(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A solid body at initial temperature T_{0}is immersed in a bath of water at initial temperature T_{w0}. Heat is transferred from the solid to water at a rate [itex]\dot{Q}[/itex]=K[itex]\bullet[/itex](T_{w}-T), where K is a constant and T_{w}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. Relevant equations[/b[STRIKE]][/STRIKE]

[itex]\dot{Q}[/itex]=K[itex]\bullet[/itex](T_{w}-T)

d(mU)_{cv}/dt=-[itex]\dot{Q}[/itex]

C_{v}=dU/dT

3. The attempt at a solution

-K[itex]\bullet[/itex](T_{w}-T)=m*dU/dt

K[itex]\bullet[/itex](T-T_{w})=m*C_{v}*dT/dt

dT/dt=K/(m*C_{v})(T-T_{w})

Now I will attempt integrating factor

dT/dt=c_{1}(T-T_{w})

[itex]\mu[/itex](t)*dT/dt=[itex]\mu[/itex](t)*c_{1}(T-T_{w})

[itex]\mu[/itex](t)=exp(∫-c_{1}*dt)=exp(-c_{1}*t)

d[T*exp(-c_{1}*t)]/dt=exp(-c_{1}*t)*c_{1}(T-T_{w})

T*exp(-c_{1}*t)=∫exp(-c_{1}*t)*c_{1}(T-T_{w})dt

T*exp(-c_{1}*t)=-exp(-c_{1}*t)(T-T_{w})

T=-(T-T_{w})

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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# Homework Help: Develop an expression for Temp as a function of time

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