# 201-Lect 12 - MECH201 Engineering Analysis(2009 p 63 10 THE...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

{[ snackBarMessage ]}

### Page1 / 7

201-Lect 12 - MECH201 Engineering Analysis(2009 p 63 10 THE...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online