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MMONISM
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Homework Statement
A warm water bath containing 10.0L of water is connected to an ice-water bath with a piece of metal of length L = 2.11 m and cross sectional area A = 1975 cm2. The metal has a thermal conductivity of km = 60.5 Wm-1K-1, a specific heat of cm = 239.7 Jkg-1K-1 and a density of ρm = 3782.5 kgm-3.The warm water bath is initially at a temperature of Th = 63.6 °C.
if the power source heating the warm water bath is switched off. In this case the temperature of the warm water bath will gradually decrease as heat is transferred to the cool water bath. We can describe the heat lost by the warm water bath and the metal rod (the average temperature is just the averages of the temperatures on either side) in time tf as:
##P(t) = -m_wc_w\frac{dT_f}{dt} -\frac{m_mc_m}{2}\frac{dT_f}{dt}##
In this equation the final temperature and power are functions of tf, the other variables are not dependent on tf. We can then differentiate with respect to time to get the expression (replacing tf with t here):
##P(t) = -m_wc_w\frac{dT_f}{dt} -\frac{m_mc_m}{2}\frac{dT_f}{dt}##
which be rearranged to give:
##\frac{dT_f}{dt} = -\frac{P}{m_wc_w+m_mc_m/2}##
what is the temperature of the warm water bath after one hour has passed.
Homework Equations
##\frac{dT_f}{dt} = -\frac{P}{m_wc_w+m_mc_m/2}##
##P = KA \frac{Th - Tc}{L}##
The Attempt at a Solution
Could someone tell me if my approach is correct? Thanks in advance.