DOT/FAA/AR-04/52 

Office of Aviation Research 

Washington, D.C. 20591

THERMOD, an Enhanced 

Thermal Model for 

Determining Aircraft 

Operational Temperatures: 

User’s Manual 

December 2004 

Final Report 

This document is available to the U.S. public 

through the National Technical Information 

Service (NTIS), Springfield, Virginia 22161. 

U.S. Department of Transportation 

Federal Aviation Administration
NOTICE 

This document is disseminated under the sponsorship of the U.S. 

Department of Transportation in the interest of information exchange. The 

United States Government assumes no liability for the contents or use 

thereof. The United States Government does not endorse products or 

manufacturers. Trade or manufacturer's names appear herein solely 

because they are considered essential to the objective of this report. This 

document does not constitute FAA certification policy. Consult your local 

FAA aircraft certification office as to its use. 

This report is available at the Federal Aviation Administration William J. 

Hughes Technical Center's Full-Text Technical Reports page: 

actlibrary.tc.faa.gov in Adobe Acrobat portable document format (PDF). 
Technical Report Documentation Page

1. Report No.

DOT/FAA/AR-04/52

2. Government Accession No. 3. Recipient's Catalog No.

5. Report Date

December 2004 

4. Title and Subtitle

THERMOD, AN ENHANCED THERMAL MODEL FOR DETERMINING 

AIRCRAFT OPERATIONAL TEMPERATURES: USER’S MANUAL 6. Performing Organization Code

7. Author(s)

Nathan Govindarajoo, Ph.D, PE.

8. Performing Organization Report No. 

9. Performing Organization Name and Address 10. Work Unit No. (TRAIS) 

1889 Old Dixie Highway, #203 

Vero Beach, FL 32960

11. Contract or Grant No.

13. Type of Report and Period Covered

 Final Report

12. Sponsoring Agency Name and Address

U.S. Department of Transportation 

Federal Aviation Administration 

Office of Aviation Research 

Washington, DC 20591

14. Sponsoring Agency Code

 ACE-120, AGATE

15. Supplementary Notes

The FAA William J. Hughes Technical Center Technical Monitor was Peter Shyprykevich.

16. Abstract

An enhanced version of a thermal analysis computer program called THERMOD has been developed for determining the 

maximum operating limit (MOL) temperatures of general aviation aircraft. This report is the second of two reports prepared in 

conjunction with this project. Enhancements included program debugging and corrections as well as development of an 

alternate implicit finite difference method, which is used in transient analysis. Numerical validation of THERMOD was 

undertaken with respect to the finite element methods. Good correlation was found. Experimental validation was also 

performed. The validated THERMOD can be used to determine the MOL temperatures of a typical aircraft that has a low wing. 

17. Key Words 

Maximum operating limit temperature, Thermal analysis, 

Conduction, Convection, Radiation, Steady State, Transient 

state

18. Distribution Statement 

This document is available to the public through the National 

Technical Information Service (NTIS), Springfield, Virginia 

22161.

19. Security Classif. (of this report)

 Unclassified 

20. Security Classif. (of this page)

 Unclassified

21. No. of Pages

 42

22. Price

Form DOT F1700.7 (8-72) Reproduction of completed page authorized
TABLE OF CONTENTS 

Page 

EXECUTIVE SUMMARY v 

1. INTRODUCTION 1 

2. A BRIEF DESCRIPTION OF THERMOD 1 

3. THERMOD GEOMETRY AND INPUT DATA 3 

4. THERMOD INPUT AND OUTPUT FILES 6 

5. REFERENCES 12 

APPENDIX A—INPUT DATA 

LIST OF FIGURES 

Figure Page 

1 A Schematic Presentation of THERMOD 2 

2 Overall Model Geometry and Typical Temperature Profiles at Critical Locations 4 

3 Input Thermal Properties for THERMOD Model 5 

4 A Typical Flight Profile During Transient Cooling 6 

5 A Sample Input File: input.dat 7 

6 A Sample Output File: summary.dat 8 

7 A Sample Output File: transient.dat 12 

iii/iv
EXECUTIVE SUMMARY

Composite aircraft structural elements are unfavorably affected by an increase in their 

temperatures due to exposure to the thermal environment. For design purposes, the synergistic 

effects of extreme ambient temperatures and the accompanying solar radiation should be taken 

into consideration in determining the maximum operating limit (MOL) temperatures experienced 

by the structural elements. Allowable design properties of composite materials may then be 

generated based on these MOL temperatures. THERMOD is a computer model that was 

developed for determining the MOL temperatures for aircraft under various paint schemes. The 

selected low-wing geometry is best suited to general aviation aircraft. 

In determining the MOL temperatures, THERMOD considers the effects of radiation, 

convection, and conduction. Radiation includes direct solar radiation, infrared sky radiation, and 

their reflections from the tarmac. Infrared emission from the surrounding structures, including 

the tarmac, wings, fuselage, and its interaction among these structures, are also considered. 

Convection due to the wind, as well as turbulent convection within the cabin is also simulated, as 

are one-dimensional conduction through element thickness. The model incorporates the effect of 

fillets at the wing-fuselage junction. Due consideration is also given to the greenhouse effect 

within the cabin, which allows for a realistic modeling of the thermal environment within the 

cabin. The effect of shade underneath the wing is also modeled. 

The soaked temperatures of an aircraft are predicted based on steady-state assumptions, giving 

conservative steady-state temperatures. Because limit loads generally occur in flight conditions, 

a transient (unsteady-state) thermal analysis is used to simulate the thermal conditions while the 

aircraft executes the following maneuvers: taxi, takeoff, climb and cruise. These maneuvers are 

cooling effects which, in addition to intentional cooling of the cabin through opening the door, 

simulate a more realistic thermal environment for predicting the MOL temperature. 

THERMOD formulates a total of 67 independent equations within a nonlinear system of 

equations. The nonlinearity is introduced through radiation effects and through convective 

properties with the cabin, which are modeled as temperature dependent. The system of 

equations is solved using the Newton-Raphson iteration technique. 

This report, the fourth of four reports prepared in conjunction with this project, will serve as an 

instructive guide for those who wish to use THERMOD for determining aircraft MOL 

temperatures. 

v/vi
1. INTRODUCTION.

An enhanced version of a thermal analysis computer program called THERMOD has been 

developed for simulating maximum operating limit (MOL) temperatures for general aviation 

aircraft. 

This report, the fourth of four reports prepared in conjunction with this project, will serve as a 

guide for users who wish to use THERMOD for simulating aircraft MOL temperatures. The 

third report, “An Enhanced Thermal Model (THERMOD) for Determining Maximum Operating 

Temperatures” [1], provides additional technical details and analytical validations. 

To help the user get acquainted with THERMOD, a brief background on pertinent thermal 

aspects of the program is presented. This is followed by a description of all aspects of the input 

data that are necessary for a successful execution of THERMOD. 

2. A BRIEF DESCRIPTION OF THERMOD.

Composite aircraft structural elements are unfavorably affected by an increase in their 

temperatures due to exposure to the thermal environment. For design purposes, the combined 

effects of extreme ambient temperature and the accompanying solar radiation should be taken 

into consideration in determining the MOL temperatures experienced by the structural elements. 

Design properties of composite materials may then be generated based on these MOL 

temperatures. THERMOD was developed for determining the MOL temperatures of low-wing 

aircraft under various paint schemes. 

In determining the MOL temperatures, THERMOD considers the effects of radiation, 

convection, and conduction. Radiation includes direct solar radiation, infrared sky radiation, and 

their reflections from the tarmac. Infrared emission from the surrounding structures, including 

the tarmac, wings and the fuselage and its interaction among these structures, are also 

considered. Convection due to the wind as well as turbulent convection within the cabin, is also 

simulated, as are one-dimensional conduction through element thickness. The model 

incorporates the effect of fillets at the wing-fuselage junction. Due consideration is also given to 

the greenhouse effect within the cabin that allows for a realistic modeling of the thermal 

environment within the cabin. The effect of shade underneath the wing is also modeled. 

In addition to the above factors, the following assumptions are made in THERMOD: 

(1) nonparticipating medium (air); (2) discretized space and time domains; (3) discretized 

elements being isothermal, opaque, diffuse, gray and characterized by uniform radiosity, 

irradiation and material properties; (4) nonopaque materials such as windows and windshields 

are considered transparent with associated transmissivity values; and (5) constant material 

properties with respect to time (and hence temperature). These assumptions are necessary in 

simplifying the complexities involved in a three-dimensional thermal problem being addressed. 

The soaked temperatures of an aircraft are predicted based on steady-state assumptions, giving 

conservative steady-state temperatures. Because limit loads generally occur in flight conditions, 

a transient (unsteady-state) thermal analysis is used to simulate the thermal conditions while the 

1
aircraft executes the following maneuvers: taxi, takeoff, climb, and cruise. These maneuvers are 

cooling effects that, in addition to intentional cooling of the cabin through opening the door, 

simulate realistic thermal environment for predicting the MOL temperature. 

THERMOD formulates a total of 67 independent equations within a nonlinear system of 

equations. The non-linearity is introduced through radiation effects and through convective 

properties with the cabin, which are modeled as temperature-dependent. This system of 

equations is solved using the Newton-Raphson iteration technique [3]. 

As previously noted, THERMOD considers all three heat transfer mechanisms normally 

associated with an aircraft parked in the open: convection, conduction, and radiation. Figure 1 is 

a schematic representation of this thermal environment showing short wavelength solar 

radiation, long wavelength sky (infrared) radiation, convective heat transfer due to the wind, and 

one-dimensional conduction (through the thickness). Also considered are reflected solar energy 

between the fuselage and wing, diffused solar energy reflected from the tarmac, and infrared 

reflections. The fillets and greenhouse effects are considered as well. Because THERMOD 

simulates a total of 67 independent equations, 67 unknowns are needed. These unknowns are 53 

temperatures and 14 radiosity functions. These 53 temperatures and their locations are noted in 

figure 1. 

FIGURE 1. A SCHEMATIC PRESENTATION OF THERMOD 

Note that not all the unknown temperatures are shown. T2, for example, is located immediately 

below T1 at the skin-core interface (see figure 2 for further clarity). Similarly, T27 is located 

immediately below T26 at the insulatory material B-composite floor interface. 

2
The geometry of the aircraft shown is low wing with sandwich construction for skins. The 

sandwich can be either foam or honeycomb. Nonsandwich skins are also admissible if the 

thickness of the core is assumed to be very small. 

A typical THERMOD analysis begins at the steady-state phase and continues on to the transient 

phase; a phase in which cooling is introduced due to aircraft maneuvers. At the point of 

application of limit or gust load, the temperatures at all 53 locations are noted. Of these 53 

temperatures, 9 temperatures are considered nonstructural. These nonstructural temperatures are 

the three tarmac temperatures (T51, T52, and T53); four insulatory material temperatures (T25, 

T26, T43, and T44); and two transparent material temperatures (T49 and T50), leaving 44 

temperatures that are structural. 

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 located at the surface (depending on how dark the paint is), inside the cabin, 

within the floor space, or located anywhere else in the aircraft. 

3. THERMOD GEOMETRY AND INPUT DATA. 

Figure 1 shows a typical cross section of a small general aviation aircraft. A simplified 

rendering of the model is shown in figure 2. Fuselage is modeled as a box and treated as an 

enclosure in modeling the greenhouse effect. The fuselage roof is treated as a surface with an 

equivalent transmissivity, allowing solar radiation to enter the cabin. The insulatory materials 

(A), (B), and (C) are simplified representations of cabin roof interior, cabin floor, and cabin side 

interior. THERMOD is built on this simplified model. Figure 2 also shows typical sections at 

critical locations. In addition to the overall geometry as indicated in figure 2, a set of input data 

characterizing each layer, as shown in figure 3, is also needed. A total of 28 distinct layers are 

considered. Each of these layers, marked P1 to P28, is assigned four properties, i.e., thickness, 

density, thermal conductivity, and specific heat capacity. The wing, fuselage, and the interior 

(B) and (C) surfaces are assigned scatter factors (sf). The aircraft is subjected to climatic factors 

that include solar radiation (Qsol), wind (Vwind), ambient temperature (Tamb), and sky temperature 

(Tsky). The layer properties include thickness, density, thermal conductivity, and specific heat 

capacity. To consider surface effects, all exposed aircraft surfaces are assigned their respective 

absorptivity and emissivity values. These values are dependent on paint color and surface 

texture. The tarmac is also considered a surface, whose surface and other thermal properties are 

required as well. THERMOD models the greenhouse and fillet effects. Transmissivity and fillet 

color information is needed to consider these two effects. The degree of greenhouse effect is 

also dependent on the percentage of transparent material. This information is input in the form

of transparent surface area in relation to the overall cabin surface area. To simulate convective 

coefficients, the kinematic viscosity and Prandtl number are needed. Because the aircraft is left 

in the open, it is subjected to the environment. The climatic data of the environment, for each 

time period, include ambient temperature, sky temperature, solar radiation, and wind speed. 

3
Lcabin

Wwing

Lledge

Lwing

Hcabin

Hwing Tarmac

d

b

a

c

Foam

Foam

Outer Skin

Inner skin

Spar cap

Air space

Spar cap

Inner skin

Outer skin

 Detail a: Wing

Outer skin F

o

a

m

Inner skin

Insulatory

material C Detail b:

Fuselage side

Detail c: Floor

Insulatory material B

Outer skin

Inner skin

Floor space

Foam

Inner skin

Outer skin

Foam

Floor

Bellypan

 Detail d: Roof

Outer skin

Inner skin

Insulatory material A

Foam

Thermal model: Top view

Thermal model: Front view

Fuselage

Wing

T1

T2

T3

T4

T5

T6

T7

 T8

T9

T10

 T43 T42 T41 T40 T39

T26

T27

T28

T37

T38

T21

T22

T23

T24

T25

T29

T30

T35

T36

Wcabin

Lflg

FIGURE 2. OVERALL MODEL GEOMETRY AND TYPICAL TEMPERATURE PROFILES 

AT CRITICAL LOCATIONS 

4
FIGURE 3. INPUT THERMAL PROPERTIES FOR THERMOD MODEL 

The final piece of information concerns cooling that takes place when the aircraft executes the 

following maneuvers: taxi, takeoff, climb and cruise. These data are furnished through a flight 

profile that is unique to a particular aircraft. The flight profile provides information on typical 

aircraft maneuvering speed with respect to time, from which time-varying convective coefficient 

is determined. This coefficient is then used in a subsequent transient cooling process. An 

example of a flight profile is shown in figure 4. 

Cooling is assumed to occur as soon as the door is opened. The greenhouse cooling duration is a 

function of the soaked steady state ambient temperature of the cabin and of tolerable ambient 

temperature desired before closing the door. This tolerable temperature, arbitrarily selected, is 

input into THERMOD as ambcabt. MOL temperature is determined at the point when limit load 

is applied as shown. This moment in time is located at 395 seconds from point V0 in the above 

sample flight profile. The ambient wind speed is added to the flight speed to give the total 

relative wind speed. THERMOD uses this relative wind speed at selected time to determine 

convective coefficient, which is a required input in the cooling process. 

5
Velocity (mph)

Time (sec)

Ambient wind speed

V0

V1 V2

V3 V4

V5

V6 V7

V8 V9

Taxi

Stop

Take-off Run

Accelerate

Climb

Accelerate

Cruise

Open door Close

door

Greenhouse cooling

Limit load application

0

 Time

V0=0 mph

V1=10 mph V0- V1=5 sec

V2=10 mph V1- V2=120 sec

V3=0 mph V2- V3=5 sec

V4=0 mph V3- V4=60 sec

V5=75 mph V4- V5=20 sec

V6=110 mph V5- V6=5 sec

V7=110 mph V6- V7=120 sec

V8=190 mph V7- V8=60 sec

V9=190 mph V8-V9=60 sec

Ambient wind

speed=9.5 mph

FIGURE 4. A TYPICAL FLIGHT PROFILE DURING TRANSIENT COOLING 

4. THERMOD INPUT AND OUTPUT FILES.

THERMOD requires one input file, which is named input.dat and is shown in figure 5. This 

information is for demonstration purposes only. It is not part of the input file. Fifteen data sets 

are required to completely define the input file. Some sets have subsets as indicated. For 

example, 28 subsets define data set 2. Data set 2 must always have 28 subsets. On the other 

hand, data set 12 may have a minimum of two subsets (data set 12-1 and data set 12-2) or as 

many as indicated by the entry in data set 12-1. A comma or a space must separate each entry in 

a data set. Once invoked (by double-clicking THERMOD), THERMOD automatically reads in 

the input.dat file and reads out two output files: summary.dat and transient.dat. The 

summary.dat file summarizes the temperature results and prints the maximum operating 

temperatures for steady state and transient state, as shown in figure 6. This file summarizes the 

analytical results generated by THERMOD. Both the steady-state and transient state results are 

presented. Results are shown for all seven periods in the example problem. The maximum

operating limit temperature is found at the end of transient analysis and it is 178.18 °F, occurring 

at location T24 and at period 3. The transient.dat file prints the transient temperatures with 

respect to time and is shown in figure 7. This file prints the transient temperature results for all 

6
periods analyzed. The truncated results are shown for five temperature locations and for period 

1 only. This type of data can be used in studying the rate of cooling at various locations in the 

aircraft. 

198e-6,0.704 Data Set 1-1 

1, 0.020,125,0.300,0.30 Data Set 2-1 

2, 0.375,4.4,0.021,0.24 Data Set 2-2 

3, 0.020,125,0.300,0.30 Data Set 2-3 

4, 0.350,125,0.300,0.30 Data Set 2-4 

5, 10.70,0.067,0.016,0.24 Data Set 2-5 

6, 0.350,125,0.300,0.30 Data Set 2-6 

7, 0.020,125,0.300,0.30 Data Set 2-8 

8, 0.375,4.4,0.021,0.24 Data Set 2-9 

9, 0.020,125,0.300,0.30 Data Set 2-10 

10,0.030,125,0.300,0.30 Data Set 2-11 

11,0.375,4.4,0.021,0.24 Data Set 2-12 

12,0.020,125,0.300,0.30 Data Set 2-13 

13,0.250,10.,0.020,0.24 Data Set 2-14 

14,48.00,0.067,0.016,0.24 Data Set 2-15 

15,0.020,125,0.300,0.30 Data Set 2-15 

16,0.375,4.4,0.021,0.24 Data Set 2-16 

17,0.020,125,0.300,0.30 Data Set 2-17 

18,0.250,10.,0.020,0.24 Data Set 2-18 

19,30.00,0.067,0.016,0.24 Data Set 2-19 

20,5.000,10.,0.020,0.24 Data Set 2-20 

21,0.020,125,0.300,0.30 Data Set 2-21 

22,0.375,4.4,0.021,0.24 Data Set 2-22 

23,0.020,125,0.300,0.30 Data Set 2-23 

24,2.500,0.067,0.016,0.24 Data Set 2-24 

25,0.020,125,0.300,0.30 Data Set 2-25 

26,0.375,4.4,0.021,0.24 Data Set 2-26 

27,0.020,125,0.300,0.30 Data Set 2-27 

28,0.250,130,0.800,0.20 Data Set 2-28 

3.5,34.0,2.5,5.5,2.2,1.1,0.3,4.0,30,24,28 Data Set 3-1 

0.70,1.0,0.999,2.0,150,0.80,0.22 Data Set 4-1 

0.3,0.9,0.3,0.9,0.3,0.9,0.1,0.9 Data Set 5-1 

3,100 Data Set 6-1 

0.9,0.9,0.0,0.0 Data Set 7-1 

1.0,0.0,0.9,0.9,1.0,0.0,1.0,0.0 Data Set 8-1 

9 Data Set 9-1 

0,10,10,0,0,75,110,110,190,190 Data Set 10-1 

5,120,5,60,20,5,120,60,60 Data Set 11-1 

7 Data Set 12-1 

60 570.67 459.67 14 330 10 10 0.36 Data Set 12-2 

75 573.67 459.67 14 355 10 10 0.31 Data Set 12-3 

90 578.67 459.67 14 355 10 12 0.33 Data Set 12-4 

75 581.67 459.67 14 330 10 10 0.31 Data Set 12-5 

60 582.67 459.67 14 291 10 10 0.36 Data Set 12-6 

45 583.67 459.67 14 231 10 10 0.44 Data Set 12-7 

30 582.67 459.67 14 160 10 10 0.57 Data Set 12-8 

1,100,180,1 Data Set 13-1 

10,1.0,1.0 Data Set 14-1 

10,10,10,10,10,10,10,10,10,10,10,10,10,10,10, Data Set 15-1 

10,10,10,10,10,10,10,10,10,10,10,10,10,10,10, Data Set 15-2 

10,10,10,10,10,10,10,10,10,10,10,10,10,10,10, Data Set 15-3 

10,10,10,10,10,10,10,10,10,10,10,10,10,10,10, Data Set 15-4 

10,10,10,10,10,10,10 Data Set 15-5 

FIGURE 5. A SAMPLE INPUT FILE: input.dat 

7
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 THERMOD: A THERMAL PROGRAM FOR AIRCRAFTS 

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 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 

 1 2 3 4 5 6 7 

 

 1 124.0 132.5 136.1 134.0 126.4 115.2 102.8 

 2 124.0 132.5 136.1 134.0 126.4 115.2 102.8 

 3 123.7 132.2 135.8 133.7 126.2 115.3 103.0 

 4 123.7 132.2 135.8 133.7 126.2 115.3 103.0 

 5 123.7 132.2 135.8 133.7 126.2 115.3 103.0 

 6 114.4 120.2 124.7 124.6 121.1 116.3 110.5 

 7 114.3 120.2 124.7 124.6 121.1 116.4 110.5 

 8 114.3 120.2 124.7 124.6 121.1 116.4 110.5 

 9 114.1 119.9 124.4 124.3 121.0 116.4 110.7 

 10 114.1 119.9 124.4 124.3 121.0 116.4 110.7 

 11 122.1 131.4 135.9 132.9 124.8 113.5 101.4 

 12 122.1 131.4 135.9 132.9 124.8 113.5 101.4 

 13 121.9 131.1 135.6 132.7 124.7 113.6 101.6 

 14 121.9 131.1 135.6 132.7 124.7 113.6 101.6 

 15 121.9 131.1 135.6 132.7 124.7 113.6 101.6 

 16 114.3 120.2 124.7 124.6 121.1 116.3 110.5 

 17 114.3 120.2 124.7 124.6 121.1 116.3 110.5 

 18 114.3 120.2 124.7 124.6 121.1 116.3 110.5 

 19 114.1 119.9 124.4 124.3 121.0 116.4 110.7 

 20 114.1 119.9 124.4 124.3 121.0 116.4 110.7 

 21 205.8 225.9 231.9 222.2 200.7 169.4 134.2 

 22 205.8 225.8 231.8 222.1 200.7 169.5 134.2 

 23 207.4 216.5 217.2 213.6 202.4 180.2 143.1 

 24 207.4 216.4 217.2 213.6 202.4 180.2 143.1 

 25 208.4 209.9 206.9 207.6 203.6 187.7 149.4 

 26 198.9 198.2 195.5 196.8 195.4 183.4 146.2 

 27 150.5 153.0 154.4 155.5 153.6 146.5 127.6 

FIGURE 6. A SAMPLE OUTPUT FILE: summary.dat 

8
 28 150.4 153.0 154.4 155.4 153.6 146.5 127.6 

 29 147.0 149.8 151.5 152.5 150.6 143.9 126.3 

 30 147.0 149.8 151.5 152.5 150.6 143.9 126.3 

 31 131.8 135.6 138.7 139.6 137.5 132.4 120.5 

 32 131.8 135.6 138.7 139.6 137.5 132.4 120.5 

 33 131.8 135.6 138.7 139.6 137.5 132.4 120.5 

 34 131.8 135.6 138.7 139.6 137.5 132.4 120.5 

 35 116.7 121.5 125.8 126.6 124.5 120.8 114.7 

 36 116.7 121.5 125.8 126.6 124.4 120.8 114.7 

 37 113.2 118.3 122.9 123.7 121.5 118.2 113.3 

 38 113.2 118.3 122.9 123.7 121.4 118.2 113.3 

 39 199.5 179.7 148.8 180.0 196.6 194.7 173.9 

 40 199.5 179.7 148.9 180.1 196.6 194.6 173.9 

 41 197.6 186.6 168.7 186.2 194.5 188.4 162.3 

 42 197.6 186.6 168.8 186.2 194.5 188.4 162.3 

 43 196.2 191.4 182.7 190.5 193.1 184.1 154.2 

 44 191.8 188.5 182.6 187.8 189.2 180.4 151.1 

 45 169.9 170.8 168.7 171.5 170.2 161.4 138.9 

 46 169.8 170.7 168.7 171.5 170.0 161.3 138.8 

 47 138.6 145.3 148.8 148.2 142.9 134.1 121.3 

 48 138.4 145.2 148.7 148.0 142.7 133.9 121.2 

 49 145.6 148.6 149.7 150.4 147.7 139.7 123.0 

 50 147.6 150.5 151.3 152.1 149.5 141.3 123.9 

 51 113.6 120.8 125.1 123.7 118.1 110.0 100.6 

 52 108.2 115.1 119.2 118.0 112.7 105.1 96.2 

 53 113.6 120.8 125.1 123.7 118.1 110.0 100.6 

 

Maximum Structural Temperature for Period 1 = 207.37 Occurring at Location 24 

Maximum Structural Temperature for Period 2 = 225.86 Occurring at Location 21 

Maximum Structural Temperature for Period 3 = 231.88 Occurring at Location 21 

Maximum Structural Temperature for Period 4 = 222.15 Occurring at Location 21 

Maximum Structural Temperature for Period 5 = 202.43 Occurring at Location 24 

Maximum Structural Temperature for Period 6 = 194.68 Occurring at Location 39 

Maximum Structural Temperature for Period 7 = 173.92 Occurring at Location 39 

 

Maximum Structural Temperature Over All 7 Periods =231.88 Occurring at Location 21 at

Period 3 

 

FIGURE 6. A SAMPLE OUTPUT FILE: summary.dat (Continued) 

9
TEMPERATURES AT THE END OF TRANSIENT ANALYSIS 

 

 TEMPERATURES IN DEGREES FAHRENHEIT

 1 2 3 4 5 6 7 

 

 1 112.0 115.6 120.5 123.0 123.2 123.1 121.1 

 2 112.3 116.0 120.9 123.3 123.3 122.9 120.6 

 3 122.6 130.8 134.5 132.8 126.1 115.7 104.5 

 4 122.9 131.3 135.0 133.0 126.2 115.4 104.0 

 5 123.2 131.6 135.3 133.3 126.2 115.3 103.6 

 6 114.4 120.2 124.7 124.6 121.1 116.3 111.0 

 7 114.3 120.1 124.6 124.6 121.2 116.5 111.3 

 8 114.3 120.0 124.5 124.6 121.3 116.7 111.6 

 9 112.2 115.7 120.7 123.2 123.5 123.5 121.7 

 10 112.1 115.6 120.6 123.2 123.5 123.7 122.0 

 11 112.0 115.5 120.5 123.0 123.2 123.1 121.1 

 12 112.2 116.0 120.9 123.2 123.2 122.8 120.6 

 13 121.3 129.8 134.3 132.1 124.6 114.1 103.2 

 14 121.5 130.3 134.8 132.4 124.7 113.8 102.7 

 15 121.8 130.6 135.1 132.7 124.7 113.6 102.2 

 16 114.3 120.2 124.7 124.6 121.1 116.3 110.9 

 17 114.3 120.1 124.6 124.6 121.2 116.5 111.2 

 18 114.2 120.0 124.5 124.5 121.3 116.7 111.5 

 19 112.2 115.7 120.7 123.2 123.5 123.5 121.7 

 20 112.1 115.6 120.6 123.2 123.5 123.7 122.0 

 21 121.5 126.7 131.9 133.3 131.6 128.9 124.2 

 22 123.1 128.2 133.4 134.7 132.8 129.9 124.6 

 23 169.8 175.0 176.9 175.1 169.9 160.9 137.9 

 24 171.3 176.5 178.2 176.3 171.1 162.0 138.4 

 25 177.8 179.5 178.8 178.5 176.0 169.0 142.7 

 26 171.4 171.6 171.2 171.5 170.8 166.5 141.6 

 27 150.7 153.2 154.6 155.6 153.8 146.7 127.7 

 28 149.6 152.2 153.8 154.8 152.9 145.9 127.4 

 29 146.9 149.7 151.5 152.5 150.7 143.9 126.3 

 30 146.9 149.6 151.5 152.5 150.6 143.9 126.3 

 31 131.8 135.3 138.5 139.7 138.1 133.6 122.4 

 32 131.8 135.3 138.5 139.7 138.1 133.6 122.4 

 33 131.8 135.3 138.5 139.7 138.1 133.6 122.4 

 34 131.8 135.3 138.5 139.7 138.1 133.6 122.4 

 35 116.8 121.1 125.5 127.0 125.7 123.4 118.5 

 36 116.7 120.9 125.3 126.9 125.6 123.4 118.5 

 37 111.9 115.3 120.3 122.9 123.4 123.6 121.9 

 38 111.8 115.1 120.1 122.8 123.3 123.6 122.0 

 39 117.4 119.2 122.0 126.5 128.2 128.6 125.9 

 40 118.8 120.4 123.0 127.6 129.3 129.7 126.5 

FIGURE 6. A SAMPLE OUTPUT FILE: summary.dat (Continued) 

10
 41 160.3 157.0 151.6 158.9 162.2 160.5 144.0 

 42 161.6 158.3 152.7 160.0 163.3 161.5 144.5 

 43 165.8 164.0 160.6 165.0 166.6 163.7 144.0 

 44 164.7 163.1 160.6 164.2 165.6 163.1 143.6 

 45 154.8 154.3 152.7 156.2 157.2 155.0 138.8 

 46 153.4 153.1 151.6 155.2 156.1 154.0 138.4 

 47 118.3 120.1 123.0 127.3 128.8 129.2 126.1 

 48 117.1 119.0 122.0 126.4 127.9 128.4 125.7 

 49 117.2 120.4 124.7 127.0 127.2 126.7 122.8 

 50 119.7 122.7 126.8 129.0 129.1 128.3 123.5 

 51 169.0 187.2 193.0 184.0 164.8 137.5 109.0 

 52 157.9 174.6 180.0 172.0 154.5 129.7 103.8 

 53 169.0 187.2 193.0 184.0 164.8 137.5 109.0 

 

Maximum Structural Temperature for Period 1 = 171.35 Occurring at Location 24 

Maximum Structural Temperature for Period 2 = 176.49 Occurring at Location 24 

Maximum Structural Temperature for Period 3 = 178.18 Occurring at Location 24 

Maximum Structural Temperature for Period 4 = 176.31 Occurring at Location 24 

Maximum Structural Temperature for Period 5 = 171.07 Occurring at Location 24 

Maximum Structural Temperature for Period 6 = 162.03 Occurring at Location 24 

Maximum Structural Temperature for Period 7 = 144.54 Occurring at Location 42 

 

Maximum Structural Temperature Over All 7 Periods =178.18 Occurring at Location 24 at

Period 3 

FIGURE 6. A SAMPLE OUTPUT FILE: summary.dat (Continued) 

11
 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 

 THERMOD: A THERMAL PROGRAM FOR AIRCRAFTS 

 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 

 

 

 TRANSIENT TEMPERATURES FOR PERIOD: 1 

 

 COOLING WHILE DOOR IS OPENED: 

 

 time(sec) 1 2 3 4 5 

 0.0000 124.0 124.0 123.7 123.7 123.7 

 2.0000 124.0 123.9 123.7 123.7 123.7 

 4.0000 124.0 123.9 123.7 123.7 123.7 

 6.0000 124.0 123.9 123.7 123.7 123.7 

. . . . . . 

. . . . . . 

. . . . . . 

 

 COOLING DURING AIRCRAFT MANEUVERS: 

 

 time(sec) 1 2 3 4 5 

 68.0000 123.9 123.9 123.7 123.7 123.7 

 70.0000 123.9 123.9 123.7 123.7 123.7 

 72.0000 123.9 123.9 123.7 123.7 123.7 

 74.0000 123.8 123.8 123.7 123.7 123.7 

 76.0000 123.7 123.8 123.7 123.7 123.7 

 78.0000 123.6 123.7 123.7 123.7 123.7 

 80.0000 123.6 123.6 123.7 123.7 123.7 

. . . . . 

. . . . . 

FIGURE 7. A SAMPLE OUTPUT FILE: transient.dat 

5. REFERENCES. 

1. Govindarajoo, N., “An Enhanced Thermal Model for Determining Aircraft Operating 

Temperatures,” FAA report DOT/FAA/AR-04/51, December 2004. 

2. Press, W.H. et al., “Numerical Recipes in FORTRAN: The Art of Scientific 

Computing,” Second edition, Cambridge Univ. Press, New York, NY. 1994. 

12
APPENDIX A—INPUT DATA

A total of 15 data sets are required to define the input file. Some data sets have subsets, as 

indicated in figure 5, and all data input is in free-format. A comma or a space must separate each 

entry in a data set. 

Figures A-1 through A-16 provide definitions of the variables involved in each data set. Data 

must be entered in the units indicated. Data format and input examples are given for each data 

set. Remarks at the end of each data set description provides further explanations on the 

variables involved in the set. 

For those interested in learning the fundamentals of heat transfer and to have a more in-depth 

understanding of the principles involved in THERMOD, reference A-1 is recommended. 

Data set 1-1 defines the air kinematic viscosity and Prandtl number. 

 ν = Air kinematic viscosity (ft2/sec) 

Pr = Prandtl number (dimensionless) 

Format and Example: 

 1 2

ν Pr 

198Ε−6 0.704

Field Contents Data Type and Value 

ν Air kinematic viscosity (ft2/sec) Real > 0.0 

Pr Prandtl number (dimensionless) Real > 0.0 

Remarks: 

1. A comma or a space must separate each entry in a row. Data input is in free-format. 

2. Air kinematic viscosity and Prandtl number are used in automatic convective coefficient 

calculations. They are temperature-dependent to a degree. It is recommended that these 

properties be based on an anticipated average temperature. 

Figure A-1. Data Entry for Data Set 1-1 

A-1
Data set 2-1 to 2-28 defines geometric and material properties for each layer. These properties 

are thickness, density, thermal conductivity, and specific heat capacity. 

t = Thickness (in.) 

ρ = Density (lb/ft3) 

k = Thermal conductivity ( (Btu/hr)/(ft °F) ) 

cp = Specific heat capacity ( Btu/(lbm °F) ) 

Format and Example: 

 1 2 3 4 5 

Layer # t ρ k cp 

1 0.02 125.0 0.30 0.30

Field Contents Data Type and Value 

Layer # Layer number Integer 1-28 

t Layer thickness (in.) Real > 0.0 

ρ Layer density (lb/ft3) Real > 0.0 

k Layer thermal conductivity ((Btu/hr)/(ft °F)) Real > 0.0 

cp Layer specific heat capacity (Btu/(lbm °F)) Real > 0.0 

Figure A-2. Data Entry for Data Sets 2-1 to 2-28 

A-2
Remarks: 

1. A comma or a space must separate each entry in a row. Data input is in free-format. 

2. There are a total of 28 layers in THERMOD. This indicates that there are a total of 28 

rows of data to be input. All 28 rows must be present. Each row must have all five 

entries as indicated in the table above. If for some reason, a layer is not needed, the row 

otherwise representing this layer must still be filled with all five entries. However, the 

thermal effects of this layer are rendered ineffective by doing one of the following: 

(1) use a relatively small value for t, the layer thickness; (2) use a relatively high value 

for k, the layer thermal conductivity; or (3) carry out (1) and (2) simultaneously. A 

sensitivity study of sorts involving the thickness and thermal conductivity of the 

nonexistent layer in question could shed light on the satisfactory values of t and k to be 

used that will not affect the overall thermal behavior of the model. It should be noted, 

however, that too low a value for these two variables may cause numerical problems. 

3. In some cases, a layer may actually be made of more than one material. In such 

instances, composite values should be used for the layer. The composite values are 

derived as follows, where n = number of materials within the layer in question. 

( )

( )

∑( ) ∑

∑ ∑

∑ ∑

∑

= =

= =

= =

=

=

=

=

=

n

i

i

n

i

p p i

n

i

i

n

i

i i

n

i

i i

n

i

i

n

i

i

c c t t

t t

k t t k

t

n

t

i

1 1

1 1

1 1

1

1

ρ ρ

4. The 28 layers must be entered in a sequential manner. 

Figure A-2. Data Entry for Data Sets 2-1 to 2-28 (Continued) 

A-3
Data set 3-1 defines the overall geometry of an aircraft. This includes the wing and fuselage 

dimensions and transparency areas such as windows and windshields. Refer figures 1, 2, and 3 

for an understanding of the variables involved. 

Wwing = Average width of the wing from the leading edge to the trailing edge (ft.) 

Lwing = Length of the wing from tip to tip (ft.) 

Hwing = Average height of the wing from the tarmac to its mid-depth when the plane is on the 

ground (ft.) 

Lledge = Average distance of the wing leading edge to the front of the fuselage (ft.) 

Lflg = Overall fuelage length (ft.) 

Lcabin = Cabin length (ft.) 

Hcabin = Average cabin height (ft.) 

Wcabin = Average cabin width (ft.) 

A1 = Total surface area of the transparencies in the cabin area (ft2

) 

A2 = Total surface area of the cabin top (ft2

) 

A3 = Total surface area of transparencies that intercept diffused solar rays reflected from the 

tarmac (ft2

) 

Format and Example: 

 1 2 3 4 5 6 7 8 9 10 11 

Wwing Lwing Hwing Lledge Lflg Lcabin Hcabin Wcabin A1 A2 A3

4.0 39.0 3.5 5.5 26.0 15.0 3.5 4.5 25.0 20.0 15.0

Figure A-3. Data Entry for Data Set 3-1 

A-4
Field Contents Data Type and Value 

Wwing Average width of the wing from the leading edge Real > 0.0 

to the trailing edge (ft.) 

Lwing Length of the wing from tip to tip (ft.) Real > 0.0 

Hwing Average height of the wing from the tarmac to its mid- Real > 0.0 

depth when the plane is on the ground (ft.) 

Lledge Average distance of the wing leading edge to the Real > 0.0 

front of the fuselage (ft.) 

Lflg Overall fuselage length (ft.) Real > 0.0 

Lcabin Cabin length (ft.) Real > 0.0 

Hcabin Average cabin height (ft.) Real > 0.0 

Wcabin Average cabin width (ft.) Real > 0.0 

A1 Total surface area of the transparencies in the 

cabin area (ft2

) Real > 0.0 

A2 Total surface area of the cabin top (ft2

) Real > 0.0 

A3 Total surface area of transparencies that intercept Real > 0.0 

diffused solar rays reflected from the tarmac (ft2

) 

Remarks: 

1. A comma or a space must separate each entry in a row. Data input is in free-format. 

2. THERMOD simplifies the complex thermal problem by approximating the fuselage as a 

box with the dimensions as indicated. 

3. THERMOD simulates the green-house effect within the cabin. The areas A1, A2 and A3

are needed to simulate this effect. 

4. The distance of the fuselage and wing from the tarmac influences the radiation effects of 

the tarmac underneath these structures. 

5. The wing width, wing length, distance of wing leading edge to the fuselage front, and the 

overall fuselage length are needed for simulating solar reflections as well as for 

simulating radiation effects within enclosures. 

Figure A-3. Data Entry for Data Set 3-1 (Continued) 

A-5
Data set 4-1 defines the transmissivity property of transparencies and the relevant tarmac thermal 

properties. 

Tr = Transmissivity of transparencies (dimensionless) 

αta = Tarmac absorptivity (dimensionless) 

εta = Tarmac emissivity (dimensionless) 

Dta = Tarmac depth (ft.) 

ρta = Tarmac density (lb/ft3) 

kta = Tarmac thermal conductivity ((Btu/hr)/(ft °F)) 

cpta = Tarmac specific heat capacity (Btu/(lbm °F)) 

Format and Example: 

1 2 3 4 5 6 7 

Tr αta εta Dta ρta kta cpta

0.70 0.9 0.9 2.0 150 0.80 0.25

Field Contents Data Type and Value 

Tr Transmissivity of transparencies (dimensionless) Real: 0.0-1.0 

αta Tarmac absorptivity (dimensionless) Real: 0.0-1.0 

εta Tarmac emissivity (dimensionless) Real: 0.0-1.0 

Dta Tarmac depth (ft.) Real > 0.0 

ρta Tarmac density (lb/ft3) Real > 0.0 

kta Tarmac thermal conductivity ((Btu/hr)/(ft°F)) Real > 0.0 

cpta Tarmac specific heat capacity (Btu/(lbm°F)) Real > 0.0 

Figure A-4. Data Entry for Data Set 4-1 

A-6
Remarks: 

1. A comma or a space must separate each entry in a row. Data input is in free-format. 

2. THERMOD models the greenhouse effect within the cabin. The transmissivity of the 

transparent materials, including the windows and the windshields and the percentage of 

their areas, influence the amount of radiation that gets into the cabin. Only one 

transmissivity value is used for the entire cabin, even though a cabin may be made of 

transparencies with different transmissivity values. It is, therefore, essential to obtain one 

representative transmissivity value for the entire cabin. This value, based on a weighted 

average, is estimated as follows, where n = the number transparency surfaces and Ai = the 

area of each individual surface. 

∑

∑ =

=

=

n

i

n i i

i

i

ATr

A

Tr

1

1

1

Figure A-4. Data Entry for Data Set 4-1 (Continued) 

A-7
Data set 5-1 defines the absorptivity and emissivity properties within the cabin. A total of four 

surfaces are involved. Three of these surfaces are attributed to the symbolic insulatory materials 

(A), (B), and (C) as indicated in figures 1 and 2, where (A) = insulatory material below the cabin 

roof, (B) = insulatory material above the cabin floor, and (C) = insulatory material on the sides 

of the cabin. The fourth surface symbolically represents collective transparencies. 

αA = Absorptivity of material A (dimensionless) 

εA = Emissivity of material A (dimensionless) 

αB = Absorptivity of material B (dimensionless) 

εB = Emissivity of material B (dimensionless) 

αC = Absorptivity of material C (dimensionless) 

εC = Emissivity of material C (dimensionless) 

αtr = Absorptivity of transparency (dimensionless) 

εtr = Emissivity of transparency (dimensionless) 

Format and Example: 

 1 2 3 4 5 6 7 8 

αA εA αB εB αC εC αtr εtr

0.50 0.90 0.60 0.95 0.45 0.85 0.10 0.90

Field Contents Data Type and Value 

αA Absorptivity of material A (dimensionless) Real: 0.0-1.0 

εA Emissivity of material A (dimensionless) Real: 0.0-1.0 

αB Absorptivity of material B (dimensionless) Real: 0.0-1.0 

εB Emissivity of material B (dimensionless) Real: 0.0-1.0 

Figure A-5. Data Entry for Data Set 5-1 

A-8
Field Contents Data Type and Value 

αC Absorptivity of material C (dimensionless) Real: 0.0-1.0 

εC Emissivity of material C (dimensionless) Real: 0.0-1.0 

αtr Absorptivity of transparency (dimensionless) Real: 0.0-1.0 

εtr Emissivity of transparency (dimensionless) Real: 0.0-1.0 

Remarks: 

1. A comma or a space must separate each entry in a row. Data input is in free-format. 

2. THERMOD simulates the greenhouse effect within the cabin. The absorptivity and 

emissivity values of the cabin interior surfaces influence the greenhouse effect, and in 

turn the temperatures within the cabin, in a significant manner. 

3. The cabin interior surfaces are symbolically represented by Materials A, B, and C as 

shown in figures 1 and 2. These surfaces are typically nonstructural in nature. However, 

their thermal properties have significant bearing on the resulting interior temperatures. 

4. The absorptivity and emissivity values for these three surfaces are estimated averages. 

The values for each surface may be obtained by employing a weighting scheme that takes 

into consideration individual areas comprising the surface and the associated absorptivity 

and emissivity values, as shown below, where n = the number of surfaces within a 

material group and Ai = the area of each individual surface within the material group. 

∑

∑ =

=

=

n

i

n i i

i

i

A

A 1

1

1

α α ∑

∑ =

=

=

n

i

n i i

i

i

A

A 1

1

1

ε ε

5. The absorptivity and emissivity of the transparent materials, including the windows, 

windshields, and the percentage of their areas, influence the greenhouse effect within the 

cabin. Only one set of values is used for the entire cabin even though a cabin may be 

made of transparencies with different transmissivity values. It is, therefore, essential to 

obtain one representative set of absorptivity and emissivity values for the entire cabin. 

This value may be obtained in a fashion similar to calculations indicated in remark 4 

above. 

Figure A-5. Data Entry for Data Set 5-1 (Continued) 

A-9
Data set 6-1 defines the paint scheme demarcation used in the fillet region of the fairing where 

the wing meets the fuselage. The fillet is modeled as a perfect quadrant. It acts as a partial 

cavity, thereby contributing to an increase in temperature in this region. The type of paint to be 

used on this fillet is governed by two entries. 

pfillet = Paint scheme demarcation code 

percent = Percentage of fillet painted with wing paint 

Format and Example: 

 1 2 

pfillet percent

3 80.0

Field Contents Data Type and Value 

pfillet Paint scheme demarcation code Integer: 1-3 

percent Percentage of fillet painted with wing paint Real: 0.0-100.0 

Remarks: 

1. A comma or a space must separate each entry in a row. Data input is in free-format 

2. The paint scheme demarcation code, pfillet, assumes one of three integers, 1, 2, and 3, 

where: 

1: the fillet is completely painted with the same paint that was used on the wing. 

2: the fillet is completely painted with the same paint that was used on the fuselage. 

3: the fuselage/wing paint boundary falls within the fillet. 

3. The percent entry = 0.0 when the pfillet entry = 1 or 2. When pfillet = 3, the percent

entry ranges from 0.0 to 100.0. For example, when percent = 0.0, the entire fillet 

assumes the fuselage paint; and when percent = 100.0, the entire fillet assumes the wing 

paint. 

Figure A-6. Data Entry for Data Set 6-1 

A-10
Data set 7-1 defines the scatter factors of the various surfaces. A scatter factor for a surface is 

defined as the fraction of reflected light that is diffused in nature. Reflected light that is not 

diffused, but rather is concentrated in a band is called specular reflection. The degree of scatter 

is a collective function of the surface texture as well as the degree to which it is polished. A dull 

surface will have a high scatter factor, while a mirror-like surface will have a low scatter factor. 

A total of four scatter factors are used in THERMOD. They are: 

sfB = Scatter factor for the floor surface of the cabin (material B) 

sfC = Scatter factor for the interior sides of the cabin (material C) 

sfw = Scatter factor for the exterior wing surfaces 

sff = Scatter factor for the exterior fuselage surfaces 

Format and Example: 

 1 2 3 4 

sfB sfC sfw sff

0.80 0.60 0.20 0.20

Field Contents Data Type and Value 

sfB Scatter factor for the floor surface (material B) Real: 0.0-1.0 

sfC Scatter factor for the interior sides of the cabin (material C) Real: 0.0-1.0 

sfw Scatter factor for the wing surfaces Real: 0.0-1.0 

sff Scatter factor for the exterior fuselage surfaces Real: 0.0-1.0 

Remarks: 

1. A comma or a space must separate each entry in a row. Data input is in free-format. 

2. A very dull surface may have a scatter factor of about 80 percent, while a polished 

surface may have a scatter factor of about 20 percent. In the first case, 20 percent of the 

reflected light is specular in nature, while in the second case, 80 percent of reflected light 

is specular in nature. 

Figure A-7. Data Entry for Data Set 7-1 

A-11
Data set 8-1 defines the absorptivity and emissivity properties of the aircraft exterior surfaces. A 

total of four surfaces are involved—one representing the exterior wing surface and the other 

three representing the fuselage surfaces. The properties are defined as follows: 

αfs = Absorptivity of fuselage sides (dimensionless) 

εfs = Emissivity of fuselage sides (dimensionless) 

αw = Absorptivity of wing (dimensionless) 

εw = Emissivity of wing (dimensionless) 

αft = Absorptivity of fuselage top (dimensionless) 

εft = Emissivity of fuselage top (dimensionless) 

αfb = Absorptivity of fuselage bottom (dimensionless) 

εfb = Emissivity of fuselage bottom (dimensionless) 

Format and Example: 

 1 2 3 4 5 6 7 8 

αfs εfs αw εw αft εft αfb εfb

0.50 0.90 0.60 0.95 0.45 0.85 0.50 0.90

Field Contents Data Type and Value 

αfs Absorptivity of fuselage sides (dimensionless) Real: 0.0-1.0 

εfs Emissivity of fuselage sides (dimensionless) Real: 0.0-1.0 

αw Absorptivity of wing (dimensionless) Real: 0.0-1.0 

εw Emissivity of wing (dimensionless) Real: 0.0-1.0 

αft Absorptivity of fuselage top (dimensionless) Real: 0.0-1.0 

εft Emissivity of fuselage top (dimensionless). Real: 0.0-1.0 

αfb Absorptivity of fuselage bottom (dimensionless). Real: 0.0-1.0 

εfb Emissivity of fuselage bottom (dimensionless). Real: 0.0-1.0 

Figure A-8. Data Entry for Data Set 8-1 

A-12
Remarks: 

1. A comma or a space must separate each entry in a row. Data input is in free-format. 

2. The exterior surface temperatures are influenced by paint color. The darker the paint, the 

higher the surface temperature, and conversely, the lighter the paint color, the lower the 

surface temperature. Note that darker paints will have higher absorptivity than lighter 

paints. Emissivity, on the other hand, serves to lower surface temperatures, i.e., the 

higher the emissivity, the lower the temperature, and vice versa. 

3. In addition to influencing surface temperatures, paint color also influences the interior 

temperatures due to the dynamics of a thermal system. 

4. Note that there is only one set of absorptivity and emissivity values for the entire wing 

(regardless of whether it is the right-hand side or left-hand side, or whether it is the top or 

the bottom). The fuselage, on the other hand, could have three sets of values—one for 

the side; one for the top, and one for the bottom.

Figure A-8. Data Entry for Data Set 8-1 (Continued) 

A-13
Data set 9-1 defines the number of time intervals needed to describe a flight profile. The flight 

profile is required as a means to generate time-varying convective coefficients. It is through 

these coefficients that heat is removed from the aircraft while it is in motion. An example flight 

profile is shown in figure 4. 

ntintvl = Number of time intervals in a flight profile 

Format and Example: 

 1 

ntintvl

22

Field Contents Data Type and Value 

ntintvl Number of time intervals in a flight profile Integer ≥ 1 

Remarks: 

1. The flight profile must be unique to the aircraft in consideration. It describes the motion 

of the aircraft on the ground and in the air. The end of the flight profile is defined when 

the aircraft is subjected to the most critical limit loads at the earliest possible time, which 

generally is while the aircraft is in the air. 

2. The flight profile is used in THERMOD to simulate time-varying convective coefficients, 

which are the transport agents through which heat is removed from the aircraft. 

Operating limit temperatures are determined at the end of the flight profile. 

Figure A-9. Data Entry for Data Set 9-1 

A-14
Data set 10-1 defines the speed at the beginning of each time interval. The speeds are needed to 

describe a flight profile. The flight profile is required as a means to generate time-varying 

convective coefficients. It is through these coefficients that heat is removed from the aircraft 

while it is in motion. An example flight profile is shown in figure 4. 

vi = Speed at the beginning of each time interval for describing a flight profile (mph) 

Format and Example: 

 1 2 3 4 5 6 7 8 9 10 

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10

v11 v12 v13 v14 ..... ..... Cont. .... .... .....

.... vntintvl vntintvl+1

0.0 10.0 10.0 0.0 0.0 75.0 110.0 110.0 190.0 190.0

200.0 205.0 220.0 225.0 .... .... .... .... .... ....

250.0 250.0 250.0

Field Contents Data Type and Value 

vi Speed at the beginning of each time interval for describing 

a flight profile (mph) Real ≥ 0.0 

Remarks: 

1. A comma or a space must separate each entry in a row. Data input is in free-format 

2. The flight profile must be unique to the aircraft in consideration. It describes the motion 

of the aircraft on the ground and in the air. The end of the flight profile is defined when 

the aircraft is subjected to the most critical limit loads at the earliest possible time, which 

generally is while the aircraft is in the air. 

3. The flight profile is used in THERMOD to simulate time-varying convective coefficients, 

which are the transport agents through which heat is removed from the aircraft. 

Operating limit temperatures are determined at the end of the flight profile. 

4. If the number of time intervals, ntintvl, is greater than 10, data are continued in the next 

row, and so on. The end of each row must end with a comma or a space, except for the 

last row. There must be a total of ntintvl+1 data points. 

Figure A-10. Data Entry for Data Set 10-1 

A-15
Data set 11-1 defines the duration of each time interval of the flight profile. The flight profile is 

required as a means to generate time-varying convective coefficients. It is through these 

coefficients that heat is removed from the aircraft while it is in motion. An example flight 

profile is shown in figure 4. 

timei = Duration of each time interval (sec) 

Format and Example: 

 1 2 3 4 5 6 7 8 9 10 

time1 time2 time3 time4 time5 time6 time7 time8 time9 time10

time11 time12 time13 time14 ..... ..... Cont. .... .... .....

.... timentintvl

5.0 10.0 15.0 25.0 5.0 15.0 10.0 20.0 20.0 30.0

30.0 30.0 35.0 35.0 .... .... .... .... .... ....

45.0 45.0

Field Contents Data Type and Value 

timei Duration of each time interval (sec) Real > 0.0 

Remarks: 

1. A comma or a space must separate each entry in a row. Data input is in free-format. 

2. The flight profile must be unique to the aircraft in consideration. It describes the motion 

of the aircraft on the ground and in the air. The end of the flight profile is defined when 

the aircraft is subjected to the most critical limit loads at the earliest possible time, which 

generally is while the aircraft is in the air. 

3. The flight profile is used in THERMOD to simulate time-varying convective coefficients, 

which are the transport agents through which heat is removed from the aircraft. 

Operating limit temperatures are determined at the end of the flight profile. 

4. If the number of time intervals, ntintvl, is greater than 10, data are continued in the next 

row, and so on. The end of each row must end with a comma or a space, except for the 

last row. There must be a total of ntintvl data points. 

Figure A-11. Data Entry for Data Set 11-1 

A-16
Data set 12-1 defines the number of requested times of the day at which thermal analysis is 

performed. For example, an analysis could be performed at 11:00 in the morning and another at 

2:00 in the afternoon. At each time, a complete set of data describing the climatic data is 

needed. 

ntime = Number of requested times of the day at which thermal analysis is performed 

Format and Example: 

 1 

ntime

7 

Field Contents Data Type and Value 

ntime Number of requested times of the day at which thermal 

analysis is performed Integer ≥ 1 

Remarks: 

1. THERMOD loops over all requested times of the day to give the maximum operating 

limit (MOL) temperature. 

2. Note that the maximum temperature in an aircraft does not necessarily occur at noon. 

The primary reason being that the maximum ambient temperature and the maximum solar 

radiation do not necessarily occur at the same time. 

Figure A-12. Data Entry for Data Set 12-1 

A-17
Data set 12-2 to 12-8 define the climatic- and transparency-related data at given times of day. 

The climatic data include the sun’s incident angle, ambient temperature, effective sky 

temperature, ambient wind speed, and total solar radiation. The transparency-related data 

provide information necessary to simulate the greenhouse effect. 

θi = The sun’s incident angle at a given time of day (degrees) 

Tambi = Ambient temperature at a given time of day (° Rankine) 

Tskyi = Effective sky temperature at a given time of day (° Rankine) 

Vwindi = Ambient wind speed at a given time of day (ft/sec) 

Qsoli = Total solar radiation (includes both direct and diffused solar radiation) at a given 

Time of day ((Btu/hr)/ft2

) 

Avi = Total projected area of the transparency that intercepts the vertical component of 

the sun’s radiation at a given time of day (ft2)

Ahi = Total projected area of the transparency that intercepts the horizontal component 

of the sun’s radiation at a given time of day (ft2

) 

FrQi = Fraction of solar radiation that is intercepted by the transparency (in terms of the 

projected areas) at a given time of day 

Format and Example: 

 1 2 3 4 5 6 7 8 

θ1 Tamb1 Tsky1 Vwind1 Qsol1 Av1 Ah1 FrQ1

θ2 Tamb2 Tsky2 Vwind2 Qsol2 Av2 Ah2 FrQ2

. . . . . . . .

. . . . . . . .

θntime Tambntime Tskyntime Vwindntime Qsolntime Avntime Ahntime FrQ ntime

60.0 565.67 459.67 10.0 250.0 10.0 12.0 0.30

75.0 570.67 459.67 12.0 260.0 11.0 13.0 0.45

. . . . . . . .

. . . . . . . .

30.0 560.67 459.67 8.0 210.0 10.0 10.0 0.50

Figure A-13. Data Entry for Data Set 12-2 to 12-8 

A-18
Field Contents Data Type and Value 

θi The sun’s incident angle at a given time of day (degrees) Real: 0.0-90.0

Tambi Ambient temperature at a given time of day (° Rankine) Real > 0.0

Tskyi Effective sky temperature at a given time of day (° Rankine) Real > 0.0

Vwindi Ambient wind speed at a given time of day (ft/sec) Real ≥ 0.0 

Qsoli Total solar radiation (includes both direct and diffused-solar 

radiation) at a given time of the day ((Btu/hr)/ft2

) Real ≥ 0.0 

Avi Total projected area of the transparency that intercepts the vertical 

component of the sun’s radiation at a given time of day (ft2

) Real ≥ 0.0 

Ahi Total projected area of the transparency that intercepts the horizontal 

component of the sun’s radiation at a given time of day (ft2) Real ≥ 0.0 

FrQi Fraction of solar radiation that is intercepted by the transparency 

(in terms of the projected areas) at a given time of day Real ≥ 0.0 

Remarks: 

1. A comma or a space must separate each entry in a row. Data input is in free-format. 

2. THERMOD loops over all requested times of the day to give the MOL temperature. 

3. Note that the maximum temperature in an aircraft does not necessarily occur at noon. 

The primary reason being that the maximum ambient temperature and the maximum solar 

radiation do not necessarily occur at the same time. This makes it necessary to loop over 

different times of the day to obtain the MOL temperature. 

Figure A-13. Data Entry for Data Set 12-2 to 12-8 (Continued) 

A-19
Data set 13-1 defines the type of finite difference method of analysis to be employed, the time 

increment involved, the tolerable ambient cabin temperature before taxi, and the amount of data 

to be printed. 

fdmethod = Type of finite difference method selected in the transient phase (cooling phase): 

explicit forward finite difference method or the implicit backward finite 

difference method. 

deltime = Delta time increment in the transient phase (cooling phase) (sec.). 

ambcabt = Tolerable ambient cabin temperature immediately before taxi, i.e., after cooling 

has taken place by deliberately opening the door (°F). 

intvprnt = The number of deltime before the next set of data is printed in the transient phase. 

Format and Example: 

 1 2 3 4 

fdmethod deltime ambcabt intvprnt

2 0.01 135.0 100

Field Contents Data Type and Value 

fdmethod Type of finite difference method selected in the 

transient phase (cooling phase): explicit forward 

finite difference method; or the implicit backward 

finite difference method. Integer: 1-2 

deltime Delta time increment in the transient phase (cooling 

phase) (sec.). Real > 0.0 

ambcabt Tolerable ambient cabin temperature immediately 

before taxi, i.e., after cooling has taken place by 

deliberately opening of the door (°F). Real > 0.0 

intvprnt The number of deltime before the next set of data is 

printed in the transient phase. Real > 0.0 

Figure A-14. Data Entry for Data Set 13-1 

A-20
Remarks: 

1. A comma or a space must separate each entry in a row. Data input is in free-format. 

2. Two types of finite difference methods have been developed under THERMOD: the 

original explicit forward finite difference method (fdmethod = 1), and the more recent 

implicit backward finite difference method (fdmethod = 2). The explicit forward finite 

difference method, while relatively less complicated in terms of coding, is not 

unconditionally stable. The spatial and time-domain discretization cannot be performed 

arbitrarily. For example, too large a time increment may render the solution unstable, 

requiring a smaller time increment. In some cases, the time increment required is so 

small that a significant amount of time is needed for a solution, rendering the exercise 

impractical. These problems have been addressed through the use of the implicit 

backward finite difference method. This method is unconditionally stable. No spatial 

and time-domain restriction exists. Too large a time increment, however, may affect the 

accuracy of the results. 

3. THERMOD models the greenhouse effect, thereby simulating high temperatures within 

the enclosed cabin. It is only reasonable, then, to take advantage of whatever relief that 

can be likely expected, short of artificial cooling through air conditioning or a fan. 

THERMOD accomplishes this by stipulating that a comfortable temperature is reached 

within the cabin before entering the aircraft. This is simulated within the program by 

opening the door and allowing the relatively cooler outside air to cool the interiors to an 

average ambient temperature indicated by ambcabt. Once this temperature is reached, 

the door is closed and the aircraft is subjected to further cooling, according to a defined 

flight profile. 

Figure A-14. Data Entry for Data Set 13-1 (Continued) 

A-21
Data set 14-1 defines the number of iterations and convergence criteria used in the thermal 

analysis. These parameters are needed because of the nonlinear nature of THERMOD. 

niter = The number of iterations allowed in the thermal analysis 

tolx = A variable-based convergence criterion (dimensionless) 

tolfn = A function-based convergence criterion (dimensionless) 

Format and Example: 

 1 2 3 

niter tolx tolfn

10 0.01 0.1

Field Contents Data Type and Value 

niter The number of iterations allowed in the thermal analysis Integer ≥ 1 

tolx A variable-based convergence criterion Real > 0.0 

tolfn A function-based convergence criterion Real > 0.0 

Remarks: 

1. A comma or a space must separate each entry in a row. Data input is in free-format. 

2. The variable-based convergence criterion satisfies the following condition. 

( )

tolx

X

abs X X

j

j

j

j j

i

i i

≤

−

∑

∑

=

=

+

+

67

1

67

1

1

1

Where: 

j ij

j j

X X

X X i

The previous value of the unknown variable

The current value of the unknown variable 1

=

= +

Figure A-15. Data Entry for Data Set 14-1 

A-22
Remarks: 

3. The function-based convergence criterion satisfies the following condition. 

( )

tolfn

fn

abs fn fn

j

j

j

j j

i

i i

≤

−

∑

∑

=

=

+

+

67

1

67

1

1

1

Where: 

j j

j j

fn fn

fn fn

i

i

The previous value of the function

The current value of the function 1

=

= +

4. THERMOD ensures that both the criteria are satisfied before a satisfactory solution is 

reached. 

5. Trial and error and by comparison to previous results determine all three entries in this 

row. 

Figure A-15. Data Entry for Data Set 14-1 (Continued) 

A-23
Data Entry: Data Set 15-1 to 

Data set 15-5 defines the initial values of the unknown variables. An initial input set is required. 

This set is made of 67 initial values, as THERMOD is a nonlinear system with 67 independent 

equations. 

Xj = Initial value of the unknown variable Xj (°F or (Btu/hr)/ft2

). 

Format and Example: 

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 

X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15

X16 X17 X18 X19 X20 X21 X22 X23 X24 X25 X26 X27 X28 X29 X30

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

X61 X62 X63 X64 X65 X66 X67

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

10 10 10 10 10 10 10

Field Contents Data Type and Value 

Xj Initial value of the unknown variable Xj (°F or (Btu/hr)/ft2

). Real 

Figure A-16. Data Entry for Data Set 15-1 to 15-5 

A-24
Remarks: 

1. A comma or a space must separate each entry in a row. Data input is in free-format. 

2. A total of 67 entries are needed. To avoid possible problems, it is recommended that 

each row have 15 entries. The row must end with a comma or a space at the end of the 

15th entry to signify the continuation to the next row. The last entry should not end with 

a comma. 

3. The first 53 entries are temperature entries and are in (°F). The final 14 entries are 

radiosity entries and are in ((Btu/hr)/ft

2

). Radiosity is defined as the total energy 

emanating from a surface that includes both the reflected as well as emitted energy. 

4. It is recommended that positive initial values be used for all 67 variables. 

Figure A-16. Data Entry for Data Set 15-1 to 15-5 (continued) 

REFERENCES. 

A-1. Incropera, F.P. and Dewitt, D.P., “Fundamentals of Heat and Mass Transfer,” third 

edition, John Wiley & Sons, New York, NY, 1990.

A-25/A-26