THERMOD, an Enhanced Thermal Model for Determining Aircraft Operational Temperatures

Transcription

1 DOT/FAA/AR-4/51 Offie of Aviation Researh Washington, D.C THERMOD, an Enhaned Thermal Model for Determining Airraft Operational Temperatures Deember 24 Final Report This doument is available to the U.S. publi through the National Tehnial Information Servie (NTIS), Springfield, Virginia U.S. Department of Transportation Federal Aviation Administration

2 NOTICE This doument is disseminated under the sponsorship of the U.S. Department of Transportation in the interest of information exhange. The United States Government assumes no liability for the ontents or use thereof. The United States Government does not endorse produts or manufaturers. Trade or manufaturer's names appear herein solely beause they are onsidered essential to the objetive of this report. This doument does not onstitute FAA ertifiation poliy. Consult your loal FAA airraft ertifiation offie as to its use. This report is available at the Federal Aviation Administration William J. Hughes Tehnial Center's Full-Text Tehnial Reports page: atlibrary.t.faa.gov in Adobe Arobat portable doument format (PDF).

3 1. Report No. DOT/FAA/AR-4/51 4. Title and Subtitle 2. Government Aession No. 3. Reipient's Catalog No. Tehnial Report Doumentation Page 5. Report Date THERMOD, AN ENHANCED THERMAL MODEL FOR DETERMINING AIRCRAFT OPERATIONAL TEMPERATURES Deember Performing Organization Code 7. Author(s) Nathan Govindarajoo, Ph.D, PE. 9. Performing Organization Name and Address 1889 Old Dixie Highway, #23 Vero Beah, FL Performing Organization Report No. 1. Work Unit No. (TRAIS) 11. Contrat or Grant No. 12. Sponsoring Ageny Name and Address U.S. Department of Transportation Federal Aviation Administration Offie of Aviation Researh Washington, DC Supplementary Notes 13. Type of Report and Period Covered Final Report 14. Sponsoring Ageny Code ACE-12, AGATE The FAA William J. Hughes Tehnial Center Tehnial Monitor was Peter Shyprykevih. 16. Abstrat An enhaned version of a thermal analysis omputer program alled THERMOD was developed for determining the maximum operating limit (MOL) temperatures of general aviation airraft. The projet was undertaken under a Federal Aviation Administration-AGATE sponsorship. This report is the seond of two reports prepared in onjuntion with this projet. Enhanements inluded program debugging and orretions as well as development of an alternate impliit finite differene method, whih is used in transient analysis. Numerial validation of THERMOD was undertaken with respet to the finite element methods. Good orrelation was found. The validated THERMOD an be used to determine the MOL temperatures of a typial airraft that has a low wing onfiguration. 17. Key Words 18. Distribution Statement Maximum operating limit temperature, Thermal analysis, Condution, Convetion, Radiation, Steady state, Transient state 19. Seurity Classif. (of this report) Unlassified 2. Seurity Classif. (of this page) Unlassified This doument is available to the publi through the National Tehnial Information Servie (NTIS), Springfield, Virginia No. of Pages Prie Form DOT F17.7 (8-72) Reprodution of ompleted page authorized

4 TABLE OF CONTENTS Page EXECUTIVE SUMMARY ix 1. INTRODUCTION Bakground THERMOD Geometry and Input Data Projet Objetives DEBUGGING AND PROGRAM CORRECTIONS BUG BUG BUG Algorithm Development for IBFDM NUMERICAL VALIDATION OF THERMOD Sample Problem Sample Problem Sample Problem Sample Problem Sample Problem Sample Problem SAMPLE PROBLEM Input Data Input File Output Files SUMMARY REFERENCES 6-1 APPENDIX A THERMOD CODE iii

5 LIST OF FIGURES Figure Page 1-1 A Shemati Presentation of THERMOD Overall Model Geometry and Typial Temperature Profiles at Critial Loations Input Thermal Properties for THERMOD Model A Typial Flight Profile During Transient Cooling Control Volumes for Developing Thermal Equations Contour Plot of the FEM Output of Sample Problem Contour Plot of the FEM Output of Sample Problem Contour Plot of the FEM Output of Sample Problem Contour Plot of the FEM Output of Sample Problem Contour Plot of the FEM Output of Sample Problem Contour Plot of the FEM Output of Sample Problem LIST OF TABLES Table Page 3-1 THERMOD Input File (input.dat) of Sample Problem THERMOD Output File (summary.dat) of Sample Problem THERMOD-Simulated Temperatures ( F) of the Right Wing of Sample Problem Finite Element Method Input File of Sample Problem Trunated FEM Output File of Sample Problem Finite Element Method-Simulated Temperatures ( F) of the Right Wing of Sample Problem THERMOD Output File (summary.dat) for Sample Problem THERMOD-Simulated Temperatures ( F) of the Right Wing of Sample Problem Trunated FEM Output File of Sample Problem Finite Element Method-Simulated Temperatures ( F) of the Right Wing of Sample Problem iv

6 3-11 THERMOD Input File (input.dat) of Sample Problem THERMOD Output File (summary.dat) for Sample Problem THERMOD-Simulated Temperatures ( F) of the Right Wing of Sample Problem Finite Element Method Input File of Sample Problem Trunated FEM Output File of Sample Problem Finite Element Method-Simulated Temperatures ( F) of the Right Wing of Sample Problem THERMOD Input File (input.dat) of Sample Problem THERMOD Output file (summary.dat) for Sample Problem THERMOD-Simulated Temperatures ( F) of the Right Wing of Sample Problem Finite Element Method Input File of Sample Problem Trunated FEM Output File of Sample Problem Finite Element Method-Simulated Temperatures ( F) of the Right Wing of Sample Problem THERMOD Input File (input.dat) of Sample Problem THERMOD Output File (summary.dat) for Sample Problem THERMOD-Simulated Temperatures ( F) of the Right Wing of Sample Problem Finite Element Method Input File of Sample Problem Trunated FEM Output File of Sample Problem Finite Element Method-Simulated Temperatures ( F) of the Right Wing of Sample Problem THERMOD Input File (input.dat) of Sample Problem THERMOD Output File (summary.dat) for Sample Problem THERMOD-Simulated Temperatures ( F) of the Right Wing of Sample Problem v

7 3-32 Finite Element Method Input File of Sample Problem Trunated FEM Output File of Sample Problem Finite Element Method-Simulated Temperatures ( F) of the Right Wing of Sample Problem Temperature and Assoiated Radiation Data Thikness of Thermal Elements of the Wing Thikness of Thermal Elements of the Fuselage Side Thikness of the Thermal Elements of the Floor Thikness of the Thermal Elements of the Roof Overall Dimensions of the Thermal Model Absorptivity and Emissivity Properties of the Exterior Surfaes of the Wing and Fuselage Thermal Properties of Various Solid Materials Thermal Properties of Air Misellaneous Properties Input File (input.dat) for the Sample Problem Sample Problem Output Data File (summary.dat) Sample Problem Output Data File (transient.dat) 4-11 vi

8 LIST OF ACRONYMS EFFDM FAA FEM IBFDM MOL SPC Expliit forward finite differene method Federal Aviation Administration Finite element method Impliit bakward finite differene method Maximum operating limit vii/viii

9 EXECUTIVE SUMMARY Composite airraft strutural elements are unfavorably affeted by an inrease in temperature due to exposure to the thermal environment. For design purposes, the synergisti effets of extreme ambient temperatures and the aompanying solar radiation should be taken into onsideration in determining the maximum operating limit (MOL) temperatures experiened by the strutural elements. Allowable design properties of omposite materials may then be generated based on these MOL temperatures. THERMOD is a omputer model that was developed for determining the MOL temperatures for airraft under various paint shemes. The seleted low-wing geometry is best suited to general aviation airraft. In determining the MOL temperatures, THERMOD onsiders the effets of radiation, onvetion, and ondution. Radiation inludes diret solar radiation, infrared sky radiation, and their refletions from the tarma. Infrared emission from the surrounding strutures, inluding the tarma, wings and the fuselage and its interation among these strutures, are also onsidered. Convetion due to the wind, as well as turbulent onvetion within the abin, is also simulated, as are one-dimensional ondution through element thikness. The model inorporates the effet of fillets at the wing-fuselage juntion. Due onsideration is also given to the greenhouse effet within the abin, whih allows for a realisti modeling of the thermal environment within the abin. The effet of shade underneath the wing is also modeled. The soaked temperatures of an airraft are predited based on steady-state assumptions, giving onservative steady-state temperatures. Beause limit loads generally our in flight onditions, a transient (unsteady-state) thermal analysis is used to simulate the thermal onditions while the airraft exeutes the following maneuvers: taxi, takeoff, limb, and ruise. These maneuvers are ooling effets that in addition to intentional ooling of the abin through opening the door simulate a more realisti thermal environment for prediting the MOL temperature. THERMOD formulates a total of 67 independent equations within a nonlinear system of equations. The nonlinearity is introdued through radiation effets and through onvetive properties with the abin, whih are modeled as temperature-dependent. This system of equations is solved using the Newton-Raphson iteration tehnique. This report is the third of four reports prepared in onjuntion with this projet. The fourth report is entitled THERMOD User s Manual, and it will serve as an instrutive guide for users who wish to use THERMOD for determining airraft MOL temperatures. The first two reports validated THERMOD analyses with test data and studied the sensitivity to input variables. The enhaned version of THERMOD was developed for simulating the MOL temperatures of general aviation airraft. Enhanements inluded program debugging and orretions as well as development of an alternate impliit finite differene method. This method for solving the transient problem is unonditionally stable over all time and spatial domain as opposed to the existing expliit forward finite differene method. Numerial validation of THERMOD was undertaken with respet to the finite element methods. Good orrelation was found. ix/x

10 1. INTRODUCTION. An enhaned version of an original thermal analysis omputer program alled THERMOD [1] has been developed for simulating maximum operating limit (MOL) temperatures for general aviation airrafts. This projet was undertaken under a Federal Aviation Administration (FAA)- AGATE sponsorship. This report is the third of four reports prepared in onjuntion with this projet. The fourth report, THERMOD User s Manual, serves as a guide for users who wish to use THERMOD for determining airraft MOL temperatures [2]. The first two reports validated THERMOD analyses with test data and studied the sensitivity to input variables [3 and 4]. To help the reader get aquainted with THERMOD, bakground information on pertinent thermal aspets of the program and input data requirements are presented. This is followed by a statement of the projet objetives. A more omplete desription of THERMOD and the development of model equations an be found in the original THERMOD doument [1]. 1.1 BACKGROUND. Strutural omponents strength and sharpness of omposite airframe are unfavorably affeted by an inrease in their temperatures due to exposure to the thermal environment. For design purposes, the ombined effets of extreme ambient temperature and the aompanying solar radiation should be taken into onsideration in determining the MOL temperatures experiened by the strutural elements. Design properties of omposite materials may then be generated based on these MOL temperatures. THERMOD was developed for determining the MOL temperatures of low-wing airraft under various paint shemes. In determining the MOL temperatures, THERMOD onsiders the effets of radiation, onvetion, and ondution. Radiation inludes diret solar radiation, infrared sky radiation, and their refletions from the tarma. Infrared emission from the surrounding elements, inluding the tarma, wings, fuselage, and its interation among these elements, are also onsidered. Convetion due to the wind as well as turbulent onvetion within the abin is also simulated, as are one-dimensional ondution through element thikness. The model inorporates the effet of fillets at the wing-fuselage juntion. Due onsideration is also given to the greenhouse effet within the abin that allows for a realisti modeling of the thermal environment within the abin. The effet of shade underneath the wing is also modeled. In addition to the above fators, the following assumptions are made in THERMOD: (1) nonpartiipating medium (air); (2) disretized spae and time domains; (3) disretized elements being isothermal, opaque, diffuse, gray and haraterized by uniform radiosity, irradiation, and material properties; (4) nonopaque materials suh as windows and windshields are onsidered transparent with assoiated transmissivity values; and (5) onstant material properties with respet to time (and, hene, temperature). These assumptions are neessary in simplifying the omplexities involved in a three-dimensional thermal problem being addressed. The soaked temperatures of an airraft are predited based on steady-state assumptions, giving onservative steady-state temperatures. Beause limit loads generally our in flight onditions, 1-1

11 a transient (unsteady-state) thermal analysis is used to simulate the thermal onditions while the airraft exeutes the following maneuvers: taxi, takeoff, limb, and ruise. These maneuvers are ooling effets that in addition to intentional ooling of the abin through opening the door simulate realisti thermal environment for prediting the MOL temperature. THERMOD formulates a total of 67 independent equations within a nonlinear system of equations. The nonlinearity is introdued through radiation effets and through onvetive properties with the abin, whih are modeled as temperature-dependent. This system of equations is solved using the Newton-Raphson iteration tehnique [5]. As previously noted, THERMOD onsiders all three heat transfer mehanisms (onvetion, ondution, and radiation) normally assoiated with an airraft parked in the open. Figure 1-1 is a shemati representation of a thermal environment showing short wavelength solar radiation, long wavelength sky (infrared) radiation, onvetive heat transfer due to the wind, and onedimensional ondution (through the thikness). THERMOD also onsiders refleted solar energy between the fuselage and wing, diffused solar energy refleted from the tarma, and infrared refletions. The fillets and greenhouse effets are onsidered as well. Beause THERMOD simulates a total of 67 independent equations, 67 unknowns are needed. These unknowns are 53 temperatures and 14 radiosites. The 53 temperatures (T1 through T53) and their loations are noted in figure 1-1. FIGURE 1-1. A SCHEMATIC PRESENTATION OF THERMOD 1-2

12 Note that not all the unknown temperatures are shown. T2, for example, is loated immediately below T1 at the skin-ore interfae (see figure 1-2 for further larity). Similarly, T27 is loated immediately below T26 at the insulatory material B-omposite floor interfae. The geometry of the airraft shown is low wing with sandwih onstrution for skins. The sandwih an be either foam or honeyomb. Nonsandwih skins are also admissible if the thikness of the ore is assumed to be very small. A typial THERMOD analysis begins at the steady-state phase and ontinues on to the transient phase; a phase in whih ooling is introdued due to airraft maneuvers. At the point of appliation of limit or gust load, the temperatures at all 53 loations are noted. Of these 53 temperatures, 9 temperatures are onsidered nonstrutural. These nonstrutural temperatures are the three tarma temperatures (T51, T52, and T53); four insulatory material temperatures (T25, T26, T43, and T44); and two transparent material temperatures (T49 and T5), leaving 44 temperatures to be onsidered strutural. THERMOD repeats its analysis over different time periods, as requested. The maximum temperature over these time periods is then reported as the MOL temperature. This MOL temperature may be loated at the surfae (depending on how dark the paint is), inside the abin, within the floor spae, or loated anywhere else in the airraft. 1.2 THERMOD GEOMETRY AND INPUT DATA. Figure 1-1 showed a typial ross setion of a small general aviation airraft. A simplified rendering of the model is shown in figure 1-2. THERMOD is built on this simplified model. Figure 1-2 also shows typial setions at ritial loations. In addition to the overall geometry as indiated in figure 1-2, a set of input data haraterizing eah layer, as shown in figure 1-3, is also needed. The layer properties inlude thikness, density, thermal ondutivity, and speifi heat apaity. To onsider surfae effets, all exposed airraft surfaes are assigned their respetive absorptivity and emissivity values. These values are dependent on paint olor and surfae texture. The tarma is also onsidered a surfae, whose surfae and other thermal properties are required as well. THERMOD models the greenhouse and fillet effets. Transmissivity and fillet olor information is needed to onsider these two effets. The degree of greenhouse effet is also dependent on the perentage of transparent material. This information is input in the form of transparent surfae area in relation to the overall abin surfae area. To simulate onvetive oeffiients, the kinemati visosity and Prandtl number are needed. Beause the airraft is left out in the open, it is subjeted to the environment. The limati data of the environment, for eah time period, inludes ambient temperature, sky temperature, solar radiation, and wind speed. The final piee of information onerns the ooling that takes plae when the airraft exeutes the following maneuvers: taxi, takeoff, limb, and ruise. This data was furnished through a flight profile that is unique to a partiular airraft and is part of the input. The flight profile provides information on typial airraft maneuvering speed with respet to time, from whih, the time-varying onvetive oeffiient is determined. This oeffiient is then used in the subsequent transient ooling proess. An example flight profile is shown in figure

13 Thermal model: Top view T1 T2 T3 T4 T5 Foam Spar ap Outer Skin Inner skin Air spae Detail a: Wing L flg L abin Fuselage Wwing L ledge Wing T6 T7 T8 T9 T1 Spar ap Foam Inner skin Outer skin L wing T43 T42 T41 T4 T39 H abin d b a Outer skin F Insulatory o material C a Detail b: m Fuselage side H wing W abin Tarma Inner skin Thermal model: Front view T26 Insulatory material B T27 T28 T29 T3 Outer skin Foam Inner skin Floor T21 T22 T23 T24 Outer skin Foam Inner skin Floor spae T25 Insulatory material A Detail d: Roof T35 T36 T37 T38 Inner skin Foam Outer skin Bellypan Detail : Floor FIGURE 1-2. OVERALL MODEL GEOMETRY AND TYPICAL TEMPERATURE PROFILES AT CRITICAL LOCATIONS 1-4

14 1-5 FIGURE 1-3. INPUT THERMAL PROPERTIES FOR THE THERMOD MODEL

15 Veloity (mph) V = mph V 1 =1 mph V 2 =1 mph V 3 = mph V 4 = mph V 5 =75 mph V 6 =11 mph V 7 =11 mph V 8 =19 mph V 9 =19 mph Ambient wind speed=9.5 mph Time V - V 1 =5 se V 1 - V 2 =12 se V 2 - V 3 =5 se V 3 - V 4 =6 se V 4 - V 5 =2 se V 5 - V 6 =5 se V 6 - V 7 =12 se V 7 - V 8 =6 se V 8 - V 9 =6 se V6 V7 Limit load appliation V 8 V9 Cruise Aelerate Climb V5 Aelerate Close Open door door Greenhouse ooling V Taxi V 1 V 2 V 3 Stop V4 Take-off Run Time (se) Ambient wind speed FIGURE 1-4. A TYPICAL FLIGHT PROFILE DURING TRANSIENT COOLING Fuselage is modeled as a box and treated as an enlosure in modeling the greenhouse effet. The fuselage roof is treated as a surfae with an equivalent transmissivity, allowing solar radiation to enter the abin. The insulatory materials (A), (B), and (C) are simplified representations of abin roof interior, abin floor, and abin side interiors. A total of 28 distint layers are onsidered. Eah layer, labeled P 1 to P 28 in figure 1-3, is assigned four properties, i.e., thikness, density, thermal ondutivity, and speifi heat apaity. The tarma is regarded as a speial layer and also haraterized by these four properties, as indiated in figure 1-3. The eight exposed surfaes are assigned absorptivity (α)or emissivity (ε) properties, depending on their surfae texture and olor. The wing, fuselage, and the interior (B) and (C) surfaes are assigned satter fators (sf). The airraft is subjeted to limati fators that inlude solar radiation (Q sol ), wind (V wind ), ambient temperature (T amb ), and sky temperature (T sky ). Cooling is assumed to our as soon as the door is opened. The greenhouse ooling duration is a funtion of the soaked steady-state ambient temperature of the abin and of the tolerable ambient temperature desired before losing the door. This tolerable temperature, arbitrarily seleted, is 1-6

16 input into THERMOD as ambabt. The MOL temperature is determined at the point when limit load is applied. This moment in time is loated at 395 seonds from point V, as shown in figure 1-4. The ambient wind speed is added to the flight speed to give the total relative wind speed. THERMOD uses this relative wind speed at seleted times to determine onvetive oeffiient, whih is a required input in the ooling proess. 1.3 PROJECT OBJECTIVES. The objetives of the projet were to Debug, identify, and orret the existing errors in THERMOD. develop and validate an alternate transient solution proess based on the impliit bakward finite differene method (IBFDM). The existing transient model in THERMOD is based on the expliit forward finite differene method (EFFDM), whih is not unonditionally stable. The solution to some problems may not onverge. An IBFDM, however, guarantees onvergene by virtue of it being unonditionally stable aross all time and spatial domain. undertake numerial validation of THERMOD, with respet to the finite element methods (FEM). The three modes of heat transfer, i.e., ondution, onvetion, and radiation, were investigated. In addition, both the steady-state analysis and the transient analysis were onsidered. solve a sample airraft problem. The problem entailed developing input data sets and determining the MOL temperature of a typial small airraft. 1-7/1-8

17 2. DEBUGGING AND PROGRAM CORRECTIONS. Several bugs were deteted in the older THERMOD program [1]. This setion disusses these bugs and presents remedial ations that were taken to orret them. 2.1 BUG 1. BUG 1 aused oasional program nononvergene. The problem was traed to subroutine updradio. This subroutine updates radiosity at eah time step of the transient analysis. Radiosity is defined as the total energy leaving a surfae, whih inludes both infrared refletion and emission. There are a total of 14 radiosity funtions in THERMOD. Two of these radiosity funtions were found to be in error. The infrared emission terms in these funtions were inadvertently left out. The subroutine is alled from either subroutine trnsien1 or trnsien2, depending on whether the EFFDM or IBFDM is seleted. Subroutine updradio in turn alls subroutines ludmp and lubksb. The two radiosities affeted were J 12 and J 14. In both funtions, the infrared emission effet of the tarma were inadvertently left out, whih are shown below: funtion J 12 : sig*t(51)**4/((1-emista)/emista) funtion J 14 : sig*t(53)**4/((1-emista)/emista) These errors have been fixed by inluding the terms in the appropriate loations. Simulation runs of sample problems were found to onverge. 2.2 BUG 2. BUG 2 oasionally produed negative temperatures in some of the 53 temperatures that defined THERMOD, whih is not neessarily an error. In the solution proess, the nonlinear solution routine determines the temperature values that satisfy user input toleranes for temperatures or thermal funtions. As long as one of the toleranes are satisfied, the solution is said to have onverged, regardless of whether there are any negative temperatures found as part of the solution. Mathematially, the solution is orret. The variable-based onvergene riterion must satisfy the following ondition: 67 j = 1 abs ( X X ) 67 j = 1 j X i + j 1 i + 1 j i tolx (2-1) 2-1

18 where: X j i + 1 X j i = The urrent value of the unknownvariable X j = The previous value of the unknownvariable X j The funtion-based onvergene riterion must satisfy the following ondition: where: 67 j = 1 abs ( fn fn ) 67 j = 1 j i + 1 j i (2-2) fn j i + 1 tolf fn ji fn + 1 j i = = The urrent value of the funtion fn j The previous value of the funtion fn j In the older version of THERMOD, when either of these two riteria were satisfied, the solution was ompleted. Beause of this approah, both the toleranes were not satisfied at the same time. In the urrent version of THERMOD, both riteria must be satisfied simultaneously before the program exits. This is enfored through the statement if(errf.le.tolf.and.errx.le.tolx) goto 2 This dual enforement removed the problem of negative temperatures for sample problems onsidered. 2.3 BUG 3. BUG 3 prevented solution onvergene for ertain laminate thikness or when too small a time step was used. This problem ours only in the transient phase of THERMOD. This problem was aused by the EFFDM. This method is not unonditionally stable and is the only method found in the original THERMOD program. An alternate finite differene method IBFDM, was developed to solve this problem. Simulation studies demonstrated that IBFDM has effetively eliminated this problem. 2-2

19 The IBFDM is unonditionally stable over all spatial and time domain [6]. This stability is attributed to the fat that, unlike EFFDM, IBFDM onsiders the transient problem as a system of equations to be solved in a simultaneous manner. This imparts stability to the solution proess and allows the use of less refined spatial disretization and larger time steps. The advantages of IBFDM over EFFDM ome, however, at the expense of more omplex programming and bookkeeping proedures. In THERMOD, for example, the development of the IBFDM neessitated the rearrangement of the transient equations, their assembly into a set of simultaneous equations, their nonlinear solution using the Newton-Raphson iteration tehnique [5], and the retrieval of the solved transient temperatures. These solution steps require a substantial inrease in omputing time. However, this inrease in omputing time may be offset by the use of less refined spatial disretization and larger time steps, exploiting the inherent stability of IBFDM. The following setion disusses the development of the algorithm for IBFDM followed by an example problem substantiating IBFDM with respet to the existing EFFDM. In addition, an example problem will be presented that will illustrate the inherent stability of IBFDM over EFFDM. 2.4 ALGORITHM DEVELOPMENT FOR IBFDM. IBFDM in THERMOD an be oneptually viewed as a proedure that reognizes the fat that the temperature of a point in spae at a ertain instane in time is dependent on the temperatures of all other points that are diretly assoiated with that point. The temperature of eah of these other points in turn has its own dependene on temperatures and other points assoiated with it, and so on. One then noties a hain-like link being formed among all transient temperatures of a system. A unique solution for this link exists for every time step, and only a system of simultaneous equations will provide this solution. THERMOD sets up these equations, assembles them, solves them, and retrieves the transient temperatures for eah time step requested. The governing equation of the transient heat transfer proess is based on the formula E in E out = E (2-3) Where E in = rate of inoming energy; E out = rate of outgoing energy; and E = rate of energy gain or energy lost. A positive E indiates energy is gained, while a negative E indiates energy is lost. A zero E implies a steady-state system where there is no gain or lost of energy in the system. Note that equation 2-3 does not inlude the effet of energy generation, whih is ertainly absent in the ase of an airraft parked in the open. The implementation of the IBFDM transient programming rests on setting up equation 2-3 for eah one of the ontrol volumes defined in the system and solving the resulting set of simultaneous equations. 2-3

20 The proedure an be demonstrated by developing the equation for ontrol volume 3 shown in figure 2-1. This ontrol volume may be viewed to be within the wing of an airraft where the surfae effets are absent. The only mode of heat transfer in this region is ondution. Only through thikness, one-dimensional ondution is assumed. Along-wing span and along-wing width ondution were negleted in THERMOD to produe onservative temperatures. Surfae d 1 Control volume 1 T 1 Area, A d 2 Control volume 2 T 2 Layer 1: k 1, ρ 1, 1 y 1 d 3 Control volume 3 d 4 Control volume 4 T 3 T 4 y 2 Layer 2: k 2, ρ 2, 2 FIGURE 2-1. CONTROL VOLUMES FOR DEVELOPING THERMAL EQUATIONS In developing the equations, it is assumed that the reader has a basi understanding of the priniples of heat transfer. For an overview of the mathematial derivation of the three modes of heat transfer, i.e., ondution, onvetion, and radiation, the reader is referred to referene

21 From figure 2-1, for ontrol volume 3 with ross-setional area A A y T T k E i i a in = where ) ( d d y + = ) ( 2 2 k k d d a d d k + + = A y T i i b out T k E = where ) ( d d y + = ) ( 2 2 k k d d b d d k + + = t T T m E i i = where: m = mass of the ontrol volume = ρ 1 A d 3 ρ 1 = density of the ontrol volume 1 = speifi heat apaity of the ontrol volume t = time step In the above equations, i+1 T represents new temperature, whih is an unknown quantity, and i T represents the previous temperature, whih is a known quantity. Substituting the above expressions for,, and in E out E E into equation 2-3 gives t T T Ad A y T T k A y T T k i i i i b i i a = ρ Eliminating A, and rearranging, one obtains: (2-4) i i i i T a T a T a T a = where: 1 1 y k a a = = t d y k y k a b a ρ 2-5

22 a 4 a 3 = k y b 2 ρ1d 3 = t 1 Note that in equation 2-4, a 1, a 2, a 3, a 4, and i i+ 1 T are known quantities. The unknowns are T 2, i+ 1 i+ 1 T3, and T 4. It is observed that equation 2-4 is impliit, in that there are more than one unknown. To solve this equation, one needs to reate another equation representing the ontrol volume adjaent to the urrent one. This equation would ontain other unknown temperatures. Thus, one needs to develop a series of equations representing all ontrol volumes that were used i+1 to define the thermal system, before a unique solution for the set of new temperatures, T ould be found. The differene between IBFDM and EFFDM an be observed by studying the derivation of equation 2-4. Had E in and E out been defined in terms of the previous temperatures, the only i+ 1 unknown in equation 2-4 would have been T, and this would have rendered the equation 3 expliit. Beause E in and E out were expressed in terms of the new temperatures, whih are unknown quantities, equation 2-4 was rendered impliit. In THERMOD, a total of 39 ontrol volumes were reated, resulting in 117 unknown i+1 temperatures, T. Note that eah ontrol volume was designed to have three temperatures, as illustrated in the derivation of the equations above. Twenty-seven of the ontrol volumes were within the struture where the only mode of heat transfer was by means of ondution, as was illustrated in the above sample equations. The remaining twelve ontrol volumes represented surfae elements where onvetion and radiation effets, in addition to ondution effets, were onsidered. The 117 equations were assembled and subsequently solved using the nonlinear Newton-Raphson iteration proedure. A desription of this proedure is found in referenes 3 and 4. Note that the nonlinear solution proedure is required for two reasons. First, infrared radiation, a apability modeled in THERMOD, follows the Stefan-Boltzman Law, whih is based on a nonlinear formulation, where radiation R = σ T 4. In this formulation, the temperature, T, is an unknown quantity. Seond, the onvetion oeffiient within the abin was modeled to be dependent on temperatures, whih again, are an unknown quantity. The transient solution is initiated by treating the temperatures obtained from the steady-state analysis as the initial set of previous temperatures. The solved new temperatures are then treated as the previous temperatures in the next time step. The analysis is repeated until a solution satisfying a required set of tolerane in temperatures and funtions is obtained. One solved, the 117 temperatures are retrieved, from whih only the 53 temperatures defining THERMOD are reported. (See figures 1-1 and 1-2 to loate these temperatures.) 2-6

23 As indiated, THERMOD is a thermal system with 67 unknowns, where 53 of the unknowns are temperatures and 14 are radiosity. Radiosity is defined as the radiative flux that leaves a surfae, and it inludes both the refleted irradiation as well as emission [6]. Beause THERMOD depends on temperatures, it follows that radiosity will hange with time in a transient environment. This hange in radiosity is implemented in THERMOD by updating its value using the new temperatures determined at the end of eah time step. The various subroutines, their definitions, and the program flow hart addressing IBFDM are ontained in appendix A. The development of IBFDM required extensive rearrangement of the transient equations, substantial bookkeeping, a nonlinear solution of the resulting simultaneous equations, and data retrieval. These extra efforts were expeted to inrease the omputing resoures. This was indeed the ase. The amount of omputing time spent was vastly different for the previous two problems. The EFFDM solution proess took less than 1 minute, while the IBFDM took about 9 minutes. As was previously noted, the primary purpose of IBFDM was to overome probable instability problems that may arise using EFFDM. The substantial inrease in the omputing effort of IBFDM, however, seems to be a worthwhile prie to pay, given that a potential stability problem was solved and satisfatory results are expeted. It is also noted that beause IBFDM is inherently stable regardless of the time step duration, the omputing effort may be substantially redued by inreasing the time step until auray of the solution is not affeted. 2-7/2-8

24 3. NUMERICAL VALIDATION OF THERMOD. This setion presents the numerial validation of THERMOD with respet to FEMs. A total of eight problems were formulated and solved using the THERMOD program and then the FEM programs. The results from both methods of analysis were then ompared. It was beyond the sope of this projet to develop a omplete FEM thermal model of an airraft. The FEM sample problems were, thus, kept as simple as possible and were yet able to apture the three fundamental modes of heat transfer mehanisms, i.e., ondution, onvetion, and radiation. The problems, where possible, addressed eah transport mehanism individually first, then progressed into more omplex thermal systems involving a ombination of these mehanisms. In developing sample input files for THERMOD, all 15 sets of data defining the entire thermal problem of an airraft must be generated and used in all sample problems. This is beause, unlike the FEM methods, THERMOD was not developed as a general-purpose thermal program. It was designed to solve a very speifi thermal problem assoiated with an airraft subjeted to the thermal environment. While the solution proess in itself is rather omplex, the inputs to the program were designed to be simple and straightforward. Beause of this onstraint, airraft problems in its entirety will be solved in THERMOD during this validation exerise. The FEM methods, however, are not subjet to this onstraint. Of the eight validation problems solved, six involved steady-state analysis. The other two involved both steady-state and transient analyses. The FEM models of the six steady-state problems were analyzed using UAI/NASTRAN software. The FEM models of the two ombined steady-state and transient problems were analyzed using MSC/NASTRAN software. The primary reason for using two different FEM pakages was that the thermal module of the UAI/NASTRAN pakage was found to be defiient in the transient analysis. It laked a timedependent onvetive oeffiient, whih is needed in analyzing transient ooling, where air is fored over the airraft while maneuvering aording to various flight profile information. The MSC/NASTRAN thermal module was found to have this apability. To have a better understanding of the sample problems presented in this setion, it is helpful to be familiar with the THERMOD User s Manual [2], whih gives a omplete desription of input data sets, variable definitions, formats, and other general information. Only information diretly relevant to the validation problems will be disussed in detail in this setion, while the disussion of other data will be kept to a minimum. It is assumed that the reader have a working knowledge of the FEM methods, suffiient to deipher NASTRAN input data deks. The relevant FEM information pertaining to NASTRAN an be found in the following douments: UAI/NASTRAN User s Referene Manual [7], UAI/NASTRAN User s Guide [8], MSC/NASTRAN Quik Referene Guide [9], MSC/NASTRAN User s Guide [1], MSC/NASTRAN Command Referene Guide [11], and MSC/NASTRAN Thermal Analysis User s Guide [12]. 3-1

25 3.1 SAMPLE PROBLEM 1. The purpose of sample problem 1 was to illustrate the ondution mehanism of the heat transfer proess. Table 3-1 shows the THERMOD input file, input.dat, of sample problem 1. As previously noted, a total of 15 sets of data must always be used to omplete a suessful run of THERMOD. These sets of data define a thermal problem of a omplete airraft. TABLE 3-1. THERMOD INPUT FILE (input.dat) OF SAMPLE PROBLEM 1 198e-6,.74 1,.2,125,.3,.3 2,.375,4.4,.21,.24 3,.2,125,.3,.3 4,.35,125,.3,.3 5, 1.7,.67,.16,.24 6,.35,125,.3,.3 7,.2,125,.3,.3 8,.375,4.4,.21,.24 9,.2,125,.3,.3 1,.3,125,.3,.3 11,.375,4.4,.21,.24 12,.2,125,.3,.3 13,.25,1.,.2,.24 14,48.,.67,.16,.24 15,.2,125,.3,.3 16,.375,4.4,.21,.24 17,.2,125,.3,.3 18,.25,1.,.2,.24 19,3.,.67,.16,.24 2,5.,1.,.2,.24 21,.2,125,.3,.3 22,.375,4.4,.21,.24 23,.2,125,.3,.3 24,2.5,.67,.16,.24 25,.2,125,.3,.3 26,.375,4.4,.21,.24 27,.2,125,.3,.3 28,.25,13,.8,.2 3.5,34.,2.5,5.5,22.,11.,3.,4.,3,24,28.7,.9,.9,2.,15,.8,.22.3,.9,.3,.9,.3,.9,.1,.9 3,1.9,.9,.9,.9.95,.47,.3,.8,.95,.47,.95,.47 9,1,1,,,75,11,11,19,19 5,12,5,6,2,5,12,6,6 3-2

26 TABLE 3-1. THERMOD INPUT FILE (input.dat) OF SAMPLE PROBLEM 1 (Continued) ,1,18,1 1,1.,1. 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1 For omparison purposes with the FEM method, the fous was on the right wing of the airraft. (The right wing is indiated by RHS in figure 1-1.) As illustrated in the input.dat file of table 3-1, the right wing was modeled as follows: Layer 1:.2-in.-thik top outer fiber glass skin Layer 2:.375-in.-thik top foam ore Layer 3:.2-in.-thik top inner fiber glass skin Layer 4:.35-in.-thik top fiber glass spar ap Layer 5: 1.7-in.-deep air spae between the top and bottom spar aps Layer 6:.35-in.-thik bottom fiber glass spar ap Layer 7:.2-in.-thik bottom inner fiber glass skin Layer 8:.375-in.-thik bottom foam ore Layer 9:.2-in.-thik bottom outer fiber glass skin The summary.dat file of table 3-2 summarizes the results of the THERMOD analysis. As observed in table 3-2, a total of seven time periods were involved in the THERMOD analysis. Steady-state temperature results obtained for time period 3 was hosen for omparison purposes with the FEM method. A total of 1 temperature loations were identified on the right wing, as illustrated in figure 1-1. These 1 temperatures from the top to the bottom are T1, T2..T1. Their values were extrated from table 3-2 and summarized in table 3-3. The results indiated that the top surfae of the wing, beause of its exposure to the sun, was hotter than the lower surfae, whih was more sheltered. There was a signifiant differene in temperature between the upper ap (T5) and the lower ap (T6). This was attributed to the 1.7 in. of air (whih has low thermal ondutivity) that separates them. 3-3

27 TABLE 3-2. THERMOD OUTPUT FILE (summary.dat) OF SAMPLE PROBLEM 1 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ THERMOD: A THERMAL PROGRAM FOR AIRCRAFTS $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ SUMMARY OF STEADY-STATE AND TRANSIENT ANALYSIS (Temperatures are shown for all time periods) TEMPERATURES AT THE END OF STEADY STATE ANALYSIS TEMPERATURES IN DEGREES FAHRENHEIT

28 TABLE 3-2. THERMOD OUTPUT FILE (summary.dat) OF SAMPLE PROBLEM 1 (Continued) Maximum Strutural Temperature for Period at Loation 24 Maximum Strutural Temperature for Period at Loation 21 Maximum Strutural Temperature for Period at Loation 21 Maximum Strutural Temperature for Period at Loation 21 Maximum Strutural Temperature for Period at Loation 24 Maximum Strutural Temperature for Period at Loation 39 Maximum Strutural Temperature for Period at Loation 39 1 = Ourring 2 = Ourring 3 = Ourring 4 = Ourring 5 = Ourring 6 = Ourring 7 = Ourring Maximum Strutural Temperature Over All 7 Periods = Ourring at Loation 21 at Period 3 3-5

29 TABLE 3-2. THERMOD OUTPUT FILE (summary.dat) OF SAMPLE PROBLEM 1 (Continued) TEMPERATURES AT THE END OF TRANSIENT ANALYSIS TEMPERATURES IN DEGREES FAHRENHEIT

30 TABLE 3-2. THERMOD OUTPUT FILE (summary.dat) OF SAMPLE PROBLEM 1 (Continued) Maximum Strutural Temperature for Period at Loation 24 Maximum Strutural Temperature for Period at Loation 24 Maximum Strutural Temperature for Period at Loation 24 Maximum Strutural Temperature for Period at Loation 24 Maximum Strutural Temperature for Period at Loation 24 Maximum Strutural Temperature for Period at Loation 42 Maximum Strutural Temperature for Period at Loation 42 1 = Ourring 2 = Ourring 3 = Ourring 4 = Ourring 5 = Ourring 6 = Ourring 7 = Ourring Maximum Strutural Temperature Over All 7 Periods = Ourring at Loation 24 at Period 3 TABLE 3-3. THERMOD-SIMULATED TEMPERATURES ( F) OF THE RIGHT WING OF SAMPLE PROBLEM 1 T1 T2 T3 T4 T5 T6 T7 T8 T9 T An FEM model of the wing was developed based on the information used in the THERMOD model. One-half of the wing was modeled as a 17-ft-long struture with layer properties as indiated above. A total of 153 elements, eah.1 ft thik, was employed, resulting in a total of 324 grid points (nodes). Material properties, representing ondutivity, were assigned using MAT4 ards. PSHELL ards were used to assign the thikness and material properties for eah of the element. 3-7

31 Boundary onditions were imposed along the outer nodes of the model. Reall that the objetive of this sample problem was to validate the ondutive aspet of THERMOD. The FEM boundary onditions were then set suh that all top surfae nodes assumed the value of T1 (136.1 F), and all bottom surfae nodes assumed the value of T1 (124.4 F), whose values were obtained from the earlier THERMOD analysis. The outer left and right most edges of the wing were assigned boundary temperatures of F, thus ompletely enapsulating the FEM model. The temperature profile aross the wing depth is now entirely due to ondution. Table 3-4 illustrates the FEM input file. TABLE 3-4. FINITE ELEMENT METHOD INPUT FILE OF SAMPLE PROBLEM 1 ID D:\Natha,MSC/N APP HEAT SOL 1 TIME 1 CEND TITLE = CONDUCTION IN RECTANGULAR ADIABATIC PLATE (t=.1ft) SUBTITLE = 1.7 in gap filled by air; k=.16btu/hr/ft/degf SPC = 1 THERMAL= ALL FLUX = ALL SPCF = ALL BEGIN BULK $ *************************************************************************** $ Written by : MSC/NASTRAN for Windows $ Version : 6. $ Translator : UAI/NASTRAN $ From Model : D:\Thermod_Validation\FEM\ondution_only.MOD $ Date : Mon Mar 6 11:27:9 2 $ *************************************************************************** $ PARAM,K6ROT,1. PARAM,MAXRATIO,1.E+8 CORD2C MSC/NC1 +MSC/NC CORD2S MSC/NC2 +MSC/NC $ MSC/NASTRAN for Windows Constraint Set 1 : steady-state_bound-temp SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC

32 TABLE 3-4. FINITE ELEMENT METHOD INPUT FILE OF SAMPLE PROBLEM 1 (Continued) SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC SPC $ MSC/NASTRAN for Windows Property 1 :.2 in glass_outer_layer PSHELL $ MSC/NASTRAN for Windows Property 2 : 3/8 in foam PSHELL $ MSC/NASTRAN for Windows Property 3 :.2 in inner glass PSHELL $ MSC/NASTRAN for Windows Property 4 :.35 in ap PSHELL $ MSC/NASTRAN for Windows Property 5 : 1.7 in air spae PSHELL $ MSC/NASTRAN for Windows Property 6 :.35 in ap PSHELL $ MSC/NASTRAN for Windows Property 7 :.2 in inner glass. PSHELL $ MSC/NASTRAN for Windows Property 8 : 3/8 in foam PSHELL $ MSC/NASTRAN for Windows Property 9 :.2 in glass_outer_layer PSHELL $ MSC/NASTRAN for Windows Material 1 : glass MAT $ MSC/NASTRAN for Windows Material 2 : foam MAT $ MSC/NASTRAN for Windows Material 3 : air MAT

33 TABLE 3-4. FINITE ELEMENT METHOD INPUT FILE OF SAMPLE PROBLEM 1 (Continued) E E E E E E E E E E E E E E E E E E

34 TABLE 3-4. FINITE ELEMENT METHOD INPUT FILE OF SAMPLE PROBLEM 1 (Continued)

35 TABLE 3-4. FINITE ELEMENT METHOD INPUT FILE OF SAMPLE PROBLEM 1 (Continued)

36 TABLE 3-4. FINITE ELEMENT METHOD INPUT FILE OF SAMPLE PROBLEM 1 (Continued)

37 TABLE 3-4. FINITE ELEMENT METHOD INPUT FILE OF SAMPLE PROBLEM 1 (Continued)

38 TABLE 3-4. FINITE ELEMENT METHOD INPUT FILE OF SAMPLE PROBLEM 1 (Continued) ENDDATA The results of the one-half FEM wing model analysis are presented in figure 3-1 in the form of a ontour plot. The ontours along the outer nodes are in agreement with the imposed boundary onditions. The ontours of the inner layers appear to be indiative of the temperature profile of the THERMOD analysis. 3-15

39 FIGURE 3-1. CONTOUR PLOT OF THE FEM OUTPUT OF SAMPLE PROBLEM 1 The trunated FEM output file is shown in table 3-5. The nodes, from top to bottom of the wing, loated at the mid-span of the model are indiated by nodes 29, 245, 226, 317, 298, 118, 137, 11, 1, and 29. The temperatures of these nodes were extrated from this file and are summarized in table 3-6. Mid-span was hosen beause it was suffiiently far away from any edge effets that might be present. TABLE 3-5. TRUNCATED FEM OUTPUT FILE OF SAMPLE PROBLEM 1 1 CONDUCTION IN RECTANGULAR ADIABATIC PLATE (T=.1FT) MARCH 6, 2 UAI/NASTRAN VERSION 2.1 PAGE IN GAP FILLED BY AIR; K=.16BTU/HR/FT/DEGF T E M P E R A T U R E V E C T O R POINT ID. TYPE ID VALUE ID+1 VALUE ID+2 VALUE ID+3 VALUE ID+4 VALUE ID+5 VALUE 1 TEMP 1.244E+2 3 TEMP E+2 5 TEMP E+2 7 TEMP E+2 9 TEMP E E E E+2 14 TEMP E E+2 18 TEMP E+2 2 TEMP 1.244E E E E E E+2 26 TEMP 1.244E E E E E E+2 32 TEMP 1.244E E E E E E+2 41 TEMP E+2 44 TEMP E E+2 47 TEMP E E+2 5 TEMP E E E E E E+2 56 TEMP 1.244E+2 58 TEMP E E+2 62 TEMP E+2 67 TEMP E+2 69 TEMP E+2 71 TEMP E+2 73 TEMP 1.244E+2 75 TEMP E E E E E E+2 83 TEMP E+2 85 TEMP E+2 87 TEMP E E