| A hot steel bar weighing 20 kg is submerged in an insulated bath holding 50 L of heavy oil. The steel
bar and the oil are
allowed to equilibrate thermally without exchanging heat with the surroundings. |
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| Initially, the steel bar and the oil are at 700oC and 25oC, respectively. Determine the final temperature of the steel bar and the oil. |
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| Data: steel: r = 8000 kg/m3, CV = 0.42 kJ/kg-K, oil: r = 890 kg/m3, CV = 2.1 kJ/kg-K. |
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| Read : |
The easiest way to
solve this problem is to choose the entire contents
of the tank, both the oil and the steel, as our system.
If we assume that this system is adiabatic and does not have any work interactions with its surroundings, then the internal energy of the system must remain constant as the steel bar cools and the oil becomes warmer.
If we further assume that the steel and oil
are incompressible,
then this is a constant
volume process. For solids and liquids it is often reasonable to assume the heat
capacity is a constant over a fairly wide temperature range. The only unknown
left in the 1st Law is
the final system temperature ! |
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| Diagram: |
The diagram in the
problem statement is adequate. |
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| Given: |
msteel |
20 |
kg |
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rsteel |
8000 |
kg/m3 |
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Tsteel,1 |
700 |
oC |
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CV,steel |
0.42 |
kJ/kg-K |
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Voil |
0.05 |
m3 |
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roil |
890 |
kg/m3 |
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Toil,1 |
25 |
oC |
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CV,oil |
2.1 |
kJ/kg-K |
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| Find: |
T2
= |
??? |
oC |
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| Assumptions: |
1 - |
- Steel and oil
have constant heat capacities. |
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2 - |
- No heat is exchanged with the surroundings by either the steel or the oil. |
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3 - |
- Steel and the oil are both
incompressible, so
this process is a constant volume process. |
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| Equations
/ Data / Solve: |
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We begin by writing the 1st Law and we choose as our system
the oil and the steel. |
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Eqn 1 |
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By cleverly selecting
our system, Q = 0 and W = 0. This makes the solution simpler. |
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Eqn 2 |
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Therefore: |
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moil |
44.5 |
kg |
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Because both oil and steel are assumed to be incompressible with constant heat capacities: |
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Eqn 3 |
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Eqn 4 |
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Now, solve for T2: |
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Eqn 5 |
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Eqn 6 |
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T2 |
80.67 |
oC |
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| Verify: |
None of the
assumptions can be verified from the data given in the problem statement. |
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Assumptions 1 & 3 are very nearly true for solids over the temperature range
covered in this problem. |
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Assumption 2 could be made nearly true with
sufficient insulation. |
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| Answers : |
T2 |
80.7 |
oC |
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