Time-dependent Boundary A slab, of thickness L, has one surface insulated and th

Question

Time-dependent Boundary

A slab, of thickness L, has one surface insulated and the other exposed to a periodic, pulsed laser beam. The irradiance of the beam, as a function of time, is given by This means that the pulse is on for t from zero to ti and off for t from ti to t2. This is a periodic pulse, i.e. G(tt2)Gt) Assuming that all of the incident flux is absorbed by the surface, and that the surface exchanges heat with the environment by convection, derive the solution for periodic temperature distribution in the slab

Explanation / Answer

Because irradiance flux is periodic, Average flux falling on slab = (Go*t1+0*t2)/(t1+t2) =Go*t1/(t1+t2)

For slab.

Total Heat fall - Total heat liberate = Heat abosrbed by slab

Let Area of slab =A

Total Heat fall = flux*Area =Go*t1/(t1+t2)*A

Convective heat transfer coefficient of air = h

Constant Atmosphere temperature = Ta

Temperature of slab at Time, t = T

Heat is liberating because of convection of air

=>Total Heat liberate = h*A*(T-Ts)

Heat absorbed by slab results in temperature rise of slab

Density of slab = P

=> Mass of slab = density*Volume = P*A*L

Specific Heat of slab = C

Therefore, Heat absorbed by slab = P*A*L*C*dT/dt

=> Go*t1/(t1+t2)*A-h*A*(T-Ts) = P*A*L*C*dT/dt

=>Go*t1/(t1+t2)-h*(T-Ts) = P*L*C*dT/dt

Integrating this,

Therefore,

t = P*L*C/h*LN(|(Go*t1/(t1+t2)*1/h-2*Ta)/(Go*t1/(t1+t2)*1/h-T-Ta)|)

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