Heat transfer by natural convection occurs when the surface temperature of the module TM is higher than the ambient temperature TA and convection is possible at least at one module surface. Convective heat flows cannot be calculated exactly by mathematics and therefore have to be computed by iteration or by approximation equations such as shown here. First, some important characteristics which are relevant for convection:
The Prandtl number Pr is a characteristic which compares two molecular transport mechanisms: impulse transfer by friction and heat transfer by thermal conduction. The Prandtl number is a function of temperature and depends on the physical properties of the fluid. For gas molecules with the same number of atoms, the Prandtl number remains constant.
The Reynolds number Re determines the heat transfer at forced convection. Its value also settles the type of flow – whether it is turbulent or laminar. The type of flow is of decisive significance: At a laminar flow no mixture movement occurs and heat transfer is low. In zones of turbulent flow the thickness of the Prandtl’s border layer depends on the Reynolds number. The “abrasive effect” of the turbulent flow core (flowing with a velocity w) tries to lower the thickness of the laminar border layer; viscosity v works in opposition to that, supported by the roughness of the wall (which is not directly included in the Reynolds number).
The Grashof number Gr provides the relation of thermal buoyancy force to inner “sluggishness.” In case of free flow conditions only natural convection is accountable for the heat transfer. The temperature difference between the wall and the average temperature of the flowing media, which causes a change in volume and therefore a change in density at some parts of the media, is the driving force for the natural convection.
The NuBelt number Nu shows the relation of the actual heat flux density to pure heat conduction through a layer equivalent to the relevant length l at which flowing occurs.
Another substantial parameter for natural convection is, beside NuBelt number Nu and the Prandtl number Pr, the Rayleigh number Ra. If Ra exceeds a certain threshold, the so called “critical Rayleigh number” Racr, the convective border layer is lifted off, which makes a case division necessary for the convective heat transfer. Ra is calculated as follows:
Ra = Pr-Gr = Pr—— (TM-TA) = Pr——(TM-TA) (102)
Gr Grashof number
Nu NuBelt number
Pr Prandtl number
g ground acceleration (g = 9.8067 m s-2)
P thermal expansion coefficient of air in K-1 ^ dynamic viscosity in Pa s (or N m-2s)
lch length of a body relevant for the flow in m
TA ambient temperature in K
TM surface temperature of the module in K
Racr can be computed by the following approximation:
Racr = 10*
with: x = 8.9 – 0.00178-(90°- /M)182
For the Prandtl number Pr:
Pr = —– S – (104)
cp specific heat capacity in N m kg-1 K-1
^ dynamic viscosity of air in Pa s к heat conductivity of air in W m-2 K-1 x auxiliary variable
Y M elevation angle of module in °
Inserting (103) into (102) using the parameters of air relevant for our problem from Table A15, page 251 (with lch = 1.3 m, ym = 45°), for Ra > Racr the following equation has to be fulfilled: TM – TA > 0.05 K, which is always true during the day, so we can take boundary layer being lifted off for granted. For the upper surface of a tilted (plane) plate the Nufielt number is for Ra > Racr (according to Fujii et al. 1972):
Nup – 0.56 4jPacr sin YM + 0.13 ( ^Ra – ^Rajj (105)
The convective heat transfer coefficient for natural convection (nc) at the front (F) surface is:
Kc. F –
In opposition to the front surface of the module a lifting-off of the boundary layer is not possible on the backside of the module, while the heated air is subject to buoyancy and can just escape along the rear surface of the module. For Nu at non-lifted-off boundary layer for yM < 90° the equation by Merker 1987 is valid, if Ra is substituted by Ra sinyM:
NuB = 0.56 ^ Ra sinj’M (107)
so the convective heat transfer coefficient for the backside (B) is:
Nu„k k 4 ,——
knc, B = ————– = 056 — JRa sil1 Ум
The main problem for further calculations is that the heat transfer coefficients are functions of the surface temperatures TF and TB. These temperatures can be computed only after the calculation of the heat transfer coefficient because it depends on the balance of heat fluxes. Therefore, an iteration procedure has to be applied. This also allows the use of the temperature of the boundary layer (average of ambient – and surface – temperature) instead of the ambient temperature TA. Estimates for starting values for the iteration process have to be given. A good estimate is TF = TB = Ta + 30 K.
Relevant properties for air (Pr, k, cp, p) at different temperatures are given in the Annex in Table A10. In a very simplified model, given by Funck 1985 (applied for the thermal layout gear boxes), heat transfer coefficients (for natural convection at small bodies) of horizontal and vertical surfaces are equalized. While the elevation angle of solar modules is between the extreme values of horizontal (PV systems at the equator) and vertical (PV facades), and further on the course of the function (elevation angle vs. heat transfer coefficient) is continuous, Funck’s approximation can be used as an elevation-angle-independent guess for heat transfer by natural convection at small solar modules:
hnc heat transfer coefficient for natural convection in W m-2 K-1
lch length of module over which convection take place in m
TA ambient temperature in K
TM module surface temperature in K
The length lch over which convection occurs does not have a significant effect on the heat transfer coefficient hnc, while its length is influencing the result only by the tenth root of its reciprocal value. This means that for all lengths of PV modules available on the market for power applications (0.5 m < lch < 2 m), the variation of the heat transfer coefficient for natural convection is only ±5 %.