Anthony J. Colozza

Analex Corporation, Brook Park, Ohio

Convective Array Cooling for a Solar

Powered Aircraft

NASA/CR—2003-212084

January 2003
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Anthony J. Colozza

Analex Corporation, Brook Park, Ohio

Convective Array Cooling for a Solar

Powered Aircraft

NASA/CR—2003-212084

January 2003

National Aeronautics and

Space Administration

Glenn Research Center

Prepared under Contract NAS3–00145
Available from

NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076

National Technical Information Service

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The Propulsion and Power Program at

NASA Glenn Research Center sponsored this work.

Available electronically at http://gltrs.grc.nasa.gov
NASA/CR—2003-212084 1 

Convective Array Cooling for a Solar Powered Aircraft 

Anthony J. Colozza 

Analex Corporation 

Brook Park, Ohio 44142 

Introduction and Background 

 The ability to fly an unmanned aircraft under solar power has been 

demonstrated.1,2 The uniqueness of a solar powered aircraft, compared to a conventionally 

powered aircraft, lends its use to equally unique applications both military and civilian. 

Most of these applications involve subsonic flight at either high altitudes or for long 

durations or both. The advantages of solar power over conventionally powered fuel 

burning propulsion systems, for these types of applications, is that a solar power system 

does not require the use of the atmosphere in the generation of power. This eliminates the 

need for multiple stage compressors that are necessary for high altitude combustion 

driven aircraft. Also since the power for flight comes from the sun, no fuel supply is 

needed. If this solar system is coupled to a regenerative fuel cell system or rechargeable 

battery system of sufficient energy density (approximately 400 Wh/kg or greater) then 

long endurance flight on the order of weeks to months can be achieved depending on the 

flight location.3–5

 Since the available power for the aircraft is limited by the solar flux from the sun 

(approximately 1000 W/m2

) any increase in the ability to utilize this available power is 

beneficial to the aircraft’s performance. State-of-the-art production solar cells operate in 

the range of approximately 10 to 20 percent efficiency, depending on the type of solar 

cell and its operating temperature. By increasing the efficiency of the solar cells on a 

given aircraft, significant performance enhancements for the aircraft can be achieved. 

This can enable the aircraft to perform missions previously not possible. The increased 

capability in both mission duration, achievable altitude and location can be seen in 

Figure 1. This figure compares the maximum achievable altitude, at a given latitude, for 

the same solar aircraft with 14 and 18 percent efficient solar cells. The 14 percent solar 

cell efficiency was selected because it represents what is achievable with present day 

silicon solar cells. The 18 percent efficiency is an estimate of what can be achieved 

utilizing the same type of solar cells but having them operate at a lower temperature. The 

data is plotted for two dates, June 21st (summer solstice) and December 21st (winter 

solstice). These two dates represent the longest and shortest days, respectively, for the 

year. Maximum altitude curves for any other date should fall somewhere between these 

two sets of curves. From this figure it can be seen that the capabilities of the solar aircraft 

increase with increasing solar cell efficiency. An example of enhanced mission capability 

can be shown by examining the highest altitude that can be achieved for a specific 

latitude. For a maximum altitude of 20 km on December 21st, the highest flight latitude 

increases from 25° to 34° with an increase in cell efficiency from 14 to 18 percent. This 

represents a 34 percent increase in possible flight latitudes, which greatly enhances the 

aircraft’s mission capabilities. These and other benefits are also shown in References 3, 4 

and 5. 
NASA/CR—2003-212084 2 

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Latitude (degrees)

0

5

10

15

20

25

30

35

Maximum Altitude (km)

 Aspect Ratio 24

50 Meter Wingspan

June 21, 18% Solar Cell Efficiency

June 21, 14% Solar Cell Efficiency

Dec. 21, 14% Solar Cell Efficiency Dec. 21 18% Solar Cell Efficiency

Figure 1.—Effect of Increase In Solar Cell Efficiency on Aircraft Performance. 

 In order to achieve a lower solar cell operating temperature, it has been suggested 

that the wing of a solar powered aircraft with an integral thermal control system can be 

used to take advantage of the airflow over the wing to assist in cooling the solar cells. 

The concept, shown in Figure 2, directs a small amount of air through a duct in the wing 

to convectively cool the backside of the solar cells. This is an additional heat transfer 

mechanism, which adds to convective and radiative cooling of the solar cells from the 

topside. This additional heat transfer mechanism can significantly reduce the operating 

temperature of the solar cells, thereby improving their efficiency, and it may be possible 

to achieve this with little or no detrimental aerodynamic drag. 

The relationship between solar cell efficiency and temperature has been well 

documented. This effect is consistent for all types of solar cells. Most solar arrays do not 

have active cooling incorporated into the array design. This is because adequate cooling 

can most often be obtained by using the large area on the backside of the solar array as a 

waste heat radiator. An additional increase in the efficiency of the solar cells could be 

achieved by incorporating a cooling system to enhance heat dissipation; however, in 

almost all applications, the cost, complexity and added mass make this undesirable. 

Therefore little research has been done on arrays that have airflow passages for 

convective cooling. However, in an application such as a solar powered aircraft, in which 

an air stream of cold air is readily available, the benefits of convective cooling can be 

substantial. By reducing the temperature of the solar cells, a significant increase in solar 

cell efficiency can be achieved. For example, a typical commercial silicon solar cell 
NASA/CR—2003-212084 3 

efficiency will increase from 13 to 17 percent when cooled from 25 to –25° C. This 

translates into a 30 percent increase in performance. The change in efficiency with 

temperature for various types of solar cells is shown in Figure 3.6

 A strong trend of 

increasing efficiency with temperature is shown for six different types of solar cells. As 

temperature goes down, efficiency improves dramatically. It is estimated that this 

increase in cell efficiency can reduce the size of the aircraft or increase the available 

power with little impact on the weight or size of the aircraft. 

Figure 2.—Heat Transfer Mechanisms for a Solar Array with Integral Cooling Passage. 
NASA/CR—2003-212084 4 

-150 -100 -50 0 50

Temperature (°C)

0

5

10

15

20

25

30

Efficiency (%)

Silicon

InP

GaAs

GaAs (concentrator)

GaSb

GaInP/GaAs

Figure 3.—Effect of Temperature on Efficiency of Various Types of Solar Cells.6 

Preliminary Analysis 

 It is a well understood principle that the operating efficiency of a solar cell is 

directly related to its operating temperature. The proposed concept is based on the ability 

to reduce the operating temperature of the solar cells by circulating low temperature air, 

available at flight altitudes, underneath the solar cells thereby cooling them. In order for 

this concept to be feasible and practical for use in an aircraft, it must be shown that, by 

circulating air underneath the solar cells, their operating temperature can be lowered 

enough to produce a justifiable increase in efficiency without affecting the performance 

of the aircraft. A preliminary analysis has been performed and is outlined below. This 

analysis was done to establish the feasibility of the concept and to estimate the 

performance increase that may be achieved. Figure 2 shows the proposed concept and the 

heat transfer mechanisms involved in establishing the solar cell temperature. The energy 

balance for the solar cells (energy in, Ein, is equal to energy out, Eout) and subsequent heat 

balance is given by the following equations where Qsolar is the heat input form the incident 

solar radiation falling on the solar cells, Qconduction is the heat transferred from the solar cells 

by conduction to the surrounding aircraft, Qconvection is the heat transferred from the solar 

cells by convection to the air stream, Qradiation is the heat transferred from the solar cells by 

radiation to the surroundings and Qelectric is the electrical power produced by solar cells. 
NASA/CR—2003-212084 5 

Ein = Eout (1) 

 Qsolar = Qconduction + Qconvection + Q radiation + Qelectric (2) 

The input solar radiation on a clear day can be assumed to be approximately 1000 W/m2

and assuming a 15 percent efficient solar cell the output power (Pout) would be 150 W/m2

. 

Based on an operating temperature of approximately 20 to 30 °C, for a solar cell installed 

on the wing of an aircraft, the main heat transfer mechanism will be the convective 

cooling over the wing surface. Therefore combining the radiative and conductive heat 

transfer terms together and calling them Qsecondary the heat transfer equation becomes; 

 850 (W/m2

) = Qconvection + Qsecondary (3) 

The convective heat transfer term is based on the solar cell temperature (T), the 

surrounding atmospheric temperature (Ta) and the convective heat transfer coefficient 

(h1). 

 Qconvection = h1(T – Ta) (4) 

 Some assumptions must be made about the aircraft and its flight conditions. These 

assumptions are based on typical missions and flight conditions for a high altitude 

unmanned solar powered aircraft. 

 Flight Altitude 18 km 

 Flight Velocity (U) 0.1 Mach 

 Wing Chord (c) 1 m 

The convective heat transfer coefficient over the surface of the wing can be estimated as 

follows. The Reynolds number based on chord (Rel) length is given by; 

Rel =UFreeStream

L

u

 (5) 

where: 

 Free Stream Velocity, Ufree stream : 30 (m/s) 

 Wing Chord Length, L: 1 (m) 

 Viscosity, u: 1.165E-4 (m2

/s) 

This produces a Reynolds number of 257,510. Since this Reynolds number value is less 

then 500,000 it can be assumed that the flow is laminar over the wing surface. For 

laminar flow, the Nusselt number based on chord length (Nul

) is given by Equation 6.7

 Nul

 = 0.664 Rel

1/2 Pr 1/3 (6) 
NASA/CR—2003-212084 6 

where the Prandtl number (Pr) is given by the following equation. 

Pr = c pm

k (7) 

 Constant Pressure Specific Heat of Air, cp = 1.006 (KJ/Kg °K) 

 Air Mass Flow, m = 1.422E-5 (Kg/m s) 

 Thermal Conductivity of Air, k = 0.0195 (W/m °K) 

Using these values Pr = 0.734 and Nul

 = 303.95 the heat transfer coefficient for the 

surface of the wing, h1, is given by the following equation, where c is the wing chord 

length. 

h1 = Nuk

c = 5.927( W

m2 °K

) (8) 

 Based on previous experience and tests conducted at the NASA Glenn Research 

Center the operating temperature of a solar cell in a sealed wing under 1000 W/m2

 

illumination can be estimated to be 293 °K.8,9 Based on this information, Equation 3 can 

be used to estimate the heat transfer due to the secondary mechanisms. Substituting in 

the values determined above into this equation yields; 

850 = 5.927 (293 °K – 216 °K) + Qsecondary (9) 

Solving for Q secondary yields 

 Qsecondary = 394 W/m2

 (10) 

Now by adding a convective cooling passage within the wing a second convective heat 

transfer path is obtained. The convective heat transfer coefficient (h2) for this second path 

must now be determined. The Reynolds number for this passage, based on the passage 

wall spacing (d), is given by the following equation where ν (m2/s) is the kinematic 

viscosity of air. 

Red =U

d

ν

 (11) 

Using a passage wall spacing of 0.009 m yields a Reynolds number of approximately 

2300. This value of Reynolds number is just at the edge of transition from laminar to 

turbulent for flow between parallel plates therefore it can be assume that the flow is 

laminar. For laminar flow in a rectangular duct with an infinite aspect ratio, the Nusselt 

number has a value of 8.23. The equation for Nusselt number based on the passage wall 

spacing (Nud) is given in Equation 12.7

 

Nud = h2

d

k = 8.23 (12) 
NASA/CR—2003-212084 7 

This equation can be solved for h2 by substituting in the values for the remaining 

parameters. 

h2 = (8.23)(0.0195)

(0.009) =17.83( W

m2

°K

) (13) 

Since the actual value of Qsecondary for an array utilizing the cooling passage is not known, 

the cell temperature with the cooling passage can be calculated for both extremes, when 

Qsecondary is equal to the value without the cooling passage and when Qsecondary is equal to 

zero. The actual cell temperature will lie somewhere between these extremes. For the first 

case, using the value of Qsecondary given in Equation 9 and the calculated convective heat 

transfer coefficients, h1 and h2, the temperature of the solar cells is given by the following 

equation. 

 850 = 5.927 (T – 216 °K) + 17.83(T – 216 °K) + 394 (14) 

Solving this for the solar cell temperature, T, yields; 

 T = 235 °K (15) 

To bound the problem the temperature when Qsecondary is equal to zero is calculated by the 

following equation. 

 850 = 5.927 (T – 216 °K) + 17.83(T – 216 °K) (16) 

Solving for the solar cell temperature, T, yields: 

 T = 252 °K (17) 

The actual solar cell temperature will lie somewhere between these two extremes. It is 

this actual temperature that will be determined through a more through analysis in the 

following section. For the purposes of this preliminary analysis the average of these two 

temperature extremes, 243 °K, will be assumed to be the resulting solar cell temperature. 

Comparing this temperature to that estimated for the solar cell array with no cooling 

passage yields: 

∆T = 298 °K – 243 °K = 55 °K (18) 

 This ∆T value represents a significant drop in the temperature of the solar cells 

compared to their temperature when no cooling passage is used. The effect of this 

temperature drop on the performance of the solar cells can be determined from the data 

given in Figure 2. For silicon solar cells, the change in efficiency corresponding to this 

∆T is approximately 3.5 percent. This translates into an estimated increase in cell 

performance of 25 percent. 
NASA/CR—2003-212084 8 

 This preliminary analysis demonstrates the feasibility of using a cooling passage 

to significantly reduce the solar cell temperature and thereby increase its efficiency. The 

next main feasibility issue is to assess the effect this cooling passage will have on the 

performance of the aircraft. This can be done by calculating the pressure drop in the 

cooling passage and determining the corresponding drag. 

 The Reynolds number for the flow through the cooling passage was given above 

as 2300. The friction factor for laminar flow between parallel plates (flam) is given by the 

following equation.10

 flam = 96/Red = 0.0417 (19) 

The pressure drop in the cooling passage (∆P) is given by the following equation. Where 

ρ is the air density at altitude,13 d is the passage spacing (0.009 m), U is the free stream 

velocity (30 m/s) and L is the wing chord length (1 m). 

∆P = ρ flam L U2

/8d (20) 

substituting the values for the variables yields 

∆P = 63.6 Pa (21) 

For a wingspan of 20 m, assuming the cooling passage extends the length of the wing, the 

frontal area (Af

) of the cooling passage is 0.18 m2. This area yields a drag (D) of: 

 D =∆P Af

 = (63.6)(0.18) = 11.45 N (22) 

 The total drag of a similarly sized, aerodynamically clean aircraft at similar flight 

conditions without the cooling passage would be approximately 150 N. This translates 

into a 7 percent increase in the drag of the aircraft. Comparing the drag increase of 

7 percent to the increase in solar cell performance of 25 percent indicates that, based on 

this preliminary analysis, this concept will have beneficial effects on the performance of 

the aircraft. 

Detailed Analysis 

 Based on the preliminary results, a more detailed analysis was warranted in order 

to fully evaluate this concept. A computer-based analytical model of the solar array with 

a cooling passage was developed. This model takes into account the heat transfer from 

the solar cells to the surroundings and the fluid flow over the cells and through the 

cooling passage. The heat transfer analysis was accomplished by setting up an energy 

balance including radiative and convective heat transfer mechanisms. Since it was 

assumed that the solar cells were isolated from the aircraft structure, the conductive heat 

transfer path was assumed to be minimal and therefore not included in the analysis. 

Solar Flux Into Solar Cells = Heat Transfer Out of Solar Cells + Electrical Energy Produced (23) 
NASA/CR—2003-212084 9 

 This analysis was used to predict the temperature of each solar cell along a cross 

section of the wing chord and determine its corresponding output power. This analysis 

takes into account the variation in electrical power output of the array as a function of 

temperature and the variation of input power from the sun, as a function of the day of the 

year, location and time of day. An example of this variation is shown in Figure 4. The 

analytical model was then used to optimize the array cooling passage in order to produce 

the greatest output power for given flight conditions and to keep the drag produced by the 

cooling passage to a minimum. The main issues examined through this analysis are the 

change in output power of the array due to the addition of the cooling passage and the 

drag produced by the cooling passage. 

0 2 4 6 8 10 12 14 16 18 20 22 24

Time of Day (hour)

Available Power

0° Lat, Date 1/21

0° Lat, Date 6/21

60° Lat, Date 6/21

60° Lat, Date 1/21

Figure 4.—Available Power Curves for Various Location and Date Combination. 

 The computer code, which was developed through this modeling process, utilizes 

various inputs that represent the configuration of the array and wing section and flight 

conditions to determine the desired quantities such as cell temperature, output power and 

drag. This analysis was done through an iterative process. A diagram of the proposed 

computer code logic is shown in the flow chart given in Figure 5. The ability to model 

the array under various conditions and configurations is necessary in order to provide a 

clear indication as to the benefits of the integral air-cooling passage on the solar cell 

performance. The following is a brief description of the analysis that was performed. 
NASA/CR—2003-212084 10 

Input 

Solar Radiation

Input Solar

Cell Characteristics

Input Array 

Characteristics

Input Flight

Conditions

Radiative Heat 

Transfer Calculation

Conductive Heat 

Transfer Calculation

Convective Heat 

Transfer Calculation

Array Cell

Temperature

Calculation

Array Output

Power Calculation

OutPut Array 

Performance

Figure 5.—Computer Code Analysis Flow Chart.

 The first step in determining the cell performance is to calculate the incoming 

solar flux and incident angle of the solar radiation on the solar cell as well as the amount 

of this energy reflected off of the cell. The solar flux (SF), which varies slightly 

throughout the year, is determined by calculating the actual distance the earth is from the 

sun, r, at a selected time of year. This is given by the following equations: 

r = rm

(1−ε

2)

(1+ε cos(α)) (24) 

α = 2π(n − 4)

365 (25) 

SF = SFmτ

rm

2

r2 (26) 

Where the mean radius of the earth (rm) is 1.496E8 km, Earth’s orbital eccentricity (ε) is 

0.017, the mean solar flux at orbit (SFm) is 1352 W/m2–4

and the day number (n) is based 

on the day of the year (for January 1, n = 1). For this analysis the solar attenuation factor 

due to the atmosphere, τ, is assumed to be 0.80. 

 The incident angle of the incoming solar radiation (θ) on the solar cell is 

dependent on the solar declination angle (δ), the latitude (φ), the orientation of the solar 

cell represented by its inclination angle (β) and azmuth angle (γ) and the time of day 

Calculate Flow field 

Properties through 

Cooing Passage 

and over Airfoil 

Calculate Pressure 

Drop through Cooling 

Passage and Determine 

Drag Input Array Cooling 

Passage and Inlet 

Characteristics 
NASA/CR—2003-212084 11 

represented by the hour angle (ω). The equations for determining the incident angle are as 

follows.3,4,11

 Earth’s declination angle (δ) varies with the day of the year (dn). This day number 

(dn) is based on the vernal equinox, so dn = 1 is March 21st. 

δ = 0.4091Sin(

2πdn

365 ) (27) 

 The hour angle (ω) is given by the following expression, where “i” is the 

instantaneous time of day in hours. The convention for ω is negative before noon, 

positive after noon and zero at noon. 

ω = 2π

(i−11.968)

23.935 (28) 

 The inclination angle (β) of the solar cell is dependent on its location on the 

surface of the airfoil and the length of the individual solar cell (l). Since the airfoil is 

curved, each cell along the chord will be at a different inclination angle. A curve fit of the 

airfoil upper surface can be used to determine the cell incident angle by calculating the 

height of the wing surface from the centerline at the beginning and end of each cell. The 

angle produced with the horizontal based on the difference of these two heights is the 

inclination angle. The inclination angle of the cells on the front portion of the airfoil will 

be positive and those on the rear portion will be negative. The determination of the cell 

angle is illustrated in Figure 6 for a NACA 0009 airfoil. It should be noted that these 

equations assume that the wing chord centerline is horizontal. If the aircraft is climbing 

or performing any other maneuver then the deviation of the centerline off of the 

horizontal must be added (or subtracted) to the inclination angle. 

 The following equations represent the curve fit for the NACA 0009 upper surface 

of the airfoil. 

 y = 0.008 + 0.349x – 1.044x2

 + 1.179x3– 0.494 x4 (29) 

β = tan−1(ya − yb )

l (30) 

 The last angle needed in order to calculate the incidence angle is the azimuth 

angle (γ) for the wing. The convention for this angle is zero due South, East is negative 

and West is positive. γ can have values of –π to +π. The diagram shown in Figure 7 

demonstrates the convention for this angle. 
NASA/CR—2003-212084 12 

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1

Series1

Centerline

PV Cell

Ya Yb Xa

Xb

Figure 6.—NACA 0009 Airfoil and PV Cell Coordinate Geometry. 

Figure 7.—Coordinate System for Hour Angle (ω) and Azimuth Angle (γ). 
NASA/CR—2003-212084 13 

Figure 8.—Reflectance of Solar Radiation as a Function of Incident Angle. 

 Once the angles described above are determined, the incident angle of a given cell 

can be calculated by the following relationship.12

 θ = acos( (sin(δ) sin(φ) cos(β)) – (sin(δ) cos(φ) sin(β) cos(γ)) + 

 (cos(δ) cos(φ) cos(β) cos(ω)) + (cos(δ) sin(φ) sin(β) cos(γ) cos(ω)) (31) 

 + (cos(δ) sin(β) sin(γ) sin(ω))) 

After the inclination angle is known it is now possible to determine the percent of solar 

energy that is reflected off of the surface of the solar cell. The percent reflectance is 

based on the type of solar cell material, surface coating and incident angle of the solar 

radiation. For this analysis the solar cell material was assumed to be Silicon with a VMI 

antireflection coating. The reflectance for this type of material is shown in Figure 8 and 

the corresponding curve fit of reflectance (RF) versus incident angle in radians is given in 

Equation 32. 

 RF = 0.99031 + 17.1θ – 91.459 θ2

 + 209.74 θ3– 201.52 θ4 + 73.893 θ5 (32) 

 Utilizing the above equations to determine the amount of solar energy entering the 

solar cell, an energy balance on each solar cell can be performed in order to determine its 

operating temperature. This energy balance is given in Equation 2 where the incoming 

solar energy (Qsolar) can be represented by the following equation. % reflectance 

Solar Incidence Angle (radians) 
NASA/CR—2003-212084 14 

 Qsolar = SF (1-RF) cos(θ) (33) 

As mentioned previously, the conduction term (Qconduction) from Equation 2 is assumed to 

be zero. The remaining terms for output power (Qelectric), convection heat transfer (Qconvection) 

and radiation heat transfer (Qradiation) are all dependent on the solar cell temperature (T). 

Therefore the energy balance equation must be solved iteratively for each solar cell along 

the chord of the wing. 

 The dependence of the solar cell efficiency and therefore electrical output power 

on temperature is shown in Figure 9 for a Silicon solar cell. The solar cell efficiency 

(SCeff) curve as a function of temperature (T) shown in Figure 9 is represented by 

Equation 34. 

Figure 9.—Silicon Solar Cell Efficiency as a Function of Temperature. 

 SCeff = 0.55977 – 0.0035564 T + 1.2178e-05 T2

 – 1.7013e-08 T3 (34) 

Therefore, based on the above relation for solar cell efficiency, the electrical output of the 

solar cell can be expressed by Equation 35. 

 Qelectric = SF (1-RF) cos(θ) SCeff (35) 

 The convective heat transfer mechanism can be broken up into two portions; the 

convective heat transfer over the upper surface of the wing, Qsurface, and the convective 

heat transfer through the cooling passage under the solar cells, Qpassage. Solar Cell Efficiency 

Solar Cell Temperature °K 
NASA/CR—2003-212084 15 

 Qconvection = Qsurface + Qpassage (36) 

Where Qsurface is given by Equations 4 to 8. Since this detailed analysis is being done for a 

general case, relations for the physical properties of the atmosphere are needed. These 

can be found in most standard atmosphere texts such as that given in Reference 13. The 

relations for the air conductivity (k) and viscosity (ν) are given below as a function of 

atmospheric temperature, Ta. 

k = 6.325e − 7Ta

1.5

(Ta + 24510

−12

Ta )4186.8

 (37) 

ν = 2.17e −5( Ta

237)

0.67 (38) 

Also, Equation 6 that represents the Nusselt number (Nu), is valid only for Reynolds 

numbers less then 500,000. For Reynolds numbers (Re) greater then 500,000 the 

following relation is used.7

 Nu = (0.037 Re0.8

– 871) Pr1/3 (39) 

The convective heat transfer due to the cooling passage must also be calculated. This heat 

transfer path is represented by the following equation, where Tai is the inlet air 

temperature in the cooling passage at the beginning of a given cell and Tao is the exit air 

temperature in the cooling passage for the same cell. 

Qpassage = h2(Tai −Tao)

log(T − Tao

T −Tai

)

 (40) 

where Tao is given by the following equation. 

 Tao = T + (Tai −T)e

− ch2

dUρc p (41) 

 The final heat transfer mechanism that must be accounted for is radiation heat 

transfer from the cell to the surroundings, Qradiation. For this analysis it is assumed that the 

radiation heat transfer takes place between the solar cell and the sky only, no radiation is 

considered between the solar cells and the aircraft structure. The following equation 

represents the radiation heat transfer. It is assumed that the emissivity of Silicon (ε) is 0.3 

and the sky temperature (Tsky) is 263 °K. The Boltzman constant (σ) is 5.67e-8 W/m2 °K 

 Qradiation = ε σ (T4

– T4

sky) (42) 

 The heat transfer mechanisms described above allow the temperature of each cell 

to be calculated for a given set of flight conditions and wing geometry. Each cell 
NASA/CR—2003-212084 16 

temperature is calculated individually along the wing chord. This is necessary since the 

air temperature within the cooling passage changes as it passes beneath each cell. 

Utilizing this approach, the input quantities can be varied and the effectiveness of the 

cooling passage on increasing the aircraft’s performance can be determined. The results 

for the above analysis are given in the following section. 

Results 

 Based on the analysis method described in the previous section, data was 

generated to determine the effect of the cooling passage on the power generation 

capabilities of a high altitude solar powered aircraft. Using a range of variables, the 

performance of the solar array with the cooling passage was determined for a number of 

flight conditions. The passage spacing was first optimized by determining the spacing 

that produced the greatest increase in solar array performance, while minimizing the 

effect of drag on the aircraft. This was done by generating data on solar cell efficiency 

and drag per unit wing length versus cooling passage spacing. This is shown in Figure 10. 

This initial optimization was performed for a set of selected flight conditions and solar 

cell and airfoil geometry. This base flight configuration was chosen as the flight 

conditions and geometry from which all variations will be compared. In other words, 

variations from this initial configuration will be used to determine what impact the 

variables have on the performance of the cooling passage. The values for the variables in 

this initial flight configuration are given below in Table 1. 

Table 1.—Base Flight Conditions 

Solar Cell (Type Size) Silicon, 6 cm × 6 cm 

Airfoil NACA 0009 

Chord Length 1 m 

Flight Date May 21 

Flight Latitude 40° North 

Aircraft Orientation During Flight (N, S, E, W) South 

Flight Altitude 20 km 

 Using the data represented in Figure 10 and the performance of the solar array 

without any cooling passage, the percent change in the solar array output and drag was 

determined. This is shown in Figure 11. Average cell efficiency without the cooling 

passage, for the base flight conditions was determined to be 7.3 percent at an operational 

temperature of 80 °C. Drag for the aircraft was estimated to be approximately 150 N at 

the flight conditions. 
NASA/CR—2003-212084 17 

0 0.01 0.02 0.03 0.04 0.05

Passage Spacing (m)

0.06

0.1

0.14

0.18

Solar Cell Efficiency

0

10

20

30

40

50

60

Average Solar Cell Efficiency

Drag per Wing Span

Figure 10.—Effect of Passage Spacing on Solar Cell Efficiency and Drag. 

0 0.01 0.02 0.03 0.04 0.05

Passage Spacing (m)

0

100

200

300

400

500

600

700

800

% Change

Change in Output Power

Change in Drag

Figure 11.—Percent Change in Solar Cell Performance and Drag. 
NASA/CR—2003-212084 18 

 Based on Figures 10 and 11, the drag of the aircraft increases significantly when 

the passage spacing is increased beyond 0.01 m. This is for an aircraft wing chord length 

of 1 m. This is due the transition from laminar to turbulent flow within the cooling 

passage. Once this transition occurs the percent change in drag is significantly greater 

than the percent change in array output power. Therefore, beyond this point, it would not 

be beneficial to utilize the cooling passage to increase the array performance since the 

additional power generated would not be capable of offsetting the additional drag of the 

aircraft. Based on these results, the passage spacing of 0.01 m was chosen as the optimal 

passage spacing for this particular airfoil and aircraft configuration. All of the following 

results will be based on this cooling passage spacing of 0.01 m. 

 Utilizing the airfoil shape given in equation 29 and the airfoil and solar cell 

dimensions given in Table 1, the operating temperature of each cell can be determined for 

the base flight conditions. These temperatures are shown in Figure 12. This operating 

temperature was based on heat transfer out of the solar cell as described in the previous 

section. The main mechanism was the convective heat transfer, which is based on the 

convective heat transfer coefficient for both the upper surface of the wing and the 

channel. For the base case, the wing surface convective coefficient was determined to be 

5.23 W/m2

°K. For the channel, it was a function of Reynolds number and is given in 

Figure 13. 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Solar Cell Number

0

100

200

300

400

Cell Temperature (°K)

Figure 12.—Solar Cell Temperature. 
NASA/CR—2003-212084 19 

0 0.004 0.008 0.012 0.016 0.02 0.024 0.028 0.032 0.036 0.04

Passage Spacing

1

10

100

1000

Convective Heat Transfer coefficient (

W/m^2 °K)

Figure 13.—Relationship Between Passage Spacing and Convective Heat Transfer 

Coefficient. 

 The effect of the cooling passage for the base case conditions, given in Table 1, 

with the optimal cooling passage spacing, 0.01 m, is shown in Figure 14. From this figure 

it can be seen that there is a significant increase in output power throughout a majority of 

the day-time period. The dip in the curve for the case without the cooling passage near 

midday is due to the increased solar cell temperature during operation. This dip is 

eliminated with the addition of the cooling passage. 

 The type and characteristics of the solar cell used impacts the overall output 

performance. However, based on the information given in Figure 3, the change in cell 

performance for a given temperature change is fairly consistent for all the cell types 

shown. This change is approximately 3 percent increase in efficiency for a 50 °C change 

in temperature. Another aspect of the solar cell that influences its operating temperature 

is the reflection of light off its surface. For a Silicon cell this is given in Figure 7. This 

reflection versus incident angle depends on the cell material and whatever anti-reflective 

coating is applied to its surface. Since the reflectance of the cell can be significantly 

reduced by the application of the anti-reflective coating, the variation in cell reflectivity 

for different types of solar cells should be minimal. Therefore, the analysis was 

performed for only Silicon solar cells. Based on the data given in Figure 3 and the antireflective coating properties, the results for other cell types should be similar. 
NASA/CR—2003-212084 20 

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Time of Day (hours)

0

20

40

60

80

100

120

Output Power (W/m^2)

With Cooling Passage

Without Cooling Passage

Figure 14.—Solar Cell Output Power Throughout the Daytime Period with and without 

the Cooling Passage. 

 The analysis was performed for only one airfoil, the NACA 0009, a symmetrical 

thin airfoil. The use of a different airfoil would change the results somewhat due to the 

difference in surface shape and thereby solar cell angle. However, the general 

performance benefit should be similar for other airfoils since the amount of energy falling 

on a wing of given chord length is constant regardless of the airfoil shape. The 

examination of different airfoils was not needed in order to determine the overall 

feasibility of the cooling passage. In general, the benefits seen in this analysis with the 

NACA 0009 airfoil should be similar to results with other airfoils. Due to this thin nearly 

flat airfoil geometry, the variation in performance with orientation of the airfoil was 

minimal. The solar cell performance was nearly identical for the four orientations 

examined (North, South, East and West). The data, both with and without the cooling 

passage, followed very closely the curves shown in Figure 14 for the solar cell 

performance. 

 The effect of the cooling passage on the power generated during flight at different 

latitudes and time of the year are shown in Figures 15 and 16. Since a change in flight 

latitude or date does not effect the Reynolds number of the flow through the cooling 

passage, the optimized passage spacing is the same for all latitudes. 

 From Figure 15 it can be seen that the addition of the cooling passage has the 

greatest impact on the array output during the midday hours, and little effect during the 

morning and evening hours. At high latitudes where the sun elevation angle is low, there 

is a peak increase of approximately 25 percent in array output. This is for the 60° case 

shown in Figure 14. However, for lower latitudes, such as the 20° case shown, the 
NASA/CR—2003-212084 21 

increase in midday output power is much more significant, an approximate 60 percent 

increase in output power. Therefore, from this figure it can be seen that the benefits of the 

cooling passage are greater at lower latitudes. 

Figure 15.—Effect of Cooling Passage at Various Latitudes. 

 Figure 16 shows the effect the cooling passage has on the output power, at a given 

latitude (40° N), at different times of the year. The dates shown are for June 21st and 

December 21st. These are the two extremes in solar elevation angle. Output power curves 

for all other dates will fall between these two sets of curves for the same latitude. From 

this figure it can be seen that the effect on the cooling passage on the output power 

reaches a peak, at the summer solstice, increasing output power about 54 percent. This 

effect diminishes to about a 12 percent increase in output power during the winter 

solstice. This figure demonstrates that at higher sun angles the effect of the cooling 

passage increases. 

0

20

40

60

80

100

120

0.5 2.5 4.5 6.5 8.5 10.5 12.5 14.5 16.5 18.5 20.5 22.5

Time of Day (hours)

Latitude 60 ̄ 1cm Spacing

Latitude 60 ̄ No Spacing

Latitude 20 ̄ 1cm Spacing

Latitude 20 ̄ No Spacing
NASA/CR—2003-212084 22 

Figure 16.—Solar Array Output at Various Times of the Year. 

References 

 1. AeroVironment, Inc., ERAST Data Memo, “Pathfinder Flight Test 95-3 Qualitative 

Report,” September 22, 1996. 

 2. Dornheim, M.A., “Solar-Powered Aircraft Exceeds 50,000 ft,” Aviation Week and 

Space Technology, September 18, 1996. 

 3. Hall, D.W., Fortenbach, C.D., Dimiceli, E.V., and Parks, R.W., “A Preliminary 

Study of Solar Powered Aircraft and Associated Power Trains,” Lockheed Missiles 

and Space Company Inc., NASA CR–3699, 1983. 

 4. Colozza, A.J., “Effect of Power System Technology and Mission Requirements on 

High Altitude Long Duration Aircraft,” NASA CR–194455, February 1994. 

 5. Lamp, T.R., Reinhardt, K.C., and Colozza, A.J., “Photovoltaic Power for Long 

Endurance Unmanned Aerial Vehicles,” Aero Propulsion and Power Directorate, 

Wright Laboratory Wright-Patterson AFB Report, 1997. 

 6. Scheiman, D.A., Jenkins, P.P., Brinker, D.J., and Appelbaum, J., “Low Intensity 

Low Temperature (LILT) Measurements and Coefficients on New Photovoltaic 

Structures,” SPRAT XIV, Oct. 1995, Progress in Photovoltaics, John Wiley and 

Sons, Inc. Publisher, March–April, Vol. 4, 1996. 

 7. Incropera, F.P. and DeWitt, D.P., Fundamentals of Heat and Mass Transfer, John 

Wiley and Sons, Inc. Publisher, 1990. 

0

20

40

60

80

100

120

0 2 4 6 8 10 12 14 16 18 20

Time of Day (hours)

6 / 21 with cooling passage

6/21 no cooling passage

12/21 with cooling passage

12 / 21 no cooling passage
NASA/CR—2003-212084 23 

 8. Scheiman, D.A., Brinker, D.J., Bents, D.J., and Colozza, A.J., “Design of a GaAs/Ge 

Solar Array for Unmanned Aerial Vehicles,” NASA TM–106870, March 1995. 

 9. Colozza, A.J., Scheiman, D.A. and Brinker, D.J., “GaAs/Ge Solar Powered 

Aircraft,” NASA CR (presently under publication) May 1998. 

10. White, F.M., Fluid Mechanics, McGraw Hill Publisher, 1979. 

11. Colozza, A.J., “Effect of Date and Location on Maximum Achievable Altitude for a 

Solar Powered Aircraft,” NASA CR–202326, November 1996. 

12. Duffie and Beckman, Solar Engineering of Thermal Processes, John Wiley and 

Sons, Inc. Publisher, 1991. 

13. Handbook of tables for Applied Engineering Science, 2nd Edition, CRC Press, 1973. 
This publication is available from the NASA Center for AeroSpace Information, 301–621–0390.

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NASA CR—2003–212084

E–13737

WBS–22–708–87–11

NAS3–00145

29

Convective Array Cooling for a Solar Powered Aircraft

Anthony J. Colozza

Solar powered aircraft; Solar arrays

Unclassified -Unlimited

Subject Categories: 07 and 34 Distribution: Nonstandard

Analex Corporation

3001 Aerospace Parkway

Brook Park, Ohio 44142

Project Manager, James Dolce, Power and On-Board Propulsion Technology Division, NASA Glenn Research Center,

organization code 5450, 216–433–8052.

A general characteristic of photovoltaics is that they increase in efficiency as their operating temperature decreases.

Based on this principal, the ability to increase a solar aircraft's performance by cooling the solar cells was examined.

The solar cells were cooled by channeling some air underneath the cells and providing a convective cooling path to the

back side of the array. A full energy balance and flow analysis of the air within the cooling passage was performed. The

analysis was first performed on a preliminary level to estimate the benefits of the cooling passage. This analysis established a clear benefit to the cooling passage. Based on these results a more detailed analysis was performed. From this

cell temperatures were calculated and array output power throughout a day period were determined with and without the

cooling passage. The results showed that if the flow through the cooling passage remained laminar then the benefit in

increased output power more than offset the drag induced by the cooling passage.