a Fortran program. This procedure is very similar to the way TSCAD originally accessed fluid properties. Program HEATER was created for the analysis and design of feedwater heaters through interactive use on personal computers. The Fortran program is driven by screen inputs, from which it predicts off-design point performance. Like TSCAD, established correlations are used to calculate the convective coefficient, h. TSCAD incorporates many of the assets which are considered the strong points of these other programs. TSCAI) uses the same effectiveness and NTU functions used by HX [11]. It can solve those problems of flow over finned tubes which involve available compact heat exchanger data, like CFHX [ll]. STAN [11] can compute the outlet fluid temperatures for shell and tube heat exchangers. TSCAD does this for concentric tube, cross-flow, and compact heat exchangers. A full expert system similar to TSCAD, is the program by Wang and Soler [12]. Empirical and scientific knowledge are used by both. Many of the modules in this program are similar to TSCAD as well, including heat exchanger type, materials, dimensions, component types, parameter determination, and calculation. Program HEATER estimates the overall heat transfer coefficient using empirical correlations, like TSCAD. There are also many facets in these programs which TSCAD does not possess. Some will be incorporated, others will not. CFHX [11] has the ability to calculate performance for flow over finned tubes. TSCAD is limited in this ability, since data for only fifteen compact heat exchangers is contained in the program. These are all of the circular tube, plate fin type. More compact heat exchanger data, as well as data for flow over banks of tubes, is a good area for future work on TSCAD. Hydraulic characteristics, such as the pressure drop in STAN, are not calculated in TSCAD. Also, the ability to change units is not present in TSCAD, as it is in MATCODE [13]. The program presently works only in SI units, but this adjustment to the program will require minimal effort. The ability to analyze shell and tube heat exchangers, present in STAN [11] and Wang and Soler's work [12], will certainly be added to TSCAD. The ability to utilize graphics in the presentation of output, as well as to assist with input is also an area for future work in TSCAD. Graphics are used by Wang and Soler [12]. The use of data bases, as in MATCODE, will probably not, be an area that TSCAD will expand into. It is felt that the 9 computation of properties using equations is a more efficient method of retrieving data. None of the literature reviewed mentioned any work in the area of concentric tube heat exchangers. The correlation for heat transfer from an annulus is unique work: in computer-aided design of heat exchangers. Also not found. in the literature is the work done in deriving temperature dependent property functions. The literature reviewed suggests some areas in which TSCAD could be improved, such as shell and tube heat exchangers and utilization of graphics. These improvements would surely enhance the performance of TSCAD. Many of the facets of the other software are already present in TSCAD, however. This fact, along with the unique components such as the annulus correlation and the property calculation, make TSCAD a valuable tool for the engineer involved in heat; exchanger analysis and design. ”l III. OVERVIE' 0' THE PROGRAM TSCAD was created with the intent of providing an expert system for the analysis and design of heat exchangers. The system can be used as a menu driven program for heat exchanger design, or as a subroutine for a larger thermal system analysis program. There are four components of TSCAD: the main program, the effectiveness-NTU calculation program, the overall heat transfer coefficient calculation, and the physical property' calculation. The main program is the menu-driven interface which collects the known data from the user. This portion of the program decides whether there is enough known information to solve the problem, and if so, calls the proper subroutine to begin the process. The effectiveness- NTU calculation portion includes the effectiveness and NTU functions themselves as well as the subroutines which make use of those functions. The overall heat transfer coefficient set of subroutines is partially menu-driven. It. calculates the overall heat transfer coefficient independent. of the effectiveness-NTU method,by making use of empirical correlations which relate the fluid dynamics, the fluid properties, and the geometry of the flow to the heat 10 11 transfer. Finally, the physical property computation consists of a set of equations, derived from curve-fits of published data, which calculate the fluid properties as a function of temperature. Figure 1 is a flowchart which shows the relationship between the subroutines in the different portions of the program. As shown, the main program receives the input and sorts for the problem type. The properties calculation portion is called by the main program. The properties are calculated by the functions CP, DENSITY, TCOND, and VISCOS. These data then are passed to the subroutines through the main program. The effectiveness-NTU program contains six subroutines: SIZING, RATES, ENERGY, ONE, TWO, and RATING: and two functions: EFF, and TRANS. These subroutines may be called directly by the main program or by another subroutine in the case of more complicated problems involving more than one unknown. The overall heat transfer coefficient calculation contains six subroutines. Subroutine OVERALL is the main program of this section, and all other subroutines are called from it. CONCENTRIC, CROSSFLOW, and COMPACT recieve data and calculate convective coefficients for the various types of heat exchangers. Subroutines RE and NU calculate the Reynold’s number and Nusselt number, respectively, for the various correlations. 12 THERMAL SYSTEM ANALYSIS PROGRAM l Heat Exchanger Design and Analysis ' Software Physical Property Data Base NTU - EFFECTIVENESS CALCULATION Overall Heat Transfer PROGRAM Coefficient Calculation Development of Ntu - Effectiveness Relationships Figure 1: TSCAD flowchart IV. BACKGROUND 4-1 W The most widely used method of heat exchanger analysis is the effectiveness - NTU method. The effectiveness of a heat exchanger is defined as the ratio of the actual heat transfer to the maximum possible heat transfer. The number' of transfer units, NTU, is a dimensionless parameter used for heat exchanger analysis. The method is based on the premise that the effectiveness of a heat exchanger is related to the number of transfer units (NTU) by some known function. Using this function, one can solve for any one unknown variable in the relation. The six operating conditions can be used to calculate the heat transfer rate and the effectiveness of the heat exchanger. If we assume that the heat exchanger is adiabatic, that is, no heat is lost to the environment, we may perform an energy balance on the heat exchanger and equate the enthalpy in to the enthalpy out. 13 14 'hih,out+‘cic,out ' ‘hih,in+'cic,in (4‘1) where Ila mass flow rate 1 = enthalpy ( )h - hot fluid ( )c a cold fluid This equation can be rearranged in the form of enthalpy change for each fluid. 'c‘ic,in'1c,out) 3 'h‘ih,out’ih,in) (4'2) For an ideal gas or an incompressible liquid, the enthalpy change can be related to the product of the specific heat and the temperature change, dT. di - cpdT (4.3) Air can be considered to be an ideal gas, and water and oil as incompressible liquids. The validity of these approximations can be shown by a thermodynamic analysis. Using equation (4.3) and assuming that the specific heat is constant over a small temperature range, equation (4.2) can be rewritten in terms of the operating conditions. (‘°p)c(Tc,in'Tc,out) ' ('Cp)h(Th,out’Th,in) (4'4) To simplify notation, we may define the heat capacity of a fluid as the product of the mass flow rate and the specific heat. C a mop (4.5) where n a mass flow rate cp = specific heat 15 Equation (4.2) can then finally be rewritten in a compact form to give an expression for the heat transfer rate q, between the two fluids. q ’ cc(Tc,in'Tc,out’ ' ch(Th,out'Th,in) (4'6) Equation (4.6) is a valuable relation, because the energy balance in equation (4.1) has now been rewritten in terms of just operating conditions and fluid properties. To calculate the effectiveness of a heat exchanger, the maximum heat transfer rate must also be known. To define the maximum heat transfer rate of a heat exchanger first consider a counterflow heat exchanger of infinite length. One of the fluids in this exchanger will achieve a temperature difference of the hot inlet temperature at one end and the cold inlet temperature at the other end. This is recognized as the maximum temperature difference possible. The fluid that will achieve this temperature difference is the one with the lower heat capacity, Cmin‘ For a given heat transfer rate q, the fluid with the maxinuun temperature difference (fluid A), must have a smaller heat capacity than fluid 8, since the two are inversely proportional. The maximum heat transfer rate can then be defined as the minimum heat capacity multiplied by the maximum temperature difference, which is the difference between the inlet temperature of the hot fluid and the inlet temperature of the cold fluid. 16 qmax ' Cmin(Th,in"Tc,in) (4.7) The effectiveness of a heat exchanger is then defined as the ratio of the actual heat transfer rate to the maxinuun heat transfer rate. E = q/qmax (4.8) where E - effectiveness q - actual heat transfer qmax - maximum heat transfer Having defined the effectiveness of a heat exchanger, the number of transfer units must now be defined. The number of transfer units (NTU) is a dimensionless parameter, developed by Rays and London, defined as the product of the overall heat transfer coefficient and the heat exchanger area divided by the minimum heat capacity. NTU = UA/Cmin (4.9) For any heat exchanger, the effectiveness is a known function of the NTU and the ratio of cmin and Cmax' a = f(NTU,Cmin/Cmax) (4.10) Thus, heat exchanger performance can be evaluated in terms of effectiveness - NTU functions, with the variables consisting of the six operating conditions, the overall heat: transfer coefficient, and the heat exchanger area. 17 4-2 9I1Il11_fl£35.11l31111_921111£113§ The overall heat transfer coefficient accounts for conduction and convection resistances between fluids separated by solid walls. To see the concept of the overall heat transfer coefficient, consider an analogy between the diffusion of heat and electrical charge. Just as there is an electrical resistance associated with the conduction of electricity, there is a thermal resistance associated with the conduction of heat. Electrical resistance is defined as the voltage drop divided by the current. For convection, the thermal resistance is similarly defined as the temperature difference divided by the heat transfer rate, q; Using Newton's Law of Cooling, Rt,conv can be written as Rt,conv - (Ts-Tf)/q - 1/hA (4.11) where Ts - surface temperature Tf - free stream fluid temperature q - heat transfer rate h - convective coefficient A - heat transfer area The associated thermal resistance for convection in a cylindrical wall is [10] where Do - outside diameter D - inside diameter L - length of the cylindrical wall k a thermal conductivity Fouling of the wall surfaces can cause another resistance tx; the transfer of heat. This process will be described in more detail later. For now, it can be defined as 18 Rt,foul ' 1"‘13/A (4'13) where R" - fouling factor - heat transfer area Assembling the different thermal resistances into an equivalent thermal circuit gives the total resistance. 1;;1 n“; 13 1% TkeTfiee , f qr-' ' WWW“. /-._-./ I'\-\ 1 L- .1. R," kh('t) R!" h. n'filuJ .H'I 0- Figure 2: Equivalent thermal cirCuito' The overall heat transfer coefficient may be defined as the inverse of the total resistance. l/UA - l/UcAc - l/UhAh (4.14) - l/(nohA)c + R"f'c/(noA)c + p‘, + R"f'h/(n°A)h + 1/(nohA)h where Rw - wall conduction resistance R": - fouling factor n - fin effectiveness 8 - area h - convection coefficient ( )h - hot fluid ( )c - cold fluid 4.2.1 Fouling The fouling factor, Rf! is a thermal resistance that results from fluid impurities, rust formation, and other reactions between the fluid and the wall material. This fouling, which results from normal heat exchanger operation, 19 can greatly increase the resistance to heat transfer between. the fluids. The value of Rt depends on the operating temperature, fluid velocity, and length of service of the heat exchanger. Table 1: Representative fouling factors [9] FLUID Rf" (m2 *K/W) —Seawater and treated boiler 0.0001 feedwater (below 50°C) Seawater and treated boiler 0.0002 feedwater (above 50°C) River water (below 50°C) 0.0002-0.0001 Fuel oil 0.0009 Refrigerating liquids 0.0002 Steam (nonoil bearing) 0.0001 4.2.2 tinned Surfaces For the case of a finned surface, the fin effectiveness. no is defined to be the ratio of the fin heat transfer rate to the heat transfer rate that would exist without the fin. The effectiveness can be calculated as no = 1 - (Ar/A) (1 nf) (4.15) where n: a fin efficiency A = fin area A = total surface area 20 For straight fins, such as rectangular, triangular and parabolic fins, as seen in figure 3 [10], the fin efficiency' I1: is Figure 4: "f - tanh(mL)/mL (4.16) where m - (2h/k.‘l:)1/2 L - length of fin k - thermal conductivity of fin h - convective coefficient t - fin thickness Figure 3: Efficiency of straight fins [10] (rectangular, triangular, and paraboIiC'prOfiles) -a..- ----—------— ----...-' ---_------..-_J — --—--—-—_—‘ - ---—-—--——---— — ———‘ m u an 23 anw‘" Efficiency of annular fins of rectangular: profile 21 For annular fins, as in figure 4 [10], the efficiency is nf = (PkAc/h)1/2(2 (rzz-r12))'1 (4.17) 4.2.3 Unfinned Surfaces For the case of an unfinned surface, the fin effectiveness is 1 and equation (4.14) reduces to 1/UA - 1/(hA)c + Rw + 1/(hA)h (4.18) Fouling effects have been neglected. For a cylindrical wall, the wall conduction can be calculated to be Rw - ln(Do/Di)/2 kL (4.19) where k a thermal conductivity of the fluid Do - outside diameter of pipe Di = inside diameter of pipe L - length of pipe For many heat transfer design problems, the cross- sectional dimensions of a heat exchanger are known, but the length is unknown. For this problem, equation (4.18) can be: rewritten as l/U = D[1/(hD)c + ln(D°/Di)/2k + 1/(hD)h] '(4.20) Equation (4.20) shows that all that is necessary for calculation of the overall heat transfer coefficient is a knowledge of the geometry (diameters) of the heat exchanger; fluid properties and the convective coefficients of the hot and cold fluids. To estimate the convective coefficient, 22 the correlations require information on the flow geometry, velocity, and fluid properties. V. NETNOD OF SOLUTION 5.1 IIIIQIIIlfllfi§_:_llfl_£AL£!LATIQNS A heat exchanger has six main operating conditions: the inlet temperature, exit temperature, and mass flow rate for each fluid stream. These six operating conditions, along with the overall heat transfer coefficient and the heat exchanger area, make up the variables that are important to the heat exchanger analysis and design programs Consider the case of any one of the six operating conditions being unknown. This condition can be directly solved for using the energy balance in equation (4.6). This: energy balance consists of only the six operating conditions and the fluid property cp. It is therefore a simple calculation to solve for one unknown condition. Subroutines ENERGY, RATING, and RATES solve for unknown inlet temperature, outlet temperature, and mass flowrate, respectively in TSCAD. ENERGY solves the simplest problem, that of unknown inlet temperature. The heat transfer rate, q, can be 23 24 calculated using the data from the known fluid. q = C(Tout-Tin) (5.1) The heat transfer rate q is then used to solve for the unknown inlet temperature. Tin 3 TOUt " Q/C (5.2) RATING solves a similar problem, that of unknown outlet temperature for one fluid. This routine uses the known heat capacity data, Cc and Ch, along with the overall heat transfer coefficient U, and the area A to calculate the NTTI for the problem. NTU = UA/Cmin (5.3) This information is then used to calculate the effectiveness; of the heat exchanger. E = f(NTU,Cmin/Cmax) (5.4) The maximum heat transfer rate qmax is a function of inlet temperatures and heat capacity, it is therefore known for this problem. The actual heat transfer rate q can be calculated from the effectiveness E and qmax' q = Eqmax (5.5) The outlet temperature is then calculated from the heat transfer rate, q. Tout = Tin - q/C (5.6) 25 The reason that RATING is not identical to ENERGY is that, in addition to solving for one exit temperature, it also is capable of solving for the exit temperature of both. fluids. RATING was designed to solve this more detailed problem. Subroutine RATES can be used to solve for the case of one unknown mass flow rate. For this case, the algorithm begins by calculating the ratio of temperature differences, which is known. It then uses the known temperatures to calculate the specific heat of each fluid. Knowing the temperatures and specific heats, the unknown mass flow rate: can be found. I( )(T-T) .c_ hcphoih= (5.7) (cp) h (To-Ti) c Each of the subroutines described above can also be used in conjunction with one or more other subroutines to solve for more than one unknown variable. The methods described above are used to solve for only one unknown operating condition. For the case of two of the operating conditions being unknown, such as both outlet temperatures, an outlet temperature and a mass flowrate, or both mass flowrates, an iterative procedure is used to calculate the unknowns. These situations were solved using RATING, subroutines ONE and TWO, and RATES, respectively. 26 RATING can calculate outlet temperatures when both are unknown. The procedure is identical to that described above, when just one outlet temperature is unknown. The effectiveness - NTU method is used to find the heat transfer' rate q. This is then used to calculate the outlet temperatures for each fluid stream, given the known inlet temperatures and mass flow rates. Since the outlet temperatures are unknown, some iteration. is required between the specific heat and the outlet temperature for each fludxi. Subroutine ONE uses an iterative approach to find the mass flow rate and outlet temperature when both are unknown: for the same fluid. The heat capacity C and the inlet and outlet temperatures of the known fluid are first used to calculate the heat transfer rate q. As a starting point, the known heat capacity is assumed to be cmin' The resulting effectiveness is then the ratio of the known temperature difference to the maximum temperature difference. The known heat capacity C is used to calculate, the NTU. A procedure then is begun which iterates on the ratio of heat capacities Cr Cr = cmin/cmax (5'8) and alternately calls the E-NTU function TRANS and checks to. see if the NTU ever becomes less than that calculated with the known heat capacity. NTU = UA/Ckno‘m (5.9) 27 If this happens, it means that the known heat capacity is the minimum. The CI. at this point is then used to calculate: the unknown heat capacity. Cmax (unknown) - Cain/Cr (5.10) If the given problem consists of an unknown mass flow rate for fluid A and unknown outlet temperature for fluid 8, subroutine TWO is called. TWO iterates between the two unknown operating conditions until the solution converges. The process is begun by calculating the known heat capacity, CB. This is the product of the known mass flow rate of fluid 8 and the specific heat of 8. This is assumed. to be Cmin to begin the iteration. Using U, A and this cmin' the NTU is calculated. Using this NTU, the effectiveness is also calculated. The unknown outlet temperature is then computed from the definition of the effectiveness. This outlet temperature is checked for convergence and the iteration is either finished or continued. The procedure used to solve for two unknown mass flow rates is very similar to that outlined earlier for one unknown flow rate. Subroutine RATES is again used. An iteration is performed by first calculating NTU, and then using the definition of NTU to find Cmin' Convergence is checked, and the procedure is either terminated or restarted. 28 The other possible unknowns, besides the six operating conditions, are the overall heat transfer coefficient U, and. the area, A. These are computed by using the operating conditions to calculate the heat transfer and the effectiveness, and then by calling the appropriate effectiveness - NTU function. NTU - r(a,cmin/cnax) (5.11) The known NTU is then used to calculate the unknown variable, U or A. U a NTUt in/A (5.12) A - NTU* in/U This is done in subroutine SIZING. The E-NTU functions are given in the functions EFF and TRNS. For the case of both the overall heat transfer coefficient and the area unknown, the effectiveness - NTU method alone is not sufficient to solve the problem. This situation motivates the need for a method of calculating the: overall heat transfer coefficient directly, which is described in the next section. Using this method, TSCAD can solve for three unknown conditions and parameters, provided that two of the unknowns are the overall heat transfer coefficient U and the heat transfer area A. For these cases, subroutine OVERALL is called and the appropriate geometric information is requested. These data are used to calculate U. Subroutine 29 SIZING then computes the corresponding area, given U. With U and A thus known, the proper E-NTU routine, as described in this section, can be used to find the remaining unknown operating condition. Each of the routines in TSCAD uses the fluid property information called by PROPS. All properties used are temperature dependent. For this reason, any routine that calculates a new inlet or outlet temperature also contains an iteration loop which computes the fluid properties at the: new average temperature, and then recalculates the unknown temperatures. 5.2 W Three types of heat exchanger configurations can be analyzed using the program TSCAD: concentric tube, cross- flow, and compact heat exchangers. For these three heat exchanger types four unique kinds of flow exist. All three: cases have a common condition of flow inside a cylindrical tube for one of the fluids. The other three types of flow are flow in an annulus for the concentric tube case, transverse flow over a cylinder in the cross-flow case, and. flow over a finned tube bank for the fin-side flow in a compact heat exchanger. The method of solution for the four types was similar. A correlation was found which expressed a dimensionless convective coefficient, such as the Nusselt number or the 30 Colburn factor, in terms of fluid properties, fluid dynamics and flow geometry. Nu - f(fluid properties,fluid dynamics,geometry) - 110/]! (5.13) The convective coefficient (h) was then calculated from the dimensionless coefficient (Nu) and used to calculate the overall heat transfer coefficient, U. 5.2.1 Internal Flow in a Circular Pipe For the case of flow in a circular pipe, correlations have been found which relate the Nusselt number to the Reynold's number (fluid dynamics), and the Prandtl number (fluid properties). The Dittus-Boelter equation [1] is a version of this correlation which is appropriate for turbulent flows, and is given as Nu - 0.023Reo'8Prn (5.14) where n-0.4 for heating n-0.3 for cooling The Dittus-Boelter equation has been confirmed experimentally for the range of conditions O.710,000 L/D>1O For laminar flows, the Nusselt number can be analytically determined for different boundary (wall) conditions. Nu - 4.36 for constant heat flux (5.15) Nu = 3.66 for constant wall temperature 31 The analytic result for a convective boundary condition: is not known exactly, but it is known that the solution for' the convective boundary condition must be bounded at the upper end by the constant heat flux solution, and at the lower end by the isothermal solution. The Nusselt number for this condition was therefore hypothesized to be the average of the constant heat flux case and the isothermal wall case. Nu - 4.01 (5.16) for convective wall condition, laminar flow Turbulent and laminar correlations thus defined, some relation valid in the transition region between laminar and. turbulent flow was needed. The solution for this problem was to use a weighted average of the turbulent solution and. the laminar solution. The transition region was assumed to begin at a critical Reynold's number of approximately 2300, and extend to about 10,000. Recr ~ 2300 ‘ (5.17) Denoting the weighting function W, the solution for the transition region was therefore Nu a “N“turb + (l-W)Nulan (5.18) where W - - (10000-2300) 32 5.2.2 A Cylinder in Cross-Flow The method of calculating the convective coefficient, h, for the outside of the crossflow condition is similar to that of the internal flow case. A single empirical correlation is used to calculate a Nusselt number from fluid dynamics, fluid properties, and geometric information. The correlation proposed by Churchill and Bernstein [2], which expresses the Nusselt number as a function of only the Reynold’s number and the Prandtl number, covers the entire range of Reynold’s numbers for which data is available. (0.62Re'5Pr'33)[l+(Re/282,OOO)'625]‘8 Nu = 0.3 + — ——————— -— [1+(0.4/Pr)°5“7]°2w5 —————— (5.19) 5.2.3 Flow Inside an Annulus The correlations for flow inside a circular tube and external flow over a cylinder are widely used and accepted. Experimental heat transfer data for flow inside an annulus is not as readily available. A theoretical correlation for' the Nusselt number at the inner wall of an annulus was derived from plots of available data collected by Rays and Leung. The data shown in table 2 were collected for an experimental condition of constant wall heat flux on the two sides of an annulus. The specific set of data used had a 33 Table 2: Heat transfer data for heating from the core tube of an annulus (D /D°-0.50) Data from Rays an Leung [4] \ Re -> 104 3 x 10‘ 10s 3 x 105 106 \ - _ ___ ____ — ___= =_=—_= Pr \ Nusselt_numbsrr 0.5 24.6 52.0 130 310 835 0.7 30.9 66.0 166 400 1080 1 38.2 83.5 212 520 1420 3 66.8 152.0 402 1010 2870 10 106.0 260.0 715 1850 5400 30 153.0 386.0 1080 2850 8400 100 220.0 558.0 1600 4250 12600 1000 408.0 1040.0 3000 8000 24000 condition of an adiabatic boundary at the outer surface, similar to the assumption for a concentric tube heat exchanger, and a given heat flux at the inner surface. This data was presented and examined to attempt to find a functional dependence on temperature. It was hypothesized that the dependence should be a power law type of function, similar to the Dittus—Boelter equation. Nu = ClPrmRen (5.20) Using the Dittus-80elter correlation as a starting point, the constant C1 which produced the best fit to the data was found. Since the data was presented for a given Prandtl number and Reynold's number, the functional dependence on either parameter could be easily isolated. The process was begun by plotting Nusselt number versus Reynold's number for each given Prandtl number. The plots were all similar to figure 5, which is for the case of Pr-0.7. 34 Using the Macintosh software Cricket Graph, a curve fit was performed on these plots. The functional dependence for the curve fit was of the form Nu - CRen (5-21) The Prandtl number dependence is absorbed in the constant c: in equation 5.21, since each plot was taken for a constant Pr. Table 3 shows the resulting values for C and n for each. Prandtl number. Nu vs. Re Pr - 0.7 120m- 1000:: 000+: E coo: 400:» 200.. 18000 30000 10.3.. 306000 1000000 Figure 5: Nu vs. Re for Pr=0.7 Plotting the two variables, C and n, revealed a definite dependence on the Prandtl number for both. 35 Table 3: Variables for Nu=CRen Pr C n 0.5 0.0198 0.7676 0.7 0.0234 0.7741 1.0 0.0258 0.7871 3.0 0.0341 0.8179 10.0 0.0398 0.8534 30.0 0.0498 0.8696 100.0 0.0652 0.8796 1000.0 0.1148 0.8851 Examination of the nearly straight line behavior of C between Pr-1.0 and Pr-100 suggests a power law relation between C and Pr, since the scale of the x-axis over this interval is similar to a logarithmic scale. To examine this theory, another curve fit was performed on the plot of C vs. Pr. The same functional behavior was assumed, and this led to the following values of C1 and m for equation 5.20. c = 0.024epr0-2174 (5.22) where C1 = 0.0248 m = 0.2174 Next, the plot of n vs. Pr was examined more closely. The plot appears to approach a value of about 0.9 asymptotically at large Reynold's numbers. An exponential relationship with the Prandtl number was assumed. n = 81 + 82exp(-Pr) (5.23) Using an ordinary least squares method of parameter estimation, 81 and 82 were found. In order to keep the 36 C auad.11 vs.lPr (NuI-(nhpn) 042v . = . 703 '01! 040" "02 DEB" '05 "015 :3 035+ a '04 03“” :0:5 '02 Oflfll "OJ : f : : : 00 ‘Qosli 03' 1 S: Pr 10 80 100 1000 Figure 6: Dependence of C and n on Pr value of the exponential to a number of significant digits that could be handled by the computer's memory, the Prandtl number was divided by 10, so that the new expression for n was The estimated parameters were 8 = 0.8833 2 - -0.1095 (5.25) The final correlation was then for the Nusselt number at the: inner boundary of an annulus is Nu - 0.024spr-2174Ren (5.26) n = 0.8833-0.1095exp(-Pr/10) 37 Figure 6 shows a comparison of the correlation in equation (5.26) with the data taken from Rays and Leung for Reynold's numbers ranging from 104 to 106. The accuracy of equation 5.26 was calculated by expressing the error between the correlation and the true value (from Rays & Leung data [4]) as a percentage of the true value. Using this method the accuracy of the correlation ranges from 11.9% to 15.3%. As a comparison, the Dittus-Boelter equation, which was used as a starting point, is accurate to between 18.8% and 35.8%. 5.2.4 Fin-Side Heat Transfer in Compact Heat Exchangers The heat transfer data for compact heat exchangers was taken from the data in Csmnast_neat_nxchansers. 1984. by Rays and London [3]. The Colburn factor jg = Stpr°57 (5.27) is plotted against the fin-side Reynolds number on a log-log plot for each compact heat exchanger considered. In each case it is simply a straight line plot. 38 Nu vs. Pr (Bscnmdo 300m: 2500:: 2000" / 1000" 1000“ 500.. / // / . __________«—<='_;_——-—-fl—*' . . . . 0 0.0010303 0.01 0.03 0.5 0.7 1 S 10 30 100 1000 Fr "'8b-t0MX) 4*‘erNXMD *tiflr-NXMDO Nu vs. Pr (Rscnmdo 25000' 20000“ 15000' 10000" 5000‘: _’;“ L 0 0.0010005 0.01 0.05 0.5 0.7 i s 10 3'0 100 1000 Pr ‘t'RUdNMXMD' 4* Rm-HNMDOO Figure 7: Comparison of Equation 5.26 with Keys and Leung Data [4] 39 0.060 0.040 0.030 0.020 0.010 in 0.“)8 0.006 1m 2 3 4 6 a m4 2 3 Reynoids number. Re Tubeoutsidediameter. D, I 16.4 mm Fin pitch I 275 per meter Flow passage hydraulic diameter. Dh I 6.68 mm Fin thickness. r I 0.254 mm Free-flow area/frontal area. a I 0.449 Heat transter area/total volume. a I 269 nfllmJ Fin area/total am. A I 0.830 Note: Minimum free- area is in spaces transverse to flow. Figure 8: Plot of Colburn Factor (j ) vs. Re From Rays 8 London, 1 4 [3] 40 These plots were used to derive analytic correlations for the Colburn factor. jH a ClRen (5.28) where C1 I intercept of plot n I slope of plot C1 and n are stored in a data statement with the rest of the compact heat exchanger information. With these, it is possible to input the calculated fin-side Reynold's number and find the corresponding Colburn factor. i11__2£2211!2_£§1231§£128 Access to the fluid properties of water, oil, steam, and air is necessary for analysis of the heat transfer process in a heat exchanger. The necessary properties include the density, specific heat, thermal conductivity, and viscosity. All of these properties vary with temperature, at least to some degree. Initially, the program was set up to access a data file which contained the pertinent property information. The process of opening and closing the data file and of A interpolating between values within the file cost valuable computer time and also occupied too much memory and disk space. These factors led to the decision to replace the data files with equations which express the properties as functions of temperature. 41 A process of parameter estimation was used to derive the property equations from the tables of property data versus temperature. The process was begun by plotting the data from the tables. The plots were then examined to try to predict the functional dependence. A least squares method was used to estimate the parameters, given the desired functional form and the tabulated data. The resulting function, using the estimated parameters, was then checked for accuracy. Standard deviation of under one percent was desired for adequate accuracy. If the plot was inaccurate, or accurate for only a portion of the range as was often the case, the process was repeated for the range of temperatures that was unsatisfactory. The procedure is outlined in more detail as follows. .A set of property data, such as that for air in Table 4, is to be replaced by equivalent functions of temperature. Figure 9 shows a plot of the tabulated values of the specific heat of air. Examination of the plot reveals that the function must be of at least fourth order, because of the three changes in concavity. But, by breaking the data up into two parts, an equivalent function can be found which expresses the behavior in three separate equations, each of which is valid over approximately one third of the domain of temperatures. 8y solving the simpler set of second order polynomial problems, a more accurate equation can be found. CF (kl/km 0P (kl/km 42 Specific Heat of Air 1.7" Lav Lb" L4" LS" L2" 14” I 1 1 “ “ .200 300 000 700 EDD lflli1300IBOOIWUOIHKKl2flN123M32500 1inanwhuu:00 Figure 9: Specific heat of air Specific Heat of Air 1r<:8001I 1.00» 1.06" 1.07: 1.000 L00? 1.040 1.00 1.02 1.01 ' 1 . : : if e : t : : : : : : I: 1‘30 160 200 250 300 360 400 460 500 660 600 660 700 750 Tinqnwwhnu Figure 10: First third of temperature domain 43 Table 4: Properties of air Temp Density Specific Viscosity Thermal Heat Conductivity [K] [kg/cu.m] [J/k9*K] [N*s/sq.m] [W/m*K] 100 3.5562 1032 0.00000711 0.00934 150 2.3364 1012 0.00001034 0.0138 200 1.7458 1007 . 0.00001325 0.0181 250 1.3947 1006 0.00001596 0.0223 300 1.1614 1007 0.00001846 0.0263 350 0.9950 1009 0.00002082 0.0300 400 0.8711 1014 0.00002301 0.0338 450 0.7740 1021 0.00002507 0.0373 500 0.6964 1030 0.00002701 0.0407 550 0.6329 1040 0.00002884 0.0439 600 0.5804 1051 0.00003058 0.0469 650 0.5356 1063 0.00003225 0.0497 700 0.4975 1075 0.00003388 0.0524 750 0.4643 1087 0.00003546 0.0549 800 0.4354 1099 0.00003698 0.0573 850 0.4097 1110 0.00003843 0.0596 900 0.3868 1121 0.00003981 0.0620 950 0.3666 1131 0.00004113 0.0643 1000 0.3482 1141 0.00004244 0.0667 1100 0.3166 1159 0.00004490 0.0715 1200 0.2902 1175 0.00004730 0.0763 1300 0.2679 1189 0.00004960 0.0820 1400 0.2488 1207 0.00005300 0.0910 1500 0.2322 1230 0.00005570 0.1000 1600 0.2177 1248 0.00005840 0.1060 1700 0.2049 1267 0.00006110 0.1130 1800 0.1935 1286 0.00006370 0.1200 1900 0.1833 1307 0.00006630 0.1280 2000 0.1741 1337 0.00006890 0.1370 2100 0.1658 1372 0.00007150 0.1470 2200 0.1582 1417 0.00007400 0.1600 2300 0.1513 1478 0.00007660 0.1750 2400 0.1448 1558 0.00007920 0.1960 2500 0.1389 1665 0.00008180 0.2220 crud/km CP (kl/km 44 Specific Heat of Air 000 < '1' < 1000 x L4" 12‘ _ _ - * ‘ ‘ - L0“ 08" 081 040 03" I I l v o t : : :7 : : :7 : : : : : e. : :as: ‘500 800 1000 1100 1300 1800 1400 1800 1600 Thuanahnu Figure 11: Middle third of temperature domain Specific Heat of Air fflXlEI< T 1.0 1.04 1.4:: _ _ 1 - - ‘ 1.2" 1.0 0.0» 0.0:: 0.41: 03:: j \ v l L A l L l L 1 l l l I l l P I U I U U U U r N U U U U U I U I Figure 12: Final third of temperature domain 45 Once the type of function to use was determined, the problem became one of estimating the parameters in the equation. The method of ordinary least squares was used for the parameter estimation. This method begins by identifying the important parameters. With the function expressed as follows, the parameters 81, 82 and 83 are to be determined. 2 cp - 31 + azr + 83T (5.29) where i? = specific heat I temperature Having identified the parameters, the sensitivity of each was determined. The sensitivity, 2, is defined as the: rate of change of the property with change in the parameter; xi = dcp/dBi (5.30) The sensitivity matrix has three component vectors, :1, x2 and 83. Expressing the specific heat data as a vector, and. the corresponding temperatures as another vector, the property function can be expressed in matrix form. cp - 31 + 32! + 83TTT (5.31) Or, equivalently, using indicial notation: 2 Cpi 3 81 + BzTi + B3T1 (5.32) The sensitivity of 81 is then the unity vector, each entry of which is one. The sensitivity of 82 is the temperature vector, T, and the sensitivity of 83 is the temperature squared vector, TTT. 46 With each of the sensitivity vectors thus defined, the ith component of the sensitivity matrix is xi - [1 Ti r12] (5.33) The parameter vector 8 is then defined to be, in terms of the sensitivity matrix, a - [x'xj‘1[x,r] (5.34) The computation of the 1’: matrix and the 1!! matrix is easily achieved by use of spreadsheet software. The x”: matrix, is T [n sum(Ti ) sum(T12 ) J r r I [sum(T 15 sum(T1 )sum(T 1)) sum(T)3um(T2 ] (5.35) [sum(Ti ) sum(Ti)sum(Ti) sum(Ti )sum Ti) 2)] where n I number of data points The corresponding 1T! matrix is .1. [ Sinai) J x Y = [sum(Yi )sum(T 1%) ] (5.36) [sum(Yi)sum(Ti) ] Each of these summations are carried out, followed by' the appropriate multiplications. The result of the parameter estimation for the first part of the temperature domain is 81 s 1040. 0 83 a 3. 838-4 For the domain of temperatures T < 800 K 47 the corresponding specific heat equation is cp - 1.0400 - 0.000217ar + 3.838-7*'r2 (5.37) The standard deviation, calculated from the tabulated data, is 0.0046. This is approximately 0.44% of the mean value for the thermal conductivity over this range of temperatures. The equations for the second and third portions of the: domain, with the computed parameters included are 0D - 951.0 + 0.1922*r - 4.83E-6*T2 (5.38) 800 < r < 1700 K cp - 2988.5 - 2.013tr + 5.916E-4*T2 (5.39) 1700 K < T The accompanying standard deviations are 0.0027 and 0.0096, or expressed as percentages 0.23% and 0.68%. Equations for the rest of the properties of air as well as the properties of water, steam and engine oil, were derived in a similar fashion. Appendix 8 has the complete set of property data for the four types of fluids used in TSCAD, along with the corresponding plots of properties as a function of temperature. The equation for each property is given in appendix A, along with the standard deviation expressed as a percentage of the mean property value over the domain, to offer an estimate of the accuracy of each function. 48 Specific Heat of Air Calculated and Tabulated LSV L0” L4” 1.8i _ GP (Id/kt!) L2" IJW 1 .-__-- 00 800 500 700 900 1uxi1300150017001em32nmizmx12500 umuian ‘”Thbdhhfl. '* Qdmfldmd Figure 13: Comparison of calculated vs. tabulated cp data VI. EEPERINENTAL VERIEICATION OP TSCAD PERFORMANCE TSCAD is a computer program that was designed for use on personal computers. It requires about 43.5 k8 of disk space in fortran form, and about 92 k8 of disk space in executable form. This is about 3.5% and 7.5%, respectively, of the space available on a 5.25 inch, 1.2 MB high density diskette. The executable form of TSCAD does not require a fortran driver, and can be used on any IBM-compatible PC. The program was tested on an IBM-compatible PC, running on a 286 processor. With the PC running at 12 MHz, the executable form of TSCAD can produce its results almost instantaneously, its computing time was estimated to be under one second. The fortran file can be used directly by utilizing an interpreter such as WATFOR77. In this form it takes longer to run, since the file has to be interpreted each time the program is used. Once the program is interpreted, the run time is comparable to the executable form of the program. Presently, the data for each test case must be entered by the user via the computer keyboard. This input time far exceeds any computing time required by the program. 49 50 This suggests a potentially valuable future improvement: to adapt the program to receive its data from data files, so that changes in one or two variables could be made without retyping the unchanged data. The experimental verification of the results obtained by TSCAD was an ongoing process occurring simultaneously with the development of the program algorithms. The data set to be presented as evidence of the program's ability was collected on February 20, 1990 and is for a double pipe heat exchanger in both parallel flow and counterflow configuration. The heat exchanger operating conditions were recorded for each trial. These operating conditions were then used to calculate the overall heat transfer coefficient U three different ways. The log mean temperature difference was used to find the ”true" heat transfer coefficient, since this method is exact for a double pipe heat exchanger. Next, TSCAD was used to calculate U by the effectiveness-N111 method and by using the correlations. 8y inputting the correct area and an unknown overall heat transfer coefficient, TSCAD calculates U using effectiveness-NTU. Finally, the heat transfer correlations within TSCAD were used to calculate the overall heat transfer coefficient. This latter approach is achieved by inputting both an unknown area and an unknown U into TSCAD. Several factors had to be considered for this particular experimental setup. First, since D/L, the ratio 51 of pipe diameter to pipe length, is relatively large (0.015), entrance effects may have a real impact on the heat. transfer. According to Burmeister [15], the average Nusselt. number for L/DI70 is increased by 10% over the fully- developed Nusselt number. Secondly, due to the constraints on the water supply, it is difficult to attain a Reynold's number in the annulus which is outside of the transition region (2300 < Re < 10000). These two constraints are recognized in the results. The experimental uncertainty was estimated to be between 5% and 10%, based on unsteadiness in the fluid flow' rates and in the inlet fluid temperatures. Four different combinations of mass flow rates were tested in both the parallel flow and counterflow configuration. A total of sixteen sets of data were then available. The operating conditions for each trial are shown in Table 5. The first eight sets of data shown correspond to a counterflow configuration, while the last eight sets are for parallel flow. The results, shown in Table 6, give the measured-heat transfer coefficient U, utilizing the log mean temperature difference method, as well as the results generated by TSCAD. These include results from the effectiveness-NTU method and results from the correlations for heat transfer' from an annulus and heat transfer from a circular pipe. Defining an error to be the difference between the TSCAD values and the actual (log mean temperature difference) 52 values, Table 6 shows the raw errors between each TSCAD method and the measured solution for each of the sixteen trials, as well as the average raw error and the average absolute error. The results show that the effectiveness-NTU method is accurate in most cases to within 1% of measured values for the overall heat transfer coefficient. The correlation results reflect the 10% increase in the Nusselt number due to the entry length effects. Also, it was realized that the linear lever rule hypothesis does not reflect the behavior of the Nusselt number in the transition region accurately. Instead, the turbulent correlation for heat transfer from an annulus was found to perform better over the whole range of Reynold's numbers. Therefore, this correlation was used for all annular flow. With these two points thus compensated for, the correlation predicted the overall heat transfer coefficient and the area with errors generally less than 10%, and averaging 6.3%. Figure 14 shows a comparison of the measured overall heat transfer coefficient with those calculated by TSCAD, both by the E/NTU method and by use of the correlations for flow through concentric tubes. The measured results are plotted on both the X and the Y axis, these data therefore produce a straight line. The TSCAD results are shown distributed about this line. The results show that TSCAD's accuracy is well within the experimental uncertainty for the double pipe heat exchanger data. As an extension of the program 53 verification, the case of fully turbulent annular flow and the effect of a thermal entry length on the heat transfer should be examined more thoroughly. 54 Table 5: Operating Conditions for Verification --Tsi ...... 392 ....... ’32 ........ '58: ....... 3’82 ....... ’31.}--- 16.59 23.31 0.1416 58.88 52.57 0.1416 10.25 16.59 0.1416 52.57 45.66 0.1416 16.07 22.02 0.1888 59.27 53.59 0.1888 10.06 16.07 0.1888 53.59 47.08 0.1888 15.49 21.28 0.2360 59.34 54.00 0.2360 10.06 15.49 0.2360 54.00 48.18 0.2360 15.23 20.66 0.2832 59.08 54.13 0.2832 10.06 15.23 0.2832 54.13 48.43 0.2832 11.31 17.23 0.1416 53.74 47.57 0.1416 17.23 21.44 0.1416 47.57 42.77 0.1416 11.04 16.33 0.1888 54.15 48.63 0.1888 16.33 20.28 0.1888 48.63 43.98 0.1888 10.71 16.07 0.2360 56.56 51.25 0.2360 16.07 20.23 0.2360 51.25 46.25 0.2360 10.52 15.70 0.2832 57.27 52.13 0.2832 15.70 20.02 0.2832 52.13 47.27 0.2832 Table 6: TSCAD Program Verification Data Log Mean Temp TSCAD TSCAD TSCAD ACTUAL Diff E / NTU Correlation Corr. U U ERROR U ERROR AREA AREA 2349.4 2363.3 0.6% 2119.5 -9.8% 0.0529 0.0474 2223.6 2206.6 -0.8% 1989.6 -10.5% 0.0526 0.0474 2654.4 2663.3 0.3% 2676.9 0.8% 0.0472 0.0474 2691.9 2674.2 -0.7% 2528.8 -6.1% 0.0501 0.0474 3153.2 3171.5 0.6% 3204.4 1.6% 0.0469 0.0474 2957.3 2942.3 -0.5% 3042.4 2.9% 0.0458 0.0474 3514.3 3535.6 0.6% 3708.5 5.5% 0.0452 0.0474 3351.0 3327.7 -0.7% 3532.3 5.4% 0.0447 0.0474 2055.7 2048.1 -0.4% 2013.0 -2.1% 0.0478 0.0474 2059.3 2033.3 -1.3% 2003.3 -2.7% 0.0477 0.0474 2358.0 2350.4 -0.3% 2549.3 8.1% 0.0434 0.0474 2371.8 2339.5 -1.4% 2535.8 6.9% 0.0434 0.0474 2776.5 2778.8 0.1% 3092.8 11.4% 0.0423 0.0474 2856.0 2813.4 -l.5% 3077.6 7.8% 0.0431 0.0474 3134.8 3136.9 0.1% 3598.7 14.8% 0.0411 0.0474 3418.6 3387.2 -0.9% 3582.6 4.8% 0.0445 0.0474 average error: -0.4% 2.4% average absolute error: 0.7% 6.3% 55 Verification of TSCAD Accuracy m" I 8600’ ““4 8200' 8000’ 230°- 230°- IIOO‘ 22°00 3°00 lJlnsunusd ' tiachud + Oil/NUS ' Iicmwnflnflons Figure 14: Comparison of calculated U vs. measured U VII. CLOSURE The work presented here, which can be used to accurately solve for up to three unknowns in the data set, was intended to offer an easy and fast method of analyzing and designing heat exchangers by making use of the advantages that computers offer. One of the big advantages of the program is speed. Iterations are performed quickly and accurately. Evaluation of property data was sped up considerably by replacing data files with equations that calculate the property as a function of temperature within the program. This computation time is considerably less than that spent in opening, closing and reading data files. The program also offers the option of calculating the overall heat transfer coefficient using geometric data and empirical correlations. This option makes use of a correlation that was derived by performing a curve-fit on a set of published experimental data. The work done in deriving a correlation to describe the Nusselt number as a function of the Reynold's number for flow in an annulus is 56 57 important in that it is not a subject for which a commonly accepted solution is available. The work has remained very compact, since the full set of required data is contained within the single program. This includes all the required compact heat exchanger data, the thermal conductivity data for different types of pipe material, and the fluid properties mentioned earlier. The work as presented does leave considerable room for additional improvement, however. One possible improvement would be expanding the capability of the system to include the ability to analyze shell-and-tube and other types of heat exchangers. Another area that the system could be expanded into is performing flow characteristic analysis. Presently, the program is concerned only with the thermal characteristics of heat exchangers. The addition of an optimization option would greatly enhance the ability of TSCAD. This would require considerable effort, but the potential benefits are great. While the addition of the aforementioned improvements could improve TSCAD, the system, as presented, can be a valuable tool for the engineer concerned with heat exchanger design and analysis. TSCAD can calculate the solution of any well-posed heat exchanger problem within its domain quickly and accurately, and it will hopefully be a valuable 58 addition to the area of computer-aided design of thermal systems. APPENDIX A: Property Functions The property functions given here were all derived by’ the methods given in chapter 5.3. The standard deviation, s, for each function is the error between the calculated function and the tabulated values, taken from [5], [6], [7], and [8]. This error was calculated as the standard deviation of the function divided by the average value of the property over the given domain. The functions for the density, viscosity, specific heat: and thermal conductivity are given below. Following the functions are tables which compare the values of the properties over a range of temperatures. 59 60 Dennis! water: T < 500 K p - 0.985 + 0.0215*exp[(T-275)/100] 500