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Saturday, 19 September 2020 1
Manish Bhasme
CFD Engineer
MTech Chemical Plant Design (NIT Karnataka)
BTech Chemical Engineer (Nagpur University)
www.manishbhasme4@gmail.com
Saturday, 19 September 2020 2
Aim : Simulate heat flow in a one-dimensional metallic rod
Distance(X)
T1 T2 T4T3 T5
1D Heat Equation
A B
u(x,t) = Temperature at space (x) and time (t)
α = Diffusion co-efficient
Saturday, 19 September 2020 3
1D Heat Equation
Discretise the equation, as follows
approximate the time derivative using forward differences
&
the spatial derivative using central differences
uk+1
n= Temperature at new time step
uk
n = Temperature at old time step
un+1= Next neighbour cell temperature
Un-1= Previous neighbour cell temperature
Saturday, 19 September 2020 4
A B
xtime
x0 x1 x2 x3
uk+1
n
Uk
n+1Uk
n-1
Δx
Δt
uk
n
Saturday, 19 September 2020 5
Saturday, 19 September 2020 6
Saturday, 19 September 2020 7
Saturday, 19 September 2020 8
1D Heat Equation
u(x,t) = Temperature at space (x) and time (t)
α = Diffusion co-efficient
replace the derivatives by finite difference
approximation
uk+1
n= Temperature at new time step
uk
n = Temperature at old time step
un+1= Next neighbour cell temperature
un+1= Previous neighbour cell temperature
Saturday, 19 September 2020 9
Thank you
for your kind attention

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1 d heat_equation

  • 1. Saturday, 19 September 2020 1 Manish Bhasme CFD Engineer MTech Chemical Plant Design (NIT Karnataka) BTech Chemical Engineer (Nagpur University) www.manishbhasme4@gmail.com
  • 2. Saturday, 19 September 2020 2 Aim : Simulate heat flow in a one-dimensional metallic rod Distance(X) T1 T2 T4T3 T5 1D Heat Equation A B u(x,t) = Temperature at space (x) and time (t) α = Diffusion co-efficient
  • 3. Saturday, 19 September 2020 3 1D Heat Equation Discretise the equation, as follows approximate the time derivative using forward differences & the spatial derivative using central differences uk+1 n= Temperature at new time step uk n = Temperature at old time step un+1= Next neighbour cell temperature Un-1= Previous neighbour cell temperature
  • 4. Saturday, 19 September 2020 4 A B xtime x0 x1 x2 x3 uk+1 n Uk n+1Uk n-1 Δx Δt uk n
  • 8. Saturday, 19 September 2020 8 1D Heat Equation u(x,t) = Temperature at space (x) and time (t) α = Diffusion co-efficient replace the derivatives by finite difference approximation uk+1 n= Temperature at new time step uk n = Temperature at old time step un+1= Next neighbour cell temperature un+1= Previous neighbour cell temperature
  • 9. Saturday, 19 September 2020 9 Thank you for your kind attention