Question

A brass bar, 15 cm in length, is welded to an aluminum bar, 10 cm in length. Both have a cross section of 3 cm on a side. If the free end of the brass bar is at 50 degrees Celcius and the the free end of the aluminum bar is at 25 degrees Celcius, what is the steady-state temperature at the junction of the 2 bars? *

Answer 1

Solution.

$l_b=0.15m;$

$l_a=0.1m;$

$d=0.03m;$

$t_b=50^oC;$

$t_a=25^oC;$

$\rho_b=8500kg/m^3;$

$\rho_a=2700kg/m^3;$

$c_b=380J/kg^oC;$

$c_a=920J/kg^oC;$

$m_bc_b\Delta T_b=m_ac_a\Delta T_a;$

$m_b=\rho_bS_bl_b; m_a=\rho_aS_al_a;$

$\rho_bl_bc_b\Delta T_b=\rho_al_ac_a\Delta T_a;$

$\dfrac{\Delta T_b}{\Delta T_a}=\dfrac{\rho_al_ac_a}{\rho_bl_bc_b};$

$\dfrac{\Delta T_b}{\Delta T_a}=\dfrac{2700\sdot0.1\sdot920}{8500\sdot0.15\sdot380}=0.5;$

$\dfrac{t_b-t_f}{t_f-t_a}=0.5\implies t_f=\dfrac{t_b+0.5t_a}{1.5};$

$t_f=\dfrac{50+12.5}{1.5}=41.7^oC;$

Answer: $t_f=41.7^oC.$