Question 8.14: An electrical circuit board contains a variety of digital lo...
An electrical circuit board contains a variety of digital logic elements. When operating, the board draws 10.0 mA at 5.00 V dc and it has a steady state surface temperature of 30.0°C. Estimate the entropy production rate of the circuit board.
First, draw a sketch of the system (Figure 8.17).
The unknown is entropy production rate of the circuit board. The system is the circuit board.
Since nearly all the electrical energy entering the circuit board is being converted into heat, we could calculate the entropy production rate of the board from its heat loss characteristics (convective heat transfer coefficient, surface temperature, environmental temperature, surface area, etc.); however, none of this information is supplied in the problem statement. We could also calculate the entropy production rate for electrical work mode dissipation directly from Eq. (7.73) if we knew how the current density J_e, electrical resistivity ρ_e, and local internal temperature T are distributed throughout the circuit board. But we do not know this information either. However, the problem statement asks for only an estimate of the entropy production rate, and we can obtain this from the special electrical work mode dissipation Eq. (7.74) if we lump all the components on the board into one uniform, isothermal, constant current density system. From Ohm’s law, ϕ = IR_e, we can write Eq. (7.74) as
(\dot{S} _P)_\text{W}=\frac{I^{2}R_e}{T} = \frac{ϕI}{T} = \frac{(5.00 \text{V})(10.0 × 10^{-3} \text{A})}{30.0 + 273.15 \text{K}}
= 1.65 × 10^{-4} \text{W/K}
This is only a lumped parameter estimate of the entropy production rate for this system. The actual entropy production rate is somewhat larger due to the nonuniform distribution of entropy-producing electrical components within the system volume.
