Problem Description

To determine the temperature profile, apply the energy equation to the window glass,

The above equation is referred to as the steady-state heat conduction equation without internal heat generation. Since the object of present study is a solid (i.e., window glass), all velocities vanish* (see note).

Further, it is assumed that the temperature varies in the x-direction only and the thermal conductivity is a constant. Then energy equation reduces to

which can be readily integrated to yield

     T = c₁x + c₂

where c₁ and c₂ are constants to be determined from the temperature boundary conditions (i.e., T = 20 ^oC = 293 K when x = 0 and T = 5 ^oC = 278 K when x = 0.02 m). Applying the conditions give

     c₁ = -750 and c₂ = 293

The temperature profile becomes

     T = -750x + 293

where T is in Kelvin (K), and x is in meter (m). Note that the temperature profile obtained in this case is not a function of the thermal conductivity (k). The thermal conductivity will affect the temperature profile if it is not a constant (i.e., say k = k(x)), or if the problem considered is not at steady-state.

According to Fourier's law of heat conduction, the heat transfer through the window glass per unit area is given by

     q/A = -k dT/dx = -k (-750)
           = (0.8)(750)
           = 600 W/m²

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*Note: In general, the heat conduction equation can also be applied to a fluid at rest. However, when the fluid is subject to a large temperature difference, then convection is induced by the density difference due to the temperature difference (i.e., the buoyancy effect).