Question #122478
Two large containers A and B of the same size are filled with different fluids. The fluids in containers A and B are maintained at 0° C and 100° C, respectively. A small metal bar, whose initial temperature is 100° C, is lowered into container A. After 1 minute the temperature of the bar is 90° C. After 2 minutes (since being lowered into container A) the bar is removed and instantly transferred into the other container. After 1 minute in container B the temperature of the bar rises 10°. How long, measured from the start of the entire process, will it take the bar to reach 99.4° C? (Round your answer to two decimal places. Assume the final temperature being asked for is reached while the bar is container B.)

t = ___ min
1
Expert's answer
2020-06-22T18:16:52-0400

The DE for Newton's Law of cooling /warming is dT/dt=k(T -Tm)T_{m}) --> dT/TTmT- T_{m} =kdt

dTTTm=kdt\therefore \int \frac{dT}{T-T_{m}}=\int kdt


=lnTTm=kt+c1ln\left | T-T_{m} \right |=kt+c_{1}


TTm=ekt+c1T-T_{m}=e^{kt+c_{1}} =T=Tm+ec1ektT_{m}+e^{c_{1}}e^{kt} =Tm+CektT_{m}+Ce^{kt}

For the first process:


Since the surrounding medium is container A->Tm=00CT_{m}=0^{0}C


Now applying the initial condition T(0)= 1000C100^{0}C we get 100=0+C=>C=100=> T= 100ekt100e^{kt}


since the temperature decreased to 90090^{0} C after one minute,

90=100ek1e^{k1} => ek1e^{k1} =90/100 => k1=ln(9/10)=-0.10536=>T=100e0.10536te^{-0.10536t}


thus after 2 minutes we have T=100e(0.105362)e^{(-0.10536*2)} =810C81^{0}C


For hte second process:

Since the surrounding medium is container B=> Tm=1000CT_{m}= 100^{0}C .

now we have initial temperature =810C81^{0}C

thus applying the initial condition we get 81=100+C => C= 81-100 =>C=-19.

therefore T=100-19ekte^{kt}

since the temperature increased by 10010^{0} C after one minute

81+10=100-19ek2e^{k2}

19ek2=9=>ek2=9/19=>k2=ln(9/19)=0.747219e^{k2}=9 =>e^{k2}=9/19 =>k2=ln(9/19)=-0.7472


Hence T=100-19e0.7472t19e^{-0.7472t}


thus for the bar to reach 99.4 we have


99.4=100-19e0.7472t=>e0.7472t=0.6/1919e^{-0.7472t} =>e^{-0.7472t}=0.6/19

therefore -0.7472 t=ln (0.6/19)


Hence t= 4.62 minutes


thus the total time from the entire process to reach 99.4 is t= t1+t2=2+4.62=6.62 minutes.


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