- #1
maverick280857
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Hello friends
I have a problem and also know its answer but I have some trouble figuring it out. Here goes
Two vessels filled with different liquids are at temperatures [tex]T_{1}[/tex] and [tex]T_{2}[/tex] respectively. They are joined by a metal rod of length l; area of cross-section A and thermal conductivity k. The masses and specific heats are [tex]m_{1}[/tex], [tex]m_{2}[/tex] and [tex]s_{1}[/tex], [tex]s_{2}[/tex] respectively. If [tex]T_{1} > T_{2}[/tex], calculate the time when the temperature difference between two liquids is halved, assuming there is no radiation loss by the liquids and the rod to the surroundings.
Answer: [tex]t = \frac{lm_{1}m_{2}s_{1}s_{2}}{kA(m_{1}s_{1} + m_{2}s_{2})}ln2[/tex]
My problem is that I keep getting a similar answer but without the ln2 because I am setting up a wrong equation. If T1 > T2, then heat should flow from m1 to m2 through the rod and so in time dt, the temperature of m1 should drop to T_{A}-dT whereas that of the mass m2 should rise to T_{B}+dT. There was another variation of the problem where the temperature of m1 was constant and for that very reason the problem was simple. Here it seems that my differential equations/reasoning are wrong. Please offer some suggestions as to how this problem can be approached.
Thanks and cheers
Vivek
I have a problem and also know its answer but I have some trouble figuring it out. Here goes
Two vessels filled with different liquids are at temperatures [tex]T_{1}[/tex] and [tex]T_{2}[/tex] respectively. They are joined by a metal rod of length l; area of cross-section A and thermal conductivity k. The masses and specific heats are [tex]m_{1}[/tex], [tex]m_{2}[/tex] and [tex]s_{1}[/tex], [tex]s_{2}[/tex] respectively. If [tex]T_{1} > T_{2}[/tex], calculate the time when the temperature difference between two liquids is halved, assuming there is no radiation loss by the liquids and the rod to the surroundings.
Answer: [tex]t = \frac{lm_{1}m_{2}s_{1}s_{2}}{kA(m_{1}s_{1} + m_{2}s_{2})}ln2[/tex]
My problem is that I keep getting a similar answer but without the ln2 because I am setting up a wrong equation. If T1 > T2, then heat should flow from m1 to m2 through the rod and so in time dt, the temperature of m1 should drop to T_{A}-dT whereas that of the mass m2 should rise to T_{B}+dT. There was another variation of the problem where the temperature of m1 was constant and for that very reason the problem was simple. Here it seems that my differential equations/reasoning are wrong. Please offer some suggestions as to how this problem can be approached.
Thanks and cheers
Vivek