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Unformatted text preview: MECH201 Engineering Analysis (2009) p. 63 10 THE PARABOLIC PARTIAL DIFFERENTIAL EQUATION 10.1 Heat conduction through a wall Section 30.1 The following problem for heat conduction through a wall of infinite plan area turns out to be an identical mathematical problem as the previous problem: the flow between two walls. Consider a flat wall, which is normal to the x-direction; that is its thickness, is measured along the x-axis. We will consider the energy balance in a thin slide of the wall of area A and thickness dx . Let q be the rate of heat flow per unit area [Js-1 m 2 or Wm 2 ] into the wall element from the left hand side. Note that q is called the heat flux. The heat flux out to the right of the wall element is ( q+ Δ q)A . The wall element has a mass of ρ Adx and a specific heat c [Jkg-1 K-1 ], will be heated up at a rate of dT Ac dx dt ρ Figure 10.1 unsteady 1-D heat transfers through an element ( q+dq ) .dy q .dy Τ, ρ C p dx dy k ρ C q(x,t) q(0,t) T MECH201 Engineering Analysis (2009) p. 64 The balance of energy, conservation of energy gives the formula (on per unit width of the wall: ( ) T dy q q qA dyc dx t T q c t x ρ ρ ∂ + Δ- + = ∂ ∂ ∂ = - ∂ ∂ Fourier’s law for heat conduction states that the heat flow is proportional to the temperature gradient. The constant of proportion is known as the conductivity which is material dependent. Thus T q k x ∂ = - ∂ note that the negative sign indicates that heat flows from high temperature to low temperature. Substitute to the first equation we have: T q T c k t x x x ρ ∂ ∂ ∂ ∂ = - =- ∂ ∂ ∂ ∂ if k is a constant this results in: 2 2 T k T t c x ρ ∂ ∂ = ∂ ∂ the constant “ k/( ρ c)” is known as the thermal diffusivity and is usually denoted by the symbol α and the unit is “ m 2 /s ” Equation of this type is also called a diffusion equation as, will verify later, the temperature, T, in the equations tend to “ spread ” over the whole domain as time increases....
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- Three '09