Find the steady state temperature Txy to the following heat

Find the steady state temperature T(x,y) to the following heat conduction problem;

Find the steady state temperature T(x,y) to the following heat conduction problem: Determine T(0,1/2).

Solution

The differential equation governing heat diffusion is: 0 dx dT k dx d With constant k, the above equation may be integrated twice to obtain the general solution: 21 )( CxCxT where C1 and C2 are constants of integration. To obtain the constants of integration, we apply the boundary conditions at x = 0 and x = L, in which case 1, )0( TT s and 2, )( TLT s Once the constants of integration are substituted into the general equation, the temperature distribution is obtained: 1,2, 1, )()( ss Ts L x TTxT The heat flow rate across the wall is given by: kAL TT TT L kA dx dT kAq ss x ss / 2,1, 2,1, Thermal resistance (electrical analogy): Physical systems are said to be analogous if that obey the same mathematical equation. The above relations can be put into the form of Ohm’s law: V=IRelec Using this terminology it is common to speak of a thermal resistance: qRT therm A thermal resistance may also be associated with heat transfer by convection at a surface. From Newton’s law of cooling, )( s TThAq the thermal resistance for convection is then hAq TT R s convt 1 , Applying thermal resistance concept to the plane wall, the equivalent thermal circuit for the plane wall with convection boundary conditions is shown in the figure below The heat transfer rate may be determined from separate consideration of each element in the network. Since qx is constant throughout the network, it follows that Ah TT kAL TT Ah TT q ssss x 2 2,2,2,1, 1 1,1, /1 /1/ In terms of the overall temperature difference , and the total thermal resistance Rtot, the heat transfer rate may also be expressed as TT 2,1, tot x R TT q 2,1, Since the resistance are in series, it follows that A