fig. 1  ventilated cabinet.
introduction
natural air convection is commonly applied as a cooling technique for electronic equipment of moderate power density such as telecommunication boxes. the main advantage of natural convection is its intrinsic reliability, because air movement is generated simply by density gradients, if an external body force field exists.
however, due to the relatively low efficiency of the cooling technique, the thermal design of the electronic equipment must be optimized, i.e. the geometrical configuration of the heat sources and the ventilated enclosure must be selected so as to generate an air flow rate which minimizes the average and the maximum temperature rise inside the enclosure itself. in particular, the thermal behavior of a case depends on the balance between the net buoyancy force and the friction losses along the whole air path. the latter takes into account the distributed pressure losses along the hot channels formed by electronic cards and the concentrated friction losses in the inletoutlet vents [12].
in this paper a practical formula for the thermal design of the ventilated enclosures is presented. the formula takes into account important parameters such as power transferred to the cooling air, chimney height (distance between the inlet and outlet vents), volumetric air flow rate, vent area, and friction losses along all the air flow pattern. application examples are given as illustrations of the practical formula used as a preliminary tool for thermal design of natural ventilated boxes.
experiments on a ventilated cabinet
the ventilated box used during the study is schematically shown in fig. 1. its dimensions are (wxlxh) 100 mm x 152 mm x 254 mm and it contains a series of equispaced, vertical heated cards (152 mm x 254 mm). the number of the boards placed in the enclosure is 6. it has been shown that this arrangement of the cards corresponds to an optimum board spacing of about 15 mm [3]. the electronic circuitry is simulated on each card by 12 rows of 12 electric resistors arranged in three equal sectors. the two vents (inlet and outlet) are placed on the same wall of the box. several experiments, performed with the same vent area for the previous tests, but with the outlet vent on the opposite wall of inlet one, showed the same thermal behavior.
the cooling air enters the case through the lower vent, removes a fraction of the heat generated inside (in proportion to the total friction losses along the air path) and exits through the upper vent. the heat not removed by the ventilation air draft is transferred to the external ambient by convection and radiation through the cabinet walls. to reduce the heat leakage through the walls and to simulate the presence of other adjacent boxes, all the walls except the one with the vents were carefully insulated.
the air temperature distribution inside the box has been measured by means of 44 calibrated ktype thermocouples (±0.1°c) arranged on each card at three different heights. the thermocouples are located near the thermal boundary layer on the heated side of the card and in the surrounding of the back side. the experiments have shown that the mean outlet air temperature is nearly identical to the integral average value at the highest elevation of the box, while the inlet temperature is equal to the ambient temperature t_{0}.
the experiments were conducted selecting very carefully the electric power dissipated by resistors in order to obtain, in steadystate conditions, a difference between the power removed by air and the desired value of ±2% (during the experiments, the ratio between the electric power dissipated by the resistors and the heat removed by air varied from 1.3 up to 1.5), and a ratio between the inlet and outlet vent areas equal to one (a_{in}=a_{out}). the latter assumption causes the optimum cooling condition inside the casing (minimum air temperature rise).
practical formula for the thermal design of ventilated castings
a simple correlation equation for thermocirculation flow rate, useful for a preliminary thermal design of cooledair ventilated enclosures, can be obtained using the model sketched in fig. 2.
fig. 2  thermal model.
in a steady state condition the heat flux p' (w) removed by the air is related to the inletoutlet enthalpy variation δh_{1,2}:

equation 1 
and the buoyancy provides the fluid with kinetic energy:

equation 2 
where v is the volumetric air flow rate (m^{3}/s), p_{0} and p are the external and internal air density (kg/m^{3}) respectively, the latter evaluated at the mean inner air temperature t (k), h is the chimney height, i.e. the distance between the center lines of the vents, k_{e} is the equivalent fluid resistance coefficient, u is the reference value for average air velocity (m/s), c_{p} is the specific heat of air (j/kgk), and δt_{1,2} is the difference between the outletinlet air temperatures (k).
assuming the air density inside the box as a linear function of the temperature and denoting by c= δ t _{1.2} /δt the ratio between the inletoutlet air temperature difference and the mean inner air temperature rise, and then combining the eq.s (1) and (2), it is possible obtain the simple formula:

equation 3 
in which the symbols represent: a the inlet or the outlet vent area (m^{2}), t_{0} the external ambient temperature, and the coefficient c can be related to the power removed by air [4]; because the c values are included in the range between 1.28 and 1.32, it is possible to assume a constant value of 1.3.
the constant 5.5 ^{.} 10 ^{5} was obtained from the following ratio 2^{.}g^{.}r_{1}/(patm^{.} ^{c}p) where g is the gravitational acceleration (m/s^{2}), and r_{1} is the air gas constant (j/kgk), patm is the atmospheric pressure (pa).
k_{e} represents an equivalent fluid resistance coefficient that takes into account the distributed pressure losses along the electronic card channel and the concentrated pressure losses for the two openings including the bending in the air path close to the vents:
k_{e} =k_{ch}+k_{con} 
equation 4 
the available correlations for evaluating the coefficient k_{ch} in natural convection are scarce and generally refer to idealized models far from the real geometry of the electronic cards. the correlating equation proposed in [5], valid for forced and mixed laminar convection flow inside channels with twodimensional obstructions, can be utilized. the results available in literature show that the relative contribution of coefficient k_{ch} is small compared to that of k_{con} .
particular attention has therefore been placed on the term k_{con} of the eq. (4). the coefficient k_{con} can be expressed as the sum of the two concentrated losses in the inletoutlet vents:
k_{con}=k_{in}+k_{out} 
equation 5 
an attempt has been made to obtain some quantitative information on k_{in} and k_{out} .
the analysis has been conducted by simplifying the assumptions:
a) 
the inner air temperature in the cabinet is equal to t 
b) 
the air flow rate through a single vent opening is equal to the total air flow rate v. divided by the number of single openings; 
c) 
since all the analysis parameters are considered average values, it has been assumed k= k_{in}=k_{out} . 
a formula for k has been obtained by minimizing the scatter between the experimental and theoretical volumetric flow rates eq.(3). to complete the analysis, the coefficient k has been put in relation with the reynolds number, based on the hydraulic diamet diameter d_{h} of the single opening:

equation 6 
in which v is the kinematic viscosity of air evaluated at t and w_{v} represents the mean air velocity through each single opening evaluated as w_{v}= v /a_{v} /n (a_{v} is the area of single opening, and n is the number of openings on the vent).
in order to analyze the thermal behavior of the cabinet, and in particular, to obtain the relationship between k and red_{h} several tests were performed varying systematically the following parameters:
shape of outlets 
circular holes, horizontal slots, vertical slots 
inlet or outlet vent area 
600 10^{6} m^{2}; 800 10^{6} m^{2}; 1000 10^{6} m^{2} 
hydraulic diameter 
3mm ; 4mm ; 5mm 
porosity 
0.2 ; 0.33 
power removed by air 
5w ; 10w ; 15w 
the porosity (of the vent is defined as the ratio between the single opening area of the vent and the surface between two adjacent single apertures of the slots or rows of holes, i.e.,ß = π d2h/4/p2t (circular holes), ß = h_{0}w_{0}/p_{t}w_{0}(horizontal slots), and ß = w_{0}h_{0}/p_{t}h_{0}(vertical slots) (see fig. 3).
fig. 3  shape of the vents.
a deeper analysis of the thermal behavior of the casing was reported in [4]. in the same paper, the relationships between k and re_{dh}, for the different shapes and porosities investigated, are shown. to make easier the utilization of eq. (3), a relationshi relationship between k and re_{dh}, valid for all the three vent shapes and the two porosities investigated, is reported below:

equation 7 
the standard deviation associated with eq.(7) is 0.23. the formula is valid for reynolds numbers re_{dh} from 30 to 120.
example of thermal design of natural ventilated cabinet
in this paragraph, the procedure for using the practical formula (eq. (3)) is reported. the equation is not explicit, because in k the volumetric air flow rate is involved.
a flow chart (fig. 4) that explains the steps of the iterative procedure is depicted above. few iterations are necessary to reach the final value of a. several iterative results are listed in table 1.
iteration number 
a_{in}=a_{out }·10^{6} m^{2} 
number of holes 
1 
400 
20 
2 
613 
31 
3 
629 
32 
4 
637 
32 
5 
644 
32 
6 
643 
32 
table 1
the simulation is referred to the following input data:
p' = 10 w, t =50 ^{o}c, h=0.2 m, d_{h}=5 mm, t_{0}=20 ^{o}c,
vent shape (holes), error admitted 1%, and as tentative value a* =400^{.}10^{6} m^{2}. the iteration number, the inlet or outlet vent area, and the number of holes.
fig. 4  flow chart of the iterative procedure (eq. (3)).
application examples are given as illustrations of eq. (3) as a preliminary tool for the thermal design of fig. 5 (overleaf), the lines represent the predicted values of the inlet or outlet vent area a as a function of the mean inner air temperature rise using eq. (3) in comparison with some experimental data (d_{h}=5mm, three vent shapes, and three values by air). there is good agreement between the theoretical results and the experimental ones. in this example, it has been assumed a_{in}=a_{out}=a.
conclusions
a practical formula for the thermal design of ventilated enclosures used in the electronic equipment is presented. the formula takes into account several parameters, such as power removed by air, air flow resistance, average inner air temperature, inlet or outlet vent area and chimney height. obviously, this simple formula must be used only as a preliminary tool for the thermal design of casings containing electronic boards. deeper analysis can be performed using other techniques such as cfd (computation fluid dynamics).
nevertheless, even though the flow pattern in a casing and along the electronic boards is very complex, the validity of the few simple formulas available in literature must be checked by the users in order to know the error between the theoretical analysis and the real behavior of the enclosures. in other words, it is very hard to make many tests to confirm the validation of the practical formulas, because the parameters that can be involved in real applications are numerous.
simple formulas in many cases can be useful only for the evaluation of the vent area.
fig. 5  inlet or outlet vent area a vs. mean inner air termperature rises for three different values of power thermocirculation p'. vents as circular holes, horizontal slots, and vertical slots (d_{h} =5mm).
mario misale dipartimento di termoenergetica e condizionamento ambientale university of genoa via all'opera pia 15/a  (i) 16145 genova, italy tel: +39 10 353 2576 fax: +39 10 311 870 email: ditec@unige.it
references
1. 
ellison, g.n., thermal computations for electronic equipment, van nostrand reinold company, new york 1984, reprint edition by robert z. kreises publishing company, malabar, florida, 1989. 
2. 
guglielmini, g., milano, g., and misale, m., some factors influencing the optimum free air cooling of electronic cabinets, in heat transfer in electronic and microelectronic equipment, a.e. bergles, eds., hemisphere publications corporations, pp. 311325, new york, 1990. 
3. 
barchoen, a. and rohsenow, w.m., thermally optimum spacing of natural convection cooled parallell plates, j. of heat transfer, vol. 106, pp. 116123, 1984. 
4. 
misale, m., electronic cabinet cooling by natural convection: influence of vent geometry, 3rd world conference on experimental heat transfer, fluid mechanics and thermodynamics, vol. 1, pp. 764771, honolulu, hawaii, usa, 1993. 
5. 
braater, m.e., and patankar, s.v., analysis of laminar mixed convection in shrouded arrays of heated rectangular blocks, int. j. heat transfer, vol. 28, n. 9, pp. 16991709, 1985 
