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Question 4. (a) Using an energy balance on a thin element, briefly derive the one dimensional heat equation: $\frac{\partial^2 T}{\partial x^2} = \frac{1}{\alpha} \frac{\partial T}{\partial t}$ k Where the thermal diffusivity $\alpha = \frac{k}{\rho C_p}$ (units = m²/s) [4 marks] x (a) For heat conduction in a semi-infinite slab, the introduction of a similarity variable $\eta = \frac{x}{\sqrt{4\alpha t}}$, reduces the second order partial differential equation in part a) above to the following second order ordinary differential equation: $\frac{d^2T}{d\eta^2} = -2\eta \frac{dT}{d\eta}$ Write down the boundary conditions associated with this equation in terms of the new similarity variable $\eta$. [2 marks] (b) The temperature T in the thick slab is given by the following equation: $\frac{T - T_s}{T_i - T_s} = \frac{2}{\sqrt{\pi}} \int_0^{\eta} e^{-u^2} du = erf(\eta)$ Use this equation and the data below to calculate the temperature of a point in the interior of the slab, at a depth of 1cm from the hot surface, after a time of 30 minutes. [4 marks]

          Question 4.
(a) Using an energy balance on a thin element, briefly derive the one dimensional
heat equation:
$\frac{\partial^2 T}{\partial x^2} = \frac{1}{\alpha} \frac{\partial T}{\partial t}$
k
Where the thermal diffusivity $\alpha = \frac{k}{\rho C_p}$ (units = m²/s)
[4 marks]
x
(a) For heat conduction in a semi-infinite slab, the introduction of a similarity
variable $\eta = \frac{x}{\sqrt{4\alpha t}}$, reduces the second order partial differential equation in
part a) above to the following second order ordinary differential equation:
$\frac{d^2T}{d\eta^2} = -2\eta \frac{dT}{d\eta}$ 
Write down the boundary conditions associated with this equation in terms of
the new similarity variable $\eta$.
[2 marks]
(b) The temperature T in the thick slab is given by the following equation:
$\frac{T - T_s}{T_i - T_s} = \frac{2}{\sqrt{\pi}} \int_0^{\eta} e^{-u^2} du = erf(\eta)$
Use this equation and the data below to calculate the temperature of a point in
the interior of the slab, at a depth of 1cm from the hot surface, after a time of
30 minutes.
[4 marks]

Question 4.
(a) Using an energy balance on a thin element, briefly derive the one dimensional
heat equation:
(∂^2 T)/(∂ x^2) = (1)/(α)(∂ T)/(∂ t)
k
Where the thermal diffusivity α = (k)/(ρ Cp) (units = m²/s)
[4 marks]
x
(a) For heat conduction in a semi-infinite slab, the introduction of a similarity
variable η = (x)/(√(4α t)), reduces the second order partial differential equation in
part a) above to the following second order ordinary differential equation:
(d^2T)/(dη^2) = -2η(dT)/(dη)
Write down the boundary conditions associated with this equation in terms of
the new similarity variable η.
[2 marks]
(b) The temperature T in the thick slab is given by the following equation:
(T - Ts)/(Ti - Ts) = (2)/(√(π))∫0^η e^-u^2 du = erf(η)
Use this equation and the data below to calculate the temperature of a point in
the interior of the slab, at a depth of 1cm from the hot surface, after a time of
30 minutes.
[4 marks]

Added by Ashley C.

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