An array of power transistors, dissipating 6 W of power each, are to be cooled by mounting them on a 25cm ×25cm-square aluminum plate and blowing air at 35°C over the platewith a fan at a velocity of 4 m/s. The average temperature of the plate is not to exceed 65°C. Assuming the heat transfer from the back side of the plate to be negligible and disregarding radiation, determine the number of transistors that can be placed on this plate

Assuming the heat transfer from the back side of the plate to be negligible and disregarding radiation, determine the number of transistors that can be placed on this plate

Solution:-

- We will perform the heat balance on the system ( aluminium plate integrated with transistors ). We will ignore the radiation effects and only consider the forced convective cooling of the transistors subjected to a stream of air forced at V = 4 m/s over the plate.

- From heat balance the rate at which heat is dissipated from the circuit -board is:

[tex]n*q = h^*.A_s.( T_s - T_i_n_f )[/tex]

Where,

h*: The average heat transfer coefficient over the edge of the plate

As: The area exposed to convective current

- We need to determine the average heat transfer coefficient ( h* ) over the flat plate for the given conditions. We know that the heat transfer coefficient is a function of a dimensionless quantity ( Nu* ), thermal conductivity ( k ) of convection fluid and edge length of the plate ( L ).

- We will first determine the thermo-physical properties of the convective fluid ( air ). Its a standard practise to evaluate these properties at the film temperature ( T ) , i.e the average of surface temperature ( Ts ) and the free stream temperature ( T∞ ). Thus,

[tex]T_f = \frac{T_s + T_i_n_f}{2} = \frac{65 + 35}{2}\\\\T_f = 50[/tex]

- Use thermo-physical tables ( A - 4 ) and evaluate properties at T = 50°C:

k = 0.02735 W /m.K

Pr = 0.7228

ρ = 1.092 kg/m^3

μ = 1.963*10^-5 kg/m.s

- The average Nusselt Number ( Nu* ) is a function of flow properties of the convective fluid namely: ( Re , Pr ). We will determine the Reynold's number over the edge of the aluminium plate as follows:

[tex]Re_L = \frac{p*V*L}{u} = \frac{1.092*4*0.25}{1.963*10^-^5} = 55629.1391[/tex]

- The Reynold's number define the flow conditions of the fluid. We see that Reynold number calculated above is within the critical Reynold's number ( Re,c ). Therefore,

Re = 5.6 * 10^5 ≤ Re,c ( 5*10^5 )

- For flows with Re < Re,c, we take the assumption of "Laminar Boundary Layer".

- The corresponding (averaged) Nusselt Number empirical relation for Laminar flow regime and constant surface temperature ( Ts = 65°C ) over flat plate ( forced convection) we have:

[tex]Nu^* = 0.664. Re_L^\frac{1}{2}. Pr^\frac{1}{3}\\\\Nu^* = 0.664. (55629.13907)^\frac{1}{2}. (0.7228)^\frac{1}{3}\\\\Nu^* = 140.5482[/tex]

- From above relation we can evaluate the average heat transfer coefficient ( h* ) as follows:

[tex]h^* = \frac{Nu_L^*. k}{L} = \frac{140.5482. 0.02735}{0.25}\\\\h^* = 15.376 \frac{W}{m^2.K}[/tex]

- Now we can use the energy balance applied to the system initially developed and solve for ( n ):

[tex]n = \frac{h^*. A_s. ( Ts - T_i_n_f)}{q} \\\\n = \frac{15.376. ( 0.25)^2 . ( 65 - 35)}{6} \\\\n = 4.8[/tex]

Answer: The number of transistors that can be placed are 5.

Note: In the above evaluation we have made an assumption that the exposed area ( As ) is equivalent to the surface area of the aluminium plate. This neglects the area associated with the thickness of the transistors. Moreover, we have assumed that the back-side of plate is thermally insulated. Also the surface temperature ( Ts ) of the plate base and the top of the transistor is assumed to be similar (if not then, we would have applied extended fin analysis ).