Temperature profile around heat source

AI Thread Summary
The discussion focuses on determining the temperature profile around a point heat source in water that emits 10 W for 10 seconds before being turned off, with the surrounding water initially at 20 °C. Participants suggest that the problem cannot be solved using steady-state assumptions and recommend using the inhomogeneous heat transfer partial differential equation for a time-dependent solution. There is a request for guidance on how to approach the transient nature of the problem. The complexity arises from the need to account for both the heating phase and the subsequent cooling phase. Understanding the heat transfer dynamics in this scenario is essential for finding the temperature profile as a function of distance and time.
Haye
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Homework Statement


A point source in water provides 10 W of heat for 10 seconds, and is then shut off. The surrounding water has a temperature of 20 °C. Provide the temperature profile as a function of distance r and time t.


Homework Equations





The Attempt at a Solution



If it were steady-state, I could use Gauss theorem to solve this problem. But since the point source is turned on and off again, I have no idea how to approach this problem.
If someone could hint me in the right direction, it would be very much appreciated.
 
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You have to use the inhomogeneous heat transfer partial differential equation.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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