A power transistor mounted on a finned heat sink can be modeled as a spatially i

Question

A power transistor mounted on a finned heat sink can be modeled as a spatially isothermal object with internal heat generation and an external convection resistance. (a) Consider such a system of mass m, specific heat c, and surface area A, which is initially in equi librium with the environment at T.. Suddenly, the device is energized such that a constant heat gen- eration Eg(W) occurs. Show that the temperature response of the device is = exp(-RC where T-Te) and Too) is the steady-state temperature corresponding to t 003, = Ti-Te ); T, initial temperature of device; R= thermal resis- tance l /hAs; and C= thermal capacitance mic. (b) A device which generates 100 W of heat is mounted on an aluminum heat sink weighing 0.35 kg and reaches a temperature of 100°C in ambient air at 20°C under steady-state conditions. If the device is initially at 20°C, what temperature will it reach 5 min after the power is switched on?

Explanation / Answer

a) let m represent mass , c specific heat, h heat transfer coefficient, T is instantaneous temperature of body, Ta ambient temperature ,

Then by energy balance equation rate of decrease in internal energy of body = heat dissipation to surrounding

m×c× dT/dt = h × A× ( T - Ta)

Where t is time required

Re arranging the terms

-m × c × dT/( T - Ta) =h× A × dt

Negative sign before m represent decrease in temperature with time

Integrating the above we get

-m×c×ln(T-Ta) = h× A × t + C1

Where C1 is constant of integration

at t =0 , T = Ti

Putting in above we get C1 = m×c× ln( Ti - Ta)

m ×c × ln[ ( Ti- Ta)/(T - Ta)] = h× A× t

ln[( Ti-Ta)/ (T- Ta)] = (h× A× t)/ m×c

(Ti- Ta)/ (T- Ta) = e^((hAt/mc)

Or

( T- Ta)/ ( Ti - Ta) = e^ (-t/ RC)

Where R = 1/hA

C= mc

T-Ta = Q

Ti- Ta= Qi

Q/Qi = e^(-t/ RC)

  

  

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