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swmmr1928
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Homework Statement
A solid body at initial temperature T0 is immersed in a bath of water at initial temperature Tw0. Heat is transferred from the solid to water at a rate [itex]\dot{Q}[/itex]=K[itex]\bullet[/itex](Tw-T), where K is a constant and Tw and T are instantaneous values of the temperatures of the water and solid. Develop an expression for T as a function of time t. Check your result for the limiting cases, t=0 and t=∞. Ignore effects of expansion or contraction, and assume constant specific heats for both water and solid.
2. Homework Equations [/b[STRIKE]][/STRIKE]
[itex]\dot{Q}[/itex]=K[itex]\bullet[/itex](Tw-T)
d(mU)cv/dt=-[itex]\dot{Q}[/itex]
Cv=dU/dT
The Attempt at a Solution
-K[itex]\bullet[/itex](Tw-T)=m*dU/dt
K[itex]\bullet[/itex](T-Tw)=m*Cv*dT/dt
dT/dt=K/(m*Cv)(T-Tw)
Now I will attempt integrating factor
dT/dt=c1(T-Tw)
[itex]\mu[/itex](t)*dT/dt=[itex]\mu[/itex](t)*c1(T-Tw)
[itex]\mu[/itex](t)=exp(∫-c1*dt)=exp(-c1*t)
d[T*exp(-c1*t)]/dt=exp(-c1*t)*c1(T-Tw)
T*exp(-c1*t)=∫exp(-c1*t)*c1(T-Tw)dt
T*exp(-c1*t)=-exp(-c1*t)(T-Tw)
T=-(T-Tw)
I would expect a the Temperature of the solid to decrease, as stated in the problem, but eventually level off at an asymptote as it approaches equilibrium.
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