Question formula only The initial temperature of an iron bar is zero (T = 0°C ) for all points on the bar, except at x = 0.0m as shown in Figure 1. The bar is heated at x = 0.0m where the temperature is kept at T = 8 °C at all time. The bar is fully insulated. Therefore, the temperature at the other end of the bar is always same with the temperature at its neighboring point = 0 at x = 1.0m). dx T-8°C T=0 °C x=0.0m x=1.0m Figure 1: Initial condition of the bar Please state the general formula to approximate the temperature T(x,t) using numerical method.

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Transcribed Image Text: The initial temperature of an iron bar is zero (T = 0°C ) for all points on the bar, except at x = 0.0m as shown in Figure 1. The bar is heated at x = 0.0m where the temperature is kept at T = 8 °C at all time. The bar is fully insulated. Therefore, the temperature at the other end of the bar is always same with the temperature at its neighboring point = 0 at x = 1.0m). dx T-8°C T=0 °C x=0.0m x=1.0m Figure 1: Initial condition of the bar Please state the general formula to approximate the temperature T(x,t) using numerical method.

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As we knowSolution.Conductance \sigma=- conductivity \times \frac{\Delta t}{\Delta x}=-k \frac{\Delta t}{\Delta x}Thus, the temperature ws a function of position and time asA(x, t)=\frac{d \sigma}{d x}=-k \frac{d}{d x} \frac{\Delta t}{\Delta x}=-k x-\frac{\Delta t}{\Delta x^{2}}=k \frac{\Delta t}{\Delta x^{2}}Thus for \Delta t=0.15-0=0.15 & \forall x=0.25 \quad \& k=1\lambda(x, t)=1 \times \frac{115}{125^{2}}=2.4000^{\circ} \mathrm{C}& For \Delta t=.025 \quad \Delta x=0.25 \quad \lambda(x, t)=1 x^{.025}\left(\frac{25}{125}=0.4000^{\circ} \mathrm{C}\right. ...