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

I haven't done ODEs in a few years and I am trying to do this equation:

[itex]m_{Hg}C_{p,Hg}\frac{dT_{Hg}}{dt} = Q[/itex]

[itex]Q = hA(T_{air} - T_{Hg})[/itex]

[itex]T_{Hg}(t = 0) = 20[/itex]

I need to find [itex]T_{Hg}(t=590)[/itex]

2. Relevant equations

3. The attempt at a solution

[itex]h, A, m_{Hg}, C_{p,Hg}, T_{air}[/itex] are all constants

[itex]\frac{dT_{Hg}}{dt} = \frac{hA(T_{air} - T_{Hg})}{m_{Hg}C_{p,Hg}}[/itex]

[itex]\frac{m_{Hg}C_{p,Hg}}{hA(T_{air} - T_{Hg})} dT_{Hg} = dt[/itex]

[itex]\int\frac{m_{Hg}C_{p,Hg}}{hA(T_{air} - T_{Hg})} dT_{Hg} = \int dt[/itex]

[itex]\frac{-m_{Hg}C_{p,Hg}}{hA}\int\frac{1}{(-T_{air} + T_{Hg})} dT_{Hg} = \int dt[/itex]

[itex]\frac{-m_{Hg}C_{p,Hg}}{hA} ln|T_{Hg} - T_{air}| = t + c[/itex]

[itex]ln|T_{Hg} - T_{air}| = \frac{t + c}{\frac{-m_{Hg}C_{p,Hg}}{hA}}[/itex]

[itex]T_{Hg} - T_{air} = e^{t/\frac{-m_{Hg}C_{p,Hg}}{hA}}e^{c/\frac{-m_{Hg}C_{p,Hg}}{hA}}[/itex]

[itex]e^{c/\frac{-m_{Hg}C_{p,Hg}}{hA}} = c[/itex]

[itex]T_{Hg} - T_{air} = ce^{t/\frac{-m_{Hg}C_{p,Hg}}{hA}}[/itex]

[itex]T_{Hg} = ce^{t/\frac{-m_{Hg}C_{p,Hg}}{hA}} + T_{air}[/itex]

[itex]T_{Hg}(t = 0) = 20[/itex]

[itex]T_{air} = -7[/itex]

[itex]h = 3.9[/itex]

[itex]A = 0.00176[/itex]

[itex]C_{p,HG} = 139.0908[/itex]

[itex]m_{Hg} = 0.05[/itex]

[itex]20 = ce^{0/\frac{-(0.05)(139.0908)}{(3.9)(0.00176)}} + (-7)[/itex]

[itex]c = 27[/itex]

[itex]T_{Hg} = 27e^{t/\frac{-m_{Hg}C_{p,Hg}}{hA}} + T_{air}[/itex]

[itex]T_{Hg}(t=590)[/itex]

[itex]T_{Hg}(t=590) = 27e^{590/\frac{-(0.05)(139.0908)}{(3.9)(0.00176)}} + (-7)[/itex]

[itex]T_{Hg}(t=590) = 8.0822[/itex]

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# Homework Help: First order ordinary differential equation

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